The International Journal of Analytical and Experimental Modal Analysis v 7 n 3 p 169-178 Jul 1992

by P. Sriram, Wichita State University; S. Hanagud and J.I. Craig, Georgia Institute
of Technology

ABSTRACT
In conventional modal testing, accelerometers are used to sense structural response data which is processed
to obtain the natural frequencies, damping and mode shapes of the structure under test. In the case of light-weight
structures like composites where mass loading and other local effects of these transducers are not negligible, opticat
instruments like the laser doppler vibrometer (LDV) are used. The availability of real-time scanning LDV's has
introduced many interesting measurement possibilities. By applying a time-domain sorting algorithm, we have
recently demonstrated the use of such a scanning LDV to simulate multiple discrete sensors distributed over the test
structure. In the method developed in this paper, we process the scanning LDV velocity output signal in the frequency
domain to obtain directly the deflection shape of the vibrating structure in a functional (series) form. The technique
is illustrated by measuring the second mode shape of a light-weight cantilever beam. A discussion of the limitations
of the method and comparisons with theoretical predictions are also included.

AJ Chebyshev coefficients of q>(x) Greek Symbols

BJ Chebyshev coefficients of 'JI(x) a relative phase between velocity samples
Tk kthChebyshev polynomial l/J(x) spatial velocity distribution function (sine
t time coordinate component)
V(t) scanning LDV velocity output VJ(x) spatial velocity distribution function
v(x ,t) space-time distribution of velocity (cosine component)
w(x) weighting function in error integral mb structure (beam) vibration frequency
x spatial coordinate along beam Wm LDV scanning frequency

The design criterion for many structural elements is based on perfonnance under dynamic conditions
and thus, accurate dynamic characterization of such structures is of paramount importance. The standard
approach is to predict the response based on a mathematical model, typically a finite element description.

P. Sri ram (SEM Member), Assistant Professor, Department of Aerospace Engineering, Wichita State University, Wichita, KS 67208.
S. HanaKud ( SEM member) and J.l. Craig, Professors, School of Aerospace Engineering, Georgia Institute ofTechnology, Atlanta,
GA 30332-0150.

An earlier version of this paper was to have been presented at the 9th International Modal analysis Conference, Florence, /taly, April
1991.
Frnal manuscript received: March 19, 1992
 Frequently, the model is required to be validated and this function is accomplished through an iterative
interaction with experiments. The use of such an approach allows the designer to observe the effects of
potentially destructive loads without destroying the structure itself. One of the building blocks in this
technique is the ability to measure the dynamic characteristics of structures quickly and accurately.
In conventional vibration testing, accelerometers are commonly used to sense the acceleration time­
histories at selected locations. The acceleration (response) and the load (excitation) signals are processed
using various analysis techniques[ I], usually involving digital signal processing, to estimate the system
dynamic characteristics. The dynamic characteristics of the system are usually described by the eigenvalues
(natural frequencies and damping ratios) and corresponding eigenvectors (mode shapes) obtained through
such analyses. The natural frequencies and damping ratios are obtained through some type of curve-fitting
procedure. The mode shape infonnation is obtained by using multiple exciters and/or sensors or by
successively moving the exciters and/or the sensors to different p oints in the structure. In the case where
there is little interaction between the modes, the structure can simply be excited at a selected resonance
frequency and the resulting response vector used as the approximate mode shape. A similar technique can
be used even in the presence of significant modal interaction by using multiple exciters and special control
logic. There are many sources for error, and in testing light-weight structures, the mass loading of the
response sensors can often distort the measurement so that one has to seek alternate non-contact sensors
like the laser Doppler vibrometer (LDV).
The LDV is based on the measurement of the Doppler shift of the frequency of laser light scattered by
a moving object. The magnitude of the Doppler shift is related to the optical geometry and the velocity of
the scattering object. Application of the LDV to the field of vibration measurements in solids was initially
in the context of rotating systems[2]. Subsequently, the technique has attained popularity in special
situations requiring the use of a non-contacting optical sensor such as in biomedical vibration analysis[3,4].
The instrument is perceived to be useful enough that now there are several commercial LDV systems
available for vibration measurements in solids, including a portable model based on a novel frequency
shifting scheme[5].
The measurement of vibration mode shapes requires the sensing of structural response at a series of
locations on the structure. When using an LDV system, the straightforward solution is to translate the test
object or the complete LDV system so that various points of interest can be probed. While this is simple
for small test objects and small LDV systems, it is not always convenient. Various modifications have been
introduced into LDV optical systems to make multi-point measurements easier, including the use of a fiber
optic link which allows most of the LDV components and the test object to remain fixed while only a few
components have to be moved. Such a system has been demonstrated for the measurements of vibration
frequency and velocity amplitude[6]. An alternate approach introduced by Bendick[7] involved the
translation of a single mirror to move the probe area along the optical axis of the LDV. There are also
commercial LDV systems specifically oriented towards vibration measurements wherein the traverse is
automatedt .
There are two main types of LDV, namely, the reference or single-beam and differential or dual-beam
arrangements[8]. A single-beam LDV measures velocities along the line of sight and introduction of
scanning causes a response due to the varying distance between the surface and the sensor. Most
commercial LDV systems intended for structural applications are of this type, apparently, since scanning
during measurements is not anticipated. On the other hand, a two beam arrangement is sensitive only to
transverse velocities in the plane of the beams and is, therefore, free of this range sensitivity effect. The
following discussion assumes the use of a dual-beam LDV. For measurements on opaque solid surfaces,
the LDV beams have to impinge on the surface and this requires the optical axis of the system to be at an
angle to the surface. Then, axial motion of the probe volume is of limited use since the probe volume will

tThe Ometron Vibration Pattern Imager VP/ 9000, Ometron Inc., Herndon VA 22070 and Polytee OFV-050, Polytec Optronics Inc.,
Costa Mesa CA 92626.
have to be accommodated either inside the solid (which is impossible since the object is opaque) or away
from the surface (in which case the velocity measured is not the solid surface velocity). The surface can
be scanned only by moving the LDV probe transverse to the optical axis or in a combination of axial and
transverse motion. A transverse scanning technique has been devised by Durst, et al [9] with scanning
motion derived from an oscillating mirror in the laser beam path. The scanning rates were limited in this
system by inertial effects in the mirror to about 15 Hz (lines/sec) for scan angles of about 10°. Chehroudi
and Simpson[ 1 0] have improved upon the concept by incorporating a commercial scanning device and strip
mirrors to obtain scan rates up to 150 Hz over linear scan lengths of about 400 mm.
A scanning LDV can cause errors in fluid flow measurements where the scattered light signal arrives
in intermittent bursts corresponding to particles crossing the LDV probe volume[9]. In the case of solid
surfaces, light is scattered continuously as long as the probe volume intersects the scattering surface, and
thus the Doppler signal is continuous, providing for good spatial resolution. Then, a scanning system can
be used effectively on vibrating surfaces to map the spatial velocity distribution (mode shapes) accurately.
The use of digital signal processing techniques means that the velocity at a given point is required only
during the sampling intervals. Between samples, the sensor system is idle. A scanning LDV can make use
of this idle time to measure the response at other spatial locations. In previous work[ 11], we addressed this
possibility and constructed a scanning LDV system that accomplishes this task of measuring the response
at multiple locations in an interlaced fashion within the sampling interval of the signal processing system.
One of the problems encountered therein was the inability to use anti-aliasing filters due to interaction with
the spatial scanning. In this paper we develop an alternate approach that harnesses this interaction to infer
the spatial velocity distribution. The theory of Chebyshev demodulation, as this approach is called, is
outlined and test data from a light-weight cantilever beam excited at its second bending mode is presented
for verification.

The experimental setup will be described only briefly here since a detailed description has been
presented previously[ll]. A TSI 9100-7 high-power two velocity component LDV operated in a single
velocity component measurement mode was used. The single component mode was enabled by operating
the laser in a single-mode configuration radiating at a 514.5 nm wavelength (green). The actual laser power
output that was used during the tests was in the range of 10-20 mW, which was sufficient to produce good
signals from the vibrating beam. Scanning capability was added to the LDV by directing the emerging laser
beams onto a scanning mechanism, as illustrated in Fig. 1. In the figure, the beam is assumed to vibrate in
and out of the plane of the paper, matching the sensitivity direction of the LDV. The scanning mechanism
consisted of a pivoted front-surface mirror driven through a lever arm by an electrodynamic shaker. The
lever arn1 arrangement converted the rectilinear motion of the shaker head into angular motion of the mirror.
A mirror position sensing transducer was also built into the mechanism though it was not used for the tests
reported here.
Using the above described configuration, the mirror scanner can be placed on a mounting independent
of the rest of the system. Thus, vibrations due to the oscillating mirror can be isolated from the rest of the
optics as well as the test structure. The working distance from the scanning mirror to the test object was
1.0 m. The optics limited the scan to a ±7 .5° range due to defocussing effects. The maximum scan rate was
restricted by inertial effects in the mirror to about 150Hz for the full range scan, with higher rates attainable
for shorter scans. These range and rate limitations can be readily overcome using correcting lenses and/or
commercial scanning devices.
The test structure was an acrylic plastic rectangular bar of cross section 26 mm x 2.8 mm. One end of
the bar was clamped between two aluminum blocks to produce a cantilever beam of length 130 mm. The

-.-L._ •- ... - ···- - .. �---' •-- --·---1 -� A--l . o�.:-

. ... 1 ---1 r- • .- - ..:-.--6 ... 1 l.A-.....1 ... 1 .
A- ... 1 ..... :,.. -4"7-4
 density of the beam material was so low (86.0 g/m linear mass density) that even a sub-miniature
accelerometer (1 g typical mass) caused significant mass loading. The beam was driven at its tip by a B&K
4810 exciter and the force input was measured with a PCB 208A02 load cell. The excitation force, scanning
mirror position, and velocity output signal from the scanning LDV were fed into a Preston analog-to-digital
converter system controlled by an HP 1000 minicomputer. The overall instrument response time was
measured to be in the range 200 - 250 Jlsec. The velocity measurement resolution and range were limited
by the LDV electronics to 0.0004 m/sec and ±0.5 m/sec respectively. In order to provide a stringent test
of the measurement system and to illustrate the effect of errors, the excitation level was set to provide only
about 0.01 m/sec peak response or 2% of the full-scale reading. In tenns of the more common acceleration
units, the measurement resolution and range were 0.05 g and ±80 g, respectively, while the peak response
signal was about 1.6 g, all assuming a vibration frequency of 255 Hz.
Initially, the tip response was sensed with the LDV operating in a non-scanning mode and the load and
response signals were routed to a GENRAD 2515 modal analysis system. The first two natural (bending)
modes were estimated to be at 23.6 Hz and 255 Hz, with damping ratios of about 0.08. These frequencies
may appear to be excessively separated considering the fact that for a cantilever beam the ratio of the first
two natural frequencies should be about 6.3. However, the test beam had an accelerometer (PCB type 303A)
mounted near the tip. It can be shown that for a cantilever beam with such a tip mass, the spacing between
the first two natural frequencies increases significantly[12]' a trend consistent with the measured data. The
well separated natural frequencies with low damping suggested that the modes could be considered to be
non-interacting. Thus, the deflection shape of the beam at resonance could be used to approximate the
corresponding mode shape. In order to sense this approximate mode shape, the beam was excited using a
255 Hz sine wave while the LDV was configured to scan the entire length of the beam. The scan rate was
set in accordance with the data analysis requirements as described subsequently. If the LDV were operated
in a non-scanning mode, the velocity signal would also be a 255 Hz sine wave with an amplitude
corresponding to the deflection magnitude at the measurement location. However, with the scanning
action, the measurement location is not fixed and, thus, the velocity signal is multiplexed or modulated by
the spatial deflection distribution. In the following section, we describe a method of demodulating this
multiplexed signal to obtain the spatial velocity distribution.

Oscillating
Mirror

LDV System

Vibrating
Beam

Fig. 1 Scanning LDV schematic

17? .l11lv 1 QQ?

 It is assumed for convenience that a beam-like vibrating structure is situated in the region -1 S x � 1.
Even in the case of complex three-dimensional structures, the method applies in the domain represented
by the scan region. The structure is assumed to be excited at one of its resonance frequencies, mb. The velocity
distribution within the domain of interested is written as

(1)

where t/J(x) and ljl(x) are mutually orthogonal components of the vibratory response velocity distribution,
allowing for the general case of complex mode shapes. If the mode shape under examination was real, these
two functions would differ only by a scale factor and the equation can be written using a single harmonic
tettn. The case where there is a non-oscillatory velocity field (for example, as in a rotating system) can be
dealt with in a similar fashion and this has been reported on elsewhere [13]. The first measurement is
arbitrarily assigned as the reference, implying a zero phase. Subsequent samples of the velocity distribution
have to include an unknown phase difference in the argument. This problem will be dealt with
subsequently. The problem posed now is to be able to detect t/J(x) and VJ(x). A non-scanning LDV senses
velocity at a given point (x = constant) where the LDV beams are incident on the structure. These beams
are now assumed to be scanned in a controlled manner by a norrnalized scan function of the fonn

X= COS (J)mt (2)

where mm is assumed as the scan rate. At any instant of timet, the scanning LDV output corresponds to the
velocity at the position x given by Eq. 2. This output, denoted V(t), is

(3)

The functionals l/J and ytin this equation are even and periodic by virtue of the cos mmt argument. Hence,
these can be expanded in their Fourier cosine series as

00

(4)
k=l

00

(5)
k=l

00 00

(6)
k=l k=l

By writing the trigonometric products in tenus of sums, we obtain

(7)
00

Bk
00

Bk
+ B0 cos robt + cos( rob + kmm )t + cos ( rob - kmm )t
k=l
2 2
k=I
.

It is to be noted that Eqs. 4 and 5 can be written as

00

1
t/J(x) = Ao + Ak cos kcos- x (8)
k=l
00

1
Bk cos kcos- x
l/f(x) = B0 + (9)
k =l

Recalling the definition of the Chebyshev polynomials Tk

(10)

it can be recognized that Eqs. 8 and 9 represent the Chebyshev series expansions for t/J(x) and 1p(x). The
quantities Ak and B k are the Chebyshev series coefficients. If a finite number of these coefficients are
available, a Chebyshev approximation to the velocity distribution is obtained. A Fourier transform of V(t)
will exhibit peaks at the frequencies ( mb±kmm). Measurement of the complex amplitudes at these frequencies
will, therefore, yield the Ak and B k coefficients which determine the spatial distributions ¢(x) and VJ(x), and
hence v(x,t). The location of these peaks in the frequency domain is illustrated in Fig. 2.
If discrete Fourier analysis is used, it is essential to perforrn the Fourier transfonn such that the spectral
lines of interest occur on the analysis spectral lines so that the required amplitudes are estimated accurately.
This can be readily accomplished in practice through a frequency matching scheme, namely, by setting the
scan rate mm to a l /(n+.5) fraction of the excitation frequency mb where (n+ l ) terms in the approximation
are sought and having an integral number of analysis spectral lines in a bandwidth of mm . An additional
advantage of this choice is that apparent velocity signals due to vibration of the scanning stage (which occur
at the harmonics of rom ) can be readily isolated from the true beam vibration signals. The need to isolate
and suppress such apparent velocity signals severely restricts usage of windowing techniques. The
frequency matching scheme will be numerically illustrated while discussing the test results.

Imaginary
(sin)

Real (cos)

Fig. 2 Frequency components of scanning LDV signal

 The foregoing procedure can be used to obtain a single estimate of t/J(x) and 1/l(x). In order to obtain
statistically significant results, many such estimates have to be averaged. This is not straightforward due
to the unknown phase difference between samples. The following error integral minimization procedure
was used to account for this. Consider two samples separated by an unknown phase difference a. Using
subscripts to denote successive samples and dropping out the functional arguments,we have

(11)

(12)

Now, v2 can be rotated to the same phase as v1 by using

(13)

where (14)

and 1J12 *
= 1Jf2 cos a + .m '�' 2 stn
•

a (15)

The rotated v2 can be directly averaged with v1 and a similar process used for each sample. The task is to
find a suitable a which is accomplished as follows. We choose a such that the integrated error between v1
and v2 is a minimum. The integrated error is expressed as

1 2 I

£12 = ( t/>1 -t/>2 cos a+ '1'2 sin ) dx

a + ( '1'1 1J12 cos -t/>2 sin )2 dx
- a a
(16)
-1 -I

By setting the first derivative of this integral with respect to a to be zero, we obtain

(tP2V't -l/JtV'2 )dx

tan a=
(17)
(tPttP2 "'1 "'2 )dx
+

where the limits of integration are (-1,1). This leads to two principal values for a which differ by 180°. From
the nature of the problem, one of these corresponds to the minimum error and the other to maximum error.
The value of a corresponding to the minimum is easily determined by evaluating the second derivative of
the error integral. The integrals can be simplified if a weighting function w(x) is used while evaluating the
error integrals. For example, if we set

1
(18)

the integrals in Eq. 17 degenerate into summations of products of Chebyshev coefficients of t/J and 1JI (due
to the orthogonality property of Chebyshev polynomials).

• • • •
.
. .•

. .

The second mode of the test cantilever beam was excited using a 255Hz sine wave. A simple analysis
showed that a five-tertn Chebyshev approximation to the second mode shape would have less than 1% error
and this was set as the target. Thus, the scan frequency mm was set to 255/4.5 56.67 Hz. Choosing to have =

six lines in the mm bandwidth leads to a required spectral resolution Llf of 9.444 Hz. For a block length of
256 samples, then, the required sampling rate is half the number of samples times Llf or 1208.9 Hz. The
sampling rate was set accordingly. Under these conditions, the first Chebyshev tenns were at mb or the 27th
spectral line with the successive terms at (mb±kmm) falling on the 21st and 33rd lines (for k=l), 15th and 39th
lines (k=2), 9th and 45th lines (k=3) and 3rd and 51st lines (k=4). The scan harmonics were at the multiples
of mm, i.e., the 61h, 12th, 18th... lines, clearly isolated from the required Chebyshev tenns.
Initia1 tests showed that the fidelity of the test data depended to a large extent on the accuracy and
stability of the frequency ratios. Ideally, one would solve the problem by deriving all signals from a
common clock source. However, it was found that two signal generators could be tuned by hand with
sufficient accuracy while observing one wavefonn on an oscilloscope triggered by the other. This was the
method used.
In order to demonstrate the overall validity of the analysis technique, the auto-spectrum of the scanning
LDV output was estimated using ten samples. This is presented in Fig. 3. A wideband spectral noise floor
of about 0.05 mm/sec is clearly evident in the figure along with several other spectral peaks. The highest
peak can be easily located at 255 Hz, corresponding to the first terms in the Chebyshev series. Further terms
in the series are also visible in the figure, along with several scan harmonics (especially the first one at 56.67
Hz), though it is somewhat challenging to identify the peaks without the aid of a clear frequency scale. In
order to easily compare the test results with theoretical predictions, the mode shape was assumed to be real
and thus corresponding l/J and 1f1 terms were combined to obtain the experimental mode shape. Both
theoretical and experimentally obtained modes shapes were nonnalized with respect to the corresponding
tip deflections.
A close inspection of the data in Fig. 3 reveals that the fourth terrn is barely above the noise floor while
the fifth tennis virtually indistinguishable. Thus, one can attach a fair degree of confidence to a four-tenn
approximation whereas a five-tenn approximation is bound to be contaminated. In Fig. 4, the experimentally
obtained three-tenn approximation is presented along with the theoretical second mode shape of the beam.

0.15 1.0
Theoretical
---

Experimental
�
.... 0.5
CJ ·-
QJ u
Ill
0.10 - Q
e �
-

I
E >
0.0 I
s '0
� I
Nl
= •
N

�
...
\
lA
·-
- -

� "' -..
CJ
QJ �
E \
"-J
�
Q, 0.05 -
VJ
r� ,/
lr'
/
\
..
-0.5
Q
-
= = �
< z ' �
' /
' ,.
.... - -

0.00 • •
I I I I
-1.0
0 100 200 300 400 500 0.0 0.2 0.4 0.6 0.8 1.0

Frequency (Hz) Spanwise Location (x/1)

Fig. 3 Auto-spectrum of scanning LDV output Fig. 4 Three-term approximation of beam mode
signal shape

176 July 1992

Clearly, the experimental data is inadequate. However, it is interesting to note that a three-tern1 Chebyshev
approximation of the theoretical mode shape has an 8% rms error while the corresponding experimental
value is 24%. This is close enough to be encouraging, considering the fact that the signal levels were only
around two percent of the full-scale values.
The four-tenn experimental approximation is compared with the theoretical mode shape in Fig. 5. Good
overall agreement is observed, validating the measurement technique. The effect of using noise contaminated
tenns is illustrated in Fig. 6, where the experimental five-term value is compared with the theoretical mode
shape. The agreement is significantly poorer than in the case of the four-tenn approximation, testifying to
the danger of using contaminated data. Mutual comparisons between the three-, four- and five-term data
also suggest a simple test to determine when contamination has occurred. While the four-term approximation
is relatively smooth (similar to the three-term value), the addition of the fifth tenn does not follow the trend,
indicating possible errors.

In this paper, we have shown the feasibility of using a scanning Laser Doppler Velocimeter to measure
the vibrating mode shape of a structural dynamic system. The technique of simultaneously gathering
vibration velocity information from multiple locations along the scan has been demonstrated. This makes
it possible to obtain the velocity from multiple points while using only a single sensor system. In particular,
the new data processing technique allows measurement of a Chebyshev approximation to the velocities of
a structural dynamic system modeled as a distributed parameter system. While not the subject of the present
paper, the basic scanning technique can also be used to simulate a finite number of discrete transducers [ 11].
The scanning LDV provides a measuring tool that is non-contacting and adds no additional mass to a light­
weight structure. The modal data extracted from measurements using the scanning LDV exhibit good
agreement with theoretical predictions. Additional work is necessary to exploit the benefits of this
development and to apply it to other important practical problems like separation of interacting modes and
non-harmonic or random vibrations.

1.0
Theoretical Theoretical
1.0
Experimental I
Experimental
---

---

� 0.5
....
�
·c
·-
u
0.5
.s -
C>
�
>
, 0.0
�
�
·­
' 1 0.0
-
' =
e '
' ; I
; -0.5 ' s.
-0.5 I
z c
z I

-1.0 -1.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Spanwise Location (x/1) Spanwise Location (x/1)

Fig. 5 Four-term approximation of beam mode Fig. 6 Five-term approximation of beam mode
shape shape

• • • •
 The authors gratefully acknowledge the support received for this research from the US Artny Research
Office as part of the Georgia Tech Center of Excellence for Rotary Wing Aircraft Technology (GT­
CERW AT). The authors would also like to thank Dr. N. M. Komerath and Dr. S. G. Liou for their
suggestions and assistance in using the LDV system.

I. Ewins, D.J. Modal Testing: Theory and Practice New York: Research Studies Press- John Wiley�
1984.
2. Davis, Q.V.; Kulczyk, W.K. "Vibrations of turbine blades measured by means of a laser." Nature
v 222 n 5192 p 475-476 1969.
3. Michelsen, A.; Larsen, O.N. "Biophysics of the ensiferan ear." J Comparative Physiology v 123
p 193-203 1978.
4. Buunen,T.J.F.; Vlaming,M.S.M.G. "Laser Doppler velocity meter applied to tympanic membrane
vibrations in cat." J Acoust Soc Am v 69 n 3 p 744-750 1981.
5. Pickering, C.J.D.; Halliwell, N.A.; Wilmshurst, T.H. "The laser vibrometer: a portable instrument."
J Sound Vib v 107 n 3 p 471-485 1986.
6. Cookson, R.A.; Bandyopadhyay, P. "Mechanical vibration measurements using a fiber optic laser
Doppler probe." Opt Laser Techno/ v 10 n 1 p 33-36 1978.
7. Bendick, P.J. "A laser Doppler velocimeter to measure 'instantaneous' velocity profiles." In:
Dowdell, R.B., (ed.) Flow, its Measurement and Control in Science and Industry. Pittsburgh, PA:
Instrument Society of America; 1974. p 1033-1036
8. Durst, F.; Melling, A.; Whitelaw, J.H. PrinciplesandPractice ofLaser DopplerAnemometry, 2"ded.
New York: Academic Press; 1981.
9. Durst, F.; Lehmann, B.; Tropea, C. "Laser Doppler system for rapid scanning of flow fields." Rev Sci
lnstrum v 52 n 11 p 1676-1681 1981.
I 0. Chehroudi, B.; Simpson, R.L. ''A rapidly scanning laser Doppler anemometer." J App/ Phys E v 17
n 2 p 131-136 1984.
11. Sriram, P.; Craig, J.l.; Hanagud, S. "A scanning laser Doppler vibrometer for modal testing." Int J
Anal Exp Modal Anal v 5 n 3 p 155-167 1990.
12. Maltbaek, J.C. "The influence of a concentrated mass on the free vibrations of a unifonn beam." lnt
J Mech Sci v 3 p 197-218 1961.
13. Sriram, P.; Hanagud, S.; Craig, J.l.; Komerath, N.M. "A new laser Doppler technique for velocity
profile sensing." Appl Opt v 29 n 16 p 2409-2417 1990.

178 July 1992