Bulletin 42

(Part 2 of 5 Parts)

THE
ir _ SHOCK AND VIBRATION
Vo BULLETIN
Part 2
Ground Motion,
Dynamic Analysis

JANUARY 1972

A Publication of
THE SHOCK AND VIBRATION
INFORMATION CENTER
Naval Research Laboratory, Washington, D.C.

1 DDC

Office of
The Director of Defense
NATIONA TIC b Research and Engineering
INFORM4ATIOtJ SERVICE
Spdngfl.Id, Va. 22151

This
67 document has /(-~
been approved for public release and sale; its distribution is unlimited.
 Best
Available
Copy
 2_ .

SYMPOSIUM MANAGEMENT

THE SHOCK AND VIBRATION INFORMATION CENTER

William W. Mutch, Director
Henry C. Pusey, Coordinator
Rudolph H. Volin, Coordinator
Edward H. Schell, Coordinator

Bulletin Production

Graphic Arts Branch, Technical Information Division,

Naval Research Laboratory

.. II for

N ............................

UST. AVAIL &U/iu VECIAL.

" - o
 IF
BOultin 42
(Par -o Prs

THE
SHOCK
. AND VIBRATION
BULLETIN

JANUARY 1972

A Publication of
THE SHOCK AND VIBRATION
INFORMATION CENTER
Naval Research Laboratory, Washington, D.C.

The 42nd Symposium on Shock and Vibration was

held at the U.S. Naval Station, Key West, Florida,
on 2-4 November 4971. 'he U.S. Navy was host.

Office of
- The Director of Defense
Research and Engineering

---------------------
 CONTENTS,

PAPERS APPEARING IN PART 2

Ground Motion
-
SINE BEAT VIBRATION TESTING RELATED TO EARTHQUAKE.RESPONSE:SPECTRA .. . 1---
E. G. Fischer, Westinghouse Research-Laboratories; ittjsburh, Pennsylvania

SEI1MIC EVALUATION OF ELECTRICAL EQUIPMENT FOR NUCLEAR POWER STATIONS ... 11

3 H. Prause znd D. R. Ahlbeck, BATTELLE, Columbus Laboratories, Columbus, Ohio

SHOCK INPUT FOR EARTHQUAKE STUDIES USING GROUND MOTION FROM UNDERGROUND
NUCLEAR EXPLOS IONS.......... ........ ...................... ......... 21
D. L. Bernreuter, D. M. Norris, Jr., axd F. J. Tokaz, Lawrence Livermore Laboratory,
University of California, Livermore, California
ROCKING OF A RIGID, UNDERWATER BOTTOM-FOUNDED STRUCTURE SUBJECTED TO
SEISMIC SEAFLOOIR EXCITATIN ...................................... 33
J. G. Hamm2r and H. S. Zwlbel, Naval Civil Engineering Laboratory, Port Hueneme,
California
DEVELOPMENT OF A WAVEFORM SYNTHESIS TECHNIQUE-A SUPPLEMENT TO RESPONSE
SPECTRUM AS A DEFINITION OF SHOCK ENVIRONMENT ...................... 45
R. C. Yang and H. R. Saffell, The Ralph M. Parsons Company, Los Angeles, California

THE RESPONSE OF AN ISOLATED FLOOR SLAB-RESULTS OF AN EXPERIMENT IN

EVENT DIAL PACK ...... ................. ..................... #........ 55
J. M. Ferritto, Naval Civil Engineering Laboratory, Port Hueneme, California

A SHOCK-ISOLATION SYSTEM FOR 22 FEET OF VERTICAL GROUND MOTION ............67

E. C. Jackson, A. B. Miller and D. L. Bernreuter, Lawrence Livermore Laborato'y,
University of California, Livermore, California

THE COMPARISON OF THE RESPONSE OF A HIGHWAY BRIDGE TO UNIFORM GROUND

SHOCK AND MOVING GROUND EXCITATION ..................................... 75
N. E. Johnson and R. D. Galletly, Mechanics Research, Inc., Los Angeles, California
DEFORMATION AND FRACTURE OF TANK BOTTOM HULL PLATES SUBJECTED
TO MINE BLAST ..... . . . . . . . .. . . . . . . . ...... .. .. 87/
D. F. Haskell, Vulnerability Laboratory, U.S. Army Ballistic Research
Laboratories, Aberdeen Proving Ground, Md.

THE IMPULSE IMPARTED TO TARGETS BY THE DETONATION OF

LAND MINES . . . . . . . . . . . . . . . . . . . . . . . ...... 97
P. S. Westine, Southwest Research Institute, San Antonio, Texas
CIRCULAR CANTILEVER BEAM ELASTIC RESPONSE TO AN EXPLOSION ............. 109
Y. S. Kim and P. R. Ukrainetz, Department of Mechanical Engineering, University
of Saskatchewan, Saskatoon, Canada
MEASUREMENT OF IMPULSE FROM SCALED BURIED EXPLOSIVES .............. 123
B. L. Morris, U.S. Army Mobility Equipment Research and Development Center,
Fort Belvoir, Virginia

iii
 Dynamic Analysis

THE EFFECTS OF MOMENTUM WHEELS ON THE FREQUENCY RESPONSE

CHARACTERISTICS OF LARGE-FLEXIBLE STRUCTURES........ ...... . ........ '29
F. -D. Day I and S. R. Tomer, Martin Marietta C-rporationDenver, Colorado

r INTEGRATED DYNAMIC ANALYSIS OF A SPACE STATION WITH CONTROLLABLE

SOLAR ARRYS. ............. .. .
J. A. Heinrichs and A. L. Weinberger, Fairchild Industries, Inc., Germantown, Maryland,
137
and Me D. Rhodes, NASA Langley Research Center, Hampton, Virginla

.. .PARAMETRICALLY-EXCITEDCOLUMN WITH HYSTERETIC MATERIAL

D. T. Mozer, IBM Corporation, East Fishkill, New York, and R. M. Evan-Iwanowski,
Professor, Syracuse University, Syracuse, Now York

DYNAMIC-INTERACTIOI BETWEEN VIBRATING CONIVEYORS AND

SUPPORTING STRUCTURE ................. ........................... 163
M. P", Professor, Civil Engineering-Departmen, University. of.Louisville,
Louisville, Kentucky; and O.'Mathis, Design Engineer, Rex Chainbelt Inc.,
Louisville, Kentucky
RESPONSE OF A SIMPLY SUPPORTED CIRCULAR PLATE EXPOSED TO THERMAL
AND PRESSURE LOADING ............. ...................... 171
1........
J. E. Koch, North Eastern Research Associates, Upper Montclair, N.J., and M. L. Cohen,
North Eastern Research Associates, Upper Montclair, N.J., and Stevens Institute of
Technology, Hoboken, N.J.
WHIRL FLUTTER ANALYSIS OF PROPELLER-NACELLE-PYLON SYSTEM ON LARGE
SURFACE EFFECT VEHICLES ..................... *.................... 181
Yuan-Ning Liu, Naval Ship Research and Development Center, Washington, D.C.

STHE DYNAMIC RESPONSE OF STRUCTURES SUBJECTED TO TIME-DEPENDENT

BOUNDARY CONDITIONS USING THE FINITE ELEMENT METHOD .................
G. H. Workman, Battelle, Columbus Laboratories, Columbus, Ohio
195

VIBRATION ANALYSIS AND TEST OF THE EARTH RESOURCES

TECHNOLOGY SATELLITE ............................................ 203
T. J. Cokoni.s and G. Sardella, General Electric Company, Space Division,
Philadelphia, Pennsylvania

FINITE AMPLITUDE SHOCK WAVES IN INTERVERTEBRAL DISCS ................. 213

W. F. Hartman, The Johns Hopkins University, Baltimore, Maryland

ACCELERATION RESPONSE OF A BLAST-LOADED PLATE .................. 221

L. W. Fagel, Bell Telephone Laboratories, Inc., Whippany, New Jersey

EFFECT OF CORRELATION IN HIGH-INTENSITY NOISE TESTING AS INDICATED

BY THE RESPONSE OF AN INFINITE STRIP.................................. o235
C. T. Morrow, Advanced Technology Center, Inc., Dallas, Texas

PAPERS APPEARING IN PART 1

Invited Papers
SMALL SHIPS-HIGH PERFORMANCE
Rear Admiral H. C. Mason, Commander, Naval Ship Engineering Center, Washington, D.C.

v2

<~.-rA~W.CJ
.'.<fl~ . . . rr,~-.'V~rt
. .- ,--. . . . .
 Specifications
SURVEY, OF VIBRATION TEST PROCEDURES IN USE BY THE AIR FORCE
W. B. Yarchb, Air Force Flight-Dynamics Laboratory, Wright-Pitterson Air Force
Base, Ohio

SPECIFICATIONS - A PANEL SESSION

SOME ADMINISTRATIVE FACTORS WHICH INFLUENCE TECHNICAL APPROACHES
TO SHIP SHOCK'HARDENING
D. M. Lund, Naval Ship Engineering Center, Hyattsville, Maryland

Measurement and Application of Mechanicil Impedance

FORCE TRANSDUCER CALIBRATIONS RELATED TO MECHANICAL IMPEDANCE
MEASUREMENTS
E. 'F. Ludwig, Assistant Project Engineer, and N.D. Taylor, Senior Engineer, Pratt & Whitney
Aircraft, Florida Research & Development Center, West Palni Beach, Florida
THE MEASUREMENT OF MECHANICAL IMPEDANCE AND ITS USE IN
VIBRATION TESTING
N. F. Hunter, Jr., and J. V. ,Otts, Sandia Corporation, Albuquerque, New Mexico
I
A TRANSIENT TEST TECHNIQUES FOR MECHANICAL IMPEDANCEAND MODA.
SURVEY TESTING
J.D. Favour, M. C. Mitchell, N. L. Olson, The Boeing Company, Seattle, Washington

PREDICTION OF FORCE SPECTRA

MOBILITY MEASUREMENT BY MECHANICAL IMPEDANCE AND ACOUSTIC
TECHNIQUES
R. W. Schock, NASA/Marshall Space Flight Center, Huntsville, Alabama and G. C. Kao,
Wyle Laboratories, Huntsville, Alabama

DYNAMIC DESIGN ANALYSIS VIA THE BUILDING BLOCK APPROACH

A. L. Klosterman, Ph.D. and J. R. Lemon, Ph.D., Structural Dynamics Research
Corporation Cincinnati, Ohio
MOBILITY MEASUREMENTS FOR THE VIBRATION ANALYSIS OF CONNECTED
STRUCTURES
D. J._Ewins and M. G. Sainsbury, Imperial College of Science and Technology,
London, England

LIQUID-STRUCTURE COUPLING IN CURI.ED PIPES - II

L. C. Davidson and D. R. Samsury, Machineey Dynamica Division, Naval Ship Research and
Development Center, Annapolis, Maryland

Transportation and Packaging

A SURVEY OF THE TRANSPORTATION SHOCK AND VIBRATION INPUT TO CARGO

F. E. Ostrem, General American Research Division, General American Transportation
Corporatic Niles, illinois

THE DYNAMIC ENVIRONMENT OF SELECTED MILITARY HELICOPTERS

M. B. Gens, Sandia Laboratories, Albuquerque, New Mexico
HIGHWAY SHOCK INDEX
R, Kennedy, U. S. Army Transportation Engineering Agency, Military Traffic Management
and Terminal Service, Newport News, Virginia

I¢
 DEVELOPMENT OF'A ROUGH'ROAD SIMULATOR AND SPECIFICATION FOR TESTING
OF EQUIPMENT TRANSPORTED IN WHEELED-VEHICLES
H. M. Forkiois and E. W. Clements, Naval Research Laboratory, Washington, D.C.
LABORATORY CONTROL OF DYNAMIC VEHICLE TESTING
J. W, Grant, U. S. Army Taik-Au"omftive Command, Warren, Michigan
IMPACT VULNERABILITY OF TANK CAR HEADS
J. C. Shang and J. E. Everett, General American Research Division,
General American Transportation Corporation, Niles, Illinois

A STUDY OF IMPACT TEST EFFECTS UPON FOAMED PLASTIC CONTAINERS

D. McDaniel, Ground Equipment and Materials Directorate, Directorate for Research,
Development, Eginegeriv an. Missile Systems Laborator, U. S. Army Missile Command
Redstone Arsenal, Aima, and R. M. Wyskida, Industrial and Systems Engineering
Department, The-Universfty of Alabama In Huntsville, Huntsville, Alabama

DEVELOPMENT OF A PRODUCT PROTECTION SYSTEM

D. E.:Yound, IBM General Systems Division, Rochester, Minnesota, and
S. R. Pierce, Michigan State University, Fast Lansing, Michigan

MOTION OF FREELY SUSPENDED LOADS DUE TO HORIZONTAL .1lP MOTION IN

RANDOM HEAD SEAS
H. S. Zwibel, Naval Civil Engineering Laboratory, Port Huenem:, California

PAPERS APPEARING IN PART 3

Test Control
ON THE PERFORMANCE OF TDM AVERAGERS IN RANDOM VIBRATION TESTS
A. J. Curtis, Hughes Aircraft Company, Culver City, California
A MULTIPLE DRIVER ADMITTANCE TECHNIQUE FOR VIBRATION TESTING OF

- COMPLEX STRUCTURES
S. Smith, Lockheed Missiles & Space Company, Palo Alto Research Laboratory,
Palo Alto, California, and A. A. Woods, Jr., Lockheed Missiles & Space Company,
Sunnyvale, California

EQUIPMENT CONSIDERATIONS FOR ULTRA LOW FREQUENCY MODAL TESTS

R.G. Shoulberg and R. H. Tuft, General Electric Company, Valley Forge,
Pennsylvania
COMBINED-AXIS VIBRATION TESTING OF THE SRAM MISSILE
W. D. Trotter and D. V. Muth, The Boeing Company, Aerospace Group,
Seattle, Washington
SHOCK TESTING UTILIZING A TIME SHARING DIGITAL COMPUTER
R. W. Canon, Naval Missile Center, Point Mugu, California

A TECHNIQUE FOR CLOSED-LOOP COMPUTER-CONTROLLED REVERSED-

BENDING FATIGUE TESTS OF ACOUSTIC TREATMENT MATERIAL
C. E. Rucker and R. E. Grandle, NASA Langley Research Center,
[F
Hampton, Virginia

PROGRAMMING AND CONTROL OF LARGE VIBRATION TABLES IN UNIAXIAL

AND BIAXIAL MOTIONS
R. L. Larson, MTS Systems Corporation, Minneapoll3, Minnesota

vi
1'
'1p

A DATA AMPLIFIER GAIN-CODE RECORDING SYSTEM

J. R. Olbert and T. H. Hammond, Huhes Aircraft Company, Culver
City, Califorila
STABILITY OF AN AUTOMATIC NOTCH CONTROL SYSTEM IN SPACECRAFT
TESTING
B. N. Agrawal, COMSAT Laboratories, Clarksburg, Maryland

Test Facilities and Technlques;

SINUSODAL VIBRATION OF POSEIDON SOLID PROPELLANT MOTORS

L. R. Pendleton, Research Specialist, Lockheed Missiles & Space Company,
Sunnyvale, California

CONFIDENCE IN PRODUCTION UNITS BASED ON QUALIFICATION VIBRATION

R. E. Deitrick, Hughea Aircraft Company, Space and Communications Group,
El Segundo, California

SIMULATION TECHNIQUES IN DEVELOPMENT TESTING

A. Hammer, Weapons Laboratory, U. S. Army Weapons Comna.-, Rock
Island, Illinois

A ROTATIONAL SHOCK AND VIBRATION FACILITY

RM T. Fandrich, Jr., Radiation Incorporated, Melbourne, Florida
THE EFFECTS OF VARIOUS PARAMETERS ON SPACECRAFT SEPARATION SHOCK
W. B. Keegan and W. F., Bangs, NASA, Goddard Space Flight Center, Greenbelt,
Maryland
NON-DESTRUCTIVE TESTING OF WEAPONS EFFECTS On COMBAT AND
LOGISTICAL VEHICLES .......................
R. L. Jo_son, J. H. Leete, and J. D. O'Keefe, TRW Systems Group, Redondo
Beach, California, and A. N. Tedesco, Advanced Research Projects Agency,
Department of Defense, Washington, D.C.

THE EFFECT OF THE FIN-OPENING SHOCK ENVIRONMENT ON GUIDED MODULAR

DISPENSER WEAPONS
K. D. Denton and K. A. Herzing, Honeywell Inc., Government and Aeronautical
Products Division Hopkins, Minnesota
DEVELOPMENT OF A FLUIDIC HIGH-INTENSITY SOUND GENERATOR
H. F. Wolfe, Air Force Flight Dynamics Laboratory, Wright-Patterson
Air Force Base, Ohio
DEVELOPMENT OF A LIGHTWEIGHT, LINEAR MECHANICAL SPRING ELEMENT
R. E. Keeffe, Kaman Sciences Corporation, Colorado Springs, Colorado

TECHNIQUES FOR IMPULSE AND SHOCK TUBE TESTING OF SIMULATED

REENTRY VEHICLES
N. K. Jamison, McDonnell Douglas Astronautics Company, Huntington
Beach, California

VIBRATION FIXTURING - NEW CELLULAR DESIGN, SATURN AND ORBITAL

WORKSHOP PROGRAMS
R. L. Stafford, McDonnell Douglas Astronautics Company, Huntington Beach,
California
WALL FLOW NOISE IN A SUBSONIC DIFFUSER
E. F. Timpke, California State College, Long Beach, Calfornia, and R. C. Binder
University of Southern California, Los Angeles, California

vii

. . . ..o.
 PAPERS APPEARING IN PART 4
Isolation and Dmping
TRANSIENTRERONSE OF REAL DISEPATIVE STRUCTURES
"1. Plunkett, UnWfisity of Minnesota, Minneapolis, Minnesota
DYNAMIC RESPONSE OF A RING SPRING
R. L. Eshleman, lIT Research Institute, Chicago, Illinois
SHOCK MOUNTING SYSTEM FOR ELECTRONIC CABINETS
W. D. Delany, Admiralty Surface Weapons Establishment, Portsmouth, U.K.
METHODS OF ATTENUATING PYROTECHNIC SHOCK
S. Barrett and W. J. Kacena, Martin Marietta Corporation, Denver, Colo.'ado

P =kGY ABSORPTION CAPACITY OF A SANDWICH PLATE WITH

CR!Tft ABLE CORE
D. iXxjcinovic, Argonne National Laboratory, Argonne, nlinois
ON THE DAMPING'OF TRANSVERSE MOTION OF.FREE-FREE BEAMS IN
DENSE, STAGNANT FLUIDS
W. K. Blake, Naval Ship Research and Development Center, Bethesda, Maryland

OPTIMUM DAMPING DISTRIBUTION FOR STRUCTURAL VIBRATION

R. Plunkett, University of Minnesota, Minneapolis, Minnesota

A LAYERED VISCOELASTIC EPOXY RIGID FOAM MATERIAL FOR

VIBRATION CONTROL
C. V. Stahle and Dr. A. T. Tweedie, General Electric Company, Space
Division, Valley Forge, Pa.
OPTIMIZATION OF A COMBINED RUZICKA AND SNOWDON VIBRATION
ISOLATION SYSTEM
D. E. Zeidler, Medtronic, Inc., Minneapolis, Minnesota, and D. A. Frohrib,
University of Minnesota, Minneapolis, Minnesota
TRANSIENT RESPONSE OF PASSIVE PNEUMATIC ISOLATORS
G. L. Fox, and E. Steiner, Barry Division of Barry Wright Corporation,
Burbank, California
EXPERIMENTAL DETERMINATION OF STRUCTURAL AND STILL WATER DAMPING
AND VIRTUAL MASS OF CONTROL SURFACES
R. C. Leibowitz and A. Kilcullen, Naval Ship Research and Development Center,
Washington, D.C.

DAMPING OF A CIRCULAR RING SEGMENT BY A CONSTRAINED

VISCOELASTIC LAYER
Cpt. C. R. Almy, U.S. Army Electronics Command, Ft. Monmouth, New Jersey,
and F. C. Nelson, Department of Mechanical Engineering, Tufts University,
Medford, Mass.
DYNAMIC ANALYSIS OF THE RUNAWAY ESCAPEMENT MECHANISM
G. W. Hemp, Department of Engineering, Science and Mechanics, University
of Florida, Gainesville, Florida

Prediction and Experimental Technlqubs

A METHOD FOR PREDICTING BLAST LOADS DURING THE DIFFRACTION PHASE

W. J. Taylor, Ballistic Research Laborator.les, Aberdeen Proving Ground, Maryland
viii
DRAG MEASUREMENTS ON CYLINDERS IN EVENT DIAL PACK
S. B. Millis, bitence Research Establishment Sdffield, Ralston, Alberta, Canada

DIAL PACK BLAST DIRECTING EXPERIMENT

L. E. Fgelo, S. F. FTields, and W. J. Byrne, General American Research
Division, Niles; Illinois
BLAST FIELDS ABOUT ROCKETS AND RECOILLESS RIFLES
W. E. Bake]r, P. S. Westine, and R. L. Bessey, Southwest Research Institute,
San Antonio, Texas
TRANSONIC ROCKET-SLED STUDY OF FLUCTUATING SURFACE-PRESSURES
AND PANEL RESPONSES
E. E. Ungar, Bolt Beranek and Newman Inc., Cambridge, Massachusetts, and H. J.
Bandgren, Jr. and R. Erwin, National Aeronautics and Space Administration, Geo'ge
C. -Marshall Space Flight Center Huntsville, Alabama

SUPPRESSION OF FLOW-INDUCED VIBRATIONS BY MEANS OF BODY

SFIRFACE MODIFICATIONS
D. W. Sallet and J. Berezow, Naval Ordnance Laboratory, Silver Spring, Maryland

AN EXPERIMENTAL TECHNIQUE FOR DETERMINING VIBRATION MODES OF

STRUCTURES WITH A QUASI-STATIONARY RANDOM FORCING FUNCTION
R. G. Christiansen and W. W. Parmenter, Naval Weapons Center, China Lake,
California

RESPONSE OF AIR FILTERS TO BLAST

E. F. Witt, C. J. Arroyo, and W. N. Butler, Bell Laboratories, Whippany, N.J.

PAPERS APPEARING IN PART 5

Shock and Vibration Analysis

BANDWITH-TIME CONSIDERATIONS IN AUTOMATIC EQUALIZATION

C. T. Morrow, Advanced Technology Center, Inc., Dallas, Texas

A REGRESSION STUDY OF THE VIBRATION RESPONSE OF AN

EXTERNAL STORE
C. A,. Golueke, Air Force Flight Dynamics Laboratory, Wright-Patterson
Air Force Base, Ohio

FACTOR ANALYSIS QF VIBRATION SPECTRAL DATA FROM MULTI-LOCATION

MEASUREMENT
R. G. Merkle, Air Force Flight Dynamics Laboratory, Wright-Patterson Air
Force Base, Ohio

RESPONSES OF A MULTI-LAYER PLATE TO RANDOM EXCITATION

H. Saunders, General Electric Company, Aircraft Engine Group, Clncinnati, Ohio
RESPONSE OF HELICOPTER ROTOR BLADES TO RANDOM LOADS
NEAR HOVER
C. Lakshmikantham and C. V. Joga Rao, Army Materials and Mechanics Research
Center, Watertown, Massachusetts
INSTRUMENTATION TECHNIQUES AND THE APPLICATION OF SPECTRAL
ANALYSIS AND LABORATORY SIMULATION TO GUN SHOCK PROBLEMS
D. W. Culbertson. Naval Weapons Laboratory, Dahlgren, Virginia, and
V. F. DeVost, Naval Ordnance Laboratory, White Oak, Silver Spring, Maryland

ix

----------
 THE EFFECT OF "Q' VARIATIONS IN SHOCK SPECTRUM ANALYSIS
* M. B. McGrath, Martin Marietta Corporation, Denver, Colorado, and W. F. Bangs,
. National Aeronautics and Space Administration, Goddard Space -'Uight Center, Maryland

* RAPID FREQUENCY AND CORRELATION ANALYSIS USING AN ANALOG COMPUTER

J. G. Parks, Research, Development and Engineering Directorate, U.S. Army. Tank-
Automotive Command, Warren, Michigan

U4VESTIGATION OF LAUNCH TOWER MOTION DURING AEROBEE 350 LAUNCH

R. L. Kinsley and W. R. Case, NASA, Goddard Space Flight Center, Greenbelt, Maryland

ON THE US E OF FOURIER TRANSFORMS OF MECHANICAL MHOCK DATA

H. A. Gaberson and D. Pal, Naval Civil Engineering Laboratory, Port Hueneme,
California
WAVE ANALYSIS OF SHOCK EFFECTS IN COMPOSITE ARMOR
G. L. Filbey, Jr., USAARDC Ballistic R13sear.h Laboratories, Aberdeen Proving
Ground, Maryland

STATISTICAL LOADS ANALYSIS TECHNIQUE FOR SHOCK AND HIGH-FREQUENCY

EXCITED ELASTODYNAMIC CONFIGURATIONS
K. J. Saczalski and K. C. Park, Clarkson College of Technology, Potsdam, New York

Structural Analysis
NASTRAN OVERVIEW: DEVELOPMENT, DYNAMICS APPLICATION, MAINTENANCE,
ACCEPTANCE
J. P. Raney, Head, NASTRAN Systems Management Office and D. J. Weidman, Aerospace
Engineer, NASA Langley Research Center, Hampton, Virginia

EXPERIENCE WITH NASTRAN AT THE NAVAL SHIP R&D CENTER AND OTHER
NAVY LABORATORIES
P. Matula, Naval Ship Research & Development Center, Bethesda, Maryland

RESULTS OF COMPARATIVE STUDIES ON REDUCTION OF SIZE PROBLEM

R. M. Mains, Department of Civil and Environmental Engineering, Washington
University, St. Louis, Missouri
STRUCTURAL DYNAMICS OF FLEXIBLE RIB DEPLOYABLE SPACECRAFT ANTENNAS
B. G. Wrenn, W. B. Halle, Jr. and J. F. Hedges, Lockheed Missiles and Space
Company, Sunnyvale, California

INFLUENCE OF ASCENT HEATING ON THE SEPARATION DYNAMICS OF A

SPACECRAFT FAIRING
C. W. Coale, T. J. Kertesz, Lockheed Missiles & Space Company, Inc.,
Sunnyvale, California
DYNAMIC WAVE PROPAGATION IN TRANSVERSE LAYERED COMPOSITES
C. A. Ross, J. E. Cunningham, and R. L. Sierakowski, Aerospace Engineering Department
University of Florida, Gainesville, Florida
R-W PLANE ANALYSIS FOR VULNERABILITY OF TARGETS TO AIR BLAST
P. S.Westine, Southwest Research Institute, San Antonio, Texas

PERFORM: A COMPUTER PROGRAM TO DETERMINE THE LIMITING PERFORMANCE

OF PHYSICAL SYSTEMS SUBJECT TO TRANSIENT INPUTS
W. D. Pllkey and 1o Ping Wang, Department of Aerospace Engineering and Engineering
Physics, University of Virginia, Charlottesville, Virginia

x
 STRUCTURAL DYNAMIC ANALYSIS AND TESTING OF A SPACECRAL T DUAL TRACKING
ANTENNA
D. D. Walters, R. F. Heidenreich, A. A. Woods and B. G. Wrenn, Lockheed Missiles
and Space'Company, Sunnyvale, California

j Ship's Problems
DETERMINATION OF FIXED-BASE NATURAL FREQUENCIES OF A COMPOSITE
STRUCTURE OR SUBSTRUCTURES
C. Ni, R. Sccp, and J. P. Layher, Naval Research Laboratory, Washington, D.C.

EQUirVALENT SPRING-MASS SYSTEM: A PHYSICAL INTERPRETATION

B. K. Wada, . Bamford, and J. A. Garba, Jet Propulsion Laboratory,
Pasadena, California
LONGITUDINAL VIBRATION OF COMPOSITE BODIES OF VARYING AREA
D. J. Guzy, J.C.S. Yang, and W. H. Waiston, Jr., Mechanical Engineering
Department, University of Maryland, College Park, Maryland

SIMPLIFIED METHOD FOR THE EVALUATION OF STRUCTUREBORNE VIBRATION

TRANSMISSION THROUGH COMPLEX SHIP STRUCTURES
M. Chernjawski and C. Arcidlacona, Gibbs & Cox, Inc., New York, New York

ix
 GROUND MOTION

SINE BEAT VIBRATION TESTING-RELATED TO

EARTHQUAKE -RESPONSE SPECTRA

E. G. FISCHER
WESTINGHOUSE RESEARCH LABORATORIES
PITTSBURGH, PENNSYLVANIA

Vibration test criteria are developed for evaluating the earthquake re-
sistance and reliability of electrical switchgear, including sensitive
control devices. A series of sine beat vibrations applied at experimentally
determined, natural frequencies of the equipment is potentially more
damaging than the original seismic motion. The test table input can be
related to the floor response spectra as calculated for a particular power
plant structure and location in an active earthquake zone.

INTRODUCTION damping for typical nonlinear systems, and such

information can be used to authenticate parallel
Itiscustomary to develo;, ,nathematical computer-aided dynamic analyses.
models of building structures l] and then make
computer-aided analyses of their dynamic re- RANDOM MOTION THEORY
sponse to seismic disturbances. Similarly,
mechanical equipments located in such Random mudon is non-periodic and,
structures may require dynamic seismic analy- therefore, unpredictable with time. However,
ses, although they become more complicated it can be described in statistical terms based
and difficult to model (2] . In the special case upon "the analysis f noise in communication
of sensitive electrical control devices subject circuits" as pioneered by Rice[3].
to possible malfunction during earthquakes (for
example, the safeguards equipment required for Figure la illustrates a random vibration
nuclear power plants), it becomes more practi- excitation which is characterized as broadband
cal to rely upon a conservative method of en- in the sense that it appears to'include all freq-
vironmental testing. Experience indicates that uencies in addition to its large fluctuations in
the inadvertent loss of principle function under amplitude. (Earthquake accelerographs of free-
shock can pose more of a design problem than field ground motions usually exhibit such char-
an obvious strength failure. acteristics.)

To obviate either difficult or question- By way of contrast, Fig. lb illustrates

able analyses, earthquake vibration test cri- the corresponding narrow-band vibration re-
teria have been developed for evaluating and sponse of a simple (mass-spring-damper)
demonstrating the reliability of electrical con- oscillator wherein the frequency is restricted
trol devices for Electric Utility Systems. In to the system natural frequency, "although the
particular, a series of sine beat v 'rations are amplitude fluctuations persist. As judged by
applied at experimentally determined natural the envelope of the peak magnitudes, this more-
frequencies of the equipment. This means that or-less harmonic motion is referred to as a
the equipment is conservatively tested by a random sine wave. (Building floor seismic re-
procedure which makes it most vulnerable. sponse records exhibit similar sine beat char-
acteristics.)
In addition, sine beat vibration inputs
during test can be related to foundation or In general, the original seismic dis-
building floor shock response spectra for turbance can be generated In terms of a non-
computer-simulated earthquakes. The test stationary random process involving damped
results provide data on natural frequencies and sinusoids with random frequencies as well as
 Envelope Curve of Response
Peak Magnitudes
: /-Excitation
Time --- ,

(b),Narrow-band Simple System Response InForm of a /

Random Sine Wave

TFrequency. u

(a)Random Vibration Excitation Fig. 2 - Frequency spectra showing the output

response of a simple oscillator for a
random vibration excitation
Fig. 1- Acceleration-time history records of
broadband excitation and narrow-band
response SINE BEAT VIBRATION

In general, when a random excitation is

random phase relationships [4]. However, the put through a lightly-damped, narrow-band-
present discussioncan be simplified if limited pass filterp the outputcresponse usually appears
to a "stationiry random process" consisting of to be a harmonic signal with a slowly varying
an ensemble of motion-time histories, the amplitude. The energy is transmitted primarily
latter considered to be stationary if the sta- in the neighborhood of the natural frequency of

I
tistical properties are not affected by a trans- the system filter. The vibration response wave
lation of the origin of time. Consequently, by (see Fig. 1b) appears to be a sine wave at a
means of the ergodic hypothesisp the required single frequency but with pulsating amplitude;
assembly-averaging of random data can be hence the so-called sine beat vibration. The
replaced by the more simple task of time- probability density of the instantaneous values
averaging over a single record of long duration, of the filtered response tends to be normal, or
Gaussian (synmetrical)(5].
Because resonant vibration buildup is an
important engineering phenomenon, it is On the other hand, the density distri-
essential to evaluate the frequency content of bution function of the amplitude variation of the
seismic vibration excitation. For this purpose envelope of the random sine wave (see Fig. 1b)
the quantity "power spectral density" (PSD) is tends to be skewed. It. can be expressed
used as a measure of frequency content of explicitly by means of the well-known Rayleigh
random-type functions. (Important frequency distribution curve, which is employd in studies
effects in earthquake free-field accelerographs of cumulative vibration fatigue. It is also
appear to be limited to a range from 1 to 25 Hz.) called the two-dimensional error distribution
with reference to the "random walk" problem,
Figure 2 shows the frequency spectra which was first solved by Lord Rayleigh(6].
chart of PSD for a random broadband excitation Strength failures caused by random ceismic
and a narrow-band response, the latter demon- disturbances involve a relatively ifw load
strating the filter action of a simple oscillator, cycles, but they can work th, iiat. rial in the
Hence, the acceleration-time histories shown elasto-plastic range of cumulative fatigue.
in Figs. la, b have now been characterized by
means of a statistical analysis as plotted in A special case of-tho fluctuating sine
Fig. 2. (Earthquake grcind motion response wave is) of course, the true sine beat vibration
spectra usually appear as broadband excitation, which can be expressed in terms of two rotating
whereas building floor response spectra corres- (acceleration) vectors as follows:
pond to narrow-band quasi-resonance buildup at
one or more natural frequencies of the building.)

LIP
 Z ZZ;
[ArI Fl,t;-
l: 0'.1
- -- ,.3,--
-. ,,,g-.j•merit[8, hock response
93.
hh::~
table motions which, in turn,eisi
o develop test
specified
spectra in the mounted equip-

F+F.F F Also, steady-state vibration testing at

equipment natural frequencies is well-known in
terms of the Navy MIL-S-167 for shipboard
environmient. However, for seismic test
criteria the sine beat moion input is much
where F - natural (test) freq. preferred because it pro 'uces only a limited
of equipment, Hz quasi-resonance magnification and less cumu-
lative fatigue.
Sw freq. of envelope, HzA novel feature introduced by this pro-
(2 beat pulses) posed sine beat vibration testing is that the
test results and parallel design calculations for
test frequency, Hz the equipment can be related to the peculi-
2 (no. of cyc/beat) arities of the foundation or building seismic
response spectra as supplied by the Electric
Utility System fur their installations.
= p c 3 ) TRANSIENT ANALYSIS OF BUILDINGS AND
EQUIPMENT
In actual testing, the frequency of the The well -known normal-mode method
sine beat vibration of the test machine imunting can be applied specifically to mathematical
plate corresponds to each natural frequency in models of buildings and equipment inseries
order ofto
the equipment as determined by a continuous determine a dynamically equivalent
sweep frequency search from I to 25 Hz. dtrieadnmclyeuvln eiso
simple oscillators in terms of the uncoupled
Hence, the equipment's capabilities are evalu- normal-mode shapes and fixed-base natural
frequencies. In turn, the motion-time history
of the earthquake disturbance can be applied to
In general, environmental testing tech- each oscillator and a summation obtained of
niques are well-established and -several the total transient response of the building or
available for reproducing actual etoipm ent
earthquakeare
machines motion-time histories [7 , or test equipment.

Normal Mode Frequencies

~2Hz
6 Hz 22 H

(a)Dynamic Model of Building (b)Equivalent Simple Oscillators

Fig. 3 - Computer-aided dynamic analysis of building subject

to seismic disturbance

3
 However, a distinction~must be made FILTER ACTION OF THE BUILDING
between the broadband seismic iexcitation at STRIICTURE
ground level and the narrow-ltand floor motion
at the various building elevations-the latter Figure 4 illustrates how (a).the broad-
resultk.ng from the filtering action of'the mech- -band seismic excitation at the base of the
anical-soil-structure system. Alsol i 'the buil dtigcan be' (b) flteredW'andgniied by
following discussion an essential distinction the building structure, anl(c) amplified by the
must be made between the customary shock equipment response at the coincident building
response spectra used in design-- and the natural frequency of 6 Hz.
original motion-time history-used as the basis
for the equivalenitsine beat vibration testing. The resulting floor motion shcwn in
Fig. 4b conslsts of various harmonic oscil-
Figure 3 shows (a) a typical mathe- lations depending upon the different paths along
matical model of a nuclear power plant building, which the ground disturbance-has been propa-
and (b) thedynamically equivalent seriis of gated. Essentially, this motion under the
simple oscillators. Only the three lowest equipment -has been filtered at 6'Hz and magni-
modes of vibration with natuial frequencies of fied 2. 0 times by the building structure. In
2, 6 and 22Hz need-be considered, 'since'they other words, it now approximatesfa random
are the only ones calctated within the most sine wave as shown in Fig. 1b, and in-the form
imp6tant earthquake hazard range ibr vibration of an, equivalent sine beat vibratlon ciabe
buildup from about 1 to-25 Hz. (Similar freq- applied to the test machine mounting~plate in
uencies found'in the equipment;model are not order to'evaluate tfie seismic capabilities of
necessarily coincident with those in -the the equipment.
building.)
JIn turn, the test machine will duplicate
It Is appreciated that in complex struc- (or exceed) the resulting equipment response
tures there may be important cross-coupling motion shown'in Fig 4c, which occuriswhen a
effects and fluctuating response motions when natural frequency of the equipment coincides
natural frequencies are close together. with a natural frequency of the building
However, this situation is usually covered by structure. At 6 Hz with.5 p'rcenT damping,
specifying an envelope-type of building floor the magnification of the' floor motion in-the
response spectra. (Similarly, when there is equipment amounts to 5. 5 times. (An equiva-
interaction of two or more equipment modes, lent sine beat vibratio,itest using 5 cycles/beat
then a more severe response condition might at 6 Hz will produce this same magnification in
have to be recognized.) equipment having 5 percent damping.)

It should be recognized that when a

S.5pek Respon natural frequency of the equipment does not
coincide with a natural frequency of the building
1structure then the quasi-resonance motion
buildup in the equipment will be much less
evident than as slzwn by Fig. 4c. (In turn,
-19' " although a sine beat vibration test would be
-(c Manifiecd qulpinenio of 5 a 69 with applied at the equipment natural frequency, the
0.} 1
0 5Sine, Beat 416 Ii
SneBealal,,
, 2 9Peak input the
amplitude should be reduced to produce
,only floor response specified at the test
o Mf frequency, which no longer coincides with a
Time building structure natural frequency. The net
-0.5 Uresult Is a somewhat conservative sine beat
2x byBuIlding Strulure test compared to using actual floor motion
inputs.)
0.29 O,139 Peak Acceleration
0 . BUILDING EQUIPMENT "DESIGN"
I
-0~g
'
lime - PROCEDURE
StcureEqutpent
(a) Rjndm'type Ground Motion Input toComputer Modl of Figure 5 shows a semi-log plot of typi-
cal floor response spectra as supplied by the
Architect-Engineer of the building. It is infor-
Fig. 4 - Typical accelerogranis of OBE hori- mation derived from a transient analysis of a
zontal motion as transmitted by pertinent building structure model with a base
dymanic model acceleration input corresponding to a

4
 horizontal motion-time history for anOperl-tting
2:0 Basis Earthquake (OBE). Presumably the
_ Fmodel
_
Damping
includes a soil-structure interaction
factor, and the OBE has been established on
1 .i25% the basis of the seismic history of a specific
1.56 L9- power plant site.
1-15 The filtering action of the building, as
previously illustrated by Fig.. 4, was based
W 1.2 upon the data to be found and interpreted from
__ _
H 'random
Fig. 5. In other words, the ground motion is
with a peak value of 0. 13 g. At an
. 0.8- upper floor level where the equipment might
I I be located, the input motion has been magnffied
11
. f
-V2 times to give apeak value of 0.27 g corre-
sponding to the horizontal asymptote for freq-
* 0.iI uencies above 30 Hz. (Compare Fig, 4b with
Fig. 5). Finally, for 5 percent damping in the
equipment at 6 Hz, which is the coincident
00 1 13 building-equipment natural frequency; the
0.30.5 1 3 to 30 maximum equipment response acceleration
Equlpmenthf'erequency, Hz becomes 1.5 g. (Compare Fig. 4c with Fig. 5&)
5- OBE horizontal floor response spectra Figure 5 also gives floor motion
Fig. 5response spectra (actually the peak acceleration
at equipment location in building measured on the equipment at each of its natu-
ral frequencies) for several values of percent
critical damping, which can be determined for
the equipment from the motion buildup during
the continuous sweep frequency test. The

S-6.0 Hz, 2.7 g Peak Respon se 2.7 g

2g =5%, Free Vibration

-29-

(b) Test Equipment Acceleration

-Ig - ,0 Hz, 5 Cycles/Beat 2 Second Pause

0.49g Peak Input, Q =5.5 x
(a) Test Table Motion Under Equipment

Fig. 6 - Sine beat motion test for DBE 1.8 x OBE

5
 maximum overall magnification of the ground-
motion into the equipment at 5 percent damping 25
is 1.5 g/0.13 g = 11.5 times, which shows that
the selective filtering action of the building Il
structure can be quite. influential. (For the Steady Stale,
L El Centro earthquake at 5 percent damping-the
shock response spectra magnification is only •

The floor response spectra shown in 10Cycles/Bea

Fig. 5 can be used for equipment design 1-
purposes as follows. The peak horizontal re-
sponse acceleration at the equipment cg will be C I I,CNles/Beat
1.5 g at 6 Hz and 5 percent damping. This g 7C\ !esI-
means that 1.5 times the weight of the equip- Y 10 ----------..
ment should be applied as an equivalent static / . Random,
horizontal load at the cg in order to verify that E I _"
stressed members are below the yield point. L:/j \ >
Also, if the equipment natural frequency is 5 -- - -

30 Hz or greater, there is no further magni- &

fication ofrthe basic floor motion which is'onlyt -I-.. I
0.27g. 5 7 '
BUILDING EQUIPMENT "TEST" PROCEDURE 2 4 6 8 10
E, percent ofcritical damping
For seismic testing of equipment, the
vibration machine input is intended to simulate Fig. 7 - Vibration magnification at quasi-
the worst aspects of-the motion of the building resonance
floor at the attachment of the equipment. A
conservative approximation of the earthquake-
induced floor motion is applied by means of the equipment natural frequency. In view of the
sine beat vibration motion, cumulative fatigue effects associated with test-
ing the equipment in three directions (East-
In addition, since the probability of an West, North-South and vertical), it becomes
earthquake worse that the OBE is difficult to apparent that sine beat vibration testing can be
agree upon, the Intensity multiplier is used to made very conservative with regard to
achieve a more safe Design Basis Earthquake, probable local seismic activity.
whereby DBE = 1.8 (OBE). (In practice, the
Electric Utility Company and the Architect- In summary, it is Important to recog-
Engineer supply the foundation and floor re- nize that for design purposes the floor
sponse spectra with a statement regarding response spectra of Fig. 5 can be applied
ODE and DBE.) directiy. Howeverj for test purposes it ,.
necessary to apply the actual floor motion
Finally, Fig. 6a s huws that a sine beat shown by Fig. 4b, or else apply the conserva-
vibration input consisting of 5 cycles/beat at tive substitute shown by Fig. 6a.
6 Hz and a peak amplitude of 0. 49 p is the
appropriate test machine input to represent the SINE BEAT VIBRATION TEST METHOD
damaging effects of the floor motion under the
equipment as shown in Fig. 4b, since Figure 7 provides the theoretical data
1. 8 (0. 27g) = 0.49 g. Fig. 6b shows that the needed for the correlation of floor response
equipment response amplitude reaches a peak spectra (Fig. 5) and test table input (Fig. 6a).
of 2. 7 g and then gradually decays as a free Based upon the transient analysis of a linear
vibration at C Hz before the next sine beat mass-spring-damper model with a sine beat
vibration is applied, base motion excitation, the vibration magni-
fication curves (so-called Q-factor at resonance)
The total number of successive beat are calculated for both 5 cycles/beat and 10
effects applied to the equipment depends upon cycles/beat over a range of damping values
the earthquake magnitude, hence the time (i. percent of critical). This elementary
duration of the corresponding strong motion computer-aided analysis is equivalent to the
accelerograph. As many as 5 successive sine normal-mode evaluation of the equipment re-
beats have been used in actual testing at each sponse (see Fig. 4c and 6b).

6
 Figure 7 also shows the Q-factors for
steady-state vibration resonance (Q = 100/2 g )
and for random (white noise) excitation equal to 50
the square root of Q. The quasi-resonance .
buildups produced by the sine beat vibrations S Cri/ 15
fall'in between the latter two extremes. By 40 - 5 Damping /
Damplin -
comparison, seismic.motion response magni- 6OIl
f-ResConace Sle
fication factors fall soniewhat below the random Tests /
excitation values. The most damaging sections 2
of earthquake oscillographs usually correspond N
to a sine beat vibration excitation of about 3 2-_
cycles/beat at various preferred frequencies 1 15
from about 1 to 10 Hz.

The previous example for selecting a 1 -

sine beat vibration test corresponding to build- 5
ing floor response spectra can be summarized 4
as follows. At each experimentally determined 3: 0 3 o
natural frequency of the equipment, the-ordi- 0 5 10 1 2 25
nates for the OBE response spectra curve Resomant Frequency, Hz
should be increased by the DBE factor (1.8
times) and then reduced by the Q-factor (5. 5
times at 5 percent damping) in order to get the Fig. 8 - Chart for determining number of off-
peak value of the sine beat input acceleration resonance tests at 1/2 peak amplitude
(0. 49 g at 6 Hz), in this example for 5 cycles/
beat. Notice that at equipment natural freq-
uencies of approximately 4. 2 and 8.4 Hz, which usually evaluated by testing at successively
are non-coincident with the building structure larger g-levelsp the danger of fatigue failures
natural frequency at 6 Hz, the response at 5 is reduced because of lower average stressing.
percent equipment damping is only 0. 75 g
instead of 1.5 g maximum at 6 Hz. Hence, the OFF-RESONANCE SINE BEAT TESTS
sine beat input acceleration peak should be only
0. 245 g at 4. 2 and 8. 4Hz instead of the 0. 49 g The computer-aided analysis used to
at 6 Hz. This example shows the need for obtain the Q-factor curves of Fig. 7 has also
adjusting both building and equipment designs to supplied data for plotting resonance curves of
avoid coincident natural frequencies, magnification versus frequency from 1 to 25 Hz.
Somewhat arbitrarily at the 1/2 peak amplitude
It should be emphasized that in the ordinate, the corresponding (2) sideband freq-
example of Fig. 4a, the peak acceleration of uencies can be determined along the abscissa
0. 13 g corresponds to an earthquake character- for resonance curves at various values of damp-
ized as VII on the Modified Mercalli Intensity ing in the simple oscillator system.
Scale (6. 5 Richter Scale), meaning damage
"slight to moderate in well -built, ordinary struc- Figure 8 shows the above information
tures." However, as shown by Fig. 6b the plotted as a family of fan-shaped curves for a
peak acceleration of 2.7 g in the power plant sine beat excitation at 5 cycles/beat. The
equipment under test amounts to an overall upper and lower sideband frequency ordinates
magnification of 21 times, indicating that sine correspond to the intersections of a vertical
beat vibration testing can be made quite severe. 'line at a resonant (test) frequency abscissa with
the two radial lines drawn for some value of
This proposed test method is conserva- percent critical damping. (Similar curves can
tive in that should either the damping of the be plotted at 10 cycles/beat and steady-state, 4
equipment be evaluated on the high side, or the wherein the sideband widths becomes increas-
input "cycles/beat" selected on the low sidej ingly smaller.)
the net result in both cases would be a smaller
Q-factor and a larger test machine acceleration. The stair-step construction shown by
Also, compared to steady-state vibration test- the dashed-lines gives the maximum spacing
ing, the Q-factor near zero damping means only between "off-resonance" test frequencies, to-
a limited resonance buildup, In other words, gether with the assurance that at least 1/2 peak
when testing unknown systems there will be no amplitude will be excited.
destructive buildup produced inadvertently. In

7I
addition, since equipment fragility levels are
 Of course, when there are many equip- REFERENCES
ment natural frequencies distributed over the
range from 1 to 25 Hz, the stair-step construc- 1. John A. Blume, et al, Design of Multistory
tion may indicate that additional off-resonance Reinforced Concrete Buildings for Earth-
-tests are unwarranted. On the otherhand, when quake Motions, Portland Cement Assoc.,
there are only one or two obvious equipment 1961.
natural frequencies, or only building structure
natural frequenciei to consider, then they can 2. E. G. Fischer, et al, "Mathematical Model
be used to start the stair-step construction in Analysis for the Dynamic Design of
both directions to establish additional off- Machinery", SESA, Experimental Mech-
resonatice test points. anics, October 1967.

The primary purpose of the previous 3. S. 0. Rice, Bell Sys. Tech. J., 23, 282
development is to avoid unwarranted cumulative (1944) and 24, 46 (1945).
fatigue and wear of equipment being tested for
earthquake resistance. (As a practical matter, 4. J. W. Miles and W. T. Thompson:
typical complex electrical switchgear systems "Statistical Concepts in Vibration," Chapter
usually introduce nonlinear effects in terms of 11, Shock & Vibration Handbook, McGraw-
snubber springs, clearances, friction, cross- Hill, New York, 1961.
coupling, etc. Equipment resonances appeai
heavily damped, but persist over a relatively 5. J. L. Bogdanoff, et al, "Response of a
wide frequency range and they can be excited in Simple Structure to a Random Earthquake-
411 three directions of testing.) Type Disturbance," Bull. SSA, 51, 2 April
1961.
CONCLUSIONS
6. G. S. Mustin, "Theory and Practice of
In general, there are insufficient Cushion Design," Shock & Vibration Info.
strong-motion earthquake accelerograms avail- Center, U.S. Dept. of Defense, 1968.
able for power plant structures, let alone
specific types of equipment, to establish "an 7. J. Penzien, "Design and Research
acceptable seismic risk". Also, it is generally Potential of Two Earthquake Simulator
agreed that an isolated peak acceleration Facilities," Richmond Fie!d Station, Univ-
response is not a reliable indication of damage. ersity of California in Berkeley.
In switchgear equipment, where possible loss
of principle function is a more important con- 8. G. Shipway, OA New Technique for Seismic
sideraton, there does not appear to be any Shock Simulation,)) Wyle Labs; Norco, Cal.
obvious correlation with field service reports. and Huntsville, Ala.

The use of computer-aided analyses of 9. E. G. Fischer, "Design of Equipment to

buildings and equipment, and the specification Withstand Underground (Nuclear Weapon)
of earthquake response spectra, all emphasize Shock Environment," 28th Shock a.d
the importance of quasi-resonance phenomena Vibration Symposium, U. S. Dept. of
as the source of damage or malfunction. Henc% Defense, July 1960.
sine beat vibration testing appears appropriate
for evaluating complex equipment where it is
most vulnerable-at its natural frequencies as
determined by a vibration sweep at a reduced
acceleration input from I to 25 Hz.

Sine beat vibration testing produces

only a limited quasi-resonance buildup and
*avoids excessive fatigue in the equipment, both
of which conditions appev' typical of seismic
disturbances. Also, sine beat test inputs and
subsequent equipment resp-,ses can be directly
related to the foundation or building floor
seismic response spectra as supplied by the
Electric Utility Company. In general, the
severity of the test can be simply adjusted and
the test results can be made conservative with
respect to any specified seismic environment.

8
 DISCUSSION

Mr. Gaynes (Gaynes Testing Laboratories): How quencies, and determine the resonance modes by
-did you-monitor the relays and-swltches to-determine moitoiligl the dtaplaceiment of the specimen. We al-
wheter they were functioning or not functioning? so have gone quite a bit further. We have gone into
the shock specters"., approach by usigrandom trn-
Mr. Fischer: Usually we had an electrical hook- sients In determining the ability of the product to
up to an oscillograph element. The definition of real- withstand vibration. We will be presenting a paper in
function is quite a touchy point. In some applications, the near future on that particular comparison, and I
such as computers for the Safeguard system, any kind think you will find that your feelings are correct. Us-
of relay flutter is not allowed at all, so one practi- bg the sine beat seems tobe a more severe test than
cally has to use solid state circuitry. Again, depend- using random transients, but If the shock spectrum
ing on the application, somebody gives you a defini- can be adequately defned, I think that random tran-
tion of what constitutes a malfunction. There are ac- sients would be v.aperkir i;y to go.
celerometers located on the structure etc, but as far
as malfunctioning, it seems to be the electrical oper- Mr. Fischer: Well it could be more authentic. I
ation that is the most significant. Of course ifthe cir- agree, but I started out by saying: "In the interests
cult breaker pops open it is pretty obvious that you of simplicity we pretty much stuck to the sine beat.,,
are Introuble. We Justify the simplicity by-saying it is a conserva-
tive test. Frankly, I will take testing with simultan-
. Heous input anytime. With actual earthquake records,
Mr. Haag,(MTh Systems Corporation): We have as I believe Dr. Piunkett pointed out tids morning,
been performing several tests in the seismic shock you just think you are pushing something In acertain
area including the sine beat, and we pretty much con- direction. It can always escape at right angles. When
firm your approach. We feel it is a severe test. To you use three dimensional testing you never test
find the resonances, we do pretty much the same as nearly as severely as we are suggesting with sine
required in MIL-STD-167, where you sweep the ire- beat testing.

* -,-- -
'-. t t.W ' V ° ' ii , ' 4, H - o *. , - ,1$---
.o='A f. .. $,

SEISMIC EVALUATION OF ELECTRICAL EQUIPIENT

FOR NUCLEAR POWER STATIONS

Robert H. Prause and Donald R. Ahlbeck

BATTLLE, Columbus Laboratories

Columbus, Ohio

A review of current seismic specifications for electrxcal equipment and

the different types of vibration excitation that might be used to satisfy
these specifications led to the design and construction of a seismic
vibration facility with the capability of reproducing typical building
floor earthquake acceleration time histories. The design of the vibra-
tion table and control system and the development of the acceleration
time-history command signal are discussed. A comparison of the acceler-
ation response spectra for the table motion and a typical command signal
demonstrates the feasibility of using this type of realistic simulation
of earthquake motions for evaluating electrical equipment performance.
Significant results from this work are included to provide equipment de-
signers with some typical experience.

INTRODUT1ION analysis whether the electrical components will

perform reliably without damage or temporary
The design of nuclear power gener- interruption of operation during a seismic
ating stations requires the consideration of disturbance. As a result of this conclusion,
many types of possible accident situations in a laboratory facility for the seismic evalua-
order to insure that the public is protected tion of Class I electrical equiptrent was de-
from potential exposure to nuclear radiation, signed and installed early in 1971 at the
The considaration oi earthquake effects is Battelle-Columbus Laboratories. The design of
particulLrly important because the forces can this facility was based on the desire to
be extremely large and all pacts of a nuclear achieve a realistic simulation of the earth-
station could be affected simultaneously, quake vibration environment.
Contrary to popular opinion, nearly all loca-
tions in the United States have a history of
earthquakes, although the frequency of occur- SEISMIC RDUIRENTS FOR EECTRICAL EQUIPM T
rence and severity vary considerably. As a
result, ail nuclear power generating stations The components and structures of a
being built in this country are designed to nuclear plant that are critical to the -hut-
withstand earthquake motions even though they down and maintenance of the reactor in a safe
may be located in regions where earthquakes are condition are designated as "Class I". These
ignored for other types of structures. must be designed to remain functional when
subjected to the Design Basis Earthquake (DBE),
Modern dynamic analysis techniques sometimes referred to as the Maximum Credible
have proved to be valuable tools for predicting Earthquake, that has been selected for the
Lhe re ,ponse to earthquake excitation of build- particular plant location. Structures and
ing structures, piping systems, and many types components whose failure would not interfere
of mechanical equipment [1]. Electrical equip- with a controlled shutdown of the reactor or
merit, however, often consists of a light frame contribute to an excessive release of radiation
strcture supporting flexible racks and panels ar generally designated as "Class II", and
to which a variety of components such as trans- these are subject to less severe seismic de-
formers, relays, switches and meters are sign requirements.
mounted. Analytical techniques can be used to
determine the adequacy of the basic structure Manufacturers of Class I electricali
of an electrical equipment cabinet to survive equipment are now being required to verify
an earthquake; however, it is considered that their equipment is capable of acceptable
impractical, if not impossible, to determine by performance during a seismic disturbance,
Preceding page blank 11

_ _ J
A basic problem faced by these manufacturers is the acceleration time histories of the floor
that only rarely are those persons responsible response could be included directly in equip-
for the design of this type of electrical ment specifications, it has been customary in
equipment also knowledgeable in earthquake earthquake eniineering to describe the equip-
engineering technology. Adding to this prob- ment vibration environment by a response spec-
le is the fact that there are no standard trum. The use of a response spectrum-has
requirements for seismic evaluation, so a particular signLfcance:as& methbdof char-
manufacturer receivesa-different set of ieis- acteriziing a complex ti.ansient vibration in a
mic specifications for each new nuclear power way that is useful lor determiniag h.
jenerating station. While-it is expected that simple structure will respond to that vibra-
the severity of the DBE wiii vary for differ- tion. In particular, the earthquike-response
ent locations in the country, there is also spectrum for a building floor ie the maximum
considerable difference in the type of speci- response of a series of 3ingle-degree-of-'
fications being used to assure compliance with freedom oscillators that are excited by the
seismic requirements. The greatest source of floor motion. Each oscillator has a fixed
confusion from nearly all types of seismic percent of critical damping, but a different
specifications is5the use of response spectra natural frequency, so the series of maximum
as the principal means of describing the earth- responses gives a good representation of the
quake vibration environment. Therefore, a frequency content of the floor motion. The
brief review of the development and implica- floor response spectrum does not describe the
tions of earthquake response spectra for acceleration versus frequency characteristic
earthquake engineering seems justified, since of the floor motion directly,,and this is the
experience has shown that this-type of descrip- principal sourie-of misun4derstanding-by those
tion often confuses even specialists in other who are familiar with the-use of "the.Fourier
1
areas of structural dynamics and vibrations. spectrum or Fourier series to describe cotp ,ex
signals.

EARTHQUAKE'RESPONSE SPECTRA

The design of each Class I building

for a nuclear power generating station requires
a detailed analysis of the response of that 0
building (deflections, accelerations, stresses,
eta,.) to a typical earthquake ground mction
selected for that location. The earthquake
ground motion will have acceleration components
in two orthogonal horizontal directions of
about equal intensity and a vertical component 2i
that usually has lower peak accelerations, _
although the energy is concentrated at higher .0
frequencies. Fig. 1 shows the component of p
ground acceleration recorded at El Centro, 0 --
California, during the earthquake of May 18, Z
1940. Although the ground motion lasted for 0-.-
about 45 seconds, the most severe part of the '$' - 4
earthquake occurred during an interval of only -'
10-15 seconds. The maximum acceleration of 0 4 8 1 16 20 32
0.33 g measured at El Centro is the strongest T*X,SECONDS
ground motion that has been recorded (the
authors have no information on very recent Fig. I - Ground acceleration record of
earthquakes such as the one in Los Angeles in North-South component of El
1971). Data of this type are often used to Centro, California Earthquake,
establish the DBE ground motion at nuclear May 18, 1940
plant sites by retaining the time history and
by linearly scaling the amplitudes to achieve
a desired maximum acceleration.

The earthquake analysis of a Class 1 0411 0711,+.o3'o . .

building requires a detailed mathematical 0
model of the building's stiffness and mass y
distributions. The predictions of the build- o I
ing floor motions in response to the DBE ground .o. . .. . .
motion are then used to establish the seismic o
requirements for equipment that will be lo- T... 4s ,
cated in that building. Fig. 2, for example,
shows a typical 5-second record from the most Fig. 2 - Typical horizontal building
severe part of the acceleration time history floor acceleration for DBE
preulcted for the floor of tie auxiliary
building of a nuclear power station. While

12
 The diagram shown in Fig. 3 of a Fig. 4 shows an acceleration response
Sseries of mechanical oscillators resting on a spectrum for the horizontal floor acceleration
building floor ha proved useful for explain- time history shown in Fig. 2. Most of the
ing the response spectirum. The floor ancel- horizontal vibratory eitergy fron earthquakes
eration time history, '(t), would be predicted occurs at frequencies below 10 to 15 Hz, and
from a structural dynamics analysis of the the largest response of the building occurs at
b-ilding response to the DBE. The equation of its natural frequencies. Therefore, the most
motion for any one of the michanical oscilla- severe vibrations will be experienced by floor-
torn is wounted equipment with natural frequencies
close to those of the building. Equipment
2 which is quite rigid (natural frequencies
n 2nzn(t) nn above about 30 Hz) will follow the floor motion
exactly. Therefore, the acceleration ampli-
where tude which is approached asymptotically at high
natural frequencies on the response spec.crum
zn(t) - X(t) - y(c) (2) is identical to the maximum floor acceleration
n+ that would be observed from the floor" accel-
is the relative displacement between the mass eration time history (see Fig. 2).
and the floor, is the undamped natural fre-
-quency of the nth oscillator, and C is a
selected value of critical damping ratio.
The'ire are many techniques available for cal-
culating the displacement response time his-
tory, z(t), and the maximum absolute value 60

-the response, Iz Imax , can be obtained for

-of
.eachoscillator. Tie process can be repeated
4
30

to determine a family of curves of 1z I ma 20

versus natural frequency, wn',or diferel .%
values of damping, and this is the displace- c _%
ment response spectrum for the floor acceler- 1%
ation, 1(t). It is a matter of choice whether Jc oGO
the maxicul response parameter of relative
displacement tz,,I x.,relative velocity 04
[in max' or absolute acceleration 1"n + max
lu 0
is plotted, because they are relAted for zero o3
damping by the following equations:
'n 0 /|I op 1I 1.1 I 1 1 1 1 I
111 1 1 I
max ax 0 03 0 O 0 1 20
03 40
30 O0 20 30 40 0

r, rvency. "erttz
NOalwaO

2 .1 (4) Fig. 4 - Floor horizontal acceleration

I n+ I
mAx
- wunIZnI
max
response spectrum for DBE

A detailed discussion of the approximations SEISMIC VIBRATION STUDIES

involved for small damping can be found'in _
Reference (2). The use of the response spectrum has
some important advantages for designing struc-
tures to withstand earthquakes. However, its
use as a specification of the earthquake
vibration environment for equipment evaluation
has some significant limitations resulting
from the absence of all phase angle informa-
tion. This makes it possible to devise any
number of vibratory motions that will satisfy
• " X1 0, X (1) X3 (t) ftWthe damping, of whilch for
many spectrum
response a have
will selected value
little of
resem-

blance to a typical earthquake floor acceler-

ation time history.
Le_ Some of the different types of
Floor vibration currently being used to evaluate
electrical equipment are decaying sinusoidal
Fig. 3 - Series of sechanical oscillators excitation, sine beat excitation, and constant
used to determine floor moLion amplitude sinusoidal excitation. All of these
response spectrum employ single frequency excitation so that a
number of evaluations at different frequencies
are required to cover the response spectrum.

13
 Also, the response spectrum amplitudes for the spectrum of horizontal acceleration, with
these motions vary more than the earthquake minor peaks at 1.5, 2.2, and 6.5-Hz. Negli-
response spectrum for changes in damping. gible response was found above 10 Hz. The
For example, the peak response spectrum ampli- vertical acceleration power spectrum showed a
tude of 1.1 g at 8 Hz for 5 percent damping predominant peak at 1.0 Hz, with lower ampli-
shoin in Fig. 4 can be satisfied by using a tude peaks from 7 to 11 Hz and from 17 to 21
0.11 g, constant-amplitude (Q - 1/2C - 10), Hz. Thecs amplitude and frequency reuirements
sinusoidal vibration at the equipment base. led to the selection of a servovalve-controlled
However, if the equipment being evaluated has hydraulic actuator to power the vibration
its lowest resonance at 8 Hz with only 2 per- table.
cent damping (Q = 25)9 the sinusoidal excita-
tion would produce a peak acceleration of 2.75 Since the type of equipment to be
g compared with about 1.6 g'Sa for the floor evaluated by this seismic vibration simulator
earthquake motion--an "overtest" of 70 p'rcent. was not expected to respond to large displace-
The use of sine beat excitation is preferable ment components at low frequencies (below I
to constant-amplitude sinusoidal excitation Hz), one of the first design decisions was to
because the variation in the response spectrum limit the total actuator stroke to 6 inches
amplitude with damping is less, but it is still (peak-to-peak). This was done in order to
different from that of an earthquake motion. minimize the entrapped oil in the hydraulic
This requires making conservative estimates actuator, thereby keeping the oil-column
of the expected equipment damping or measuring stiffness high and maximizing the high-fre-
equipment damping at resonances-- a procedure quency response capability. To keep the
which is both expensive and possibly inaccurate commanded displacement within this 6-inch
because of nonlinearities. limit, a high-pass filter was used In series
with the acceleration comand signal to
The alternative of using the maximum attenuate frequency components below 1.5 Hz.
flooracceleration amplitude to test at all Analog acceleration signals were recorded on
frequencies may also severely overtest equip- FM tape by digital-to-analog conversion of the
ment. Furthermore, equipment with several original digital computer data for use as
resonant frequencies below 30 Hz will not simulator command signals. Command acceler-
respond to single frequency excitation in the ations and the resulting filtcred displaciments
same way it would respond to an earthquake used as a basis for the facility design are
motion, and the large number of evaluations shown in Fig. 5. Peak accelerations of 0.37 g
required might produce fatigue failures that in the horizorital and 0.25 g in the vertical
would never occur during an earthquake, direction were recorded during five seconds
of the floor vibration response data for the
For these reasons, it seemed desir- DBE.
able to develop a more realistic simulation of
the earthquake vibration environment. The The design of the seismic vibration
principal goal was to be able to evaluate facility included consideration of several
equipment efficiently while reducing the tradeoffs necessary to achieve a "cost-effec-
possibities of subjecting it to vibration tive" facility. Simultaneous vibration in
which might be considerably less severe or more vertical and horizontal directions, while
severe than necessary. It would be quite dan- desirable for achieving the most realistic
gerous to "undertest" the equipment, while simulation, was not incorporated because of
overtesting might result in an unnecessary its considerably greater cost and complexity,
increase in the equipment cost. The coupling effects of simultaneous vertical
and iorizontal floor motions during an actual
earthquake ure reduced in importance by their
SEISMIC VIBRATION FACILITY DESIGN considerable difference in both amplitude and
frequency content. It is quite unlikely that
The first task in designing the the maximum accelerations will occur simmlta-
Battelle-Columbus seismic vibration facility neously in time for the two directions.
was to deffne the amplitude and frequency
response require-ents needed to reproduce the Physical Nize of the simulator was
typical floor acceleration time history shown based on an estimate of equipment sizes to be
in Fig. 2. Preliminary procesaing of the evaluated in the foreseeable future. T Is
acceleration data consisted of numerical limit was chosen to beobout 3000 pour .sin
integration to obtain velocity and displacement weight with maximum base dimensions itabout
signals. The horizontal displacement reached 80 by 48 inches.
* a peak amplitude of 12 inches, with a predom-
inant 0.3-11z frequency plus some additional To provide a table wit's minimum
lower frequency component (;he displacement weight and maximum stiffness, o equilateral
did not return to the initial position within triangle of steel I-bears with three angle-
5 seconds). Acceleration power spectral den- bisccting beams In the center was designed.
sity curves for the 5-second signals were The single hydraulic actuat-.r is attached below
generated by Fast Fourier Transform techniques the table centroid, and the actuator can be
to provide some idea of frequency content, rotated 90 degrees to provide motion in either
Large peaks were found at 0.3 and 0.9 Ilzin vertical or horizontal directions. Normal

!4
 [ Rqproduced 'fn
: modes of vibration, of the table when actuateit O
lvailable COPY-
at the centroid vere calculated by means of a

finite-element digital computer program, and

.ll natural frequencies were predictad to
exceed 50 Hz. Support and guidance of the
table in the horizontal mode are providcd by
Thomson "Roundways", while in the vertical
mode a combination of Roundwtys and linear ball -
bushings is used for guidance at the three -

corners of the table. Three Firestone "Airlde'

springs connected through a pressure regulatev
to shop air are used to support the static
weight of the table and equipment for excit,-
tion in the vertical direction. Fig. 6 shris
the seismic vibration table pos.tioned for
vertical motion.

,4 ,qo . _. F!g. 6 - Battelle Seismic Vibration

a. Hofqlato Floor Acceletolson Table In vertical mozde

Conflicting requirements of stiffnes

(oil-column resonance) and maximu oil flow
t rate at peak vclocity dictated the choice o
actuator size. A -1/2-inch bore, I-nhrd
93
on. I , double rod-end cylinder was chosen to rate with
V r an available 3000-psi, 25-gpm hydraulic supply
and loog 76-104 (15-gp) flow control servo-
b Horizontol
FWoo Oocement valve. For this size of actuator, a miim.z
oil-colu=n resonant frequency of 22 11zwas
estimated based on an assumed effective oil
4 . bulk modulus of 100,000 psi (in tests this
m , * .
' I ' &" " .
IO •Since the hydraulic power supply is
I • 25Gmnomum 414"
9. .. located some distance from the simulator, a
1-gallon bladder accumulator, 10-micron fi!ter,
c Verhicol Floor Acceleroton and solenoid shut-off valve are located at the
. ,base of the vibration table, with short lengths
j of hose supplying fluid to an actuator-mounted
47,n . manifold. The servovalve and crossover relief
valve are attached directly to this manifold
j j toiprvd side
"active" minu enrpe
of the i
servovalve. onSafety
the fine-
-:. H tions (over-travel limit switches and servo
Floor msvoement
d veW,cot over-coand voltebe level) deenergize the
shut-off valve in the event of a system fail-
Fig. 5 - Acceleration and filtered ure and stop the table with a maximum qccel-
displacement time histories oration of about 0.75 g.
of bilding floor response
Design of the servo controller was
An I-beam frae is used to provide based on the salient fact of life of a
a base for accurate alignment of the guideways hydraulically actuated inertial load: the
and actuator. This base is fastened to a oil-column resonance (3). This resonance
heavy "strongback" formed of 36-inch-deep I- results from the mass of the table and equip- +
beans located beneath the floor level of the ment oscillating on the effective stiffness
laboratory. The strongback, fastened to a of the entrapped oil, piston red, and actuator
heavy concrete slab, provides the necessary support structure. Without compensation, the
seismic mass against which the simulator can oil-column resonance adds a pair of lightly
react, damped cnaplex poles to the root locus plot.
Several methods can be erployed to control the

15

L *
 effects of oil-column resonance [4]: accelera- Fig. 8 shows the control system in
tion feedback, pressure feedback (inconjunc- block-diagram-form with linearized-transfer
tion vith a pressure control servovalve), or functions for the major cimponents. Primary
controlled bypass flow, for example. servo-system feedback is derived from a post-
tion transducer (DCDT) mounted on the actuator.,
Adjustable bypass flow using a nee- The positin feedback signal is subtracted from
die valve and small tube across-the actuator the filtered, double-integrated acceleration
is an effective, yet inexpensive, method of command signal to provide the error-signal to
c~ntrolling the resonance. The effects of the servovalve. A &odifiedlacceleration feed-
introducing bypass flow (based on a linear back signal from a strain-gage accelerometer
estimate of system response) are sketched in mounted on the table is also summed with the
the root loci of Fig. 7. For comparable error signal to improve system response in-the
values of forward-loop gain (the parameter Ka 10 to 20-Hz range.
represents servo amplifier gain in amps per
volt), the system with bypass flow is far more
stable--this is shown by the reduced angle of a
vector from the origin to the gain-dependent .. ,'.
root. One disadvantage of bypass flow is the
lower frequency of the resonant peak for desir-
able levels of damping.

Rod 20/
Sec 1200
OO
X×z .0 Fig. 8 - Seismic vibration facility
Jos control system

1100
SEISMIC VIBRATION FACILITY PERFORANCE
at 5 05 0o Closed-loop frequency respons:
=Rod 'oo .1"' evaluations were conducted with a '1500-pound
sec dummy equipment load on the table at normal
acceleration amplitudes. A constant-amplitude
iloo voltage command to the double integration
0 No bypol flow circuit wgas varied from 1 to 40 Hz, and the
resulting table acceleration was monitored
using a Quan-Tech Model 304TD-wave analyzer to
0 track the fun "uental component of acceleration
with a 1-lz bandwidth filter. The results of
the sinusoidal closed-loop frequency response
evaluation are shown in Fig. 9. A second-order
•Rod resonance with a + 3-dB peak ic evident at 10
See Ht, and the effective system bandwidth (-3 dB
Zo 2oo point) is about 14 Hz for the command signal
1amplitude of 0.2 g.

0While the closed-loop frequency

o)
response of oftheaccurately
Icapability control system determines
reproducing the
a complex

"a o o0oo ,0 0 o2 X$o, acceleration time history,

eration determined the maximum
by the actuator accel-
stroke,
e oo flow rate, and force limitation is the most
significant parameter for any type of sinusoi-
,'altesting. Fig. 10 shows the maximum table
,,o accelerations that have been measured for the
byaw frequency by the
are limitedrange
bctuntor of maxiatum
1 to '5 lci
. Accelerations
X flow rate of the
S lservovalve at frequencie from abo t 1 to 10
hydruicsly and by the taximum
actuator force of about
r , ld7500 pounds
maximn, floorataccelerations
higher frequencies. Since the
seldom exceed 0.5
i Fi. 7 Th effct f acuatr byassg for a DBE, the acceleration capability is
Fig.7
efec ofactato
Te byassadequate for frequencies above 2 liz. Lower
flow on root loci of the lower- frequencies are of little practical interest
frequency tormst hydraulic eye- because the lowest equipm~ent resonant fre-
tew With Inertial load quencies are seldom below 5 Itz,

16 *

. iU
 ance by comparing the time histories of the
comand and output signals. However, a com-
parison of the response spectra, Fig. 12, for
0the floor-acceleration command signal and the
o* resulting table acceleration indicates accept-
04. able agreement. The amplitude of the tabl e.
01
2 0 comand ,al acceleration can be increiiid above the 0.37-g
maximum accelerution for the DBE so that. the
response spectrum of the table motion exceeds
o-the
0 required spectrum at all frequencies where
any type of equipment resonance might occur
t (i.e., above 4 to 5 Hz). The increase in amp-
3 litude required to compensate for filter char-
01
oacteristics can be reduced by using an in-
0001 creased rate of filter attenuation (sharper
filter) in the control system or by-preprocess-
00 Equgment Weight- 1500:b ing the original floor-acceleration time his-
05. Mtories with digital computer filter algorithms
o
04 prior to obtaining an analog signal. High-
i ~03 ! ' I actuatori
frequency noise can be reduced by using an
S3 actuator with low-friction seals and extending
requency, orti, the control system bandwidth by improved com-
Fig. 9 -Table acceleration response pensation for the oil-column resonance.
to sinusoidal, constant-
amplitude acceleratiorn comoand
signal

; ~~~p [ ri, ccor -- -

6.0- 0. Horizontol Floor Accsleration

(Al . .. .. . . . . ---
. - - --- --
2.0-
3.0-

06 b. Horizontal Toble Accelerotion

0.6 Fig. 11 - Comparison of horizontal table

Equipment acceleration with DBE floor
O Curve Weight, lb acceleration comnand signal
o A 1500
o 3000 SEISMIC EVALUATION RESULTS

The seismic vibration facility hap

I 0I I 111 I I I I I been in operation at Battelle-Columbus since
1. 2 3 4 6 a 10 20 30 40 60 February, 1971, end seismic evaluation pro-
Frequency, Hertz grams have been completed for a variety of
Class I electrical equipment. Results from
Fig. 10 - Maximum acceleration this work are cuumrrized briefly to provide
capability of seismic equipment designers with some typical data
vibration table for on the natural frequencies, damping factors,
sinusoidal excitation and electrical performance of this type of
equipment. Fig. 13 shows a typical electrical
Figures 11 and 12 show the capability equipment cabinet mounted to the seismic
of the vibration table to reproduce a floor- vibration table and being subjected to earth-
acceleration time history. The results In quake excitation in the front-to-back hori-
Fig. 11 show that the use of a high-pass filter zontal direction. The equipment is energized
to attenuate low-frequency components and the electrically during the seismic evaluation
presence of some high-frequency noise makes it and all critical electrical performance para-
difficult to evaluate control system perform- meters are monitored continuously.

17

LI
 VIBRATION CHARACTERISTICS

It is a general practice to conduct

a brief vibration survey of each equipment
40: ~unit in order to identify major natural fre-
20% rCo.. d-,i quencies pr'or to evaluating its electrical
.10
1100o eqpmtft e.qhl purformance inder specified seismic conditions.
• oTable 1 lists the natural frequencies which
have been measured for a variety of electrical
requipment. These natural frequencies-were
1° ,eonse
To*I- determi.ed by comparing the acceleration amp-

jO 0540esonso046 litudes from accelerometers located at the

i -base and near the top of the equipment cab-
- 0376 inet. While this procedure is satisfactory
03o for determining the major structural natural
02 frequencies of the cel,'4 -?s, the identifica-
02 tion of local panel resonances and resonances
of components mounted inside the cabinet
0 02O
0 1 b 1SI
, ,, 2I F040i0tnc,
0at,r3l3 4 ,,requires
6 Hes
6 -0 20 30 40 60 more
practical. extensive an
Therefore, measurements than are
evaluation procedure
which is based on exciting-the equipment cab-
Fig. 12 - Comparison of vibration table inet with sinusoidal vibration only at natural
horizontal acceleration response frequencies has the risk of missing a reson-
spectrum with floor response ance of some small component that could cause
spectrum for DBE an electrical failure.

TABLE I
Typical Natural Frequencies of
Electrical Equipment Cabinets
Equip. Natural
Item Equipment Wt., Frequencies, Hz
No. Size, in. lb Horiz. Vert.

i -
:•,1 . , 36 24 90 2350 4.5,
5.0 >25

2 36 24 qO 700 9.5, >25

10
3 60 20 90 3500 7.5,
7.5
4 36 24 56 1200 11, 20
12.5
5 30 30 75 800 10.5, >25
.. ... 11.5
6 60 30 75 2000 9.5, >25
10.5

The data in Table 1 indicate that it

is unrealistic to expect equipment of this
5_ type to be designed to have horizontal natural
frequencies above the maximum excitation fre-
quency for earthquakes (about 20 l1z). Natural
frequencies shoikld be increased as much as
possible during design by locating heavy com-
por.ents near the cabinet base and sizing the
structural members at the cabinet base to
transmit the inertia: loads to the mounting
bolts without exceesive deflections. However,
experience indicates that it is even more
Fig. 13 - Time exposure of typical equip- important for designers to consider the de-
ment during horizontal vibrations tails of the mountings used for the components
inside the cabinets. A brv..en mounting bracket
for a transformer weighing about 100 pounds is
the only structural failure of any type that

18

4 I, . ... .. , . , , ,,.. .... . ::< _., , . , :.., . . . .

 has been observed. However, even this failure CONCLUSIONS
occurred only.after extensive exposure to
sinusoidal excitation at resonance caused a The resultw'which.have been achieved
fatigue condition that would not be expected to date indicate the'feasibility of using the
froman earthquake, actual floor-acceleration time-historiespre-
dicted for a DBE to evaluate electrical.equip-
The data in :able 2 show the appar- ment performance. The use-of tlhs-type of
ent magnification factors and corresponding realistic excitation substantially reduces the
critical damping ratios measured while a typ- possibility of overtesting or undertesting the
ical equipment cabinet was excited at reson- equipment. Furthermore, the vibration evalu-
ance at different acceleration amplitudes. ation portion of a -sismic evaluation program
The selection of an appropriate damping factor can be reduced substantially, since a prelim-
is oue of the most difficult and important inary survey for natural frequencies is un-
judgments that must be made for any type of necessary and the total exposure to the earth-
dynamics study. This type of equipment usually quake environment can be limited to about 60
has damping factors no lower than about 3 per- seconds in each direction.
cent, and in most cases the damping is in the
range of 5 to 10 percent of critical. However, It is expected that the appropriate
the increase in damping with vibration ampli- acceleration time histories can be supplied
tude shown in Table 2 can be expected for this by the nuclear utilities once the advantages
type of equipment, indicating that nonlinear of using a more realistic earthquake simulation
effects are quite significant, are recognized. However, there are-some cases
in which the use of an artificial earthquake
signal has some important advantages. This
TABLE 2 type of signal can be generated using digital
Effect of Acceleration Amplitude On or analog techniques to modify wide-band noise
Damping or ATpcraloElctritce
Damping For A ypical Electrical EEquip-- with appropriate filters to produce a time-
history signal having any desired response
ment Performance spectrum characteristic. Battelle's digital
Equip. Equip. computer facility has been used to generate
an artificial signal having a relatively flat
Base Top Magnifi- Damping, (constant amplitude) acceleration response
Accel.,
Acce.* Accel.,
cc8 cation
Factor Percent
Critical spectrum over the frequency range of 2 to 20
F C Hx. This type of excitation is desirable for
evaluating equipment that will be installed
0.11 1.2 11 4.5 in several different nuclear power generating
028 6stations where the seismic requirements are
0.25 2.1 8.4 6.0 similar but where the particular time his-
tories and response spectra depend on the
0.39 2.6 6.7 7.5 different building natural frequencies and
location of the equipment in the building.
0.67 2.9 4.3 11.6

ACKNOWLEDGMNT
* - Constant-amplitude horizontal
sinusoidal vibration at 8 1z The authors are grateful to the
natural frequency. assistance of Mr. Charles Rodman and the other
members of the Mechanical Dynamics Division at
Battelle who have contributed to the design
EQUIPMNT PERFORMANCE and operation of the seismic vibration facili-
ty. They also wish to express their apprecia-
It is encouraging to report that most tion for the cooperation and encouragement of
of the electrical equipment that has been sub- Mr. Julius Tangel of the Public Service Elec-
jected to vibration simulating the DBE for tric and Gas Company.
nuclear power generating stations has performed
(satisfactorily without any significant design
changes. No examples of cabinet structural REFERENCES
1'damage to the main structural members or hold-
down bolts have been observed. The few elec- 1. John A. Blume & Associates, "Summary of
trical failures that have occurred were usual- Current Seismic Design Practice for
ly caused by high-voltage arcing or relays Nuclear Reactor Facilitios", United States
which malfunction. Failures have been ob- Atomic Energy Report TID-25021, Sept. 1967.
served in some meters which are often mounted
on a flexible door panel, but a failure in 2. R.D. Kelly and C. Richman, "Principles and
this type of component seldom effects the Techniques of Shock Data Analysis", SVM-5,
primary functional performance of the equip- The Shock and Vibration Information Center,
ment Naval Research Laboratory, Washington, D.C.,
1969.

19
 3. V.H. Larson, "The Control of Acceleration
by Electrohydraulic Shaker Systems", 145
Systems Corp., Technical Bulletin 840.00-1.
4. L.H. Geyer, "Controlled Dampin Through
Dynamic Pressure Feedback",, Moog, Inc.
Technical Bulletin 101.

20

S..' V t J%<AC ~I A.m

4 ~~ 4,~ "~ Y .--
 SHOCK INPUT FOR EARTHQUAKE STUDIES USING GROUND MOTION
FROM UNDERGROUND NUCLEAR EXPLOSIONS*

D. L. Bernreuter, D. M. Norris, Jr., and F. J. Tokarz

Lawrence Livermore Laboratory, University of California
Livermore, California

Comparisons are made between the ground motion from earthquakes

(EQs) and underground nuclear explosions (UNEs). It is shown that
peak g-levels and response spectra form a reasonable basis for com-
parison. Several approaches which attempt to characterize the time
history are also discussed. It is shown that (1) the peak g-levels-
from UNEs have magnitudes comparable to those estimated for the
strongest EQs, (2) the spectra from UNEs are similar to those from
EQs, and (3) a time history comparison shows that both the duration
of strong motion and the number of near-peak g-level cycles for
UNEs fall within the'range established for strong EQs. Based on
these results it is concluded that ground motion from UNEs can pro-
vide an EQ-like environment for testing full-scale structures.

INTRODUCTION rent methods of formulating mathematical

models of real structures are not yet developed
The recent San Fernando earthquake to the point where they can readily produce ac-
(M = 6.3) provided the first real test of curate models for most structures. This is
earthquake-resistant design as practiced in true even for very low level seismic input, as
California. Many structures failed the test. shown by Tokarz and Bernreuter [1). It has
This earthquake caused 64 deaths and an esti- been very difficult to correlate results obtained
mated $553 million damage. The major cause from analysis with actual dynamic behavior of
of death was catastrophic collapse of multi- full-scale structures, principally because not
story buildings. much detailed data exists on actual dynamic
behavior of structures during earthquakes.
Earthquake-resistant design criteria
used in California building codes are based on Dynamic tests have been conducted on
static approximation of the dynamic loads pro- some large-scale structures in order to de-
duced by earthquakes. These criteria require velop more accurate mathematical models [2J.
that the structure be capable of carrying a set However, because of the power and mass lim-
of static lateral loads whose magnitudes and itations of existing vibration generators, the
distribution are chosen so as to approximate amplitudes of vibration in these tests have not
the effects of the dynamic loads an earthquake been large enough to cause substantial inelastic
might be expected to produce. In view of the deformation. The dynamic response informa-
damage caused by the San Fernando earthquake, tion is therefore valid only for small ampli-
it is clear that this static approximation tudes of vibration. Since substantial inelastic
method isinadequate, deformation usually occurs prior to cata-
strophic failure, it is most important to treat
The design of structures to withstand EQ the vibration problem at the relevant ampli-
(earthquake) forces requires a more detailed tudes. This highly nonlinear behavior makes
theoretical approach-one involving thorough it extremely difficult to apply the data that can
dynamic analysis-to guarantee basic surviv- be obtained from model tests [21. The cost of
abTlity. Significant capability for such anal- building a shaker facility large enough to test
ysis now exists with the computers and soft- full-scale structures has been estimated to be
ware presently available. Computer methods approximately $20 million [31. Even then the
produce a dynamic analysis by idealizing the size and type of structure would be severely
real structure into a mathematical model and limited.
then determining the response of the model to
some prescribed ground motion. However, due An alternative approach would be to use
to the lack of sufficient experimental data, cur- the ground motion from an underground nuclear

Work performed under the auspices of the U. S. Atomic Energy Commission.

21

?4
 explosion (UNE) to excite properly located In addition, no clear relation between EQ
structures to EQ-like ground motion. Several magnitude and peak g-level has been established.
major advantages of using the ground motion Ibis is shown in Figs. 1 and 2. Figure I gives
from UNEs are: (1) the ground motion is avail- a correlation-of g-level vs distance from the
able free as a by-product from nuclear-tests, closest point of observed faulting. The EQ
(2),there is no limit on the size or type of magnitudes range from 5.5 to 8.3. It-should be
structure that could be tested, and (3) true noted that the peak g-level from the San
soil-structure interaction would be achieved. Fernando EQ-was around 1-lg. Figure 2 gives
a correlation by Houcner (111 relating EQ m ag-
The ability to test any type of structure nitude to peak g-level and range. Also shown
and the achievement of true soil-structure . in this figure are the Parkfield EQ (M = 5.7)
interaction are important. For example, under- and the San Fernando EQ (M = 6.3). As can be
ground nuclear reactors and storage containers seen, these clearly do not fit Housner's pro-
can be tested. In fact, for marr types or large jected correlation.
structures this technique of subjecting the full-.
scale structure to ground motion from a UNE The response spectrum has been pre-
seems to be the only practical way to investi- ferred for structural engineering studies of
gate the structure's response to an EQ. strong-motion earthquakes, because it com-
bines both the representation of the exciting
CHARACTERIZATION OF EQ GROUND force and the response calculations. It thus
MOTION lumps together under one representation the
major parameters of interest to the structural
The feasibility of simulating EQ ground engineer. The major disadvantage of the re-
motion with UNE- induced ground motion de- sponse spectrum is that only peak response is
pends on the similarity of the two phenomena. determined. The numbei of near-peak response
The mechanism of energy release is much dif- cycles, which are important for studying post-
ferent in a UNE than in an EQ. There are con- yield behavior of structures, is lost.
siderable similarities and also considerable
differences between the induced ground motions The Fourier spectrum of an input function
depending on the criteria of comparison. The shows directly the significant frequency charac-
choice of criteria will depend upon the purpose teristics of the function, and -from it-the time-
for making the comparison. For example, cri- histcry response of the system can be computed.
teria used by seismologists to compare UNE However, in the study by Jenschke et al. [51 it
and EQ ground motion from the viewpoint of was found that results obtained using the
seismic detection of underground testing [4] are Fourier spectral method were not satisfactory:
quite different from criteria used by structural the sine and cosine Fourier transforms showed
engineers whose basic concern is the response high irregularities of the same order as the
of structures. ground acceleration function, and no correlated
characteristics could be found. These high ir-
In order to show that UNE-induced ground regularities make it difficult to use the sine
motion is similar to that from EQs it is neces- and cosine Fourier spectra for comparison
sary to establish a means of properly charac- purposes. Hudson [6] showed that the Fourier
terizing EQs from the point of view of struc- amplitude spectrum is much more regular and
tural response. Many studies have been made is very closely related to the relative velocity
to develop the best means of characterizing EQ spectrum. It has no advantage over the rela-
ground motion [5-101. However, this has tive velocity spectrum and in fact is inferior in
proved difficult to accomplish, one reason being that the peak levels may be lost; time-history
that only a few records of strong-motion EQs aspects are also lost.
exist (here strong motion is arbitrarily taken to
be a peak g-level greater than 0.1 g). These The power spectral density apprcach is
records show a large variation in pulse shape, attractive in that it allows a probabilistic ap-
time of due'ation, peak g-level, and number of proach to be used. However,. Jenschke
cycles. Some of the complex causes of these et al. [5] found that this approach was inade-
variations are discussed in a paper by quate since ground motions produced by EQs
Trifunac 181. and UNEs are essentially nonstationary phe-
nomena, even for dynamic systems having nat-
The most generally used criteria to ural periods considerably shorter than the du-
characterize EQs for structural studies are ration of the ground motion. This is a serious
(1) peak g-level, (2) response spectra, problem when dealing with the ground motion
(3) Fourier spectra, (4) power spectral density, from UNEs because the duration of shaking is
and (5) time history comparisons. None of quite short-abt.ut 5 to 15 sec as compared to
these criteria are completely satisfactory. 30 sec for the El Centro EQ.
Peak g-level has the advantage of being Several investigators have made attempts
extremely simple to compare. It is also phys- to generate artificial accelerograms using
icclly meaningful. Nevertheless, peak g-level various nonstationary processes. But as
alone is not adequate to characterize EQs (7]. pointed out by Trifunac 181 the models generally

22

Lkp.. Q 'AAA<C ,t.- - .~.. -

 5000i I I a 1 ! 1 1 1 1 1-lol

£ Pacoima Dam (San Fernando epicenter distance)

a Boxcar

Mississippi (a) Lo (a/g) 3.5-2 log (D + 80)_

00Lg 10 (a/g) =3.0-2og (D +43) \ .1

8

1I
Pld.
o,,ld Calif. 27J.one 19(6-M.). iI. Hegnloire, Moto 17A 09 . 195(M-. 7). \ 6 6.\
.C 1940(M -7.1).
2. El Centro, Calif. I8 Maky 22. SonJbt., CostaRica 5sOct. 1950(M 7.7).\
10 3. PugetSound, Wash. 13Apr. 1949(M "7.1). 23. Me,.ico City, Mjsxco I I May 1960(M - 7.2). \, •6
• 5. Longkoch, Calif. IMar.
1 1933(M 6.2). IS. SonJose, CasioRica lB8Nov. 1945 (M -7.0)
6. Kern Sounty,Calif. 21 July 1952(M -7.6). 16. Sishop,Calif. lOApr'. 947 (M. 6.4).
7. El Contre, Calif. 30 D¢c. 1939(Me 6.5). 27. Limo, Peru 17 Ocr. 1966 (M *7.5).\ i
8. Logan, Utah 30 As.g. 1962 (M * 5.7). I8. Ltuyo Say, Aloslr 10 July 1958 (M *7.8).
9. Portland, Oregon 6 Nov. 196?(M -*4.8). 19. PrinceWliam Soun, Alaka 27 Mar. 1964 (M c 8.3).
20.sontlag, oi . et, 1945 (\ ,7,)
0.01 0.1 1 00
Distance -miles

Fig. 1 - Plots of g-level vs distance from closest point of observed faulting. From Cloud and
Perez 010]

used are too simple. This results from a the time histories relative to post-yield be-
lack
strong knowledge
of motion of the actualAtcharacter
accelerograms. of
the present havior of structures.
i time these investigations have not progressed Trifunac (81 suggests using the response
Sfr enough to make use of here, envelope spectrum which is a three-dimensional
plot of magnitude of response of one-degree-of-
Direct
ally not madetime-history comparisons
except to obtain qualitativeareinsight
usu- freedom oscillators vs time and frequency.
Only some accelerograms from the 1040 El
into the nature of the ground motion. Some gen- Centro EQ have been studied in this way; hence
eralized time-history characterization of EQ at the present time this approach-while inter-
ground motion is needed to study the post-yield esting-does not offer a meaningful character-
behavior of structures,. No such characteriza- ization of EQs.
tion exi5ts. Cloud and Perez 1101 suggested
comparing total time the acceleration
y1he was above above discussion summarizes the ap-
a certain g-level Figure 3 shows such a cm- proaches generally suggested to characterize
parison for the Parkfield and El Centro EQs. It EQ ground motion for structural response
is not at all evident from this figure that the studies. It is clear that no adequate single
El Centro EQ was by far the more damaging. characterization exists. Based on this discus-
sion we plan to establish a similarity between
numberOther investigators
of cycles of a given suggest counting
g-level. Both the
this EQ and peak
UNE g-levels
ground motion by showing that
() the from UNEs are within the
behvio
suggestion and ofstrctues
the previous Nosuc chracerza-
one (total time Izaionof23s
range estimated for the strongest EQs. (2) the
above a certain g-level) give some useful infor- response spectra from typical UNEs aresimilar
mation, but they do not properly characterize
tion~~~~ exst.
ugese lod to those of strong EQs. and (3) the duration of
ndPeez11]

copain toa ieteaclrto a bv h bv icsinsmaie h p

gd
 1.2 * Figure 4- is a plot of peak g-level versus
range for "typical" nuclear explosive yields of
I0, 100, and1000kt(kilotons). For ranges
less than I DOB the valke plotted is the initial
1.0- NI Pokfield (M=5.7) acceleration. In this regime, the motion is
such that the top several hundred feet of earth
a San Fernando (M=6.3) spalls away from.the lower layers, often caus-
Ing a larger peak'g-level when the spall gap
closes. It would be undesirable to locate test
0.8- _ structures within this spall regime.
It is clear from this figure that the peak
g-level of the strongest EQ can be easily du-
*plicated, or exceeded (if desired), by a UNE.
-- 0.6 See Fig. I for estimated EQ peak g-levels.

Response Spectra romparisons

E Various types of respobse spectra can be
"r 0.4 - generated. These izclude relative displace-
=8 ment, relative velocity, and absolute accelera-
0 tion spectra. For low assumed values of vis-
% 7 cous damping (less than 10% critical) all of the
0 X7
Nv " above spectra are related 16]. Since all of the
. .- *% spectra to be considered for comparison have
goo low values of damping, only the velocity spec-
0 trum will be used.

L.- ~ ~No standard EQ spectrum has been pn-

0 20 40 60 so erally accepted for comparison purposes [71

Hous
Distance to fault - miles

Fig. 2 - Plots cf earthquake magnitude and

peak g-levels vs range.
(IlV]

g-levels fall within the range established

major EQs.
by
From
IN
650 0.5
0.4
0.

•0.3
N

Vt
(a) Parkfield, California "
Station 2

.2 0.2
COMPARISONS OF UNE AND EQ GROUND
MOTION "u 0.1

Peak g-Level Comparison 0

Most of the published correlations of UNE 0.01 0.1 1 10
ground motion data deal with peak g-levels be- Duration of acceleratlion - sec
low those of interest [13,14]. To extrapolate
this data to close-in distances leads to gross
errors. The most comprehensive dircusslon of * 0.4- () El Centro, California
surfacc ground motion from UNEs Inthe strong N
N -S
motion regime is that by Bernreuter et al. [12].

The magnitude of the peak g-level at the "-

ground surface from a UNE depends on several r 0.2
factors [121. These include (1)yield, (2) the
depth of burial (DOB), (3) distance from surface a
ground zero (range), (4) geology around the ex- 0.1
plosive, and (5) geology through which the wave
Is transmitted. For a given yield the peak sur- 0 . ....... I . ... ,,,,I , , ,,_,_ ,
face ground motion can vary by a factor of 20. 0.01 0. 1 10
But since devices of a given yield are usually
buried in similar geologies (and at similar Duration of acceleration sec
depths) it becomes more meaningful to talk
about general curves of peak g-level versus Fig. 3 - Comparison of Parkfield and I Centro
yield and range. earthquakes

24
 10 . t (although several have been proposed). Com-
parison herein will be made relative to the El
Centro (1950, NS), Olympia (1949, N8OE), and
Taft (1952, N21E) spectra; these are the spec-
tra most used for design studies.

There is substantlal variation in spectra

m i % - between different
EL recorded EQs andstations.
between the same
at different Figure 5
%
% t shows a comparison of the above-mentioned
EQs. Figure 6 shows two stations, from the
-Parkfield EQ, both having a peak acceleration
10 level of 0.5 g. A comparison of Figs. 5 andi 6
shows even greater differences between the'
same EQ at different stations than between dif-
o ferent EQs.

•! 0.1 - 0kt Spectra of the UNE ground motion re-

corded off NTS have been extensively
studied [15,16]. However, extrapolation of
these studies to locations much closer to
ground zero Is not valid.

Spectra of the close-in ground motion

from UNEs have not been extensively com-
0.n puted or studied. This is because most per-
10,000 100,000 manent NTS structures are qimple, one-story,
strongly built field-type structures located out
Range from surface ground zero - ft of the very strong motion regime. Also, for
close-in survivability, the peak g-level and
Fig. 4 - Peak radial acceleration for typical change in kinetic energy are the important
UNEs as a function of range for sev- parameters. This limits (at this time) our
eral yields choice of UNE ground motion for comparison.

6.28 3.14 2.09

El Centro EQ (48 km)

70- Olympia EQ (72 km)

-,"•,
60 e

*6 40
0

10/
0
20
-
2 3

" "-Taft EQ (64 kin)

0 I 2

Natural period - sec

Fig. 5 Comparison of E1 Centro, Olympia, and Taft EQs

25
 - ,. ~. - ~ "L - 777- 4. *7 4- *.,. * - --.

60 _ 1 1 Table I lists four events for comparison

Parkflstd EQ, Cholame Shandon purposes. These events were chosen since
Array,' Station 1o. 5 N85E they cover a wide range in yield and geological
50 A Sconditions. Figure 7 compares Mississippi (a)
3.3 miles from fault and El Centro. Figure 8 compares Missis-
Peak g-level, 0.46 sippi (b), Aardvark, and Olympia. Figure 9
.40 compares Blanca and Taft, and Fig. 10 com-
pares liallbeak and El Centro.
30-" -A Study of Figs. 7-40-shows that the re-
sponse spectra from typical UNEs are similar
: 20 to those from strong EQs. Furthermore, the
variations between the UNEs and EQs are no
greater than the variations befween the EQs
themselves, as shown in Figs. 5 and 6. Carder
and Cloud [17] also noted this similarity be-
tween EQ spectra and UNE spectra.

0 1 2 3 4 Time-History Comparisons
Figure 11 shows accelerograms from
k00 E m three.UNEs, namely Aardvark, Mississippi,
Parkfield EQ, Cholame and Boxcar (1.2 Mt). (No accelerogram was
- Shandon Array, Station available for Halfbeak; the spectrum was com-
[ 80 - No. 2 N65E - puted from a velocity transducer.) The dura-
C 270 ft from fault tion of strong phase motion (peak g-level
60 Peak g-lvel, 0.5 greater than 0.1 g) is approximately 2 sec for
60- - Aardvark and over 7 'sec for Boxcar. This fig-
ure shows in addition that the duration of strong
"* motion from a UNE increases with larger yield
and also with increased distance from ground
40- zero.

20-,
Figure 12 shows the accelerograms from
the El- Centro, San Fernando, and Parkfield
J
EQs. The duration of strong shaking for these
EQs was Z0, 10, and 1.5 sec, respectively.
The number of near-peak g-level cycles of
ground motion ranged from 3 for the Parkfield
0 1 2 4 EQ to over 10 for El Centro.
Period - ec
The accelerograms of the EQs and
UNEs
shown are quite different. Nevertheless, the
Fig. 6 - Response spectrum from Parkfield EQ duration of strong motion and the number of
(June 1966) at two stations cycles of near-peak g-level for the UNEs fall

TABLE 1
Sources of UNJ, Spectra
Yield Recording Rango Peak
Event (kt) Geology (it) g-Level Ref.

Mississippi (a) 20-200 'ruff overlaid with 3280 0.98 [5]

deep alluvium
Mississippi (b) - 6560 0.45 [51
lialfbeak 300 Saturated tuff over- 7000 - (19]
laid with dry tuff
.Tlanca 19 Tuff 5310 0.5 (171
Aardvark 38 Tuff overlaid with 3600 .74 (5]
deep alluvium

213
 I'
-Mississippi UNE (Ikin)

.ii

2 ~ '
iv , . ~
El Centro
EQ (48 km)

0 20 40 60 so 100 12
Undamped natural frequency --rad/sec
Fig. 7 Mississippi UNE v El Centro EQ. Relative velocities normalized by dividing original
velocities by 0.1 g

1.5

R Olympia EQ (72 km)

"10 , Aardvark UNE (I.I km).

0.5 , \...u,,
10 A•
I-j..'

0 2. 40 6 80 t20
0

0 20 40 60 80 100 120
Undamped natural frequency - rod/sec

r'ig. 8 - Mississippi and Aardvark UNEs vs Olympia EQ. Relative velocities normalized by dividing
original velocities by 0.1 g

27
 I~~~~ _ _ _ _ _ _ _ _ _I_

1.2 __________
0 80-
Aardvark UNE
.60 -.
* . -Taft EQ (N21 E)
00

2Og4-erlanca UNE -0.

Natural period - sec

0.6-
Fig. 9 - Blanca UNE vs Taft EQ O . issip N

*J0.2
~90
- ~80 -Halfieck UNE0

* c7 0 ~ ~ ...- El Centro EQ .<-2

k-5 V- -0.4
840- I ' -Cs.6 1 I1 I
> 0 Af 0 1 2 3 4 5 6

1~0 0.4-

0 1 2 3 0.2-24,0 f
7Natural period -sec 0
-0.2
Fig. 10 -lialfbeak UNE vs El Centro EQ
-0. 4
2 3 4 5 6 7 8 9 10
within the range establisHed by the EQs. It is
therefore noncluded that from the standpoint of The- see
a time history comparison, the ground motion
from any of these UNEs could be considered as Fig. I I - Accelerograms from Aardvark,
being representative of a future major EQ. Mississippi, and Boxcar UNEs.
SUMMARY
The oreoingcomarisns how hatnear-peak g-level cycles for UNEs fall within
7be oreoingcomarisns how hatthe range established by strong EQs.
from the structural engineer's viewpoint,
close-in ground motions from UNEs are sim- From these results it is concluded that
liar to those produced by major EQs. More typical UNEsof generate
specifically: lent to that a possibleground
futuremotion
major equiva-
EQ.
(1) T7he peak g-leveis from UNF~s are Therefore, by predetermining the yield and
location of UNEs (relative to the location of a
compar., -? to tho~se estimated for the strongest test structure), a test structure can be excited
HQs. by grouand motion comparable to that of a future
major EQ.
(2) The upectra !rom typical UNEs are
slnr~ir to tliis,2~r EQs (i.e., the envelope However, to study post-yield behavior of
UCVelOpcir ypical UNE spectra would
.. structures it will be necessary to generate UNE
match an envelope from strong EQs). ground motion with longer pulse durations and
also more cycles of near-peak g-level. This
(3) A time history comparison of ground can be accomplished by sequentially firing
motion from typical IJNEs shows that both the IJINEs or by subjecting test structures to ground
duration of strong motion and the number of motions from UNEs fired over a large time

28
 0.4 Parkfield EQ, 6-27-66, 2026 PST-
0.2 Cholme Shandon No. 2

- , _V PA, %A %&-- - I
0 2 6 8 10 12 14 16 18 20 22 24 26 2830
El Centro EQ, 5-18-40, 2037 PST
0.2 -. 1 Strong motion

0 4 6 8 10 12 14 16 18 20 22 24 26 28 30

I 1.2 -San Fernando EQ

Pacoima Dam S160E

J
*~1.0-

0.8-

0.6-

4 0.4
0.2 -

-0.2 -

-0.4 --

-0.8

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Time - sec

Fig. 12 - Accelerograms from Parkfield, El Centro, and San Fernando EQs.

span. Both of these possibilities need much One important fact to note is that lower
further study. yield explosives can be used. The only reason
for using very large yield explosives is to in-
crease the duration of shaking. This is of
In order to properly locate structures a some importance since a large number of lcwer
study must be made to correlate the ground yield explosives are detonated as compared to
nuction
and depthspectra fromofUNEs with yield, geology, the very high-yield explosives. Springer a--'
of burial the explosive. Once this Kinnamannj 181 give a list of U. S, UNEs. This
is accomplished it would be possible to subject list indicates that more than a sufficient num-
test structures to increasingly strong motion ber of UNEs are exploded every year to supply
ranging from elastic response to fi.allysevere the required ground motion for any type of test
damage, program.

29
REFERENCES 4th World Conf. Earthquake Eng., Vol. I,
pp. A2-119-A2-132, 1969
I. F. Tokarz and D. L. Bernreuter, "Com-
parison of Calculated and Measured Re- 11. G. W. Housner, "Intensity of Earthquake3
sponse of a High-Rise Building to Ground Ground Shaking Near the Causative Fault,"
Motions Produced by Underground Nuclear Proc. 3rd World Conf. Earthquake Eng.,
Detonations," Lawrence Livermore Labo- Vol. I, pp. 111-94-111-115, 1965
ratory Rept. UCRL-50977, Dec. 23, 1970
12. D. L. Bernreuter, E. C. Jackson, and
2. D. E. Hudson, "Dynamic Tests of Full A. B. Miller, "Control of the Dynamic
Scale Structures," pp. 127-149 in Earth- Environment Produced by Underground
quake Engineering (R. Wiegel, editor). Nuclear Explosives," in Proc. Symp. on
Prentice-Hall, Englewood Cliffs, N.J., Eng. With Nuclear Explosives (Las Vegas,
1970 Nev., 1970), U. S. Atomic Energy Com-
mission Rept. CONF-700101, Vol. 2,
3. J. Penzien, J. G. Bouwkamp, R. W. pp. 979-993, May 1970
Clough, and D. Rea, "Feasibility Study of
Large-Scale Earthquake Simulator Facil- 13. J. R. Murphy and J. A. Lahoud, "Analysis
ity,' Earthquake Engineering Research of Seismic Peak Amplitudes from Under-
Center, University of California, Berkeley, ground Nuciear Explosions," Bull. Selsm.
Rept. EERC-67-1, Sept. 1967 Soc. Am., Vol. 59, pp. 2325-2341, 1969

4. J. R. Evernden, "Identification of Earth- 14. W. V. Mickey, "Seismic Wave Propaga-

quakes and Exlosions by Use of Tele- tion," in Proc. 3rd Plowshare Symposium
seismic Data,' J. Geophys. Reg.. Vol. 74, (Davis, California, 1964), U.S. Atomic
No. 15, pp. 3828-3856, 1969 Energy Commission Rept. TID-7695,
pp. 181-194, 1964
5. V. A. Jenschke, R. W. Clough, and
J. Penzien, "Characteristics of Strong 15. R. D. Lynch, "Response Spectrum for
Ground Motions," Proc. 3rd World Conf. Pahute Mesa Nuclear Events," Bull. Seism.
Earthquake Eng., Vol. 1, pp. 111-125- Soc. Am., Vol. 59, pp. 2295-2309, 1969
111-142, 1965
.udson, "Some Problems in the 16. R. A. Mueller and J. R. Murphy, "Seismic
6. D. E. Hudon o m robems to Spectrum Scaling of Underground Detona-
Application of Spectrum Techniques to tos"EvrnetlRsac op
Strong-Motion Earthquake Analysis," Bull. tions," Environmental Research Corp.
Seism. Soc. Am., Vol. 52, pp. 417-430,
1962
17. W. K. Cloud and D. S. Carder, "Ground
7. D. E. fludson, "Design Spectrum," Motions Generated by Underground Nu-
pp. 93-106 in Earthquake Engineering clear Explosions," in Proc. 2nd World
(R. Wiegel, editor). Prentice-flall, Conf. Earthquake Eng., Vol. III,
Englewood Cliffs, N.J., 1970 pp. 1609-1632, 1960
8. M. D. Trifunac, "Response Envelope 18. D. Springer and R. Kinnamann, "Seismic
Spectrum and Interpretation of Strong Source Summary for U. S. Underground
Earthquake Ground Motion," Bull. Seism. Nuclear Explosions 1961-1970," to be
Soc. Am., Vol. 6, pp. 343-356, 1971 published in Bull. Seism. Soc. Am. See
also Lawrence Livermore Laboratory
9. S. C. Liu and D. P. Jhaveri, "Spectral Rept. UCRL-73036, February 1971
and Correlation Analysis of Groured-Motion
Accelerograms," Bull. Seism. Soc. Am., 19. If. F. Stevenson, "Structural Response
Vol. 59, pp. 1517-1534, 1969 to Close- in Horizontal Ground Motion
from Underground Nuclear Tests at Pahute
10. W. K. Cloud and V. Perez, "Strong- Mesa," Holmes and Narver, Inc., Las
Motion Records and Acceleration," Proc. Vegas, Nev., Rept. IIN-20-1039, 1969

30
 DISCUSSION

Voice In terms of the duration at certain am- soil conditions in the surrounding areas. For in-
plitudes in those plots that you showed, what was the stance, the acceleration levels in a particular earth-
origin of that.data? quake may vary from place to place depending on the
type of soil or rock encountered. Are you aware of
Mr. Bernreuter: That was from a paper by Cloud any studies that have been done?
and a coauthor in the fourth world earthquake confer-
ence. He was looking at how long the level remained Mr Berreuter: Yes, there have been quite a few
above a certain value; for example,how many seconds studies. Harry Seed, University of Southern Calif-
it remained above 1/2 g or 1/4 g for different earth- ornia has made studies tryiny to relate ampliflca-
quakes. tions of base rock motions to soil structure. There
are also several other papers on the subject.
*yjjThen this was not the actual time he
dwelled at any particular level?
Mr. lUernreuter: No. Mr. Haag: Have they been conclusive inany way?

Voice Was it the time after the initiation of the Mr. Bernreuter: I do not really feel that they
event? have been conclusive. One main reason being simply
the lack of recorded motion. Once the San Fernando
Mr. Bernreuter: Yes. records are studied in detail, one might be able to
understand this better. This is the first time wbhave
Mr. Haag (MTS Systems Corporation): I am in- had so many recording stations for a given earth-
tcrcsted in knowing if any one has done any work In qtake over varied geologies. So there is some hope
relating the earthquake accelerations to the type of that this may be clarified.

31
 ROCKING OF A RIGID, UNDSRWATER BOTTOM-FOUNDED
STRUCTURE SUBJECTED TO SEISMIC SEAFLOOR EXCITATION

J. G. Hammer and H. S. Zwibel

Naval Civil Engineering Laboratory
Port Hueneme, California

This paper describes in analytical study of an early-generation seafloor struc-

ture subjected to severe seismic excitation. The structure Is assumed to
consist
pads on of
the asea
spherical
floor. pressure
Frictionhull supported
between on legs,
the pads which
and the in turn soil
underlying rest and
on

the shear resistance of the soil prevent sliding of the structure, but the
structure tends to rock rigidly in a manner that compresses the soil beneath
it and may cause lifting off.

The problem is formulated by second order differential equations containing

non-linear terms. Effects of drag, added mass, and foundation reaction are
considered. A numerical methud of solution is followed that generates solu-
tions at successive time intervals that are automatically adjusted to conform
with preestablished accuracy criteria. The behavior of the foundation depends
on the vertical forces and the load history. The ground excitation is a simu-
l.ated earthquake motion generated from a random process of prescribed power
spectral density, multiplied by appropriate envelope factors.
An example structure is analyzed and its response computed for three different
support conditions. The results are shown in terms of displacement and accel-
erations of the structure and shearing force and vertical reactions of the
supporting medium.

INTRODUCTION

A structure placed on the floor of the

ocean must be adequate for the loading condi-
tions expected during its useful lifetime. One
of the most severe conditions, and one that is
not too well understood, would occur when there
were heavy seismic disturbances in the ocean
floor. This paper describes a study and analy-
sis of the behavior of a particular kind of
structure under such loading.

The structure considered is an early- h

generation type that would be constructed out
of the water and lowered into position on th(
ocean floor. It might be a self-contained,
unmanned scientific or observation station; cr
it could be a manned station complete with life
support systems and means for ingress-egress.
As envisioned, it would consist of a negatively.
buoyant spherical pressure hull supported by
framed legs, which in turn rest on footings.
The footings would rest on or in the bottom
material, possibly with additional fasteners to
prevent sliding. The structure could look like Fig. 1 - Simplified structore
the simplified model shown in Fig. 1.

Preceding page blank 33

 The structure is assumed to be in moderate described by the holding forces to prevent
to deep water. The spherical pressure hull is sliding relative to the ocean floor and by the
designed for the hydrostatic pressure at that vertical forces acting on the footings.
depth. The footings are proportioned so as to
support the weight of the submerged structure
without exceeding the bearing strength of the PROPERTIES OF THE OCEAN FLOOR
bottom material. The mass of the structure is
assumed to be concentrated at the center of the Probably the largest unknown factor in
spherical hull. The main drag effects of the the prol.lem being considered is the nature of
water are assumed to be those acting on the the material on which the structure is likely
spherical hull. The position of the structure to be placed. Submarine soils apparently vary
at any time is given by two coordinates as as widely as do terrestrial soils. One of the
shown in Fig. 2. most coinplete surveys of available data is Ref.
[1], which considered about 250 reports and
abstracted data from the most relevant of them.
The actual samples taken from the deep ocean
floor are very few. Ref.[l] reported that only
300 or so cores have been obtained from the
North Atlantic, for example, which averages
only one sample for each 30,000 square miles.
/ The quantity of data from Pacific Ocean sampl-
T -/ ing.is even less representative.

/ What is believed, however, is that the

floor of the Pacific Ocean is covered chiefly
with elastic, red clay deposits consisting of
volcanic and terrigenous colloidal matter.
The predominant sediments of the Atlantic and
-.___ -__ I Indian Oceans are calcareous oozes, with an
my °ooze being defined as a material containing
.greater than 30% organic material. Closer to
- the land masses, terrigenic deposits that have
come from rivers and currents cover the ocean
- ' floor on the continental shelves. The portion
of the ocean floor that is solid rock is
believed to be a very small percentage.
Fig. 2 - Coordinate system
Ref. [1] gives the following estimates as
to the composition of the ocean floor. About
The rotation clockwise from its initial posi- 82 of the area of the sea floor is terrigenic
tion is given by 0, and the vertical downward shelf sediment in an average water depth of
displacement of the center of mass is given by 100 meters. About 46% is an ooze and 28% is a
y. The horizoraal position of the center of red clay. These latter two compositions occur
mass relative to the base of the structure is in the deep ocean and are pelagic sediments.
proportional to 0 for small rotations since The remaining part of the ocean floor is
the structure moves as a rigid body. Its total believed to be hemipelagic muds, plus the small
displacements are the sum of the relative move- amount of rock outcropping previously mentioned.
ments and the movements of the supporting
medium. These ocean bottom sediments have accumu-
lated over long periods of time. The rates of
The loading situation is as follows. The deposit are of the order of a centimeter or
ocean floor undergoes a violent horizontal less per thousand years. Based on this kind
shaking, which in turn shakes the footings and of rate, the total depth of sediment would be
the rigid structure that they support. The expected to be a few kilometers. However,
system moves in a combination of rocking, hori- seismic surveys indicate a sediment thickness
zontal translation, and vertical settlement of of only 0.1 to 0.5 kilometers. It is believed
the footings. Forces acting on the structure that below this depth gradual hardening of the
are the inertia of its mass and the added mass sediment has taken place so that the layers are
of the water, the drag forces caused by the intermediate in composition between the upper
motion of the water relative to the structure, sediment and the basaltic bedrock underlying
gravity forces due to the submerged weight of it.
the structure, and the reactive forces of the
ocean floor material on the footings. All of The core samples that hav been ,aken of
the motion is assumed to occur in the single the uppermost few feet of the ocean bottom
vertical plane that contains the horizontal sediment indicate very low shear strengths.
axis of movement of the base excitation. The These range from 0.5 psi to 2.5 psi. The
response of the structure is described by time water content varies from 80% to 150% in the
records of accelerations, velocities, and dis- majority of samples. The porosity was in the
placements. The foundation requirements are range 60% to 80%.

34
- --Ti,, ., .-2 .e. .. '- >..,, , = . .... . .. . . i

The shear strength seems to increase almost classical way, and solutions usually used are
linearly with depth below the water-sediment those due to Rankine and to Coulomb (Ref.[7J).
interface. The void ratio apparently does not It would be possible, therefore, to make a
decrease with depth. This might indicate that rough estimate of the horizontal restraint on a
the older, lower deposits have become stronger sunken footing, given the properties of the
through age rather than by consolidation. At soil material and the structural forces and
the soil-water interface, however, one can con- geometry. Lacking specific information about
elude that the material supporting the struc- the sediment properties, it is convenient to
ture will be very soft with a low shearing and assume that sliding is prevented, and to com-
bearing. In fact 89% of the samples reported pute the horizontal restraint required during
in Ref.1] had bearing strengths in the range the response. In a specific location the capa-
0.5 to 2.5 psi. bilities of the bottom material could be com-
pared with what is needed to prevent sliding.
When a negatively buoyant structure is
'placed on the ocean floor, there will be some The thick layer of soft material over-
imnediate settlement due to an almost elastic laying the bedrock will tend to alter the
distortion of the bottom material. Following nature of seismic disturbances occurring in
this, there will be a much slower, and greater, the bedrock as they are transferred up to the
settlement duc to consolidation of the material sediment-water interface. Studies have been
under the additional weight of the structure, made of layered soil masses to show the effect
An equilibrium is eventually reached, and this on a seismic disturbance applied at the lowest
condition is assumed at the time the seismic layer and transmitted by shear from horizontal
disturbance occurs. layer to layer until it reaches the uppermost
layer (Ref.[8]). These studies seem to show
If a footing has settled into the sediment that an input earthquake such as that recorded
it will resist being pulled out because of at El Centro in 1940 will emerge with similar
suction formed beneath it. These break-cut low frequency components but with attenuated
forces are commonly experienced in anchor and high frequency components. Of interest is the
salvage work. It is believed that pullout fact that the peak accelerations of the move-
finally results when a failure mechanism ment do not change appreciably since they seem
occurs in the soil material surrounding the to be associated with the lower frequencies.
footings. It is sometimes considered that the The particular structural size ad type that is
average foundation stress resisting pullout of of interest here responds primarily to the
a footing is comparable to the ultimate bearing lower frequencies, and it is therefore assumed
capacity of the soil material. This was that ignoring the moderating effect of the
assumed in Ref.[2], and was substantiated by layers of sediment would not alter the struc-
actual experience reported in that study. tural response significantly if the bedrock in-
put resembled a large land earthquake. The
It is believed that the rate of loading problem, of course, is that one cannot predict
has an effect on the static resistance of the what the disturbance will be, even in the
soil mass. There are several studies (Ref.[3], bedrock.
[41) that have looked at the dynamic properties
of soil under footings of structures on land. What can be done is to take a record of a
Ref.[51 assumes a homogeneous, isotropic, elas- strong typical earthquake and use it for study
tic medium and derives expressions for a soil purposes as though it were applied to the foot-
resistance function involving a linear combi- ings of the structure. The analytical proe-
nation of the deformation and the velocity of dure established for this can be refined as
deformation. The deformation is multiplied by information concerning sediment properties and
a coefficient K, and the velocity by a coef- depths and bedrock motions becomes available.
ficient C. K and C are functions of the shear-
ing modulus of the soil, its mass density, the
size of the footing, Poisson's ratio of the EQUATIONS
soil, and the frequency of the periodic force
that is loading the soil. Ref.[6] uses a simi- Fig. 3 shows the system of forces acting
lar soil resistance function to study dynamic on the structure when it is displaced so that
field tests on a soil-pile system, with the 0, y and xb are all positive and the motion is
additional assumption that C is also propor- such that the three accelerations 0, y and xb
tional to the depth of the hole. are also positive.

Another possible motion of the footing The center of mass is accelerated horizon-
(assumed to rest on or in a horizontal ocean tally by the combination of two motions: a
floor) is horizontal sliding. Sliding motion motion relative to the base as the rigid struc-
would be resisted by friction and by the shear- ture rotates, and the motion of the base itself.
ing resistance of the sediment material. It is The total horizontal acceleration of the mass
believed that the shearing resistance manifests is thus h0 + xb, and the inertia force is
itself in a way that offers passive resistance
to slipping within the soil along some interior
=
surface outward from the footing. This kind of F 0N(hO + xb)
passive resistance has been studied in a

35
 '_'

*y

Fig. 3 - Force system (OyxbOyi b positive)

where H is the mass of the structure and 8 is a with H the mass, a the radius, and Y the proper
multip)ier that incorporates the added mass of coefficient so that YMa 2 is the mass moment of
the water, inertia of the spherical structure about its
own diameter.
The structure is considered to move through
the water with a velocity resulting from the RL and RR describe the static and dynamic
same two motions: A horizontal component of resistance of the soil beneath the footings.
the rotation about the base, plus the base They are assumed to have the following form
motion. The total velocity uf the structure
through twe stationary water is then h6 + ;b,
and the drag force acting on the sphere is RL = RS(L) + t LAL)

FD - 1/2 CD Tra2p ~h6 + )(h5 + 1)RR " RS (R )+Ri("R'4R)

where A. and A are the vertical movement of

where CD is a drag coefficient, ra2 is the. the fooings and AL and AR are the accompanying
cross section presented to the flow, and p is velocities. From the geometry of Fig. 3
the mass density of the water. The quantity in
the second parentheses is shown as an absolute b
value so that the velocity squared will have AL - y- 0
the same algebraic sign as the velocity; and
the drag force will have a direction consistent - y + b0
with the direction of the flow. R 2

These horizontal drag and inertia forces

acting thrcugh the center of mass are opposed The static resistance of the soil material is
by a force FS acting through the base in a given by RS(A), in which the resistive force
horizontal direction opposite to the algebraic depends on the vertical compression; and
sum of the inertia and drag forces. RD(Ali) is the additional effect caused by the
rate of loading A and the depth of the hole A.
There is an inertial resistance to rota-
tion of the sphere about its own diameter. The other vertical forces act through the
This resistive tcrque is center of mass. 'Teyare the inertia force

10 yHF - OMY

36

? a ~ aaA~a' c ~ ~ a
and the submerged weight Eq. (3) is a set of simultaneous nonlinear
first order differential equations. The ini-
tial conditions at time zero are

Xl(to) =X2(to) "X 3 (to ) -X 4 (t o ) " 0 (4)

where Mg is the weight in air and a is a multi-

plier to Cive the weight in water.
While Eqs.(3) and (4) cannot be solved
From the force system shown in Fig. 3 directly, they can be handled readily by numer-
equilibrium equations can be written. Summing ical methods. The procedure used here is to
the moments about the center of mass begin a step-by-step solution using the Runge-
Kutta method to predict values of each of the
Xn coordinates at the end of successive time
-M2 -M~b + _ ) R, + h)- -O intervala. The values for any given time are
2 hsubstituted into Eq. (3) to find the time
derivatives.

-1/2 CDra20h(hi + + :t) (1)

hb)(h6 After the Runge-Kutta pronedure has given
points for four times and the corresponding
first derivatives have been obtained, Haming's
modified predictor-corrector method is used to
advance through succeeding intervals of time.
This method is a stable fourth order integra-
(2) tion procedure that requires approximately half
BH4y- c~g - RL - RR the calculations per step that other methods of
comparable accuracy require. It has the addi-
tional advantage that at each step the calcula-
where RL and RR are funecions of B,y,i and ; as tion procedure gives an estimate of the local
previously explained, truncation error. The procedure is therefore
able to choose and change the size of the time
Eqs.(i) and (2) can be transformed into interval based upon a pre-established accuracy
four first order equations by making the fol- criterion. The particular subroutine used is
lowing substitutions given in Ref.[91.

X Y/h AN EXA1PLE STRUCTURE

In order to demonstrate the application
X2 = of the procedure, a number of assumptions were
made as to values of the parameters, and a
particular structure was analyzed. As far as
possible the assumptions relating to the prop-
erties of the soil media are consistent with
the reported data of Refs.[J1, [2] and 17]. No
X4 h attempt was made to make a detailed design of a
4g structure, nor was an optimum configuration
sought.
These equations can then be arranged in the It is assumed that a steel structure Is to
form be placed on the level ocean floor at a depth
of 6,000 feet. The structure resemb as that
f1 (X3) shown in Fig. 1, with a radius a of 10 feet, a
I base dimension b of 30 fect, and angle 0 of 60*.
If the spherical hull is designed for the hydro-
2 f2 (X 4 ) static pressure at 6,000 feet and an estimate
made for the weight of the legs, footings and
interior loads, the whole structure is assumed
=
f 3 kXI X2 'xy x 4 ) to weigh 418,000 pounds in air and 148,000
pounds In the water. For symmetry, it is
assumed that the structure has four legs and
4 4four footings, with each footing about 12 feet
square and having a bearing pressure of about
1.8 psi when resting in the static position.
where b and ;b are the acneleration and valo-
whee x an th
x ar acelraton nd .Il- The added mass of the water Is assumed to
city of the
ration support medium, and g is the accel-
of gravity. b one-half
be e ate mass of
the mass of the voe of
the volume of ater
water
displaced by the spherical hull. The drag

37
coefficient is assumed to be unity. The mass numerically equal to one-tenth the slope k of
moment of inertia of the structure about a the static resistance. This static slope was
horizontal diameter of the spherical hull was assumed to be 6,000 lb/in, for both types (b)
assumed to be 2/3 Ma2 with no added mass effect and (c); and the coefficient c was therefore
when the sphere rotates. taken as 600 lb-sec/in. 2 .

Three types of support media were con- The above assumptions imply that the struc-
sidered. The first was an almost rigid rack- ture placed on type (a) material will not settle.
like surface that has the static resistance When the seismic loading occurs, the footings
function shown in Fig. 4(a). The second was a will tend to rock or slide, but will not depress
the supporting medium.
In type (c) material an initial settlement
of

, (Ws4)(l/K) - \ """'oo
4 b"
A 6,000 Wi/n..n Z 6 in.

(a) Rigid bottom material was assumed under each footing. In type (c)
material an initial settlemint of 6 inches was
assumed due to elastic distortion, and an addi-
tional settlement of 18 inc'ies was assumed due
r barltin to consolidation.
uliabearinh
CaacitY
{ Type (a) material was assumed to have un-
limited bearing strength. Types (b) and (c)
were assumed to have an ultimate bearing
strength three times the static bearing pres-
(b) Soft eiato-plastic bottom sure. The pullout strength of type (c) mater-
ial then becomes the force necessary to lift
the footin a distance equal to the consolida-
tion settlement against a resistance of 6,000
lb/in, plus the addition velocity dependent
viscous force.
ultimate
bearin rle
i structure-soil-water system is then
capacity assumed to be excited by a horizontal motion
of the base resembling a strong earthquake on
pullout land. This obviously will not be the same
for, motion as occurs in the bedrock below the sup-
porting material, except perhaps for material
type (a); but indications are that the layers
of sediment will have a small effect on the
(c) Soft llterir exhibiting frequency component that affect this structure
the most. Using the base motion
as though it
Fig. 4 - Static resistance functions were applied to the base of the structure is
assumed for seafloor material therefore not considered unconservative; and
of course one is not sure anyway of the nature
of the movement in the bedrock. It is pos-
soft material that deformed elasticially under sible to model the effects of the sediment
the dead load of the structure but did not con- layers on the motion passing through thcm, but
solidate. When loaded further it followed the this could be done if more specific information
pattern shown in Fig. 4(b). The third type were available about a particular structural
deformed initially under the dead load both by site.
distortion and consolidation. It follows the
pattern of Fig. 4(c). An additional property The input used in this example is an arti-
of type (c) material is that it offers a ten- ficial accelerogram, taken from Ref.[10], that
sile type resistance to having the footing lift is believed to contain all the characteristics
out. of a very strong earthquake, stronger than that
for which recorded data exist. The earthquake
The type (a) bottom material does not Is of 120 seconds total duration and is designed
change with rate of loading so its resistance to represent an upper bound for the ground
does not depend upon A. Types (b) and (c) are motions to be expected in the vicinity of the
assumed to be velocity dependent and to have a causative fault during an earthquake of magni-
resistance function of the form R - kA + cLA. tude 8 or greater.
The coefficient c was asaumed to have a value

38
 Ref.[i0] followed the following procedure. having the same support materlal, did not lift
An approximation to white noise was passed off at all. It instead moved horizontally with
through a filter to give the process the desired the ground.
frequency content as determined by the power
spectral density. The resulting stationary In Fig. 6(b) a departure from 6 - 0 does
Gaussian process was then given the desired non- not necessarily mean that lift-off has occurred
stationary properties by multiplying by a suit- because the rotation may be due to vertical
ably chosen envelope. The record was then movement of the footings. Lift-off did occur,
processed and corrected to filter the spurious however, for this case on both sides. This can
long period componets. Response spectra were be seen from Figs. 7(a)and 7(b), which are plots
then calculated and compared to those of re- of the force exerted by the left and right foot-
corded motions. Undesirable frequency compo- ings as a function of time. It is interesting
nents were removed by filtering again. Finally to note that the reaction force can momentarily
the accelerograms were scaled to the appropriate exceed the static bearing capacity becLuse of
intensivities of shaking. Fig. 5 shows the the dynamic effect. Comparing the final por-
resulting simulated records, which are used as tions of the curves in Fig. 6(b) and Fig. 7
the input for this structure. shows that the structure is settling into a

i" h1k& .6 Ig h"Iip

10

- "
t
IN''
. .M 25 A 5M 4 L. . 0.j
. . W

,, tx, V

Fig. 5 - Assumed base excitation

RESULTS

Fig. 6 shows the displacement xr of the canted position even though the foundation
center of mass of the structure relative to ith reactions are fairly balanced.
base and the angle of rotation in degrees, both
as a function of time for the first 80 or 100 In Fig. 7(c) tie response is less because
seconds of the earthquake. Fig. b(a) As the of the resistance to uplift of the footings.
response if tie supporting material is type (a), In this particular case, none of the footings
Fig. 6(b) is for type (b), and Fig. 6(c) is for broke loose to lift off.
type (c).
In computing the response the assumption
In Fig. 6(a) any departure from 0 - 0 means was made that the structure was restrained from
a footing has lifted off. The maximum lift at horizontal sliding. The- results in Figs. 6 and
0 - 7* is about 3-1/2 feet. The same structure 7 do not include any sliding relative to the
responding in air to the same excitation, and support material. The total horizontal

39
 2.0

2.0

I II Iq I I I I I I I

0 0 20 to w0 2 W ~ I of02

6(b) tYr-9b)

1.0

2.0

-2.0L

0 201 0 20 40 50 40 0 70

0 typo Wp

Fig. 6 - Response of sample structure

40
 0 20 3 0 50 6 0 4
o¢
lo

02.

1.

I I V a I I I

2.0
0

,AA
1O 20 O0

hmAAAh AA.^. .^.41OV

Q0
t
50
(coo)
GO 70 O
s0 00

' Z

0t

Fig. 7 - Vertical reactions on footings

restraining force required to prevent slid'ng (c) would have a very low scatic shearing
was calculated, however, and the values were as strength, possibly not exceeding 1 psi. It
follows: would appear that types (b) and (c) might de-
velop these forces under the footings assumed
if all the footings were acting. The structure
Table I on the type (a) materiel would have to develop
its horizontal restraint principally through
friction between footings and support material.
Type of Bottom Maximum FS if The force required would be nearly equal to the
Material no Sliding submerged weight of the structure, so it ap-
pears doubtful that the required frictional
(a) 130,000 lb force could develop. Some kind of shear fas-
(b) 111,400 teners might be used to hold the footings
(c) 104,600 against shear. These fasteners would of course
affect the tendency toward lifting off, so the
problem would require re-analysis.

These horizontal forces must be resisted Finally, the peak values of absolute ac-
by friction between the footings and the bottom celeration of the center of mass of the sphere
material, and by the dynamic shearing resis- were computed during the response time in eaci
tance of the material. The horizontal force of the three cases. The peak acceleration was
would probably be shared unequally by the four C.250 g for the structure on type (a) material,
footings if none lifted off; and by perhaps two 0.202 g for type (b), and 0.189 g for type (c).
of them if lift-off occurred. Whether or not The Input acceleration of the base had a peak
horizontal1 resistances of the m.ignitude shown value of about 0.44 g that did not coincide
in 7able I could be developed is;,ot certain, with the peak response-; of the structure.
It is certain that materials of type (b) and

41
 F I,

CONCLUSIONS [6] K. W. Korb and H. H. Coyle, "Dynamic and

Static Field Tests on a Small Instrumented
The kind of approach and solution des- Pile," Research Report No. 125-2, Texas
cribed here could be extended and applied to Transportation Institute, Texas A&M Univ.,
the analysis of a great number of situations. Feb. 1969.
The method itself is not limited to simplified
abstractions, and if more accurate information (7] K. Terzaghi and R. H. Peck, Soil Mechanics
were kno,.i about a particular structure and in Engineering Practice, John Wiley & Sons,
location, a more realistic analysis could'be 1948.
made. The purpose of this paper has been to
discuss some of the problem elements and to [8] I. H. Idriss and H. B. Seed, "Seismic Re-
demonstrate a method of solution. sponse or Horizontal Soil Layers," J. of
Soil Mechanics and Foundation Div., Pro-
Even such a simplified analysis, however, ceedings of the ASCE, Vol. 94, Jul. 1968.
should be useful. It can permit a rapid pre-
liminary analysie of any structure being [9] IBM System/360 Scientific Subroutine Pack-
planned for the ocean floor. From the analysis age (360A-CM-030 Version III, Programmers
rough design criteria can be formulated for the Ma nual, p. 337, Fourth Ed. 1968.
strengths required in the various structural
components. The possibility of a failure of a [10] P. C. Jennings, G. W. Housner and N. C.
footing in bearing or shear might be predicted Tsai, "Simulated Earthquake Motions,"
and avoided. In some cases, failure by gross Earthquake Engineering Research Laboratory,
movement in overturning or sliding might be California Inst. of Technology, Apr. 1968.
indicated if additional restraints were not
provided. The predictions of peak accelera-
tions might show the need for shock mounting
delicate equipment. The maximum displacement
and final displacement might be useful if pre-
cise positioning were a requirement for the
structure.

Continuing the approach described here to

study structures of varying size and mass ought
to provide considerable additional insight Into
the behavior of ocean floor structures in gen-
eral when they are subjected to earthquake
forces.

REFERENCES

[I] W. E. Schmid, "Penetration of Objects Into

the Ocean Bottom," Final Report, Contract
N62399-68-C-0044 for Naval Civil Engineer-
ing Laboratory, Mar. 1969.

[2] W. D. Liam Finn and P. M. Byrne, "Analysis

of Ocean Bottom Sediments," 1971 Offshore
Technology Conference Paper No. OTC 1471,
Apr. 1971.

[3] R. V. Whitman and F. E. Richart, "Design

Procedures for Dynamically Loaded Founda-
tions," J. of Soil Mechanics and Founda-
tion Div., Proceedings of the ASCE, Vol.
93, Nov. 1967.

[4] R. T. Ratay, "Sliding-Rocking Vibration of

a Body on Elastic Medium," J. of Soil
Mechanics and Foundation Div., Proceedings
of the ASCE, Vol. 97, Jan. 1971.

[5] T. K. Ilsiech, "Foundation Vibrations,"

Proceedings of the Institution of Civil
Engineers, Vol. 22, May 1962.

42
 NOTATION

Symbol Meaning

a radius of spherical hull

b base dimension of structure

c velocity coefficient in soil resistance

g acceleration of gravity (32.2 ft/sec 2 )

h height of center of mass above base ,f structure

k displacement coefficient In soil resistance

q ultimaate bearing stress of soil

t time

xr motion of center of mass relative to base

xb , xb, xb displacement, velocity, and acceleration of base

excitation

y, vertical displacement, velocity and acceleration

t
of center mass

CD drag coefficient

FD drag force

F% inertia force

FS composite force acting horizontally through base

of structure'

.q mass of structure

RI,,RR left and right vertical reaction forces on footings

Ws submerged weight of structure

coordinates of transformcd equations

a ratio of weight of structure in water to that in air

0 ratio of effective mass of structure and water to rhe

mass of structure
2
y coefficient of MIa in expression for mass moment of
inertia

Al, AR vertical displacement of left and right facings

AL, AR vertical velocity of left and right footings

a 0 0 angular displacement, velocity, and acceleration of

rigid structure

p mass denqlty

base angle of structural configuration

43i

I ,
% --. . '
 DISCUSSION

Voice: Did you include any kind of damping to Mr. Zudanz (Franklin Institute), The model
account for the soil radiation? looked to me like you had a weightless type of found-
ation at each support. 'You did not have any coupling
Mr.' Hammer: We have run cases where we cou- between different support points. Is that correct?
pled the damping of the soil with cquivalent struc-
tural damping to get a new equivalent factor, but in
this study we just assume 5% critical damping for Mr..Hammer: No. There were only soil springs
each mode. to ground and not soil springs from support to support.

44
 DEVELOPMENT OF A WAVEFORM SYNTHESIS TECHNIQUE -

A SUPPLEMENT TO RESPONSE SPECTRUM AS A

DEFINITION OF SHOCK ENVIRONMENT
Robert
TheC. YangM.
Ralph HerbertCompany
andParsons R. Saffell
Los Angeles, California

A procedure is developed for synthesizing a time-history

to describe a transient disturbance such that the
response spectrum of the disturbance matches a given
spectrum and the amplitude ratios and phase relationships
of its frequency components remain within assigned ranges.

INTRODUCTION In most analyses and in all tests,

therefore, the disturbance must be de-
Analyses and tests to determine the scribed by time-histories. If the
responses of systems exposed to transient strength of the disturbance is bounded
disturbances often employ a response by a spectrum, however, spectra of the
spectrum as a means for bounding the synthesized time-histories must closely
severity of all probable disturbances, approximate the design spectrum.
As the response spectrum describes only
peak responses to the disturbance of Peak responses of any system except
linear, undamped oscillators expressed an ,ndamped single degree-of-freedom
as a function of their natural frequen- linear system to a motion time-history
cles, the spectrum does not define whose spectrum matches the design spec-
unique amplitudes and phase angles of trum may differ significantly from those
the frequency components of the disturb- bounded by the spectrum. The magnitude
ance itself. Au infinite number of of these differences will depend both on
disturbances, each with different corrpo- the characteristics of the system and on
nent amplitudes and phase angles, can the amplification ratios and phase
generate peak responses in the linear relationships of the frequency compo-
undamped oscillators which are essen- nents of the disturbance. If all
tially identical. Indeed, as transient responses boundcd by the spectrum are
disturbances are rarely unique, this the result of disturbances which must be
generality of the response spectrum has considered in design, strict compliance
been one of the factors which has with the spectrum would require that an
prompted its extensive use as a means infinite number of time-histories be
for defining shock severity. ccnsidered in analyses and tests, each
with frequency components whose ampli-
The response spectrum alone as a tudes and phases were different.
criterion of input motion in analyses or
tests of many practical systems, how- Practical considerations, however,
ever, may not be sufficient tr. pvoperly limit to a very few the number of time-
define system responses. First, as the histories which can be considered in
spectrum bounds the peak motions only of analyses or tests, sometimes to only a
li:tear, undamned systems, responses of single one. Thus, in selecting a few
systems with other characteristics may specific time-historiles to represent the
be significantly different. Second, entire family bounded by the spectrum,
while in some respects the generality of the characteristics of the system which
the spectrum may be desirable, in others will be exposed to the motions must be
it may be too broad in that other infor- examined and the parameters of the fre-
mation concerning the disturbance may be quency components of the time-histories
available which might be cmployed to chosen so that, hopefully, critical
reduce the size and perhaps the severity responses of the system will be
of the family ci possible disturbances. generated.
Third, of course, shock test machines
must generate motions which are describ- The family of motions bounded by a
able in time rither than In frequency. spectrum can be reouued in size if an

45
 --. -;7T
77 77,~~~ 7 X 7,;

examination of the basic phenomenon H

which resulted in the spectrum indicates W(t) = mA
m (t) (2)
that limits can be established, even m
grossly, for the ranges of possible
amplif-ication ratios and phase relation- where Am are constant coefficients de-
ships. For example, all possible ining the amplitude of the function
disturbances might be describable as r(t) In synthesizing a time-history
S "pulse-like "oscillatory" ratios
implying that ortheas amplification the
functions, = l,2,. a
select
is to (m
problemfro(t), sequence
..M), ofM
with asso-
ciosd fiets m s tht the
for the former will be low while for the
latter they will be higher. Or, perhaps, response spectrum of the composite wave-
fcrm not only matches i. points on a
it may be known that all the energy is
given response spectrum but at inter-
introduced at essentially the same time,
such as might occur impact, thus
' mediate frequencies as well. To match a
providing some insight into the phase spectrum exactly at all frequencies, an
relationships. infinite number of terms would be
required. However, as the response
Synthesis of a time-history of a spectrum to be matched usually involves
realistic disturbance thus involves not some measure of uncertainty, small
only matching the design response spec- deviations may be accept.ole in fre-
trum but also incorporating specified quency regions lying between the
amplification ratios and phase relation- selected frequencies wn so long as they
ships for each frequency component. do not exceed prescribeC bounds.
Response spectra, in nondimensional
form, for pulses, such as a sine, versed An analytical sclution of Equation 1
sine, ramp, terminal peak, square, and for the forcing function fo(t) which would
others have been treated extensively in reduce the problem to an explicit
the literature, and little problem is solution of a set of M simultaneous
usually encountered in synthesizing a algebraic equations for M unknown value08
family of such pulses which will satisfy of A. is not possible since an infinite
these three requirements. Where the number of such functions can, be found
disturbance Is more oscillatory, how- whose spectra will closely match the
ever, the problem of matching closely required spectrum.
not only a response spectrum but also
the amplification ratios and phase Forcing Function. If a unique
relationships of the original disturbance forcing function is to be defined,
is more difficult. therefore, it is necessary to postulate
a function fr(t) such that its amplifi-
This paper describes a technique for cation ratio and component frequencies
synthesizing oscillatory waveforms such may be varied to satisfy the equation.
that their response spectra closely Further, since the function will be used
match a given spectrum and the amplifi- to represent a real transient event, its
cation ratio and phase relationship of initial conditions must be zero and Its
each of its frequency components approx- terminal values of both acceleration and
imate specified values, velocity must vanish. For some phenom-
THIEORETTCAL FORMULATION ena, displacements must also terminate
at zero.

Princioles. The equation of motion In constructing such a function, con-

of a simple linear oscillator subjected sider first the case where all frequency
to base excitation is: components are in phase. EHquation 3
describes one forcing finction which
d2
4tY y =
2
W(t) (1)
satisfies all specified conditiors when-
ever N, the number of half-cycleb, is
any odd integer except unity.
where y is the absolute acceleration of
the mass, w. Is the natural frequency of fn(t) = sin 2nb t sin PrNmb t 0 < t<T
( )
the system, t Is time, and W(t) is the
base acceleration. 'rThe response spec- = 0 t > 'I'
m
trum of the excitation function W(t) Is
a plot of the maximum responses of the The frequency of each component of
system defined by Equation 1, as a con- the forcing function isN!mb and the fre-
tinuous function 01 w,. quency of its envelope 1, , where bm is
expressed In liz. Then Tm , the duration
w(t) can be expressed as a linear of the forcing function, is T.1 = I2b m '
combination of a;celeratton functions This forcing function where N = 5 is
r(t), such that shown in Figure 1.

46
 Solution of Equations. The solution
of Equation I (Reference 1), substituting A --

£ Equations 2 and 3, is

where
Figure 1. Selected Acceleration
Forcing Function, N - 5

2 - )b t ]
21nt cos n(l
c;2 -cos

C(N )22 -O;

co 2Z t cos 2nnt- Con [ n (t - (T)

J+(iN 2b2 - 2 (TM t)2(5)

= 24 , and H is the leaviside Equation 5 is determinate for all

function. The first term of Equation 5 values of' t, , and bm except when
defines the principal responses of the bm= ;n/(l - Nr)"and b= n/(i + rm). As bm
oscillator to the forcing function while approaches either of these values, how-
the second term defines its residual ever, the limit of Qm can be obtained by
responses.
(Equation applying L'Ilospital's rule. Thus,
5 may be rearranged

N -1)2 os 2nt - cos 2n(--t [[+ -

[ao sin
M-1n I~~~ It tn
_N4 2.i) +6 - .+.)2(.( - t)(

+ Nl(U-.1),, [1+' 6(1m-1)] f6t (N -1)2 cos 2nr t + 2 sin 2,,

it] (
+ M 14[2N+ 6(Nm+l)2(Nm -)] 1r m

where 6 = Wnl.N5 - b) and 161<<i, and

t
(11M+l)2 1cos 2 rnt - cos 2 7, -+ 16( m+a ) 1 n ]

lJ-211+4(11_1) 2 (Ifm+1)

- ,-5 ["(i+i)t cos 27nTt + sin 2r.r,,t!if (' - t) (7)

c~s
Ct + ? sn 2ntj
+1 (N +1) n[1+26 (u1 +1)j[6,(1m+1) 2

+ ~~1-21;
f +6 2Ift T

where 6 n and j6 <«1.

47

='
 Amplitude Coefficients. From the Thus, at each iteration step, the
response spectrum to be matched, M variation of Am is a single valued
numbers of Yk maximum responses corre- function of Yk and the process converges.
sponding to frequencies 1 (k - 1,2,...M) To generate large diagonal elements in
can be selected. If the response spec- [Qkm] , the value of b. for a given Nm
trum is to be matched closely at must be selected so as to produce a near
intermediate frequencies, the choice of resonant response; that is, the maximum
the number and relative 'values of the response at a selected system frequency
frequencies is not completely arbitrary should be dominated by one frequency
nor is the selection of component fre- component of the forcing function and
quencies bm. Procedures for establishing none of the other frequency components
these parameters are discussed later. should contribute substantially to it.

Assume that values have been assigned Amplification Ratio. The amplifi-
to Yk, Zk, and b, and that a trial set of cation ratio of each individual frequency
amplitude coefficients Al, A2,,... based component of the waveform can be deter-
on the desired amplification ratio, have mined from its normalized response
been selected. Either Equation 5, 6, or spectrum (Figure 2). Maximax responses
7 may be searched numerically for the are indicated by the solid lines while
maximum response occurring during the the residual responses are shown by the
time period o Z t V (Tm + 1/ 2lTk) and for dotted and dashed lines.
the time tk at which the maximum response
occurred.
A set of matrix equations can be 00- I N. -
written - L; _TII]I
6.[-MINM SECIRA "- " V 1

URk Q
= X
Lkm] I'm k,m =1,2,3 ... 1 (8 4.0
RSIDLAI.SPCIU.--9---.

where k is the maximum value of AQ,, 3z.0 : -*

corresponding to Zk , and Qkm is evaluated /
at t=tk.. - ... t " -I
Usually, the maximum response 9k at -- h
frequency k calculated from the first N1
trial values of Am will not coincide .6 - .
with the maximum responses indicated on
given spectrum at the corresponding -
frequency. A new set of Am values can . 11 ...... II-i 1:0
then be calculated as . 4

jAm) = j k' k,m = 1,2,3 ..... M (9)-

Using the modified coefficients, a .01 .0? .0 .6 .0 .1 .Z .4 .6 .8 1.0
new set of values for 5k and t k can be ,oIZED
F0(%,C RATO Rb./
determined. This process may then be
repeated until the calculated maximum
responses are approximately equal to the Figure 2. Normalized Maximax and
response values indicated on the response Residual Response Spectra
spectrum; or lYk- 50 < 9 1 where S is an of the Selected Waveform
arbitrarily small number. Component for N = 3, 5,
7, and 9
The iteration procedure employed to
obtain coefficients Am is not necessarily It may be noted from Figure 2 that
convergent for any value of waveform the maximum normalized response amplitude
component frequency bm. From Equation 8 ratios for components where N is 3, 5,
it may be noted that if the diagonal 7, or 9 are very nearly equal to the
terms of the square matrix (Qkm] are value of 11for that component. Where Ii
large compared to tLe off diagonal trms is equal to 3 or 5, the value of the
of the same rcw, the diagonal terms of normalized frequency parameter bm/; is
the inverse matrix [Qkm] - ' will also be equal to 0.28 and 0.19 respectively.
relatively large. it each iteration For N = 7 andN = 9, the normalized fre-
step the new value of coefficient Am of quency at which maximum response occurs
each row is influenced principally by is even closer to 1/lj. Thus, if the
the required maximum response value Yk forcing function frequency Nmbm is
where In and k have identical index selected to correspond to a proper ratio
numbers and correspond to one system of the selected system frequency wn the
frecuency k on the given spectrum. amplification ratio at the frequency wn

48
 in equal approximately to the selected To demonstrate the method consider
number of half-cycle oscillations the trapezoidal spectrum shown in Figure
N. 3 and the three collinear points which
are to be matched. The problem is to
select the three frequencies, ;1, ;2,
As an example, assume that it is and ;3 such that both the solution will
desired to match a given spectrum at a converge and, In the region of the spec-
frequency of 10 Hz with a single fre- trum between these frequencies, the
quency component with an amplification spectrum of the synthesized wave will
ratio of 7. The selected component lie close to the line which includes the
should have N n 7 half-cycle oscillations three points.
at a frequency of bm a 10/7 - 1.43 Hz. How-
ever, if the forcing function is to
include components at other frequencies,
convergence of the iteration procedure 0
can be assured only so long as the 0
response of any single frequency compo-
nent is not influenced substantially by C-3
those due to the remaining components. 0
Note also, that the amplification ratio 2
of a single component can only be an odd t
integer and that the minimum amplifica- 0
tion ratio is 3 in order to satisfy the 0
requirement that all motions vanish at LOG FREQUENCY
t = Tm .

Selection of Frequencies. If the Figure 3. Three Selected Match Points

iteration procedure converges, the spec- on a Trapezoidal Response
trum of a waveform synthesized as Spectrum
described above will match M points on
the given spectrum and the specified In determining the frequencies, two
amplification ratios will be approximated adjacent match points are considered at
at these points. The problem which a time. In the region between two match*
remains is that of selecting frequencies points, the spectrum is influenced more
wk such that they are sufficiently by the component at the higher frequency.
separated to ensure convergence of the Thus, in the procedure, if the components
solution but not separated so far that for the two match points do not have the
at intermediate frequencies a close same value for N, the value corresponding
match between the two spectra cannot be to the higher frequency component should
obtained. be assumed to be applicable to both
points.
The complexity of the equations
describing the response motions precludes If the match points lie on a line of
the use of a closed form solution to constant relative displacement, the
identify optimum match points. However, stipulation can be made
a semi-empirical approach has been
developed which, although perhaps not Yk
yielding optimum values, has been found -
to give good results and is simple to k-1 -I)
apply. While if they lie in a line of
Frequently a design response spec- constant pseudo-velocity
trum, when plotted on four coordinate
oaper (such as shown later in Figure II) Yk Wk
is trapezoidal with a low-frequency Y-=-'1
range of constant relative displacement,
a mid-frequency range of constant pseudo-
velocity, and a high-frequency region of If they lie in a line of constant
constant absolute acceleration. While acceleration, of course Yk =Yk-i'
spectra of few, if any, physical phenom-
ena are of this exact shape, spectra of Using these relationships, a large
many occurrences of a phenomenon are number of cases were calculated in which
frequently bounded in this manner. the separation ratio a, where a n k]Zk.1,
While this spectrum shape is not and the parameter 11,were varied.
essential to the technique for selecting Optimum separation ratios for different
apnropriate frequencies, it does sim- values of N arc shown In Table 1.
pilfy the selection.

49

L
 TABLE 1 (Reference 1) Dr (2Nmm) td, (M+1), M
OPTIMUM SYSTEM FREQUENCY RATIOS FOR
DIFFERENT VALUES OF N Note than when Dr ; N, the time-
.. histories of the two frequency components
N 3 5 7 9 11 13 occur consecutively. For match point
frequency ratios shown in Table 1, a
Wi correlation with the desigh spec-
c1.35cose
S=1 1.35 1.6 1.3 trum in the region between these
k_1 frequencies cannot be achieved where the
time delay ratio exceeds N.
Thus, in the example, if II = 5, 12 = 7,
3 = 5,
and N1 and assuming , = 1.0 Hz, ;2 Ranges of time delay ratios for
should be 1.6 liz and w3 = 1.6 x 1.35 = 2.16 values of N from 3 thru 13 which will
Hz. minimize deviations from the design
spectrum in the region between match
Phasing of Waveform Comnonents. point frequencies are shown in Table 2.
Phasing of the waveform components can These ranges were calculated for three
be represented by specifying a delay in frequency components whose frequency
the starting time of each waveform com- ratios wer, ai indicated in Table .1,
ponent. Mathematically, the waveform and whose values of N were equal. For
with phased components can be written other frequency ratios and combinations
similar to Equation 3. Thus of N, of course, other ranges of time
delays might yield equally acceptable
N correlations with the design spectrum.
m=1 TABLE 2
fm= 0 0 < t < tdm RANGE OF ACCEPTABLE TIME DELAY RATIOS
= sin 2i, bm(t-td) sin 2, Hmb(t-t) (10) FOR DIFFERENT VALUES OF N
RANGES OF TIME DELAY RATIOS, Dr
tdm i t <tdm + T
m

where td5 Is tne time of inititation of '.--.3 '., .

the mth waveform component from time ". .
zero.
Equations b thru 7 remain valid even
when the time delay is included in then. MWAVSYN Computer Program. A computer
Thus, the time delay term will have program, designated MWAVSYN, was devel-
. little effect on the responses at the oped to solve Equations 5, 6, and 7 for
spectrum match points. In the range amplitudes Am and to perform the iter-
between the match points, however, the ations necessary to ensure that the
effect of the time delay can be much response spectrum of the composite time-
more pronounced and match points history match the design spectrum within
selected to minimize error in the inter- an assigned tolerance 5.
mediate ranre are no longer valid for
all time delays. Input data for the program consists
of a definition of the required trape-
Usln' the searatton rrenuencies zoidal response spectrum, amplification
ratios shown in Table ], responses at ratios and time delays as functions of
the intermediate frequencies were calcii- frequency, and the accuracy tolerance
lated for various time delays of any imposed on the spectrum of the synthe-
three adjacent frequency components and sized waveform. Using the optimum system
the ranves of time delays aere identified ratios given in Table 1, the program
in which deviation was found to be a selects the system frequency match
minimum. Tf tdm is the time delay of points on the spectrum. An Interpolation
the mth frequency component measured subroutine then establishes the required
from time zero, the time delay between response amplitude, amplification ratio,
the (M +l)th and the Mth components is and time delay for each frequency compo-
td, (4+1), 1. The time delay ratio Dr Is nert. Amplification ratios are reduced
,leefnqd as the ratio or the time delay to Integers which are nearest to the
of two idjacent component: to the period odd-numbered half-cycle oscillations.
or a hile-cvcle or the comoonent at Based on the haif-cycle oscillations at
frequency NMbm. ''hujs the selected system frequency points,

50
the time delays are forced to stay with- -

in the rangps shown in Table 2. A

minimum accuracy tolerance of 15 percent
has usually been found to be easily,
achievable.

Output data consists of time-motion

histories of the composite waveform
expressed in terms of acceleration, |
velocity, and displacement. Time-motion
histories canobe punched on cards and/or
plotted by a CALCOMP machine. A detailed
description of the program is presented
in Reference 1.
EXAMPLE
Dynamic analyses of structures sub-
jected to earthquake or nuclear weapons
effects involve the calculation of time-
motion histories at many interior
locations. Owing to the many assumptions Figure 4. Comparison of the Vertical
involved in these analyses, such as the Spectrum of a Typical
detail nature of the external loads, the Structure and the Spectrum
soil and site properties, and the Generated by the Synthesized
dynamic characteristics of the structures Waveform
themselves, responses calculated at any
given location can be regarded as only
one of a large number of possible
resoonses. To account for possible
differences in the details of the -
responses due to possible variations in - .
the parameters, the shock environment
within various zones of structures are
frequently bounded by spectra enveloping
the spectra of the calculated responses
of all noints lying within that zone.
-- l eQUenD AIR SPECTRUM
For example, an envelope of spectra I ( I
of predicted vertical responses of a -- SNTHESITZo WAVCFOM
typical structure is shown in Figure 1; AIR CT UM I I
by the dotted lines. In the design of 01 05 ,o so 100 200
shock isolation s'ystens supported by the ,RIQU1NCV. HZ
third floor, time-histories of the
disturbance were required. As no single Figure 5. Ccmparison of the Required
time-history of the predicted responses Amplification Ratio Spectrum
had a snectrum which matched the and the Amplification Ratio
envelope, and as it was not feasible to Spectrum of the Synthesized
investigate the behavior of the shock Waveform
isolation system to all time-histories
whose spectra were included in the
envelope, it was desirable to synthesize
a time-history which would incorporate the low-frequency responses of the
"worst" conditions, structure were the first to be energized
by their exposure to an outrunning
Based on a consideration of the fre- ground motion, that as the airblast
quencies of the modes of tht shock pressure wave approached closer to the
isolation system, it was concluded that structure higher-frequency modes were
It would be necessary to match the energized, and that the arrival airblast
design snectrum only within the range of pressure wave at the structure was
frequencies from 0.1 to 20 Hz. Time- accompanied by the highest-frequency
histories of the calculated structural responses.
resnonses were then filtered within this
frequency bandwidth to determine their Input requirements based on these
amnlilication ratiosestimated
r' lationshIrn were (Figure 5).
f'rom Phase
an estimates and ofthethecalculated
coefficients amplitude
synthesized waveforn
examination of the nredicted time- are summarized in Table 3. The response
histories a' the loading rhenc:,ena spectrum, generated from the synthesized
which nroduc Pd them. It was noted that waveform is shown by the solid line in

51
Figure 4. As noted, the accuracy is
within +15 percent at frequencies of . ..-. -
lower than 0.3 Hz and better than +10 _
percent at higher frequencies. The
required amplification ratio spectrum
and the amolification ratio spectrum of - - _,
the synthesized waveform are compared
in Figure 5. Acceleration, velocity, .i 1-4t i I-
and displacement time-histories are
shown, in Figures 6 thru 8. 47
SUMMARY
A technique has been developed which -d d
simplifies the synthesis of an oscil- .1i il
latory waveform incorporating specified r
amplification ratios and time delays and - T
matching a given response spectrum. The . . . ... .
nature of the basic synthesized waveform
limits the frequency components which
can be included in the waveform and Figure 7. Velocity-Time Function of
requires that their amplification ratios the Synthesized Waveform
be equal to odd integers greater than
unity.
Desnite these restrictions, however,
the technique is a significant improve-
ment over the older cut-and-try methods
and, in addition, has the important
advantage of oermitting the systematic
variation of amplification ratio and
phase, two parameters of vital importance
to the responses of most practical
systems. ,,

FT
.~-
A-
,--- !..,~ ....
_JaJ.
_
~,. .
IrIT
.... ..
I
-
a. F...
-
,
i
T
....

_i _I
j-

~A;WII -M4- .~ W_.;4

Figure 6. Acceleration-Time Function of Figure 8. Displacement-Time Function of
the Synthesized Waveform the Synthesized Waveform

52
 TABLE 3
SYSTEM FREQUENCIES, COMPONENT FREQUENCIES, TIME DELAYS, AND
CALCULATED AMPLITUDE COEFFICIENTS OF THE SYNTHESIZED WAVEFORM

NUMBER OR SELECTED NUMBER OF

FREQUENCY SYSTEM COMPONENT HALF-CYCLE TIME AMPLITUDE
COMPONENT FREQUENCY FREQUENCY OSCILLATIONS DELAY COEFFICIENT
Sm1z bm Hz Nm tdm, sec Am in./sec2

1 0.100 0.028 3 0 0.37

2 0.160 0.023 7 0 0.614
3 0.232 0.026 9 0 1.02
3 0.336 0.037 9 0 1.71
5 0.1188 0.054 9 0 .1143
6 0.706 0.078 9 0 7.51
7 1.025 0.1111 9 0 16.42
8 1.1187 0.165 9 0 31.90
9 2.152 0.239 9 0.925 76.70
10 3.120 0.3147 9 1.563 153.111
11 11.530 0.503 9 2.005 219.115
12 6.110 0.555 11 2.889 23'.05
13 7.950 0.612 13 3.707 2116.13
1l 10.310 0.793 13 4.006 312.37
15 13.1130 1.033 13 4.006 280.70
16 17.48 1.3115 13 4.006 250.35

REFERENCES
1. R. C. Yang, Modification of the
WAVSYN Computer Program, Document
No. SAF-U2, The Ralph M. Parsons
Company, 30 April 1971

I
 THE RESPONSE OF AN ISOLATED FLOOR SLAB-RESULTS
OF AN EXPERIMENT IN EVENT DIAL PACK (U)

J. M.Ferritto
Naval Civil Engineering Laboratory
Port Hueneme, California

This paper outlines atest of a horizontal cylinder covered with an earth berm subjected to
the pressure and drag forces in the 300,psi overpressure region from the detonation of the
500-ton high-explosive shot of Event DIAL PACK. The objective of this test was to obtain
information on the response of an isolated floor slab placed on asoil fill inside the concrete
cylinder. Data from seventeen channels of active instrumentation, composed of a pressure
cell, velocity gages and accelerometers, were recorded. Reduction of the data was made.

INTRODUCTION Shock Isolation

Objective An essential element in the design of protective

structures is the provision of a reliable shock isolation
The prime objective of this project was to obtain system for personnel and equipment. The levels of shock
information on the response of an isolated floor slab inside to which a system acting at its designed structural capacity
a horizontal cylinder covered with soil and subjected to pres- may be exposed can be very high, with ground accelerations
sure and drag forces in the 300-psi overpressure region. The over a hundred g's and ground displacements over two feet.
specific objectives were: Very few devices arc capable of withstanding shocks of this
1. Determine the absolute motions of the floor magnitude without serious damage. A more serious limita-
slab caused by the blast loading. tion is the peak shock which an unsupported human being
can tolerate. Yet the unimpaired functioning of both per.
2. Determine the motions of the slab relative to sonnel and equipment immediately following an attack must
the cylinder, be preserved if the facility is to fulfill its intended mission.
3. Evaluate the survivability of the structure In the design of shock resistant equipment, the required
including the performance of the retaining strength of the equipment is controllkd by its response to the
wall. shock produced by the dynamic loads on the structure.
Expensive conventional shock isolators could increase the
Background cost of the structure to an extent that might be unacceptable.
An alternate approach is to isolate the floor slab by "floating"
There is a requirement to store sensitive equipment if) it iii a layer of soil! within the cylinder. The cylinder is a
aboveground hardened shelters. The equipment must be closed structure capable of being constructed economically
shock isolated from the shelter. It is important to limit the and supporting high overpressures with minimum disturbance.
blast induced motions transmitted to the floor slab to reduce The floor slab must be strong enough to carry the equipment
load and resist the longitudinal and torsional motions from
the shock. The motions of the floor slab are related to the
crete slab on asoil fill inside ahorizontal concrete cylinder motions of the surrounding concrete cylinde, transmitted
is being considered as a possible means for diminishing the through the soil fill. However, as the cylinder is displaced
shock motion. This structure, if satisfactory, will provide downwardh theiefi owevisolated floir slab tdnas to
simple, inexpensive shelters which can be rapidly constructed reduce the peak accelerations felt on the floo.
and will eliminate the cost of providing expensive shock iso-
lation platforms.

Preceding page blank

$ 55
EXPERIMENTAL PROGRAM Instrumentation
Test Structure Seventeen channels of active instrumentation consisting
of eirht structure velocity gages, eight accelerometers, and
The isolated floor slab test was planned as part of Event one pressure cell were used. The layout of the instrumenta-
DIAL PACK [1], a high explosive field test of 500 tons con- t.ca is shown in Figures 6 and 7. The first letter, A or V,
ducted at the Dcfence Research Establishment Suffield indicates an accelerometer or a velocity gage; the second
(DRES), Ralston, Alberta, Canada. The test structure, letter, V or H,indicates vertical or horizontal orientation.
located at an azimuth of 120 degrees 270 feet from ground The data were conditioned, amplified, and recorded on 32-
zero, was at an anticipated side-on pressure range of 300 psi. track tape recorders located in a bunker 2,700 feet from
The structure location and construction detcils are shown in ground zero. An inter-range instrumentation group timing
Figures 1 and 2. It consisted of a 6-foot-inner-diameter 7- system was recorded on one track of each tape recorder and
inch-thick horizontal right-circular cylinder aligned later used in the data reduction. Timing from an NCEL-
perpendicular to the direction of propagation of the blast designed timing generator was also recorded on one track
wave. The invert of the cylinder was 1-1/2 feet below the of all recorders. A detonation zero pulse provided by DRES
natural grade. The cylinder was covered with 2-1/2 feet of was recorded as received directly onto the last track of each
compacted fill forming a tapered earth berm extending along tape recorder to provide a reference for data located on dif-
the sides of the cylinder and one end at a 3:1 slope. A 2- ferent tape recorders.
foot-thick retaining wall held the other end of the earth The gages were mounted on steel plates cast in or bolted
berm in place; a bolt-on steel closure plate was incorporated to the structure. The pressure ceh was installed in a specially
in the retaining wall to provide access into the cylinder. One designed concrete mount which was cast in the wall with the
and one-half feet of compacted soil was placed inside the heat shield flush with the face of the wall. Passive instrumen.
cylinider, and a 6-inch concrete slab was cast over the soil. tation consisted of tree orthogonally oriented reed gages
A spring-mass system, Figure 3, was installed 4 feet from mounted on the slab and a scratch gage (Figure 7) mounted
the rear end of the slab to simulate equipment on the slab. between the retaining wall and the cylinder.
Figure 4 shows the structure under construction, and Figure
5 shows the completed structure.
:=
" 53' - 2'-,-

berm 2'I_\&
6" - cylinderc
wallZ

"B' 420" closure

Figur 1T 1''6
,_ .....
... _ _T_
_
Nt.SeNoe:Stelrenfrcmetomttendaing.
renoreenomtte
l ndaing.1.. Section, B..-B u ,-
Jectionoam

19'A 6i8' urY1.' 6onstrucionnplan

F3 pie cyinde
I..- -- ' o"
 270ft

ground zero I C L

270 ft

53 ft x 42 ft (berm dimensions)

120-degree azimuth Figure 2. Layout of Project LN322.

A..
1/4.in.-thlck end plates welded to beams
7"ItF 1/4.ln. end plate"

~~~ 3/4-In. plywood

bled to flanges
~2 x4 2 angle
0 2 x 2 x 1/4 angles, equally spaced,
welded to beam flange

Spring-mass system filled with SecionAd

bags to a total weight of 4,OU0 lb. Fgr .Srn~asla ytm

Figure 4. Test structure under construction.

597I
 !P7. -

figure 5. Completed test structure prior to, test.

groujnd zeroj

AVi
andAH
V4ad V

VH57 andVH

AAV1 and AH6

Figur 6.Lgaigefsei ags

AV3 and 58
 ~AV1 and AH2

AH 4

Figure 7. Test gages.

The Pressure gage Was Set in the outside (aec of the RESULTS AND DISCUSSION
retaining wall to measure thc fre.icld sidc-on overpressure.
Horizontal and vertical accclcromcters wcrc set on thle cvI Observed Damagc
indcr ncar each cnd to measure thc horizontal and vertical
components of tlic cylinder motion.. The -lab instrumnenta- Thc detonation of the 500-ton highi-explosivc chiarge
tion was placed in tlac same cross-sectional planes as thc occurred on 23 July 1970. At D+lI hour project personnel
cylinder instrumentation to evaluate the relative motions returned to 6r.L LN322 site. The fireball Wablackened the
of the slab with respect to the cylindei. A vcrtici'l and two retaining wall and the berm with anlayer of carbon dust.
horizontal acclerMCM' etesre set at the center of the floor Twelve inches of crater ejccta were deposited in front of
slab to record floor accelerations in three orthogonal direc the ground zero end of the retaining wall. This level was
tions. The two vertical velocity gages at eath end were reduced to about 2 inches in)front of the closure plate.
installed near opposite edges of the floc. slab to record IThe maximnumn size of the ejrcta was estimated to be 8
rotational motions of the floor slab. The three orthogonal inches. Approximately 18 inches of the berm on the
reed gages were used to provide records of horizontal and groueMd Meo side were compressed and/or blown tway.
vertical shock spectra. The scratchi gage was used to record
relative motion between the retaining wall and the cylinder.

59
 A pattern of major diagonal cracks was noted on the Postshot measurements revealed that the slab rotated
wall. S,,vceal of these cracks were observed on the top sur- clockwise approximately 3 degrees. The permanent relative
face of the v'all and extended completely through the wall. horizontal translation of the slab was 1/2 inch away from
The wall had about 1.degree permanent rotation into the the ground zero side of the cylinder. The center of the slab
berm. Some cracks in the concrete were noted around the remained at the same elevation relative to the cylinder. The
closure plate. At D+I day the closure plate was removed permancat horizontal translation of the cylinder with respect
revealing additional cracking in the wall. These cracks to the retaining wall was 3-5/16 inches away from ground
appeared to go completely through the wall. zero.
The concrete cylinder sections suffered significant
dam. 6 v. A horizontal line of compressive failure was located Reed Gages
at about 30 dcgrees from the top of the cylinder on the lec-
ward side of all the sections. A region of major apalling and The plates from the recd gages were removed, and
tensile cracking was noted at about 90 degrees from the com- measurements of the traces were taken and converted to
pression zone (60 degrees from the top of the cylinder) on displacements. These values, plotted in Figure 9, give the
the windward side of all of the sections (Figure 8). The first shock frequency spectra of the slab.
three sections had been unintentiorally oriented so that the The reed gage consists ot'a number of masses on a rigid
splice in the circular reinforcing steel was located in the bs,. The response of equipment mounted on the slab to a
region of the tension failure. Th6 splice appeared to have given shock can be determined fr3m the measured response
failed, and a section of the pipe was pushed inward approx- of a reed having the same frequency and damping as the
imately 4 inches along this line. The floor slab was littered equipment to the shock. Thus, the shock spectra diagram
with spalled concrete and sand. can be used to determine the shock isolation requirements
of the equipment.

1 , -Active Instrumentation

All of the active channels of instrumentation functioned

I satisfactorily. The analog data tape was returned to NCEL
where the data were converted to digital form using the
S, , NCEL Analog-to-Digital Converter. Accelerations were sam.
(: .... pled at increments of 0.1 msec for 200 resec, and velocities
• . - were sampled at increments of I insec for 2,000 msec. No
filtering was used. The data were automatically plotted using
a CDC 6600 computer.
Figure 10 is a plot of the side-on pressure data recorded
' at the
The peakmidpoint of recorded
pressure the retaining
wai wall
317 above theshock
psi. The closure plate.
wave

4 had an arrival time of 32 msec and a total duration of the

positive phase of 103 nsec. The actual duration of the
i
positive phase was longer than the expected value of 60
nmsec.
AIn several radial horizontal velocity plots, the velocity
at late time did not retun to 7ero, indicating permanent
tilting had occurred. Visual measurement made after the
shot indicated a permanent tilting of the slab of about 3
degrees clockwise when viewed from the retaining wall
looking into the structure. Correction factors were sub-
tracted in an attempt to remove the tilting effect. The
accelerations and velocities were integrated. A summary
of the peak values is presented iuTable I. Positive values
I.s indicate motions down. away from ground zero. and to the

Figure 8. Postshot view of interior of structure. right looking at ground zero.

60
 11000Shock
Spectra

0~~~
HorzotalRaia

9W
ElHorizontal Transverse
SVertical

011 10 100100
-4
Frequency (cps)

Figure 9. Shock spectra of floor slab.

350-

-70'
Time (msec)

Figure 10. Pressure-tirre curve, gage PSI.

61
 Table 1. Peak lnstrumer.tation Values

Peak Peak Peak

Type of Gage Acceleration (g) Velocity (fps) Displacement (in.)
Instrumentation Negative I Posit iti Nive Positive

Cylinder
19.6 146.8 2.3 20.1 0.0 8.5
Vertical accelerometer AVI
Vertical accelerometer AV7 112.8 173.9 7.6 22.4 2.4 5.6
Horizontal accelerometer AH2 90.2 52.6 9.9 5.0 2.3 3.2
Horizontal acceleromctar AH8 72.4 86.0 6.3 11.2 0.4 13.9

Floor Slab

Vertical accelerometer AV3 23.7 28.5 4.8 10.7 0.0 4.9

Vertical accelerometer AV4 15.5 11.3 1.1 10.5 0.0 5.5
Vertical velocity VV1 5.4 11.3 0.0 5.3
Vertical velocity VV2 2.0 9.2 0.0 5.4
Vertical velocity VV3 1.0 9.1 0.0 4.8
Vertical velocity VV4 2.0 7.4 0.0 6.6
Horizontal transverse accelerometcr AH6 3.0 2.2 0.0 .5 0.0 .4
Horizontal transverse velocity VH6 0.7 .9 0.4 .6
Horizontal radial accelerometer AH5 13.5 19.4 0.0 3.8 0.0 2.8
Horizontal radial velocity VH5 1.0 3.4 0.6 3.8
Horizontal radial velocity VH7 1.3 3.9 0.3 4.0

Spring-Mass

Horizontal radial velocity VH8 0.7 3.2

gage data taken for 2,000 msec.
Note: 1. Accelerometer data taken for 200 msec, and velocity
2. Positive direction is downward, away from ground zero, and to the right facing ground zero.

Cylinder Displacement

The horizontal and vertical displacements obtained fromi

double integrations of cylinder accelerations were used to -31
produce a plot of cylinder movemcnt (Figure 1I).Although -1 ground zero
there may be some error associated with this procedure, it is -

believed to be accurate enough to relate direction of motion

and approximate orders of magnitude of movement. Similar -3 down
plots, Figures 12 and 13, were made for the slab movement
using the horizontal and vertical displacements ubtained
from the integration of the velocities. T =5
lmsec
10
! '1
> -7

-9
Horizontal Displacement (in.)
Figure 11. Displacement of cylinder.

62
 -

i -
1 1
II I I I ' I I l l I
-1 1 3 -1 1 3 5
ground zeroundro

down

0-U

>

-5

Horizontal Displacement (in.) Horizontal Displacement (In.)

Figure 12. Displacement of floor slab, windward side. Figure 13. Displacement of floor slib, leeward side.

Figur:- 11, 12, and 13 indicate tie motion of the inches. The amount of rotation shown is about 3 degrees,
structure was initially downward and away from ground confirming the ficld measurement. From Figure 14, the
zero, chcn downward and toward ground zero. Subsequent rotation of the slab began at about 125 msec and rotated
motion at about D+100 msec was upward and away from the full amount at about 200 mscc.
ground zero. The motion toward ground zero is quite Generally the data obtained in the first few hundred
unusual and unexpected, but is believed to be valid. Pos. milliseconds are very reliable. After this period secondary
sible sources of error such as gage rotation, reversed effects may adversely affect the data. The peak values and
calibration, or an erroneous gage were considered and wave shapes in early time (several hundred milliseconds) are
found not to have been present. This effect was obse'ved quite reliable and are relatively the most important. Accel-
on four independent sets of instrumentation. Ovaling of crometers are usually more reliable than velocity gages when
the cylinder was also dismissed as a possible explanation rotation of the gage is suspected. Vcloc.ity gages experiencing
because the direction of ovaling near the gage location rotation as slight as three degrees may erroneously indicate
was inward rather than outward. The initial downward apparent motion of 100 percent of the anticipated value in
movement was caused by the direct compressive wave. A magnitude. Evidence that this has occurred is noted when
possible explanation of the horizontal movement toward velocities at late times (2 seconds) fail to return to zero.
ground zero is the reaction to the magnitude shear wave Accelerometers are relatively insensitive to rotation and are
which enveloped the berm producing planes of opposing in.;uenced only by the prc luct of the sine of the angle of *
shear forces. These forces gave rise to motion as slippage rotation and the component of acceleration iii the perpcn-
occurred. The reverse movement of the structure was dicular direction. Thus, for small rotations the influence
caused by a combination of rebound and direct induced is negligible.
ground shock arriving about 100 mscc after detonation. Integrations of acceleration data give reasonable
The time to peak positive accelerations of the cylinder indications of wave shape and velocity. l)isplacenents
was about 12 msec after the arrival of the blast wave- how- from integration of accelerations should be capable of giving
ever, the time to peak rositive acceleration of the floor slab an order of magnitude of niovemnent and direction. The inte-
was about 110 insec after the arrival of the blast wave. The gration of data represents a second level of confidence.
magnitude of the floor slab accelerations was substantially
reduced compared to that of tile cylinder. Once set in Shock Isolation
motion, the movement of the floor slab was independent
of the cylinder. The slab appears to have remained station- Usiiig all isolated concrete slab "flo.itinmg' on sand within
ary during the initial movements of the cylinder, amid then a horiLontal cylinder reduced the peak a.celcration of 174g's
moved downward until coming to rest on the soil. on the cylinder to about 2 8 g's on the floor slab. The 22-fps
Figure 14 shows the relative vertical displacenient of peak vertical vclocit) of the cylinder was reduced to about
two velocity gages located on tl,e slab aid separated by 36 iI fps. Table 2 compares tihe peak motions and shows the

63

Ai
shock isolation of the isolated floor slab. Table 2. Comparison of Peak Motions

CONCLUSIONS(p)(g) Velocity Acceleration

(fps) WgS)
The objectives of this project were satisfied. The !
motions of the floor slab were determined, and the surviv- Vertical
ability of the structure evaluated. All of the instrumentation Cylinder 22 174
functioned satisfactorily. Data were obtained to compute Floor slab 1I 28
the shock isolation requirements for equipment to be stored
in the shelter. The significant findings and conclusions arc: Horizontal
1. The isolated floor slab significantly reduced the Cylinder 11 90
transmitted motion. Floor slab 4 19
2. Shock spectra data for designing the required shock
isolation system has been| determined.

REFERENCES

1. NCEL Technical Report R-726, "Dynamic Response of

an Isolated Floor Slab-Results of an Experimental Test in
Event DIAL PACK." by J. M. Fcrritto, May 1971, Port
Huenene, California.

6-
VV2

~VVll

1.000 2.000

-2
Time (msec)

Figure 14. Rotation of floor slab.

64

~I,
 DISCUSSION

Voice: Concerning the measurement on the floor about 4,800 degrees of freedom, so it is quite a large
slab, does this refer to the isolated mass sitting on problem. We are attempting first to look at it struc-
top of the springs, or ig the slab sitting on top of the turally, and the time step is made as large as possi-
sol? ble keeping the economics in mind in order to be able
to run the problem.
Mr. Ferritto: All the measurements were made
on the actual concrete floor slab. The spring mass Vcice: Do you solve 4,000 dynamic degrees of
system shown in the slides is simply used to provide freedom?
an equivalent equipment on the floor slab.
Mr. Ferritto: Yes, We are running approximate-
Voice: Is it correct that you were not trying to ly 300 time increments to approximate about 100 mil-
establish the effect of isolation using a layer of soil? liseconds, I believe.

Mr. Ferritto: Basically that is correct. The Voice: What is the highest frequency of the mod-
springs of the mass and the weight of the mass were el you are putting together?
selected to provide the frequency that we would ex-
pect from a typical piece of equipment placed on the Mr. Ferritto: It is a very complex problem be-
slab. It actually represented what we tried to approx- cause the loading is quite complicated. We have a
imate in the model, traveling wave. We have a very complex soil-struc-
ture interaction. The extent to which the berm partic-
Mr. Keen (Bell Telephone Laboratories).: The ipates in the problem still remains an unknown. This
high accelerations ii your last slide uould indicate is one rescn we are using a plane strain, finite ele-
the presence of very high frequency data. You also ment analysis with a quadrilateral element rather
mentioned that you planned to use nonlinear finite el- than another type of modeling procedure. The period
ement analysis techniques. Would this predict any of of the cylinder by itself in compression is about 2
the high frequency phenomena which I assume to be milliseconds. The period of the cylinder in flexure is
present? about 11 milliseconds.
Mr. Ferritto: To this date we have been running Mr. Zudans (Fraklin Institute): I am interested
a structural analysis primarily interested in the op- in your 4,000 dynamic degrees of freedom. Are you
timization of the structure itself without looking at modeling it as a two-dimensional infinitely long type
the isolation characteristics. We have made several of strip?
runs. The time step and other information we have
used in sizing the finite element mesh has been scl- Mr. Ferritto: Yes, it is a plane strain model.
lected, basically, to satisfy the structural character- The quadrilateral element that we are using has 12
istics. The structure, as we are now looking at it, has degrees of freedom and approximately 400 elements.

65
 A SHOCK-ISOLATION SYSTEM FOR
22 FEET OF VERTICAL GROUND MOTION*

E. C. Jackson, A. B. Miller and D.'L. Bernreuter

Lawrence Livermore Laboratory, University of California
Livermore, California

Shock isolation of fragile equipment from severe ground motion induced by

underground nuclear detonations requires special techniques for inexpen-
sive, reliable performance. Two shock-mitigation systems that have been
used successfully for the past several years are described. These systems
have allowed equipment to be located closer to the explosive source,
resulting in considerable savings in diagnostic cable costs. A-new
system has been designed for even more severe ground motion. The new
design, the instrumented testing program, and test results are discussed
in this paper.

INTRODUCTION REQUIREMENTS AND ENVIRONMENT

Experimental data from underground nuclear For maximum utilization and mobility, most
detonations are conditioned and recorded by electronic equipment is housed in truck-trailer
sensitive electronic equipment. Normally, the vans or portable buildings on skids. Most
experiment requires the equipment to be located trailers are conventional highway type, but a
relatively close to the explosive source. In few have been fabricated for higher shock
many cases, ground motion induced by these requirements (7 g vertical). Loaded weights
nuclear explosions requires shock-isolation vary from less than 10,000 lb to 72,000 lbs for
systems capable of supporting heavy instrumen- trailers and up to 140,000 lb for skid
tation vans and isolating them from a very buildings. There is a large variety of
severe three-dimensional dynamic environment, electronic equipment and accessories. Shock
fragility levels vary from 1/2 g to more than
During the past several years we have 50 g. When properly mounted, most equipment
designed, developed, and fielded several falls within our medium fragility range - 6 g
different shock-mitigation systems. Two basic vertical.
systems have been standardized and are discussed
in this paper. These standard designs have The ground motion induced by an underground
allowed us to reduce signal attenuation and to nuclear detonation varies considerably,
reduce very large cable costs by locating depending upon yield of device, geology, and
equipment close to the source. Shock-isolation location of interest. The optimum location,
costs themselves have also been reduced, and with respect to reliability and overall costs
overall reliability has been increased by this for diagnostic and other portable instrumenta-
stat.dardization. tion bunkers located on the surface, regardless
of yield, is safely outside the subsidence
To meet requirements for even more severe crater, but not more than half the depth of
ground motion, we have designed a shock- burial from surface ground zero. In this area
mitigation system to withstand vertical ground the surface usually spalls and follows a
motion up to 32 ft/sec, or about 22 ft ballistic path. Figure 1 is a time history
displacement. The new system is a modification curve of surface ground motion for a large
of one of our standard designs, with the usual event, but the characteristics are typical of
crushable materials replaced by a columnar the spall region. The vertical slapdown
energy absorber. This energy absorber allows a acceleration pulse is usually followed by a
longer stroke without an increase in payload horizontal radial pulse, which can be either
initial height. away from or toward surface ground zero.
Horizontal tangential pulses are also occasion-
In this paper,
the instrumented we discuss
testing the and
program, new the
design,
test ally significant.
results. Work performed under the auspices of the

Preceding page blank U.S. Atomic Energy Commission.

67
 The basic requirement of all shock-
mitigation systems is controlled relative
displacement and force transmission between
the shock input and the package or system to
be isolated. The fundamental classification
of shock and vibration systems is the manner
in which it stores, absorbs, or dissipates
energy. A great many different materials and
methods are used in shock and vibration
isolation. A complete discussion of all

.
AV --,., pare-eters involved in shock mounting items for
underground detonations would be quite long;
therefore, in this paper we will describe only
'dII 0 h....,l a few methods that we have been using.
" :,4 In some cases, accurate final position or
.1 V alignment of the equipment with respect to the
ground is important and must be considered
in the overall design. However, in most cases
position and alignment are iot important. This
paper concerns only these cases. This
variance allows us to design around the three-
--------.....------ %,6 dimensional ground-motion environment by
decoupling the horizontal shock components.
Decoupling is achieved by placing the system
on surfaces with very low friction. Vertical
accelerations are isolated by constant force
vs displacement energy absorbers. Ideally,
these absorbers should have negligible rebound
characteristics. With this condition tha
conventional analysis is relatively simple
o-.(see Ref. 1). Accurate measurement of th6 pay-
load weight and center of gravity and the
"-" appropriate sizing of the energy absorbers
minimize the amount of differential vertical
displacements (tilting) of the system. When
required, excessive horizontal displacements
are controlled by nylon tethers. Energy-
Fig. I Typical time history of ground motion absorber deceleration set values are based on
within the spall region. equipment fragility levels and a structural
amplification factor of 1.75. This factor
accounts for the elasticity of the shock mount
DESIGN METHODS structures,
trailer equipmentDynamic
structure. mounting brackets and the
effects of impact
velocity on energy absorber forces are
The overall problem blends together: (1) considered separately.
definition of input, i.e., ground shock
parameters; (2) fragility level or shock
sensitivity of equipment to the input and We have developed several shock mount
1..,ose (i.e., does it record or transmit systems varying in complexity and cost. The
through shock arrival time?); and (3) design selection of a system for a given event depends
of a fail-safe shock mount system to modify primarily on the estimated maximum vertical
the shock environment when required. ground motion as shown in Table 1. The energy-
absorber system does not necessarily have a
The reliability of a system is strongly factor of safety greater than unity with these
dependent on a foreknowledge of the input maximum estimated input conditions, except for
conditions. Prediction of surface ground the incorporation of fail-safe features. The
motion for our purposes relies heavily on respective nominal ground motion values are
empirical data and methods (see Ref. 1). The much less.
shock-isolation design for a given event is
based on nominal and maximum ground motion
estimates in order to obtain a factor of FOAM AND REUSABLE CRIBBING SYSTEM
safety based on energy. The maximum estimates
are based on maximum credible yield and maximum For many events the estimated maximum
scatter of applicable empirical ground-otion ground motion is less than 10 ft/sec and we use
data. Whenever possible, the design is based the foam and reusable cribbing (F&RC) system
only on peak input parameters since detailed for trailers and vans. It includes permanent
time histories of ground motions are very reusable wood cribbing with an angle iron base
difficult to predict.

68
 TABLE
I
Shock-Mitigation System Limits

[Maximum
System
Vertical Ground Motion
Velocity, ft/sec Displacement, ft

FFoam and reusable cribbing

Universal guided column
Full-stroke guided column
10
24
32
nu2-1/3
" 12
% 22

tied to the trailer (Fig. 2). This forms a

solid extension of the trailer structure to the
normal suspension height and is capable of with-
standing the horizontal loads. A simple foam
and plywood crush pad is bolted under the
cribbing. Styrofoam Is used because for the Um
required heights and surface areas, it is
strong enough to withstand the horizontal shear
forces without additional guide structures,
The crush strength of the polystyrene foam is
constant
increases for the first
gradually to 35%
aboutof 140%
deflection and
at 65%.: .'.',, ...
deflection. Because very-low-friction surfaces . . ..- ',
are not required, the crush pads sit directly - - .
on plywood ground pads.
Fig. 3 Typical installation of Universal
UNIVERSAL GUIDED-COLUMN SYSTEM Guided Column (UGC) system.

The universal guided-column (UGC) system

was designed for ground motions exceeding the
F&RC system capabilities and up to a maximum The bottom af the column is attached to a metal
of 24 ft/sec or about 12-ft displacement. A disk by means of a pivot joint. The crushable
typical installation is shown in Fig. 3. It is material sits between the metal disk and the
an energy-absorption system that can use almost beam as shown in Fig. 4. All horizontal loads
any crushable material. Commercially available are transferred to the metal disk and therefore
polystyrene rigid foam has been the most into the metal guide column. Horizontal loads
inexpensive reliable material. The vertically into the disk are controlled by antifriction
controlled crush load is transferred to the surface pads made of properly sized Teflon,
trailer via a cross beam. Each end of this grease, and acrylic. The pad design is based
beam contains a guide in which a column is on many laboratory friction tests and field
inserted. This column extends through Jhe experience. Measurements of the actual full-
crushable material down to the surface pad. size system have consistently indicated a 0.06
coefficient of friction. The horizontal force
is transferred to the beam by means of the
metal column, which imposes a twisting moment
on the beam. Because the beam is a box-type
structure, it can withstand the torsional
loads. The magnitude of this twisting moment
is a function of the coefficient of friction
/for the surface pad dnd crushable material
-- height and crush load.
c1 We have found that the maximum horizontal
0 displacements occur after slapdown and after
the crushable material has compressed;
Fig. 2 FARC shock-mitigation system: (A) therefore, the surface pad Is designed to have
replaceable foam pad bolted to very low friction for a limited horizontal
cribbing, (B) turnbuckle tie, (C) displacement while the structure is mounted
wood cribbing with plywood scabbing, high and then after slap-down the column
(D) angle iron frame, (E) reusable moment arm is smaller and the system is
cribbing assenbly. allowed to slide onto higher friction

69
 A primary difference in the system is the type
and location of the ene.rgy absorber (see Fig.
5). All remaining components function the same
as the UGC system.
B
The constant-force columnar energy
due to
Momrent absorber is called a TOR-SHOK and is manufac-
hrnl o f tured by A. R. A. Products, Inc. A single
stage of this device consists of two concentric
tubes with a coil of ductile wire forced
between them. The interference fit between
oC the wireand
sliding andtotheforce
tubestheis sufficient to prevent
wires to rotate.
Rotation of the wires is similar to rotating"
the ring of a torus inside out. The resulting
tensile and compressive strains are in the
F plastic range. Contracting or extending the

I-Veitical force

Fig. 4 UGC characteristics: (A) column, (B)

crossbeam, (C) crushable material,
(D) pivot joint, (E)metal disc, (F)
Teflon, (G)anti-friction surface pad,
exploded for clarity.

surfaces. Nylon rope tie downs are also used SECT A-A
to prevent excessive horizontal displacement
and to absorb some of the energy. Colun

This system has been used reliably on more

than twenty occasions with actual ground Rubber
motions exceeding 12 ft,'sec or 36 in. displace- isolator
ment. Vertical energy-absorption strokes have
reached 40 in. and horizontal displacements up
to 8 ft have occurred.
Energy
absorber
FULL-STROKE, GUIDED-COLUMN SYSTEM

A new shock-mounting system has recently

been designed to meet requirements for very
severe vertical ground motion up to 32 ft/sec
or about 22-ft displacement. This design is a
modification of the UGC system, with the crush
materials replaced by a full-stroke, constant-
force proprietary
materials energy
(honeycomb, absorber. Forthe crush
Styrofoam,
energy-absorption stroke relative etc.)
to the Crossbeam
original height is designed to be about 35% for
nominal design conditions and/or 65% for
maximumi credible conditions, depending upon pua e
which case prevails. The crush materials
bottom out at 65 to 75% deflection. The full-
stroke system allows tl! original height to be Crushable
reduced by one third, and therefore the material
structural requirements for some of the basic
components are also reduced by one third.

The structures for the full-stroke, / / -§//'-

guided-column (FSGC) system are similar to the
UGC system except that they are stronger. The Fig. 5 Characteristics of FSGC system.

70
 tubes apart axially forces the wires to rotate ratings and cluster arrays of energy absorbers
many times. The repeated cyclic plastic were tested. At some locations the clusters
straining results in almost constant energy were arranged to impart very high eccentric
absorption per cycle of rotation (or inch of loadings into the columns. The FSGC system set
linear stroke) until eventual fatigue failure. up before drop testing is shown in Fig. 6.
( Before incorporating TOR-SHOKs into a Test instrumentation consisted of acceler-i
shock-mounting system we purchased some for ometers and high-speed movies. Thirteen
static and dynamic tests of individual units accelerometers were placed on the top of the
in both the extension and contraction columns, on the FSGC beams and on the equipment
directions. During the static tests loads were inside the trailer. All channels were recorded
uniform and increased breakaway loads occurred on magnetic tape, with seven selected channels
only a few times. Impact acceleration pulses on m scietigraph , f i cev aluatio n
(three times the average deceleration values)
were measured on the payload in about half of
the extension and compression drop tests. The The drop tests are summarized in Table 2.
axial component of the lateral resonant Tilting occurred during the first three tests,
vibration was quite apparent during the but averaged deceleration strokes agreed with
compression drop tests. In one case, this was test parameters. An erroneous center of
a function of two stages and not the overall gravity location, which was discovered before
length of the column. In all tests drop height, testing began, caused the tilting. After the
stroke, and the average deceleration values
agreed very well. There were no changes in load
during the tests, even though one unit
experienced five complete strokes.

In order to use the full-stroke capability

of the guided-column system, we attach the
energy absorber between the top of the column
and the beam, as shown in Fig. 5. We are using
three-stage energy absorbers with a compressed
length of 58.5 in. and 8-ft stroke capability
with capacities up to 12,000 lb. Each column
can accommodate up to four energy absorbers
with a total deceleration load capacity of
48,000 lb. Rubber isolators are used in
mounting to insure that no bending moments are
transferred into the energy absorber.

DROP TESTS
A series of full-scale drop tests using a
30,000 lb trailer was conducted. Various load
Fig. 6 Prototype FSGC system before testing.

TABLE 2
Drop-Test Results - Average Values

Deceleration
Drop Velocity Desi!n Conditions Actual
Test Height at Impact Force Stroke Stroke
No. (in.) (ft/sec) (g) (in.) (in.)

1 21.5 10.7 2.7 7.9 7.62

2 38 t4.3 2.7 14 13.5
3 74 19.9 2.8 26.4 26.9
4 12 8.0 3.0 4 4
5 180 31.0 3.17 57 55
77 --right
left side
side

t Gravity force not included L

Three energy absorbers failed and system bottoned oqit on right side.

71

)-
 third test the energy absorbers were reset
(compressed) and redistributed in accordance
with the correct center of gravity location.
Figure 7 shows the trailer after the fourth
test during preparation forthe 15-ft drop of
the fifth test. Fifteen energy absorbers were
used during the fifth test, and three of them
failed. They represented 26.6% of the total _

deceleration force. Two failed on the side 'q

that bottomed (see Fig. 8). They represented
34% of the deceleration force on that side..
Peak accelerations inside the trailer, when the
system bottomed, were 12 g (see Fig. 9). None
of the equipment inside the trailer was damaged
(including the fluorescent lights) and the only
damage to the trailer was a wrinkled skin in
one area requiring replacement of a few rivets.
Motion of the trailer during the third, fourth,
and fifth tests is shown in the movies.

Typical unfiltered acceleration response

of equipment inside the trailer is shown in
Fig. 9. Transducers 12 and 13 are at opposite
ends of the trailer. An initial peak pulse
occurred only at one end during the second test
and at both ends during the fifth test. These
initial peaks occurred randomly during the
testing. Impact acceleration at the top of the
columns was 100 to 300 g.

Failure of the three energy absorbers

during the fifth test occurred when individual
stages ,;2 ced the end of their stroke and the
spot-welded bands failed to retain the wires
(see Fig. 10). Modified energy absorbers were
then successfully tested. Several units were
statically loaded to 25,000 lb (twice the Fig. 8 Right side of FSGC system after
maximum absorption load), and one unit was 15-ft drop test, showing two
separated energy absorbers.

A AA

A ff~T7~7T~N
A, 72

Fig. 7 Prototype F5GC system ready for Fig. 9 Response of equipment inside trailer
15-ft drop test. during tests 2 and 5. ,
impact-tested to failure. This unit finally pads. These are under the protective polyeth-
failed at a swaged tubing flange. ylene covers at the base of each column in
Fig. 11.
Our comprehensive test program identified
discrepancies in the as-built FSGC system.
These were corrected before using the system ACKNOWLEDGMENT
under actual severe ground-motion conditions.
Normally we desire fail-safe shock-mitigation The authors would like to acknowledge
systems. In actual field applications of this the cooperation and assistance provided by
system, we have installed rigid-foam backup Holmes & Narver, Inc., (Nevada Test Site) in
connection with the F&RC and UGC systems and
EG&G, Inc., (Las Vegas) in the drop test
portion of this program. In particular, the
efforts of Mr. E. Fuller and Mr. H. Montalvo
,,Wer Nbof H&N and Mr. P. Hulhall and Mr. R. Nakanishi
of EG&G are gratefully acknowledged.

: REFERENCE

1. D. L. Bernreuter, E. C. Jackson, and

'o A. B. Miller, "Control of the Dvnamic
Environment Produced by Underground Nuclear
Explosives," in Proc. Symp. on Eng. With
Nuclear Explosives (Las Vegas, Nev., 1970),
U.S. Atomic Energy Commission Rept. CONF-
700101, Vol. 2, pp. 979-993, May 1970

II

Fig. 10 Cross section of energy absorber at

end of extension stroke.

I"

Fig. 11 Field installation of FSGC system.

73
 DISCUSSION

Mr. Fox (Barry Wright Corporation): How much

deflection did you get out of the foam itself? Was Mr. Fox: You were only concerned about the
inital shock"blast. You did not care what happened
most of the deflection taken up under the shock? after theocital blast? Is that correct? h e

Mr. Bernreuter: Yes, the actual extensionof the

Mr. Bernreuter: Yes.
TOP SHOCKS themselves took up most of the shock.

Mr. Fox: Was that fifteen feet? Mr. Peralta (Bell Telephone
sizing the shock absorbers, do youlaboratories):
do any kind of
In

Mr. Bernreuter: No, I think their stroke was calculation as to how much energy they should be
able to absorb?
designed to be twelve feet capacity. In this case they
stroked 12 feet because they bottomed out. In fact, a
Mr. Bernreuter: Yes we do quite a bit of calcu-
couple of the units failed. We picked the trailer
feet off the ground up 15
and dropped it, so it had a velocity lation based on the weight of the trailer, the expected
level, etc. The formulas are worked out,
of about 28 feet per second when It struck the ground, and we have
performed a compute*, analysis.

74
 THE C014PARISON OF THE RESPONSE

OF A HIGHWAY BRIDGE TO UNIFORM GROUND SHOCK

AND MOVING GROUND EXCITATION (U)

Neil E.Mechanics
Johnson and Robert D.
Research, Inc.Galletly

*1 Los Angeles, California

This paper compares the dynamic response of a highway bridge structure

subjected to:
* The Uniform Ground Acceleration Time History for the May 18.
1940 El Centro Earthquake
* Moving Ground Acceleration
9 An Average Acceleration Shock Spectrum for Strong Ground Motion
An analytical procedure is !outlined for predicting the response of a
highway bridge structi.rewhose ground motion varies between its supports
and is dependent upon seismic wave propagation characteristics.
A typical six span highway ibridge is presented. A finite element model
A of this bridge is develope4 including representation of soil/foundation
*1 stiffness and damping characteristics. The response of this structure
is evaluated for the various ground motion conditions using normal mode
techniques. The dynamic analysis of this st'ucture was predicted using
the tMHI/STARDYNIE Structural, Analysic System.

INTODUCTION seismic wave train must be connidered as it

traverses the structure the assumption of uni-
Considerable interest is currently being ex- form ground motion for the structurallarsembly
prorced so to the behavior of civil structures can nn longer be Juatified. If it car. be antic-
uzidr thf. effects of seluinic excitation. Tra- ipated that the ground motion may differ In
,iior,.l analyris techniques applied to such phase and magnitude among several structural
:'tr et.ure include the use of quani-statie load supports then the resulting structural response
rar'tors, n:hock spectra and, in some eases, must be reviewed carefully. This paper Rd-
tr .e,1rration tlm,- histories. This paper ad- drersen n hypothetical case where a strong: -el-
'Ir,~"':,'.a ,;pecial casve of time history analysis mic disturbnce propagates along the prounl
-'aer,'n th" propalration velocity of a seismic supportino, a typiclu two lane, six span, sinrh,
watvr 'movinfg !on(itdinally along the structure ped^ntal r:upport highway bridge structur, nt .A
1con::idered. liRults from this moving groun, relatively slow, uniform velocity.
motion nnalycsi technique are compared with more The method of:inalysin utilired for attacKin-
trc-
Thepro othe lya ire ne or' aa jtrue-
tra,,itionil
niuen timethehistory
through and shock
application of all three tech-
spectra the problem of the 4lynamic response of
tenique toh theiplicatinh ofa three tural system to moving ground motion Ir eutlin,4
techniques to an idealizrd hirhway bridge, in detail. Particular emphaiiis IN placel upon
For .tructural nnalv-' the ,tsnumed ground utilizing standard atructural coessoitia nrd
motion is often given in the f'onri of an accel- analytical techniques. The recults ofr thi.
eration timn history recorded durini a neinmic -ffort were then compare-. to re.,ultn eonidrit,
,rint FI or an artifJ'Inlly i.enerated time uniform ground motion for tl,- -amo ieismic di.;-
hi,'. 'y with certain nimrlni qn,1 statictical turbance and to reculto conil srldeng a typlenl
qurntitJ,:; rettined [,I. For purpo.es of truc- average sheock speetrn. for wtrone, ,round mot.n.
t;ural 'voluqtion thin g:rond ttotion if, generall:Y A discuarion of ths fridpo mn-I I r jv"%,, This
" rpplre,-to the foundation and 'rfonol suppoktr inelmior the geotrmry qn .e.rnie.irrlptleo of
Jn n unif'orm manner, that is.with thr' diftlr- the bridge ansembly. A 1trler ummary ir %r'
bnlriv(! of the prond exactly the cam, tit each aseumetd aoil properti Is lndleate'. Pte
rcura. porint. 11,,r n,.h time. .tmutrel was %Aste.ll1n I urrtel - fvn',i-
ur Qro with ltuc * '. tri lo'
e h,.,e r .. ' , a1., m
When tho ,ho ar't,,rirtit- iengt,, of a
I';Inri, ,,noii, '1 that the popoWf'tflrn t, imn ,1 tte'
r) frxs'1 pre' - l

75

COPY
 2 23

Nodes with Lumped 2

Coordinate System

Pig. 1 " Blridge Physical Characteristics and Idealized

)'thematica. M4odel

A finite element model of the bridge struc- to minimize the number of dynamic degreos of
tural system was developed. The natural fre- freedom for the problem. Note that a much more
quencies, mode shapes and related properties detailed model might be desired if an actual
arc identified. Normal mode techniques were highway bridge were to be analyzed using the
utilized in the problem formulation. The techniques described herein. Such a model
STARDYNE Structural Analysis System and its would probably include a more extensive three-
auxiliary subroutines were used exclusively for dimensional model considering both beam and
the modal vibration analysis and the dynamic plate finite elements and a more detailed de-
response computations. scription of soil/structure interaction includ-
BRIDGE MODEL ing pile foundations, if used. The assumed mass
~ asue.
tion ~ ~ ~SpotclmsaBrid
~ Fgwa scaphrct
ge risisan The isazdYN
distribution summarized in Ta.,le 2.
rgadvlpdb
A mathematical model of a typical 5 ix-span, The roadway superstructure and support columrns
singe
sppot
pdesal hghwa brdgewaswere modeled as a series of beam mrembers. Si.c-
developed to illustrate the analysis techniques tion properties evaluated for these beams are -
described herein. The bridge, wnich is shown
i~
Fi.
n Fg.
Fi. Fg.2
9 m~a:d
is
isann iealzaton
dealzaton ff agiven
aassume in Table 3.as These
concrete section
the basic properties
material with
typical structure and is patterned after typical aprpit adfctosfrrifrigsel
~~~~two-lane
utiled ihighway
the bridge fyeo
prolme designs.
raion. The seldtehniued. prpit oilain
Teodes her o enocn
i.Sch a meogeelte.: l
Using the finite element model described
Thew lnidealized bridgeconsiststh of six 120 foot above, the natural frequencies and normal modes -
long toln spn wih heroadway superstruc- of' the bridge were determined through the use of
ture placed approximately 50 feet above the teSAON tutrlAayi ytmcmue
ground urface. Reinforced concrete construe- thegram.eTNE STA l Anas Semomputer
ens ae assumed. toe supporte bridge
sv pa- and in use by a large number of engineering firm
tpe fodane a s, worldwide through CoNtrol Data Corporation Data

typefoudatins.Centers,
Foundation/soEf interaction was idealized as dy is a large
oamic
analysis (up based
system to 6000onDOF) itatic and
the finite ele-
a series of linear support springs. These ment, normal mode bethodof analysis.
springs have been designed to give reasonable The lowest sixty (60) normal modes were
translational and rotational stiffness proper-dermnd Ths oswihcnb gnral
sri te aSsrued nye ob e
Ta soi classed as either horizontal bending, vertical
spxilaring
r srutne s i eT dex bending or longitudinal are summarized in Table
The example bridge was modeled as a lumped . The generalized
s
weight
h
and modal particpa-
ascmtrilwt
n s atredatrtyia asm onrt
tyia srcue
mass, finite element ap shown in Fig. 1. screi
tion factors correspnding to each o these c
This modtl consists of 45 unconstrained nodes mldes are alsoI givenm. in Table A brief de-
each having 6 degrtes of freedom. Only 2 n es scriptono f the normlr ode ethod an the ters
were selct.ed as m,ss poi locations
dt in order in Tble T is given in Appendix A. e

76
 Due to the symmetry properties of the ideal- Table4 were used in the various dynamic response
ized structural system, it should be noted that analyses described below. A modal damping fac-
only a portion of the normal modes listed in tor of .05 was used for all response analyses.

17"-0" TA= I
SUMIARY OF SOIL SPRING CONSTANTS

8 4'-6" Direction Soil Spring Rate

pr[ 121"
X) 1.2P'5 x 106 lb/in

61J X2 1.215 x 106 lb/In

Assume 2 sq. in.

steel per foot X3 2.1450 x 106 lb/in
of length

,4 9.1475 x 109 in lb

10"
S6 x 109 in ib
9.)47,5

SUMARY OF MASS PROPERTIES

t Element Mass Property

Roadway Superstructure 84;76.32 lbs/ft

Fig. 2 - Typical Cross-Section Support Columns 23414.8 !bs/ft
Pad Foundations 201;,282 b.; Each

R3,AM PROPERTI ES

Beam Area J 12 I3 Shear Shape Factors

Type In in in in 2

Roadway 8111,, V2,71 ),60 h, )71,662 -0,o5",240

0.35) .

Supp__rt
Colu-n 23114 ,522, , 1,21l.44
1i3 ].2.
.,, ,l'h.

77
 TABLE 4
RESULTS OF THE MODAL ANALYSIS

Frequency Genralized Weight Modal Participation Factors o

Mode tbde De-cription
No. (cps) (kips) x x3

1 0.577 3360 0 1.247" 0 Horizontal Bending

2 1.41 9 3h27 0 0 0 flor zontil Bending
3 2.005 6557 i.0414 0 0 Longi tul nal
4 2.717 3570 0 0.50,)5 0 llcrizontal Bending
5 3.021 17211 -0.1081 0 0 Vertical Banding
6 3.037 1396 0 -).6009 Vertical Bending
7 3.185 165) 0.0588 0 0 Vertical Bending
8 3.319 2669 0 0 0.);782 Vertical Bending
9 3.1431 1958 0.0240 0 Vertical Bendingt
10 3.502 '°84 0 0 1.57)1 Vertical Bending
11 3.997 2557 0 0 0 Horizontal Bendin
4.15) 43 , ) 0. 3618 0 nori zontal
13 1.454 3204 0 0 0 iforizonzal Rotation
11h 5.058 15 A4 1) 0.5317 0 Hori zontal Pending
15 ).4O6 1811O) 0 0.181,3 0 Horizontal Bending
i( 5.731 2188 0 0 O. 111 Vertiva! Bendin.1
17 u.058 3412 . 170 0 0 Vertical Ber.ding
38 6.449 1773 0 0 0 Horizontal Bending
]( '.186 ,827 0 00 Horizontal Bending
20 6.512 3207 0 0 -0.0062 Vertical Bending
21 6.981 58O 0 -0.P120 0 Vertical Bending
22 7.013 35P2 0.033, 0 0 bongitudi hal
23 7.1)0 753 0 0 -0.03b2 Vertical
?)l 7.20) 585 0 .0,413 o ';orlzontal
25 7.2141 1227 0.8))5 0 0 Longiltudinal
26 7.256 620 0.0160 0 0 Longitutdinal
27 7.259 5)2 0 0 0 Horizontal
28 7.259 701 0 0 -0.0183 Longitudinal
29 7.278 761 -1.0444 0 1lon1i tudinal
30 7.434 15)2 0 0 0.218P Vertical Bendin,-
31 7.492 5 0 1. 00.56 0 Horzontal
32 7,572 1250 .Y17. 0 0 Vertical Pending
3 7.676 3178 f 0 .(P3% Vertical Bending
34 7.709 27 0 0 0 'ors i on
35 8.4 0 38. ) -. (7)4 0 dor zontal
i 8.511 1 3. 2 0O. ,, * Ion,,it-dinal
R,.7 i -0. )! ,2 0 'iorni.on'al BLndin
18 10"' 0 0 0 iorl zo:It-]
3'j i.7? 2 -o. ,10, 0 10orn4(".t't
ho 11.73 ]80 0 0 *¢:2 on tai 1;ending
,I 12.3 32 1 82 0 0 !aoOh
42 13.176 1188 0 O. "17-3 0 8or! zontal
4 13.88 1053 0 0 0 "orizontal
14 1;. 315 10, 0 -0. ) 00 I 8ori zoistal Bendi n,
45 I4.4hil 15 ,(l')5c 0 ,ori.zontail iending
40, 14.77 * 15 Y) 0 0 0 orizontal
47 14 .e1 5 1, 0 -Q.j'-7 0 orl zont i]
48 15. 0'0,'43 oQ.t )~f on1:t'l i'll
10.
I 17. V1' 0 ) ori,'ontal Bendin.:

51 18.718 i -,. , Ion, "i .'dinal

'13 1 .774 1 .' 'id, i rwtl
"'.0.
54 1.13 1,1 -'. )' 0 e I
"0I. 2') "'4 - ,. )' or:or i,'c
,l Be,,W r,

f 5'0 .IA> -' " - ).,) P!I',tP

I121

78
 PRESENTATION OF THE METHOD OF ANALYSIS Rearranging Eq. (2), the equations of motion
The earthquake response of structures is de- become
scribed in many sources (5, 4]. A variety of I cl'
N 4I 1xl - Wflyl + (5)
computational tools are available to the struc-
tural analyst. Both transient response tech- Just as the terms on the left hand side of the
niques (5] and shock spectra methods (6] have equations can be considered as the forces acting
achieved wide acceptance in predicting the be- upon the system in the absence of ground motion
havior of structural systems subjected to ground (lyl =jW =(Y' -0), the terms of the right hand
motion. This paper addresses the problem of the side can be viewed as those forces Induced upon
response of structures with large characteristic the system by the ground motions lyf when the
length and considers the case where unattenuated mass points are constrained (lxi =.il = x] =O).
strong ground motion traverses the supports of a This provides a straightforward way for construc-
structure at a uniform velocity. The method is ting [c')and [').
developed In a straightforward manner and is out- For seismic analysis the ground motion is
lined below. often described in terms of ground acceleration.
Consider a system with n degrees of freedom Eq. (3) may be rewritten in the following manner
and r resilient supports to ground. Fig. I shows to accommodate these terms.
a typical system. Assume that the equations of Introducing a new setof auxiliary coordinates
motion are to be written in center-of-mass Jz(, the forces induced by ground motion on the
coordinates and the mass matrix of the system constrained system (right hand side of Eq. (3))
Nt] is diagonal. For simplicity, assume the
damping is small and proportional to the stiff- may be written
ness. This is done for ease of the present K]IZI+ [CilCI =']lyl +['Jlfl ()
analysis; other restricted types of damping
could be considered. Modal techniques will be Substituting the transformation
introduced, modal damping will be utilized and
modal coupling due to damping will be neglected. Izi = [T)JYj (5)
In the absence of ground motion the equations
of motion are: where
r
rmIYJI+CI1 -, [K] XI=0 0) niT] is a transformation of ?onstant
wcoefficients;
where n1
n nIz I is an auxiliary set of coordinates
nNq is the diagonal mass matrix of the into Eq. (4), it is seen that
system;
nn [K][Tflyl + [C][Tfl I. ,Jjyl + ,]iJi (61
n[C) is the damping matrix of the system; [

nn Comparing like coefficients in Eq. (6) it may

[K) is the stiffness matrix of the system; be written that

n[x] describes the absolute motion of the [KI(TI=I]:

mass elements of the system.
For ground motion of the supports the forces in- and (7)
A duced upon the system are described in terms of'
relative motion between the mass elements and [C][TJ= !c')
the support elements. Considering this effect,
the equations of motion can be written Eqs. (7) may be rewritten to give

~,~(
'I[C]l -l c'IJ] +[K]ix!-
PkI+'I'h o (P) (T) -- Kf 1 f'C[c'1 (8)
where The two right hand portions of Eq. (0S) are
r identical expressions since it was previously
n[kI] is a rectangular array of stiffness assumed that
coefficients relating ground attach
nodes to adjacent internal nodes of [C) =o{KI and 'c'] =a .'J (9)
the system;
r Furthermore, the existenc- of iK1 is assured
n , a rectangular array of dampini: since !;,] repre-ents the stiffness %atrixof a
coefficients relating ground attach constrained zyster.
nodes to adiacent internal noles of Next, u.'in F1. (',, F'. (1) be writken
h..ay
tree system;
1 j'zx -. C , i*JK ;W 'KJlr7 + ( 1l )
r ly! d'scribes thp absOlute ivround1 motions
(displacements) nt each of the r hoarrn{ in,'rrr, Fh. (10, .eco.es
supports.

79
 N4I'i+[C(ikl-fli)+[K(txl-Izi),o (11) Then
Defining y,(t) (Y(t) t ) I
IeI =xl-f, 0 )]lY'(t){ Y(t- 2

and subtracting 1JYiJ from both sides, Fq. (11)

may be written
ii= Y~) =
1
Ytt)(8

I'v~~~~~~~~~lY
KII: I' cIIi ,(t - tr)
Yi'
Noting that
Fig. h shows the time histories 6ira ground dis-
"I =[T] I= Ill
f[K (1i) turbance for the special case of several evenly
spaced colinear ground points.
Eq. (13) may be written

I! +{c]I +[K]I f -Nt [KM"I [k'1 J (15)

4.,
or 4 C

N14l +[C]II +IKJIB -[D][k') l (16) 0 P4

where (D] =-N[K I1 is the dynamical matrix for
the constrained system. H

For the case of moving ground motion, it can

be assumed that the basic ground disturbance has
a plane wave front and a time history given by
5'(t). Further, let the velocity of propagation
for the disturbance be v and the time the distur-
bance encounters the first ground point be t D-. , 4-1
Then, the disturbance will arrive at successive 4 .
ground points at a time a
die i
tdcosoiI
0 VVV
t v (see Fig. 3) (17)

where 04)

t is the time of arrival of the disturbance

at the ith ground point;
di is the distance from the first ground
point to tie i t h ground point; 43

O is the angle between a line adjoining the

first ground point and the ith ground LAAA AA
point and the direction of wave propaga-
tion; 0s..
v iz the velocity of propagation. 0
$s4
Plane Wave Front ' -
Time
tr Fig. 4 - Time History for Moving
Ground Motion
; t
13- Orcuind
Point
ith Ground Consider a solution to Eq. (16) -f the form
VPoint 'f' 'Ii (19) r

where
(j is an arraj of mode shapes o& the un-
di damped constrained system;
IIin a set of Penerallzerl (modil) co,)rdi-
Fig. 3 - Wave Train Propagation nates.

80
-

Substituting and premultiplyine, by [O]T, Eq. (16) that for the May 18, 1940 El Centro earth-
becomes quake. The vertical component of ground
acceleration equal to 0.6 times the hori-
S- V zontal component (Case 2);
0) . The horizontal component of the ground
°
acceleration in a direction 45 to the
On defining, in the usual manner, the generalized longitudinal direction of the bridgc with
parameters a magnitude equal to that for the May 18,
1940 El Centro earthquake. The vertical
[¢]T N([I = r'-aj component of ground acceleration equal to
I JT(CJ( 4],= 0.6 times the horizontal component (Case3).

RESULTS
J I¢J=X Using the loading conditions described above,
,JI I (' ' = a series of nine independent analyses were per-
formed. Each of these analyses resulted in the
The equations of motion become computation of displacements, velocities and
accelerat.,ns at the various nodes on the I leal-
~JI'i + + (21) ized bridge structure. These results can be
further extended to include equivalent nodal
After solving Eq. (21) for I1, the absolute forces and internal member loads and stresses.
accelerations of the system become To briefly illustrate some the results ob-
tained and to provide a basis for comparing the
' ] = li! - IZ - (TJYJ three methods used, the following representative
data are presented.
and 72) Acceleration responses of a point on the road-
way at the midspan of the bridge are presented
I~x!
4,1 J*.fTHY
Iin Fig. 7 through 12. Figs. 7, 9 and 11 show

INPUT DATA representative acceleration time histories using

the Uniform Ground Motion (UGM) :ethod for each
For this study three methods for solving the of the three load chses. Figs. 8, 10 u,!. 12 show
dynamic response of a structural system to seis- similar acceleration time histories using the
inic inputs were used. For each method, three Moving Ground Motion (MG4) method.
loading conditions were considered. The methods
wereFigs. 1, 14 and 15 show the distribution of
maximum values of acceleration for each load
* Transient response of the structural sys- condition as a function of bridge span (see Fig.
tem subjected to the uniform ground accel- 1 for span locations). Shown on each figure are
eration time history for tL. 'ay 18, 1940 results from the Uniform Ground Motion, Moving
El Centro earthquake (North-South compo- Ground Motion and Shock Spectra (SS) analysis
nent); methods. It should be noted that the accelera-
* Transient response of the structural sys- tion values shown on these figures for various
tem subjected to moving ground motion with span locations do not necessarily occur at the
a constant propagation velocity of 400 ft/ same point in time. Consequently, some care
see and an acceleration time history given should be used when interpreting these analyis
by the May 18, 19110 El Centro earthquake results.
(North-South component); CONCLUSION
9 Hespon~e of the structural system subjected
to the average acceleration shock spectrum A number of conclusions may be drawn from a
for strong ground motion 153. review of the analysis results presented above.
Some of the more significant of these are de-
The time history of the May 18, 191.0 El Centro scribed below.
earthquake (North-South component) is shown in
Fig. 5. The average acceleration shock spectrum It is seen from Figs. 7 through I that the
for. strong grage mo eeratiosn shon sp. time range of peak response of the bridge is
for strong ground motion is shown in Fig. .generally of longer duration for the Moving
The loading conditions considered were: Ground Motion cases than for the Uniform Ground
" The Lorizontal componcnt of the ground Motion cases. This phenomena would increase the
acceleration in the lateral direction of probability ofachleving n extreme peak response
the bridge with a nagr.it We~ equal to tat due toa fortuitous combinationof modal responses.
for the May 18, 19O F.lCentro earthquake. An examination of Fig. Is show. that for Uni-
The vertical component of ground sccelera- form Ground Motion, the longitudinal response is
tLion equal to 0.( times the horizontal Aignificintly higher than for Moving Ground Mo-
component (Case I); tion. This is due to very largerol.alparticipa-
* The horizont-! component of the ground tion of" the first longitudinal mo'le of the ;yrt-n
acceleration in the longtudinal direction when both end supports have the identical ground
of the bridge with a magnitude equal to excitation. For the Movinv Grounli c-ase, a reIur-

81
 .50

0215

0.5 -

0 5 10 Time, shc
15 20 25 50

Fig. - North-South Coirponent of Horizontal Ground

Acceleration of El Centro Earthquake May 18, 19140.

0The response spectrum selected for this con-

parison was a composite average of acceleration
: .8.. ,0
shock spectra for several strong ground motion
i events [3. Since this spectrum does not bound
- Percent of Critical Damping that derived from the ground motion time history
~utilized for this problem the resulting response
does not necessarily encompass the results ob-
.6 tained from time history studies. For a shock
O-spectra analysis the phase relationships are lost
= .oo nd a pessimistic summation of modal response is
I assumed. For a given ground motion time history
o4 / 2a respo~nse derived from shock spectra analysis
'-.4 will bound the response developed from a time
-- '- .... history analy'sis for either the Uniform or Moving
e Ground Motion method.
.In summary, it can generally be concluded that"
.c =0. an evaluation of Moving Ground Motion effects
Sresult in different, structural resp-nse than
.would be computed from a Uniform ground Motion
o analysis. The probblity of higher responses
tends to increase as the number of supports in-
_ _creases. Consequently, it can be recommended
that Moving Ground Motion should be considered in
0 0.8 1.6 2.4 the seismic design of multiple support brde .
Undamped Natural Period, Sec It can be further concluded that time history
response analysis of either the Moving or Unifor
Ground Motion types will give moretmehnondful
to ..6response predictions than those obtained from
>Shock Spectra analysis when many modes contribute
Spectrum Curves to the response of the uystem. For thi reason,
realistic and economieal highway bridge design
01 practice Ghould
consider tine history effects.
,i.n in th2.ptrticipation of the first longitu- REFdRet .
cinal mode w observed when the ground excita- Efh.
tinn r 'ch nndupportwa distintly different. 1. Robert L. Wiegel, Earthquake Fgineerin,
ue to 'he' bridpe symmetry, the resiponses to Prentice-Hall, New eiersey, 1970
1.h,. Uniform G;round Motion exhibit :.;iraitry prop- P.M. Amin A. Ang "Nonstationary Stoehn
tnd Ic
,rties ,ihout theei'pan of' .he briIe. ThiS Models of Earthquake Motion", o rn l oftib
prnpnr'y n tte. reSponse rersult doot occur Applied Mechsnics Division, ASCE, Vol. 94
f'or '-i Mwvlni' g;round M'tion method. This ii due No. M, Procurement hpr a0y . April si
S,* v'rltpian .oon ceround
in it time h eects.
orides
* u[,,"-
I ,'. ,t+h,'
,r p , enerally, of the rosp travets's
1. mif dusturhane,, on gothe V'. rl M. Harrnd,
n: Voals ,.'raw-le, h~ 'n
*wn m'd wiffer
b n t hethev
' produein hcit
tiron ofc opnde l,pen upon both the loafflnt. 1..Robert. Hurtyand
ieg, heF. F.tinst in, :)ynai
, , I1.Vr*an
.. - 'in- lre tlon consid-red. )~,trtures, Hll, Prentlce-ePt
New 970 .Jesrsy,

s 82
 -V

__..1
". 0

-. 23. -Ae AWAA

Z .25

Time.moo Time. see

flt. 7 - Hid Span'Horitontal Acceleratioo History Fig. 8 Kid Span Horizontol ActeleratlonHistory
thiforn Oroua Mtion Grond lotlon
ov4'rg

Case 2 UGH Came 3M

-|

0-

0 2 3 4 50 1 3 4 3
Tie*, see Time see
Fig. Span Longitudinal Acceleration History
i -d1 Fig. 10 M
kid Sjpn Longitudical Aceleraton History
UniformGround Mtion Moving GrOund W.tio

,oNc Came3
vvvv 3CM

.3 8.2

.22

7v 0

v. -. 2-

-.2

0 1 2 3 4 5 0 1 2 3 4
Tio'. see
Ti- , e
Figure It Kid SpunVertical. Acceleration History ?Ig. I 1 Mid~f
4 lpnertical" lraln.
Wlifors Crownd Flotion ivll ud tin
-.
1

1- UGM
1.0 A- lMM
o a- SS

. ..o..
.. o
It
.- ".-1-'d ",- o\--'."
",I,,
\--,.,-.0*<. , "0 , 10%" % -.-
< --
, .. .0- --

ot m
I I , I I I , i ,I ..
1 3 5 11
7 13 915 17 19 21 P3 25
Span Location
Fig. 13 - lMhimun Horizontal Acceleration Distribution
(Case I)
195

---
O...
-O ...
<)--
O --<'- -O ... - 0- - -0 - - - - 0O-
--O0

,4
~' 1.0 A- MGM
[3- SS
.94

.5
.. ..---. -- GA . --.-
-- A--.-o-A-. - -

01 I I I ,, a I ; I I 2; 2
1 3 5 7 9 !I1 1 7 9 >
Span iocation
Fig. 1, - 1aximt1. Lonitudinal Acceleration Dintributon
(Case ")
1.0 0- UG'
A- IJM

0 V
43/" .1/ / r%, /1\a-
//
l0.5 / .

o - _

1 3 5 9 11 15 1 1) 1 19 21 25 25
Span Location
Fi . 1) - Maximum Vertical Acceleration Distribution
(Case ")

84
5. Richard Rosen, et al, STARDYNE: User's Manual, Note that modal participation factors are com-
Mechanics Research, Inc. and Control Data puted for each of the threetranslational degrees
Corporation, 1971 of freedom.
6. Richard T. Haelsig, DYNRE IV User's Guide,
Mechanics Research, Inc., 1968

APPENDIX A: DEFINITION OF ANALYSIS TERMS

The STARDYNE System is formulated around the
"Stiffness" or "Displacement" method of struc-
tural analysis. In using this method, the struc-
tural system can be represented by the following
equations of motion (for free vibration):
r?4Ji 'j + [KF]Ix =1.0 (1)

where

M4 is the diagonal mass matrix;

[Ki is the stiffness matrix;

jxj is the system coordinate.

The cigenvalues (natural frequencies) and

eigenvectors (normal modes) of the system can be
extrar'ted from the following reformulated version
of Eq. ()

K 1,( =j
0~ (2)
where
2
o is the 1.th elgenvalue;
th
(r) is the r eigenvector.
The generalized mass of the structure is given
by

The generalized weight is given by

(4)

where

g s tho Gravitational constant.

The rodal participation factorr :j~r) are "a-

fined to be
,mil'ir

Mr
where

Pi(r) is the modal participation fact-r for

'he rth mode in jth translatIona direc-
tion;

m] is the oecntL of rars .- soeiati with

Ith d'oreo 'f freedon;

'1ir aie tho component

srocinte'l
of rth 7o1al vector
with i t !' dereop -f frre'ioq

r11r is the r'enerali7e'] mafrr for rth mo,1.

'C ~85 -

p.
 i1'

DE:OI1MATION AND FRACTURE OF TANK BOTTOM

HULL PLATES SUBJECTED TO MINE BLAST

Donald F. Hlaskell
Vulnerability Laboratory
U.S. Army Ballistic Research Laboratories
Aberdeen Proving Ground, Md.

An analytical study has been performed to develop a method for pre-

dicting the deformation and fracture characteristics of flat
rectangular tank bottom hull plates subjected to blast attack from
shallow buried mines located under the plate's center. The analytical
method developed does not require electronic digital computers in its
application, and is an easy to use, accurate, and directly applicable
tool for vulnerability assessments and engineering design. The method
of analysis is based on large structural plastic deformation, a semi-
inverse energy method and a reasonable description of material behavior
including a static stress-strain curve and strength failure criterion.
Blast loading is characterized by the energy associated with the blast
wave. Plate deformation is calculated by equating blast energy
imparted to the plate to the strain energy absorbed by the plate in
reaching its final deformed shape. These formulations developed correlate
within an average error of 7% with available aluminum and steel plate
mine blast test results.

The results of the effort are presented in simple graph and nomogram
format for rapid armor areal weight determinations and mine blast-tank
bottom plate evaluations. This method of analysis will facilitate vuln-
erability assessments and engineering design of armor subjected to mine
blast attack.

LIST OF SYMBOIS U plate strain energy, in-lb

•1 mine explosive charge weight, 11)
A maximum transverse plate deformatinn i.e.
amplitude of displacement w (sec Fig l),in a,b width an'Jlength respectively of rectan-
VE reflected2 mine blast energy flux density, gular plate [see Fig I), in
M in-lb/in e base of the natural system of logarithms
Fy plate tensile yield strength, psi el: plate tensile no,'mal fracture strain.

K plate fracture criterion constant with the h plate thickness, in

following values: 1.315 - mean fracture; h no-fracture thic,ness i.e., plate thick-
0.835 - sure fracture (95% probability of ness with a 951 probability that fracture
fracture); 1.791 - no fracture (951e witl h oca . upoamit hat frtactr
probability that fracture %ill not occur) will not occur upon mine blast attack, ip
M typi-al measurement I sure-fracture thickness i.e., thickness
with 95: probability of fracture, in
mean value of tyi ical measurement.
Ih mean thickness at hlich fracture occurs,
I standoff (plate-bottom to mile-top i
distance i.e., see Fig 1), ii H number of mcastirement-;
S r 1.(1 * ) FYeF: plate strain energy p normally reflected pressure from the blast
absorbed to fracture wave, 1'si
per ui;it plate ,,rea, in-lb/in'
11 plate strain energy of deformation, -n-b/ standard deviation of the
ntc tI average relative error.

Preceding page blank 8

8"7

: _2
,7 ".,-

INTRODUCTION under the test plate in a predug hole leaving a

distance of 17 inches between the top surface
The vulnerabi!ity analyst, munitions de- of the mine and bottom surface of the test
signer, and armored vehicle designer need a plate. The mine was then covered with approxi-
methodology for determining the effects of mine matel:' 3 inches of loosely packed soil. After
blast attack on the integrity of armored vehicle detonation of the mine, deformation and the
hull bottom plates. To be efficient and effec- extent ef cracking, if any, were recorded.
tive, this methodology should be reasonably
accurate and account for the various factors The A-my BDARP mine category data collected
that influenL, tank bottom plate integrity in described damage to the equipment including such
actual mine blast encounters. In addition, it factors as damage location, systems affected and
should be simple i. form so practical answers threat involved.
can be easily obtaied in a short time. Such a
methodology has been unavailable up to the pre- Tables I and II list the deformation and
sent time. In the ast the information required fracture test results of rolled homogeneous
to assess mine-tank bottom encounters had to be armor steel plates and 5083 aluminum plates.
obtained by rather expensive and time consuming Charge weight ranged from 2.5 to 24 lbs with a
test programs - analytical methods to develop constant stan'off and burial depth of 17 inches
this information were not available. This and 3 inches, respectively. Since mechanical
situation was caused by the very complicated properties test results for cach of the plates
nature of the problem, tested were not available, estimates of the
necessary parameters were obtained from the
To avoid these complicationa, an approxi- material specifications. Yield strength and
mate method of aiialysis was selected to treat failure strain for the 5083 aluminum alloy
the subject problem. The result of this treat- plates here obtained from MIL 111)K 23 specifica-
ment is a simple method for rapidly predicting, tions. In regard to the steel armor plate,
with reasonable accuracy and without recourse measured or specified hardness values were con-
to a high speed electronic digital computer, the verted to yield strength and failure strain by
deformation and fracture cl-aracteristics of neans of well known relations (Ref. i).
armored vehicle hull bottom plates attacked by
blast type mines. The particular equations The steel plates ranged in thickness from
developed are applicable to flat rectangular 3/4 in. to 1 1/2 in. while thickness of the
plates of uniform thickness with fully clamped, aluminum plates ranged from 3/4 in. to 3 inches.
or built-in, edges that are subjected to blast All plates here essentially the same size -
from shallow buried mines centrally located either .10 in. by 65 in. or 44 in. by 75 in. All
tunder the plate (see Figure 1). These equations the aluminum plates vere apparently of the same
should be particularly useful for quick vulner- strength and ductility whereas the teel plates
ability estimates and in the initial design ranged in strength and ductility from 90,000 psi
phases of land mines and most armored vehicle to 172,000 psi and 0.15 in/in, to 0.23 in/in.
design projects. respectively. Correlation betheen these test
results a:d predictions of the analysis i!
"ILST I)hSCRltIP'ION discussed in the next section.
All the data presented in this report has CORIU:I.A'IION 01' ANALYSIS A.I) TF.ST RE.SIULTS
obtained from available test results and battle
damage records. The data consist of restuIts leformation and Fracture Relations. The
from armor plate - mine blast tests performed deformation and fracture analysis developed
at Aberdeen Proving Groutd as well as combat yields the foliloing relations for plate defor-
damage data collected by the BDAI6P (Battle nation and fracture thickness:
Damaqe Assessment and Reporting Program) teams I laml, I2
F 1/2
operating in South Vi.tnam. A - F ()

In the APG tests 'he mines included

pancake-type service and experimental cased
mines as well as experimental bare pancake K I.m
charges. The test procedure was as follous. A h1 - m (2)
test plate a show in ijure 2 wa, placed on (l4) I' .
top of two steel beams measuring apt ro~imate ly
" 1
twelve inches square by sixty inches in length where 0.23 -0A4I
wi tir the beams resting on the ground. A rec- -- 1.35 ; e (3)
tangular Otccl frame heighing three and a half Ii
tong and an 8 ton plate for a total hold-dohn- 1.31.5 for the meedian fractur(
to-test plate height ratio of 28 were then (
placed on top of the test plate. ght ie.S hold-
(Tis may I 8, for
or susure fracture, i.e., 95'.
down-to-test plate weight ratio may be compared k probaih, lit ot fracurt
to the typical tank-to-botton plate height
ratio of 24.) 1is fixture provided eseit ally 1.791 for no fractur(, that is, 9.'
clamped (fixcd) boundary conditions all around Iprobahii : that fracture hill not
the plate. A mine %as then positioned centril) ,,Ikor

88

2 55
Deformation Correlation. Table I lists theoret- action. Iowever, until this analysis is
ical results for plate deformation calculated by improved to bring the theory into closer agree-
use of Eq. 1 as well as the deviation of these ment with test, adequate fracturc predictions
predictions from the test results. As indicated, may be made by using the least squares data fit
the average positive error is 14% and the nega- equation as the fracture criterion. The aveiage
tive error is 18.1% with the overall average absolute error between this least squares fit
error -6.4%. The average absolute error is equation and the plate test data is 13.2%.
16.6%. These test and analytical results are Besides, as indicated by Figure 4, the actual
plotted in Figure 3. In this figure the test tank mine damage data points which were not
data has beei. plotted as reduced deformation used in arriving at the least squares equation
amplitude A defined by: and its 2a limits fall well within the 2a prob-
= 2A a 2 b 2 F hI11/2 ability limits of the least squares fit equation.

Ia
2 2
b Y DISCUSSION
versus mine blast energy flux density Em . The General. The deformation relation, Eq. 1, was
test data is as follows: circles represent 5083 derived from first principles without making use
al-iminum and the squares class 2 rolled hon.o- of test data to develop the equation. First
geneous steel armor. Lach data point represents principles were also used in developing the form
a single test plate or the average of either two of the fracture relation, Eq. 2, although the
or three tests. A least squares fit of the data :xact value of the fracture criterion constant K
is shown as the solid (- ) line with the 2o, was obtained from a least squares fit of test
or 95% probability of occurrence, limits as the data.
dotted (----) lines. The dashed line is the
theoretical curve. As indicated, the theoreti- A relatively simple and straightforward
cal curve is slightly below the least squares means for calculating the thickness of a given
curve. Consequently, plate deformation pre- plate that will fracture when attacked by mine
dicted by the present theory is,on the average, blast is presented by the nomograms of Figures
6.4% lower than would be obtained in practice. S through 7. These nlomograms have been prepared
If a conservati'.e method for predicting deforma- from the fracture thickness equatio)n (Eq. 2).
tion is desire(* tile equation corresponding to Figure 5 i. a nomogram for mean fracture thick-
either of the 2o limits could be employed. For ness bosed on Eq. 2 with K r 1.315 where this
example, use of the upper 2o curve to predict value of K has been determined by a least squares
deformation would yield results that 95% of the fit of the plate fracture data. As indicated
time are higher than would be obtained in actual previously, the average error of this relation
practice. is -0.6% and the average abiolute error is 13.2%.

Figure 6 presents the conditions for plate

Fracture Correlation. Plate fracture data thickness with a 95% (or + 2u) probability of
li.sted in Table 11 is plotted in Figure 4. In no fracture, hlr. That is, plates of this
this figure plate strain energy absorbed to the thickness have a 95% probability of retaining
point of fracture per unit plate area S is thines a e a m natty This
plotted versus mine blast energy flux cisity their integrity after a mine attack. This
.m according to Lq. 2. Along with the test data nomogram is based on Equation 2 with K 1.791.
points this figure shows the least squares fit Plate thickness that will be breeched by 95%
curve of the test data (equation 2 with k of the mine attacks may be obtained from Figure
1.315), 2o probability limits (equation 2 with 7. This figure is a nomogram for sure fracture
K equal to 0.835 and 1.791) and the theoicti-al thickness, |1., i.e., it represents 9;% proba-
fracture Iine. As in Figure 3 the circles and bility of fracture. It is based on Eq. 2 with
squares represent 5083 aluminum and class 2 K = 0.835, the equation for the -2o limit curve
rolled homogeneous armor steel plates respec- of Figure .1.
tively. Tile triangles correspond to actual tank
damage from nine blast in South Vietnam. Through five scales, these nomograms
a
in
The theoretical criterion for fracture is ) tough
elsentially ness II, fracture thickness hi. , scaled standoff
1 '3
* , and mine charge weight h. In adit ioi
there are two unlabeled scales used in the
whereas the l-ast
"estsquaiesdta
fit equation for the
iscalculation%. alc tonl
:est data is
S - 0.533 I. TLe procedure for using the fracture notno-
m
That is, the -lope of the least squares data f:t grams is as follows. The sure-fracture nomogram,
is s g~lht ly hl~'ler than half that of the theorct- Figure 7, will be used as an example. To cal-
titan hal tht
is~~~~clt ohi th teoi thihtlthii%
culate the thioknes, h" at %illch a plite of
iL i1 phte fracture critorio. here may be two a
posible reasons for this diserep.iny between width-to-length a:pect ratio with train energ)
theory and test results, namely, inadequacy of
the tr.icture cri terion and neglect of bending R

89
 11

with 95% certainty under attack from a mine of

charge weight W at a standoff R:

1. Locate the aspect ratio on the 9 scale,

point 1, the plate toughness on the U, scale,
point 2, and draw a straight line through these
points intersecting line a at point 3.
2. Similarly, locate the charge weight on
the W scale, point 4, and the scaled standoff on
the R/W scale, point S. Draw a straight line
through points 4 and 5 intersecting line b at
point 6.

3. Draw a straight line through points 3

and 6 interesecting the sure fracture thickness
scale hsF at point 7. This va!ue is the thick-
nes which fracture occurs.

Two equations have been developed, one for

deformation and one for fracture of flat,
rectangular, clamped plates subjected to blast P
attack from shallow buried mines. The average
error between predictions of these equations and
test results, including both plate and tank
data, is less than 7% with the average absolute
error less than 17%. From the fracture criterion
a set of plate fracture nomograms have been pre-
pared that yield plate thickness for 95% prob-
ability of fracture under mine blast attack, 95$
probability that the plate will not fracture,
and mean fracture thickness. Fracture thickness
and areal weight can be determined from these
nomographs in little more than the time it takes
to draw three straight lines on the nomograph.
REFERENCES

I. T. Lyman, Ed., Metals Handbook, pp. 108-

110, 8th Ed. Vol. 1, Properties and Selection of
Metals. American Society for Metals, 1961.

IV)

Figuue t - Mnme-R~ctongulor Plate confhqu¢,tlon

90
 PLAN VIEW
OF
TEST PLATE

HOLD-DOWN
t PLATE

FRM

TEST PLATE

MINE

SIDE VIEW OF TEST ARRANGEMENT

Figure 2 -Typical Plate Test Arrangement.

300 f
2o

200- 0 0 01

'11/ 'LEAST SOS. FIT -

0L6bI9.oil 0 '4

00
100

010 20 30
E m~ los IN-LB/IN!

Figure 3 -Plate Deformation; 0: 5083 Aluminum Test Data, 0:-Rolled Homogeneous Armor
Steel Test Data,-: Least Squares Fit of Data,--: Theory

91
 .. 4).* .~.a* *q00O4
4Jq t .4 -1% N'. SNO) O 4 (1
Nq .4C;-1( 4
+ I. + 1 +I
4 +4 +I + II

0 m wo"l '., ~N
m N'ae m

4J 11 4.+1.44 In .11 .

*m
0
~ e N. -N
-q ~Or )
r,
r tfN
N a) N
0
ID t,
-.4
0 0) r,
1"NL) -too l
It in Go N In -TQO

F-.0

03 ~ ~~
CL-. - -

cl-

CC 9

W 00

4l o '4o
CL

41

i-in

CSA4r-NrtCD(
rt,(1 I I 00 CS92a0
 LIN

Ic

Table II. Fracture Data

Stand-Off 17 In. (This Includes 3 In. Mine Burial Depth)

W h a a Fy ef Deviation
Test Charge Plate Plate Plate Yield Failure From
Case Weight Thickness Width Length Strength Strain Least Sqs
No Material lb in in in KSI in/in

1 5083 Al. 4.0 0.75 44 65 33 .16 +25.7

2 6.5 1.25 44 65 33 .16 -0.2
3 12.0 2.0 44 65 33 .16 +15.6
4 11.9 2.0 44 65 33 .16 -14.6
5 11.0 2.0 44 65 33 .16 - 6.2
6 15.9 2.5 44 65 33 .16 -21.7
7 19.9 3.0 44 65 33 .16 -25.2
8 14.5 2.5 44 65 33 .16 -10.6
9 17.S 2.5 44 65 33 .16 -27.0
10 Steel 20 0.75 44 60 135 .17 -16.6
11 20.7 1.SO 44 60 112 .20 +55.8
12 20.2 1.0 40 65 137 .175 + 7.6
13 20.2 1.0 40 65 116 .22 +14.4
14 21.4 1.0 40 65 161 .155 + 5.5
15 20.1 1.0 40 65 163 .1S - 0.1
16 21.9 1.0 40 65 137 .17 - 4.6
17 20.8 1.0 40 65 160 .16 +11.S
18 17.6 0.75 40 65 172 .13 -14.0
19 24 1.25 40 65 1FO .16 +13.6
20 19.9 1.0 44 65 90 .225 - 4.0
21 19.9 1.0 44 65 90 .23 - 2.4
22 19.9 1.0 44 65 114 .20 + 7.8
23 19.7 1.0 44 65 137 .17 +10.2
24 19.9 1.0 44 65 110 .21 - 2.8

Average * -0.6%
Error
Average
Absolute = 13.2%
Error

30~

2-
/0 -
20 ,.7e / rI~

. o).4 -

Ri

~
0 ~ 0
¢394

Fqgure 4- Plate Froture. 5083 Aluminum Test Dot 0. Po'led homoi$ ecs Armor Steel T¢il
T
00!0: 0, Ton Test Ooto6 LeOst .u' s rI Ot "ti - - t :yt -

93
S) co00 0 0 In 0 In 0 0
. 11 . . 1 1. y .... T., A ~ .L .

C0C

00
2 cr 0n
0 w 00 q 0 0 <

£0 _j3 W NC

W>-
-an
W
00 0 W0smco

___ ___F__ - I*J<~ z

x < FW
cI F-
w-

U; %on<-.
94
OD -

No,

0
CL

K 0
11 E
0

o
LL
cm-

00
Ir CL

0 fa i

-j Zt

0 0 t w"ifu- :E
40(4M

00

- N F)
0 - ~~ ~ ~ ~CK) IDI
~ n ( j2
'2 40 0

95~
on InJi
-'I

?. 0.20

E
0

22

LL

w0

00

U,, w0

0-*- 0
Z I

962
 ,I
THE IMPULSE IMPARTED TO TARGETS BY THE
DETONATION OF LAND MINES

Peter S. Westine
Southwest Research Institute
San Antonio, Texas

Numerous expensive testing programs have been conducted to deter-

mine how land mines damage vehicles ox' armor plates; however,
very few analytical studies'have been conducted because the load im-
parted to a structure from a land mine explosion has never been
determined. The few analytical calculations which have been made
assumed that the target was loaded by an air blast. In this paper,
the writer shows that the impulse imparted to the target is not en-
tirely an air blast phenomenon. The impulse is caused principally
by the momentum in the soil particles surrounding the buried explo-
sive. A technique is developed for predicting the impulse imparted
to targets such as wheels and armor plates. Several Illustrative
calculations are compared with experimental test results to establish
the validity of this procedure for e.stimating the impulse imparted to
a target from a land mine explosion.

INTRODUCTION study; however, the authors did recognize the

limitations of using air blast data directly and
Although the effects of land mines on com- attempted to account for the effects of shock
bat vehicles have constituted a serious problem wave reflections from the ground plane by
for a long time, very few theoretical or care- doubling the explosive charge weight. A lim-
ful experimental studies have been made of this ited set of experiments measuring blast loads
problem in the past. Almost all past efforts from buried 5-lb TNT explosive charges was
have been purely empirical; that is, one simply conducted by Aerojet General Corporation 14]
blew up vehicles or armor plates with land and compared with air blast data. The con-
mines, and determined what thicknesses of clusions made from these experiments were:
various armor materials would be necessary to
"defeat" given mines. Numerous testing pro- I) The impulse from an explosive charge in
grams of this nature are listed in a recent the ground is a function of charge weight,
bibliography [1]. stand-off distance, and depth of burial.

All of the analytical studies which have 2) Buried charges demonstrate a greater
been conducted assumed that the loading is an efficiency in transmitting impulse to a
air blast phenomena. Personnel at Cornell target than does a blast wave from a
Aeronautical Laboratories [2] used 13RL data spherical charge in air.
on reflected impulses and pressures from ex-
plosive charges in air to estimate loads on In this paper, the author will demonstrate
tank hull bottoms from mine detcnation. These that the impulse imparted to a target by deto-
air blast data were applied without considering nating a land mine is not an air blast phenome-
ground reflection factors or accounting for the non. The basic premise behind these earlier
effects of burial. In another study conducted studies, chiefly that the loading wab caused by
at the Cleveland Army Tank Automotive Plant, a shock wave propagated through air, was in-
AllIson Division of General Motors 131, the correct. Aerojet General's conclusion that
same BRL data for impulsive loads from air buried charges are more efficient was a cor-
blast were not properly used as in the Cornell rect observation; however, they were

97
incorrect in comparing the loads to air blast spheric density, or velocity of sound in air,
ones. The impulse imparted to a target from can be found in Eq. (1), The specific impulse
a land mine explosion is caused by the momen- imparted to the body being loaded is caused
turn in the explosive products from the charge primarily by the momentum of the explosive
and primarily from the soil encasing the products rather than being caused by a shoc.k
charge. This paper considers only the i- wave propagated through air. Although this
pulse, i.e. the area under the positive rres- equation is Zor a spherical explosive charge in
sure history of the load imparted to a target air, a land mine represents a very similar
from a land mine explosion. Peak pressures circumstance, One can think of the ground
and transient loading histories are not in- which surrounds the mine as a charge sur-
eluded, and cannot be discussed until direct rounded by a weak case of soil. The mass
measurements are made of these effects, term, M, is thus the mass of the mine and an
These limitations will not affect structural equivalent mass of soil. The philosophy rep-
studies of hull plate response or vehicle over- resented by Eq. (1) is applicable to a land mine
turning provided the response of the system explosion because the mass of the engulfed *i-
being stiadied falls within the impulsive loading at standoffs typical of mine attacks is very
realm. T41.4 paper will indicate how specific small relative to the mass of the explosive
impulse is di-tr* ,ted over the surface nf products. Ifwe assume that a 20-lb land mine
several targets, particularly wheels and flat is covered by 3 in. of earth and that the hull of
plates. a structure is 17 in. above the center of the
mine, thin we can demonstrate the validity of
LOADING FROM A MINE BLAST this claim. The density of air under ambient
sea-level atmospheric conditions is 0.0765
The nature of the loading applied to P lb/ft3 and the volume of this air enclosed within
structure auch as the hull of a combat vehicle a sphere 17 in. in diaimeter is 11. 9 ft3 . Thus,
from a land mine explosion involves very com- the weight of an equivalent amount of engulfed
plex phenomena. A shock wave propagated air is 0.0765 times 11.9 or 0. 91 lb, which is
through air is only a minor cause of loading very small relative to 20 lb of high explosive.
and not the principal one. Rudimentary calcu- Actually this ratio of air mass relative to mass
lations indicate that the mass of earth and ex- of explosive products is smaller than 0.91 to
plosive products impinging upon the floor plate 20. 0, because the effective weight of soil (ap-
contribute considerably to the loading. There- proximately 100 Ib) is 5 times greater than the
fore, the loading is a very complex wave form weight of the charge in this illustrative example.
that differs considerably from the classical, Later discussions will show howto estimate the
exponentially-decaying wave that is usually effective maos term, M, in Eq. (1).
associated with blasts.
Jack and Armendt [6] have measured tran-
Baker [51 has shown that very close to the sient pressures at the surface of a plate from
charge the normally-reflected specific im- spherical pentolite charges detonated in a
pulse from a spherical charge detonated in air vacuum. This loading is believed to be some-
equals: what analogous to loads caused by land mines
because the pressures are primarily caused by
I/ 2
(2 ME) the explosive products. A typical pressure
4iRS21 () trace as recorded by Jack and Armendt may be
seen in Fig. I. Observe that this normally-
where M = total mass of explosive and reflected pressure history differs markedly
engulfed air from conventional air blast waves. Two shock
fronts may be seen at A and B in the pressure
history in Fig. I. The rest of the wave has a
S = stand-off distance from center gradual rise time and a rounded shape. The
of charge gradual rise time and rounded shape are
caused by the mass of the explosive products
Sreflected specific impulse, impinging upon the plate and the pressure
This equation is applicable whenever the mass transducer in it. Theoretical studies such as
of the engulfed air is less than approximately those by Lutzky [7] indicate that explosions in
one-tenth the mass of the explosive products. an absolute vacuum would have these charac-
If the explosive charge represented by Eq. (1) teristics. Jack and Armendt feel that the ini-
is encased, the mass, M, represents the mass tial, sharp-rising shock front at point A is an
of the case and explosive. Observe that no Initial reflected air blast wave that would dis-
parameters defining ambient atmospheric con- appear entirely were they to have had a corn-
ditions, such as ratio of specific heats, atmo- plete rather than partial vacuum. They offer

98
 * Pressure3

Time
Fig. 1. Normally- Reflected Pressure History Taken
In Vacuum Near a Spherical Pentolite Charge

no opinion as to the cause of the second shock so that the spLCific normally reflected pulse,
front at location B ini Fig. 1. The second shock ipR, can be computed, we will consider how
front may be a secondary shock (,sometimes Eq. (1) can be applied to real targets which
called a "pete" wave by other investigators). possess complex configurations. Eq. (1)
Ordinarily, these shocks-are less intense than allows one to compute the normally reflected
the first shock; however, under these circumn- impulse per unit area at various standoff dis-
stances the wave has an atmosphere of gases tances. This equation does not tell one how
from the explosive products through which it to calculate the total impulse, 1, imparted to
can expand; whereas, the Incident wave has
almost no medium through which it can be con-
ducted. A land mine blast will probably give
a loading somewhat similar to that seen in
a wheel or to a plate where the impulses im-
parted to differential areas on these targets
are not normally reflected. Total impulse
can be deterrilned by knowing the projected
I
Fig. 1; however, the relative magnitude of area of a target and the peak normally reflect-
first shock, explosive products, and second ed impulse imparted to this area :)y using
shock wave pressures will vary considerably. Eq. (2).
One can observe in Fig. I that the impulse
(area under the pressure history) is princi- IR =iR Af (2)
pally caused by the explosive products and
that air shocks produce very little impulse. The term I in Eq. (2) is a shape factor which
Probably in a land mine explosion, the initial is a function of target shape and standoff con-
peak would be larger because of the pre- -ditions. This shape factor will be calculated
sence of an atmos.-here, but the secondary for wheels and rectangular plates, both of
shock would be greatly diminished, because of which are common targets.
reflections being transmitted far into the
earth and because the cylindrical shape of a SHAPE FACTOR FOR A WHEEL
land mine would disperse reflections rather
than focus them as in a spherical charge, A To calculate the shape factor for a wheel,
strong possibility exists that a peak pressure consider encounter conditions as presented
will occur which is a shock rather than being in Fig. 2a. The wheel being considered is
*caused by momentum from soil flung upward located directly over the center of the explo-
by a land mine explosion: however, the vast sive charge. We determine the total vertical
majority of the impulse is caused by momen- impulse imparted to the wheel by considering
tum in soil products. the specific imnpulse imparted to a differential
area located on the rim of the wheel. The
LOADING IMPARTED TO COMPLEX total impulse is then computed by integrating
TARGETS the differential areas around that portion of
Before ,iscsslng wat num,,,ecal valuethe rim being loaded. Fig. 2b shows the 4
Befoe dsct~sin
wha nuericl vluespecific impulse being applied to a differential
should be assigned to the mass, M, in Eq. (1) area, dA.

99
 r cote '
rfl-cosO)N rsinO~

L._

(a) (b)
Fig. 2. Wheel Traversing a Land Mine

If the wheel is a thin strip of thickness, h, This impulse may be divided into its vertical
then the differential area, dA, equals: and horizontal components. We are only in-
terested in the vertical component as all hori-
dA = hrd0 (3) zontal components cancel because of sym-
metry. The vertical component, dl, of the
The specific impulse, i, as shown in Fig. 2b impulse equals:
at the location of the differential area equals
after substituting for S in Eq. (1): dl = i cos (* + e) cos e dA (8)

(2 ME) 1/2 The total impulse imparted to the wheel is ob-

2
= 4 nr(l-cos e) + d (4) tained if one integrates Eq. (8) over the loaded
L cos segment of the wheel. Because this loading is
caused by soil particles impacting the wheel,
Geometric considerations which are seen in all segments of the wheel are loaded until the
Fig. 2a indicate: impulse vector, i, becomes tangent to the
wheel. The impulse vector which is parallel
r sin5 to a line from the center of the charge to the
r(I - cos 0) d differential area becomes parallel to the wheel
when tan€ is a maximum. This occurs at:
The cos ?/ can be calculated from the tan t and
equals after substituting Eq. (5) for tan t: d
(tn 0
r(l-cosO) +d
Cos*-r n (.oer (6) .rcose (d+ r-r cos 0)+ r 2 Vin? =0

or at
We will assume that only that component of the
specific impulse which is perpendicular to theCo(9 cos 8 -r (9)
differential area imparts momentum to the r -, d
target. This assumption would be invalid were
this to be an air shock; however, it is valid Let us define a nondimensional quantity, 6,
because we are considering momentum from suoh that
moving particles of soil. The impulse which
is applied to the differential area equals: 6 1 +A (10)

diA i cos I dA Substituting for dr in Eq. (9) and rearranging

terms indicates that the limits of integration
are:

100
 e arc coo (-) () The differentl:n area, dA, equals:
dA:. dy dx (17)
Because of symmetry, the total impulse may
be obtained by multiplying Eq. (8) by 2.0 and Eq. (1) indiciaes that the specific impulse
integrating the resulting expression from 6 directed at the differential area is:
equal to zero to the limit expressed by Eq. (11).
This integration gives: i (2ME)I/ (18)
2
e 41R
I =fzi coso( e) cosO dA (12)
0 Once again we assume that the component of
this impulse which is tangential to the differ-
Substituting Eqs. (3) and (4) into Eq. (12) and ential area does not load the differential area,
recognizing that 2 rh equals the projected and that all of the impulse imparted to this
area, A, of the wheel yields: differential area is caused by the normal com-
0 ponent of the impt,.se, i, expressed in Eq. (18).
J= (2ME) 1 2 A f
_d2 cos 2 $cos(*+0)cos0 dO The impulse imparted to the differential area
4TS 0 (d+ r-r cose)2 equals:
'A S2
= /X 2 7 $ ( 19 )
But Eq. (13) is Eq. (2) provided the shape
factor, 0, for a wheel equals:
0 One calculates the total impulse imparted to
Sird)2 f cos2# cosO cos(*+9) do the entire plate by substituting into Eq. (20).
(d+ r-r cos 0) 2 X Y
If 4f f aAdA (20
Eq. (14) for the shape factor may be simplified 0 0
further if the appropriate substitutions from
Eqs. (5) and (6) are made into Eq. (14). Making Substituting Eq. (16) into Eq. (18), Eq. (18) into
these substitutions and gathering terms yields: Eq. (19), and Eq. (19) and (17) into Eq. (20)
e (6 Cos
2 -1) cos 0 d9 yields:
0= 62 (15) x Y
f6+1-26os3= f f (2ME) _/4Sdxdy (21)
0 0 4r(x2+y2 5 2 )3 / 2
Although this integral is a very difficult one to
compute in closed form, a numerical solution If one multiplies and divides Eq. (21) by S2 XY
is easily obtained on a computer. The results and recognizes that 4 XY equals the projected
from a numerical investigation are presented area, A. then:
in Fig. 3 where shape factor 0 for a wheel is
presented as a function of 6. 2 s
i (2ME) 1/ 2 A $- r Sdxdv
SHAPE FACTOR FOR RECTANGULAR 4n S2 00 (x 2 -+y2 S2)3/2
PLATES
But Eq. (22) is Eq. (2) provided the shape fac-
To calculate the ahape factor for a rect- tor. 0. for a rectangular plate equals:
angulhr plate, consider a plate with a rect-
angular x-y coordinate system having an origin rS3 X dy
at the center of the plate. Because of the sym- E
J
r f0 x
d3
y 2 + $2)3/2
metry which exists when the charge is buried
directly under the center of the plate, we will
consider the vertical impulse imparted to a After performing this integration, one obtains
differential area located in only one quadrant of for the shape factor of a rectangular plate:
the plate. The plate will have half spans of
X and Y. The slant range, R. from the center ! a (Y/S) (XIS) 1 2
2
of the charge to the differential area equals: (X/S)(Y/S) a iY/S)(IS) 4 1] i

R F/(x24y2+S2) (16)

101

Ye,
 4-

~7/4- -- '/
1 2 4 6 8 10 20 40 60 80 100
(I + dIR 3061

Fig. 3. Shape Factor For a Wheel

MODEL ANALYSIS TO DETERMINE crater produced by detonating a buried explo-

EFFECTIVE MASS sive, indicates what soil properties should be
selected. In order to determine the impulse
If one knows that value of effective mass is from the detonation of a buried explosive, we
appropriate for a given size of explosive charge are interested in the momentum imparted to
and depth of burial, he can compute either spe- soil particles. The explosive cratering analy-
cific impulse or total impulse by selecting the sis must concern itself with both the momen-
appropriate shape factor and substituting in tum imparted to soil particles and the influ-
Eqs.(1) and (2). Effective mass, M, will be ence of gravity on the trajectory of these par-
determined by using the analytical results ex- ticles. Reference [87 uses the bulk density of
pressed by Eqs. (1) and (2), conducting a simil- the soil, p. the seismic velocity of the soil, c,
itude analysis to interrelate charge weight and and the acceleration of gravity, g, to charac-
depth of burial, and applying some experimen- terize soil properties. In this study, we will
tal results to devise a numerical relationship use two of these three parameters, the density,
between the various parameters. p, and the seismic velocity, c; however, we
will delete the acceleration of gravity, g, from
To conduct a similitude analysis, one be- our analysis. Deleting this term from the
gins by listing the relevant parameters. We analysis is equivalent to assuming that gravi-
wish to measure specific impulse, i; this para- tational effects are insignificant. Such an
meter is our response parameter. The im- assumption is appropriate as the gravitational
pulse is created by detonating an explosive field of the earth is not strong enough to appre-
charge of weight, W, which was buried at some ciably reduce the momentum in soil particles
depth, d, from the surface of the soil to the in the relatively small distance between most
center of the charge. Impulse is to be mea- land mines and targets. The six paramete's
sured at some standoff distance, S, from the which have just been presented as defining this
center of the charge. The most difficult as- problem of the impulse from the detonation of a
pect in conducting this similitude analysis is land mine are summarized in Table A together
the selection of appropriate parametc rs for with their fundamental dimensions in the engi-
characterizing the soil. Fortunately, a recent neering system of force, F; length, L; and
analysis [8] of a similar problem, the size of time, T.

102
 F -'r

TABLE A I fI '6
Parameters For Determining Impulse d 73)
,1
From A Land Mine Explosion
Eq. (26) defines a three-dimensional space.
Fundamental This space can be reducedi in a two-dimen-
Symbol Parameter Dimensions sional space by squaring the second pi term
and multiplying the result by the first pi term
i specific impulse FT/L 2 to form a new dependent parameter. After
W charge weight FL performing this operation, Eq. (26) may be re-
d depth.of burial L written as Eq. (27).
S standoff distance L
p density of soil FT 2 /L 4 iS2 = f"(, (27)
c seismic velocity L/T d3 d3

Eq. (27) is Eq. (1) without an explicit expres-

Many different textbooks present several sion for the appropriate functional format.
different procedures for developing the three Eq. (1) states that impulse varies inversely as
nondimensional ratios called pi terms from the square of the standoff, S. Inasmuch as the
the six parameters listed in Table A. In this effective mass, M, in Eq. (1) annot possibly
paper, we will omit the algebraic procedures be a function of S, Eq. (1) ir.diated that the
and present the results. Table B lists three three-dimensional space expi essed by Eq. (25)
pi terms which can be obtained, should be rewritten as the two-dim-nensional
space, Eq. (27).
TABLE B
Pi Terms - Impulse From Eq. (1)does not tell us how effective mass,
Land Mine Explesion M, relates to charge weight, W. and depth of
burial, d; thus, it does not furnish the de-
i sired functional format for predicting impulse
'r d Scaled impulse from a land mine explosion. Eq. (27) which is
Eq. (1) expressed in a slightly different form
r Geometric similarity does not provide the desired functional form
d either. The functional format relating scaled
W
Tn3 .Scaled charge weight charge weight, W/d 3 . to scaled impulse,
2 3
pc
2
d3 (i S )/(d ), must be obtained from experi-
. . . .. . ... ........ .. mental test data. Experiments must be con-
ducted so the measured scaled impulse,
The first pi term is normalized or scaled (iS 2 )/(d 3 ) can be plotted as a function of
impulse. This pi term io a unique function of scaled charge weight, W/d 3 . Before making
the other two pi terms, a statement of geo- this plot, Eq. (27) will be modified by substi-
metric similarity and scaled charge weight. tuting Eq. (2) to form Eq. (28) which is an ex-
We write this functional relationship as Eq.(25). pression for total impulse rather than for
specific impulse. Analyzing test data in terms
(f3 W (2) of total impulse rather than in terms of spe-
pcd d pc2 d 3 / cific impulse is more convenient.
2
The soil parameters P and c can be deleted IS f.(2
3
'from Eq. (25), because they are essentially Ad (d 3 )
constants. Soil d,.ity varies very little over
a wide range oi soil conditions. Seismic velo- SCALED IMPULSE AS FUNCTION OF
city for soil in a -olid state does vary over a SCALED CHARGE WEIGIIT
limited range- however, the soil which pro-
vides most of the impulse from a land mine Reference 141 describes a limited series
explosion acts more like a fluid than a solid, of tests in which 5.0 lb charges were buried at
We will assume that the seismic velocity of all various depths and a rigid mass with an ex-
soil "fluids" is corstant. A careful perusal of posed surface area of 50 in. 2 suspended at
A Reference [8] would show that relatively accu- various heights directly over these charges.
rate predictions of crater size can be made The masses were not constrained or prevented
when P and c are treated as constants. De- from moving vertically; however, they could
leting P and c from Eq. (25) because they are not rotate or translate horizontally. By mea-
constants yields Eq. (26). suring the displacement of the mass for dif-
ferent depths of burial and standoff distances,

103

'~' g
normally reflected total impulse could be The wheels were held by a yoke whose other
easily calculated from Eq. (29). end was attached by means of a torsio.a spring
to a rigid wall. This test configuration is
V 2gh(9 effectively a spring-loa~ed pendulum with a
2 Vwheel as the ball of the pendulum. By mea-
surIng the maximum rotation of the yoke and
Table C contains the test number, depth of wheel system as a result of the detonation of
burial, air gap b~tween ground and bottom of various size explosive charges, the impulse
the mass, and the calculated total impulse for imparted to a wheel could be calculated. The
these tests. The depth of burial was mea- explosive charges used in this test varied
sured from the ground to the top of the charge from 0. 106 lb to 0. 468 lb of pentolite. The
rather than to the center of the chrge. *By as- charges were rectangular parallelepipedes
suming that the 5.0 lb charges were cylinders with a constant surface area of 2" by 21" and a
with diameters of twice their thicknesses, thickness that depended upon the size of the
depths of burial and standoff distances fromthe explosive charge. 1/2" of soil was placed
center of the charge were estimated. Because over the charges. Steel wheels which were 7"
the masses being loaded by the explosive in diameter and 1" thick were in direct con-
charge and soil products are very small, the tact with the ground. On a few occasions the
shape factor § is essentially equal to 1.0 in yoke held only one wheel which was centered
these tests. The final two columns in Table C directly over the mine; however, most ex-
present scaled impulse and scaled charge periments were conducted on a 3-wheel array
weight. In subsequent discussion, these two with a 1. 75" center-to-center spacing between
quantities will be plotted to determine the successive wheels. The middle wheel in any
functional format for Eq. (28). The quantities array was always over the center of the land
(IS 2 )/(A~d 2 ) and W/d 3 are dimensional be- mine. In a 3-wheel array, the impulse was
cause the soil constants have been deleted applied to all 3 wheels. To estimate the im-
from this analysis. Throughout the rest of this pulse imparted to the center wheel only, the
paper, the units for I will be (lb-sec), A will following equation was applied and solved for
be (in 2 ), S will be (ft), d will be (ft), and i the impulse imparted
will be (psi-sec).

TABLE C
I S2- W
Versus - Using Rigid Mass Test Data
A§d3 d

Soil Cover IS2 J W

To Top Total A§ d3 d3
t
TestNo. Of Charge Air Gap Impulse d S /Pss S.\ I lb
(in) (in) (lb- sec) (ft) (ft) it /

101 12 4 123.5 1.125 1.46 3.68 3.51

102 12 0 166.0 1.125 1.125 2.95 3.51
105 4 12 80.3 0,458 1.46 35.5 51.8
106 8 8 118.3 0.791 1.46 10.2 10.1
107 12 4 154.2 1,125 1.46 4.60 3.51
108 12 8 90.9 1.125 1.79 4.07 3.51
109 12 0 198.9 1.125 1.125 3.54 3.51
111 4 16 39.5 0.458 1.79 26.3 51.8
113 8 12 75.5 0.791 t.79 9.68 10.1
114 4 8 177.5 0.458 j. 125 46.6 51.8
115 8 4 202.9 0.791 1.125 10.3 10.1
117 4 8 170.7 0.458 1.125 44.6 51.8

A second group of experiments which mea. ic + 2i c sin 0 13 (30)

sured impulse imparted to targets from land
mine explosions are from a series of unreport- where tan (L d-
ed model tests conducted by Mr. Bruce Morris 1.75)
at MERDC, Fort Belvoir. In these experi-
ments, wheels were placed over buried charges. to the center wheel, ic . The experimental

104
data indicate that Eq. (30) proportions the im- A straight line fits the experimental data
pulso appropriately. In developing Eq. (30), we presented in Fig. 4 very accurately. The
assune that the impulse imparted to the out- equation of this line as obtained by a least-
aide wheels is applied parallel to.a line from squares fit to the data in Fig. 4 gives Eq. (31).
the center of the charge to the bottom of the
wheel, and that the upward momentum of out- I W0 . 72 d0 . 84
#A = 1. 725 s2 (31)
side wheels is caused by the vertical compo-
nent of this impulse. Table D contains the
total impulse measured by this test arrange- Eq. (31) is the explicit expression for either
ment, the number of wheels in the test array, Eq. (27) or Eq. (1). If one substitutes
the equivalent impulse imparted to the center 1.4 x 10+6 W for E in Eq. (1) and equates the
wheel only, the charge weight, depth of burial; left hand side of Eqs. (1) and (31) after making
and standoff distance. Before computing the the expressions dimensionally consistent, he
total impulse imparted to a center wheel, one obtains the effective mass term, M. M when
must look up the shape factor for the wheel in expressed in slugs is given by Eq. (32).
Fig. 3. Both 6 and # are listed in Table D
for each test configuration. The dependent M(slugs) = 3.49 W0 44
d1 . 6 5 (32)
quantity, (IS 2 )/(At d3 ), and independent quan-
tity, W/d 3 , are computed from this informa- Usually the reader prrers to think of mass in
tion and listed in the last two columns of units of pounds. Mass in pounds is given by
Table D. Eq. (33).

TABLE D
I S2 W
A Versus - Using Wheel Data

IS 2 W
Total Impulse on AI d3 d3
Test Impulse No. Center d S 6 f (psi-sec
No. (lb-sec) Wheels Wheel (ft) (fit) \ft/ ft3
2 23.5 3 12.1 .0832 .3749 1.285 4.3 98.3 400.
3 32.6 3 16.1 .0930 .3847 1.319 4.0 105.5 353.
4 18.6 3 10.7 .0607 .3524 1.208 5.45 155. 475.
5 27.7 3 14.6 .0778 .3695 1.266 4.55 131.5 425.
6 28.5 3 14.1 .0930 .3847 1.319 4.0 92.6 353.
8 53.3 3 24.2 .117 .4084 1.402 '.25 110.5 261.
9 28.1 3 13.9 .0930 .3847 1.319 4.0 91,2 353.
10 43.2 3 20.1 .110 .4017 1.378 3.4 102. 286.
11 47.1 3 21.4 .117 .4087 1.402 3.25 97.5 261.
13 71.4 3 31.7 .125 .4167 1.430 3.1 130. 240.
14 13.7 1 13.7 .0930 .3847 1.275 4.0 90. 353.
15 15.9 1 15.9 .103 .3947 1.354 3.6 90. 311.
16 23.9 1 23.9 .113 .4047 1.387 3.4 113.5 274.

Tables C and D contain all of the ea- 1

surements of impulse imparted to objects M (lbs) = 112.5 W0 . 44 d I 68 (33)
from land mine explosions that are known to
this writer. In spite of spending many years The author had substituted into Eq. (33) before
and dollars on explosive plate bulge tests, no making his earlier statement that the most
one has measured the impulse imparted to the significant mass in the explosive products
plates in such experiments. In Fig. 4, one from a land mine explosion was the mass of
sees a plot of scaled impulse, (IS 2 )/(A§ d 3 ), the soil products.
as a functlon of W/d 3 for the data contained in
Tables C and D. These experimental data ap- Fig. 4 also shows that experimental re-
pear to collapse into a unique function as pre- suits agree well with Eq. (31). The scatter is
dicted by Eq. (28). small considering the inherent scatter in ex-
perimental test results. Although data from

105
 600 ' ' ' ' '' 1 ' '

400 x IMPULSE ON THREE WHEELS

+ IMPULSE ON ONE WHEEL
200 - 0 IMPULSE ON BLOCKS

100 X _

60-
is' 40 r

10 m7
f \ . . , WO-7d o.8+
4-"4 " --"s" -+
1.725
44

2
I Ih II I I ii,1l I1

2 4 6 10 20 40 60 100 200 500

W Il-b"'

Fig. 4. Scaled Impulse Versus Scaled Charge Weight

only two types of experiments are used in the ground. On the other hand, if the charge
developing Fig. 4, a significant number of is located at the surface with no cover over
data points (25 points) from a variety of test it, the impulse begins to be an air blast phe-
conditions is used to develop this curve. The nornenon. Three data points were not includ-
charge weights in the wheel experiments were edt in Table C, because in those rigid mass
varied by a factor of 4. 0. Depths of burial on tests the top of the charge was flush with the
the rigid mass tests ranged from 4" to IZ" of stil'face of the ground, and no soil covered the
soil cover, and the air gap varied from zero charge. Because the center of the charge was
to 16". By combining these facts with the ob- below the surface of the ground, W/d 3 equaled
servation that theae data involve two different a finite value of 2570. Extending the least-
types of targets, one can see that 4, W, d. and squares fit in Fig. 4 would predict that scaled
S have all been varied in Eq. (31). inipulse, (I S)/(d3 ), should equal 490. Using
Eq. (t) as applied to air blast with a ground
SUMMARY reflection factor of 2. 0 predicts that scaled
impulse, (I SZ)/(d 3), would equal 4745. The
Fig. 4 and Eq. (31) do have limits of W/d 3 average of the three tests on the rigid masses
for which they apply. Obviously, if the depth was a scaled impulse, (I S2 )/(d 3 ), of 3227.
of burial is too great, the detonation of a Obviously this test was in a transition range
buried explosive will not disturb the surface of where the impulse was changing from being

106
caused by momentum in soil particles to being The author is indebted to Mr. Bruce
caused by momentum in the explosive products Morris, the Army's technical monitor on this
from the charge. Some finite soil cover is re- project and Mr. Alexander Wenzel, SwRI
quired for Eq. (31) to be valid. Provided project leader, for being given the opportunity
scaled charge weight, W/d 3 , falls between of probing into this problem. In addition,
1 and 1000, the analysis procedure recom- Dr. Wilfred E. Baker of SwRI is hereby
mended in this report should be valid, thanked for reviewing this text and making
several helpful suggestions.
The other major restrictions to this analy-
sis procedure is that the standoff distance to REFERENCES
the target must be sufficiently small so that the
weight of an equivalent sphere of air must be 1. J.Sova, "Summary of Armor Materials and
less than 0. 1 times the weight of the explosive Configuration Tests at Aberdeen Proving
products given by Eq. (33), For a 20 lb charge Ground, "Combat Vehicle Mine Protection
buried 3 in. deep, this observation means that Conference (U), U.S. Army Weapons Con-
the standoff distance must be less than 38 in. mand, 28 June 1967 (Confi( -itial Report).
for this approach to be strictly valid, Pro-
bably this standoff distance can be doubled 2. J.K. Cockrell, R. Anderson, et al., "Phase
without causing serious error. Most targets III Parametric Design/Cost Effectiveness
are much closer to the ground than several Study for a Mechanical Infantry Combat
feet. This final restriction does not appear to Vehicle (MICV), "Cornell Aeronautical
be very restrictive. Labs., Report 6M-2144-H-4, 20 February
1968 (Confidential Report).
In this paper a procedure has been devel-
oped for predicting either the specific impulse 3. A. B. Wenzel, R, C. Young, and C. R.
or the total impulse imparted to any target ex- Russell, "Structural Response and Human
posed to a land mine explosion. We have seen Protection From Land Mines (U), "Allison
that the impulse is not primarily an air blast Division of General Motors Corp., Cleve-
phenomenon. The loading iscaused by mo- land Army Tank-Automotive Plant, TR 3481,
mentum in the enplosive soil products. To June 1968 (Secret Report).
calculate the impulse imparted to complea
targets, one must determine a shape factor 4. W. L. Kincheloe, "Reduction of Blast Ef-
which Isa function of the geometry associated fects, " Final Quarterly Report, 0477-01
with encounter conditions. The shape factor (04)FP, Contract DA-44-009-ENG-4780,
for a wheel may be obtained from Fig. 3 and May 1962.
the shape factor for rectangular objects such
as plates may be calculated using Eq. (24). 5. W. E. Baker, "Prediction and Scaling of
These shape factors are substituted into Eq. Reflected Impulse From Strong Blast
(31) to compute the total impulse imparted to Waves, " Int. Jour. Mech. Sci. , 9, pp. 45-51,
any target. The specific impulse at any loca- (1967).
tion on a target may be estimated by substi-
tuting Eq. (33) into Eq. (1) and taking the appro- 6. W. H. Jack, Jr. . and B. F. Armendt. Jr.,
priate component of the resulting specific "Measurements of Normally Reflected
impulse. Shock Parameters From Explosive Charges
Under Simulated High Altitude Conditions,"
ACKNOWLEDGMENTS BRL Report No. 1280, Aberdeen Proving
Ground, Maryland, April, 1965.
This paper is a direct outgrowth of con-
tract DAAK 02-70-C-0579 between the U.S. 7. M. Lutzky, "Explosions in Vacuum,"
Army Mobility Equipment Research and NOLTR 62-19, White Oaks, Maryland,
Development Center, Fort Belvoir, Virginia, November 1962.
and Southwest Research Institute, San Antonio,
Texas. Under the terms of this contract. 8. P. S. Westine. "Explosive Cratering, ",
Southwest Research Institute is to design, Journal of Terramechanics, Vol. 7, No. 2,
develop, and test a new mine clearing roller 1970, pp. 9-19.
system. A rational design of a new roller
system required that we determine the nature
and magnitude of the loads from a land mine
explosion.

107

[I
 CIRCULAR CANTILEVER BEAM
ELASTIC RESPO1JF4 TO AN EXPLOSION

Y.S. Kim and P.R. Ukrainetz

Department of Mechanical Engineering
University of Saskatchewan
Saskatoon, Canada

The response of a circular centilever beam subjected to a plane

transverse air blast was obtained. From this response, the drag co-
efficients of the circular cylinder under the unsteady flow conditions of
an air blast wave were determined using the domain conversion method.
Using this method, a conversion of the response was made from the time
domain into the frequency domain and then from the frequency domain back
to the time domain.

Two circular aluminum cantilever beams of lengths 5 ft. and 2 ft.,

and diameters 3 in. and 2 in., respectively, were tested at the 11 psi
nominal overpressure location in a 500 ton TNT field explosion (Dial Pack)
From the measured response, the drag force was obtained, and using the
empirical equation for dynamic pressure, the drag coefficients were
obtained in the region of Reynolds number 7.81 x 105 to 3.7 x lO5 and Mach
number 0.41 to 0.23.

INTRODUCTION efficients under unsteady-state conditions such

as those from the air blast wave have only
Baker [1] obtained maximum responses of recently begun to be investigated (3][4].
rectangular cantilever beams subjected to an air Bishop [3] obtained drag coefficients of a
blast wave. The loading was separated into a circular cylinder and pressure coefficients
diffraction phase and a drag phase, each having around the cylinder when subjected to an air
pressure decreasing linearly with time to zero. blast wave of nominal pressure 20.7 psi. The
In considering the drag phase, a mean drag co- obtained drag coefficients were in the range of
efficient was used for its duration. This Reynolds number (Mach number) from 3.75 x 106
duration was determined from the linearly decay- (0.6) to I x 106 (0.25). The results agreed
ing time function which maintained the same drag roughly with steady state values. Mellsen [4]
impulse as that obtained from the Brode [2] measured drag coefficients of circular cylinders
empirical relation for dynamic pressure. For a by the free flight method at 12 psi and 8.5 psi
stiff beam, the predicted and the experimental nominal pressure locations. The blast wave
results showed good agreement. However, for source was a 500 ton TNT field explosion. The
slender beams, poor agreement was obtained, obtained drag coefficients were in the range of
Reynolds number (Mach number) fiom 1 x 106
The elastic response of a circular canti- (0.433) to 0.44 x 106 (0.205). The drag co-
lever beam subjected to a transverse plane air efficients oscillated about the mean values of
blast wave was studied in this investigation. 0.67 and 0.48 for the 12 psi and 8.5 psi locat-
The loading was separated into two phases as was ions, respectively. It was mentioned that
done by Baker. However, the drag phase analysis Reynolds number and Mach number appeared to vary
considered the actual drag loading which was ex- jointly in such a way that the drag coefficient
pressed in polynomial form. Drag loading is did not change with the time varying flow of the
drag pressure times projected area and drag blast wave.
pressure is drag coefficient times dynamic
pressure. In this work, the drag loading (drag co-
efficients) from an air blast wave was obtained
The drag coefficient will be a function of using the elastic response record of a circular
time due to a continuously changing Reynolds cantilever beam subjected to the air blast wave.
number and Mach number during the air blast To obtain the actual drag loading, the elastic
loading. The drag coefficients of a circular response of the beam was separated into a quasi-
cylinder under steady-state conditions are static response and a sinusoidal response. by
already well known. However, the drag co- expressing the drag loading (forcing function)

Preceding page blank 109

 in a polynomial form, the relation between the in the wake flow.
forcing 'fnction and the quasi-static r6sponse J
wasobtained. Thus the forcing function was Mellsen and Naylor [6] arrived at a non-
determined from the "known" quasi-static res- dimensionalized time for double vortices to
ponse. The quasi-static response was obtained fully form and start shodding. It was photo-
by converting the time domain response into the graphically obtained from the interaction of the
freouency domain and then from the frequency air blast waves (oC peak ovcrpressures 10 to 17
domain back to the time domain. iii)-with circular cylinders. This time is
given 'by
BLAST LOADING

The overpressure-time record at some dis- - 2.S (4)

tance from an open air explosion source can be- D
represented by Friedlander type decay [5]'which
is given by where U is shock front velocity, D is diameter
of the cylinder, and T, is the time for double
p(t) a P (I e-Kt/T. vortices to fully form and start shedding. This
is then the start of steadily decaying flow.
where P is peak overpressure, T+ is the posi-- As already mentioned, the net transverse
tive duration of the overpressure and K isa loading on a circular cylinder as a result of
decaying constant. For low values of P -(about the air blast wave is separated into the dif-
10 pAi), the overpressure can be represented fraction phase and the drag phase. Fig. 1 shows
with reasonable accuracy by setting KR: 1. this type of loading. In the approximate pro-
- cedure for determining the net transverse load-
The dynamic pressure in the air blast wave ing during the diffraction phase, the net trans-
is given by the emprical relation [2] verse overpressure is considered as a pressure
t -t/T d decaying linearly from the normal reflected peak
q(t) - q0 T1
d (2) overpressure at time t = 0 to the drag pressure
d at time t =T I . The time T1 is cbtained from
(4). The drag phase then follows the diffrac-
where q is peak dynamic pressure which can be tion phase.
obtaine3 from the Rankine-Hugoniot relation
~17.S.
q o X P, (2.S -- (3)

where r is the ratio of the peak overpressure -

to the ambient pressure. In (2), T is the 2

duration of the dynamic pressure an Is usually
slightly larger than T. due to the inertia of
moving air. For low values of qo, the dynamic
pressure can be expressed-with reasonable ac- .
curacy using 8 a 2.

When the shock front hits the leading edge

(stagnation point) of a circular cylinder, the
pressure rises instantaneously to the noirmal
~over reflected pressure. regular
the cylinder, As the reflections
shock front occur
passes Fig. I
until the angle between the shock front and the
interacting surface reaches a critical value
which depends on the shocX strength and the RESPONSE IN TIM DOMAIN
radius of curvature. At this stage, Mach stems Small deflection, linear elastic theory
arising from the incident and reflected waves (the Euler beam equation) will be used for pro-
begin to forn at each side of the cylinder.
~These Mlach stems cont-Inue to envelope the sur- dicting the response of a circular cantilever
face until they cross over at the trailing edge beam subjected to a transverse plane air blast
and start to go back around the cylinder. For wave.
small shock strengths tipto about S psi peak The solution of the Euler beam equation
overpressure, the waves continue right back to with respect to deflection is represented by
the front of the .ylinder and croQ4s over once
again at the leading edge. After this, the
waves become very weak and are rapidly dissipat- y(xt) - nl Xn(X) gn(t)
ed. In the case of stronger thocks (for example
about 20 psi), the waves, on passing back across where X (x)are normal mode functions and g (t)
the cylinder, form their otn comfplex Mach stems. are time functions which can be determined ?rom
Thes2 never really reach the front of the cy- Lagrange's equation. When the beam is subjected
linder because they break up and are dissipated to a uniformly distributed forcing function

110
p(t) a P ff(t), the time function equation CM = H - m if (H-m) is even
becomes
PO = M- m - if (H-m) is odd.
in(t)+ Co )2gnCt) nft 6
n (12) can also be expressed as

where w are natural frequencies, . a%6 modal 113 i

participation factors, and Mn are generalized n ml i-0,2,4,..c-1) ) (13)
masses. Dots in the expression represent time "(n)
differentiation. m * 0,1,2.

The solution of (6)with nonvanishing Rewriting Gn(t) of (11) gives

initial conditions becomes
g (~)~ = ( nGn(t) gn*(t) g*(O) cos wnt - sin wnt
gn n
(tPo _ (0Gnt*ld
(wn)'Mn(14) n n n si wn
An (o)
+ gn(O)cos W t + An sin wnt (7) g*(t) An sin({nt n) (IS)
nnn'
where
where T is a dummy variable. get) -rO +M + C-tt + 6Mt
The strain (for zero initial conditions) at
the stagnation point at distance x from the root a f(t) 1 2) 1f()t.
(t ) .
of the beam becomes t n)nf ) M+ -. +
)
n ~
M2f(M)(t)
e(x,t) - P J, En(x).G (t) (8)(14/ H
4D2 tn dn(0) 2h
2 2 n
2
where En(X)n = 4 (V n X (9) A_ ([g(O)J + [----
n X (L) 2 Tx1
K) nO
W g*(O)
In (8), Gn(t) is the nth mode response time On = tan k;n (17)
function; in (9), D is diameter, E is modulus of n
elasticity, I is moment of inertia, L is length
of the beam, Pn are constants of the normal In (15), G (t)was separated into two parts which
modes X (x), and Kn are related to the roots are g*(t) and a sinusoidal response. gnCt) will
AfKO of the frequency equation, be called the nth mode quasi-static response
n time function. This quasi-static response is
The normalized (inmagnitude at t = 0) diffe-ent from the static response which can be
forcing function f(t) is a polynomial of the Mth obtained from the Euler beam equation by setting
order in time. the inertia term equal to zero.
t2 + The strain corresponding to g,(t) becomes
f(t) - a0 + ait + a2 to becme

for 0tsT (10) e*(x,t) p En(X)

P g (t) (18)
=0 for t>T
g(t) can be expressed Inmatrix form as
For this forcing function, the nth mode response
time function Gn(t) for 05t!T becomes g*(t) = (t)T (C (19)
n~- d )si (Cn) (19)f
t n
nCt) = onwheesint n (t-Tdsiwhere
0
{tT = [, t, t'M. .
= CO + c~t + C t2 .. + CrV
n n n n n ,
-"Cn cos wnC - CC
- sin wnt el1) C
n

where the coefficients are given by n

m ('1)1/2 (mmi)l,* a +i (1
n i=.0,2,4,. * mle
l

m 0,1,2,..,M e
In (12)
The constant colunn vector (Cn) can be expressed

111
 Zl4

by substituting fjTxof (10) and the infinite serife

expansion of en nI into (23). The coefficients
[Cn ) [Sn . (A) (20) are thus

here (A) is a constant column vector of the n (.)i / 2 i( )u l (29)

(,2,4.
l'ocin fuctin
ft) nd S ]is 'NxH upper n &,,..__
triangular square matrix which can be obtained
from (12). Then substituting (19) and (20) into In (29), A are the constants of f(T)e nw
(18) gives expanded irin infinite series. These are given

e*(x,t) - P (t)r nl -n(x)[Sn]'(A) (21) by 0 (Cn'n)(m*i'j)

In (21), P (t)T(A) is the forcing function P(t). m~i * Jo ( j (30)
Thus if e*(x,t) is known, the forcing function
P(t) can be obtained from (21) by expressing
e*(x,t) in polynomial form. where aj 0 for j > M.

So far dmping has not been considered. With the consideration of damping, the nth
However, in the actual case there is a slight mode quasi-static response time function,
damping in the response due mainly to internal g e'¢nunt,
-4(t) can be expressed in an infinite
damping and aerial damping. When damping is 9sIe as
considered, the strain becomes
1 2 2--~t
*(t)en nt c0 + t+ C ~ 2+
tc + c' t
ed(xt) P n l End(x) Gnd(t) (22) gn +C ne ne ne
+... (31)
where End(x) ' En(X)/[-(Cn ) 21
In (22), G (t)is the daired nth mode response where
time funct gn which corresponds to Gn(t) in the m (-rnwn)(
1
case of no damping. n cm E 32n
ne j=o (Mj) CndJ 32)
Gnd(t) • 0nd 6t f(,)e-4nns(t-r)in wnd(t.i)d
By taking terms up to the M'th order (!'Q M) in
(23) (31), gnd(t) e-Cnwnt can be expressed in matrix
0
*d( ) form as
[g d(t) - g d(O) cos wdt - - - n t
gd(t) e = (t)T(Cne (33)

sin wndt]oennt (24)

In (33), (Cne =(Bn](C nd (34)

(t)eCWt -
where [B I is a M' x M' lower triangular square
e-cnwn t matrix which can be obtained frol (32) and (Cnd)
is a constant column vector of C d (j = 0,1,2,..
•gd(t) e'nn *
t + *d
Set .-Ac nd'oS(wndt )
* ').(Cnd) can be expressed as

e'%nJnt (25) (Cnd) = ISnd]D](A) (35)

where ni " n (26) where ca RSd

which ] is
be an upperfrom
obtained triangular
(29), andsquare
ID ] matrix
is a

+ Ct + C2dt2+ (27) lower triangular square matrix which ca be

nd nd nd obtained from (30).

2 2 (0) } .non~hen
cd the strain corresponding to g*d(t).
becomes
And '([gnd2(O)J + 2

(
-I Wnd g9nd0) C -1(x,t) ) • NOT} T F Dn)(A) (36)
nd = tan ( n),d tan (l/tanend Cn(x t) = P't) xn '

(28) Thus if the quasi-static response es(x,t) and

Coefficients CO Cd . the damping ratios Cn are known, the forcing
nd' Cd'nd. are obtained by function P(t) can be obtained from (36) by

112
expressing e*(xt) in polynomial form. For this where 4
case, an error will result due to truncation of
the infinite series when taking terms up to the • )n(w) Rf(w) (45)
N'th order only. However, if N' is taken suf- n)2 -?
ficiently large, the error can be minimized.
RESPONSE IN FREQUENCY DONAIN
(46)
Taking Fourier Transform (FT) of (7) with
zero initial conditions gives Taking IFT of a (w)using the real part only as
given in (44) yfelds

Pn PO 2
an W)Mwnfiw)Jf(w) (37) M - n [g~)# (t)] (47)
n n gn (tan)2% wnn n~
Substituting (37) into FT of (5) yields a(t) and g2(t) are time functions obtained from
Px n((w) and d2CW), respectively.
n i)( P(2) (38) n
2
where F1(w) is FT of impulse response function
sin w t of the beam. This impulse response
functdon is causal, i.e. sin w t is zero for g(t) J 0[Xf(w)6(w-tn) - Xf(w)6(w+wn)].
t cO. Thus
Wn j
cos Wt dw
fn (w)= F{sin wnt u(t) "
a (0 ~ = ~ 2 - f(wn) cos nt (9
)
[6(w + ton 6(w- n)] (39) If the response in the frequency domain
n(w) is known, the response in the time domain
where 6(w) is the delta function, can be obtained from (48) and (49). Also, if
the response in the frequency domain is known,
Forcing function f(t) is also zero for M(w)
can be readily obtained because _2(w) has
t < 0. Let FT of f(t) be delta functions only. Knowing (hw),
Af(w) can
be obtained from (45). Then by taking IFT of
No= f + jf (40) Rf(w), the forcing function f(t) can be obtaine.
If the imaginary part of %n(w) rather than the
real part is used, a parallel analysis will give
Substituting (39) and (40) into (37) will give the same result.
an(w) which has real and imaginary parts.
The domain conversion is made possible
FT of a real causal time function h(t)(71 using the Fast Fourier Transform (FFT). lowever
Is FIT does not have the characteristic property of
FT of a causal time function. Thus the respons,
() + jX( )
M(w) (41) likewise the forcing function, in the frequency
domain obtained from the above analysis and that
where obtained using FFT will be different. Actually,
R( NO cos w obtaining the forcing function directly from
(45) is possible In theory only. In practice,
R) h(t) sin wt dt (42)' the quasi-static response can be obtained using
the characteristics of FFT. The forcing functon
can then the
be quasi-static
obtained from response
the derived
Inverse Fourier Transform (IFT) of 11(w) is between and relation
the forc-
h(t) ;2 o. R(w)cos wt dw ing computing
of function. FFT Is simply an efficient method
the Discrete Fourier Transform
2 RMX sin wt dw (DFT) [8).
(43)
As shown in the Appendix, a sinusoidal
As indicated by (43), the causal tine function function (sine and/or cosine) can be represented
can be obtained using only the real or thefrequency domain. This Is
cainary t d u g oy taccomplished by taking the sampling duration as
imaginary part of FT. an integer number of periods corresponding to
The real part of Gn(w).is the frequency of the function. Impulses on the
p n imaginaiy axis correspond to a sine function and
R (w))
a n (c.)J (44) Impulses on the real axis correspond to a cosine
) } )
n {wn n n function.

113
 Cutting (or smoothing) thase impulses in The FT of this response was obtained using
the frequency domain corresponds to removing the 1024 sampling points (N) for the duration of
sinusoidal function in the time domain. This eight times the fundmental period (the same
"cutting impulse technique" can thus be used to sampling duration which was used for the compil-
obtain the quasi-static response of the beam. ation of experimental results for theS ft.- 3 in
The sinusoidal response is removed by making use beam). Since no damping was considered, sampi-
of this domain conversion method. Ing was started at t = 0 (with sample intervzl
being AT'. The result is shown in Fig. 3. (FT
A slightly damped sine function can still is shown for positive w only in all Figures).
be represented by impulses at the corresponding In Fig. 3, an impulse at wl on the imaginary
frequency on the imaginary axis, except that axis corresponds to the sine function and an
there will now be a small side lobe across the impulse at wI on the re-al axis corresponds to
impulses. The sampling duration must be the the cosine function. Since the sampling dura-
same as for the case of no damping. Thus, using tion is not an integer multiple of the period
the imaginary part of FT only (refer to the corresponding to w2 , the response in the frc-
Appendix), the cutting impulse technique can quency domain at w2 is not of impulse form.
still be used to remove the damped sine function.
In this case an error due to the existing small
side lobe will result. However, the error is
quite small as shown In the Appendix. 1500 ..........- REAL
MGNRPPRr

A slightly damped cosine function will have ... IMAGINARY PART A

impulses and small side lobes on the real axis M.1024
in the frequency domain. The cutting impulse 21 &
technique can also be used to remove this func- 00a Nt
tion, this time by using the real part of FT I./NAT
only.
Thus, when slight damping exists in the re-
sponse, the sinusoidal function must be made a s0
sine or a cosine function only with a zero phase -
angle in order to use the cutting impulse tech- 2
nique. This is possible by starting the sampl-
ing at time z fI
C o . . .. ..

TS nd
n_ or T. nd a
'nd : 'nd in (25) / -
o /
- Soo0
In Fig. 2 the response (strain) of an o0 4 0 600
aluminum cantilever beam of length S ft. and FRoUECY Ia fa
diameter 3 in. (which was also tested in a field
explosion) subjected to a uniformly distributed
exponential forcing function is shown. Here -,CC¢
modes up to the 2nd were taken into account and 499 4
damping was neglected. The units of the respore Fig. 3 Response in Frequency Domain of Circular
are psi, which is the result of dividing the Aluminum Cantilever Beam Subjected to Uniformly
strain by n W.l istributed Exponential Forcing Function e-kt
Dn(x)"
(W1 1 4S.5 rad.sec., k=20, L=5 ft,D:3 in.)

" Cutting the impulses at w 1 by linear inter-

polation using adjacent points implies removing
" ,, the sinusoidal response corresponding to w! in
the time domain and also implies that the quasi-
static response in the frequency domain has a
/ value at w1 which is the average of the two ad-
, .-. jacent point values. After cutting the impulse.%
IFT was taken using both real and imaginary
parts. The result is shown in Fig. 2 and Is
called the modified response. As shown In Fig.
2, the fundamental mode is absent from this re-
Pig. 2 Responses of Circular Aluminum Cantilever sponse. From this modified response the quasi-
Beam Subjected to Uniformly Distributed Exponen- static respnnse can be readily obtained and Is
tial Forcing Function e-kt (k * 20, 1, 5 ft,, also shown in Fig. 2. If the sampling is taken
1) 3 in.) fine enough such that an integer number of
periods corresponding to w2 can be obtained for
another sampling duration from this IFT, the

114
sinusoidal response corresponding to w2 can also mately l.S%.
be removed if needed.

Instead of using both real and imaginary

parts for the IFF, the real part or imaginary REAL PART
part only can be used. After cutting the im- - AGINARY PART
pulses, IFT was taken using only the imaginary N , 1024
part of FT. The modified response is shown in OT. t.-
Fig. 4. This modified response is different 1000 W, N
from the modified response shown in Fig. 2 due f . I/N.T
to the folding effect refer to the Appendix).
The forcing function e - Kt is around 0.001 at the
end of the sampling duration. Thus at t - 0, -
the error in the quasi-static response due to Soo
the folding effect is around 0.1%.

4 0.0
,.....
..
... .........
.....
.!
10 1 40*',.OU

+ o. o o
co.+,,.,,<,, 1, ..,,,'++I
o€* '¢u i 200 400 60

- -- I"t011.0PO
o

Fig. 5 Damped Response in Frequency Domain of

-- ................
c. 0Uniformly Circular Aluminum Cantilever
Distributed Beam Subjected to
Exponential Forcing
-
,t,, D, l Function e kt (w, ' 145.5 rad./sec., k = 20,
Fig. 4 Responses of Circular Aluminum Canti- = 0.0074, L - 5 ft., D = 3 in.)
lever Beam Subjected to Uniformly Distributed
Exponential Forcing Function e'k t (I.- S ft., To avoid ambiguity, the exponential forcinp
D 3 in., k = 20) function and the quasi-static response obtained
from the modified response considering damping
For the S ft.- 3 in.beam, a damping ratio were not shown in Fig. 4.
of 0.0074 for the fundamental mode was observed
from initial laboratory tests. The beam res- EXPERIMENT AND EXPERIMrNTAL RESULTS
ponse was obtained again, this time taking the
damping into account. To make the sinusoidal Two aluminum cantilever beams, eiic of
functioa a sine function (with zero phase angle) length 5 ft. and diameter 3 in. and the other of
only, sampling was started at time Ts - 0.96 ms. length 2.5 ft. and dihmeter 2 in., were tested
The FT is shown in Fig. S. After cutting the at the expected 12 psi nominal overpressure
impulses at w1 (of the imaginary part), IFT was location in a field explosion of 500 tons of TNT.
taken using the imagi nary part only. The result This test was conducted on JuJy 23, 1970 at
is shown in Fig. 4. The differences between the Defence Research Establishment Suffield, Ralstog
two modified responses of Fig. 4 are due to: Alberta. Circular aluminum rods of smooth sur-
face were fixed to steel baseswere
using
thena shrink
fixed in Z
1. Sampling starting time T . fit method. rbe steel bases

2. Existing damping effect on the modified the field by bolting then to concrete bases.
Four strain gauges (one facing charge one on the
response. Damping effect on the quasi-static opposite side, and the other t4o similarly
response is very small and the difference located, but higher on the beam) were Installed
due to damping was estimated to be only about on each beam. Two strain gauges were used for
0.3% each strain output. The straip gauges were
connected through bridges to r magnetic tape re-
3. The remaining sinusoidal difference cor- corder which was located in . bunker.
responding to 1. This is duo. to the exist-
ing side lobe in the frequency domain when An overpressure rec,,d obtained from a
damping Is considered. The maximum differ- piezo-electric gauge at lhe expected 12 psi
ence is about 2% (refer to the Appendix). nominal overpressure lo~ation Is shown in Fig.6.
From this record it coa be seen that the actual
The similarity between the exponertial peak overpressure is ipproximately 11 psi and
forcing function used and the quasi-static re- the duration is ahoyo. 0.23 sec.
sponse was examined. The quasi-static response
was smaller than the forcing function by approxi-

115

All A
 Predicted(rad/sec) Actual(rad/secl

S ft.- 3 in. 145.5 144.0

a 2.5 ft.- 2 in 388.1 3955

!....? .TABLE 1 Fundamental Frequencies

, , To make the fundamental mode of the sinu-
soidal response a sine function only, sampling
Overpressure obtained from 500 Ton TNT was started for the 2.5 ft.- 2 in.beam at Ts =
Field6 Explo)sion
Fig. at the Location of Expected 16 AT and was carried on for the duration of
12 psi Nominal Overpressurc, twenty-two times its actual fundamental period.
For the 5 ft.- 3 in. beam, sampling was started
Strain outputs from the 5 ft.- 3 in.beam at Ts s 33 AT and continued for the duration of
at the location 1.0 in. from the root and from eight times its actual funJamental period. Then
the 2.5 ft.- 2 in. beam at the location 0.75 in. a conversion into the frequency domain was made
from the root are shown In Fig. 7. The responses using FFT. After cutting the impulses (on the
show that the ground shock arrives at the beam imaginary axis) corresponding to the fundamental
before the air blast wave. Due to the excita- frequency wl, IFT was taken using only the imag-
inary part to get the modified response. The
small free oscillation of the beam, mainly of result for the 5 ft.- 3 in. beam is shown in
fundamental modethetype, is initiated. Whenatthe Fig. response
modified(which
8. Theresponse har *odes a
up of than
is madehigher
shock front of air blast wave arrives sinusoidal
the beim, the initial conditions of the beam 2nd) superimposed on the quasi-stat.c response
respOhseare no longer zero because of these (dotted line). The actual quasi-statlc response
small'free oscillations. (up to 65 as) was obtained by eye from the quasi-
static responses (which were obtained by taking
two more and two less sample points at the
_____________-I Ibeginning)- ithe using This
Appendix). an "averaging"
response ismethod in Fig. to
shown (refer
by a dashed line. The actual quasi-static re-
sponse does not have any sinusoidal response
remaining In it.
SMA Of LtI6TM * V nD SD4WTII *I

---- ACTUAL OVASI.STATIC IaCSPOS

D WI~t~MACi1YIIMA
- - -o..,,,.. C

6 8IA Of LIOTP $FT AND 1AMIT

Fig. 7 Response (Strain Output) of Circular

Aluminm Cantilever Beams Subjected to Air
Blast Wave of Peak Overpressure 11 p.,i
3 4% -

I
The beam response from the air blast wave 0 o
A t: &a
. G ©-4
- .o

consisted of a slightly damped sinusoidal re- TIE lSt€oftsi

sponse (mainly of fundamental and 2nd mode type)
superi mposed on the quasi-static response. The Fig. 8 Responses of Circular Aluminum Canti-
domain conversion method and the cutting impulse lever BELan of Length 5 ft. and Diameter 3 in.
technique can be used to remove the sinusoidal at II psi Location (AT = 0.125 ms., N = 2790)
response and so yield the quasi-static response.
The modificd response and the actual quasi-
The response was sampled by digitizing static response for the 2.5 ft.- 2 in.beam were
using a sample interval (AT) of 0.125 ms. The obtained using the same 1.-ocedure and are shown
time t I 0 was set to be the time when the shock in Fig. 9.
front arrived at the beam. Fundamental frequen-
cies (w ) of both beams were obtained by taking Several attempts were made to remove the
FT of the responses after 0.23 sec. (slightly sinusoidal response corresponding to the 2nd
damped free vibrations now exist). The actual mode (..2)from the modified response using the
fundamental frequencies obtained, as well as technique described previously. However, due to
predicted values, are shown In Table 1. the difficulty of f'l=ing an integer number of

116
 either of the two parts of (50) and (51) in each
case. However, differences will be small be-
cause the slope differences at the intersection
h ~ -QASI-STATIC
------ACtuAL os stATIC RsPt
R[SIPONS[ NO1 If-V94~
points are small.
o PRIsIIIR
DAAG-- Actual drag pressures were obtained from
(50) and (51) using (21). In (21), modes up to
A the 6th were taken. Since slight damping
exists in the actual response, (36) should have
S 10
been
modesused.
higherHowever,
than the because
second damping ratios
were very hard of
to
1[ i iobtain and the damping effect was in fact very
small,(21) was used instead of (36). The drag
os /UIIIJ'I1IM pressures obtained are shown in Figs. 8 and 9
ind can be represented by the following:
ocL0 ,-----.--
01 00 003 0
I
~ 00 -; 1-- --
coo0co
5 ft.- 3 in.beam
•IM, IsEco*os 0 - 35 ms Pd(t) = 1.396-24.986t-S87.10t2+
26127.7!"3 (52)
Fig. 9 Responses of Circular Aluminum Canti- 2( 2
+ lever Beam of Length 2.5 ft. and Diameter 2 in. - Pd(t)
535ms 55ms
Pdt) -5.625-404.82t-7990.08t
-5625-074.8t379.t

at 11 psi Location (AT = 0.125 ms., N a 2796) 3

+50734'8t
2.5 ft.- 2 in.beam
cycles of the 2nd mode from the modified response,
it was not possible to construct an impulse at 0 - 15 ms P 1.630-42.55t(4774523t2+
w2 in the frequency domain. This difficulty
arose as a result of the relatively large sample 15 - 35 ms Pd(t) a -11.265+S72.18t-22080.6t 2
interval for U2 and the accumulation of error in .267967.9t 3
obtaining FT and IFT using FFT.
As mentioned when dealint with response, the
An empirical relationship was fitted to the existence of the intersection region introduces
actual quasi-static response time data (data an error. The error will be small, however,
were obtained using a time increment of 1.25 ms). because the intersection region is very short.
The type of curve fitted was a polynomial obtain- Bec'use this error will accumulate as more parts
ed using polynomial regression. Due to the are taken into account, the drag pressures (and
"bumps" (a fairly rapid rise followed by a also the actual quasi-static responses) were not
gradual fall) in the responses, the relationship obtained after 65 ms for the 5 ft.- 3 in. beam
for each beam was obtained in parts as follows nor after 35 ms for the 2.5 ft.- 2 in. beam.
(actual quasi-static response Ps in psi and time
t in sec): Drag coefficients can then be obtained from
Pd(t)
Cd(t) = (t
5 ft.- 3 in. beam 2
0 - 35 ms: Ps(t) 1.446-31.643t-S87.1Ot +
26127.7t 3 (50) For approximately the first 0.5 ms the flow
2
3S 6S ms: Ps(t) =-.369392.87t-7990.8t is conditioned by reflection and diffraction of
+50734.8t 3 the incident shock and hence there is no true
drag coefficient until a steadily decaying flow
2.5 ft.- 2 in.beam is established around the cylinder. However,
2
0 - 15 ms: P (t) 1.687-53.006t-4774.23t + the "equivalent" drag coefficient, as it may be
299549.3 3 (51) termed, is useful for the purpose of construct-
2
15 - 35 as: P (t) *3.571+563.53t22080.6t ing the loading configuration on the beam as
s- 76.90
-267967.9t3 shown in Fig. 1.
Drag coefficients obtained from the responses
Expressing the actual quasi-static response of the two beams as a function of Reynolds
in parts using polynomials gives rise to the number and Mach number are shown in Figs. 10 and
difficulty in matching the magnitude and the II.
slope (rate of change) of the responses at the
points where two parts meet. With regard to (50) DISCOSSION
and (51), there is a slight difference in the
slope at the point where the magnitude of the Drag Coefficients
responses Is made to coincide.
Drag coefficients obtained from the response
A smooth curve was drawn through the inter- of the S ft.- 3 in. cantilever beam for the
section point to connect the two parts for each duration of the initial 65 ms are for Reynolds
case. The regions of inaccuracy resulting (intc' number 7.81 x 105 and Mach number 0.41 (at time
section regions) are actually very short. The
intersection regions cannot be expressed by
t 0) to Reynolds number 4.1 Y 105 and Mach
0
tumber 0.23 (at time t 65 ms). Mach numbers t
117

,- I-+* . .. -1 <' - - r' = .- . -+

 differences were observed:

1. The upper critical Reynolds number is higher

in the case of air blast drag coefficient values
2. The drag coefficients above the critical
Reynolds number range are a little higher com-
pared to the steady state values even consider-
Ing the Mach number effect.
,a-3. The levelling off of air blast drag co-
efficients at Reynolds number around S x 105 was
not expected. Of course this region is in the
critical region where flow conditions vary con-
tinuously. Thus, any small shift in the flow
0conditions can cause considerable changes. This
may be the explanation for the "levelling off"
011- of the drag coefficient at Reynolds number aroud
5 X 105. Variations in the drag coefficient
04 (the same kind of phenomenon as noted above)
were also observed for the 2.5 ft.- 2 in. beam.
0, 045 J5 oI 0J 06 07 0,5 08 Drag coefficients obtained from the 2.5 ft.-
Ol4l 10|45 0o4 10101 01 I111 f# IIIIo1|r0
40*1.I 45,I 103011 1614S.61 Wo 4838 2 in. beam response for the duration of the
REYNOLOS No..,- initial 35 ms. are for Reynolds number 5.2 x 105
IAcH and Mach number 0.41 (at time t = 0) to Reynolds
Fig. 10 Drag Coefficient vs Reynolds Number and number 3.7 x lO5 and Mach number 0.3 (at time t
Mach Number obtained from Elastic Response of 35 ms). Drag coefficients as a function of
Cantilever Beam of Length 5 ft. and Diameter Reynolds number and Mach number varied between a
3 in. at 11 psi Location maximum value of around 0.61 and a minimum value
of around 0.41 for the range of Reynolds number
and Mach number observed. It appears that the
drag coefficient increases considerably before
os - levelling off as Reynolds number decreases below
3.7 x 10
07 -Elastic Response and Maximum Strain

0The response of a cantilever beam subjected

to an air blast wave will exhibit a side oscill-
2 ation perpendicular to the direction of blast
05- wave travel due to vortex shedding. This side
0oscillation is very small compared to the re-
0sponse in the direction of the blast wave. Since
IX04 the shear Centre and centroid coincide, the bend-
ing vibrations about the two perpendicular axes
are independent of each other. Thus, the ob-
03 tained strain outputs from the strain gauges
(placed in the direction of the blast wave) did
not record any strain from the side oscillation.
03 035 04 045 05 055
,. , . , ,It"o, ,o40 The maximum displacements at the mid-sectior,
6
REYNOLDS No (. 10- ) of the two beams were:
(MACH No 5 ft.- 3 In. beam 0.01 ft.
Fig. 11 Drag Coefficient vs Reynolds Number and 2.5 ft.- 2 in. beam 0.004 ft.
Mach Number Obtained from the Elastic Response As shown above, the displacements of the.
of Cantilever Beam of Length 2.5 ft. and Dia- beams were very small and hence use of the Euler
meter 2 in.at 11 psi l.ocation beam equation is justified.

are lower than the critical Mach number (which Since the response was mainly of fundamental
Is approximately 0.41). Thus with regard to the mode type, the maximum velocity at midsection
steady state values of the drag coefficients, t was obtained using the maximum displacement at
Reynolds number effect on drag coefficients is midsection and the fundamental frequency. Thus,
expected to be dominant in this Mach number r.ngc the maximum velocities were:
5 ft.- 3 in. beam 1.55 ft/sec
Drag coefficients of a circular cylinder
subjected to an air blast wave generally follow 2.5 ft.- 2 in. boam 1.45 ft/sec
steady state values. However, the following The velocities of the beams are very small com-

118
pared to the air particle velocity during the The forner is symmetric about the folding samp-
blast loading (at t a 0, the air particle vel- ling du.ation tf(k z N)and the latter is anti-
ocity was 504.S ft/sec). Therefore, the effect symmetric about tf.
of the velocity of the beams on Reynolds number
of air flow around the beams can be neglected. Expanding R(x(k)) for a point k* (k I N
For both beams, maximum strains were ob- gives
tained near the quarter point of the first cycle R(x(k*)) * [R{(i)cos WI-I{ (i))sin 2 i]
of each fundamental frequency. Analytically, i-0
maximum strains were computed using the loading _N
configuration of Fig. I together with initial 1 N-i N-i 2-k* 21
conditions. For the drag loading, mean drag kEO iN
coefficients of 0.48 and 0.562 for the S ft.- 3
in. beam and the 2.5 ft.-"2 in.beam, respect- 1EI I(x(k)csN-i sk* 2k
ively, were used. Mean drag coefficients are N k-0 icO
the mean values of the drag coefficient for the
duration of a quarter of the first fundamental I N-I N-1 2k* 2k
period of each beam. Computed and measured N kE0 R(x(k)1ii0 sin -- 'sin -W- i
maximum strains are shown in Table 2.
I N-I N-I 2vk* 2%k
Computed(uin/in) N kWO-
k4p I
S ft.-3 in. 'beam 653
(A.3)
2.5 ft.- 2 in.beaml 490 510 N-i 21 k* 2lk•
In (A.3), lie cos -W--i-cos N izO Ni
TABLE 2
Experimental and Computed Maximum Strains 2r Nl
sin 2k- , N 27Tk*. .21Nk i Ni
sin -ix*sn - and t=0sin
The computed maximum strains show close
agreement with the experimental values. Instead 2rk* 2o1
of using mean drag coefficients with dynamic N are analogous to the integral
pressure of Friedlander decay type as was done forms
above, maximum strains aere also obtained using
the actual drag pressures of (52) and (53). The c s
maximum strains so found were very close to the 0o 2"k*y'cos 2wky dy, Icos 2wk*sin 2wky di
strains determined using mean drag coefficients. s 2ak*ysin 2wky dy and 1sin 2aky.cos 2xkydy
The good agreement between computed maxi-
mum strains and experimentally obtained maximum respectively. Thus R(x(k*)) becomes
strains implies that the obtained drag coeffic-
ients up to at least 4 of the period of each * N-I
fundamental frequency are correct. Also, the 2-Nk=O (k))}kk**
good agreement suggests that the loading con-
figuration used is justified. I R{x(k)N- 6
+N k=O k** kk**
where 6kk,, is the Kronecker delta
k*i
The Fourier Transform pair for continuous
signals can be written in the form k** Is k* and N - k*
X(W) z .ftx(t) e'jwtdt rk** ' I when k** is k*
2 (A.1l) k
x(t) z f' A(l))Jwtdw = -1 when k** is N -

The analogous Discrete FoL.,ier Transform pair In (A.4), the first term resulted from
to (A.1) is ~ 2nk*
N-1 R{X(i)}cos -r - and the second term from
X(i) j 2lI.)0
N-
NI x(k) 2A kkF i-
1
N~i)
N=- 1 x~k) e N (A.2) N1-
1s2 2ak F ,{Rqi)j
the same
N 2
3
x(k) = i X(I) -N result will be obtained. Therefore, it can be

where N is number of sample points, seen that T0 R((i))cos -- i is a symmetric

When X(i) and x(k) of (A.2) are separatcrd expression of R(x(k) ) folded about the folding
into real and imaginary parts, it can be shown
that R(x(k)) is composed of two parts, sample duration tf and - i=O l(X(1))sin i is
N-i 2Nk N-I 2Nk
1 R(X(i))cos L i and iO l(X(i))sin L i. an antisymmetric expression of WR(x(k)) folded

119

t! 2,
about tf. Thus R(x(k)) :an be obtained using
only the real part or the imaginary part of X(i)
If Rfx(k)) is finite up to tf. The same is true NN
for I(x(k)). 40

Thus, when x(k) is real, R{R(i))is symmet-

ric and IT{(i)) is antisymmetric about the fiding 400 j
frequency ff (i N).
o0 - REAL PART
Bt taking IFT using R{((i)) only (setting IMAGINARYPART
I((i)) equal to zero), a modified h x(k) which ,.oo074
is symmetrically folded about tf in the real 300 ,.1455
pazt will be obtained. Likewise, by taking IFT N • 1024
using I({(i)) only, the modified h x(k) which is - - T. -
antisymmetrically folded about t in the real W,WI
part will be obtained. Since R(f) is inter- - , II8T
preted as being periodic from the sampling 200 S

theorem, the Fourier coefficients between N/2

and N-i can be viewed as the "negative frequency"
-1. Likewise, the
harmonics between -N/2 and
last half of the time function can be interpreted
as negative time. ,oo

The DFT of a sinusoidal function depends o ,

upon the sampling duration taken. If the samp-
ling duration is taken as an integer number of
periods of the corresponding frequency, there 0 ,10)
will be impulses at the corresponding plus and ,t
minus frequency points iiithe frequency domain. .so
Impulses will be obtained on the imaginary axis
for a sine function and on the real axis for a Fig. A.I Slightly Damped Sine Function in
cosine function. The magnitude of an impulse is FrAquecy Dmin
.f the sampling duration is not an integer
imber of periods, taking DFT will not produce
impulses. The fact that the occurrence of im- UNI

pulses depends on the sampling duration of the

DFT can be explained by the sampling theorem 171.

Whlen damping exists, the sinusoidal function 40 j , .00o74

cannot be represented with impulses only in te ,.,4s
05
frequency domain. Sin )1t e4 1 lt with w1 = N .1o24
145.5 rad/sec and 41 = 0.0074 is shown in Fig. [ ,L2 *
A.I. There are impulses at ± w I and small side N
N,

lobes around ! wI in the imaginary part while the 2 NOA

real part is far different from that when there /ME. , N-
is no damping (the real part is zero when there W M"0
is no dimping). Cos w t e' l1lt will have the "2"
same FT as sin Wlt C
e "-1lt except that the real s #
part and imaginary part will be Interchanged.
In Fig. A.2, the imaginary parts of sin wlt. ,

e "41W lt are shown for different sampling durat- "

ions. The sampling duration of I is eight times 0 L (-Si
the period of w1 , the sampling duration of I1 Is
shorter than that of I by AT,1 , and the sampling
duration of Ill is longer than that of I by TI " ,o 0
As shown in Fig. A.2, taking more or less samp
points than the exact number makes the side lobe
change sign across the corresponding frequency. Fig. A.2 Imaginary parts of Slightly Damped
After cutting the impulses, IFT was taken using Sine Function Obtained with Different Sampling
only the imaginary part for each case. Fer I, a Duration
function very close to a slightly damped sine
wave (with zero phase .ngle) of initial ampli- cases I1 and III, if the middle of the two re-
tude 0.02 was recovered. For II and III, func- covered functions is taken (an "averaging" used),
tions very close to slightly damped plus and the error dtict, the existing damping can be
minus cosine waves (with zero phase angle) of minimized
initial amplitude 0.02 were recovered, respect-
ively. Thus, for these cases, the error due to
the existing side lobe was 2% at most. For

120
 From the above discussion, it is evident
that when slight damping exists, the sinusoidal
a co-
function must be a sine function only or
sine function only to make use of the cutting
impulse technique; while in the case of no dampA
ing, the sinusoidal function does not necessar-
ily have to be a sine or a cosine function only.

REFERENCES
1. W.E. Baker, W.O. Ewing, Jr., J.W. Hanna,
and G.W. Burnewitch, "The Elastic and
Plastic Response of Cantilevers to Air
Blast Loading", Proceedings of the second
U.S. National Congress of Applied Mechanics,
1962, pp. 853-866.

2. N.L. Brode, "Numerical Solutions of Spheri-

cal Blast Wave", Journal of Applied Physics,
June 1955.

3. V.J. Bishop and R.D. Rowe, "The Interaction

of a Long Duration Friedlander Shaped Blast
Wave with an Infinitely Long Right Circular
Cylinder. Incident Blast Wave 20.7 psi,
Positive Duration 50 ms, and a 16 cm Dia-
meter Cylinder", AWRE Report No. 0 - 38/67.
April, 1967.

4. S.B. Mellsen, "Drag Measurements on Cylind-

ers by the Free Flight Method-Operation
Prairie Flat", Suffield Technical Note No.
249, Jan. 1969.
5. S. Glasstone, "The Effects of Nuclear
Weapons", Published by the United States
Atomic Energy Commission, 1957, Revised in
1962.
6. S.B. Mellsen and R. Naylor, "Aerodynamic
Drag Measurements and Flow Studies on a
Circular Cylinder in a Shock Tube", Suffield
Memotandum No. 7169, May 1969.

7. A. Papoulis, The Fourier Integral and Its

Application, McGraw-Hill Book Co., Inc.,
New York, 1962.
8. G.D. Bergland, "A Guided Tour of the Fast
Fourier Transform", IEEE Spectrum, July 1969

ACKNOWLEDGMENTS
The work reported herein was supported by
the Defence Research Board (Grant No. 1678-09)
and the National Research Council (Grant No.
A-3384). The authors wish to thank Defence
Research Establishment Suffield personnel for
their helpful suggestions and for the oppor-
tunity to participate in event Dial Pack (500
ton TNT field explosion).

121
j MEASUREMENT OF IMPULSE FROM SCALED BURIED EXPLOSIVES

Bruce L. Morris
U.S. Amy Mobility Equipment Research and Development Center
Fort Belvoir, Virginia 22060
I-dimensional anaiysiswais peorfoR-d to determine the physical scaItng
parameters governing the response of wheels to blast loadihg. Hopkinson
scaling was used to determine the proper charge size and location for one-
iquarter scale blast tests. The total energy imported to the test wheels
by the detonation was determined, and the scaled specific impulse was
1clwculated. The test and calculation procedures are descrbed.J
INTRODUCTION IT tLl/2p 1/2 Time scaling
.* The utilization of mine neutralization
hardware require that this equipment operate
under the intense pressure of near-field ex- T2" .L3..
plosive detonations. Since accurate theoretical "0P
knowledge of this explosive-target interaction Soil conditions
is limited, designers have had in the past to 113. cM1/2
resort to full-scale explosive tests to evaluate 3-02P/
and prove their designs. This process is ex-
pensive and time-consuming, so this Center
elected to use scale models to evaluate materi- 7r4- rt
als and configurations for mine clearing roller
wheels capable of withstanding the blast effects Irs
5 9 . Initial conditions
of 30 lbs of explosive. Zp and restraints

DERIVATION OF SCALING LAWS fZ

Inorder to correctly interpret the experi-
mental results, itwas necessary to determine z7=E
the dimensionless products, or Pi terms, PO
governing the interaction between explosive and .L
-target. Since the target wheels are in contact /f8 = aM
with the ground, parameters describing ambient 2 Response scaling
air conditions were omitted from the analysis.
It isbelieved that a major portion of the im-
pulse imparted to the target iscaused by the
soil being thrown out o? the, crater and im-
pinging on the target. Westine [1] concludes
that density and seismic velocity, rather than
a stress parameter, best describe the soil con-
Tig=

/i0=
1 2
IL 1
p1/2 M112
Q
P
Lm=,
I
ditions. These parameters, along with others For replica models, E /7where m and
governing this phenomena, are listed inTable I p denote model and prototype respectively. If
along with their
time (FLT) system.dimensions ina force-length- the same material MM
isused in the model and
prototype wheels, I? = X 3." Ie assume equality
Ten dimensionless products, or Pi terms, of blast pressure, ie,Pm Pp. These con-

I
can be formed from these 13 parameters. There straints are then applied to the above Pi terms
are many techniques for creating this list of to establish the scaling law below.
terms, and no matter which method isused, the
analysis isnot modified as only the algebra is
deleted. Listed below isone set of dimension-
less products or Pi terms.

Preceding page blank 123

•; .
A7_
_ ----
_ _ _ _ _ _ - - - '; - " - . i
Table I. Physical Parameters Governing Explosive-Target Interaction

Symbol Description Units

P Blast pressure FL- 2

t Time T
2
p Mass density of soil FL- 4T

c Seismic velocity of soil LT-I

L Characteristic length L

ri Shape of System -------

2 -
M Mass of target FL']T

g Acceleration of gravity LT-?

f Total load on wheel F

E Energy absorbed in wheel system FL

a Acceleration of wheel under blast load LT-2

t Impulse applied to wheel FT

<- Stress in target - FL"2

(T1 tm=t P Time scales as length TEST EQUIPMENT AND PROCEDURES
ratio
The test wheels were placed in a 3.5by
V2 p m/Op Sam soil for model 2.5 by 2.5'foot dirt-filled test rig as seen in
and prototype Figure 1. The wheels were connected with a
773 cm= cp common axle and secured to the test box by
rubber torsion spring with a total 'calibrated
rr4 (ri)m=(ri)p Geometric similarity rotational stiffness of 1180 inch-pounds per de.,
gree of rotation.
fl5 gm I g Since model and
5 prototype tests are The explosive charges (2 inch square blocks
conducted in same of C-4 explosive cut to provide desired weight)
gravitational field, were placed under th- center wheel with one-
this term is dis- half inch of soil cover over the Lharge. The
torted as an engin- charges were-detonated by M,.6 blasting caps,
eering judgement. and the spring rotation wai measured by-,the
scribes in Figure 1.
16 fm=Xfp Loads scale as square
of length ratio to
provide equal
stresses.

17 7 Em- AE p scribe1.
f8 m=l a p springs
ykes
kmYl orVeel
fr-0 Tmp 0 cag
Hopkinson has shown that blast pressure is a
function of stand-off distance R and charge
weight W as P=f(R/WI/ 3 ). Thus, if Rm=ARp,
ur= A.JWp to produce equalblast pressures.

Figure 1. Test box 5howing rubber torsion

springs, scribes, and test yokes.

124

5l
IMPULSE CALCULATIONS -(M
o0+ -- 1
For computational purposes, the wheel
torsjon-s rtfig system is as shown in Figure 2. 0 + ( K )Q= "( - ) (5)
-

Boundary conditions on eq. (5) at t-O are O-O

and 0=orig. Thus,
I borig t

%% i5orig"

1.i For small angles (sin 0 0), the solution of

eq. (5) is given by
moFigure 2. Q= A cos w t + B sin w t - Mo "(6)

o=-A w sin w t + B w cos w t

0 prelcad angle where

9
0max maximum rotation
K spring constant Substituting the boundary conditions into eqs.
(6)and (7)yields
I total impulse
total energy A= Mo
K and B= Ii

Mo initial spring moment .Omax occurs at a time tma x when 0= 0, or when

A length of yoke tan w tmax = B/A

mass moment of inertia Sin w tmax and cos w tmax are calculated from
T torque tan w tmax , and the results are substituted into
Ignoring gravitational effects, the total energy eq. (6) to determine Qmx,
absorbed in the test system is given by

E= ForceX Distance= [k(Op ) X Omax

X ]
x A 1 +O V
Y (B/A)2 K

E- K(Qp+ max). (1) Substituting for A and B and solving for i

2- yields
From conservation of energy, K.E.= P.E. and
V= 2E/H (2) i= 2K1 max I + 2 Mo

From the impulse-momentum relationship,

But Mo=K Op and the above equation reduces to
I a mv VIEm. (3) eq. (4).
VT Equation (4) gives the impulse imparted to
5ubstituting eq. (1) into eq. (3)and realizing the three-wheel test rig, but the impulse on the
that Izintlyields center wheel alone is desired. This is approx-
1-= VYK-1
i_ meimated
Qma, A1+ 2W (4) oy

.ma x ic + 2 ic sin $ = i3 wheel

Equation (4)can also be derived using a where ic = impulse on center wheel
conservation of momentum approach. Here,
T= 1 0 where 0 and 0 denote derivatives with
respect to time. d = depth to center of charge in feet

125

~!
 Westine [2) has developed a method of REFERENCES
calculating the impulse imparted to a target
from a land mine detonation given by [1]. P.S.,Westine, "Explosive Cratering,"
J. Terramichanics, Vol. 7, 11o.2, pp. 9
I= A0 to 19, 1970.
heeII -
where a specifi impulse
total impulse []
[2). by Westine, of
P.S.Detonation "Impulse ImpartedintoSympesium
Land Mines," Targets
I - specific impulse
A - projected area of target 7 in2 for on tn Detection and Neutralization,
these tests March 24-25, 1971, Vol 1 of 2, Fort
0 = shape factor which is a function of Mc Nair, Washington, D.C.
target shape and standoff conditions [3]. W. D. Kennedy, "Explosions and Explosives
Shape factors are determined for the various in Air, " in Effects of Impact and Ex-
configurations tested under this program. These plosions Volume I, Summary Technical
shape factors, together with the total impulse Report, NDRC, Washington, 1946.
as calculated from eq. (4), are used to cal-
culate the specific impulse generated by the
detonations. This impulse is transformed into
scaled specific impulse by dividing by the cube
root of the charge weight. These scaled
specific impulses are presented in Table II
along with the scaled distances and other data
items and are compared to previous extrapolated
data [3] for TNT in Figure 3. The TNT data has
been adjuated to C-4 explosive. Data generated
from these small charges is thus seen to fall
within the limits of data generated using ex-
plosive charges of up to 550 lbs.
Table II. Scaled Specific Impulse

Shot Charge wt. Scaled Dist. Impulse on 1 Wheel 9 Specific Imp Scaled Specific
Number (lbs) (ft/lb" 3 ) (lb-sec) (psi-sec) Impulse
pst-msec
lbl/3

2 .229 .068 12.1 4.30 .402 656

3 .285 .062 16.1 4.00 .575 875
4 .106 .088 10.7 5.45 .281 594

5 .200 .072 14.6 4.55 .457 782

6 .285 .062 14.1 4.00 .504 766

8 .420 .056 24.2 3.25 1.063 1420

9 .285 .062 13.9 4.00 i .496 755
10 .381 .058 20.1 3.40 .845 1165

11 .420 .056 21.4 3.25 .942 1260

13 .470 .054 31.7 3.10 1.460 1880
14 .285 .062 13.7 4.00 .489 744
15 .342 .060 15.9 3.60 .631 900

16 .395 .057 23.9 3.40 1.005 1370

126
 2000 - iii

1000 *
800 4+
600 +

140 -

03 20B %"L .""-

o1SYBL SOURCE TYPE OF CHARGE 42 4

V) 8 A BRL Bombs, all sizes -

6B8 RRL-ARD 8-550 LB bare charge
D DATA SHEET Bombs, all sizes
P PUS 0.5 LB rect blocks
U UERL 10 LB bare charges A
+ MERDC Current tests P
2 \

1 11111II
I lI I III I !
.06 .1 .2 .4 .6 1 2 4 6 10 20 40
3
SCALED DISTANCE (FT/LB1/ )

Figure 3. Scaled Specific Impulse vs Scaiea bistance.

127
 DYNAMIC ANALYSIS
THE EFFECTS OF MOMENTUM WHEELS ON THE FREQUENCY RESPONSE
CHARACTERISTICS OF LARGE FLEXIBLE STRUCTURES

F. D. Day III and S. R. Tomer

Martin Marietta Corporation
Denver, Colorado

A computer program using existing mathematical techniques has been

developed for use In performing linear frequency response analyses
of large elastic systems that contain axisymmetric momentum wheels
(gyros). The program incorporates the state vector form of the
dynamic equations of notion, and uses complex eigenvalue/eigenvector
transformations to yield a set of uncoupled equations that are easily
solved for the steady-state responses of the system. The structural
characteristics are input In the form of normal modes of the system
with nonrotating momentum wheels (i.e., all translational Inertias
and the rotacional inertias about the axes normal to the spin axis
are included). These system modes may be derived directly from a
finite-element model or through the modal coupling of a number of
subsystems. The program was used to calculate transfer functions on
the Skylab Apollo telescope mount both with and without the control
moment gyros. A comparison of these transfer functions is presented.

INTRODUCTION

The generally accepted method for deter- equations of notion only when modal damping or
raining transfer functions on large systems is damping proportional to either the mass or stiff-
based on calculating the normal modes of the ness matrix is assumed. The introduction of the
system and using them to uncouple the equations momentum wheel terms, however, recouples the
of motion. This method is based on the assump- modal damping matrix, and a second eigenvalue
tion that the modal damping matrix is diagonal. s3lution must be determined. This is done using
There are times, however, when the assumntion of the state vector form of the equations of motion.
a diagonal damping matrix is not valid; for ex-
ample, when a structure has gyros or momentum This meLhod was used to calculate transfer
wheels. The introduction of the momentum term functions for the Skylab Apollo telcscope mount
produces a skew-symmetric, rather than a diago- (ATM) in order to determine the effects of the
nal, damping matrix. In the past, analyses of control moment gyros (CMGs) on the structure's
this type have neglected the effects of the mo- behavior. The importance of this example and
mentum wheels. The purpose of this study is to the reason for choosing it lie in the fact that
include the momentum term and develop a method these same transfer functions are used In de-
to facilitate the calculation of the transfer signing the experiment poitting control system
functions so that their effects on the structure (EPCS) for the ATM. Because of the proximity of
can be evaluated. the COGs (which are used for overall cluster con-
trol) to the ATM, their effect on the EPCS son-
The method makes use of existing mathemati- sors could be significant. Transfer functions
cal techniques to develop the state vector form with and without the C1Gs were computed. The
of the dynamic equations -i motion. From these results are compared in this paper.
equations, complex eigenvalue/eigenvector trans-
formations are then generated and used to un-
couple the equations of motion. Once the equa- THEORETICAL DEVELO.-NT
tions are uncoupled, they are easily solved for
the harmonic steady-state response. The derivation of techniques used to gener-
ate acceleration responses or transfer functions
Undamped normal modes are used to describe for large flexible structures with momentum
the characteristics of the flexible structure, wheels will be developed in four parts:
These modes are sufficient to uncouple the

Preceding page blank 129

 9~~- -- ~ 7~ 17-777.W,~

1. A brief review of the equations of and for steady-state responses,

siotion for the elastic structure with-
= e
out momentum wheels; {4(t)) jqo} ift, (4)

2. Equations of motion for a spinning

rotor as a free body; where the modal coordinate vector 1 qol consists

3. Equations of motion for a flexible of complex numbers yielding magnitude and phase
structure with spinning momentum wheels information.
or gyros; Substituting Eq. 3 and 4 into Eq. 1 yields,

4. Procedure used In the digital computer when the normal modes are ortho.-ormalized on the
program for uncoupling and solving the mass matrix,
response equations.
(j~
- ~] i 2CUwoRJ) io}
1. Equations of Motion for Large Flexible . 1T o (5)
Structures (0 (5)o
since
Response analyses of large flexible struc-
tures generally employ component modal substi- - ol0
tution or modal coupling [Ref 1] techniques to [meqJ
reduce the number of equations-required for the when the modes are orthoncrmalized on the wass
solution. These techniques rely on the ortho- matrix. Solving for 4qoj
gonality of the undamped normal modes of the
structure or substructure to produce diagonal
equivalent mass and stiffness matrices. In ad- lqol (I2r21 2 +I r2~c-fl1)
dition, modal damping is usually assumed ao that
the final equations are uncoupled and readily 10]T f (6)
solved. T{o")

In terms of the generalized or modal coor- Substituting Eq. 6 into Eq. 2 yields a ma-
dinates, the uncoupled equations are of the form trix equation relating responses to forcing
functions in discrete coordinates:
ml()+ [rw1 { . 2a ](q) -1
q P 0eq eq
- iF(t)) (1) {Xo}= [41 - I + i 2;. o

where 01 Tfol" (7)

{q} - vector of modal coordinates; Fcr iirgle-unit input forces, Eq. 7 describes
m [4 1 T a 4,j the steady-state transfer function.
[(in]
- generalized mass matrix; The uncoupled set of equations formed by
Eq. 6 and 7 can be solved quite efficiently by
2 1 ,T r1digital computers. The solution involves only
Smeq t [k matrix multiplication and the inversion of a
- generalized stiffness matrix; diagonal complex matrix. As will be shown in
the following
mentum wheels sections, including
fully couples rotating
the modal mo-
equations
4W m 0Tc 0
o eq] of motion, greatly increasing the computer cost
involved in determining responses from this for-
generalized damping mulation of the problem.
T
(F(t)) - {
(f(t))
- generalized force, 2. Equations of Motion for a Spinning Rotor as
a Free Body
and the discrete coordinates, (x), are related
to (q) by the modal coordinate substitution Consider a rigid body spinning about one of
its principal axes of inertia. If the coordinate
(x) - [] {q). (2) system is fixed In the body at the center of
mass and aligned with the principal axes of in-
Now, for a harmonic forcing function, ertia, Euler's moment equations can be expressed
as follows:
+
(f(t)) - fo e t; (3) Io 0 I 0 0 O;

x x y y

130
 mI + - In the previous section, the equations of
yV X X notion in modal coordinatei,'Eq. 6, were un-
z I +6 1y - y 1X (8) coupled and could be-readily solved for re-
sponses. Equation 12, however, is fully coupled
in the velocity coefficients. If one pvocaeds
For an axisymetric rotor (Ix -y = Is) with as in section I ari lets
constant angular velocity X about the z axis, int
Eq. 8 becomes (f(t)) = If t e ;
M. -XI x Ox +
6y ( Iz " 1s) X;
(F(t)) - IFo1 e ift ;

y - -Ox
60 - Ao
; (9) (q~t)) -ilol
z 0. for steady-atate response to a harmonic input,
the solution to Eq. 12 would be in the form
Since the inertias about any axes in the x-y T
plane are the same, the coordinates can be fixed - - 02 + i + []
at the center of mass with the z axis coincident [r10 j onj
with the spin axis and the x and y axes nonrota- (13)
tin&. The angular velocities and accelerations, Cr1 [ ] / fFo}.
6 and Oi' respectively, are then descriptive of The solution then involves invertiog a fully
the coordinate system's notion, while the spin coupled matrix that is also dependent on the
rate, A, relates the body motion to the coor- frequency of the forcing function. Since the
dinate system. inversion must be performed for each forcing
frequency, the determination of responses
from
In matrix form, letting y = I z - Io A, Eq. 13 is very time-consuming and costly. How-
~z ever, these equations can be uncoupled by re-
Eq. 9 becomes writing them in state vector form and generating
complex eigenvalues and eigenvectors. The pro-
1. 0n
00
0
00

L
F 000
( 10) c

4.
t b f

Procedure
e

Used to of
System Equations
ul

Uncouple
Motion the Combined
matrix is in-
The acceleration coafficient
dependent of the spin rate and can be included Rewriting Eq. 12 as
in the system inertia matrix when the momentum
wheels are coupled with the elastic structure. {q) + [B) {) + [E] (q) (F)
( (14)
The velocity coefficient matrix, however, .is
dependent on the spin rate and will couple the where
system modal equations. _
B [ + ]T[] [w;

3. Equations of Motion for a Flexible Structure [E]-

with Spinnip Momentdm Wheels t @
Wriig the equations of motion for the equations of motion can be expressed in terms
irtng teeutosomoonorthe of a state vector by making the olon sub-
combined system in discrete coordinates, stt [ef. 2 a: following

m] (x) + (c] + [r]) fx)+ (k] (x) - (f(t))

q+

where [m) now contains the rotor inertias about -

.'-s nn.-mal to the spin axis, and [F] contains 1
the skew-'ymmetric rotational-dependent terms, -
y, transformed to the system coordinates. JR] - 1--.0,0 j(s
;
A rvird. nate transformation, (x) 10)€(q), E __ !
uz:6 "1v,undamped normal modes yields (15)

30 -1I

(q)+ ] [,IT
+2W [,F]) ;) I

131
The modal equations of notion can then be ex- -
pressed as hC -3 ItoI 17 till

JR) {b + [HI Z} = ID). (16) or

Preaultiplying by -[R] yields =

-o a -170
-iR 3 ] Jol. (24)

!
-(i) -[R]-i [H] {z} -JR]- (D) This formulation of the equations of motion
involves the inversion of a forcing frequency-
or dependent matrix that Is diagonal. If responses t
or transfer functions are to be generated for
JU] (Z) - {7) - (J), (17) more than just a few forcing frequencies, it be-
comes much more efficient to generate the com-
where

[Rh
[0
L-.-
,.0
"'-.-i,
] plex eigenvalues and eigenvectors once and to
invert the diagonal matrix r'o - i0.
than to invert the coupled matrix In Eq. 13 for
each time

11.o.0 -B each forcing frequency.

which must exist, and Substituting back to discrete degrees of
freedom from Eq. 12 and 17 into Eq. 24 yields

-0](H
[UB -B
-0 -------

-JR] - -B -10 (f(t))$

The homogeneous form of Eq. 17 is the standard and, since

formulation for an eigenvalue problem. Now,
letting (Z) = q} €

{z) =( 0 eat, (18) q

the homogeneous form of Eq. 17 then becomes and

(Jul - arl-o.]) {(o - (o), (19) {)-[ q H{)
the discrete coordinate accelerations can be ex-
where the set of values nj are complex eisen- pressed as functions of the discrete forces by
values with corresponding eigenvectors . -I
Assembling the eigenvectors into a matrix xi - ia (j 1f 21 -a - n.]

[Y], the coordinate substitution (

(z - [V] (0} (20) [

into Eq. 17 produces T
- (21) where 172] is the lower half of the matrix
Ju] I) w 1 .j.

Since, for the elgenvector matrix [7, [ -- .

Jl IT] - (v) t ,.J,

premultiplying Eq. 21 by uncouples the

1[]- SAMPLE PROBLEX
system of equations In the form The following example shows how the method

-1 Was applied to include moentum wheels when de-

1mJ (F,]- (.) - [] - (J). (22) termining the frequency response of the Skylab
ATH and associated substructures pictured in
For r harmonic input and steady-state responses Fig. 1. The ATM is the experlment pertion of
Skylab that houses the photographic and tele-
i scopic equipment.
(J) - J Odt;
e
{(J-} 4¢o e (23)

ao that Eq. 22 becomes

132
I
FIGURE 1. ASSEMBLY,SO0ARARRAYS
ATh RACK, DEPLOYMENT

Figure 1 shows the ATM rack, deployment The design of the EPCS depends on transfer
assembly, and ATM solar arrays. The rack is the functions calculated for various points on the
main support, and houses the structure contain- ATM system. The goal of this task was to de-
ing the experiments; in addition, it houses the termine the effect of the spinning CHGs on the
CMGs, which are used to stabilize the entire EPCS transfer functions. Although the cluster
Skylab cluster in orbit. These three MGs, control system, of which the CHGs are a part,
which have a spin rate of 9300 rpm, are the mo- and the EPCS are independent control systems,
mentum wheels considered in this analysis. the elastic motion of the CMG support structure
Figure 2 shows the orientation of each of the will cause the wheel (gyro) to generate a torque
three CMGs to the Skylab cluster, that may be picked up by the EPCS sensors and
cause the system to respond.
Figure 3 shows the ATM spar, canister, and
gimbal ring assembly (GRA). The spar is the Using existing models of the various ATM
structure on which the various experiment pack- substructures for the vibration analysis led to
ages are mounted. The GRA is used to aim the a total structural model with 1321 degrees of
various photoeraphic experimenLs. The GRA is freedom. Two factors associated with this study
controlled by the EPCS, which has motion sen- dictated that the size of this model be reduced:
sors on the spar. first, the computer cost to obtain an eigenvalue

I; - -... ............. ............ ......

TY.
,-/ ,.

eTZ TX+Z

FIGURE 2. CM, SPIN AXES

133
solution to this large a model was excessive, To perform the transfer function analysis
and second, the current transfer function pro- on the ATKA force input points and acceleration
gram could only accommodate 50 modes due to output points were selected on the basis of their
computer storage limitations. In light of these effect on the EPCS. The input points selected
two factors, we decided to use the constrained were the GRA flex actuator torque motors and the
component mode substitution method [Ref. 3], or rotor imbalance moments on the CMGs. The output
inertial modal coupling method, as it is more points selected were the fine sun sensor and the
commonly called, to determine the modal proper- EPCS rate gyros (both of which are on the spar),
ties. To use this method, the ATH system was the flex actuators, and the CMGs.
broken into three substructures, consisting of
the spar and GRA, the ATM rack, and the canis-
ter. For this analysis, the deployment assembly RESULTS AND CONCLUSIONS
and ATh solar arrays are considered part cf the
rack. The results shown in Fig, 4 thru 7 indicate
that the CMGs on the ATh have a distinct effect
By using this method, the eigenvalue solu- on the ATM transfer functions. In general, the
tion for the final modes and frequencies was effect is random: that is, the amplitude of the
performed on a 288-degree-of-freedom system, transfer function for a given input/output com-
rather than on a 1321-degree-of-freedom system. bination may remain unchanged. increase, or de-
These 288 degrees of freedom represent component crease, depending on the frequency range in
modes of the various substructures selected with question, Prcdictably, the greatest effect
a frequency cutoff criterion. The frequency of occurs when either the input or output point is
the modes obtained ranged from 0.009 to 68 Hz. near the CMGs, as can be seun in the figures.
*E--5

Despun CMGs
- I-I Spun-up CMS.
:III.; ; IIIII I
Iii II
-u -_-_=

FREQUENCY
(HZI i FREQUENCY IHZ
FIG. 4.-FINE SUNSENSOR
VS. FLEX ACTUATOR FIG. 6- SPARCENTER V1. CMG

Z-6

FREQULNCY IZ FREQUENCY (1111

5-USCMSO VS.
FI./-FIN. FLEX
ACTUATOR ~~E-6 FIG. - CMNTGVS. CG
E- - - E- 1 4
-i ..
--... .....----- ---

- . UiI/I -4

13

-Rvv ULjJY 6lR RQUENCY

HlZ) (ttzI

FIG, 5- CMGVS, FLEX

-aR
ACTIU 0 FIG. 7- CMIGVS. CMG

134
 Although the results of this study indicate
that including momentum wheels or gyros in the [Ri -
r ..--
I-
j1.0 I
[ = state-vector velocity co-
calculation of transfer functions for structures
containing them can influence the magnitude of
[1.0J 0 efficient matrix
EI
the the
on transfer functions,
magnitude of theirthe
effect
governing
are not clearly
factors [= -----|0 state-vector
cotaefictoi mti paeent
displacement
understood. Limitations on the scope of this 0 -1.0] coefficient matrix
study prevented investigation of some key fac- -
tors, such as the proximity effect exhibited in -[R] [H] - state-vector characteristic
the example and the effect of higher-mode trun- matrix
cation on the validity of the transfer functions [I] - matrix of complex eigen-
near the truncation frequency. vectors
[a J = diagonal matrix of complex
NOMENCLATUREegnaus
- matrx of udanpednormaleigenvalues~f[~
1711 - upper half of [Y)

s[2] - lower half of [I]

[) =matrix of undamped normal Vectors
modes, either generated di-
rectly from a discrete coor-
dinate model or indirectly (q} - vector of normal mode coordinates
through component mode ayn- (F(t)) - vector of generalized forces in normal
thesis mode coordinates
[m) n mass matrix in discrete co- T
ordinates - Ii MW
(ki = stiffness matrix in discrete (f(t)) - vector of discrete forces
[k) - damings matrix in discrete
coordinates (x = vctrodireefceapius
vector of discrete coordinates
[c) - damping matrix in discrete. .fol
- vector of discrete force amplitudes i
coordinates
d aeqo- vector of modal coordinate amplitude/
m diagonal generalized mass phase coefficients
N Jq - atrix
- [1.0J if the normal modes { - vector of discre.e coordinate ampli-
are orthonormalized on the tude/phase coefficients
mass matrix {O] - vector of moments in rotor coordinates
2m - diagonal generalized stiff- (O - vector of rotational degrees of freedom
_ j ness matrix in rotor coordinates
- diagonal generalized damping [ol - v2oW
vector
o of normal mode coordinate gen-
0.q matrix eralized force coefficients
[r] - matrix of assembled rota- Z} - state vector
tion-dependent terms
No. of Rotors

[ (D) = generalized forces in state-vector for-

J -l mulation of the equations of motion

Y0 0 1[T tJ)l -[RJ (D) {2}-

- 0 0 ('*T - complex eigenvector

] -- vector of complex mode generalized

coordinates
[T] = coordinate transformation rt
matrix relating the rotor 1 ol - coefficient matrix in (JI o e
local coordinates (0) to the
system coordinates xl Scalars
[B] - coupled velocity coefficient f = circular frequency of forcing function
matrix for inclusion of mo-
mentum wheels W - undamped natural circular frequency of
_ T th mode

matrix
[E] 0 - generalized stiffness

135
 I -mass inertia about principal axis

e - rotational degree of freedom about axis

j
- constant angular velocity about rotor
spin axis
y = angular momentum

a complex eigenvalue

Symol diagonal matrix

[ I transpose of matrix
-1
- inverse of matrix
[--] - partitioned matrix
- partitioned vector

REFERENCES

1. W. C. Hurty, "Dynamic Analysis of Structural

Systems Using Component Modes," AIAA J.,
Vol. 3, No. 4, pp. 678-685, April 1965.

2. P. W. Likens, "Dynamics and Control of Space

Vehicles," NASA TR 32-1329, Rev. 1, Jet Pro-
pulsion Laboratory, Pasadena, California,
January 15, 1970.

3. W. A. Benfield and R. F. liruda, "Vibration

Analysis of Structures by Component Mode
Substitution," AIAA/ASME lth Structures,
Structural Dynamics, and Materials Confer-
ence, Denver, Colorado, April 22-24, 1970.

ACKNOWLEDGEMENTS

The authors wish to acknowledge the efforts

of Mr. Jack Nichols and Mr. Wayne Ivcy, ShE -
ASTN-ADS, Marshall Space Flight Center, for
their aid in obtaining permission to use the
Skylab ATH modal data obtained under Contract
NAS8-24000.

136

T
 i'a

INTEGRATED DYNAMIC ANALYSIS OF A SPACE

STATION WITH CONTROLLABLE SOLAR ARRAYS

Joseph A. Heinrichs and Alan L. Weinberger

Fairchild Industries, Inc.
Germantown, Maryland

Marvin D. Rhodes
NASA Langley Research Center
Hampton, Virginia

An integrated dynamic analysis and corresponding digital computer simulation

for application to a space station with controllable solar arrays are presented.
The analysis and simulation have been developed for the primary purpose of
evaluating dynamic load Interactions between the solar arrays and the space
station which can result from orbital perturbations of the combined system.
Integrated into the analytical formulation are the dynamics associated with the
space station, the solar array flexibilities and their respective control systems.
Application of the simulation is made utilizing present concepts of a space
station with large area arrays and of typical control systems. A structural
analysis of Ahe flexible solar arrays Is initially required to provide
modal data for the simulation; and analytical results for an array concept are
given.

A verification of the structural dynamic methods used Inthe simulation is

presented. This verification is accomplished by the application of the simu-
lation to a problem of known solution, a uniform beam subjected to a unit step
load applied at mid-span.

INTRODUCTION design results, and the primary array frequencies

may fall within the control system bandwidth.
The solar cell and battery system has been A digital computer simulation for evaluating the
successfully used on many small spacecraft; dynamic interactions of large solar cell arrays
however, space stations of the future will have power and orbiting space stations has been formulated
requirements which are much larger thin those and considers the dynamic characteristics of
within the present design experience of solar cell the array structure and the required systems for
systems. Therefore, the solar cell arrays used attitude and orientation control. The objectives
on future spqce stations must be relatively large of this simulation were to (1) provide an auto-
and be capable of tracking the sun in a manner mated methodology of Interaction loads analysis
that does not restrict the desired space station for use as a design tool, (2) analyze present array
orientation. This Is usually accomplished through structural concepts which are to provide 100 KW
the use of an orientation control system (OCS) for of electrical power to future space stations and
the arrays. A potential problem exists duo to (3) obtain an indication of space station stability
undesirable Interactions between the solar arrays by its real time motions.
and space station caused by required control and
stabilization forces combined with external portur- The equations of motion for an orbiting space
bations. station with attached controllable arrays have been
generated and were digitally programmed for solution
Spacecraft Inmtabilities have been observed by numerical integration techniques. In the develop-
in the past when flexible appendages are par' of the ment of the system's motion equations, a modal
satellite structure. This past experience is sum- synthesization technique was employed whereby the
marized by Likins and Bouvier (1) . Because of the elastic characteristics of the arrays were described
requiremento imposed upon large area solar arrays, by a finite set of orthogcnal cantilever ndes.
a weight-efficient design rather than a stiffness Only rigid body motions of the space station were

137

'p ~~~~"'
~ ~~ ~ I ~ 4 ~4 C~ ~ ~ ~ .k ~ ~ ~ Lt a-C2JSr~s<~
 considered. The rigid body/flexible body Interface
was described by space station acceleration in terms
of Induced inertial loads forcing the flexible arrays;
in like manner the flexible array root forces and
moments acted as forcing functions on the space
station. Structural mode descriptiona of the arrays
were required as Input to the simulation; therefore,
a structural analysis of the elastic system was
synthesis analysis of the array structure was
performed by a stiffness matrix method utilizing
equivalent discrete element structural models
representing 600 Inertial degrees of freedom. The
modes utilized in the simulation were chosen on
the basis of a significant percentage of load
participation in Interface force and moment.
Simulations were performed on the orbiting
t
I
required prior to the performance of this simulation, structural system perturbed by Initial attitude
Provision was made In the simulation for closed errors and external forces representing docking.
loop attitude control system dynamics of the space
station and OCS dynamics for the solar arrays. The To demonstrate the adequacy of the method-
latter control system provided the desired orientation elegy which has been simulated, a problem of
of the arrays with the sun by controlling the rotation known solution was selected --- the uniform free-
about the orbit-adjust and seasonal-adjust ax.. free beam planar response to a unit step load
Outputs of the simulation include interaction forces applied at mid-span. The flexible appendage solar
and moments, magnitudes of all motion variables arrays were represented as cantilever uniform
and control parameters as functions of time. beams, having the first five bending modes as
flexible degrees of freedom. When coupled
The formulated simulation has been applied Inertially with the rigid body translatien mode, the
to an extendible solar array structural concept cantilever mode solution yielded results for freq-
and space station which are presently undergoing ueny and loads which compared favorably with the
separate engineering evaluations. Also, candidate exact free-free beam solution.
array orientation control and space station attitude
control systems have been mathematically described CONSIDERED PARAMIETERS
and digitally programmed for this application. Two The presented analysis and corresponding
attitude control systems were provided for the space simulation Is intended to be applicablo to future
station; they are the reaction jet and control moment space stations with controllable solar arrays such
gyro (CMG) systems. The necessary modal as that shown In Figure 1. Structural concepts

II

Fig. 1. Space Station and Solar Array Structural Concepts

138
2,of arrays and space stations as shown and associated liroviding sun aignment within a specified'time
control systems are presently undergoing separate after leaving tie earth's shadow. In addition,
engineering developments without regard to a total the OCS mus, meet accuracy requirements
system dynamics criteria for minimizing inter- despite experienced space station disturbances
action loads. One objective of the present analysis andf provide minimum dynamic excitation to the
is to assist In the development of dynamics arrays. Two generic types of OCS drive systems
criteria for each of the cormponent structures from have been considered in the simuiation and ares
analytical results of total system characteristics. he continuous and non-linear drive system,s.'
The continuous-type drive system employs either
In order to account for all of the significant a DC torque motor or a variable-frequeny
dynamic influences upon the space station and synchronous motor as its drive element. A
solar array load interactions, the following block diagram of the cortnuous-drive-OCS model
parameters were considered as basic and are contained In the simulation is shown In Figure 3.
accounted for in the simulation
Conhns.aio "4colr

0 Solar array flexible body dynamics in __

terms of generalized modal coordinates.

* Space station rigid body dynamics. nik M

* Space station control system dynamics

including guidance and command.

* Solar array OCS dynamics Including

guidance and command.

It was assumed that the space station 71 - m or lmecon.st

structural frequencies are significantly higher than Kn " r co:fic.,.,
the solar array frequencies and are decoupled from IA- sr(, ,ttis biourotation axis

the control system. The analysis considerations K g..?-&W for lopcorn;timtIon

ti.s vm.,stanM
can be best described by the block diagram shown K - 8E gearr4o
In Figure 2. This illustration shows the breakdown 0, - sSe nfiu
of the basic parameters together with the considered Fig. ., Contiruous Drive OCS Model
counling paths.
The non-linear drive OCS Is similar to the contin-
a Auous drive system with the exception that the
or.ynarl control logic of the non-linear OCS is operated
in an on-off manner. When the array error
exceeds some preselected threshold value, the
motor Ia turned on until the array Is driven to
Control a null position at which point the motor is rwitched
,Sp¢ eIatlomi. off. This threshold value has been made a user
option in the simulation so that its affect upon
loads can be evaluated by parametric studies,

Two generic types of control laws/orquers

J:,,i
Cenerator i i .Solar Array
OrlevvIaln
Driver "
have been Incorporated into the simulation for
attitude cotrol of the space station and are the CM, ,
and the reaction jet control system (ICS). The
Analysis ConsideratIons CMG control system is used for precision attitude
stabilization against eyclic disturbance torques
without the need for propellant exr'enditurc. It
The constraint placed upon the representation consists of three two-degree-of-freedom control
of the array dynamics Is that they bp described in moment gyros with parallel outer gimbals and
terms of their orthogonal cantilever modes. The with their momentum vectors Initially equally
orientation and space station control systems spaced In the orbit plane (Figure 4.). This
that have been Incorporated Into the simulation partict,'ar CNIG configuration permits simple A
are those designed to complement the present steering laws and a planer, rather than three
space station and array configurations. The dimensional, enti-hangup law. The CMG control
primary requirements of an OCS are maintaining dynamics included in the simulation (2] ha? a
a desired accuracy with the sun vector and system frequency of 1.4 IN and a damping ratio

139
 Z Axis C Outer Gkim Spbas
X Aded ot lomon

Odter
Gimbal
Axis
KKsu)
_°
1+
MO Axisl Sc Attitude Anigle
a Commended
0,; - Attitude Artle: late
IVASr K*, Kh Attitude Argle Galn. Rate Gala
Gimbl P v Control LoopDeadbad
Axis
T a RCS Torque L l
I w Sp-e Station Moment of tnertla

Fig. 6. RCS Model

ANALYTICAL FORMULATION
Fig. 4. 3 PM Centrot Moment Gyro Array
Space station and solar array motion equations
were formulated together with the Interactive
of 0. 707. The RCS Is primarily used for dynamics provided by the respective attitude and
reference attitude acquisition maneuvers and the orientation control systems. The method given
momentum desaturation of the CMG system. It by Likino [4] has been used as a basis for the
is also an alternate to the CMG for controlling methodology provided in the simulations. The
attitude of the space station. The RCS is depicted in simulation developed from the employed mathe-
the sketch of Figure 5. It is comprised of 4 matical models has been successfully run on the
sets of quad thrusters providing redundant control CDC 6600 computer. The simulation model
about the pitch, roll and yaw body axes. All utilized for the mathematical system is described
maneuvers using the RCS are performed by firing below.
the thrusters in pairs. The RCS model included
in the simulation is depicted in Figure 6 and * The space station and the two arrays are
the corresponding control equations are given In each modeled as interconnected bodies
Reference 3. with each of the arrays permitted
controlled rotations about the spacecraft
In order to account for space station disturb- attachment points. The allowed axes
arices such as crew motions and docking forces, of rotation consist of those parallel to the
provision has also been made In the simulation to space station roll axis and the array
allow for the application of time dependent forcing vane axis.
functions.
* The flexibility of the solar array Is
modeled by means of a truncated set of
+yau Z cantilever modes which Is excited by
the acceleration of the array support.
+ A difference equation technique [5J is
utilized to obtain the modal response.

* M~aneuver and attitude control of the space

station together with the solar array or-
ientation control are modeled In terms
%, of the transient forces and torques pro-
duced by closed loop guidance equations.
Both the space station attitude and array
* l'itch orientation guidance commands are
computed external to the structural
dynamics section. These provide the
space station with a fixed orientation
relative to orbit coordinates and point
Fig. 5. RCS ,Jet Location/Function the solar arrays at the sun.

140
 The array driver gear train for the. axis The following notation applies
parallel to the roll axis Is modeled as
an ideal mechanical transformer. The J - lndex.of solar array, J is equal
vare driver axis is directly driven, and to 1 or 2
either or both motions about these axes
may be rigidly constrained. FAj TA " qare
the transient forces and
J torques produced by the flexible
9 The simulation orbit generator uses array dynamics
Lyddane's method (6) for near earth
orbits. The generator is included In T - torque exerted on space station
the simulation to provide a reference for by hinged body along constrained
guidarnco commands. A block diagram axes
representation of the simulation program
is presented in Figure 7 where Impor- mT - total system mass
tant logical switches and function Inter- Tr
connection have been clearly delineated. 0 - Newtonian reference point

RIGID BODY AND FLEXIBLE BODY DYNAMICS 0 - Spacecraft reference point

FORMULATIONS B

Salient equations and techniques used in the CS. S. Spacestationcenterofgravity

simulation are shown below. A complete fornu- CG - Center of gravity for the entire
lation c! all equations may be found in Reference 7. system
The space station and array are modeled as a TH
subsystem of Interconnected bodies whose motion CGj - Center of gravity for the J
Is described by the Newton-Euler equations of hinged body
motion. RO Radius vector In ECi
The connected space station and solar array R
bodies In the Earth Centered Inertial (ECI) FR External force applied to space
coordinate frame are show,, In Figure 8. The statioui
following three vector equations are utilized. sai
System Force Li Angular momentum of J array
body abouts Its CG f
2 - L Angular momentum of space
2
mT d (Re+C = + FR (1) station about Its CG

h - Vector from space station

reference point to space station/
array interface
Space Station Moment
-jvector from the space station/
d (( )x 11+ 'xI ri array interface
the jTH to the
array body (onCG of
~--( 0 ) jstation) space

it - vector from point of external

+ T -r FA -r- (2) force anplication to space
Ai J rej ' IA - station CG
- vector from space station
+T +T reference to system CG
AC CNIG
Tl
T - Interface torque on .J array
Jth Array Body Moment body produced by OCS

dt (J d -T - rj x(30._ r' - (fo- h ) force moment arm

(jL) T -rj xFFI F - Interface force exerted by space
I station on jTlI array body

141

. n-
, ~r~~i&r
- ~ Va
~ ~ z.~ 's4
.t .~ ra~n~z n~z ~ MudS~aas 3I~....Eg ai!
 -777"N

d reference frame. This requires the updating of

implies differentiation w. r. t. specific direction cosine matrices during the
t an inertial reference frame simulation. The principal coordinate frames and
. -direction cosine identities utilized are as follows:
T CMG -control torque exerted by the
control moment gyros (or s Lo X where iS the vector
reaction jets) T In the (ECI) coordinates,
IX [ {X5 J Xs Is the vector in space
T 1station coordinates, and
Inittatisalion IXA Cj X I XA is the vector In solar
2 array coordinates.

The direction cosine matrices are calculated in

dit Routine
Ei Rupdated the simulation in terms of Euler angles and are
periodically by the following equation:

IC
Miajor Cycle Furtiona i
o Appendage Equation Update
o space Station Guidance '113 it
o Solar Array Guidance
Intermediate Step 2 0mi
411. 12 i
r 3 Vwherec l, w12, W13 are the rotational rates about
Dynamica Equationa the ith coordinate frame axis. The rigid body
o CNIG/teaction Jets
o Appendage ynamics scalar equations derived from those presented
o Rigid Bodies Dynamics above reduce to the matrix form shown in Figure
9. The submatrfees (Aij represent the linear
Blk 4 i term coefficients, Vol and ol0represent the rigid
Inisgration Packsage 1ody translational and rotational accelerations
respectively and LAI the rigid array rotational
accelerations relatiA to the space station
Fig. 7. Simulation Flow Chart coordinates for the two unconstrained axes of
rotation. The right hand side of the equations
represent the applied forces, torques (control
Space sttion torques Included) and all non-linear terms.
Controlling torque profiles are computed in the
- CG simulation at designated time increments by
0S Space the appropriate control equations and are used to
SeCO staion force the above matrix equations. The array and

r RF space station attitudes are referenced to their

respective coordinate systems and are periodically
"- updated during the simulation.

42.4 ' The flexible array dynamics formulations are

o adapted from the flexible appendage equations
;0 09 developed by Likins [4. Shown in Figure 10
is a sketch of the flexible array geometry utilized
in the analysis. it is assumed that the particle
masses of the array have negligible inertias and
ON deflections are sufficiently small that linear struc-
tural analysis it valid. The force on the ith mass
Fig. 8. Rigid Body System Geometry Is given In Equation b.
d2
F, = in1(1 +1 -

Equations 1-3 were formulated Into a set F

of matrix equations for the facilitation of tare defined in Figure 10.
The terms used
digital computations. The system force and space
station moment equations were formulated with
respect to the space station reference frame and
the array body equations were written In th array

142

V V.

LPVZVL
- t>V-, ~ t.I'V~t4I
- -V
 -- -- -- -- M is the mass matrix

[A] [ 5] [,] [A,3] ,.) K is the symmetric stiffness matrix

Do(3 L is the matrix of rigid body inertial loads
(Reference 4)

i i i0 ri (I B and G ace force coefficient matrices which

are dependent upon rotation rates (Reference 4)
1 2 6 1[ ] 14 ] ! 2 F2
0o3

- (4) The order of the above equations of motions are

3N where N is the number of discrete masses
, comprising the dynamical system. Following the
[A 3 ] [ 7 It 11 [I, I ,A
1 method of Reference 4, It Is of convenience to
LA transform Equation 6 Into orthogonal coordinates
representing cantilever modes of vibration. The
system of equations in orthogonal coordinates cfhn
then be truncated on the basis of some chosen
criteria or engineering experience. This procedure
1 A4
4]
1
i
1 L-
1A 1
4A2
*A21
Z522
F4
F4 permits orders of magnitude reduction in the number
of equations describing array flexibility. The
discrete coordinate equations can be transformed to
normal coordinates by orthogonal transformations
Fig. 9. Rigid Body Matrix Equations prodticed by most automated matrix methods of
structural dynamics analysis. The transformed
equation then becomes

where Il, 7, arc N1X 1 matrices and M is the

hlef. AqI. number of cantilevered modes utilized. Note that
Iling' It""% a modal damping term (2ecr) has been arbitrarily
inserted in the classic manner of structural

analysis.
I'lexillet Arrav
+ 7 The assumption has been made in going from
Pl.~l I 4r Ir Equation 6 to Equation 7 that the motion dependent
i
If- - e'ctot i,.hn of IT t I , matrices which are functions of rigid body rotation
mat particle of flexlii h ass 1'oin rates, are small and have a negligible effect upon
asv,;r pirn. the resulting transformation procedure. Without
corsion o T i t this assumption the simulation would be required
falp pnrice for urml.,rmed to he performed in discrete coupled coordinates
aippervIlsge
- t ior ileflhtlion of iast with resulting manipulations of large order
porticle matrices.

Fig. 10. Flexible Array Geometry The left hand sides of Equations 4 and 7 are
constructed in the simulation from computed
direction cosines, the rigid body inertia tensor,
Substitution of the appropriate direction cosine center of gravity and appendage attachment locations
matrices and consilderailon of the appropriate In the space station coordinate frame and modal
properties resulting from elastic deformation properties of flexible appendages. The latter
gives the following: includes deflection coefficients, frequencies,
damping coefficients and masses, for a chosen
[MNl*q +[K~q -[G~q -[11)4 + 1, (6) nunlber of modes.

where q "
ri21 1
3
2
1
2
2U
2
U1
N Rloth the rigid body andi elastic equations
are solved sequentially employing a finite
N N] difference method. A change to the Integration
U2 U procedure in the simulation is presently being

1
143

4.4
made and incorporates the simultaneous solution exercised independently and the results correlated
of the equations. Excitation of Equation 7 is with known data. At that point, the complete
accomplished by the internal loading acting on each program was checked for continuity and a problem
of the discrete masses resulting from the trans- of known solution was then executed to verify
lational ani.rotational accelerations of the rigid the structural dynamics methodology contained in
bodies. The rigid body equations are in tirn the simulation. The problem ---a free-free beam
forced by the application of the flexible body with zero damping subjected to a concentrated
interaction loads, control system torques and force at mid-span ---was selected because it
external forces. The simulation computes was considered to be a good test for solution
interaction loads from the following definitions, convergence. It also provided information
concerning the accuracy of the analytical approach
F = m Transient and programming techniques. Closed form solu-
J
A 1. 71 force tions for the modal response of free-free beams
subjected to concentrated forces are provided by
T r in O " Transient Leonard (8J. In addition, two other solutions
A [ moment for the response of a free-free beam were obtained
using numerical integration methods. One
solution was obtained by the method provided in
defined by aReference 5 and the other by an independently
derived method using a variable order Adams
F -F + C F integrator. These solutions provided a basis
INTj II
J A
Aj for verification of the simulations.

T +J Itwas necessary to approximate the free-free

TINT = H + j TA (CjJ
JrC beam configuration in the simulation by two canti-
lever beams, attached at their points of constraint.
where
Fw Figure 11 shows a sketch of the cantilever geometry
is the total force exerted by the th and associated coordinates. The "space station"
INTflexible array/hinged body combination mass has been set to zero and the "solar arrays"
on the space station. 0. Symmetric About
TT total moment exerted by the J Ttflexi-
ble array/hinged body combination on x,
the space station. '

FIf , T It are the hinge forces and moments A

*J Jon the jTiI rigid hinged body. _
and
Cj is the appropriate direction cosine matrix Z
(internally coniputed). F)
T(t) - ao
The above completes the outline description
of the formulations and the computations used Fig. 11. Cantilever Beam Simulation of a
in the simulation. The simulation computes all Free-Free Beam
transient variables at specified tine increments
with the option of automatically plotting these represented by cantilever-free beams. An analysis
variables with the Calcomp plotter. As prey- was initially made for system elgenvalues and
lously mentioned, a basic assumption of rigid elgenvectors of the two-cantilever arrangement in
space station dynamics Is made in the mathematical order that comparisons could be made with
dev'elopment. Work I; in progress for modifying corresponding theoretical free-free modal
the present simulation to include spqce station properties. These comparisons were necessary
flexibility in terms of free-free modes of vibration, to determine if cantilever beam modes could he
used to accurately represent free-free beam
SIMULATION VEJITICATION modes. Motion equations for the system of two
connected cantilevers were derived to facilitate
the frequency analysis: these equations are
Verification of the simulation program was presented below.
accomplished in two distinct parts. Each
functional subprogram ---Lyddane's orbit N
generator, CMG space station control, OCS linear A X*(t)+ i i (2 (i
and non-linear solar array drives ---was NA O

144
 MEQ 1 + M 2 derived and are shown below. The subscript "F"
1(1) EQ (a refers to free-free in these equations.

+ () X (t) = 0 1 2 (9) X M(14)

*.Ft)( 2)
2
where MIA

i I, 2,3... N Fi
2#1 it) 0 F
F EQF 2 o (15)
and

MA = Maass of the cantilevered beam 2F40 L

V1 ( ~(t)in the fTtt
ModalL shear force coef. Fl t) 00 12 (1-coss t1) (16)
cantilever mode at mid-span hEQ fl2Fi
of the simulated free-free beam Ft

F0 applied step load at mid-span It should be noted that anti-symmetric modes of the
free-free beam were not excited due to the
X(t) = coordinate for rigid body motion positioning of the disturbance force at mid-span.
hEQ = generalized mass of the ITil Modal data for both the free-free and canti-
I cantilever mode lever uniform beams were obtained from stzndArd
reference tables (9] in order to compare the
o = iT1 cantilever mode natural cantilever and free-free beam formulations.
frequency (uncoupled) Numerical data assumed for this comparison were:
MA = 5,0 slugs, Fo = 0. 5 lb. I 1 = 12. 566 rad/sec.
(t) = generalized modal coordinate The evaluation of the numerical coefficients of
for the ITI1 cantilever mode Equations 8-16 was based upon a 25 point
discretized mass representation of both the canti-
= coupled frequency of O.bratior. lever and free-free beamt. The frequencies
of the system, approxlmates the obtained by an orthogonal coordinate transformation
iTil free-free frequency analysis of Equations 8 and 9, for the cantilever
beams, are given in Table 1. These frequencies,
The solutions of Equations 8 and 9, using only along with corresponding free-free beam
the fundamental cantilever mode, are given by the frequencies have been normalized with respect to
following: the frequency of the fundamental free-free beam
mode. The rate of convergence in this frequency
F2 comparison is demonstrated by the successive
X (t) " - + _ cos g t (10) number of modes used. Similarly, the degree of
A A ( A Q1 I
Vcorrelation and convergence in the modal
~amplitude domain can be seer. in Figure 12
which is based upon use of three cantilever modes.
F The comparlson, show that the cantilever beam
-2 formulation can be used to accurately descrbr
(AMEQ -l ) free-free beam modal properties If a sufficient
numiber of cantilever modes are usedl.

VIF Although motion histories of the flexible

o (1-cos Qt)(12, appendages were of Interest, the primary
I p2 (MA I FQ 2 tijectlive of the slmulation was to obtain Inter-
1 actio. loads between the space station and solar
array fo desln evaluation. Therefore, for
purposes of :',rification with known solutions
where I A N5,
EQ 1 cmae 8] shearh foi hae histories
ifr atbandfo
the one-quarter
h s
ft - -21
I
ca) 13,span
(1) beam locatlotm
compires the shear hior were obtained. Figure 13 use
"tainel Arom the

To A - ) of one cantilever mode It'he simulation with the

shear calculated using the n."dal acceleration
method and one free-free mode. Excitation at
Similarly, the corresponding analytical the base of the cantilever beams b. a zero nass
solutions for the perturbed free-free beanm were space station posed no problem for ti, simulation.

145
 T'ABLE 1
F'requency Comparlisan of (Uniform Beam) Cantllor~r + Rigid Body Mode Representation
of a Free-Free Uniform Beam
Frequency Ratios: f ,
n I Jrei-free

Reference Feeqiency Ratios Calcilated Coupled JM1ede Frequerncy Ratios

I Mode 2 'Mode 3 Mode- 4 Mode 5 Mode

Uncoupled Canti- Canti- Canti- Cant[- Canti-
Symmet ric Free- Canti- lover lever lever lever lever

NorFe Beam 11~nm

oe Rigid
em+Body Ba Body
Rigid Rigid.
em+ Body Ba Body
Rigid en Body
Rigid

1 1.000 0.632 1.0101 1. 0006 1. 0003 1.0000 1.0000

2 5.404 3.958 --. 5.548 5.420 5. 401 5. 411

3 13.344 11.0114 1--- -13. 7*t 13.410 12. 367

4 24.814 21.652 --- -- 25.584 24.965

5 3,q.812 36.861 -- - ----- 41'.010

1.03

Note Obse CalculaW.Us.ingMcdal Dipac fl I.*..

-. 33r 3. 0341 at (i

'L4o
Sm.0.t abyrn-rI ______.__._ _1_If__1

.6-ree "s

3. ..
FilM. 13Mise313t
Timel.V
.. . 0

ZdSymomerc Mod. Figt. 13. Com parison of Cantilever & Free-Free

1.0 Sm. shut C eam Shear tit 114 Sr~aai vs. Time for n Unit Step
Force~ Applied at riMI-Spart.

Figure 13 ase contrastr thr, modal sr-cratIon Anx]

mods~i di.%place-ncnt methiods o)f computing totol
.6 . 4 0 3.31 loadr. The calculation of force hiatcrv co:-rep-
pondi-ig to the Modal diaplaecmrent Method was
-. 11 !>a Ptat . mzari'-?Indtl ndcntly of the oyrautatlon. at The
-1.6on-quarter .'p21, the' ditferevcet In loadt front
the two tnds were not ovignttica.-A. With
Fig. 12. Com~parison of Mode Shapes for a1 respect to computing total (quasi-dtcady plus
Frce-Free Uniform Bean) and a 3 Cantilever dynimir! ioads, Bimpiinghoff (1io,Pointe out
Mode Approximation the dert uilitv of the modal accelerr Pion metho'I

146
 which Is the method used in the simulation. .3
The comparisons given by Figure 13 show an .2
excellent agreement between the simulation results .1 . 2 .0 .06
and those obtained by an independent method
0 I.r
utilizing the varlable order Adams numerical -. 1 . ... I-
Integration technique. .3

Figure 14 presents a comparison of simu-

I. ~ ~~lation
results using five cantilever modes with ?7 -- ,os,
independently calculated results using free-free ,ith

modes. The low frequency waveform and magni- NI]2 Ro.. ts.uise frst rl,,c,-tlvr wed"
tude in both histories is seen to be in good agree- .. S

Scontent ment. between

The
the degree ofsmall histories
thedifferences
convergence are attributable
in the theto
high frequency
In approximating h.' :",. .l , ,. . .
Tie I.lase

given number of free-free modes by the same

number of cantilever modes. -,4

Variable Order AdamsNumericul Integration Reults VaiI the r

Frat fo mntc Fr
rn ree- r.. Beem Modes
._.2. -. 1 (I/4, t)- -M (i/4) 9 (1- S. V (IA) Yl (0

' ~ ~ ~ ~ l 06f/o,
.08 .,o fIt.,, .,6 .1, .10 .11
'14-.6-e *Fig. 15. Uniform Beam Comparisons of Shear
... __ . istory @1/4 Span for a Unit Step Force Applied
..4 at Mid-Span

" A mode-by-mode comparison of the maxima

shear force components derived from the simu-
lation and those given in Reference 8 is presented In
in Figure 16. Again, the differences Indicated

simlaltion Ietutt. teg firr ite Carfiler .ode. are attributable to the degree of convergence in
approximatltug the given number of free-free modes
by the same number of cantilever modes.
l08 .o 0. 12. 14. 16. 18. 20. 22.,
The above presented comparisons between
Time.e cantilever an free-free beam response re-
suits demonstrated that the structural dynamics
methodology contained in the simulation was
sufficient for the accurate evaluation of Inter-
action loads.
-. 8 . I t -Il-M 11/4) 9(1) - Z V 41/4, I )

Fig. 14. Uniform Bean Comparisons of Shear Maximum her at Span x. Nlde
listories @ 1/4 Span for a Unit Step Force Applied 1 Mro
at Mid-Span icw00 Ilekrence 8
90 Figure 9

80 Sim latlon
Figure 15 presents a comparison of simulation antlr
C70 Aroximaion
reflults using five cantilever modes with results
obtained using tho Nastran "Direct Transient 40
Itesponse Method" [5]. The free-free .
30
beam representation was discretized Into 40
masse; for use i this method. Modal truncation 20
was not considered in this method; each of the o
0 2 3
F4 5
discr2te masses was allowed two Inertial degrees
of freedom co.'responding to planar bending of l(ilHfd-Dlmly) symmetric hlode Number. I
the beam. In general, good agreement exists
between the frequency content of the two shear Fig. 16. Comparison of Modal Shear Force
histories. Also, the maximum shear force Participation for a Uniform Beam Subjected
given by the hitorles is in excellent agreement. to a Unit Slop Load Appli.d at Mid-Span

1147
i'
•4
 TYPICAL SIMULATION RESULTS 3 degrees of freedom. The stiffness of the array
membrane was considered as a function of applied
An extensive analysis of the parameters that tension loading, and the central boom was modeled
influence the structural dynamics of solar arrays as a beam column. A detailed description of the
is planned. This analysis will form the basis of structural modeling and resultant modal data is
a generalized dynamic design criterion for solar reported In Reference 3. Selected frequencies are
array structures. Ithough this analysis has not listed in Table 2 in terms of in-plane and out-of-
been completed, some preliminary results have plane modes.
been obtained.

t
The selections of modes to be used In the
The solar array/space station conceot being simulation was made on the basis of those contri-
evaluated is shown In Figure 17. The space station biting a large percentage of load participation.
is a rather stiff structure (modeled as a rigid Load participation for symmetric modes is
body) which contains 96. 5%of the total mass. evaluated by calculating the shear at the array
The solar array is a large flexible structure attachment point due to a base translational
composed of membrane strips stretched between acceleration. The participation is equal to
the Inner and outer structural support members. (Z mi 0i) 2 /Rn Zmi where mI is the discrete mass,
An extendible boom in the center of each vane 0i is the modal deflection coefficient and Rn
applies a tension load to the membrane strips. is the modal mass. Load participation for anti-
The solar cells and associated interconnects are symmetric modes is evaluated by calculating
cemented to the membrane substrate and generate .the moment at the array attachment point due to
the power required for space station operations, a base rotational acceleration.2 This participation
This array was designed Iy the Lockheed Company function is equal to (Zmo i ri) AZn mr12
under Contract NAS9-11039 and is reported In where r[ is the distance from the vane axis of
Reference 11. the array to the mass point. A typical mode shape
seais shown in Figure 18 for an out-of-plane anti-
Outter Strxtursl SRtdst) symmetric mode. The load participation factor
Support for this mode was 67. 7%. The results of the
. 2! analysis also Indicate that modes with a significant
load participation ;all within the frequency band-
width of the space station attitude control
system. Therefore, coupling of the attitude
.. control system with array modes can be expected.
'y 7

Selected results from one simulation are

M ram presented and correspond to the configuration of
@triph Figure 17 with small Initial attitude errors. The
Inner Structural 1104" s space station was perturbed by a force history
srepresenting the docking of an orbiting module. The
disturbance force is directed along the X axis
of the space station. The continuous OCS drive
Solar Arrays Fully Deployed
Sollup system was used In this simulation for array
(Flexible Body) control.
|
'4(10,000 PT Array WetSs - 7,890 Ib.
StationWeight -710,090 Tb.
Space Figure 19 is a graphical series of results
space Station Inrtias from this simulation. The docking load profile
and resulting roli axis acceleration, is given in
1, 2.4x 10 UK_-Fj.a Figure 19a. It Is seen from Figure 19a that
I, " 5.4,iX I s lug-Pt.:
2
I, • s.s x 1o sut.rt. the magnitude of feedback force from the flexible
arrays upon the space station is small because of
Fig. 17. Space Station/Solar Array Concevt the non-oscillatory nature of the acceleration history.
for Dynamic Interactions Study (Rollup Flexible Figure 191) shows the Initial array orientation errors
Array) with respect to the sun vector as a function of
time. Figure 19c shows the time variations of
modal acceleration for two array modes - an
A structural analysis was performed on these out-of-plane mode forced primarilly by OCS
solar arrays prior to simulation to obtain required torque and an in-plane bending mode forced
modal data. The matrix method of structural primarily by space station acceleration. Space
analysis considered in Reference 12 was utilized, station control torques were not initiated during
The array was modeled using a discrete mass this simulation since the angular dead-band
technique with 200 mass points, each mass having contained within the reaction jet control system

148
 TABLE 2
Frequencies and Modal Participation Factors of Selected Roilup Array Modes
Out of Plane
_______ Symmetric Anti symmetric

Frequency Percent Frequency Percent

(Hz) Priiton(Hiz) Participation

6 .0734 50.4 6 .0557 67.7

36 .203 12.5 31 .162 5.4
56 .287 3.5 57 .318 7.3
62 .340 7.7

In-Plane
Symmetric Antisymmetric

Mode Frequency Percent Mode Frequency Percent

(Hiz) Participation (Hlz) Participation
2 2.26 57.0 1 .291 55.0
2 1.94 3.1
7 2.71 6.5
8 4.12 2.7

2500

2000 Applied Docking Force and Space Station

500 Acceleration istories (Holt1Axls Direction)

L 1 1h<'

1000Foc

Force

1000
I~~

- Accelerationx

- 500

0 0.5 .0 1.5 4 6 0 10
Timne (Second@)
"NC IFig. 19 (a). Simulation Rlesults for an Externally
Applied Docking Force

was not exceedled. Figure l~ci presents the timc

variation of interaction force along the space
stati-m X axis; and Interaction moment sbout the
space station V axis. liiitia! high frequency tran-
Freq~vocy - 0.0557 Ha sient loadhi are seen to decay rapidly. This Is
X2 Particiption - 67.7% caused by a relatively high value (0. 05) of the
Inpuit modall damping parameter. These limited
oo results which arc presented show the type of design
X3 data that can he obltained from this dligital siniu-
lation of the integrated dynamic analyis. A
Fig. 18. Out-of-Plane Antisymnietric Motion complete set of (data, corzrespondling to variations

149
in initial parameters and basic structural data, interaction Force ad Moment Histories
will provide the basis for ihe derivition of a 200
otructural design critetion.
.1 Solar Array Anglar Error Hitorlos

Seasoal Adjust Axis

.05 asOrbit Adjust Axle~

-
0

-. 05 - --

02 4 6 10 10

Fig. 19. (d)

Time (Seconds)

Fig. 19. (b)

Modal Coordifte Acceleration Histories CONCLUSIONS )

le Mio- c e An Integrated dynamic analysis method has
(Acceleration x 10) been developed and implemented in a digital
computer program for eimulating the structural
0 dynamic interactions between a space station and
controllable solar arrays. Simulation verification
lot Antlaymetric In-Plan studies demonstrated that the analytical formulation
Bending Mode and the modal synthesization technique employed
provide an accurate method for evaluation of
dynamic Interactions. In addition, the verification
studies showed the programming of the almulation
S-2 to be correct.

-3L

0 2 4 6 a 1O
Time (Secors)

Fig. 19. (c)

150
REFERENCES 7. "Structural Interaction Simulation System",
Technical Report R104, Revision 1, Wolf
i. P. W. Likins and H. K. Bouvier, "Attitude Research and Development Corporation, River-
Control of Non-rigid Spacecraft," J. Astronautics dale, Maryland.
and Aeronautics, May 1971/Vol. 9 No. 5 Pg.
64-71. 8. R. W. Leonard, "On Solutions for the Transient
Response of Beams", NASA Technical Report
2. "Preliminary Synthesis and Simulation of the R-21, 1959.
Selected CMG Attitude Control System," General
Electric'Report EL-506-D, 5 March 1970, 9. D. Young and R. P. Felwar. Jr., "Tables of
General Electric Company, Binghamton, New Characteristic Functions Representing Normal
York. Modes of-Vibrition of A Beam," University of
Texas Publication No. 4913, 1 July 19)49.
3. "Interim Report, The Study of Dynamic Inter-
actions of Solar Arrays with Space Stations and 10. R. L. Bisplinghoff and H. Ashley, Principles of
Development of Array Structural Requirements" Aeroelasticity, pp. 344-350. John Wiley and
Fairchild Industries Report 8581R-1, February Sons, Inc., 1962.
1971, Fairchild Industries, Germantown,
Maryland. 11. "Evaluation of Space Station Solar Array
Technology and Recommended Advanced Develop-
4. P. W. Likins, "Dynamics and Control of ment Program,", First Topical Report LMSC-
Flexible Space Vehicles" Jet Propulsion Labor- A981486, Lockheed Missiles & Space Company,
atory Technical Report 32-1329, Revision 1, December 1970.
Januay 15, 1970.
12. "Stardyne User's Manual", Mechanics Research
5. "The NASTIRAN Theoretical Manual", NASA Inc., Document, Los Angeles, California,
SP-221, Section 11. 3, 1970, Office of Technology January 1971.
Utilization, NASA, Washington, D. C. %

6. R. If. Lyddane, "Small Eccentricities or

Inclinations in the Brouwer Theory of the
Artificial Satellite," Astronomical Journal,
Volume 68, No. 8, October 1963, p. 555,

DISCUSSION

Mr. Zudans (Franklin Institute): When you indi- Mr. Mains (Washington University): You said at
cated on one of these diagrams the docking force in- one step that you were using a 600 degree-of-freedom
troduced into the system, how was it introduced rel- system for analysis. Is that correct?
ative to the mass-center of the system? You seem to
be getting the moments, but there were no indications Mr. Weinberger: That is correct.
on the slide as to how this was done.

Mr. Weinberger: In this particular case the Mr. Mains: Did you use a direct integration
docking force was applied at the aft end of the space technique to get those response curves that you
station along the axis of the space station. There showed?
were some out-of-plane motions. These motions
were due primarily to a slight misalignment of the
solar arrays that were active with a linear control
system. This means that even 0.1 of a degree atti- Mr. Weinberger: Yes we did.
tude error would cause some motion of the solar ar-
rays, and hence some torques Into the system. But
the docking force was applied along the axis of the Mr. Mains: How do you have any handle on the
space station. There was no eccentric force or load- meaning, the reliability, of a solution of that size for
Ing. this kind of problem?

151
 DISCUSSION

Mr. Weinberger: The 600 degrees of freedom Mr. Weinberger: We are examining this problem
simply referred to the structural analysis model that in connection -with the tension loads that are trans-
was developed. From the structural analysis model, nritted through the boom and so on. This is one of the
the generalized mass and modal coordinates were areas in which we have made certain assumptions in
used for the modes that were selected. In this case linearizing. The validity of these assumptions has not
we used 12 modes, so the 12 elastic modes were been established at this point, although we do have
used for the solar arrays plus the 6 rigid body de- some preliminary test data from Lockheed on the ar-
grees of freedom for the space station. We did not ray of this type. This was a Lockheed array geometry
have a 600 degrees of freedom model for direct into- that we were studing. From what we have been able to
gration. ascertain, we have chosen the model In such a way
that the results agree fairly well with the test data.
XrMjps Did you then do an elgenvalue solu- We get z correlation in that respect. This is the only
tion on the 600 degrees of freedom? way that we can have any confid ence in the lineariz-
ation of the array.
Mr. Weinberger: Yes. We did It to get the fre-
quencies and the mode shapes and the generalized
masses. Mr. Clevensen (Langley liesearch Center): If I
understood you correctly, you used NASTRAN for
Mr. Mains How do you have any handle on the verifying some of your results?
reliability of an elgenvalue solution of that size?
Mr. Weinberger: That's right.
Mr. Wr¢nberger: I think the reliability of struc-
tural models of that size is fairly well documented, Mr. Clevensen: Why could not you have used
for example, in the NASTRAN program and users NASTRAN exclusively and saved considerable work?
manual.
Mr. Weinberger: Of course one of the problems
Mr. Mains: You might be surprised if you would that we looked at was the coupling of the control sys-
check the orthogonality of the vectors sometime. tem. I am not familiar with the NASTRAN program
with regard to the demap instructions and the auxil-
iary useage of NASTRAN, other than for structural
Mr. Zudans: I would like to comment more on and vibration analysis. I assume that you might be
these questions, because 600 degrees of freedom dy- able to code subroutines which represent the orient-
namically today Is nothing. You can handle 3,000 and ation control system and the rigid body mechanics.
there is a perfcct orthogonality. The NASTRAN pro- One thing that comes to mind immediately is the size
gram and many other programs use the invrse iter- of NASTRAN. We have been able to modularize this
ation routine with spectral shapes and it is. very, very program in such a way that it uses much less digital
reliable. However, I wanted to ask a question. How computer time in core than the NASTRAN program
did you account for obvious nonlinear atti'.udes in would use. We felt it was more efficient in develop-
your solar array? It is such a flexible st,ucture that ing our own program, rather than to resort to the
it could not be handled as a linear one. NASTRAN program.

152
 PARAMETRICALLY EXCITED COLUMN WITH HYSTERETIC MATERIAL PROPERTIES

. T. Mozer
IBM Corporation
East Fishkill, New York

and
R. M. Evan-Iwanowski, Professor
Syracuse University
Syracuse, New York

An investigation is performed to determine the effects of hysteretic material l.

properties on-the dynamic response of parametrically excited systems. Expres-
sions are developed for stress distributions, forces and moments for a-para-
metrically excited column. Column response is detemined for-the stationary
case with-additional assumptions on material properties and stress distributions.

INTRODUCTION

This paper is related to the description, Pit) P0 + P1 (t) come(t)

simulation and means of minimizing the effects
of mechanical vibration on technical hardware by
means of a theoretical investigation of the ef-
fects of hysteretic material properties on the
response of parametrically excited systems. I
If an initially straight column, Fig. 1, is
subjected to a periodic axial load, P(t) = Po + u(x,t)
PI(t) cose(t), and if the maximum amplitude of
the load is less than that of the static buck-
ling (Euler) load, PE, then the column experi- x
ences only longitudinal vibrations. There are,
however, certain relationships between the dis-
turbing frequency 6(t) = v(t) and the natural
frequency n of transverse vibrations of the col-
umn for which the straight column becomes dynam-
ically unstable and lateral vibrations occur. u
For a sufficiently small value of Pl(t) one such
relationship between b(t) and n is 0 - 2n. The
response which occurs under these conditions is Fig. 1 - Column Configuration
called parametric resonance. This phenomenon of
parametric resonance may be expressed more gen-
erally as v(t)/2l = 1/.n where m = 1,2,3,-.. obeys Hooke's law. It is well known that few
solids actually obey Hooke's law very closely.
For increasing values of P1 (t)the points Of these few solids, many are used in structural
of instability mentioned above cover instability members. The principal reasons for their use
regions. That is, there exist continuous ranges are: (1) they display nearly Hookean behavior
of the values of v(t) for which the column will to a relatively high stress level, (2) over-all
be unstable, in (v/24, P1/2(PE-Po)) space. structure cost, (3)over-all structure weight.
The largest of these instability regions, In many instances, however, it is desired
and thus the most important region, Is the first to have machine or structural members perform
region characterized by v = 2o. Consequently multiple functions. For example it may be de-
this paper is concerned with the nature of vi- sirable to have a column perform both as a col-
bratory motion of the column related to this umn and as an electrical or thermal conductor.
region. In this case the selection of the material will
Our discussion thus far has implied, among be based on some compromise between its load
other things, that the material of the column carrying capabilities and its electrical or

153
thermal conductivity. Such situations may pre-
sent the necessity to use materials whose me-
chanical properties are non-Hookean over-all but
the smallest stress levels. Once we leave the
domain of the Hookean solid elastic column, a
large field opens up even to the most casual ob-
server. The number and complexity of the mater-
ial properties, along with their relatively
crude rheological models becomes overwhelming if -co
one attempts to apply them with rigor to the
problem of the parametrically excited column, co c
(an exhaustive discussion is given in Ref. 1).
Several authors have considered material
properties other than the simple Hookean case in
the analysis of the arametrically excited col-
umn. K. K. Stevens [1), for example, solves for
the cases of the Maxwell Element and the Three-
Parameter Model as material properties for the Fig. 2 - DaVidenkov's Model
stationary case. The case of a simple viscous
damper in parallel-with a spring (Voigt Element) cross section at any time will be developed from
is shown in references treated for stationary the stress distributions. On other instances
and nonstationary modes. Later K. K. Stevens expressions for the temporal part of the motion
and R. M. Evan-lwanows'd [2] introduced the corn- of the column are derived.
plex modulus material property representation to
the stationary response of the parametrically Analysis of the system configuration yields
excited column. It is important to note here the partial differential equation
that in this analysis energy dissipation occurs
only due to the bending of the column and not a2 t 2 t a2 U
due to its axial compression. V. V. Bolotin (3] x JR) + Pit)2u+m...ZO (1.2)
considers amplitude dependent damping forces due a xa x a t-
to viscous and dry friction at an end support of
the column, but these forces do not arise from where M is the moment at any cross section and
the material properties of the column itself. M = fA a y d A. We have made the usual Bernoul-
Detailed discussion is also given by Mozer (4]. li-Euler assumptions, and the deflection u is re-
garded as small.
This paper deals with determination of re-
gions of stability and instability as well as The strain dt any point is
the lateral amplitude response of the axially 22 U
excited column whose material properties are of C = C + Y2 c + y K = E + y 2 (1.3)
the pointed hysteretic loop variety. The Davx- a X
denkov model is used to represent this material
property. So far the treatment is for any material proper-

1. BASIC RELATIONSHIPS ty.

Consider now the stresses in the column
Davidenkov (5) developed relations to rep- whose material property is such that it is char-
resent the behavior of metals in the form acterized by a pointed hysteresis loop. ..e know
Snalready that in the principal region of insta-
a E(c t n (o + C)n - o
cnol (1.1) bility the frequency of lateral response of the
Ccolumn ishalf that of the excitation frequency.
Thus the period of the bending stresses will be
These relations depend only on the amplitude of twice the period of che periodic portion of the
the strain. It is graphically represented in axial stresses. In addition these two stresses
Fig. 2. will in general be out of phase. A typical
stress strain diagram for a fiber at some dis-
The Davidenkov expressions have been used tance from the neutral axis is presented in
extensively by many authors, e.g. [6] to solve Fig. 3.
several classes of problems in which the rater-
ial is assumed to have hysteresis loops of the In Fig. 3 the small loop is due to the per-
form represented by (l.1). iodic axial load and the larger loop is due to
the slower periodic bending. The period of the
Stress Distributions in Parametrically Excited total loop is the same as the period of lateral
Column. motion of the column. The loop contains four
distinct branches. If the bending is reduced
It is the objective of this section to de- the picture is altered to look somewhat like
velop stress or strain distributions for the that shown in Fig. 4. If the bending is re-
parametrically excited column. In some instan- moved completely, the two loops converge into a
ces the expression for the moment on any column single two branch loop, the period of which is

154

-j
the same as the period of P(t).
0

AlI

Fig. 3 - Stress strain in a fiber Fig. 5 - Stress strain in a cross

of a column with hyster- section of a column with
etic material. Large hysteretic mterial.
moment.
The stresses on a cross section of the col-
umn are such that at certain times three branch-
es of the four branches of the loop are present.
Thus for a material having a pointed hysteresis
loop, we require several equations to represent
the stress across the column at different times.
Each equation contains two or three terms cor-
responding to the different branches of the hy-
steresis loop present in a cross section.
The use of the Davidenkov relations in the
case of the parametrically excited column where
axial stresses are taken into account is in gen-
- eral not possible since the constants in the re-
4 lations will not always allow the loop to close
after a complete cycle of the column. However,
if one assumes that the strains due to bending
and the strains due to the axial load are in
phase, then it is possible to write equations
for a four branch loop based on the Davidenkov
relations. The evaluation of the constants in
the resultant expressions is difficult since
they depend on the distance from the neutral
axis, y.

2. RESPONSE OF COLUMN WITH POINTED

LOOP MATERIAL HYSTERESIS
We make the additional assumption that the
Fig. 4 - Stress strain in a fiber axial stresses are small and will not contribute
of a column with hysteretic appreciably to the column response. This assump-
material. Small moment. tion is a fairly good one for the slender col-
um. Under the assumptions made, moment acting
The stress strain distribution at a cross on a cross-section is
section of the column will be somewhat like that a2u
shown in Fig. 5. M E I32u+ 2 () (2.1)
3 X

155
E<< 1Eand
where iswhere
a small parameter such that E > 0,
her)
(lnd 2 i
2n-l(a u)n yn+l d A (2.7)
(a,
X) x2 max
aX
is the dissipation functional depending only on Comparing (2.7) with (2.1) we deduce that
the curvature and material constants. The ar-
rows above t represent the branches correspond- E f [( _U a2 u n -

ing to increasing and decreasing curvature. n [(- u) .

Substituting (2.1) into (1.2), we get A x max ax
4.t
E I " u + Z -I-" (; 2 ) +- " 2n-l (12u n )yn+l d A (2.8)
ax ax ax ax max
+ P(t) + 0 ( ) Substituting the value for u from (2.3) in-
a x2 t2 get differentiating twice with respect
to X, we and
to (2.8)
We seek a solution to equation (2.2) of the form

u(x,t) - f(t) Sin (2.3) L-~~~~~a

- E
2 = f._ n 2
nl in2I
ax A L
Substituting these values into (2.2) we get
- n Sin n [(f(t)m 4 f(t))n -
E 12-4Sn
Z-E(t) + Z -2-2 (3 2 3] "
-

a x2 . 2n-I (.l)n f(t)n ]} yn . y d m(

A (2.9)

- P(t)f(t)!gin x+ m i(t) Sin tx- 0 (2.4) where fm(t) is the maximum value of f(t). It
can be shown that the expression + Ey/n{ )yn(
Inorder to make any headway toward solving in (2.9) is always odd about y = 0 regardless of
the value of n. Thus the total integrand of
(2.4) we need to know more about the functional (2.9) is always even. Due to this property we
,. We rewrite the equation for Davidenkov's may integrate the above expression from zero to
-model h/2 and multiply by two instead of integrating
from -h/2 to +h/2. This removes the necessity
-[(co
- )n 2n-I - on)) (2.5) to keep track of the signs of y and c. Thus
n4 h/2 2
__2____ -
n'
and the definition for the moment a X2 ,= ; 2 EYiW"0 2 L2 n [(n-1) Sinn
". ydA ax2 L
and also noting that .nSin n ix (I)n+2 [(-f(t)m f(t)n

a2 u
Y -y . 2n-l (.
1 )n f(t)nm)) y
yn+l d
dy
ax
in the case where the axial strains are neglec- where W indicates the width of the column. In-
tegrating the above, we obtain
ted. We may now substitute the above value for
c from (2.5) and substitute the resulting ex-
pression for the stress into M we get a2 2 E W n+44 2
C 322 2 WnIT ( [(n-1) Sinn n -
ax 1. 1?
2
E 2y ~ U[,2U +
a x, d A + x2 max - n Sinn x [(-f(t)m ;f)) n
- 2

+ 3 u)n
2
yn+l d A (2.6) f(t)nmn] (h)n n (2.10)
ax a X max
We consider the special case where n 2
2 2
=E I -u -:EX [(,2 2U
)M +. u n
a x 2 n a x max2 - =2
ax

156
 4form

3.2 -Ey W W6 h4' 2

1rXx- 2)
C 2 6 {[l 2 Sin--- + n2 f t [f2 t 2ff .m+
2 + Z2n 2 u Cos o f (2.14)
E(f(t) + 2 f(t) fit) + f(t)-

2 m 6
- f(t)m]} (2.1l) where Es 96L mn
96
Equation (2.11) represents the distributed lat- and the term 2n2p has been replaced by i2n2 p.
eral load due to dissipative material properties. Note that if (2.14) were expanded we would get
We now substitute (2.11) into (2.4) and apply the term + cB 2fmf. Itwould appear at first
Galerkin's method by multiplying the resultant glance thit the coefficient + Z02fm could be
by sin vx/L and integrating, we get combined with n2 on the left hand side of (2.14).
It must be noted, however, that this term is ac-
, L .2 L tually a variable coefficient of f due to the al-
2El 24L f(t) - 2 P(t) f(t) Z- + ternating signs and we are thus justified in
L 1 leaving it on the right hand side of the equa-
+ 2m)it! TEy Wh4s r5 . [ f(t)2 tion. We represent (2.14) in the form
48 L f + n2 f = ;F (f,o,r) (2.15)

]
i2f(t)2 f(t) + f(t) 0 (2.12) where

Remembering that P(t) +

P0 P, cos and f F m
.2 + 2f12 lpfCos 0
E L 2We seek a solution to (2.15) in the form
is Eulers' buckling load and fit) =_f = aft) Cos (0 + (t)) (2.16)

w2 El
L'2V( We have assumed analysis of the first instabil-
ity region only by using the ang,,lar displace-
is the transverse natural frequency of the col- ment term o/2 in (2.16). The terms a(t) and
umn without axial load and the transverse natu- t(t) are to be determined from the usual rela-
ral frequency of the column loaded by Po is tions,
a = EA1 (r,a, ) (2.17)

1E 41v(t) + cBl(Tpa,, ) (2.18)

We also denote the loading parameter v as After some calculation we find:

P1 (t) E yW h 6 (4a 2 _a2 )

2
= (E - P) E" 9 L6 m 2n a
Substituting these relationships in (2.12) and
rearranging, we get,E
y W h - P
aCos2 (2.20)
+ 2(1-2 Cos O) f Ey96WL6h4S
m . 96 L6 ma
Stationary Response
[f 2 !2 f- 2
f 0 (2.13) We now proceed to determine the stationary
response of the column by investigating (2.19)
Equation (2.13) represents the temporal equation and (2.20) in a special case. Stationary re-
of the motion of the column, and is the subject sponse is defined as the case where no system
of the following asymptotic solution, parameter changes with time. Explicitly we
write v(t) = constant, u(t) - constant which
Asymptotic Solution implies that a = 00. Under these conditions
(2.19) and (2.20) become
Only the first approximation will be sought
in this analysis. We rearrange (2.13) in the

157
 : 2The region of stability is distinguished from
T 2 S_0 (2 the region of instability by the curve repre-
4 2
-ITO( am -a ) - Sin 2¢u.O (2.21) sented by
V (2.28)
. n B~nam n-- Cos 2p- 0 (2.22) a=n - ).¢v

where called "stability curve". The stable and un-

EyWh4 W6 stable branches of the phase angle ' are found
S L6 (2.23) to be
96 L m (n-V)/d < 0 (2.29)

Eliminating € from these equations while noting

I that am a a constant for the stationary case, Hence the solution stable for v ; 20 and d/dv >
Sweget the following 0, otherwise it is unstable.
2
a 2r2 p {2- [(2n- v) 3. RESULTS AND CONCLUSIONS
81(4 + 9) Discussion of Material Properties and Stress
2
2) 04 U )p)
2 Distributions.
42 2 (2.4)The significance of the harmonic axial load
on column response is demonstrated. Stress dis-
The t sign denotes the possibility of two solu- tributions in the column are also illustrated
tions for a. for the pointed loop hysteretic material.

We may now determine the stationary phbse Stationary response curv for the parame-
angle o from (2.21) a using the value of a from trically excited column whose mwiterial may be
(2.24) represented by the Davidenkov Model with the
special assumption that the axial stresses are
1
[ negligible are plotted in Figs. 6-10. Fig. 6
= Sin- [2A - v and Fig. 7 show the effect of varying the load
2nP(4 f2 + 9) parameter P on the column response, Fig.8 shows
the effect of varying the Davidenkov model pa-
[(2 -f-V2 2 -4
2
V 2+ 9 rameter y, while Fig. 9 shows the change of the
) - V)2 - first instability region with amplitude of re-
sponse. Fig. 10 shows the instability region in
three dimensions.
- P-t] ] (2.25) It is seen from Fig. 6 and Fig. 8 that the
point defined by da/dv = 0 separates the region
for which non-trivial stable solution can exist
The so-called "backbone curve" of the sta- to the right and the region for which no non-
tionary response is that curve lying halfway be- trivial stable solutions can exist to the left,
tween the two values of a obtained from (2.24). n, u, P0 and y being constant. This means that
It is no non-trivial solutions exist for y/2aless than
2___f?(2ov - Eywh4V 6 that at which da/dv = 0. This result has not
a, Z = E wh 6 been frequently observed in the literature, al-
si(4n2+9) 96L m though it can be shown that the response curve
of a parametrically excited column having non-
The boundaries of instability zone are: linear damping of the form dff 2 where d is the
nonlinear damping coefficient and structural
V [111 2p] nonlinear elasticity does close at the backbone
curve in a similar manner to the present case.
determination of
steps in we
known bnownhes,
Applying nwellsable It is further noted that as long as y is posi-
,saly oteratn heof- tive the backbone curve leans toward decreasing
stable or unstable branches, we obtain the fol- .frequencies, characterizing soft systems.
lowing: For d ao/dv > 0 the solution is stable (Negative y implies that the material property
if is such that it generates energy over a cycle).
Pisarenko [6] arrives at dynamic response curves
8, ao for various problems Using the Davidenkov rela-
- V 2 - 4 - > 0 (2.26) tions which have also soft characteristics, but
beyond this, comparison of results loses most of

and for d ao/dv < 0 the solution is stable if its meaning since Pisarenko analyzes systems
near dynamic resonance, and the present work is
a, ao 2 concerned with systems near parametric reson-
2a - v - 2 - 4 V3 <0 (2.27) ance.

158
 It is seen from Fig. 9 that the curve de- second frequency to the response of the model
fining the region of instability for zero ampli- does not undully complic2tesubsequent use in
tude of vibration is the same as that for the the governing equations then such work would
perfectly elastic case, The authors believe constitute a valuable 6ontribution. It may also
that this is due to the fact that the axial be possible to incorporate into.this model'the
stresses were considered small. The coupling of nonlinear elastic effects without difficulty.
the bending and the axial stress probably would Such nonlinear elasticity could be obtained from
have provided a more significant dissipation experimental data on the maximum point locus
term for a 1 0 and thus the curve would have curve.
been shifted slightly to the right from u - 0;
v/2n - 1 and would have been rounded somewhat, REFERENCES
similar to the case where linear velocity de-
pendent damping is included. In the case con- 1. K. K. Stevens, "On the Parametric Excitation
sdered the energy dissipation per cycle is pro- of a Viscoelast'ic Column," A.I.A.A, Journal,
portional to y a3 and so the curves in Fig. 9 Vol. 4, No. 12, Dec. 1966.
defining the region of instability shift to the
right and become rounded with increasing ampli- 2. K. K. Stevens, R. M. Evan-Iwanowski, "Para-
tude. metric Resonance of Viscoelastic Columns,"
•The downward shift in the curves defining int. J. Solids Structures, Vol. 5, pp. 755-
765, 1969.
the region of instability with increasing ampli-
tude may be due to the nature of the maximum 3. V. V. Bolotin, The Dynamic Stability of
point locus curve for the Davidenkov model: Elastic Systems, Holden-Day, Inc., San
Francisco, London, Amterdam, 1964.

tEco n 01 4. D. T. Mozer, "Parametrically Excited Column

with Dissipative Material Properties," M.S.
This equation contains a nonlinear component Thesis, Syracuse University, Syracuse, N.Y.,
proportional to co2 which is symmetric, due to 1969.
the alternati:g sign. Lazan [7J indicates that
the maximum point locus curve may be an appro- 5. N. N. Davidenkov, "Energy Dissipation in Vi-
priate one to use for the calculation of stored brations," J.-Tech. Phys., 8, No. 6, p. 483,
energy in material. In our case this "nonlin- 1938.
ear elastic" term is probably the most appropri-
ate one to use in explaining the nonlinear char- 6. G. S. Pisarenko, Dissipation of Energy in
acter of the column response. From Fig. 9 one Mechanical Vibrations, (inRussian), Kiev,
can further note that in the present case re- Izd-Vo Akad Nauk, USSR, 1962.
gions of instability exist for large values of a
which do not exist in the linear elastic damped 7. B. J. Lazan, Damping of Materials and Mem-
case. For example, in Fig. 9 let our system be bers in Structural Mechanics, Pergamon
described by state v/2a - 0.9 and u = 0.13 (for Press, Inc., 1968.
the elastic case, our system is stable). Let
the system now be perturbed such that ah/L > 8. T. J. Mentel, C. C. Fu, "Analytical Formula-
.0013. Under these conditions the system finds tion of Damped Stress-Strain Relations Based
itself in a region of instability and a large on Experimental Data with Applications to
amplitude results. Vibrating Structures," ASD Technical Report,
p. 61-63, 1961.
One must approach the problem of selecting
a mathematical model to represent the real ma- LIST OF SYMBOLS
terial of the parametrically excited column (or
in fact any vibratory system) with extreme cau- A Area of cross section of column
tion. If one attempts to use a strictly linear E Young's Modulus
mathematical relationship to represent a materi- h Depth of column in y direction
al whose characteristics are essentially nonlin- I Moment of inertia about Z axis
ear, large errors may arise in the prediction of K Curvature I/
system vibratory response even though care is L Length of column
taken
is the tosame
see asthat
the energy dissipationseeperMentel
cycle m Mass per unit length
real material; r..o Material constants
and Fu (8], for example. R Euler's Load
P t) Axial load Po + PI(t) Cos 0(t)
Biot's linear hysteretic model is perhaps Po Static compressive load
the best linear hysteretic model to extend for PI(t) Amplitude of dynamic load
use with multi-frequency excitations as encoun- W Width of column
tered in the parametrically excited column or x Axial coordinate end
plate. It possesses the characteristic of amp- y Distance from neutral axis.
litude depe;odent damping found in most engin- Z Coordinate
eering materials and it results in a well de- c, Lo Total normal strain, Maximum strain
fined mathematical problem at least for the Ca Normal strain due to Po 4 P1 Cos 0
single frequency case. If the addition of a Small positive quantity

159
 Radius of curvature C1f~j

O4 Anua dslcmetla

All Frqunc .1.0;,(

LoaOraet /(P P

FTig. Effect of Varying on Amplitude

8 '

oftVayn
6 - Efftaictt Fig. -n-mltd

Fig. - Cangeof
Intabiity i icreasing

Fig. 7 -Effect of Varying uon Phase Shift -p

Fig, 1 l0 - Three Dimrensional Representa2tion of

instability Region

160
 DISCUSSION

Mr. Zudans (Franklin Inetitute): You surprised Hertz. It was in this general frequency range that we
me with very poor orthogonality. Was that mainly be- were concerned. We had a number of modes which,
cause of the use of experimental modes wihich hadnot through increased growth of the spacecraft, had grad-
been orthogonalized before their usage? ually crept down into the pogo frequency range. We
were also concerned with some of the large amplitude
Mr. Stahle: The orthogonality referred to the modes like the first longitudinal which actually went
abbreviated model test. The criterion that we had set as high as 55 Hertz.
up was that the measured experimental mode would
check within 10 percent -no' the analytical modes. The Mr. Schrantz (Comsat Labs): Did you couple
problem is the very limited amount of instrumenta- your model with the Thor Delta to check out the
tion used on the solar array panels. I think our main responses?
confidenco was gained from the fact that the frequen-
cies matched up very well, and that the main struc-
tural modes agreed relative to the modal admittance
through the base shear. Mr. Stable: This is done by Douglas personnel.
The model that we have been using is the model that
Mr. Prause (Battelle Institute): What are some I presented here. Essentially it Is a modal model us-
of the important frequencies? We saw a Ioc of natural ing modal coupling techniques to marry the spacecraft
frequencies in the presentation but what rre the con- back to the launch vehicle. It follows the basic iner-
trol system frequencles and what are the pogo fre- tial coupling procedures of component synthesis dis-
quencies for this type of space stations? cussed in the literature to couple this analytical rep-
resentation of the spacecraft back to the luanch ve-
Mr. Stable: This is the Earth Resources Tech- hicle.
nology Satellite which is a fairly small, 2,000 pound,
space craft going up on the Thor Delta. The main po- *The paper was presented and discussed by C. V.
go frequency varies somewhere between 17 and 23 Stahle for the authors.

161
 DYNAMIC INTERACTION BETWEEN

VIBRATING CONVEYORS AND SUPPORTING STRUCTURE

Mario Paz
Professor, Civil Engineering Department
University of Louisville
Louisville, Kentucky
and
Oscar Mathis
Design Engineer, Rex Chainbelt Inc.
Louisville, Kentucky

The dynamic analysis of the conveyor-structure system is pre-

sented using the stiffness method and an iterative scheme in
which the structure and the vibrating conveyor are analyzed
successively taking into account interacting effects. An
example of a structural truss supporting a vibrating conveyor
is given.

INTRODUCTION the supporting structure is rigid, the

dynamic analysis of the conveyor is
Vibrating conveyors are widely performed. Then the structure is ana-
used in industry for conveying granular- lyzed under the effect of the reactive
type material (1]. Depending on spe- forces at the points of support of the
cific applications, these conveyors conveyor. From this analysis the mo-
could weigh from a few hundred pounds tion of the points of support is deter-
to several tons. The main components mined and used as boundary conditions
are the pan or trough where the mate- in repeating the analysis of the convey-
rial is conveyed, a driving mechanism or. This iterative scheme cintinues,
which produces rectilinear motion at an and the new reactive forces are applied
angle with regard to the horizontal di- in repeating the analysis of the struc-
rection, and a system of springs which ture, and after a few cycles, the final
connects the trough to a base, absorb- results are reached.
ers, or boosters, depending on the type
of conveyor (2). ANALYSIS
Usually the conveyors are support- The vibrating conveyor and the
ed to a rigid foundation through isola- supporting structure are analyzed inde-
tion springs, but, in some cases, they pendently by the dynamic stiffness meth-
are suspended from or supported by a od where the system is divided into a
structural system. When vibrating con- number of members, each having known
veyors are supported by a frame or elastic and inertial properties.
truss, the dynamic forces which are
transmitted to the supporting structure Dynamic forces and moments at the
result in vibration of and interaction ends of each member are expressed in
between the structure and the conveyors, terms of the displacements and rotations
at the ends, giving the so-called ecl-
The dynamic analysis of the con- ment dynamic stiffness matrix [31.
veyor-structure system is presented Basically, this result can be achieved
using an iterativr scheme where the by solving the appropriate equation of
three dimensional structure and the two motion. The nystem dynamic stiffness
dimensional conveyor are analyzed suc- matrix is assembled from element matri-
cessively taking into account inter- ces using the conditions of continuity
acting effects. First, by assuming t' between elements and equilibrium at the

Preceding page blank 163

joints; the latter will include any The drive unit houses two motors, one
applied external force and the forces above the other, with double extended
from all the elements forming the shafts to which eccentric masses are
joint. Equilibrium conditions at the affixed. The motors rotate in opposite
joints result in n equations relating directions producing a horizontdl hat-
the applied external forces to the monic force.
independent joint displacements of the
system. The n equations may be written The dynamic boosters are tuned at
as a natural frequency in the neighborhood
of the operating frequency of the mo-
[S]{uJ = {FJ tors. The boosters function as dynamic
(1) absorbers (4) for the component of the
impressed force along the direction of
[S] is an nxn symmetric matrix com- the boosters. As a consequence of this
posed of terms derived from the dynamic absorbing action, the trough vibrates
stiffness matrices of the component harmonically along a direction approxi-
elements of the system,(uJis a vector mately normal to the orientation of the
of the n independent joint displace- boosters.
ment in the system, and{F is a vector
of the external forces which are excit- In the analysis of the conveyor
ing the structure at frequency w. the boosters are treated as rigid
bodies connected elastically to the
Vibrating conveyors are manufac- trough. The trough is assumed to be
tured in a variety of types, ranging composed of continuously connected
from a single moving deck to three or beams with distributed and concentrated
more vibrating masses. In this pre- masses. The drive unit as well as the
sentation the so-called "dynamic boost- connecting leaf springs is considered
er conveyor" is described in relation to be a special beam element, and the
to the dynamic problem originated by isolation springs are treated as mass-
the interaction between vibrating con- less elastic members.
veyors and aupporting structures. As
shown in Fig. 1, the dynamic booster Diagrams for the basic elements
conveyor consists of a trough supported of the booster conveyor are shown in
by isolation springs, a drive unit con- Figs. 2 through 4, and the correspond-
nected through leaf springs to the ing dynamic stiffness matrices are given
trough, and a series of spri ng-mass in the appendix. The coordinates indi-
assemblies known as dynamic boosters. cating joint displacements at the ends

A A A A A
C B B a B B

Oe G G G( G NG

x
'/ D GLOBAL

COORDINATES

Fig.l - Dynamic booster cenveyor

164
of the elements are numbered consecu- In general, any member of the
tively. truss, although assumed to be ideally
pin connected at its ends, may undergo,
under the action of inertial forces,
El flexural deformation in each of the two
A 5 principal planes in addition to the
2 A extensional deformations along the Ion-
mgitudinal axis. It is assumed that for
each individual element of the system
-1 4 these three deformations are uncoupled;
thus, the dynamic stiffness matrix for
the element of the truss is obtained
L independently for the two flexural de-
formations in each of the principal
planes and for the axial deformations.
The dynamic stiffness matrix for a
uniform pin-jointed bar element shown
in Fig.5 is obtained by solving the
5 corresponding Bernoulli-Euler differen-
tial equation for flexural deformation
and the wave equation for the axial and
by introducing the appropriate boundary
conditions. The dynamic stiffness
6 matrix
is givenforinthe pin-jointed bar element
the appendix.
2 5

ELATOER -4
1~

9 L

Fig.3 - Dynamic booster element Fig.5 - Pin-jointed bar element

EXAMPLE
A dynamic booster conveyor support-
5 ed by a truss type bridge between two
64 buildings is presented to illustrate
the interaction analysis.
Fig. 6 shows the schematic diagram
2- | of the space truss supporting the con-
3veyor. As explained above, the convey-
or is analyzed initially under the
action of the driving force and the
condition of zero displacements at the
supporting points. Then, the reactive
Fig.4 -Isolation spring element forces are applied in performing the
analysis of the truss to obtain the
first approximation for the displace-
SUPPORTING STRUCTURE ments at the points of support of the
conveyor. In the next cycle these
installations of conveyor support- dynamic displacements are imposed as
insionaltres anr complex three external actions on the conveyor. The
di m inple
dinbui
rfuefls or
a ndyst omtr asim
sses to exs
ums. m-- f irst four cycles of the interacting
supporting structure
ple building floor systems. A space effects between are shownandinthe
the conveyor
truss is used in the interaction exam- Table I.
ple presented.

165
 Ph, -------- 7: ,-A j -, -t- 4-1--
*I

BWOTTO DIAGONALS

Plan View

BOOSTERSRO-

SPIG "TOOL ' Ix STAGGERED

SPRIN To,/

' 7 /2' 2 End View

8 Spaces at 30-240"

Elevation View

Fig.6 - Example space truss supporting vibrating conveyor

TABLE I
Interaction Results
(a) Amplitude of truss vibration at conveyor suports fin.)
Suppot Support 2 support suport4
Cycle Supo 3
-t - up
1 y lorizontal
0 Vertical
0 Horizontal
0 Vertical
0 - Horizontal
0 Vertical
0 Horizontal
0
Vertical
0
2 -0.0077 -0.0370 0.0022 -0.0830 0.0082 -0.0998 0.0389 -0.0480
3 -0.0365 -0.0710 -0.0198 -0.1560 -0.0045 -0.1890 0.0600 -0.0770
4 -0.0390 -0.0730 -0.0200 -0.i590 -0.0053 -0.1940 0.0620 -0.0800
5 -0.0390 -0.0730 -0.0200 -0.1580 -0.0052 -0.1940 0.0620 -0.0800
(b) Amplitude of conveyor reactive forces at supports (lbs.)
Support I
Cycle iorizontal Support 2 Support 3 Support 4
Vertical Horizontal Vertical Horizontal Vertical Rorizonta Vertical
1 339 -172 203 -T6 175 -33 139
2 153 -123 162 -111 178 -62 204
3 133 -132 153 -110 173 -48 237
4 136 -132 153 -110 173 j -48 239

166

*'4
NOTES ON COMPUTATIONAL METHOD CONCLUSIONS
A computer program in Fortran IV An iterative method for the analy-
is developed for the analysis of the sis for vibrating conveyors mounted on
space truss, the dynamic booster con- supporting structure has been presented.
veyor, and the initerative procedure The two systems are analyzed separately
as described in this paper. The flow using as boundary conditions the de-
diagram of the computer program follows. flections and forces developed at the
points where the conveyor is supported
Flow Diagram by the structure.
Machine-Structure Interaction
Several testing problems as well
as actual cases of installation of con-
START veyors supported by structural systems
were analyzed by the iterative method.
1(l) READ AND PRINT INPUT DATA This study of the interaction of a vi-
brating conveyor and the supporting
2) SET DEFLECTION AT CONVEYOR structure indicates that the method
SUPPORT POINTS EQUAL TO ZERO presented requires four to eight cycles
4to converge to the final solution.
(3) LOOP I = 1 CALL
CONVEYOR ANALYSIS PROGRAM APPENDIX
(4) SET REACTION ON TRUSS EQUAL The dynamic stiffness matrices for
TO FORCES DETERMINED IN (3) the basic elements of the booster con-
iveyor and for the pin-jointed bar ele-
(5) CALL TRUSS ANALYSIS PROGRAM] ment of the truss shown in Figs. 2
TO DETERMINE DEFLECTION through 5 may be written as follows:
AT CONVEYOR SUPPORTS
Ia.- Isolation Spring
1 (6) IS THERE A SIGNIFICANT CHANGE
IN DEFLECTIONS AT CONVEYOR SUPPORTSI Kt 0 -Kv -Kt 0 Kv
NO YES 0 Ka 0 0 -Ka 0
vSET NEW -K 0 K Kv 0 KR/2
DEFLECTION VALUES
-K t 0 Kv Kt 0 K
rCALL CONVEYOR PROGRAM TO t v t v
IDETERMINE MOTION AND FORCES 0 -K 0 -Ka 0
a
CALL TRUSS POGRAM TO -K 0 Kv 0 KR
R
DETERMINE MOTION AND FORCES
IPRINT OUTPUT MOTION AND FORCES
~where:
STOP
Ka = Axial spring constant

The prescription of imposed displace- Kv = 1/2 KtL

ments at the points of conveyor support
may be programmed for a digital com- KR = 1/3 KtL2
puter such that there results a reduc-
tion in the total number of equations L = Length of the spring
to be solved. In order to avoid re-
arrangement of computer storage, how- Kt = Transverse Spring Constant
ever, it is often more convenient to
proceed with the direct solution by
implementing a computatiunal device due
to Payne and Iron and referred to by
Zienkiewicz(5). The diagonal coefficient
of the dynamic stiffness matrix cor-
responding to the equation with imposed
displacement is multiplied by a very
large number (say 1021); at the same
time the corresponding force term of
this equation is replaced by the product
of the nevly formed diagonal element and
the prescribed displacement.

167
 A

b.- Conveyor Beam Element

E Nb iw2
o DaQ-Mit2
Symmetric
o DS DP-J.W 2
-RA0UDaSaHsa
-EABUb 0 0 EABNb-afW 2
o "Da (S+Hsa) D (Ca-Hca) 0 Da2Q-mIW2
-he: Da (Hca-Ca) D(Ha-Sa) 0.aSaHsa DP-Jf 2
where:

E = Modulus of elasticity
Ha = eosh(aL)
A = Cross-sectional area
Hs sinh(aL)
B =(m2/EA)
D -Ea/(l-Ca ca)
w Angular velocity of forcing
frequency S - SIN(BL)
b
m Mass per unit length =
Cb COS(BL)
L a Length of beam
Ub - COSEC (EL)
Nb = COT(BL) Q aHsa+Salca
M i - Concentrated mass at left end of
beam P Salca-CaHsa
Mf = Concentrated mass
at right end of
a = (mw/EI) 1/4 beam
C s= Concentrated
mass moment of
I Cross-sectional moment of inertia inertia at left end of beam
a = COS(aL)
Jf Concentrated mass moment
of
Sa = SIN(aL) inertia at right end of beam

c.- Booster Element

C2 K c +21 M2
s-

SC(Kc-Ks) SSKe+CCKsmW2
Symmetric
-Cx c - i K, (1+e/l 2)-j
sSC(K-K) CK 1C 2
Cse+Sc
K+S2X
SC(K-K)(SKcC 2 s)S1cl
SC(K _- (SI +C KCS SC(K-Ks) S2 Kc.C2Ks
0 0
sc "ejc/12
c s 0
c(cK8)S 0+I s e2 / 2
where:
C = coso
m = mass of booster
S = sinO
j= mass moment of inertia
about
Kc = Spring modulus-compression Point 0 (iq. 3)
ESpring moduluse.
s =Sprig
moulu-she~r ,= , 0 (See Fig. 3)

18
d.- Pin-jointed Bar REFERENCES

(1) Paz, M. - Conveying Speed of

A1 0 0 A2 0 0 Vibrating Equipment
Publication No. 64-WA/MH-I -
0 Q(Z) 0 0 P(Z) 0 ASME 1965

0 0 Q(Y) 0 0 PY) (2) Hinkle, T. Rolland - Design of

Machines
A2 0 0 Al 0 0 Prentice-Hall, Inc. New Jersey
1957
0 P(Z) 0 0 Q(Z) 0
(3) Przemieniecki, J. S. - Theory of
0 0 P(Y) 0 0 Q(Y) Matrix-Structural Analysis
McGraw-Hill, New York 1968

where: (4) DenHartog, J. P. - Mechanical

Vibratioas
Al f EA~cot0L McGraw-Hill Book Company, Inc.
1956
A2 -EAcosec8L 1
(5) Zienkiewiez, 0. C. and Cheung,Y.K.
a1EIr The Finite Element Method in
Q(r) = -- r (cOtrL-cotha rL) Structural and Continuous Mechanics
McGraw-Hill Company, London 1967
acEI
P(r) - (coeecharL-CosecrL)
2
r - (r -y, Z)
Ur r DISCUSSION

S= M-- 2Mr. Zudans (Franklin Institute): I do not think I

AE understand your model. My impression is that dy-
namically you only considered the conveyer and that
NOMENCLATURE the truss was taken as a simple static structurewith-
out any dynamics considered In the process of analy-
A = Area sis. Is that correct?

E = Modulus of elasticity
L, l,e,h = dimension Mr. Paz: No, sir. The forces coming from the
conveyer to the truss were dynamic forces and the
G = Modulus of elasticity in truss was analyzed as a dynamic problem. The truss
shear members had distributed mass. The paper shows the
dynamic stiffness matrix for a truss which, although
I,yIy, I z = Cross-sectional moment of it is pin-connected at the ends, still has bending due
inertia to the inertia effect.
JiJf = Mass moment of inertia Mr. Zudans: Did you have your truss represent-
with dynamic
systemdegrees degrees of
J'Jxk = Polar moment of inertia
kfreedom and notmass
ed as a lumped only static of freedom?
kaikc,k s = Spring constant
Mr. Paz: Actually this was done by the co-author,
Mimi'm j = Mass but this is not the case presented here. This is the
distributed mass case It has a finite number of de-
grees of freedom because of the masses of the ma-
0 = Angle trix method of structural analysis. But the equation to
determine the dynamic stiffness for each element
w = Angular velocity takes Into account the distributed mass and elasticity.

169
 RESPONSE OF A SIMPLY SUPPORTED CIRCULAR PLATE

EXPOSED TO THERMAL AND PRESSURE LOADING

J.E.Koch
North Eastern Research Associates, Upper Montclair,N.J.
and
M.L.Cohen
North Eastern Research Associates, Upper Montclair,N.J.,and
Stevens Institute of Technology, Hoboken,N.J.

Using classical plate theory, equations are derived for

the response of a simply supported circular plate exposed to
time dependent pressure and thermal loading. Results based
on these equations are presented for several loading condi-
tions and plate'geometries. One interesting effect is the
presence of thermally induced vibrations having rather sub-
stantial amplitude in some cases. In addition it was ob-
served that the- thermal stresses may be compressive at both
the front and rear surfaces, and tensile in the center.

INTRODUCTION are time dependent but assumed not to

vary in the in-plane directions. Regard-
There has been, in the recent past, ing the heat conduction problem as un-
a sustained interest in the consequences coupled, the temperature distribution
of exposing a structure to time-varying and history are computed using a finite
thermal as well as mechanical loading difference technique.
conditions. One may readily envision
situations in which this type of loading Results computed on the basis of
might occur. For example, in the envi- these equations are presented for
ronment of a nuclear event, a period of several loading conditions and plate
rapid and intense heating would soon be geometries. One interesting result is
followed by a shock wave. Whereas the that in some cases the rapid heating of
effect of a shock wave on various struc- such a plate by exposure to thermal
tures has recieved considerable study, radiation induces vibrations of substan-
both analytical as well as experimental, tial amplitude compared to the quasi-
the effect of rapid heating has been static displacements. Another result of
givin less attention, the computations is the observation that
the thermal stresses may be compressive
The object of this study is to on both the front and rear surfaces and
determine the response induced in a tensile at the center. In effect there
circular plate simply supported on its appear to be two stress-free surfaces.
boundary, when exposed to such loading
conditions. The regimes in which the GENERAL FORMULATION
dynamic effect and the thermal loading
become important may then be identified. We wish to determine the displace-
ments in a simply supported circular
The analysis is based on the plate subjected to a uniform time-
thermbelastic equation of motion for dependent pressure p(t) and a time-
flexure of a circular plate subjected to dependent temperature field T(z,t) which
time dependent lateral loads and temper- varies depth-wise through the plate. The
ature distribution. Classical modal problem shall be formulated along clas-
series methods are used together with sical plate-theory lines wherein, briefly,
the Mindlin-Goodman procedure for the displacement w must satisfy the well
treating time dependent boundary condi- known [I] thermoelastic-dynamic plate
tions. The elastic properties have been equation together with the appropriate
assumed to be uniform and not temperature boundary and initial conditions.
dependent. The applied pressure loading
and the temperatures within the plate

Preceding page blank 171

I
 kv 117P- ,

Furthermore, we impose the condition

that the resulting displacements and
4 stresses be finite at r-0, the center of
the plate.
Since no generality is lost if the
effects of arbitrary initial conditions
" are not included, we choose the following
conditions to apply at t-O for all values
- of r:
T W=0

DYNAMIC DISPLACEMENTS
We seek a solution to the governing
equations together with the boundary and
initial conditions in the following
general form
t= j,)T* a,, ,.()
%4 M21 6 (7)
Figure 1 - Typical Circular Plate
in which Rm(r) are the normal functions
arising from the free vibration problem,
Consider the thin circular plate of i.e.;the homogeneous solution. gl(r)
thickness h and radius a, as shown in and g2 (r) are to be selected to satisfy
Fig.l, with its median plane in the r-B the inhomogeneous boundary conditions.
plane and with z denoting the distance fl(t) and f (t)represent the time
from this plane. Within this plate the dependency 8f the boundary conditions (2].
displacement w must satisfy the govern-
ing differential equation 1. Free Vibrations

( The displacements of a plate vibrat-

-a r () ing freely satisy the homogeneous form of
where e -- the governing equation (3).

A 1 Choosing a product solution of the form

NMT = ) T* (2) we•?Rr) e
Since the temperatures as well as tbe
pressures are independent of r and l, we find that the normal function Rn(r)
this must be true of the thermal param- is composed of Bessel functions of the
eter MT as well. Thus we may say that first and second kinds [3,41:
T:
VIMT=O in the above equation and omit
derivatives with respect to) :

,)r- rdr where:,, 4.-Dycr,

For the simply supported case we require 4 ff%
that the moment Mr and the displacement w (9)
vanish on the boundary of the plate.
Thus, on r=a for all t>0: The condition of finiteness of displace-
/- t)a)0 (4) ments at the center of the plate (at r=0)
requires that the coefficients Cn and Dn
r& r )] 0r vanish. Applying the homogeneous form
Or .of boundary conditions (4)and (5) we
or, alternatively: obtain the normal function
d
,- (5) (10)
and the frequency ecuation

172
 ,7

The terms on the right side of this equa-

' ,t4) ',) tion are expressed in infinite series
2-. (11) form:
I-LV
c( being the successive eigenvalues of £
thys equation. .
_____________________
2. Orthogonality Of Normal Modes
-~t fjc ,(4) ",,-) (16)

The orthogonality condition among R.-C')

Ra(r)
normal modes may be deduced from Clebsch's q
thereom E5], stated below for the rth These series expressions may be sub-
and sth distinct normal modes: stituted into equation (15) to obtain:

(12), -G.)~ R Q,(+

In 'e case of a circular plate, if u, v where: Q() P(0t m(t) -4 (t)
and w are the radial, circumferential, Equating the coefficients of Rm on the
and lateral displacements, respectively, left and right sides of this equation, we
then we may write find that Tm must satisfy

from which the solution for Tm is readily

Ws,;: Rs e- it found to be

(As -~ o57 - r Boundary Conditions

- r= 0 The technique of Mindlin and

Goodman [21 is used to treat the time
Substituting these displacements into dependent boundary conditions which appear
equation (12) we obtain the desired in this case in connection with the ther-
condition of orthogonality among normal mal loading. This is done by selecting
modes: the arbitrary functions gl(r) and g2(r)
in such a way as to transform them into
6 -
R, Rj'rd 0- = (13) homogeneous conditions.
4L S Substituting equation (70 into the
boundary conditions (4)and (5) yields,
3. Response To Thermal And Mechanical at r=a:

Making use of the homogeneous solution

and the orthogonality conditions we may j- f&)
now turn to the response of the plate to .0
the actual time dependent thermal and
mechanical loads. Substituting into the
governing equation of motion (3) the
modal series representation of the dis- Noting that the normal function Rn
obtain: w given by equation
placement (7), we satisfieithe homogeneous boundary
-
obtain: conditionh: these equations are equiva-
lent to the following set of conditions
SR(14) on gl and g2 at r=a:
-Pa~ -
•-_-DEV"',,(. fkl ,t 3z;' atI
/

From the homogeneous equation (8) we see ( I

that DV7R, - pA'q9,'?m . Making use of or
this to eliminate V , equation (14) . r-
becomes: r 0

173
 Initial Conditions
Making use of the assumed dis- (Ihd, i .. 1
placement form (7) we find that the -1 4))o/'L' T Tr
initial conditions (6) at t-0, for all -,
values of r, become O&

(19)
0 Thus the coeficient may be computed
Te R-'+
according to the equation
The second term of each of these equations
may be expanded in infinite series form,
thereby simplifying these equations: Yr -)fk

Similarly

Thus i we note from equation (17) that ,) "14 10

"[drJ"i
"- ~ TM(0)-A m and ar,,,/,l-'=-., and substitute WoL _,Jr,
0A".
equatioiis (20) into (19) we may ej-ulre. i1 , , ,
to
the zero A and N: of Rm toobtain-
valuestheofcoefficientsm &17 61 R-.rT , ]
42

Am =Em (22)
S* where
Bm EmIf-II L i' L
' r.
The formal solution for w may now be -- IL
written: Finally, using the actual values p=p(t),
9
f =0, f = "Ir/(,-4), gl=l, and g =r0 A(,,L,)
/ (d.)1"z.) I,)
. m) aAd carrying out the integratigns these
become:

,¢ .. . ,-

-'?' t-1. (21)

an Evaluation Of Coefficients E' EM (23)

and Qm, etc. , P 50-0

The coefficients contained within - _n 14 T(o)

equation (21) may be computed in detail DT,-O)3,)
D.0=
by applying the orthogonality condition where
(13) to the infinite series expansions
(16) and (19). For example, from the K,K I ',/
]rka') -_ J(rL)r
Jr
first of equations (16) the expression
for p is

to both sides of this equation, inte--

grating with respect to r and making use

of the orthogonality condition: J(- 2 -• "1}

[7121 Or i J Using these displacements w may now be

174
 wr

written as: RESULTS

Making use of equations (24) for the
-" dynamic case and (25) for the static case,
-.
[the displacements were computed for the
M,_ Lo w., center of an aluminum-magnesium alloy
.-LM(C) Los&4 .. - 1
4K)I ob plate exposed to two radiant heating
pulses. These pulses rapidly rose to
-- ,,e-)d
-.. peak intensities, which occured at 0.1
trJ a seconds and 0.34 seconds. The t~tal
thermal exposures were 71 cal/cm and
145 cal/cm 2 respectively. PlAtes of a
number of thicknesses and diameters were
studied.
fp(r) I AnuW -)dtJ The displacement-time history of
those plates having the greatest funda-
Pk kL mental periods (ranging from 0.62 seconds
z(8*O i L) to 0.16 seconds) are presented in Figures
2 and 4, in which we may clearly see
This may be further simplified by thermally induced vibrations of a sub-
carrying out the integration of T and stantial magnitude. Dynamic displacements
observing that were observed to range as high as 1.7
times the corresponding maximum static
displacements. We see also that for
is the series representation of (r2 - a22) pulses having a rise time fairly close
deduced in the first of equations (20): to the fundamental period, the magnifica-
cc) tion is greatest.
Figures 3 and 5 summerize the
j_ - Jiau.I_ r/ results of these calculations as well as
f. (v),44-O1zIC (24) those for one other thickness. Strictly
speaking, these other plates may not be
.,l compared simply on a fundamental period
, - ) basis because the variation in thickness
l) does not merely change the period, but
0 Pr),4)t-u,/ renders the plates somewhat d±simildr
in thermal response as well. Note,
QUASI-STialC DISPLACEMENTS however, that for a given thickness, the
The quasi-static counterpart of dynamic
becomes effects diminish as the period
substantially smaller than the
equation (24) in which inertia terms are heating time.
neglected is readily shown to be: v,
,, (.rL flr 2"On the basis of some preliminary
411- L 6755
1 3 .;,147 stress calculations, another interesting
(25) phenomenon may be observed. The stresses
26 Q within the plate may be compressive at
,1. --
p" ") both the front and rear surfaces, while
they are tensile in the central region.
where{, are the quccessive roots of the This is clearly seen in Figure 6, which
equation ( . presents the static and dynamic stress
distribution in a thin plate shortly
STRESSES after the thermal radiation peak has
occured. The static distribution is
almost precisely that described above,
the displacements (24) or (25) by sub- and we see that there are now two
stituting into the thermo-elastic stress- "neutral" surfaces for stress. From the
displacement relations (1): nearly symmetrical stress distribution
_ .7 Av/"r _e in this case we also see that the static
./ - J' 4-s ) ,, stresses induce no net moment in the
(26) plate. The dynamic case is somewhat
different. While still satisfying the
Yr
I;_
.- AlL r condition of no net axial force in the
,3L---' plate, a substantial moment is induced.
Where Clearly this unusual stress distribution
arises because of the sharp thermal
H . /gradients and should be expected when
NT . rapid heating is involved, or during the
2.

175
early stages of the heating period.

ACKNOWLEDGEMENT
This work has been supported by the
United States Navy under Contracts
N00140-70-C-0019 and N60921-71-C-0197
monitored by the Ntval Applied Science
Laboratory, Brooklyn, New York, and the
Naval Ordnance Laboratory, Silver Spring,
Maryland.
REFERENCES a
I - Boley, B.A. and Weiner, J.g., Theory ,
of Thermal Stresses, Wiley, New"York,

2 - Mindlin, R.D. and Goodman, L.E.,

"Beam Vibrations
Boundary With Time
Conditions", Dependent
J. App. Mech.,
377
Vol.72, 1950, p.
3- Jahnke, Emde, and Losch, Tables of
Higher Functions, McGraw-Hil,New
York, 1960
4 -Watson, G.N., Theory of Bessel
Fucin, CambridgeU-niv. Press,
Lodn 966

5 Love, A.E.H., Theory of Elasticity,

Cambridge Univ.Press, London, 1927,
p.189

#4 '##*S'-''t'''C. 9

176
 iti
1 HiI!~~~~llll
-1IYI

49VINUIX2
fiur 2010 esponse ofT 1/8i ! Thick lat
NYin To ANC71 cal cm Radan Pule
Having ACPeak
InMnit At 0.i

71 cal/cmiRadiant Puls

177I',

A''' PH
 tfI
I'~~i 1 il 11i

HavWi A PekintniyAt03le

111
4 Ti! IJil ........ ll! I~......... fil

it H)etr

.1 14 ca/m Ra iant il
Pule 1ni ll

Ii fil 1i -1il 1!: !] ,1 ll di Il i vpIN p 1"! 1 111178l I

 l-fki iiI 1 i

VI: 1iIff: iI'%VI11H;I

-CC(
-,2t0 -'itM o ~T4C'~ C''
Srs
Ffue6 itiu i t Cne Of /8 i lat !i Atl im 0 0 e
1!!o*n Pulse1
Afe ExpsurAI71 cal/cm Rad

;M wl Ii .

179.
 DISCUSSION

Mr. Mercurio (Sperry Gyroscope Company): I Mr. Zudans (Franklin Institute): I would like to
would like to commend you on a very interesting refer to your last slide. Something confused me, and
paper. I noticed that you bad fluxes of 71, and I be- I want to clarify it. You showed the static stressesas
lieve, and 145. being perfectly symmetrical.

Mr. Koch: Yes, these happen to be particular Mr. Koch: No, not quite --- almost.
weapon pulses.
Mr. Zudans: How can that be explained? Was
Mr. Mercurio: What were the associated pres- the heat flux coming to one side of the plate or both
sure loadings? sides of the plate?

Mr. Koch: I think they were 10 psi. They would Mr. Koch: One side of the plate only. It happen-
have wiped out these plates compietely. I had to get ed, I suspect, that the temperature distribution was
rather flexible plates in order to show the dynamics, such that, if you just turned the plate a bit, you get
and I was constrained to use particular pulses by my something that looked symmetrical. It was not actu-
contract, ally, it just looked like it.

Mr. Mercurio: You have not shown the pressure Mr. Zudans: How could the temperature distri-
stresses on the plate. Is that correct? bution be anything near symmetrical If you bad heat
flux from one side only. This is what I do not under-
Mr. Koch: Yes stand.
Mr. Mercurio: Have you done any work in this Mr Koch: If you shifted the distribution curve
area? I would like to know, because some of te com- you would have something that would appear to be
mon materials that we are dealing with get into nearly symmetrical by coincidence. Accidentally this
problems when you apply both the thermal and pres- looked symmetrical, but the actual numerical value
sure loads simultaneously, was such that it was not actually symmetrical.

Mr Koch: We are getting into it right now. In Mr. Yang (University of Maryland): I have two
fact in about a months time I am due to have a report questions. On one of your curves you show various
on this subject. We are also attempting to answer the diameters of 1.5, 2 and 3 meters. Since the rise time
question: "What actually happens if the properties is dependent upon the diameter, I wonder why the rise
vary with temperature, let's say, through the thick- time for a diameter of 3 meters is the slowest, where-
nesis?" The frequency changes and the modulus as the time for 1.5 meters Is in the middle and for 2
changes, so the whole dynamic picture might change. meters is the fastest?
Mr. Koch: I think I may have flashed the curve
Mr. Mercurio: There is also the problem of the too fast. I do not think that actually happened. The 1.5
properties changing over very short time durations meter one was quickest.
where very little irlormation is available.
Mr. Yang: The second question is, lave you
Mr. Koch: I have access to some data which is done any work in the thermal stresses of composite
in a range less than one second -- perhaps in the materials?
millisecond range. I have not looked at it, but I was
promised the data. I agree there is some lack of data Mr, Koch: No, however I hace done some non-
on materials in the short time range. thermal work in the vibration of layered materials.

+ 180

1
 WHIRL FLUTTER ANALYSIS OF PROPELLER-NACELLE-PYLON

SYSTEM ON LARGE SURFACE EFFECT VEHICLES

Yuan-Ning Liu
Naval Ship Research and Development Center
Washingtonr D.C. 20034

A typical propeller-nacelle-pylon configuration installed

on large surface effect vehicles is analyzed for dynamic sta-
bility in normal operations. A whirl-flutter theory deveAoped
by the aircraft industry for propeller-rotor dynamics is used
to establish stability boundaries for variations of physical
parameters of a propeller-nacelle-pylon system. The stability
boundaries are presented in chart forms which can be used as
design guideb. The uppermost and lowest bounds for stability
are shown on these charts. Classical whirl-flutter theory,
which considers ideally rigid blades, defines the most con-
servative design and the uppermost bound. The lowest bound
is found when the blade fundamental bending frequency is 4/10
of the propeller rotating speed. This is considered to be an
optimum value in the sense of greatest freedom of choice of
nacelle mounting stiffness for dynamic stability.
Although direct comparisons of experimental data with
these analytical results are not possible in the present anal-
ysis, reported observations on the investigation of nacelle
vibration on existing hovercrafto indicates that whirl-flutter
could properly explain the results. Further studies are
recommended.

I. INTRODUCTION the understanding of this phenomenon.

This is attested to by the fact that on
In the aircraft industry, the pea- some V/STOL configurations, propeller
sible occurrence of whirling-type insta- whirl-flutter stability is one of the
bility on propeller-driven aitcrafts was main considerations in design [3-10].
recognized as early as 1938 by Taylor
and Browne [1]. This type of instabil- On a large existing surface effect
ity received a great deal of attentions vehicle (SEV) such as SR.N4 [11], the
in 1960 with the impetus of two early propeller-nacelle-pylon system is very
failures on the Lockheed "Electra" air- similar to the propeller-rotor system
crafts [2]. Suboequent investigations on a typical V/S'OL [3,10]. It seems
revealed that propeller whirl flutter logical then, for a future large SEV
could cause total destruction of the equipped with a similar propulsion
propeller-nacelle system if the nacelle unit, that studies be made to determine
was not mounted with sufficient rigid- the controlling parameters of whirl-
ity. In the past decade, large hseli- flutter to assure an adequate design.
copters and vertical and short takeoff The prdsent analysis was under-
and landing aircrafts (V/STOL) came taken for the purpose of gaining a
into existence, at least partly, due to general knowledge of the propeller

181
 whirl-flutter phenomenon as
related to
the design of the propeller-rotor k8 Effective blade bending
sys- stiffness constant at
tern for a SEV. Design criteria may be hub-center 1
developed in order to define
a stable
Equivalent mass matrix
propeller-nacelle-pylon system mb Blade mass per unit span-
ae func-
tions of design parameters,
such as
wise length
nacelle mounting stiffness,
nacelle
inerti~s, etc., and blade vibration m Nacelle (include shaft)
mass per unit length
characteristics. The overall design N Number of blades
for SEV must also consider the
vibra-
tion characteristics of other
mcchiner- n Blade designation number
ies and the platform,
(n - 1,2,3, . . . , N)
R Propeller radius
Re(s) Real point of s
II. NOIENCLATJRE

a oa lift
Local r sr Spanwise distance along
curve
blade section slope of a
the blade measured from
hub-center to a blade
ACJ
Equivalent damping
matrix
section
s Laplace transform variable
CdC Sectional blade airfoil
drag and lift coefficients,
t Time
respectively
Blade chord T Kinetic energy
y
c ,c Effective viscous damping U Resultant air velocity on
Ax y constants on nacelle pitch-
ing and yawing motions, a blade element
V Proveller advance speed
respectively
Dissipating fucinx Normalized spanvise dis-4
tance, r/R
D, dL Blade sectional drag an d
lift, respectively 4 41
Xb, s Position vectors for a
E Potential energy blade sectin and a
nacelle se'tion,
h Nacelle length respectively
ea
Instantaneous
I Blade moment of inertia blade sec-
about hub-center tion angle of attack
A'
Im(s) Imaginary part of s ao Initial
at angle
a blade of attack
section
1I11 Nacelle moment of inertid
A including the mass of an Effective flapping angle
pro- i
of the nth blade at its
peller in pitch and yaw,
first bending mode,
respectively
referred to the plane of
rotation

[K] Equivalent stiffness

Eualet , Propeller
yaw angles,disc pitch and
respectively,
referred to
the plane of
Kdol%,Kd Drag factors rotati on
2 1 Y Blade lock number,
k Effective
k nacelle spring
x y constants acR 4/b
in pitch and
yaw, respectively
Exponential decay factor

182

~AAA
IA- .~
-~. ~
A-: AAA
A~A __-__________________4_
 Nacelle damping ratio For a forward moving propeller-rotor in

A Propeller inflow ratio, level flight, the resulting flow of air

V/RO normal to the propeller plane is gen-
p Air density erally symmetrical and produces a sym-
metrically distributed propeller thrust.
4X, y Nacelle instantaneous The pitching and yawing rotations of the
pitching and yawing
angles, respectively propeller disc due to the nacelle flex-
Azimuth position angle of ibility destroy the central symmetry of
the nth blade the flow in a continuously varying man-
ner and create additional undesirable
$1 Propeller rotating speed
moments which may lead to unstable
Nacelle natural frequency nacelle circular motion in the direc-

W Nacelle natural frequen- tion opposite to that of the propeller

x y cies in pitch and yaw, rotation, i.e., the backward whirling.
respectively r i
The aerodynamic source of this insta-
W Non-rotating blade funda- bility accounts for the usual designa-
mental bending frequency a
tion "whirl flutter," which is to be
,q n Nacelle frequency ratios distinguished from the purely
x y 0 4 /a) and (w y/),
X y mechanically-induced whirling motion
respectively in a rotating shaft s)stem.

nB Blade frequency ratio

(.) Dot over a quantity indi- A. DYNAMIC STABILITY

cates differentiation with
respect to time t, or, dot
between two vector quan- When considering the dynamic sta-
tities indicates dot- bility of a propeller-rotor system a
product
irore realistic representation of the
(') Superscript indicates dif- system would be to take into account
ferentiation with respect
to the time variable Qt the flexible motion of the blades as
well as that of the rotor. The coupled
('b) Tilt over a quantity indi-
cates Laplace transforma- motions of the blades and rotor form the
tion theoretical background of this analysis.
(4) Arrow over a quantity
indicates a vector Fig. I is a sketch which Illus-
trates the configuration of the
III. THEORETICAL CONSIDERATIONS propeller-nacelle-pylon system on
SR.N4. A schematic representation of
The classical whirl-flutter phe- this system or any other propeller-
nomenun, in which the propeller blades nacelle-pylon configuration for the
are considered infinitely stiff, is mathematical model is shown in Fig. 2.
briefly explained as follows. The The mathematical model used tn this
motion of the rigid-bladed propeller on analysis is essentially that of a four-
a flexibly mounted nacelle consists of degree-of-freedom system. The steady
thd pitching and yawing rotations of aerodynamic loads and vibration theory
ti e nacelle combined with the corre- used in this analysis are considered to
sponding motions of the propeller plane, be linear. It was demonstrated by

183

I.{
 FLAPPING
HANGE

. AXISOF LADE

C-- SHAFT AWI

SOTATING

47

• L.6'96AD[ MEJINT

PIT01 AXIS YAWARlS

Fig. 2a - Propeller-Xotor Structural

Mode!

JL

Fig. 1 - Propeller-Nacelle-Pylon System PARAtLTO I[ELANEOOfATOP

on the SR.N4 Kovercraft
Fig. 2b - Aerodynamic Forces on a Blade
Element

Lytwyn [7] that the nonlinear effects Fig. 2 - Schematic Representation of

were small on the variance of stabil- Mathematical Model for a Propeller-
It is felt that omit- Rotor System
ity boundaries.
ting higher order effects is justified
and the stability boundaries resulting dssptopteiawrenedad
frod this analysis are valid. p
were formulated as follows.

The nacelle is allowed to move in T T + T

pitch and yaw directions and its flex- (shaft)
ibility is assumed to be concentrated fR
at the root of the nacelle. The indi- E (bb n d (la)
2' JI
n-l bb dmb s
vidual propel,-r blade is allowed to
0
flap in its fundamental bending mode_1

and an equivalent system is f rmed such + (- s( x'8

0 s ) dma
that the blade Is considered rigid but
with an elastically restrained flapping
at the center of pro- E - kk 2 + 2 ky
hinge located
peller disk (see Fig. 2). y (lb)
+ E k 82n
principle was used n- n
The variational
to derive the governing differential
equations. Hence, expressions for D c ;2 + Ic (2c
kinetic energy, potential energy and 2 x 2 4y y

184
 Utilizing the Lagrangian equations, the
equations of motions of the dynamic
system shown ia Fig. 2 are given by

d ' E + (2) 8" c

dt(~ q aqj C j

Silaerodynamic + ilexternal if +
oill (5

is the generalized aerodynamic or the y y

external applied force associated with
the generalized coordinates qi" The
generalized coordinates used in this
analysis were 8c' 8, #,xf and *y. A The elements in the matrices [H], [C],
constant rotating force is on and [K] are formed from the coefficients

the shaft so as to produce a constant in the governing differential equations

driving torque and this was the base in of motions, again, they may be found in
the derivation of e rnalRef. [71. By taking Laplace transform
th deiato Q5) Theome Eof itxtrl
aerodynamic forces Qi aerodynamic on Eq. (5), it becomes

involved in the above derivations were

based on the quasi-static aerodynamics. foci
The sectional lift and drag acting on
the blade element are expressed in the 2
form (see Fig. 2b) (s[H] + s[C] + [K - 01 (6)

dL iPcU C~dr (3a) y

1 2 This represents a typical complex eigen-

dD - cU Cddr (3b) value and eigenvector problem. The

characteristic roots a are those which

where C£ and Cd are assumed in the form make the determinant of the coefficient
metrix in Eq. (6) vanish. Each elgen-

[Z . a a (4a) value s, or the characteristic root,

a corresponds to a mode of oscillation

- CCK K
Kd + K d a + K d2a (2
(4b) and can be expressed in the form
d
d0 d1 2
a - Re(s) + jIm(s) (7)

The variations of Re(s) and Im(s) of

The detailed analytical derivations all the modes as a function of differ-
for the mathematical model shown in ent configurations may be expressed in
Fig. 2 may be found in Ref. [7]. The root-locus plots on a complex plane.
governing equations from the derivations Instability is considered to occur
may be expressed in matrix forms, whenever any of the characteristic

185
 _______________ ~ ~ ~ ~ 4 .- -7-T.p r, 771- 7,17- -,.

roots enters the positive real axis *A_______

plane. In physical terms, the damping //MMI .
value associated with a particular mode 0.05 OARHRLN

hsbecome negative. The roots which VNIMM/

MSCKVRD11IN
are located on the imaginary axis are
corresponding to neutral stable con- 0.0) ~ / / X-"s

variation of Re(s) alone assa function N

of different configurations is enough YK Z ''~/ ~ ""~c ~~/
to define the stability boundaries. aK YON1'1/"""01/1
However, occasionally the variatio~n of IACILL
S -0.01 NCLLPIH
Im(s) is also needed in order to define FREQUENCYU#,4R
whether the stability is a dynamic or 00
astatic divergence case, i.e., when____ ___

both Re(s) and Im(s) become zero.

STBL
Fig. 3 shows a typical root-locus plot
for blade bending frequency ratio _O.O
11 1. Furthermore, sufficient vari- j00
ations in parameter for structural con-
figurations are necessary in order to Fig. 4 -Exponential Decay Factor 6
define accurate stability boundaries. as a Function of Nacelle Pitch
Frequency for Various Blade
These are shown in Figs. 4 and 5 for Bending Frequencies
different n1 values for illustrating
* purpose. Any particular configuration
then may be checked for stability in
the preliminary design stage by uti-
lizing these results. 0.10_________________

0.0FORWARD WHIRLING0.
0.00 BACKWARD)
WHIRLING
________0.0

0 0046

~ Is 000
02s1

STA LE o UNSTABLE ___

04 0.1 0.001 .4

Fequency
-. Bending Rati PITCH
FREONCY 0orjir0 u

Fig. 3 -Root-Locus Plot for Blade Fig. 5 -Nacelle Frequency as a Function

(W 0 /In) -1.0BldBednFrqncs

18j
 A digital computer program 112] zero or by omitting time-dependent terms
developed for the Naval Ship Research in Eq. (5). One obtains
and Development Center (NSRDC) was used
to solve Er,. (6).

B. SPECIAL CASE - IDEALLY RIGID BLADES [K] S f)(8)

10)

Ideally speaking, an infinitely y

rigid blade is not possible in practice.
However, this condition provides us the
uppermost bound in defining whirl-flutter
stability boundaries. A necessary and sufficient condition to
have a nontrivial solution for the above
Referring to the set of governing equation is that the determinant of the
differential Eq. (5) one may reduce these coefficient matrix [K] is identically
four equations to a special set of two equal to zero, i.e.,
equations for the case of ideally rigid
blades. This may be accomplished by I[K]I = 0 (9)
assuming that the blade root constraint
stiffness or the blade fundamental bend- Eq. (9) was utilized in the process of
ing frequency is infinite. These two mapping the static divergence bounds-
equations of motion represent the "clas- ries.
sical" whirl-flutter phenomenon of the
two-degree-of-freedom propeller-nacelle
combinations. By studying the character- IV. PARAMETRIC STUDIES
istic roots of these equations, explicit
expressions for stability boundaries may A practical design for propeller-
be obtained. This was done by assuming rotor systems would have physical prop-
that the mass moment of inertia of erties such that their parameters were
nacelle was the same in both pitch and in the region bounded by the uppermost
yaw directions. and lowest boundaries in the stability
plot. The limiting boundaries have
already been discussed as "Ideally Rigid
C. SIATIC DIVERGENCE Blades" and "Static Divergence." There-
fore, as a design guide, parametric
Static divergence of a propeller- studies were made In order to define
rotor system may be considered as a these whirl-flutter stability boundaries.
limiting case of whirl-flutter phenome- The ratio of natural blade bending fra-
non by conaidering only the static quency to propeller rotational speed,
restoring ability from the structure. i.e., n,, and its variations form a
This may be visualized by letting the family of stability curves that are the
"natural frequency" of the system become results of such studies.

~187
 The propeller-roto'r-system on SR.N4 ,1 one can evaluate the variation of
represents a large high-soced SEV'system, eigenvalues of different configurations
and its structural configuration will and hence define the stability bounda-
probably resemble one of an anticipated ries. The results are shown in Fig. 6.
design. It is believed that design
values for the propeller-nacelle-pylon I
system on a larger and faster vehicle .-- ,
would not vary too much from those of SIATIC VIIGCCOOUNARI$ /
SR.N4. Therefore, a set of typical input CA.-...,._.._.

values resembling the propeller-nacelle- STABLEi

tTABLE
pylon configuration on SR.N4 is used in 0.s -
this report to define whirl-flutter sta- i

bility boundaries and are summarized in ley

Ta b l e 1 . 1.4 T)

TABLE 1 0. -- 0
Nacelle and Blade Parameters Used in
Defining Stability Boundaries -

Parameter Value "

UNSTABLE STATC
DIVERGNC * -
8W~ RO
T1 Variable 0.: N T1EI~i IC6N(atN~tC

Variable z

0 01 0.2 CO. 04 0.) 0.A 01?

Ia ye l PITCHFREQUEFNCY
NACELLE ftATIOq.~

N4
h/R 0.83 Fig. 6 - Propelle..-Rotor Whirl Flutter
Stability Boundaries for Various
0.2 Blade Bending Frequencies
" 0.3
'1 3
K 0.0087
The stability boundaries for the
K -0.0216 case of ideally rigid blade and static
d1 divergence were also evaluated and pre-
Kd 0.4 sented in Fig. 6. By examining the

a 5.73 result, an optimur. value for blade

a 5.73bending frequency ratio n is found to
a 4 deg be approximately 0.4. In a similar

0 propeller-rotor configuration with a

value of ng less than 0.4, a sudden
Note: Assume 1 -1 -I reversal from the backward whirl insta-
X Y bility to a forward whirling would be

developed. The stability boundary curve

Using the input values from with n, . 0.4 encloses almost the larg-
Table 1, one may form the matrix equa- eat sti-ble region and allows a designer
rion shown in Eq. (5). Solving this thp .ayvmum freedom to fulfill nacelle-
equation as functions of n8 , r , and sup, .in& 3tiffneas requirements.
x

188

'1 -I l yrk -
V. TRANSIENT RESPONSE ANALYSIS VI. COMPARISON OF THEORY WITH AVAILABLE
EXPERIMENTAL INFORMATION ON SURFACE
A physical interpretation of a EFFECT VEHICLES
propeller-rotor configuration for dynamic
stability is the transient response due Since test results and structural
to some initial disturbances. For a sta- information on existing SEV are not
ble system, the amplitude of response readily available, it is very difficult
would damp out in time, but in an unsta- to make any rational precise experimental
ble system the dmplitude increases with evaluation of the applicability of the
time. A special case is the neutral theory to propeller-rotor systems. How-
stable condition, i.e., the response due ever, with the limited structural infor-
to any initial disturbance could be rep- mation available for SR.N2, SR.N3 and
resented by some harmonic functions with SK.5 al- cushion vehicles, the following
constant amplitudes. Numerical evalu- evaluations are made. The propeller-
ations were performed on a PACE nacelle-pylon on SR.N2 is essentially the
Model-2312 active analog computer for same as the one on SR.N3. The values of
five different propeller-rotor config- nacelle pitch and yaw natural frequen-
urations to illustrate the above cdes, and the propeller blade funda-
phenomena. mental beniing frequency onf SR.N2 or
SR.N3 were obtained from Ref. [I]. The
Referring to Fig. 6, use n, 1 to values of the same parameters on SK.5
define the stability boundary and pick were obtci!,ed from a vibration shake
=
five points along the n /,no I line. test performed by NSRDC.*
x y
In practice, this line represents an
The stability boundaries of the
isotropic mounted nacelle. These five
pointspropeller-rotor configurations on SR.N2,
y) SR.N3, and SK.5, shown in Fig. 8, were
0.4, 0.3, 0.2825, 0.25, and 0.2. The obtained from the interpolation of those
first two cases are in the stable shown in Fig. 6. This was based on the
region, the third one is on the neutral assumption that the nondimensional param-
stable boundary, and the last two are in ters used for stability apalysis for
the unstable region. An initial angular SR.N2, SR.N3, and SK.5 were the same as
velocity of 0.01 in the pitch direction those shown in Table 1. This may not
was assumed throughout. The results necessarily be true; however, for the
were expressed through the usage of an purposes of preliminary evaluation, the
X-Y plotter and are shown in Figs. 7a-7e. results should be close e:..ugh to give a
These figures clearly show the backward designer a rough guidance. Nevertheless,
whirling phenomena as well as the degree experimental confirmation of the vklidity
of stability for each configuration. In of the stability boundaries shown in
these plots, the X-axis represents the Fig. 8 as well as those shown in Fig. 6
pitch response and the Y-axis represents is still lacking. The only experimental
the yaw response. Heuce, these figures
actually represent the locus of the * A ]lagcn at NSRDC performed the
motion of a Propeller hub. SK-5 vibration shake test in
March 1971.

189
 4C

- 0.

-- - - - - - - - 4 I

$ .1 ~
4-- _ __ ~0
(44

I 14.0 1

I ION SPaY Ma CC

A4 0

__ _ 44 a

-4 u

4w 4.

a 4

v 190
 T -and the condition is not serious . . .
STAIC
DYEUN~CII UIS 1 (Ref'. (111, p. 219.)
uSTABLE A
toe ALLY
RIMPKIAN
. ~e . The above quoted statement clearly

/ describes the propeller-rotor flutter

whirling phenomenon and the actual occur-

rence on some SEV. Fig. 8 shows the

same behavior, i.e., SK.5 has a very sta-
02 __ \.?L hise propeller-rotor conilguration and

UNSTABLE* ." II SR.2 or SR.N3 has a ilightly stable one.

TDIVIRCCSCE
0S.T -AR 1$7 How stable a system io can be judged by
~Ithe distance between the data point and
_ its corresponding stability boundary
a . 02 03 . S CA .

NAC PITCHREQUENYRAIO
icurve. As one can see from Fig. 8, the
data point associated with SR.N2 or SR.N3
is very close to the neutral stable line
Fig. 8 - Propeller-Rotor Whirl Flutter
Stability Boundaries for SK.5, SR.N2, and nacelle vibratory motion on SR.N2 and
and SR.N3 Air Cushion Vehicles SR.N3 were observed.

results the author could find were some

observations on propeller-rotor vibra- In order to have a better feeling
tion in Ref. 111]. of the actual physical behavior of the
propeller-rotor system on SR.N2 or SR.N3,
an evaluation of transient responses due
The pylon situation is fur- to an initial disturbance and the per-
ther complicated to the extent that not formance ot a response analysis due to
only will it have modes in the lateral some random excitations were undertaken
plane but also in the vertical plane. In through the use of analog computer simu-
some of both the lateral and vertical lation. Fig. 9 shows the transient
modes the angular deflection at the pro- responses in both pitch and yaw direc-
peller is considerable. As the propeller tions due to an initial angular veloc-
is rotating at high npeed,, angular ity 0.01. Fig. 10 shows the responses
motion, or precessio, in one plane will in the same directions due to some ran-
induce a force in the plane at right dom excitations. Here the random exci-
angles, due to the gyroscope effect . . . tations were generated by attaching a
Gaussian noise generator to the analog
The propeller is obviously the main circuit at the point associated with the
source of forcing in these modes . . . nacelle pitching angular velocity.
Fig. 10 also includes this particular
Experience so far has disclosed noise output which has a rus value of
appreciable vibration only in the low- 3.16, frequency range 0 to 50 Rz, and
eat modes. Both SR.N2 and SR.N3 pylons 3 JB cutoff. Fig. 9 shows the rela-
can be seen to nod gently when running tively small amount of damping in the
at idling conditions. The forcing at system and Fig. 10 shows the actual
these conditions is very low, however, response due to some random excitations

191i
 NACELLE FREO'JEOCaz
PITCH q#0.1
MACILL
I YAW
FREQUENCYs
II AA^
Il-

Fig. lOa - Random Exciting Forces

10 to 0 30 4 0 0 i I F -10

Fig. 9a- Nacelle Pitch Response _..1 tIj

0.04
ili
- 0 t0 20 0 0 0 to go
10 to

-0.011 Fig. lOb - Nacelle Pitch Response

Fig. 9b - Nacelle Yaw Response 0.

Fig. 9 - Analog Simulation of Propeller

iub Motion due to In4.tial Disturbance 0 10 20 0 30 60 ?0 lb go 10
for SR.N2 and SR.N3 Air Cushion , .mUI0,,i-.,i'a[4.4.t'.,,,,
VehiclesIi1 L L i
which always exist in reality. Both Fig. 10c - Nacelle Yaw Response
Figs. 9 and 10 clearly illustrate the
relatively large oscillatory motion in Fig. 10 - Analog Simulation of Propeller
Hub Motion due to Random Excita-
the nacelle pitch direction in compari- tion for SR.N2 and SR.N3 Air
son with yaw. These results agree with Cushion Vehicles
the observations in Ref. [11] quoted
above, confirm the theoretical computations;
however, some observations on SR.N2 and
VII. DISCUSSION SR.N3 tend to agree with theoretical
predictions.
A propeller-rotor whirl flutter
theory has been applied to pylon-necelle- As indicated in the evaluation of
propeller configurations on surface stability boundaries, blade fundamental
effect vehicles. Stability boundaries bending frequency plays an important
based on a typical propeller-rotor system role since it affects the stability
on large SEV were evaluated for the pur- mapping. The classical whirl-flutter
pose of preliminary design and guidance. theory in which blades are assumed to be
No experimental data were available to infinitely rigid gives the most rigid and

192

NI.. Wt..1~tt~ N'~fl ~ l.N..

overly conservative stability criteria, design guidance. Direct experimental
leading to over-design and possibly confirmation is not possible at this
uneconomical construction. An optimum time, however, tht following statemenhts
value n, a 0.4 was found, and when the can be based on this analysis.
blade bending frequency is below this 1. Observations of the nacelle
value, forward whirling as well as motions on existing SEV tend to confirm
backward whirling'could develop. How- the theoretical analysis used in this
ever, the optimum value for blade bend- report.
ing frequency is restricted in the sense 2. Optimum fundamental blade bending
that it would allow a designer the most frequency with respect to propeller
freedom to choose nacelle mounting rotating speed should have e, value of
stiffness in order to avoid whirl- 0.4 in order to impose the winimum
flutter instability. Other dynamic con- requirements on nacelle supporting
siderations should be also included in stiffness for a stable proptller-rotor
overall design. As implemented in configuration.
Fig. 6, increasing nacelle mounting 3. Classical whirl-flutter theory
stiffness to increase nacelle natural gives overconservative stability cri-
frequencies is another way to move a teria which impose the most severe
particular propeller-rotor configuration design requirements on the naceile
from the unstable region into the stable supporting stiffness.
zone. 4. Caution should be taken in the
consideration of fatigue failure even
when the propeller-rotor system is
Transient analysis by analog com- operating i the stable region.
puter simulation gives a direct physical
interpretation of the actual whirl-
flutter phenomenon. Transient responses IX. RECOMMENDATIONS
due to initial disturbance and responses
due to random excitations can be con- In view of the importance of a sta-
sidered as the results of gusts and tur- ble propeller-nacelle-pylon system on
bulence in the atmosphere or due to the large high-speed SEV, and insufficient
irregularity in the ground/water surface. experimental data to substantiate the
They are in connection with dynamic loads theoretical analysis of such a system,
and fatigue failures. These loads may be the following efforts are recommended.
aignificant for fatigue failure criteria I. Perform experimental evaluations
even though the system is operating in by using a model with the same nondimen-
the stable region. Observations on SR.N2 sional parameters used in this report.
and SR.N3 tend to strengthen this 2. Perform a vibration shake test and
consideration. underway test on a large SEV, and compare
the test results with the theoretical
computations in this report.
VIII. CONCLUSIONS 3. Extend the existing theory by
including blade first torsion mode,
Analysis performed by applying blade second bending mode, and unsteady
the whirl-flutter theory on a typical aerodynamic loadings.
propeller-rotor system of a large SEV 4. Study the effects on the change of
resulted in a parametric chart for stability boundaries as functions of

193
important nonClmensional parameters such 6. Niblett, T., "A Graphical Repre-
as nacelle damping, nacelle length, sentation of the Binary Flutter Equa-
tions in Normal Coordinates," Royal
blade inertia, propeller power setting, Aircraft Establishment Tech. Rept. 66001,
Jan 1966
etc.
5. Study the effects on the change of 7. Lytwyn, R. T., "Propeller-Rotor
stability boundaries due to the overall Dynamic Stability," The Boeing Co.,
Vertol Div., Tech. Rept. D8-0L95, 1966
structural dynamic characteristics of
surface effect vehicles. 8. Edenborough, H. K., "Investigation
of Tilt-Rotor VTOL Aircraft Rotor-Pylon
Stability," Journal of Aircraft, Vol. 5,
No. 6, Mar-Apr 1969

REFERENCES 9. Brindt, E. E., "Aeroelastic Prob-

lems of Flexible V/STOL Rotors,' paper
presented at 34th AGARD Flight Mechanics
1. Taylor, E. S. and Browne, K. A., Panel Meeting, 21-24 Apr 1969
"Vibration Isolation of Aircraft Power
Plahts," J. Aero. Sci., Vol. 6, No. 2, 10. Gaffey, T. M., Yen, J. G., and
pp. 43-49, Dec 1938 Kvaternik, R. G., "Analysis and Model
Teats of the Proprotor Dynamics of a
2. oubolt, J.C. and Reed, W.H, III, Tilt-Propotor VTOL Aircraft," Paper
"Propeller-Nacelle Whirl Flutter," presented at the Air Force V/STOL Tech-
I.A.S., pp. 333-347, Mar 1962 nology and Planning Conference,
Las Vegas, Nevada, Sep 23-25, 1969

3. Reed, W. H., III and Bennett, R.M.,

"Propeller Whirl Considerations for 11. Elsley, G. H. and Devereus, A. J.,
V/STOL Aircraft," CAL-TRECOM Symposium on Hovercraft Design and Construction,
Dynamic Loads Problems Associated with Cornell Maritime Press, Inc., 1968
the Helicopters and V/STOL Aircraft,
Buffalo, N.Y., Jun 26-27, 1963 12. Peterson, L., "SADSAM V User's
Manual," MacNeal-Schwendler Corporation
4. Reed, W. H., III, "Propeller Rotor Project Report, 1970
Whirl Flutter: A State-of-the-Art
Review," Symposium on the Noise and Load-
ing Actions on Helicopter, V/STOL and
Ground Effect Machines, Southampton,
England, Aug 30 - Sep 3, 1965
5. Hall, W. E., Jr., "Prop-Rotor Sta-
bility at High Advance Ratiou," J.A.H.S.,
Apr 1966

DISCUSSION

Mr. Gayman (Jet Propulsion LaboratorX): I ask Mr. Liu: Perhaps I did not make the point very
for a point of clarification. Early in your presenta- clear. We do consider the blade ae an elastic blade
tion you discussed the degrees of freedom you were but restrict it to the first bending mode. I made an
admitting to the problem in reference to blade bend- equlvelant system by considering the blade as rigid,
ing. Did you not mean blade flapping as a ridgid mo- but hinged at the hub. The system also included an
tion? equivalent rotation spring at the hub, and the bending
or flexing frequency was the same frequency as the
Mr. Llu: Yes, I meant the flapping motion of the first bending mode.
blade, but only restricted to the first bending mode.
Mr. Zudans: (rranklin Institute): Because of the
yaw and pitch of the nacelle, the blade plane moves as
a rigid body. Was M'at motion included in your hydro-
Mr. Gayman: That is associated with the oscil- dynamic forces?
lation of the plane ofthe propeller disk, Is it not? The
blades themselves are treated as rigid, are they not? Mr. Liu: Yes.

194
 THE DYNAHIC RESPONSE OF STRUCTURES SUBJECTED TO TINE-DEPENDENT
BOUNDARY CONDITIONS USING THE FINITE ELEMENT METHOD

George H. Workman
Battelle, Columbus Laboratories
Columbus, Ohio

The dynamic-matrix equation of motion characteristic of structures modeled

by the finite element method of analysis is vritten in general form. This
matrix equation is then rearranged and partitioned to separate constrained
and unconstrained displacement degrees of freedom.

Once the general matrix equation has been properly partitioned, then by
standard matrix manipulations, the original mixed boundary value problem
is transformed to a modified force motion problem.

The dynamic response of a bellows subjected to dynamic edge displacements

and internal pressure, as determined by this approach, is presented. This
example is used as a vehicle to demonstrate the versatility and effective-
ness of this solution technique.

INTRODUCTION discretized continuum elastic undamped dynamic

problem becomes the solution of the hiatrix
Over the past decade and a half, with equation characterized as:
the advent of high-capacity, high-speed digital
computers and the increasing needs of the aero- [N) (A(t)) + [K- (A(t)] - (F(t)) (1)
space industry, the finite element method has
emerged as a powerful tool for the structural where
analysis of large, complex structures. Evidence
of this is the large number of general purpose IN) = mass matrix
finite element computer programs in use today.
roC - stiffness mtrix
The capability of this approach to
obtain the dynamic response of compcC struc- (F(t)) - nodal time-dependent forces
tures to known forcing functions and base mo.ion
problems is well documented. This paper pre- (A(t)) - nodal displacements
sents a straightforward extension of the finite
element approach,to solve dynamic response prob- -At)
nodal accelerations.
lems having time-dependent displacement. T
The theoretical background and &ssufnptions
MErHOD OF ANALYSIS leading to the development of Matrix Equation
(I) through the use of the finite element meth-
In the finite element method, the odology is well known and not presented herein.
continuum is separated by imaginary lines or Two excellent books describing the finite ele-
surfaces into a number of "finite or discrete ment method have been authored by Zienkiewicz (1 )
elements". These elements are assumed to be and Przemieniecki( 2 f.
interconnected only at a discrete number of
nodal points situated on their boundaries. The Each row of Matrix Equation (1) repre-
displacements of these nodal points are then the aents a particular degree of freedom that, for
basic unknown parameters of the problem. The this study, can be described as being associated
form that these discrete elements takes depends with an unconstrained or constrained displace-
on the type of structural behavior assumed and ment. Displacements as used herein represent
the form of the approxication to that behavior, both translational and rotational motions.
Constrained displacements are those degrees of
Once nodal points and structural freedom for which the motion is known as a
elements have been defined, the solution of this function of time. This includes both zero and

'195
 iC,

nonzero motions. 'Therefore, accelerations and BELLOWS ANALYSIS

velocities of theme consuriined degrees of
freedom are also known. Unconstrained displace- A dynamic anilysis of a bellows was
ments are those degrees of freedom for which conducted utilizing the method of analysis
the applied forces are known functions of time given above. The bellows serves asa pressure
and whose resulting displacements are not seal between a vessel and a piston. The top
prescribed. edge of the bellows is attached to the station-
ary vessel and the bottom edge is attached to
Matrix Equation (1) can be rearranged the moving piston used to generate a pressure
such that those degrees of freedom associated pulsation in the vessel.
with the unconstrained displaceQents and those
associated with the constrained time-dependent Figure I shows the bellows cross
displacements are partitioned as shown below: section. This shape was modeled by seven

1
11 J -1 +
toroidal parts. Each part was described by its
major and minor radii and the angular coordi-
nates, tp, so that the exact shape of the bellows
in its neutral position was reproduced quite
Lii ~J accurately. The bellows is of uniform thick-
ness, t - 0.062 inch, and is constructed of
K IK Fsteel sheet -: E a 29 x 106 bin and
(2) (2) poisson's ratio - 0.3.
L;]i J] J
K is
where subscript i is associated with the uncon-
strained displacements and subscript J is
associated with the constrained displacements,. -- '

atrx Eqution (2) can then be L

expanded by rows and rearranged to yield
[Mid ] (Ai) + [K id [L) . 4P

[Fi) -Mj [ - [Kij3 (A

J). (3)

andIh

Fj = I + [M 3 j + k&W*
[Kj1) [Ai + CKj)IAj1 (4) N'ID

Matrix Equation (3) is in the standard form

associat.d with the elastic undamped multi-
degree-of-fre-dom dynamic problem with a slight-
ly modified right-hand side to include the
effect of the constrained time-dependent dis-
placements. This system of equations can be- -
solved by a variety of methods depending on the-
requirements of the problem. Three excellent
books giving a number of these teihniques have
been authored by Hurty and Rubinstein(3),
Biggs(4 ) and Bisplinghoff, Ashley, and
llalfman().

Matrix Equation (4) yields the re- Fig. I - Cross section of bellows used for
active forces at the constrained displacement. vibration analysis (neutral position)

Once the t - history solution of Three finite element models of the

tiatrix Equation (3) been accomplished the bellows were created. These models contained
nodal displacements of the discretized problem 20, 30, and 40 conical elements, respectively.
are known. The resulting time history of the The basic conical element used in this analysis
stresses and strains in the individual elements was similar to thet developed by Grafton and
6
can be determined from the assumed behavior Strome( ) except numerical integration was em-
pattern of the individual elements. ployed to generate the element stiffness matrix.
An earlier study of this bellows was conducted
utilizing a continuum axisymmetric shell program
as the basic analysis tool. This program uti-

196
 I

lized numerical integration of the governing TABL9 2

continuum thin-shell ecustions. Unfortunately, Natural Frequencies of the 30-Element Model
the modulus gf elasticity was taken to be 27.4
x 106 lb/in.4 for ;he bellws material and was Mode No. Frequency, cpa
assumed to 29 x 100 lb/in.4 for this study. 44.96

The meas of the bellows was luIsped at

the nodal points for the finite element model 2 318.38
and was evenly distributed along the length for
the continuum model. The two lowest axisym- 3 354.20
metric natural frequencies of the bellows were
calculated for the four models assuming both 4 376.17
boundaries fixed. Table I gives the results of
the natural frequency prediction along with the 5 398.58
frequency determined experimentally for the
first mode. The finite element models predicted 6 440.96
natural frequencies slightly higher duc to the
difference in the assumed modulus of elasticity.

TABLE I
Natural Frequencies of Bellows Model

Finite Element Muel

20-Element 30-Element 40-Elemen]t Continuum Model Experiment

Mode 1 45.03 44.96 44.95 43.2 44.14

Mode 2 318.44 318.38 318.49 311

Utilizing these natural frequency data

it was decided to use the 30-element model for
the dynamic analysis of the bellows. eigure 2
shows the node locations and elements for the
30-element model. Figures 3 and 4 give mode
shapes for the first and second natural fre-
quency, respectively. Also, the next four natu-
ral frequencies of the 30-element model were
determined and are given in Table 2.

4 # :

2197

004~

R4Cor,*.. \h
S, \30

21

204

I 7 7 4 7 24-
4 4 ?
24 0

Fig. 3 -Modal deflection shape for first

Fig. 2 -30 element model of bellows . axisymmetric mode of bellows

197
 Two different techniques were emp!oy-
ed for the solution of Matrix Equation (3).
to One was the straight numerical integration of
Matrix Equation (3) by a f urth-order single-
step Runge-Kutta method.( 7 The other was the
, modal acceleration method.(5 ) Within the modal
"- ' acceleration method only the first mode was
employed. The static displacement component
~~4 wasn determined utilizing standard static. finite
/3 4 element techniques. By comparing the numerical
:s "0 integration solution and modal acceleration
% /solution with only the first mode employed, it
4
\.14
%
Ithe was clear that, for engineering purposes, only
first vibration mode makes any appreciable
15s contribution to the dynamic solution.

116 Figure 6 shows the time history of

I the maximum stress occurring at the top aide ot
I7 the bellows; this stress ocurred at Node 5 ir
/ the 30-element model. Figure 7 shows the tim,
gel, / 4, history of the maximum stress occurring at th
bottom side of the bellows; that occurred at
12 Node 77 of the model. The numerical integra-
tion solution is-plo2.ed ".n Figures 6 and 7
2o along with staLic stressesi. The modal accel-
21/ eration solution was not plotted because It
.
N.__ 2Z-" /0
__ould
20o_ er not be discernible om
notediscrnibl from te
the numerical
nuecaelt
A.S.. , integration solution.
25 2
24Z The dynamic magnification of the
statically calculated stresses is quite clear
when examining the plots given in Figures 6
and 7.
Fig. 4 - Modal deflection shape for second Figure 8 shows the time history of
axisymmetric mode of bellows the vertical displacement of Nodes 8, 16, 24,
The dynamic problem was formulated and 31.
along standard finite element techniques for the
forming of Matrix Equation (1). The constrained
displacements are that the radial and vertical This paper has given and demonstrated
displacements and rotation of Node I are zero,
the radial displacement and rotation of Node 31 a very straightforward and effective method of
are zero, and the vertical displacement of Node determining the dynamic response of structures
31 is constrained to move downward as
o ovedowwad a a
31 s cnsraied a veraine
erin subjected to time-dependent boundary
suete dtoime-ependnty bounariy condi-
acod-
with a period of 33 milliseconds and a maximum tions. This capability can be easily incor-
porated into an existing finite element con-
amplitude of 0.80 inch. The internal pressure pote proa eady p iie emn dnmi
)f the bellows, initially at 80 psi, is in phase puter program already progrotoed fo dynamic
%*ith the edge displacement, dropping to 10 psi force motion problems.
as a versine. Figure 5 shows the internal pi.s-
sure and edge displacement as a function of REFERENCES
time. The matrix rearrangement and partitioning
going from Matrix '"quation (1) to Matrix Equa- 1. Zienkiewicz, 0. C., The Finite Element
tion (3) is very straightforward when accom- Method in Structural and Continuum
plished on the digital computer. The additional Mechanics, McGraw-Hill, London, 1967.
time, both engineering and computer, required to
incorporate the dynamic mixed boundary value 2. Przemieniecki, J. S., Theory of Matrix
problem as formulated herein as compared with Structural Analysis, Mcrraw-llill, New York,
that for a typical force motion problem for a 1968.
similar application is negligible.
3. lurty, W.C., and Rubinstein, M. F.,
One pulse of the piston was examined Dynamics of Structures, Prentice-llall,
and no damping was included in this analysis. Englewood Cliffs, New Jersey, 1965.
Experience has shown that, for cnalysis of this
type, modal .-itical damping in the range of 0.5 4. Biggs, J. H., Introduction to Structural
to 1.0 percent gives realistic results. This Dynamics, McGraw-Hill, New York, 1964.
small amount of damping has a minimal effect for
the first few cycles and therefore was neglected 5. Bisplinghoff, R. L., Ashley, H., and
in this preliminary study. llalfman, P. L., Aeroelasticity, Addison-

198
 Wesley, Reading, Hassachusetts, 1955 2342-2347, 1963.
6. Grafton, P. E., and Strome, D. R., "Analysis 7. Ralston, A., A First Course in Numerical
of Axi-Symmetric Shells by the Direct Stiff- Analysis, pp 191-202, McGraw-Hill, New York,
neas Method", Journal A.l.A.A., Vol. 1, pp 1965.

leo

000

Tim milliseconds
g-0 10 / 39 40 so s

i5

0. I

Fig. 5 - Time history of piston displacement and vessel pressure

40-

/ Tkne,nilsecods

"X_' D-ynomlc

-40

Fig. 6
- '!."'' -ia' , -'
..¢.' "".',-'. , , '"-,* **" " '',
Time history of maximum
~d,t "". !'" l" L' ' ''' '' ' ' stress
"'"
at Node 5
'" ... '' *' :Y'rv :/d;t'/"d r

199
 ICI

D.-€-

Fig.
7 - Time h~stor fmaiu trs t oe2

Time, miliseconds .

\% \-,-DiNe R1

-% Node 16
-a-Nod 24-

Fig. 8- Time history of vertical displacement

4of Nodes 8,16,24 and 31

200
 DISCUSSION

Mr. Zudans (Franklin Institute): You mentioned Mr. Schrantz (Comsat Laboratory): You said
comparison between modal solutions and you aid that you used conical elements to define the bellows?
the response was entirely in the first mode. Yet the
mode you showed was incompatible with boundrydis- Mr. Workman: Rightl I used a series of conical
placement, so you must have done something in add- elements.
ition to that.

Mr. Workman: No, the third slids ,,tA.ed the

equation after It is rearranged t, get it into the form Voice: Is this a toroidal bellows? What form
that is solved. If you use the modal method on that is it?
equation, that boundary appears to be fixed. Then the
displacement is used in the static solution. That mode Mr. Workman: Yes, actually it is a torold a-
finds the dynamic component, not the static displace- round a circular piston rather than a flat bellows.
ment. It is a shell of revolution.

201
I VIBRATION ANALYSIS AND TEST OF THE

EARTH RESOURCES TECHNOLOGY SATELLITE

T. 3. Cokonis and G. Sardella

General Electric Company
Space Division
Philadelphia, Pennsylvania

This paper presents a unique approach used for the launch vibraticn analysis
of the Earth Resources Technology Satellite (ERTS) and compares the analy-
tical results with experimental measurements. The ERTS is basically a mod-
ification of the Nimbus vehicle with solar arrA7 paddles unchanged. The com-
plex paddle system could best be represented by measured data obtained from
previous Nimbus modal testing. The successful extraction and subsequent re-
coupling on ERTS of the solar array paddle modes from the original Nimbus
experimental mode shapes is given. The analytical model is described along
with its verification by an abbreviated modal test. Good correlation between
test and analysis was evidenced by frequency and mode shape comparisons.
Some areas of discrepancy in the analytical model were uncovered which were
subsequently modified to improve the analytical representation of the spacecraft.

INTRODUCTION in mounted on an adapter which is bolted to the

adapter
The Earth Resources Technology Satellite launch vehicle
separation adapter
occurs lowerSpacecraft
at thering. ring flange of
(ERTS) system (Figure 1) is being developed by the sensory ring.
the Goddard Space Flight Center (NASA) for ob-
taining data from near earth orbit immediately appli- During the evolution of the Nimbus space-
craft from a 900-pound configuration to the t
cable to the determination of earth resources
and their management. The spacecraft config- 2000-pound ERTS configuration, the dynamic
uration will be launched on a Thor/Delta N-9 representation used for loads and clearance
booster. The structural subsystem is based on studies was based on an extrapolation of
the flight-proven Nimbus satellite using the same modal test data on Nimbus A. Analytical c" m-
philosophy of design. The greater weight of the parisons with test results indicated tbat the
ERTS spacecraft and specific ERTS mission extrapolated modal model no longer predicted
requirements necessitated structural modifica- the dynamic behavior of the spacecraft with the
tions and some redesign. required degree of accuracy. This was evi-
denced in the flight loads analysis for the POGO
The ERTS spacecraft structural segments condition using the extrapolated model which
are shown in Figure 2. These segments are: predicted extremely high loads due to high
1. The Attitude Control Subsystem (ACS); 2. coupling between the lateral modes and the
The Solar Array Paddles; 3. The A-Truss axial excitation. To overcome these deficien-
Structure; 4. The Sensory Ring; and S. The cies, a new analytical model was developed.
Launch Vehicle Adapter. Structural differences
with respect to the Nimbus were primarily in For an analytical model c the spacecraft
the center section of the sensory ring assembly. the representation of the solar array becomes
New structure was designed to accommodate critical. Thir complex solar array was found
the ERTS payload and other new equipment, to dominate all of the spacecraft modes. If
Sensory ring and truss structure was modified finite element representation was used for the
for increased stiffness and strength. In the solar array, it would be difficult to derive and
launch configuration, the solar array is folded its accuracy would be questionable. This con-
along its longitudinal axis to fit within the sensory sideration was avoided, however, with a modal
ring envelope and is secured to the sensory ring coupling approach using measured mode shapes.
structure by a latch mechanism. The spacecraft This required the extraction of 19 solar array

Preceding page blank203

L a
 complete spacecraft (less solar paddles) matrix.
To complete the model, the paddles were added
by the modal coupling technique. Nineteen
paddle modes and 11 spacecraft modes (less
paddles) were coupled to produce the final set
of complete spacecraft modes and frequencies.

I. Spacecraft. The launch vehicle adapter

k' structure was represented using over 160 in-
ternal joints in the MASS digital computer pro-
gram (reference 1). These joints connected the
S b-am and panel elements used to represent the
adapter structure. The resulting stiffness
matrix was reduced to nine external joints which
were selected to coincide with the sensory ring
external joints. The sensor-, ring structure in
Figure 3 was modeled also using the MASS pro-
gram. This representation consisted of over
200 joints connecting the beam and plate elements.
The resulting stiffness matrix was reduced to 18
external joints common to the adapter, the truss,
and other mass points of interest.

In a similar manner, the stiffness matrices of the

truss and the ACS structures were also devel-
oped. The complete spacecraft structure, less
paddles, was then assembled by the stiffness
coupling of each segment. The fixed-free natu-
Figure 1. ERT8 Spacecraft Vibra, in Test ral frequencies and mode shapes were calculated
using this stiffness matrix and the mass matrix
normal vibration modes from Nimbus spacecraft given by the distribution shown in Figure 4.
modal data. Such an approach is valid since
ERTS uses a solar array which is structurally 2. Solar Array. A very accurate representa-
unchanged from Nimbus. Paddle frequencies tion of the solar array paddles was possible
(starting at 13 hz. ) and mode shapes having 154 since measured mode ohapes could be utilized.
coordinates were utilized for the new ERTS ana- These measurements were taken from a previous
lytical model. Other major substructures were modal survey of NIMBUS A (4) according to the
derived by a conventional finite element stiffness pattern given in Figure 5. Since the ERTS space-
routine known as MASS (1). These substructures craft uses the NIMBUS solar array with minor
were then combined by modal coupling through modifications, these data were considered to be
a statically determinant interface in a manner appropriate for the ERTS model,
similar to that used by Hurty and Hou (5, 8).
The extraction of the 19 paddle modes from the
The following sections of the paper describe NIMBUS modal survey started with the formula-
the formulation of the ERTS spacecraft analy- tion of the characteristic matrix vibration equa-
tical modal and Its verification and evaluation by tion:
modal testing. Much of the modal dynamic analy-
sis and testing activity conducted on ERTS!Nim- n xn n x m
bus programs parallels the well-documented [K) ] -
Mariner Spacecraft Programs (5, 6, 7) and have n x n n x m m x (
had a similar degree of success.

DESCRIPTION OF ANALYTICAL MODEL with [01 normalized to,

A, The analytical nodel of the spacecraft was [9] [H) (9] = [tJ
developed by considering the major spacecraft
segments separately. These structural seg- where
ments are identified in Figure 2. Stiffness (K) is a n x n square symmetric matrix of
matrices were developed for the launch vehicle stiffness coefficients
adapter, the sensory ring, truss, and ACS, and [H) is a n x n square symmetric matrix of
each segment was then combined to form the inertial mass items

204
 so: :.:: .. L AC
ACS

ERTS/ACS

INTERFACE
ASSE4BLT

L/V IhTERFACE

Figure 2 E'ITS Structures Subsystem

A is a scalar of resonant frequency squared - 2

[9] is a n x m modal matrix of eigenvectors (mode shapes) listed columnwise tiaere m < n
[A3 is a m x m diagonal spectral matrix of eigenvalues (A, resonant frequencies squared) listed
diagonally where T [

jIJ is a m x m identity matrix

m refers to the number of modal degrees of freedom (modal courdinates)
n refers to the number of generalized displacement coordinate degrees of freedom

Equation I can be manipulated to derive the stiffness matrix if the modal, spectral and inertial ma-
trices are known (2, 3).

Post multiplying by: T

C*] [14]

[K] [9]-(4 T [M) - [M4] [ 4 [Aj (M) (2)

and utilizing the normalization

(9 I"[t [I] (9JJ - [9] 11 - [4] (3)

regrouping

205

2i
 [] 1 T [H]J it)I 1#1
gives the identity
!.] []T [H] E [I]

which simplifies equation 2 to

n x n n xn nxm mxm mxn nxn

[K] = (J [*] [,HA( [MJT (H] (4)
Utilizing equation 4 nineteen constrained solar array paddle modes were determined from the original
measured modes of the Nimbus Spacecraft by applying constraints at all non solar array coordinates.

The derived stiffness matrix was partitioned as folloit:

KI K2 XI 0 M O XI
[21 1
X1 ,
K22 X2 [ 22 2

Let = Y4
where: K1 1 is the desired cantilevered paddle stiffness matrix

X is the generalized physical displacement vector

( )1 coordinates being retained; i.e., solar array

K21
I 1

K22]
12 ,
-o
[
)2 coordinates being restrained, i.e., spacecraft

] s]
I 1 I [1
[
,T
T
T
2 T
rl
I o
(I
(6)
2'2 ] 0 2J
01
=

F0
N
M2 1
01
M2 j
[1
h
.t2 -;
A
o T
i *1
*
1¢ 1
H
0 A
l
A
1*T~.
2 1 h 0T~ T
2M
2 2
MM2 (7)

T --- (8)
2 H " 2 2 A

Transforming Equation 5 in expanded form

[ 1 A] A i 0T i
1 Ml A1 H1 A * 2 [1
122 A 1T MIV2 42 A 02 T 2 0

#
*T N1 * A ~T m
1 *Jy I ht M
1 A (9)~j
solving the above elgenvalue problem yields Y , and transforming

back to obtain the paddle e'ode rhapes I#ade I I# Y1j'~ (10)

206
'Llak4, 4L.-1-o Y O
 ... . .. ,-. , , ,= ,.f ........... ,,

-- I',
n ,z

"t I
'NI
I I

I * I

FIGURE 3 - MASS COMPUTER MODEL -

SIMULATION OF SENSORY
RING STRUCTURE (Cross-
beam omitted for clarity)

3. Modal Coupling. Having obtained the con-

strained modes, the statically determinant inter-
face consisting of three translations and two ang-
ular reactions at the Solar Array Paddle Shaft "
and a latch line vertical tie to the sensory ring
was released in the modal coordinate. This ob-
tained the free modes. The free solar array and
analytically derived ERTS spacecraft structures FIGURE 4 - SPACECRAFT ANALYTICAL MODEL
were then ready to be modally coupled to obtain
the desired ERTS spacecraft vibration mode
shapes and resonant frequencies.
part of the structural verification test. The

Basically, a simplified Hurty (5,8) attachment primary objective of this test was to obtain
technique was used which entailed free mode fixed-free modal vibration data suitable for
coupling at statically determinate interfaces, evaluating the analytical model of the space- I
The resulting eigenvalue equation form consisted craft for use in the flight loads analysis. This
of a coupled generall -,,d mass matrix and dia- required the measurement of data sufficiently
gonal generalized stiffness matrix with 30 de- detailed to enable the response in the funda-
grees of freedom. Eleven modes of the struc- mental pitch, roll and yaw modes and the lateral
ture and nin, een modes nf the solar array modes in the frequency range of the POGO exci-
were utilized in developing the complete system tation to be evaluated.
modes. The first twenty-four complete space-
craft modes ranging from 13 to 85 hz were sub- Sine sweeps for modal definition were
sequently correlated by vibration testing. performed with low level base excitation at a
rate of 8 minutes per octave. A total of ninety
MODAL TEST DESCRIPTION accelerometer channels were recorded during
the resonant dwells. These were located on
Modal testing of the ERTS full-scale the spacecraft structure in triaxial and biaxial
structural dynamic model was performed as a groups at points corresponding to mass points

4 207

........ ........... ,
 - p~il .I
I Illel

10161 TaftS pool$ W SMOM

aIl l~~* I
4'4
4 J-- I

4 I + 4 1

I A I
I I I

14ACJ4V5W"A.4.

FIGUR 5 - ACCELEROMETER LOCATIONS FOR PADDLE

MDAL SURVEY -NIMBUS

Kof the analytical model. The paddles were in-

strumented at its extremities only since modal
terms for the nine spacecraft modes was 9. 4
percent. The test plan set a goal of + 1076 maxl-
data for this solar array had previously been mum coupling as evidenced by the off-diagonal
obtained. (Previous Nimbus modal testing had terms for successful mode shape measurement.
recorded ZOO channels of data with over 100 data The greatest deviation from the criterion was
points on the solar paddles), In-phase and quad- found in the higher paddle modes. This result
rature response plots referenced to the input was anticipated since very few accelerometers
acceleration were obtained according to the were available for paddle response measurement.
phase separation technique of Reference (9) for This became significant in the higher paddle
particular accelerometer locations. These lo- modes where the necessary coarse mass distri-
cations were selected based on analytical pre- bution was inadequate.
dictions of the paddle latch, ACS and sensory
ring responses. Frequencies of interest were A comparison of test and analytical rca-
identificd for resonant dwells. Since the test- onant frequencies is given in Table 1. In general,
ing was limited to the fundamental mode in each the measured resonant frequencies are in good
axis and ihle cross-axis modes in particular agreement with the corresponding calculated fre-
frequency tanea. not all resonances had modal quencies. This is particularly true for the funda-
dwells. Modal identification was established by mental pitch and yaw modes, the ACS roll mode,
4 i2 1 i 2" '6r
and the-''insecond1 paddle
•n T ,,,
plots of the quadrature comppaent of accelera- roll mode at 18.81 hz.
tion. These plots were of the type given in Other frequencies were considered to be reason-
Figure 6 and they were made for all of ,.jp dwell ably close to the corresponding calculated values.
frequencies. From this group, nine significant Resonant frequencies for which dwells were not
mode shapes were determined. These modes made were estimated from total response plots
were then used for direct comparison with the and are noted with an asterisk in the tabulation.
analytical mode shape.
Structural damping was determined by the
The measured responses for the nine method of Reference (9). This required the
modes obtained were considcred to behave as measurement of the frequency of the peak in-
natural modes. This was bared on resonably phase component of acceleration occurring just
good orthogonality shown by the evaluation of before and just after the resonance. The damping
the generalized mass matrix which was calcu- values obtained ranged from g =0. 032 to 0. 12.
lated using the normalized experimental mode This corresponded to the range of magnification
shapes. The average value of the off-diagonal factors obtained in the high level vibration test.

4 208 E

4'V
A constant damping coefficient of g 2 C/Cc
0. 10 was assumed for all modes in the analysis,

ANALYTICAL MODEL EVALUATION

The analytical model was evaluated on the

basis of frequency, mode shape, and amplitude
correlation with the analysis. Considering each
area of comparison it was concluded that the
analytical model was a good representation of
the ERTS spacecraft. The basis of this conclu-
sion is summarized in Table 2. Confidence in
the model was supported by the good frequency
agreement and the orthogonality checks.

In addition to frequency comparisons and

orthogonality checks, another evaluation of
mode shape was made. This was accomplished
by the calculation of the modal shear for both
test and analysis. The results here showed the
agreement between test and analysis predictions
to be within + 29 percent for the primary re-
sponse axis. There were larger deviations for
those modes where the quantity of paddle mea-
surements is critical. No criterion for this
evaluation had been previously set, but the re-
sults were considered reasonable.

Improvements in the amplitude response

were suggested by the damping evaluation where
a range of values were measured for individual
modes. A need for variation of the damping
coefficient was indicated, especially in the
lower frequency modes. Other possible areas
of improvement considered as a result of the FIGURE 6 - ERTS MODAL TEST PLOT
modal testing were in the mass distribution of FREQUENCY: 15.29 Hz
the sensory ring and the representation of the
launch vehicle adapter stiffness. Refinement
of the analytical model was obtained without made during the modal dwells restricted the
significant changes in the frequency correspon- scope of the analytical model evaluation. Where
dence. This resulted in better modal shear deviations from the criterion were large, as in
correlation with test within + 16 percent. Also, the orthogonality check and modal shear coin-
better amplitude correlation'was obtained by parison, they could be traced to lack of p-,lle
varying modal damping coefficients, measurements and not lack of validity of the
mode. It was possible, however, to establish
CONCLUSIONS confidence in the model and to uncover some
r.-as of discrepancy. This resulted in subse-
Analytically derived and test correlated quent modification and improvements in the
mathematical models providc the most desirable model. Improvement was made in the mode
data for use in ascertaining structural integrity shapes which was reflected in better correla-
of a spacecraft. Limitations in schedule and tion of the modal shear. Also, better response
cost, however, frequently force the dynamicist amplitudes were obtained by the selection of
to resort to abbreviated test and analysis pro- damping for individual modes. The adjustnemts
cedures. The ERTS/NIMBUS experience has in the model were accomplished with insignifi-
shown that satisfactory vibration models of sub- cant changes in the frequency correspondence.
systems or complete spacecraft can be estab-
lished by analysis, test or a combination of both. REFERENCES

I
The extraction technique was successfully dem-
onstrated by the correlation with the modal test. I. Beitch, L., "MASS System - The Computer
Program for General Redundant Structures
The limited quantity of measurements With Vibratory and General Static Loading,

209
 TABLE 1.
COMPARISON OF TEST AND CALCULATED NATURAL FREQUENCIES

CALCULATED TEST
NATURAL FREQUENCY NATURAL FREQUENCY
,NU.]BER CPS- MAJOR RESPONSE REGION CPS

1 13.39 PADDLES X-AXIS 13.72

2 14.13 PADDLES X-AXIS 15.29
3 18.48 PADDLES X, Y-AXIS 18.81
4 19.57 ACS X, Y-AXIS 17.47
5 21.81 PADDLES X, Y-AXIS 21.00'
6 24.03 ACS" Y, X-AXIS 24.25
7 29.29 ACS Y-AXIS 28.50*
8 31.09 PADDLES Z-AXIS 30.22
9 33.26 SENS.RY, PADDLE Y, X-AXIS 34.50*
10 35.77 PADOLL, SENSORY X, Y-AXIS 39.23
11 38.53 PADDLE,T,IST Y, X-AXIS 37.00*
12 40.24 SENSORY, PADDLES Y, X-AXIS 35.27
13 43.71 SENSORY, PADDLES X, Y-AXIS 39.00k
14 47.42 SENSORY X, Y-AXIS 41.00'
1s 48.92 SENSORY Y-AXIS 42.50*
16 53.57 PADDLES, SENSORY Y-AXIS 50.00*
17 55.34 SENSORY, PADDLES Z-AXIS 53.00*
18 58.24 PADDLES Z-AXIS 55.00*
19 58.76 ACS Z-AXIS 59.35
20 62.38 SENSORY, PADLdLES Z, X-AXIS 63.00*
21 63.54 PADDLES, SENSORY Z-AXIS 65.00*
22 66.24 ACS Z-AXIS 68.00*
23 76.68 PNEUIATICS Y-AXIS 79.50*
24 85.29 PADDLES Z-AXIS 90.00'

* APPROXIMATION FRO4 TOTAL RESPONSE PLOT; ALL OTHERS DETERNIfiED PRECISELY

FRO1M
QUADRATURE
PLOTS.

TABLE 2 - ANALYTICAL MODEL EVALUATION

BASIS ANAL. & TEST REMARKS

a. Frequency Very good Correlation within less than 12% in most modes.

b. Orthogonality Good 50% of all off-diagonal elements meet +10%

criterion; 80% meet 415% criterion; overall
average is 9.4%.

c. Modal Shear Good Primary axis of response less than 16% deviation.
Those in excess lack paddle representation.

d. Amplitude Fair Measured daping ranges from 0.03 to 0.12 accord-

ing to mode; analysis used 0.10 for all modes.
Better correlation obtained by varying modal
damping coefficients.

210

- - ...- ~ .IVII
 General Electric Co., TIS R66FPD172, Sep-
tember 13, 1966.

2. Cokonis, T.J., - "Structural Dynamics

Modal Matrix Methods for the Coupling of Space-
craft/Launch Vehicle Systems, " General Elec-
tric Company, TIS 68SD325, September, 1968.

3. Berman, A. and Flannelly, W. G. -

"Theory of Incomplete Models of Dynamic
Structures," AIAA Journal, Vol. 9, No. 8,
pp. 1481-1487, August, 1971.

4. Smith, F., Freelin, T.R., Romano, R.,

and Hutton, F., - Nimbus Spacecraft Modal
Survey Vibration Test Final Report, General
Electric Company, TIS Report 68SD281, June,
1968.

5. Hurty, W. C. - "Dynamic Analysis of

Structural Systems by Component Mode Synthe-
sis," JPL Technical Report 32-530, Jan., 1964.

6. Holbeck, H.J., Arthurs, T.D., and

Gaugh, J.J. - "Structural Dynamic Analysis
of the Mariner Mars '69 Spacecraft, " 38th
Shock and Vibration Bulletin, Part 2, 1968.

7. Freeland, R.E. and Gaugh, W.J. -

"Modal Survey Results from the Mariner Mars
1969 Spacecraft, " Shock and Vibration Bulle-
tin 39, February, 1969.

8. Hou, S., "Review of Modal Synthesis

Techniques and a New Approach, " Shock and
Vibration Bulletihj No. 40, Part 4, Dec., 1969.

9. Stahle, C. V., Jr., "Phase Sepax.tion

Technique for Ground Vibration Testing,"
Aerospace Engineering, Vol. 21, No. 7, July,
i 1962.

211
2W,

FINITE AMPLITUDE SHOCK WAVES IN INTERVERTEBRAL PISCS

William F. Hartman
The Johns Hopkins University
Bal timore, Maryland

The nonlinear deformation of intervertebral discs is discussed. The

upward turning stress-strain curve implies that the discs will tend
to shape pulses having sub-millisecond rise-times into shock waves and
that shock inputs will propagate as shocks. These implications are
explored for axial compressive impact of the spine, such-as is incurred
during aircraft-pilot ejection or during-a fall onto the buttocks.
Correlation with experimental results suggests that the application of
finite amplitude wave theory to the shock loading of the spine should
be further investigated.

INTRODUCTION structure consisting of an elastic fibrocarti-

laginous envelope, the annulus fibrosus, and a
A knowledge of the mechanical properties of fibrogelatinous core, the nucleus pulposus,
human vertebrae and intervertebral discs is fun- which contains water in a mucopolysaccharide
damental to understanding body mobility and collagen framework[2]. The disc is capped
spinal injuries and to designing protectors and above and below by bony end-plates which are
protheses for these spinal components. The dy- distinct from the vertebral bodies. This
namic response of an intervertebral disc must be structure of the disc suggests a possible appro-
studied in order to predict the vertebral priate mechanical model to be a sealed elastic
stresses that are caused by rapid motions or cylinder containing a viscous fluid. However,
impact. Most measurements of the dynamic mechani- not enough is known about the geometry or the
cal properties of the disc have been restricted material properties to warrant such modelling.
to small-amplitude deformations which are as- The disc seems mechanically symmetrical about
sumed appropriate for the perspective of linear- its vertical axis, as can be inferred from the
ized response. I here describe some of the compression measurements of Brown et al.[3].
large amplitude non-linear characteristics of Since the only data, which I shall discuss here,
the intervertebral discs and discuss the impli- pertains to compressive loads and the resulting
cation of these on the dynamic response of the average strain, I shall assume the disc to be
spine, uniform, having the average mass and size of the
actual disc-complex.
The human spine is a nonuniformly curved
column normally consisting of 7 cervical, 12 Herniation of the intervertebral disc
thoracic, 5 lumbar vertebrae and the sacrum- results in a loss of material fro the nucleus
coccyx structure. Each vertebra has a cylin- pulposus. This can be caused by trauma or
drical anterior part called the vertebral body disease and is one of the frequent etiological
and posterior arch-like parts called pedicles factors associated with back-pain and, in
and facets. The latter guide and restrict rela- severe cases, sciatica[4,5). Furthermore, 60%
tive displacements between the vertebrae for of all disc protrusions are caused by injury;
bending and torsion deformations but take up one-third of these by a fall onto one's feet
only 20% of axial loads[l]. The vertebrae are or buttocks, resulting mostly in injuries to
soft, cancelluus bones, decreasing in height the lumbar spine[6]. During static compression
upwards, from about 1.2 in.for the fifth lumbar of the lumbar spine the vertebral endplate
to about 0.5 in. for the third cervical vertebra. frequently fractures, allowing the nucleus
Adjacent vertebral bodies are separated by an pulposus to protrude into the spongiosa of
intervertebral disc which in the lumbar region the vertebral body[7. Often this causes the
is approximately 0.4 the height of the cancellous bone to collapse and continued
vertebrae. loading produces a comminuted fracture of the
vertebral body.
The intervertebral disc is a nonuniform

Preceding page blarn

I2.
 Hyptheses about Ehe type and mechanism STRESS-STRAIN CURVES FOR INTERVERTEBRAL DISCS
of dyn.mI1callY Induced injuries could result
from impact experiments or they may be formu- Load-deflection data for the compression of
lated from an extension of the available lumbar intervertebral discs are given in [3,10,
static data. Of course, viscous effects are 11]. The compression tests of both Brown et al.
naturally anticipated for the deformation of a [31 and Virgin[lO] employei conventional testing
structure having the material composition of mechines and used specimens which had the
the..disc. Although the viscous properties of posterior facets and pedicles removes. The
vertebral bodies and intervertebral discs have dimensions of specimenfs were not reported, so I
not been documented for the large non-linear calculate stress by dividing axial load values
deformations which precede failures, the damp- by the average area of the appropriate lumbar
Ing properties have been measured for small disc as given by Perey[7]. Strain is calculated
deformations. Fitzgerald and Freeland[8J have by dividing the deflection by the nominal
shown that the damping of fresh canine discs height[12] of the corresponding disc. For the
decreases with increasing frequency, while data of Virgin and Brown, this results in the
Markolf and Steidel[9] report the damping of stress-strain curve-shown in Fig. 1. Virgin's
the entire intervertebral joint is small for data is reproduced from his "typical" load-
the compression-tension mode of deformation. deflection curve while Brown's is computed using
These results suggest neglecting viscous effects the averaged values of four compression tests
in a first analysis of the stresses caused by which are each described in [3]. The two curves
compressive impact. It would be overly pre- are more than similar for both are proportional
sumptuous to expect that accurate results could to the-squared strain and the proportionality
be obtained for a large. range of impact speeds; constant Is the same for each. This suggests
but any-analysis using only-the data which is that the non-linear response of the disc can be
available is worthwhile if only to point out well defined, for these two sources of data were
the necessity for specific additional experi- obtained by-different investigators in different
mental programs. Inwhat follows I shall countries using different testing machines and
estimate the impact response of intervertebral probably different loading rates. The tests of
discs using static stress-strain data obtained Virgin were conducted 20 years ago, those of
from the literature. Brown et al. 14 years ago and yet, to my best
knowledge, I present here the first correlation
of any independently performed tests of this

0.30-

- -
E 9 kg/mme e *
E
E0
0'0

II II A

*0

00 0.05 Strain 0.15

0 0.005 (Strain) 2 0.015
Fig. 1. Stress-strain curves for lumbar discs obtained from the data of Virgin, o,
and Brown et al.,A. Filled-in symbols are stress plotted against squared strain.

214

Y4,
 0.05

?[ '
I+
ft
i: 0.04 1

,E, Imm
14 kig/mm 2

I2
0.02- /9 kg/mm2

0.01 .- z:T.- kg/mm 2

00
O0 ,'"i ,i i
0.01 0.02 Strain 0.04 0.05 0.06
I I I I I
0 0.001 0.002 (StraIn) 2 0.004

Fig. 2. Stress versus strain, o, and squared strain s - from fourth lumbar disc data of
Hirsch[ll]. The near-zero modulus for canine disc[8] isshown as ---.

kind. Furthermore, since neither [3] nor [10] specimens contained half the upper and half the
report the cross-sectional areas of the speci- lower vertebral body and the corresponding
mens, the shift in the curves of Fig. 1 might intervertebral joints. The posterior pedicles
be partially due to the use of inappropriate and facets have been shown to take up approxi-
areas for one or both sets of data. mately 20% of the axial load[l]. Therefore in
calculating the stress on the disc from Hirsch's
Reproducibility and consistency can be data, I use 80% of the load values. The
found inother mechanical properties of spinal longitudinal strain is calculated from the
components. For example, the average vertebral reported lateral bulge of the disc, assuming
fracture stress calculated from the data of uniform incompressible deformation there. This
Perey[7J is .34 kg/mm 2 , which is the same value isa poor assumption for large values of stress
obtained from averaging compressive fracture because Nachemson[13] has shown that the lateral
strengths given in [3]. Of course, itis bulging varies nonlinearly with axial strain and
reasonable to assume that a general material the disc's volume isknown to decrease[3]. How-
description will aoply to vertebrae and discs ever, it should hold approximately for small
only for the averaged data of several specimens values of stress and give at least an estimate
from several bodies. Variations are due to for moderate stress. This p)-codure results in
differences inage, size, sex, disease, injury, the stress-strain curve of Fig. 2. The initial
and specimen preservation. Normal deteriorations nonlinearity is precisely the same as that of
are not yet well documented and certain patholo- Fig. 1. The change in the slope of the stress-
gical disorders certainly go undetected. Never- squared strain plot might correlate with the
theless, the degree of consistency noted above inapplicability of the incompressibility
suggests that gross material characterization assumption. A qualitative correction, based on
isboth sensible and ultimately useful. Nachemson's measurements, would increase the
strains for increasing stress, thereby making
The compressive deformation of interverte- the slope more akin to the initial behavior
bral discs is also reported by Hirsch[l1]. His which agrees with the data of Fig. 1.

215

4 4.
WAVE PROPAGATION INTHE SPINE
A distinguishing feature of the stress- fracture stress -" -"
strain curve for the compression of interverte-
bral discs is its concavity away from the strain 0.30
axis. Such material behavior is typical of many 4
biological materials but here it has interesting
implications regarding stress-wave propagation --

along the spine. E

The one-dimensional finite amplitude wave b E
theory as developed by Karman and Taylor is
given in (14]. Some pertinent results of that
theory, which isapplicable for nonlinear k
strain-rate independent elastic or plastic 2
deformations, are as follows. If the stress- 0.1
strain relation iswritten as 0.10-
a= *() (1)
where a is the longitudinal stress, c the longi-
tudinal strain,'and #the response function, 0
then the speed of propagation, C(c), of any 00
strain level is 0.05
2 (2) Fig. 3. Stress versus strain for a lumbar
C(C) (/p)l/
vertebral body - from the data of [7].
where p isthe density of the solid. The
particle velocity, V(c), associated with any e
or-p) can be written distinctions. The vertebrae, especially for
large stresses, behave linearly. Also, the
strength of the disc isgenerally assumad to be
V(e) = c !dc' (3) greater than that of the vertebrae. A per-
spective retaining the individual material
properties of these components will result ina
If0(c) isan upward turning function then rather complicated picture for impact problems,
*'(c) increases with strain and so the wave especially since the sizes of the discs are such
speeds given by Eq. (2)increase with strain, that individually they are not rods, their dia-
An interpretation of this result is that if a meters being larger than their lengths. Since
unit-step shock input isapplied to the one- I have already indicated that I will ignore the
dimensional material speed it
the then the remain
ofwill maximum hesitate tostructure
nonuniform assume that the discs, I do wave
of one-dimensional not
shock-like since
stress is the greatest speed. On the other propagation can be used to obtain reasonable
hand, if the material issubjected to a estimates.
gradually risig pulse, then this pulse as it
travels through the material will become con- Using the data described thus far, and
tinuously steeper since the larger strains Eqs. (2)and (3), I consider: (1)the pulse-
propagate faster. Ifthe material is suffi- shaping performed by the discs, (2)the
ciently long, every pulse regardless of its prediction of the impact response of a preloaded
shape will eventually develop into a shock. vertebrae-disc-vertebrae specimen
parison with the experimental
and a
results com-in
given
Although the spine has the appearance of a (11), and (3)the prediction of the impact speed
curved rod, its axial dynamic response could be which would produce fracture in the fifth lumbar
properly described as one-dimensional only if vertebra. In all cases only compressive axial 4
its major components, the vertebral bodies and deformations are considered, the spinal curva-
the intervertebral discs, were dynamically ture, bending, twisting and viscosity are ignored.
similar. The vertebrae
as rigid bodies in comparison with theassumed
are sometimes discs for The gross weights and dimensions of fresh
dynamic analysis[12]. Such an assumption does canine tntervertebral discs are given in [8].
not square with the stress-strain relation shown Using these, a bulk density for the disc is
in Fig. 3. This was constructed using the data calculated to be 2.22 g/cm . Taking the average
on (7] for the compression of vertebral bodies. curve of Fig. 1 and the above value for P,
Both the deformation and the slope of this Eqn. (2)results in the curve of Fig. 4.
stress-strain curve are comparable with those of Nachemson and Morris[15] have reported invivo
Figs. 1 and 2. Therefore, since the densities measurements of intradiscal pressures foFi'al
differ only by a factor of 2, the spinal com- loads such as those occurring during standing
ponents must be considered dynamically similar. and sitting. The corresponding axial stresses
Nevertheless, there are important subtle range from .04 to .08 kg/mm 2 . The average

216
 stress isapplied sufficiently rapidly, such as
a stress pulse whose length is less than the
average pore size, then the effective cross-
sectional area is reduced and failure will occur
at a lower nominal stress. Thus the shock wave
formation as described here could account for
100 the fact that vertebral fractures occir in-the
thoracic vertebrae during pilot ejection even
[12]. no damage is sustained by the lumbar spine
when

Some impact experiments on spinal segments

E were performed by Hirsch[1l]. A lumbar disc and
half of each adjacent vertebra including the
posterior processes were preloaded in compression
and then subjected to impact by allowing a weight
50 50. to fall through a known height onto the loading
lever of the testing machine. Since Hirsch did
not report the dimensions or the raterials of
.10 a .30 the testing apparatus, the effective impact speed
(kg/mm 2 ) at the vertebra cannot be calculated. Neverthe-
mless, his recorded values of the dynamic bulging
of the disc together with the stress strain
Fig. 4. Variation of wave speed with stress. curves of Figs. 2 and 3 and Eqn. (3) permit a
check on the consistency of the present mechani-
cal perspective. Assuming continuity of stress
X-L5 X=Li XaT! and particle velocity at the vertebra-disc
interface the following Jump equations apply for

- ~~a
shock input.+(R (]
- [a ll + [ RI - 1 T]
T(4)

.08.8-"10.5 m sec 1- i 1 + [VR i VTJ

where 1, R label the incident and reflected shock
Fig. 5. Example of shock formation by discs. in the vertebra and T labels -the transmitted
shock in the disc. Using a constant vertebrae
impedance, the linear shock equations, the
vertebral fracture stress, which cin be obtained stress-strain curve of Fig. 2 and Eqn. (3), the
from the data of [7], is .35 kg/mW . Therefore, response curve of Fig. 6 is obtained. The
Fig. 4 shows that the wave speeds double over experimental points shown in the figure are
the range of probable dynamic stresses. plotted taking the equivalent impact speed to be
twice that of the drop speed. The agreement with
Fig. 5 illustrates the pulse shaping which the theory further suggests that it might be
can occur as a stress pulse propagates along the rewarding to pursue experimental studies of wave
spine. In this case a linearly rising pulse propagation in the spine.
increasing from the normal sitting stress to a
value which is less than the fracture stress Using Eqns. (4), the speed sufficient to
steepens from a 0.5 irsec rise time et the fifth cause vertebral fracture from impact onto an
lumbar vertebra to a partial shock an 0.1 msec infinite impedance may be calculated. The
tail at the first thoracic vertebra. Of course resulting expression is:
any pulse whose initial rise time is less than
0.5 msec would become a more fully developed
shock; while pulses whose rise times are longer 2[Vii = [aD] + $C(C')dc
than several msec would not be noticeably shaped. +z--(
Rise times less than.1 msec seem plausible for c
dynamic stressing
!iaircraft-pilot such as those occurring during
ejection. where Z. is the impedance of the -.rtebral body.

The porous structure of the soft cancellous In the above, let [a n * 0.26 and
bon2 of the vertebrae make them inherently c, c = 0.17, 0.05, respectiely. This
weaker in tension than in compression. During orr sornds to a dynamic jump from an tnilial
sufficiently gradual application of compressive stress of .04 kg/mm2 to a valie,.30 kg/mmz.
forces the deforming vertebrae uniformly distri- which is 90% of the fracture stress. The
bute the load effectively over the entire croEs remaining stress is easily accomplished through
section due to the collapse of voids and the the multiple reflections dt the infinite m-
reduction of porosity. However, if compressive pedance. Using the curve of Fig. 4, this gives

217
j
 6- 6- Calculated
A Measured (-Hirsch 1955)

E4

0.04 0.05 0.06

Strain increment

Fig. 6. The calculated additional strain increment due to impact is nonlinear with the speed.
The case shown is for a lumbar disc-vertebrae segment prestre-sed 0.005 kgria2 .

an impact speed in the vertebra of 12.2 m/sec, Surg., Vol. 39-A, No. 5, pp. 1135-1164,
which corresponds to a free terrestrial fall Oct., 1957.
from approximately 7.5 meters. Since buttock
and pelvic elasticity have been ignored, this (4] D. C. Keyes and E. L. Compere, "The Normal
value is surprisingly large and certainly does and Pathological Physiology of the Nucleus
not seem to agree with ordinary experience. On Pulposus of the Intervertebral Disc,"
the other hand, impacts which are truly axial J. Bone and Joint Surg., ol. 14, pp. 897-
are seldom achieved and the consideration of 938, 1932.
bending and shear would reduce the injury speed.
(5] S. Fribert, "Low Back and Sciatic Pain
c ile mchspeculative
of what I have discussed will Caused by Intervertebral
be considered because it is based Acta Ortho. Scand., Suppl.Disc
25, Herniation,"
1957.
upon minimal appropriate data, the existence
of finite amplitude shock waves in nonlinear (6] J. E. A. O'Connell, "Protrusions of the
intervertebral discs has been shown to be Lumbar Intervertebral Discs," J. Bone and
plausible and its study is potentially Joint Surgery, Vol. 33-8, pp. 8-30, 1951.
important in understanding dynamically
induced spinal injuries. [7] 0. Perey, "Fracture of the Vertebral End
Plate In the Lumbar Spine," Acta Ortho.
REFERENCES Scand., Suppl. 25, 1957.
(8] Edwin R. Fitzgerald and Alan E. Freeland,
[1) Af. Nachemson, "Lumbar Intradiscal "Viscoelastic Response of intervertebral
Pressure," Acta. Ortho. Scand., Suppl. 43, Disks at Audiofrequencies," to appear in
1960. J. Ned. and Biol. Eng.
(2] M. B. Coventry, "Anatomy of the Interverte- [9] Keith L. Harkolf and Robert F. Steidel,
bral Disk," Clin. Ortho. and Rel. Res., "The Dynamic Characteristics of the Human
Vol. 67, pp. 9-15, 1969. Intervertebral Joint," ASME publication

[3] Thornton Brown, R. J. Hansen, A. J. Yorra, 70-WA/BHF-6, 1970.

"Some Mechanical Tests on the Lumbosacral
Spine with Particular Reference to the
Intervertebral Discs," J. Bone and Joint

218
[10) W. J. Virgin, "Experimental Investigations
into the Physical Properties of the
Intervertebral Disc," J. Bone and Joint
Surg., Vol. 33-8, pp.607-611, 1951.

[11] Carl Hirsch, "The Reaction of Interverte-

bral Discs to Compression Forces," J. Uone
and Joint Surg., Vol. 37-A, pp.1188-1196,
1955.

[12) David Orne and Y. King Liu, "A Mathemati-

cal Model of Spinal Response to Impact,"
J. Biomechanics, Vol. 4, pp.49-71, 1971.
(13) Alf Nachemson, "Some Mechanical Properties
of the Lumbar Intervertebral Discs," Bull.
Hosp. Joint Disease, Vol. 23, pp.130-143,
1962.

(14) H. Kolsky, "Stress Waves in Solids,"

Oxford University Press, 1953.
(15) Alf Nachemson and James M. Morris, "In
Vivo Measurements of Intradiscal Pressure,"
J. Bone and Joint Surg., Vol. 46-A,
pp. 1077-1092, 1964.

219

219

a't

a'4~I
 ACCELERATION RESPONSE OF A BLAST-LOADED PLATE

Lawrence W. Fagel
Bell Telephone Laboratories, Inc.
Whippany, New Jersey

A solution for a simply supported plate loaded by a step-function

pressure is closely examined to determine contributions to acceleration
the plate's higher modes of vibration. Plate dimensions are as-
sumed to be such that classical bending equations apply, and it is estab-
lished that the peak acceleration response can be as much as 2.6 times
the peak resnse of a one-degree-of-freedom analog. When damping is
incorporated in the solution, the peak values are attenuated and much of
the very-high-frequency response appears to dissipate rapidly. In prac-
*.ical situations where the damping ratio will be at least 1 percent, nine
modes should adequately represent the plate's true response; however,
computed accelerations may be nonconservative b up to 40 percent if
only four modes are considered and by more than 100 percent if only one

INTRODUCTION MOTIVATION
A commonly used technique for calculating Structures which are designed to withstand
the approximate response of a plate subjected nuclear-weapon effects are often either shell-
to blast loading
equivalenit is to consider the plate
to a one-degree-of-freedom to be
spring- like structures
tective shell-likeorstructurei
are encapsulated within
whibh, for pro-
design
mass system, the frequency of which corre- and analytical purposes, are sometimes re-
sponds to the fundamental frequencyof the plate. garded as composites of plate elements. More-
simply supported plate acted upon by a uni- over, for structural-motion-response studies,
form step-function pressure is analyzed here to it is a usual inherent requirement to consider
determine the validity of this approximation. these plate elements to be externally loaded by
Classical bending equations for plates, that is, blast-induced overpressures* which may in
plane-stress equations, are assumed to be ap- some cases be approximated by a step function
plicable and these lead to a time-dependent, for determining early-time responses. In this
double infinitetrigonometric series solution in regard, a prevalent practice for analysts is to
which each term of the series represents the construct a mathematical model of the structure
transverse vibrational response of a different assuming that an entire panel can be repre-
mode. Comparison of the relative amplitudes sented by a single degree of freedom.t A direct
of each modal response demonstrates that the consequence of this approach is that the calcu-
first mode predominates for the dynamic re- lated peak acceleration response at the center
sponses of displacement and stress; hence the of a square plate element exposed to a step
one-degree-of-freedom approximation is ap- overpressure is
propriate for these quantities. However, the
amplitudes of some of the higher modes of ac-
celeration response are significant compared Peak Peak Overpressure
that these
to those of the first mode, indicatingsolution Acceleration 5Mass per Unit Area of Panel
in-
should not be ignored. Because the
volves an infinite series in time, the maximum (1)
amplitude of response is not obvious from the
solution expression. Acceleration responses
are plotted as functions of time to determine
the contributions associated with frequencies ;Pressures in excess of ambient.
higher than the fundamental. Peak acceleration
response for an undamped plate appears to be tThis practice is recommended in "Design of
about 2.6 times the first mode response i.e., Structures to Resist the Effects of Atomic
what would be calculated using the one-degree- Weapons," U.S. Corps of Engineers Manual
of-freedom analog. EMl110-345-110, 15 March 1957.

Preceding page blank 221

 4
(1.5 is approximate to about *10%; the value w = -- ^' w
of this coefficient depends on edge-support NxAcxoy
+Y
conditions.) An examination of a plate-
vibration-problem solutionfundamental,
that includesindi-
modes go
which are higher than the
cates that Eq. (1) characterizes the response in ij(t)
the first mode; however, the amplitudes of ac- i= j=I
celeration responses in modes other than the
first, although also significant, have been sin I= sin (4b)
ignored. a b
In situations where structural responses are
computed for the sole purpose of specifying Expanding q(x,y,t) in a double Fourier series
shock environments for acceleration-sensitive involving x and y,
components within the structure, it is of course
necessary that all modes which contribute sig-
nificantly to acceleration response be accounted 0 40
for. This apparently reasonable requirement Is j-r
unfortunately somewhat ambiguous, and theee- q(x,y,t) = L1 Pij(t) sin -i1*sin L-T-.
fore there is perhaps a need for a more quanti- J=I a b
tative definition of the number of modes which,
for the stated practical purpose, should be con-
sidered. The intentions of this study are to de- The Fourier coefficient Pr(t) is evaluated as
termine an upper limit of acceleration response
values resulting from the consideration of
higherSignificantly
which modes, to ascertain thetonumber
contribute of modes
accelerationi
response, and to establish a quantitative corn- Pi(t)
!b fa q(x,y,t) sin i.x sin 15 dd
dxdy.
parison between one-mode and many-mode ac- 0 0
celeration response solutions by considering a
simply supported plate loaded by a step-function
pressure. For uniform-loading cases where q(x,y,t) varies
with time but is constant with respect to x and
ANALYSIS OF A SIMPLY SUPPORTED PLATE y, i.e., q(x,y,t) = q(t),
The response solutions for an undamped sim-
ply supported plate acted upon by a step-function b a
{
overpressure are directly derived as follows. (t) 4q(t) I sin sin dxdy
Assuming that the classical bendings, i.e., plane ii ab J f a b
stress equations apply, the governing differen- 0 0
tial equation [11 is
Pi(t) = q(t) ocs

Vh w + Eh V4 w =q(x,y,t). (2) ab a 0, J
T
0
g 12(1- 2
v ) (
Therefore,

A deflection which conforms with the simple-

support boundary conditions at x = 0 and a, y =0 Pi (t) = 0 for even values of I or J, and
and b is
Pij(t) = -6- q(t) for odd values of i and J.
w= E E sin'a'snJT. (3
Jla
1= b
i**1 J=l If q(t) is a Heaviside step function of amplitude
Po, i.e., q(t) = PoU(t), then Pij(t) = 16 PoU(t)/
The derivatives of w are Iij f for odd values of I and J. The expression
for q(x,y,t) then becomes
y
t)Sil)in Sin Ly 16P Ut '0 sin sin).!
w J ;j a b (xv y t) 16P 0 .t (5)

(4a) i=1 j=l

222
Eqs. (3), (4), and (5) substitute into Eq. (2) to The initial conditions are 0i(0) .
0
form Solving Eq. (8),

i_ (t) sin i
sin 2 2N
g " a b S +W
(9)

Eh
121- (ifr)2 + (ig)2]2 t1i(t) =2(I - Cos coj t)

sin--sinsi a b
0 ~t Eq. (9 susiue intE._3)yild

n in nsin for odd-I

1=1 a b values 2 (10)
j1of I l
and j,

and each model differential equation becomes

w.L o
Using the6 stress-str'an relations for plates:
sin k inI

Ii 2 2,2 Ez, ( 2w 1, 2w\

for odd value ofi and e i-i 2,(b 2 1Y~-ow

0 for even values of i and j, (6a) 2 W/g - a b

- 16PoU(t)
Mryy +.U
IL-6I Eh-- ++
2 a b (11)

ij-16 'o Ez +V
with
displacements for evenbeing
all initial conditions values of I Modal
zero. and j the
are trivially equal to zero by Eq. (6a). For odd
values of i and J the equations are in the form Ez 2w 2w

2 16 o U(t)= AijU(t) (7)

ij2 2(a b t

wihaliiilcniiosbigzr.Mdlt
The Laplace transform of Eq. (7) is (l 7), (Lb
~j fr evn vaues f i223
dispacemnts
Salu(0s
for o ) i(+ ijd(S) bi sin irx sin ! y I - co s Wijt) (
are rivallyequl
6a) teoForodd2 tozer byEq.+ L
valus frm
i ad jthea quaionsarein
Ez
-------------------------------------
--------------- W2 - - -
 Ez. 2
w where I and j can only be odd values. The rela-
._ ... tive amplitudes of higher-modal values com-
xy = -pared
MY to the respective amplitudes associated
with the fundamental mode are plotted on Fig. 1.
Information from these graphs concurs with the
( 00 O0 philosophy that displacements and stresses re-
2 sulting from mode shapes other than the funda-
1G Po Ez
",2W/g1+v
+and
I mental are negligible
stresses
compared to-displacements
associated with the fundamental
I=1 jl mode. On the other hand, the contributions to
acceleration from some higher modes appear to
be significant. Just how the higher-mode accel-
({I- Cos it) eration amplitudes supplement the acceleration
COS cos jXy . (13) response of the first mode is not obvious from
a b 2 Fig. I nor from Eq. (14); therefore this phe-
a j nomenon will now be more closely examined.
Interest is focused at the center of the plate
The acceleration response is where the single-degree-of-freedom analog had
been presumed to be applicable. At the midpoint
sin Go the acceleration is
16 o a b COS Wijt . 0 0 sin -sin
,r2 6/go16
W/g~ J4-I Wmidpoint 2 W/g L4 i

(14) I=1 1=

In Eqs. (10) through (14) 1 and J are odd nd Cos Wit (15)

#h
ff 3ij\ i + -(. where l and j are oddJ. At t 0, ros wit I1
v 2) \2 2 for all i and J, and Eq. (15) degenerates to
a2

Comparing higher-mode amplitudes to " 16 E E 2o 2

fundamental-mode amplitudes for displacement, wmidpoint - W/"g - i J
stresses, and acceleration, we have: t=1 j=1

Displacement: where i and j are odd. Note that

S wi, 00o in

,(first mode) 2' 2

2 (-I)k 1
i (2k + 1)
izl,3,5,etc, k--0O
Stresses:
_, xxtlj tan-1I
4 j j4
ax (first mode) -/ 2
j (+j)
: (I+ ) so thata t ,

, (first mode) 1 )2 ,
16 PO (g (_)r

k n~p IF 2W/ig k r (2k +I' (2r + 1)

Accelerg.tion:
_i 16 P'O IT2 P0
"
w(first mode) iJ' - Wg - =W'

224

~ £
 '. 1 7.,7//777/7T. .
. . . . . . . . . . . . . . . . .. . .. . . . . . .
•~~~~~~~~~~~~ " . . . . . . . . .

9 . . . . . . . . . . . . . . . . . . . . . . . . . * . " . ". '. "..'

.
.. . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . .
. . . . . . . . . . . . . . 1. . . . . . . . . . . . .
II.. .
ISl. . . . . . . . . . . . . . . . . 15 . . . . . . . . . . . . . . . .
.
I17.. . . . . . . . . . . . . . . . IT . . . . . . . . . . . . . . ..
19 . . . . . . . . . . . . . . . .. it . . . . . . . . . . . . . . . . .

21 . . . . . . . . . . . . . . . . al . . . . . . . . . . . . . . . .
2$ . . . . . . . . . . . . . . . . 3- . . . . . . ... . . .J . . . .
25S.. . . . . . . . . . . . . . . . 113. . . . . . . . . . . . . . . ..
27 .. . . . . . . . . . . . . . . . T- . .. . . . . . . . . . ..
.S. . .. .. . . . . . . . . . .. .. . . . . .' . . . . . . . . . ..
1
o
• • • • • •I ~O .. . . . . . . . . . . . . . . "
.

RELATIlVE AMiPLITUDErSOF MODAL. DlISPLACEMENTS RErLATIVE AMPiLITUDESl Of MOAL NORMAL STRE~SrS

IT- 7-.4

5-. .

Ai

, ........ .. . .. . . . . . .........
... . . . . .. . . . . . . . . . . . .

. . . . .. ... . . . .7 . . ' , . . . . .
. . . . I . . . . . . . .

.. . . . . . . . . . .I- .. . . .. . . . . . . . . . . . .
J
1.. . . . . . . . . . . . . . .. . II. . . . . . . . . . . . . . . .
9 ........................................ 1 - j' . . .. . . .
. . . . . . 19 -I . .. . . . . . . . . . . . .
. . . . . . . . . .
I..I ................................................ _j
i2, . . . . . . . . . . . . . .
21 .. . . . . . . . . ...
23. . . . . . . . . . . . . ... . -J . . . . ... . . . . . . ....

27 .. . .. ....... ..... . .........

. ............................. - J
I- ................. 2P . ..............
S.?...... . . . . . .......
. . .. . .. . . I . . . . ......
€
RELATIVE AM/PLITUDES OF MO&.AL$HlEAM S."lESS RELATIVE AMdPLITUDES[
OF MW AL ACCELEPAITIONS

Fig. I Comparisons of the Relative Amplitudes of Modal Responses

225
 By-comparing this with the first-mode contri- they are based on damping forces defined'In a
button at t =0 [(16/) Po/(W/g) 1.6 Po/(W/g) manner which preserves the-linear nature of
the modal differential equations of motioi by
which approximately agrees.with Eq. (1)], it ntrodicing a viscous damping term. Damping
appears that the one-mode-only representation is thereby defined so that Eq. (7) is repFaced by
of response Is reasonable and conservative for
the initial value of acceleration. At t> 0 how-
ever, it is not'obvious that the absolute value 2 16' Po (t)
of-the-sum-.of-the contributlonsafromuall middeb
should remain bounded by the amplitude the 4'j + 2(3 wI 4,Ii + W4'=1
;,Iof i Mi-
I Ii 0
i t U~ ,
response of the first mode, and in fact, plots of (17)
normalized acceleration versus normalized
time show a contrary trend. That is, the
equation where 2i w-0 is the modal viscous damping
force,and iJ is the p~rcent-of-critical viscous
m n sin Lr sin Jr damping ratio. When the governing partial
Wmidpoin 6 2 2 differential equation i. definedso that Eq. (17)
Po/(W/g r2 I j results the total response w and its time-
*
P( ) J=l derivatives are summations of the modal values
oij and time-derivatives of OiJ respectively.
cos wijt (I and j are odd) (16) The solution to Eq. (17) in

plotted versus wot/2v on Fig. 2, indicates that

=
A1 1 e (ji
the acceleration response increases substanti- 1ij'(t) I - sin 1olI3 t + ,
ally when higher modes are included in the 2
2o i
calculation. These time-varying plots of Eq. (16)
for increasingly larger values of m and n
demonstrate the effect of considering higher
frequencies in the acceleration response. The
peak values from each of these graphs are 0 tan "I "
plotted in Fig. 3 as a function of the number of 1 9I
modes, and this curve appears to be asymptotic li
to about 4.12 Po/(W/g) as m and n become
large, that Is about 2.6 times larger than the and the modal acceleration is
one-m~ode-only representation of response. Up anthmolacertoni
to and including 225 modes are considered in
computing these values even though it is recog-
, tary
t
nizedinertia
i that Eq.is (2),
probably
which not appropriate
neglects shear for
and the
ro- 4'iJ = A e-lij=oijt
iji + (4
very-high-frequency mode shapes. Nonetheless, 2
this simplifying assumption permits the calcula- i
tion of an approximate upper value, and as will
be shown, th consideration of damnping can
greatly reduce the contributional effect of these in iJ t +0
very-high-frequency modes.
DAMPED RESPONSE / '
The previous discussion pertained only to - kj
Wicos -3 1 t+0
undamped vibration; therefore, computed accel-
erations must be conservative since all me-
chanical vibrations, certainly all structural For small values of PiJ 0 v/2 and the acceler-
vibrations, are impeded by damping. An effort
to consider structural damping is presented be- ation can be approximated by
low; however, some discussion regarding this
topic is appropriate.
The phenomena Involved in damping of struc- Ajt el-ijwijCt 16 P 1
tural vibrations is complex and to the author's ij e
=A(t) coB - - -
knowledge, no unified mathematical treatment V g
of the problem exists which exactly defines the
physics of such damping. It Is generally con-
ceded to be a combination of air damping and
ctructural damping and although approximate c 1 1
ij1 CO t
solutions exist which incorporate these effects,

226
 PLATE VIBRATION, 1 MODE CONSIDERED
MI n 1 0 PERCENT DAMPING
S 6.000 -

a.0o

1 0.000 0.200 0.400 0.600 0.800 1.000

CT/FUNDAMENTAL- PERIOD)
1.200 1.400 1.600 1.800 2.000

PLATE VIBRATION,9 MODES CONSIDERED

Mz5 nlz 5 0 PERCENT DAMPING
S6.000

00

.00

~V-

0.000.0.200 0.400 0.600 0.8010 1.000 1,200 1.400 1.600 LOW0 2.000
(TI FUNDAMENTAL PEPIOD)

PLATE VIBRATION, 8I MODES CONSIDERED

inn17 n 17 0 PERCENT DAMPING

600

40000 .0 0. .. 0.60 .0 . .0 .0 .0 .0 . .0
(TFUDAETA0PRID

Fig. 2 - oprsnotceeainRsos
Cosierin One and Many Mode

4 2.220

'.5 S w,4 ~s. . ~ '..4~. '~4(5, s~ '. s~x. ,.sYU.-i s~s'

 PAKACCEM[RATION AT THE MIDPOINT OF AN
UI09D SIMPLY-SUPPORTED PLATE SUBECTED TO A
STEP-FUNCTION PRES*UK. VALUiS ARE CALCULATED
FROM- TN[E-[XACT 9OUyION:.
ACcL[ERATION . isf11 S IN F
14 SW I
Po/(WI) 711 .IJul i i €O*4 lIt'
WHERE THE SUMMATION LIMITS ARE REPLACED BY
FINITE VALUES, OF m AND a FOR I AND j REAiCVILY.

tis i e n e l x ep bn
tI' it It is w i g
INDE[X OF m AND

1 4 9 16 29 3 49 44 IN 10 111 144 149 196 22

NUtM[M OF MODES$ €ONS1IIO
RAr

tFig. 3 -Peak Acceleration at the Midpoint of an Undamped Simply Supported

Plate Subjected to a Step-Function Pressure
Thus a damped modal vibration can be con- damping in real structural vibrations has been
sidered the same as an undamped modal vibra- the subject of extensive experimentation, and
tion except thati it Isenveloped by an exponen-
w i t.
there is apparent world-wide disagreement in
Ualdeay e tisepszdthths the interpretations of results from such experi-
reslteca ared aJ't isemofhadsie thtohi ments asevidenced by a sample of some of the
result is arrived at because of a desire for literature on the subject. While testing build-
mathematical simplicity rather than from an ing structures Alford and Housner [21 found that
exact definition of the physical phenomena for the damping ratios for the higher modes were
energy dissipation. The damping forces in the same as for the fundamental mode; and
structural vibrations are not, in fact, known to Sesan, Crongradi Diaconu and Strat [31 con-
be viscous; however, this simplifying assump- curred with this finding in a separate study
tion provides a means to an end in that it yields On the other hand, Kawasumi and Kmi[41
approximately correct dynamic responses, and claim to have experienced proportion:l damping
it is a standard method for evaluating damped by doing similar experiments. Nielson's [51
multimodal vibration responses. experiments showed that the same damping co-
efficient was evidenced at different modes during
Regarding values for modal viscous damping tests on one structure; however proportional
ratio iJ, the literature on structural damping damping was experienced while testing a differ-
contains different opinions regarding this sub- ent structure. Kimball [61 found that internal
ject. If the viscous-damped assumption were damping is neither proportional to frequency
completely true, a Kelvin (alIso known as a nor constant, but rather that for most materials
Stokes) model could be assumed for the mate- it increases to a maximum value at some fre-
rial (stress is proportional to strain rate as quency and then decreases for all subsequent
well as strain), the governing biharmonic equa- frequencies. Mindlin, Stubner and Cooper.[7[
recommend using a constant damping ratio.
tion would be modified by adding a (/ t) V4 w Adamson's [8) observations are similar to
term, and modal Eq. (17) would evolve where Kimball's; however, Adamson has a compro-
P J is proportional to wij. Whether or not the mise recommendation to use proportional damp-
modal damping ratio is indeed proportional to ing if only a few modes are inherent but a

228

'I * .,a %j.,t - 1 Adt3,'.~~~~xstkvt~ c A c-. - - ~ ~- 'ir. .! .

constant ratio if many modes of vibration exist sidered, most materials (even steel plates)
in the response solution, probably have minimum equivalent damping
ratios of about ['percent. The nine-mode curve
There exists, therefore, a variety of opinions on Fig. 5 indicates only about a 15 peicentd is-
regarding how equivalent modal viscous damp- agreement with the 225-mode response at 1 per-
ing ratios vary as a function of modal frequency. cent damping; therefore nine-modes should, for
The advice of Mindlin and others who recom- practical purposes usually be adequate for dy-
mend the use of a constant damping ratio, inde- namic response calculations involving plates
pendent of frequency is used in the investiga- subjected to impulsive loads.
tion below to-determine the effect that damping
has on the acceleration response of a vibrating If fewer than nine modes are used to repre-
plate subjected to a step-functi'n overpressure. sent plate-responses in a dynamic-analysis
If Pij = 3 for all values of I and j and j << 1, study, the calculated peak acceleration re-
sponses will possibly be nonconservative by an
the acceleration response at the midpoint be- amount dependent on the percent of critical
comes damping and the number of modes considered.
If the excitation is impulsive (blast loading),
the curves on Fig. 5 are indicative of the possi-
m n sin ir sin--! ble amount of nonconservatism which should be
W16 expected. For example, the peak acceleration
midpoint LL
P W i=1 j=1
-

I J response calculated for a 2-percent-of-critically

damped plate that is approximated by using only
one mode and considered to be subjected to an
impulsive load, could be nonconservative by a
of two.
e iJt CosWfactor If the excitation is oscillatory the above
recommendation is not intended to apply. If
dominant frequencies in the forcing function
where I and j are odd. The effect that P has coincide with frequencies of the plate's higher
on response is demonstrated in Fig. 4 which modes, these frequencies will contribute more
shows the midspan response considering 225 significantly to acceleration responses (dis-
modes with and .without damping. The very- placements and stresses also) than if the struc-
high-frequenicy characteristics quickly dis- ture were impulsively loaded. For these situ-
appear and the peak responses are attenuated. ations it most certainly would be prudent to
Of particular interest is the comparison of the have the mathematical model of the plate contain
amplitude of the damped-acceleration peak re- modes which at least include the dominant fre-
sponse considering few and many modes of vi- quencies of the forcing function.
bration. This is shown on Fig. 5 where it is
inferred from the asymptotic nature of the CONCLUSIONS
curve on Fig. 3 that the response considering
225 modes is equivalent to the response con- The contribution to peak acceleration re-
sidering an infinite number of modes. sponse of a blast-loaded plate from modes
other than the fundamental can be significant;
The curves on Fig. 5 suggest that larger therefore modal values associated with frequen-
values of critical damping ratio produce two cies above the fundamental should be incorpo-
desirable effects from a structural analysis rated in blast-response computations. For
viewpoint. They of course reduce the peak ac- engineering purposes where at least 1 percent
celeration responses; however, the error intro- of critical damping may be assumed, it is
duced by using only a few rather than many recommended that the mathematical represen-
modes to represent dynamic response is also tation of the plate contain at least nine modes
reduced. For materials which have three or if blast-loading conditions exist and accelera-
more percent damping, the difference between tion responses are desired. Computed acceler-
the peak responses using 9 and 225 modes is ations may be nonconservative by up to 40 per-
indistinguishable. From a practical standpoint, cent if only four modes are used and by more
when both air and structural damping are con- than 100 percent if only one mode is used.

229

-f1
 PLATE VIBRATION, 225 MODES CONSIDERED
mx29 n:29 0 PERCENT DAMPING
S6.000
. . . . . . . . . . . . . . . . . .°

3R° 4.000. . . . .--

-00
0 0.000

Z -4.000 K. .....
....... .

0.000 0.200 0.400 0.600 0.00 1.000 1.200 1.400 1.600 1.600 2.000
(T/FUNDAMENTAL PERIOD)

PLATE VIBRATION,225 MODES CONSIDERED

mz29 nz29 I PERCENT DAMPING
6.000

-. 2 000 -
-...... ...............

inA
: 0.000 AA /
1 - -- I-
0.000 0.200 0.400 0.600 0.800 1.000 1.200 1.400 1.600 1.900 2.000
(T/FUNDAMENTAL PERIOD)

,PLATE VIBRATION, 225 MODES CONSIDERED

m s 29 n 29 3 PERCENT DAMPING
6.000

4.00
2.000

S-2.000 .. V* o .

-4.0001 . . . H.... -
-6000 , i
0000 0.200 0.400 0.600 0.800 1.000 1200 1.400 1.600 1.800 2.000
(T/FUNDAMENTAL PERIOD)

Fig. 4 - Comparison of Normalized Acceleration Response Withot jid With Damping

230
 U )

to NOmsECONSINNRED
9 AIOSS CONSINERI
i i ! I I i I I
0 I 2 3 4 1 6 7 S
PIERCENT OF CRITICAL DAMPIIiN

Fig. 5 - Ratio of Peak Acceleration Response Considering

225 Modes to Peak Acceleration Response
Considering Fewer Modes

231
 APPENDIX I - REFERENCES

1. Timoshenko, Vibration Problems in Engi- 5. N. N. Nielson, "Damping in Multistory Build-

neering, Van Nostrand, January -1955. lngs Determined from Steady-State Vibration
Tests," ASCE Structural Engineering Ccn-
2. B. J. Alford and G. W. Housner, "A Dynamic ference, January 31, 1966.
Cf a Four-StoryReinforcedCo
lTest ncrete 6. A. L. Kimball, "Vibration Problems, Part V -
Building," Bulletin of Seismological Soiciety, Fito n ienl ~ Junlo
J 1Applied Mechanics, Vol. 8, 1941.
3. A. Sesan, I. Crongradi, D. Diaconu, and 7. R. D. Mindlin, F. W. Stubner, and H. L.
L. Strat, "Experimental Determinations of Cooper, "Response of Damped Elastic Sys-
Natural Periods and Dami in Buildings," tems to Transient Disturbances," Proceed-
Buletinul Institutului Polite cniciTomul X ings of the Society for Experimental Stress
(XIV), Fasc. 3-4, 1964. Analysis, Vol. 5, No. 2, 1548.
4. H. Kawasumi and K. Kana, "Small Amplitude 8. B. Adamson, "A Method for Measuring Damp-
Vibrations of Actual Buildings," Proceedings ing and Frequencies of High Modes of Vibra-
World Conference on Earthquake Engineer- tion of Beams," Publication of International
ing, 1956. Association for Bridges and Structural En-
g.nering, 1955.

2
 APPENDIX II - NOTATION

ab = plate dimensions in x,y directions w = transverse displacement

Alj = modal coefficient of forcing function w = transverse acceleration
E = Young's modulus W = weight per unit area =yh
g = gravitational constant x,y,z = plate coordinates
h = plate thickmess y = density
i,j = modal indices V = del operator
Pi (t) = Fourier coefficient =iJjjth frequency
Po = overpresmure 0o--- fundamental frequency
q = pressure function
= Poisson's ratio
S = Laplace transform variable
t = timeq = normal stresses

U(t) = Heaviside unit step function y = shear stress

xy

233
 EFFECT OF CORRELATION IN HIGH-INTENSITY

NOISE TESTING AS INDICATED BY

THE RESPONSE OF AN INFINITE STRIP (U)

Charles T. Morrow
V , j
Advanced Technology Center, Inc.
Dallas, Texas

(U) A narrow strip (bar or ribbon) is taken as a theoretical test case for
the realism of high-intensity no4 se testing in much the same way that a simple
mechanical resonator is commonly taken as a theoretical test case for more
conventional shock and vibration testing. It is shown that in an effort to
design a realistic test, one must consider tile point-to-point correlation of
the applied field as well as the sound pressure level, even when the damping of
the strip is large enough to prevent significant retu:-n reflections from the
ends. Three types of correlation are investigated in particular--complete cor-
relation at the coincidence angle for the given frequency, and independent ex-
citations at the different antinodal regions. With minor reinterpretation, the
formulas remain applicable when, in addition, an exponential decay of correla-
tion with distance in either direction along the strip is introduced.

(U) This type of analysis is also potentially useful in the prediction of

responses. The prediction method that has received the most intensive devel-
opment, Statistical Energy Analysis, is conceptually suited primarily to the
reverberant field, since normal modes are assumed in the exciting field as
well as in the structure. It may eventually be possible to calculate response
corrections for reverberant versus flight fields.

INTRODUCTION will necessarily do the same in any item of more

complicated internal dynamics. Yet, if the
(U) Environmental testing is intended to pro- responses were not similar in the simple res-
vide an evaluation, with a minimum of computa- onator, they would have almost no rhance of be-
tion, of whether a test item would survive and ing similar in the more complicated item of
function as intended, in practical use. To ac- equipment. In spite of the limitations of the
complish this instead by theory, for an item of simple test case cnu the compromises that are
equipment subject to shock or vibration, would necessary to achieve a practical environmental
be time consuming and inaccurate, and some modes specification, it is generally possible to pre-
of failure might be overlooked. The same would scribe a useful test.
be true for a complete space vehicle excited by
turbulence or by high-intensity noise. (U) Turbulence and rocket noise are distributed
excitationc. So is the high-intensity noise
(U) No environmental simulation is ever com- usel frequently now to simulate them. It is not
pletely realistic. Nevertheless, shLck and possible to obtain vaid criteria for realism
vibration testing can be a useful engineering by a test case that involves only a single-
tool when it is planned intelligently on the point input. The simple mechanical resonator,
basis of suitable criteria for realism. These even if the excitation iq conceived as a force
criteria are obtained by calculation of the re- acting on the mass, i-"inadequate.
sponses of a simple structure as a test case.
The time-honored test case, more than any other, (U) The test case ,ilized in this paper is
for vibration and shock, is the simple mechan- a simple strip of structure, narrow by compar-
Ica] resonator. For vibration, the amount, of ison with the lateral correlation distonces of
dazmp'ng is an important parameter. For shuck, the exciting field. The velocity of transverse
U .age potential is less critically depen- vibration along the strip is assumed to vary
dent n damping and the resonator is frequently with frequency, possibly according to the square
assumeL to be undamped. It is not true that root, but the precise relationship and the
two shocks or two vibrations that produce prac- fundamental theory behind it. are not criticaQ
tically identical .'espon3es in a simple resont, or lo the discussion. The strip could Le general-

Preceding page blank

 ized to two dimesinns, but the ultimate purpose (U) It is well knovn that the power spectral
Js inference about practical situations. As density for the sum or difference of two randon
with shock and vibration, the simplest test case signals is given by
yields mucb of the information one needs to know
for-this. For the present study the strip will
be assumed infinite in both directions. Damping, wA+B = wA + wB :2wAB
or, more properly, attenuation with propagation
distance, will be an important parameter. As w 1!2
+
we are concerned only with relative responses WA B WB) AB (1)
to differez,t fields, the mechanical impedance of
the strip on aisabsolute scale is unimportant.
The important field characteristics will turn where wA and wB are the individual power spec-
out to be not only the power spectral density of tral densities,
the sound pressure as a function of position and
frequency but the point-to-point correlation as correlation density, co-spectrum or real part of
well. the cross-power spectrum, and cAB 03 the nar-
row-band correlation coefficient.

(U) At one time, I carried out a simplified (U) That c is of magnitude not greater than
analysis of the infinite strip by assuming that unity msy be proved by squaring the sum and dif-
* excitation took place only at the antinodes, as ference of two random signals A and B, normal-
in Figure 1. This is a convenient simplified ized to unity variance by division by the re-
* model to visualize and to use for inferences in spective standard deviations oA and OB, and
advance of any calculation. For a completely averaging over time
correlated wave at normal incidence, the trans-
verse waves propagating in either direction from
alternate antinodes should be of opposite phase - -
2
and tend to cancel each other. The lower the A 2 2
2 2
damping, the more complete the cancellation. At (A/ I B/OB = A2/O 2AB/OA B + B =
the other extreme, for a completely correlated
wave incident at the coincidence angle so that 2(1 > 0. (2)
its trace velocity equals the velocity of pro- -AB
pagation of transverse waves in the strip, the
transverse waves propagating from the antinodes Hence
should all be directly additive. The lower the
damping, the more antinodes contribute signif-
B i
icantly to the summation. For completely un- IcAB- (3)
correlated excitations exhibiting no dominant
phase angle, the waves propagating from the
antinodes should combine as the square root of (U) In the analysis to follow, an expression
the sum of the squares, producing an inter- analogousof to equation (1) must be derived for
mediate behaviour. the sum an infinite number of random signals,
subject to a phase reversal according to whether
(U) Although the analysis based on this aim- the integer numbering a particular antinode is
plified model does not permit accurate prediction even or odd, and subject to attenuation of the
of the response of the strip to continuously dis- transverse waves on their way to the observation
tributed excitation, it does provide insights point, which will be chosen as the origin.
into the nature and effect of correlation. It
is therefore worth summarizing here as a pre- (U) Let a sinusoidal pressure p ut the i'th
liminary exercise. antinode result in a transverse 4elocity
8
vi = i P"*|
V, pi

i=-2 i=-1 0 i=1 i=2

k=-2 k=-1 0 k=l k 2

FIGURE 1 EXCITATION ONLY AT THE ANTINODES OF AN INFINITE STRIP.

236
at, the origin, with a real quantity. Simi-
larly, for the k in antinode, wI = (
Vk = Ok Pk' K5 1 (10)

(U) Now assume random pressures p if(t) and

Af. The and
Pkf(t) within a narrow bandwidth l
corresponding total velocity at the origin is K (l)ke Jk $ (11)

v Af(t) SpiAf(t) 0k PkAf(t) (6) where K is a constant of proportionality, 6i

i=-M k=-w a decay constant, and the choice of sign for the
exponents i and k has no numerical effect and

(U) The square is will merely produce a form more similar to that
of a derivation to come. Equation (6) becomes
(t ) (t )
V f2(t) = IP~ " Pf
p8
2
wf) =K w (f -k
i e-"- IiI+IkI)
pci(f). (12)
Sl
I k PiAf (t) PkAf(t) (T)

I- k=-w (U) For complete correlation at normal inci-

dence, all cp(f) = 1. By expressing Equation
pik
(U) Average over time and let Af approach zero (10) as the product of two summations in i and
to obtain the power spectral density of the k and applying the expression for the sum of a
total response binomial series and the definitions for the hy-
perbolic functions, we obtain

wv(f) L I k wpik(f) wvn(f) = 2wp(f) tanh 2 (-/2) (13)

i=-- k=--
1/2 (U) For incidence at the coincidence angle,
pik M
Fik [wpi(f) Wpk (f) (f)Ic
YQ (8) cpik(f) = (-I) i-k,and we obtain
i=- k=--

at the origin. All possible power spectral Wvcf = 2p(f)/tan h 2(=12)

densities (ik) appear In the summation, and
all possible co-spectra. The latter are shown (U) Finally, if the excitations are uncorrela-
also as geometric means of all possible pairs ted Fillt f hicions r bu
out,
of power spectral density, multiplied by the Cp (f) =fI when I = k. We obtain
corresponding narrow band correlation coeffi- pik
cients. w(f) = K2wp(f) tanh (15)
(U) If all i = 0k = 1, this would be ., direct
generalization of Equation (1), taken witi, the (U) These results are consistent with the
positive sign for the power spectral density of trends predicted in advance but are not accu-
a sum. If, for any pair of I and k, 8 8 k = -1, rate for a strip excited all along its length.
indicating a phase reversal before combination
at the origin, it would be a generalization of EXCITATION AT ALL POSITIONS
the power spectral density of a difference. (U) We will now turn our attention from this

(U) For the problem at hand, we will assume preliminary exercise to the analysis of a more
a constant power spectral density wp(f), a realistic model excited continuously as a
function of position. Except in special cases,
phase reversal when the integer i or k is odd, the summations become integrals. As phase
indicating an odd number of half wavelengths shifts in propagation are not limited to 0
spacing from the c;ign, and a transverse wave and 1800, the quad spectrum or imaginary part
propagating in either lirection from each anti- of the cross-power spectrum of the sound pres-
node and decaying exponentially. In short, sure has an effect.

237
 (U) In Appendix 1, the derivatior of a gcneral (U) For a completely correlated field incident
equation for the response of the continuous strip at the coincidence angle such that the trace
and expressions for three specific cases is car- velocity equals the velocity of propagation of
ried out in detail. transverse vibration in the strip, the final
# result is
to Equation (8)
(U) The equation corresponding
~is
i ) X2H2W(r) a2+T
cf.
22 ~v p 2
(02+h,2) (8
X(f)
Hwp (f) d a(Igl+lhla)

(U) Up to this point, the fields considered

0)dh, (16) have been completely correlated except for phase
(f)cos(Igj-Ihj- effects. We would like an indication of what
Lppens to the response as the correlation dis-
tance of the pressure field approaches zero.
(f) describe the correlation Let Cpg(f) = +1 and 0 IgI - i for g -Ag
where 0 and C
charactelistlg of the incident field, H is a I h < g + Ag, and Cpxy(f) w 0 for all other
constant, and g and h are continuous variables pairs of x and y.
along the strip, not confined to the integral 2
values i and k, and A is the wavelength of A2 (f) +1g
transverse vibration in the strip. The express- w (f) - -P dg
ion v qIhd g+Ag

(")akc i c( Igl+lhl )dh, (19)

is replaced by the continuous expression which approaches zero as Ag approaches zero.

(U) On the other hand, If we take the antinodes

C (f)cos(ig-jhj-0) to be the centers of half wavelength segments of
pg h the strip, as in Figure 2, and excite each sag-
ment by an independent normally incident wavo no
again dependent on both the transmission char- that the excitations of different segmantgP aro
acteristic and the correlation of the incident uncorrelated, the final result is
fiela. dlI2wp(f) 2 U .3
wvu M P '
completely
(U) For a (e.g. correlated field at normal
incidence incident normally from a single a2 + 2 1-u GU)
distant source), the final result is

X2 H2w(f) low, to weigh the realism of exiLting a lti'1i1-

_ ture by a field of one charuIcterintie at, I1
= 2 imulation of tile effect of field (l' it 111'"
vn a (17) ferent characteristic, we aru partiulaPrly In

-2 -1 0 1 2
0 0 0o 0 1 0° I
-2 -1 0 1 2
11- 2yfl

FIGURE 2 ANTINODES AS CENTERS OF HALF-WAVELENGTH SEGMENTS,

238
i.
terested in ratios of response. It follows CONCLUSION
immediately that

W(f) "2a
(U) The effect of the type of correlation of
vn l-e the correlation of the pressure field on
Rn =- 2 ctructural response is by no means negligible
Wvu(f) 2+3e'a-e
3
, (21) even for infinite structures of typical Q's or
structures long enough so that return reflect-
ions are negligible. Introducing an exponential
is the ratio of normal incidence to uncorrelated decay of correlation along the strip has the
response. The ratio of coincidence to un- same effect on response as increasing the inter-
correlated response is nal attenuation. If the analysis given-here
were extended to cover a strip of finite length,
it would also show a marked difference in re-
W (ff le.2a sponse to the different fields according to
vc • (a (22) whether the number of antinodes is odd or even.
2 2
wVU(f) a2 (a +4V ) 2+3e'o-e'3a (U) It follows that the correlation of the
pressure field as well as its sound pressure
(U) We can find an equivalent Q corresponding level must be considered in establishing a
to the decay constant a in the following way. simulation. As usual, however, realism must
Consider a finite strip starting at g=O and be a compromise limited by practical constaints.
ending at gun, an integer. The return wave When it is not possible to coritr0l the correla-
from the m'th reflection at the far end, after tion closely, it may be desirable to introduce
stoppage of the excitation, is proportional to corrections in the SPL to compensate. Analyses
such as the one given here may be useful in
suggesting the amount of correction to apply.

ACKNOWLEDGMENT

I am indebted to my colleague,
is The time of arrival of the m'th return wave Warren A. Meyer, for verifying the derivation
of Equation (20).
t = mn/f,

so that

mn=ft.

(U) A simple resonant system decays according

to

(U) Consequently, the equivalent Q is

Q = /2u (23)

(U) The two ratios are plotted against both a

and Q in Figure 3, bracketing the possible re-
sponses of an infinite strip or a strip with
enough internal attenuation tc make returns from
,.nd reflections negligible. Quite large effects
of the type of correlation occur for Q's that
are typical of airframe structure.

239

- ,- ' - - 2' ."" ... ... . , - ........ .., .... , . . , , o- '

 11.
0.2 0.4 0.6 0.8 1.0 a
50 -

10

S1 " Rc FOR COMPLETELY CORRELATED

|1- - . EXCITAlION AT THE COINCIDENCE ANGLE

20 10 5 0 2

o.1
7 € ". EXCITATION AT NORMAL INCIDENCE

0.1
0 0.2 0.4 0.6 0.8 .

FIGURE 3 RATIO OF RESPONSE P.S.O. TO RESPONSE P.S.D. FOR EXCITATION

CORRELATED ONLY WITHIN HALF WAVELENOTH SEGMENTS.

240

- V
 APPenIDIX I
ANALYSIS OF AllINFINITE STRIP

EXCITED AT ALL POINTS

(U) The equation very simply.

+ 2 ft
(U) The analysis to follow will be expendited
e n = cos 21ft + J sin 2nft, (24) by using the complex exponential.

(U) Imagine a loop of magnetic tape, with a

where j is a unit vector along the axis of sample of random pressure signal, played back
imaginaries, expresses a rotating unit vector continuously so as to generate an arTificially
as a complex sum of projections along two periodic function, with energy at the funda-
orthogonal axes. The symbol e is the base of mental frequency and at each harmonic. Let us
natural .logarithms. It follows that examine the n th harmonic, which is among those
within a small bandwidth Vf. Let the pressure
at position x

eJ 2 11f t f+ eJ2sft j (27ff- )

=
2 (25) p P MP (30)

at the k th antinode result in a transverse

or more simply, the cosine function can be velocity response
extracted from Equation (24) by taking the real
part. J (2rfm-lxm)x
(U) A phase shift 0 can be expressed as com- &vm a P e AX (31)

plex multiplier:

e. o
J
j2
ft
e(2tft-0) at the origin. The yXM is taken to be complex,
e e for all possible phase shifts can occur in
propogation Prom various x to the origin, but
it is assumed constant over Af, equal to
cos(2,nft-0) + J sin(21ft-0) Yxfo
(U) The velocity produced by all pxn within
Af is
or
f-I" AvJ(2nf -0)
mAV
mxm
oe 0 J2ift . ej(2,7ft-)
Af
cos(2ft-0) - J sln(2dft-0) (27)
•f x J2nf 2JI x~
e-x ( 2

so that e Pxm e Ax (32)

t~
+Let
J(2nf-0)
0)
cos(2nft-0) ej(2nt- ) + e- J (21ft -
2 (28) "Jexm (33)

Note also that (U) The total complex velocity at the origin
is
d j2nft J2nfr j2 n ft
dt ' (29)

241

/V
 F-i1

the transmission characteristic of 'the strip,

Yxm P m J (2i~f
) dx 0 are alwy andsimple functions of frequency and
position way be assumed constant over Af
Vxf Af xm as follows:

I * J(2f"-Q-0=)d
j (20 2021 X
f xn xn e dx G G (38)
Af(34)
x=e
X (39)
(U) The total real velocity at the origin Is 0
eyn y (40)

VA f G P cos(2 1vf-e -0 )dx "

eI x y eJ("
-' +)

xn Pyn e 0 dy (4
"J" Gxmui

-j (211fro"
Ox0-+xC)]dx,+e-XW E • dy
+x •M (3)A xn Pyn (l

xm~d (35)

which can be written in terms of y as the in-

dependent variable and n as the subscript. Ob-
2 f
f "=
J
tain the square by multiplying the integrals in
x and y together, and average over time

P P OS 0-0 yn)

+ sin ( Iy

V f 2 dx f Gx G P PAf
xfm yn xm yn
+ 2f
r-j 2
+ ___ -0y)- As(
f
fe xy mon+ e xw yn xn yn Idy PP
exm yn sInn(mxm-y
n dy h3)

• dx P P
dx G Pxn Pyn
Pxn
2yn

(U) 11ow let the length of the sample increase

coS(xm.Oxn+0n )dy"
y(36) (36) beyond limit and Af approach zero, more slowly
so that the number of spectral lines within Af
also increases beyond limit. The power spectral
thlePxn a
Now, te
(U) (U)No, Pn are random quantities,
and nnaerdo density of transverse vibration at the origin is

The 0 anid 0 are also random, although their

xn
difference mayynoradj
may not be a simple function dx
of frequency and ponition, depending on the de-
Wv(f) GxGy e0O W (f)
M + jqp
j (f)j
JJ.L P Y PXY
gree of correlation of the incident wave. flow-
ever, Gxm Gyn n0 and 0 , characterizing the
(hi)
y xn yn

242
 e dy
+ eJ [Wpxy(f) - Jqpxy ()]1 1

where w is.the co-spectrum or real part of J(0-)

dy f d x GXY
the cross-power spectrum and x is the quad- "/
spectrum or imaginary part. It follows that 1w/() c
Wp(f)I2 o"d

)k(f xy ) - f - '€
0 (45)
wLPC u(f
PY J .PY COSW(0)dy ,
(5)

where tow w ) +[ xx( ]l/

z( r) let g = 2x/A sa
PXY
PX +q XY(46) and
adh - 2y/A (53)
where A is the wavelength of transverse vibration
qpxy(f ) in the strip.=
Wx
= 2wf.2
(f f)]/2)52 6f,G(jn
(fk- f d9)Wvf

(U) Further, the magnitude W y(f) is obtained ) ]1/h2g~

Cxyfcs(~id
by mutltiplying out the complex conjugates and 9 hW9ph CpYf)o(OOd
taking the square'root. (5")

Wpx(f) " PxM P o~lmn Let

WpXyf - Cos(00 ) Lt = He-Igl (55)

___ COfl Xyf

Px nG h0)+E
Ieah
xm yo(9f P (56)
sln((xm. yn)0 l2o:(19l-lhl) = (57)
yn(D"-' 2 Sin(0m-0 and
fW ((f)
9f) = wplf), (58)
(U) It is important to note that within etch
product the m's and n's of one summation are a constant with distance.
independent of those of the other. Without ac-
tually assigning new symbols to one pair of sub-
A2 11
2
scripts and multiplying out, we can see that the w (f)
result would reduce to a single summation of w (
4 d
.
I .fe-a(Igl+lhl)
terms each involving four pressure amplitudes v
multiplied by the squared cosine of a conpond
difference angle, and, if all these difference
angles should be zero, reaches a maximum value
Cpgh~fes( gl.l -)h

Max W (f) Wpx (f) w (f) W f) 1/2 , (19) which is the same as Euation (16).

w (U) Consider first a field completely correlated

when Af approaches zero, which can also be in- so that Cpxy(f)
ferred from Equation (3). X =1 for all x and y, and normally
incident so that 0 o.
(U) If we define a new coefficient

Cpx(f)
p
*1(f)/
Wpxy
Wpx(fW
(OW y M ,px
f)
Mv/2- A2CWP f ,
ea(Igl+lhl)
" (50)

243

- ,
r
P e-0 d9H~
+f -(.
g+2rgdg
2
)2 2w M
(f
0 Jc(g+h)4
4
-aO) Iffo
f&5h-Jguhdh]

A2H2W M xc-~i ~zJu

2 U0 5120 1

A2 ~gh)+ (f) ,g9h)J Finally cosie a4on il uhta

X2H2W (f)
is the
Eqaton(1).p~
which~~5 12n~ .~

poiie2 in 2 2
2 22
A H~w
I (f
''gjIhI)A24 f
-i
we )-ajrhd3 - '-Jdg e~ Boh)r
4o whic istesaea qato 1)

(f)
A2HHVw d e1d U ialcnie
1_gj on il uhta

k4 k+-2 PX

excep
haeefetHs
fo ht 1 (f) w 2

24

X2H2W (f
cos~gj--jhjh~d +2 _c(j~jjhj+J,,j~jjhj
8 gf ed

k- k-

~j - [ r~ii
-, -jh~j)Jv 'j h 2 , ma~ja aS
 - 9.chl+jwilhlh
e )22H2W -2a)- 3c -2m +e:c
+f . dik
. , - 2+2e-2t e 2
+j~1
* 2 e +0 +0
2 2 1a 2 2
2 [1- 3 2a
f ) 2H2w (fM
~~2H2w 3
- (f +e~ a
I .1 (. 2 2) l~e-2a
edg dg 2 ejIhIJ
eI dh

kf-ki
2uw k-l which is the same as Equation (20).
2 , 2

2 2
- HW 5 -cg+Jirg
ef~ 2-ah-juh
Mdg f e dl'

,k k4
+ I. fT IF C-ag+J1Tg d +2 -ah-ith dh

2 2 _g e +2~

I a- I

2
2 4- j (k+-i-)
+~~~~[.ky- + e

p +C -A2[J /2 + j k4)]

k-a + C-0 jJ

- AH~wV)~ + e - +e/ +~I/

e

~2+~.2

12

245