CSI Reliability Week, Orlando, FL October, 1999

INTRODUCTION TO OPERATING DEFLECTION SHAPES

By
Brian J. Schwarz & Mark H. Richardson
Vibrant Technology, Inc.
Jamestown, California 95327

ABSTRACT TWO TYPES OF VIBRATION

Mode shapes and operating deflection shapes (ODS’s) are All vibration is a combination of both forced and resonant
related to one another. In fact, ODS’s are always meas- vibration. Forced vibration can be due to,
ured in order to obtain mode shapes. Yet, they are quite
• Internally generated forces.
different from one another in a number of ways. In this
paper, we will discuss ODS measurements, and their rela- • Unbalances.
tionship to experimental modal parameters. • External loads.
• Ambient excitation.
INTRODUCTION An operating deflection shape contains the overall vibration
In another article on operating deflection shapes [2], the au- for two or more DOFs on a machine or structure. That is, the
thor made the following statement, ODS contains both forced and resonant vibration compo-
nents. Other the other hand, a mode shape characterizes only
"Operational deflection shapes (ODS’s) can be measured the resonant vibration at two or more DOFs.
directly by relatively simple means. They provide very use-
ful information for understanding and evaluating the abso- Resonant vibration typically amplifies the vibration re-
lute dynamic behavior of a machine, component or an en- sponse of a machine or structure far beyond the design levels
tire structure.'' for static loading. Resonant vibration is the cause of, or at
least a contributing factor to many of the vibration related
What is an Operating Deflection Shape? problems that occur in structures and operating machinery.
Traditionally, an ODS has been defined as the deflection of a
structure at a particular frequency. However, an ODS can be To understand any structural vibration problem, the reso-
defined more generally as any forced motion of two or more nances of a structure need to be identified. A common way
points on a structure. Specifying the motion of two or more of doing this is to define its modes of vibration. Each mode
points defines a shape. Stated differently, a shape is the mo- is defined by a natural (modal) frequency, modal damping,
tion of one point relative to all others. Motion is a vector and a mode shape.
quantity, which means that it has location and direction asso-
ciated with it. This is also called a Degree Of Freedom, or UNDERSTANDING RESONANT VIBRATION
DOF. The majority of structures can be made to resonate. That is,
Why Measure ODS’s? under the proper conditions, a structure can be made to vi-
Measuring ODS’s can help answer the following vibration brate with excessive, sustained, oscillatory motion. Reso-
related questions, nant vibration is caused by an interaction between the inertial
and elastic properties of the materials within a structure.
• How Much is a machine moving? Striking a bell with a hammer causes it to resonate. Striking
• Where is it moving the most, and in what direction? a sandbag, however, will not cause it to resonate.
• What is the motion of one point relative to another (Op-
erating Deflection Shape)? Trapped Energy Principle
One of the most useful ways of understanding resonant vi-
• Is a resonance being excited? What does its mode
bration is with the trapped energy principle. When energy
shape look like?
enters a structure due to dynamic loading of any kind, reso-
• Is there structure-born noise?
nant vibration occurs when the energy becomes trapped with-
• Do corrective actions reduce noise or vibration levels? in the structural boundaries, travels freely within those
boundaries, and cannot readily escape. This trapped energy
is manifested in the form of traveling waves of deformation
that also have a characteristic frequency associated with
them. Waves traveling within the structure, being reflected
off of its boundaries, sum together to form a standing wave
of deformation. This standing wave is called a mode shape,
and its frequency is a resonant or natural frequency of the
structure.

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Another way of saying this is that structures readily absorb

Time Domain ODS
energy at their resonant frequencies, and retain this energy in
An ODS can be obtained from a set of measured time do-
the form of a deformation wave called a mode shape. They
main responses,
are said to be compliant at their natural frequencies.
Why then, won't a sandbag resonate when it is struck with a • Random.
hammer? Because energy doesn’t travel freely within its • Impulsive.
boundaries. The sand particles don't transmit energy effi- • Sinusoidal.
ciently enough between themselves in order to produce • Ambient.
standing waves of deformation. Nevertheless, a sandbag can Figure 1 shows the display of an ODS from a set of impulse
still be made to vibrate. Simply shaking it with a sinusoidal response measurements.
force will cause it to vibrate. Sandbags can have operating
deflection shapes, but don't have resonances or mode shapes.
Local Modes
Energy can also become trapped in local regions of a struc-
ture, and cannot readily travel beyond the boundaries of
those regions. In the case of an instrument card cage, at a
resonant frequency of one of its PC cards, energy becomes
trapped within a card, causing it to resonate. The surround-
ing card cage is not compliant enough at the resonant fre-
quency of the card to absorb energy, so the energy is reflect-
ed back and stays trapped within the card. The card vibrates
but the cage does not.
Many structures have local modes; that is, resonances that
are confined to local regions of the structure. Local modes
will occur whenever part of the structure is compliant with
the energy at a particular frequency, but other parts are not. Figure 1. Time Domain ODS From Impulse Responses.

VIBRATION MEASUREMENTS Frequency Domain ODS

An ODS can also be obtained from a set of computed fre-
The vibration parameters of a machine or structure are typi- quency domain measurements,
cally derived from acquired time domain signals, or from
frequency domain functions that are computed from acquired • Linear spectra (FFTs).
time signals. Using a modern multi-channel FFT analyzer, • Auto power spectra (APS’s).
the vibration response of a machine is measured for multiple • Cross power spectra (XPS’s)
points and directions (DOFs) with motion sensing transduc- • FRFs (Frequency Response Functions).
ers. Signals from the sensors are then amplified, digitized, • ODS FRFs.
and stored in the analyzer's memory as blocks of data, one
Figure 2 shows the display of an ODS from a set of FRF
data block for each measured DOF.
measurements.
ODS MEASUREMENTS
An ODS can be defined from any forced motion, either at a
moment in time, or at a specific frequency. Having acquired
either a set of sampled time domain responses, or computed
(via the FFT) a set of frequency domain responses, an operat-
ing deflection shape is defined as:
Operating Deflection Shape: The values of a set of time
domain responses at a specific time, or the values of a set of
frequency domain responses at a specific frequency.

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Figure 2. Frequency Domain ODS From a Set of FRFs al speed (order) of the machine can acquired with a single
channel analyzer, but only if the operation is repeatable.
FRF values at a frequency, taken from two or more FRFs,
is an ODS.

TESTING REAL STRUCTURES

Real continuous structures have an infinite number of DOFs,
and an infinite number of modes. From a testing point of
view, a real structure can be sampled spatially at as many
DOFs as we like. There is no limit to the number of unique
DOFs at which we can make measurements.
Because of time and cost constraints, we only measure a
small subset of the measurements that could be made on a
structure. Yet, from this small subset of measurements, we
can accurately define the resonances that are within the fre-
quency range of the measurements. Of course, the more we
spatially sample the surface of the structure by taking more
measurements, the more definition we will give to its ODS’s Figure 3. Repeatable Operation.
and mode shapes.
Steady State (Stationary) Operation
DIFFICULTY WITH ODS MEASUREMENTS Steady state, or stationary operation can be achieved in many
situations where repeatable operation cannot. Steady state
In general, an ODS is defined with a magnitude and phase operation is achieved when the auto power spectrum (APS)
value at each point on a machine or structure. To define an of a response signal does not change over time, or from
ODS vector properly, at least the relative magnitude and measurement to measurement. Figure 4 shows a steady state
relative phase are needed at all response points. operation. Notice that the time domain waveform can be
In a time domain ODS, magnitude and phase are implicitly different during each sampling window time interval, but its
assumed. This means that either all of the responses have to auto power spectrum does not change.
be measured simultaneously, or they have to be measured
under conditions which guarantee their correct magnitudes
and phases relative to one another.
Simultaneous measurement of all responses means that a
multi-channel acquisition system, that can simultaneously
sample all of the response signals, must be used. This re-
quires lots of transducers and signal conditioning equipment,
which is expensive.
Repeatable Operation
If the structure or machine is undergoing, or can be made to
undergo, repeatable operation, then response data can be
acquired one channel at a time. To be repeatable, data acqui-
sition must occur so that exactly the same time waveform is
obtained in the sampling window, every time one is acquired.
Figure 3 depicts repeatable operation. For repeatable opera- Figure 4. Steady State Operation.
tion, the magnitude and phase of each response signal is
unique and repeatable, so ODS data can be acquired using a For steady state operation, ODS data can be measured with a
single channel analyzer. An external trigger is usually re- 2 channel FFT analyzer or acquisition system. The cross
quired to capture the repeatable event in the sampling win- spectrum measurement (XPS) contains the relative phase
dow. between two responses, and the auto power spectrum (APS)
of each response contains the correct magnitude of the re-
Order Related Data
sponse. Since the 2 response signals are simultaneously ac-
A single channel analyzer can also be used to acquire ODS’s
quired, the relative phase between them is always main-
using a tachometer pulse as the trigger. In this case, spectral
tained. No special triggering is required for steady state op-
magnitude & phase data at any fixed multiple of the rotation-
eration.

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forms of excitation are either unmeasured or un-measurable.

ODS’s FROM FREQUENCY DOMAIN MEASUREMENTS
On the other hand, ODS’s can always be measured, no matter
Any set of vibration data taken from a structure is the result what forces are causing the vibration.
of applied excitation forces. Whether it be operating data,
caused by self-excitation, or data taken during a modal test Transmissibility
under tightly controlled excitation conditions, the operating Transmissibility measurements are made when the excitation
deflection shapes are always subject to both the amount and force(s) cannot be measured. Transmissibility is a 2-
location of the excitation. channel measurement like the FRF. It is estimated in the
same way as the FRF, but the response is divided by a refer-
Linear Spectrum ence response signal instead of an excitation force. Phase is
This frequency domain function is simply the FFT of a sam- also preserved in Transmissibility's, and a set of them need
pled time domain function. Phase is preserved in the Linear not be obtained by simultaneously sampling all of the time
Spectrum, so in order to obtain operating deflection shapes domain responses. Each response & reference response pair
from a set of Linear Spectra, either the measurement process must be simultaneously sampled, however.
must be repeatable, or the time domain signals must be sim-
ultaneously sampled. Since the Linear Spectrum is complex As with FRFs, a set of Transmissibility's contain both magni-
valued (contains both magnitude and phase information), the tude and phase at each frequency, so ODS’s obtained from a
resulting operating deflection shapes will also contain magni- set of Transmissibility's will also contain correct magnitude
tude and phase information. and phase information. The units of the operating deflection
shapes are response units per unit of response at the refer-
Auto Power Spectrum ence DOF.
The APS is derived by taking the FFT of a sampled time do-
main function, and multiplying the resulting Linear Spectrum An unexpected drawback of Transmissibility measurements
by the complex conjugate of the Linear Spectrum at each however, is that each resonance is represented by a “flat
frequency. Phase is not preserved in the APS, so a set of spot” in the data instead of a peak. This is shown in Figure
these measurements need not be obtained by simultaneously 5. The top curve in Figure 7 is a response APS showing 4
sampling all of the time domain responses. Since phase is resonance peaks. The Transmissibility below has “flat
not retained in these measurements, operating deflection spots” (no peak) in the frequency range where a resonance
shapes derived from them will contain only magnitude, and peak occurs.
no phase information.

FRFs
The FRF is a 2-channel measurement, involving a response
and an excitation signal. It can be estimated in several ways,
depending on whether the excitation or the response has
more measurement noise associated with it.
The most common calculation involves dividing an estimate
of the cross power spectrum (XPS) between the response and
excitation signals by an estimate of the auto power spectrum
(APS) of the excitation, at each frequency. Averaging to-
gether of several XPS’s and APS’s is commonly done to re-
duce noise in these estimates.
Since a set of FRFs contains both magnitude and phase at
each frequency, the operating deflection shapes derived from
a set of FRFs will also contain both magnitude and phase
information. The units of the operating deflection shapes are
acceleration, velocity, or displacement per unit of excitation
force at the reference DOF.
Difficulty with FRF Measurements
FRF measurement requires that all of the excitation forces
causing a response must be measured simultaneously with Figure 5. APS & Transmissibility.
the response. Measuring all of the excitation forces can be
difficult, if not impossible in many situations. FRFs cannot
be measured on operating machinery or equipment where
internally generated forces, acoustic excitation, and other

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equal to the mode shape. This concept becomes clearer

ODS FRF when sine wave excitation is considered.
An ODS FRF is a different 2-channel measurement that can
also be used when excitation forces cannot be measured. USING A SINUSOIDAL ODS AS A MODE SHAPE
The advantage of the ODS FRF over the Transmissibility is
If a single sinusoidal force excites the structure, its steady
that the ODS FRF has peaks at resonances, thus making it
state response will also be sinusoidal, regardless of the fre-
easy for locating resonances.
quency of excitation. However, the ODS that is measured
Like Transmissibility, an ODS FRF also requires a reference also depends on whether or not a resonance is excited. In
(fixed) response measurement along with each response order to excite a resonance, two conditions must be met:
measurement. Each ODS FRF is formed by replacing the
Condition 1: The excitation force must be applied at a
magnitude of each XPS between a response and the refer-
DOF, which is not on a nodal line of the mode shape.
ence response with the APS of the response. The phase of
the XPS is retained as the phase of the ODS FRF. Condition 2: The excitation frequency must be close to the
resonance peak frequency.
This new measurement contains the correct magnitude of the
response at each point, and the correct phase relative to the If both of these conditions are met, and the resonance is
reference response. Evaluating a set of ODS FRF measure- "lightly" damped, it will act as a mechanical amplifier and
ments at any frequency yields the frequency domain ODS for greatly increase the amplitude of response, or the ODS. Con-
that frequency. Figure 6 shows the display of an ODS from a versely, if either condition is not met, the mode will not par-
set of ODS FRF measurements. ticipate significantly in the ODS.
All single frequency sine wave modal testing is based upon
achieving the two conditions above, plus a third,
Condition 3: At a resonant frequency, if the ODS is domi-
nated by one mode, then the ODS will closely approximate
the mode shape.
If Condition 3 is not met, then two or more modes are con-
tributing significantly to the ODS, and the ODS is a linear
combination of their mode shapes.

EXCITING RESONANCES WITH IMPACT TESTING

With the ability to compute FRF measurements in an FFT
analyzer, impact testing became popular during the late
1970s as a fast, convenient, and relatively low cost way of
finding the mode shapes of machines and structures.
To perform an impact test, all that is needed is an impact
Figure 6. ODS Displayed Directly From ODS FRF Data. hammer with a load cell attached to its head to measure the
input force, a single accelerometer to measure the response at
MODE SHAPES FROM ODS’s a single fixed point, a two channel FFT analyzer to compute
FRFs, and post processing software for identifying and dis-
We have already seen that ODS’s are obtained either from a
playing the mode shapes in animation.
set of time domain responses, or from a set of frequency do-
main functions that are computed from time domain respons- In a typical impact test, the accelerometer is attached to a
es. In addition, modal parameters (natural frequency, damp- single point on the structure, and the hammer is used to im-
ing, & mode shape) can be obtained from a set of FRF meas- pact it at as many points and as many directions as required
urements. In general, the following statement can be made, to define its mode shapes. FRFs are computed one at a time,
between each impact point and the fixed response point.
“All experimental modal parameters are obtained from
Modal parameters are defined by curve fitting the resulting
measured ODS’s.”
set of FRFs. Figure 7 depicts the impact testing process.
Stated differently, modal parameters are obtained by post-
processing (curve fitting) a set of ODS data. In other words,
a set of FRFs can be thought of as a set of ODS’s over a fre-
quency range. At or near one a resonance peak, the ODS is
dominated by a mode. Therefore, the ODS is approximately

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placement or acceleration responses. (The peak values of

the real part are used for velocity responses.) All of these
very simple curve-fitting methods are based on an analytical
expression for the FRF, written in terms of modal parameters
[3].

ODS’s AND MODE SHAPES CONTRASTED

Even though all experimental mode shapes are obtained from
measured ODS’s, modes are different from ODS’s in the
following ways,
1. Each mode is defined for a specific natural frequency.
An ODS can be defined at any frequency.
2. Modes are only defined for linear, stationary structures.
Figure 7. Impact Testing. ODS’s can be defined for non-linear and non-stationary
structures.
Curve Fitting 3. Modes are used to characterize resonant vibration.
In general, curve fitting is a process of matching an analyti- ODS’s can characterize resonant as well as non-
cal function or mathematical expression to some empirical resonant vibration.
data. This is commonly done by minimizing the squared
error (or difference) between the function values and the da- 4. Modes don’t depend on forces or loads. They are inher-
ta. In statistics, fitting a straight line through empirical data ent properties of the structure. ODS’s depend on forces
is called regression analysis. This is a form of curve fitting. or loads. They will change if the loads change.
5. Modes only change if the material properties or bound-
ary conditions change. ODS’s will change if either the
modes or the loads change.
6. Mode shapes don’t have unique values or units. ODS’s
do have unique values and units.
7. Mode shapes can answer the question, “What is the rela-
tive motion of one DOF versus another?” ODS’s can
answer the question, “What is the actual motion of one
DOF versus another?”

CONCLUSIONS
Operating deflection shapes were defined for both time and
frequency domain functions. We saw that ODS’s can be ob-
tained from a variety of both time and frequency domain
functions, but restrictive assumptions must also made with
each measurement type.
We also discussed ODS’s and modes shapes, and made the
Figure 8. Curve Fitting FRF Measurements. statement that, “All experimental modal parameters are
obtained from measured ODS’s.” In spite of this close rela-
Estimates of modal parameters are obtained by curve fitting tionship, we contrasted ODS’s with mode shapes and pointed
FRF data. Figure 8 depicts the three most commonly used out seven ways in which the two are different from one an-
curve-fitting methods used to obtain modal parameters. The other.
frequency of a resonance peak in the FRF is taken as the
modal frequency. This peak should appear at the same fre-
quency in every FRF measurement.
The width of the resonance peak is a measure of modal
damping. The resonance peak width should also be the same
for all FRF measurements. The peak values of the imagi-
nary part of the FRFs are taken as the mode shape, for dis-

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REFERENCES
[1] Potter, R. and Richardson, M.H. "Identification of the
Modal Properties of an Elastic Structure from Measured
Transfer Function Data" 20th International Instrumentation
Symposium, Albuquerque, New Mexico, May 21-23, 1974.
[2] Døssing, Ole "Structural Stroboscopy-Measurement of
Operational Deflection Shapes" Sound and Vibration Maga-
zine, August 1988.
[3] Richardson, M. H., "Modal Analysis Using Digital Test
Systems," Seminar on Understanding Digital Control and
Analysis in Vibration Test Systems, Shock and Vibration
Information Center Publication, Naval Research Laboratory,
Washington D.C. May, 1975.

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