RESPONSE OF STRUCTURAL ELEMENTS
TO RANDOM EXCITATION
By
MARC ROGER PIERRE TRUBERT
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
June, 1962
ACKNOWLEDGMENTS
The author takes this opportunity to express his sincere
gratitude to Dr. W. A. Nash, Chairman of his Supervisory Commit
tee, for the constant guidance he provided to him during his entire
graduate work.
The author wishes to express his appreciation to Professor
W. L. Sawyer, Head of the Department of Engineering Mechanics,
Dr. T. C. Huang, Associate Professor of Engineering Mechanics,
Dr. C. A. Sciammarella, Associate Professor of Engineering Me
chanics, and Dr. C. B. Smith, Professor of Mathematics, for serv
ing on his Supervisory Committee.
Finally, the author expresses his appreciation to the Air
Force Office of Scientific Research for their sponsorship of the work,
and to the Ford Foundation for its financial assistance.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS . .
LIST OF FIGURES . .
NOTATION .......
Chapter
I. MATHEMATICAL BACKGROUND
* C S S S C
1. Introduction .. .
2. Methods of Analysis . .
3. Determination of the Response .
4. Random Loading . .
5. Correlation Function and Power Spectral
Density ... .
6. Response to a Random Loading .
7. Discrete Point Loading .
8. Particular Case of TwoPoint Loading .
9. Determination of the Transfer Function
II. EXPERIMENTAL INVESTIGATION ..
1. Scope of the Study .......
2. Experimental Apparatus .
3. Testing Procedure .
4. Data Reduction ............
5. Numerical Computation of the Power
Spectral Density ....
III. INTERPRETATION OF THE RESULTS .
Transfer Functions ........
Loading of SeriesI ........
S 54
S 60
. 54
. viii
. 0 0 .
. 0 .
. .
...
* w
* *S 4
TABLE OF CONTENTSContinued
Page
3. Effect of Correlation . .. 80
4. Series II Loading . 84
5. Conclusion .... ........ .90
LIST OF REFERENCES ....... ...... 91
BIOGRAPHICAL SKETCH .............. ... 94
LIST OF FIGURES
Figure Page
1. Load on a Structure .. ...... .. ... .. 5
2. Experimental Arrangement .. 22
3. Beam, Suspended Coils and Accelerometer 23
4. Beam, CoilMagnet and Stand . .. 25
5. Location of Acceleration Measurement and
Forces on the Model .. . 26
6. Flow Chart for Loading and Measurement 27
7. CoilMagnet System . 30
8. Calibration of CoilMagnet System by Static Test 33
9. Verification of CoilMagnet Calibration .. 37
10. Typical Force Signal (Points 1 and 2) ..... .39
11. Typical Signals of Acceleration (Point 0)
and Force (Point 1) ........ ...... 39
12. Analog Correlator . .. 47
13. Typical Squaring of Random Signal .. 48
14. Typical Product of Random Signals .. 48
15. Recopying the Delayed Signal on the Loop .. 50
16. Fourier Transform Integration . 52
17. Autocorrelation of the Force for the Determination
of the Transfer Function .. . 55
LIST OF FIGURESContinued
Figure
18. CrossCorrelation of Acceleration and Force for
Determination of Transfer Function H'( .
19. CrossCorrelation of Acceleration and Force for
Determination of Transfer Function H() .
20. Modulus of Transfer Functions H(f)and H )
21. Phase Angle of Transfer Functions HQ({) and H (f)
22. Transfer Coefficients T, and Ta .....
23. Transfer Coefficients T, and T .
24. Typical Autocorrelation of Force for a TwoPoint
Loading (Point 1, Case 1) . .
25. CrossCorrelation of Forces at Points 1 and 2,
Series I . . .
26. Autocorrelation of Acceleration at Point 0 for
TwoPoint Loadings, Series I . .
27. Power Spectral Densities of Forces, Case 1 .
28. Power Spectral Densities of Acceleration, Case 1
29. Power Spectral Densities of Forces, Case 2 .
30. Power Spectral Densities of Acceleration, Case 2
31. Power Spectral Densities of Forces, Case 3 .
32. Power Spectral Densities of Acceleration, Case 3
33. Power Spectral Densities of Forces, Case 4 .
Page
. 56
57
. 58
. 59
61
S. 62
. 63
. 64
. 65
. 67
68
. 69
S. 70
. 71
. 72
. 73
LIST OF FIGURESContinued
Figure Page
34. Power Spectral Densities of Acceleration, Case 4 74
35. Power Spectral Densities of Uncorrelated Forces,
Case 5 . . 75
36. Power Spectral Densities of Acceleration, Case 5 76
37. Effect of CrossCorrelations of Forces on the
Experimental Accelerations . 82
38. Computed Acceleration, Correlated and
Uncorrelated ., ..... 83
39. CrossCorrelation of the Forces at Points 1 and 2,
Series II .. . ... 85
40. Power Spectral Densities of Acceleration, Case 6 86
41. Power Spectral Densities of Acceleration, Case 7 87
42. Power Spectral Densities of Acceleration, Case 8 88
43. Power Spectral Densities of Acceleration, Case 9 89
NOTATION
d d~
t
t
u(F1)
CA)
g
F,
T
h v( S, o
H F6, u)
H
I.
Position vectors on the surface of the structure
Element of surface area
Time
Time delay
Lateral response of the structure
Lateral load on the structure
Crosscorrelation of two time random functions,
(rkt)and y(,t) at points i and I respectively (8)
Cross power spectral density for these same
functions (9)
Angular frequency in radians per second
Frequency in cycles per second
Time length of a record
Real and imaginary parts of the cross power
spectral density (13)
Response to the unit impulse
Transfer function (Sec. 2, Chap. I)
Denotes complex conjugate
Discrete representation of the transfer function
Modulus of the discrete transfer function (33)
viii
e6
r
j
A( Bs
P
N
Phase angle of the discrete transfer function (33)
Imaginary unit j =
Discrete notation for 'X(,t) and y(T, )
Real and imaginary parts of the transfer function
Acceleration of gravity
RMS value of the acceleration
RMS value of the random input forces
Pressure signal used as pilot signal
Subscript denoting normalized quantity
CHAPTER I
MATHEMATICAL BACKGROUND
1. Introduction
Although, in many instances, random problems have reached
a standard level in communication engineering and have been investi
gated for some time in the theory of turbulence and in several other
fields, it is only recently that the systematic study of mechanical
structures under random loadings has been undertaken [1 through 10].
There has been a fair amount of work done on theoretical grounds,
most of it with the hypothetical white noise input, but relatively little
attention has been given to the actual testing of structures under actual
distributed random loadings [1, 3]. The experimental aspect of the
problem will be emphasized in the present investigation. An attempt
will be made to verify the validity and the feasibility of a numerical
prediction of the response of a structure to an actual spacetime ran
dom input. Both the characteristics of the structure and the random
input will be obtained experimentally.
Experimental measurements on random processes make in
tensive use of the basic concepts of generalized harmonic analysis.
Consequently, although the study reported here is experimental in na
ture, the mathematical point of view of the problem will be reviewed
in this chapter and presented for a direct application to the subsequent
experiments.
2. Methods of Analysis
We shall consider the spacetime response, UL(Fi ) of a
linear structure, S subjected to a lateral spacetime load, p(,) ,
per unit area distributed over the entire surface of the structure. The
nature of the load will not be specified for the first part of the follow
ing analysis, and it can be deterministic as well as random. The re
sponse can be either a displacement, a velocity or an acceleration.
Two methods can be used to predict the response. The first
is the normal mode approach which requires the knowledge of the
eigenfunctions and the eigenvalues to describe the characteristics of
the structure, i.e., mass, rigidity and damping distribution. This
method was developed by several authors [4, 7, 8]. The response in
volves a series representation.
A second method can be developed as an extension to the
multidimensional case of the wellknown harmonic analysis for the
onedimensional electrical systems. In this analysis the character
istics of a linear system are known either by the response, h(t) of
this system to the unit impulse or by the Fourier transform of this
response,
S h(l)e(th (1)
The function H(o) is called the system function or frequency response
function [11, 12, 13, 14]. The extension to multidimensional problems
of mechanics has been investigated by several authors [6, 8]. The
function h(t) now becomes a function of two space coordinates, i"
and s namely, h( ,,t) which is called response to the unit
spacetime impulse or simply response to the unit impulse as before.
Similarly, the system function H ()) becomes H(i',C,) which
will be called the "transfer function" in this study. I In the two func
tions, (Fr,?,t) and H (f',~j) the first vector, F de
notes the point of the structure where the response is measured, and
the second vector, the point of the structure where the unit im
pulse is applied. Evidently, H (F,4 y.o) is still the Fourier trans
form of h (",,',L) according to equation (1).
The interesting feature of the second method is that the struc
tural characteristics are lumped into a single function rather than
IBeer [6] calls H (?', t) the "influence function, the
term "transfer function" being reserved for the multidimensional
spacetime Fourier transform of h ( ',?, t)
being represented by an infinite series as in the first method. More
over, the transfer function H (Fr,c3) is susceptible to an experi
mental determination as will be shown later. This is the second
method that will be reviewed here.
3. Determination of the Response
Let us assume that the response to the unit impulse, h(i,,t),
is known. Let hd denote the displacement corresponding to the unit
impulse, h the corresponding velocity, and h1 the acceleration.
Then we have
h =A(h,) (2)
dV
h A %_ (h) (3)
dt'
Taking the Fourier transforms of (2) and (3), one readily obtains the
transfer functions for the velocity and the acceleration:
H, = jo H (4)
H, =Ha (5)
This shows that once the response to the unit impulse or the transfer
function is known for one type of response, then it can be readily found
for the two other types.
0
Figure 1. Load on a Structure
Let us now consider the response du (i) of the structure
to a load p( St) d9 distributed on an elemental area d cen
tered at point 3 as shown in Figure 1. From the convolution theo
rem relative to the time variable, we have
du(?b) = ^ (FA) [pc(t.o d 5] dv (6)
where I is a dummy variable of integration. The total response
U (r' t) due to a load (P' t) per unit area distributed over the en
tire surface of the structure can be obtained from the superposition
principle, since the structure is assumed to be linear. Integrating
over the whole surface, we have
u ,k) = // h(6,.) p(,t d) d j (7)
4. Random Loading
We shall now consider a load p ( ,t) which belongs to a
random process. Let us recall that a random process is represented
by the ensemble of all possible time records of a given phenomenon.
We call (, ) ,_o < L <+ oo one particular record of the
random process. Letting k= I,2,3,... for the same ? gives other
records of the process. The ensemble,
I b
represents a random process. A characteristic feature of a random
process is that only statistical properties can be determined.
We shall now restrict the random process to a particular
class, namely, a stationary and ergodic random process. Many phys
ical phenomena belong to this class. Moreover, this class is the most
readily measurable. In a stationary random process the ensemble
averages over all possible records are independent of the fixed time 1,,
at which these averages are taken for each record. The ergodic prop
erty means that a time average over one single record is equivalent to
the ensemble average. This last property is certainly the most inter
esting, since a time average is easily realizable electronically and
since one single record of the process is sufficient to represent the
entire random process. On the contrary, the experimental determi
nation of the ensemble averages, required for a more general process,
would become rapidly impractical. Verifying that a random process is
stationary and ergodic is not an easy matter. To be rigorous, ensem
ble averages should be taken over a large number of records as stated
above and then compared to time averages. This procedure is often
bypassed and only partial checks are made. It is simply verified that
time averages over some records give essentially the same results.
We will follow this procedure here.
5. Correlation Function and Power Spectral Density
Both correlation function and power spectral density give a
partial statistical representation of a random process. It should be
pointed out that they are in no way sufficient to determine completely
a random process for which the probability density function also has
to be known. However, as insufficient as they might be, the correla
tion function and power spectral density are useful from an engineer
ing point of view, for which root mean square (rms) value and
frequency power distribution are of interest. Indeed, if the random
process is assumed to have a Gaussian probability density function,
then only the second moment is required for the determination of this
function. Therefore, the knowledge of the crosscorrelation function,
which is the second moment, will be sufficient for a complete statis
tical representation of a Gaussian random process. Finally, let us
remark here that the output of a linear system excited by a Gaussian
random load is also Gaussian in nature.
The correlation function and power spectral density can be
defined independently of each other and are related through the Fourier
transform pair [11, 12]. For a pair of time random functions, 'X (, )
and y( '() located in space at t and s respectively, the cross
correlation function for an ergodic process is defined by
+T
xpY T,,z) Oxr r (g( c)dt (8)
TO 0 I T Z _T
where T is a time delay. The crosspower spectral density, also for
an ergodic process, is defined by
/T +i
# ej )ca dt y(r, t) e" dt
(9)
SP c,. ) = li T
STwo T
where C) is the angular frequency.
Between I and we have the Fourier transform relations:
+o00
aO
Since in the subsequent analysis the functions 'X and y will take dis
crete values, we shall introduce the following notation:
(12)
lt (V,6, to)  Y ( W)
r,s = 1,2,3,...
In (9) through (12) the subscripts indicate which functions are consid
ered in the correlation or the spectra. The vectors r and S indi
cate where the functions % and y respectively, are located on the
surface of the structure. The superscripts P and s are used for the
case of a discrete loading at and S The autocorrelation is ob
tained by setting X= y and r or r= We recall that (?*,?,T)
is real, but that ,,) or I (:c) is complex, i.e.,
f( 'Gs rs
Y
written in the discrete notation.
6. Response to a Random Loading
We shall apply formula (9) to the responses UC(1,t) and U('L,)
at two different points, i and viz.,
4T Tt
(liA = tim u t)e dtot t4 e7 )e dt (14)
T< 1T
Before using equation (7), some remarks will be made [11, 13]. Be
cause of the random nature of the loading and the response, the upper
limit of integration for the variable 9 in equation (7) will be extended
to + oo since initial conditions and the corresponding transient are
not considered in a random process. The load is assumed to have
been acting for a sufficient time such that the transient part of the re
sponse has died out. The lower limit of (7) can be extended to o0 ,
since the response to the unit impulse h(F',t) vanishes for negative
values of time t Equation (7) is now
t = hC , ) P(,a) d d P (15)
which is more symmetrical for the Fourier analysis.
Substituting equation (15) into equation (14), and rearranging
terms, we have
L + @
+T
p(",t,,) ejot't,
(16)
. p(a, P ta ) eletdd ,d dJ ddi,
T
We now make the substitution
L = 6 + 0,
b,6= 6,+ 3
Using an abridged notation for p and h we obtain
Sr I+TJ,
0 [ =T ,
, e l d ,h
][+0"
)=, L
:,e de 6,
= T
lirm
To T
[
(17)
e *d
(18)
e d c ds. 8d a
The curled bracket is simply the definition of the crosspower spectral
density for the loads p(',t) and P(i ) at the two points i and .
The second square bracket is the Fourier transform of the response to
the unit impulse, i.e., the transfer function H (, ,(a) The first
square bracket is the complex conjugate of the transfer function
M
H (9., ) Rewriting (18), we have
r{ )c F (i,? ,o ) RH(,,) dpp,,, ), (19)
Equation (19) shows that in order to determine the crosspower spec
trum for the response ) at two points r and r one
needs to know the crosspower spectrum for the load p (u.,,}
at two points and 1, and the transfer function H (F S, c) and
its complex conjugate H (F'?, )) At this time it is worthwhile to
compare (19) to the wellknown case of a onedimensional system for
which we have
uu () =HrCU) O (20)
In (20) all quantities are real and are not functions of space so there is
no integration. In (19) the crosspower spectra and the transfer
function are complex and are simultaneously functions of two space
coordinates ? and T They have to be determined for all possible
pairs of points over the whole surface of the structure. This makes
the problem much more involved. We will now turn to the case of a
discrete loading which can be more easily investigated experimentally.
7. Discrete Point Loading
Let us assume that the structure is loaded at r discrete
points IS (t= i,2,s...,n.) by concentrated forces pj(t). The load
per unit area can be written as
p(^t) (21)
where 0o 4) represents the Dirac delta function in twodimensional
space. Substituting (21) into (19) where 5 and Sp have been replaced
by 6~ and for convenience, we have
i., k
Sk ssie (22)
Then, since for a function ( )
ST(r) (l?)di = C( ) (23)
5T
we finally obtain, using the discrete notations for the crossspectra,
H() H(z) =
5959 (z4)
5s) a
r, )r, = 1, ,3, ...,m.
S, S = ,,3.. n.
where Hs(W) is the discrete notation for H(F,'c) Also, it has
been assumed that the response is measured at h locations that may be
different from the number of n loading points.
Equation (24) can be put into a matrix form which is better
adapted to highspeed computing machines. We write
H = H.:(w) : mx n complex matrix of the transfer function
I
[p :
e=^[O"
^=[CH
conjugate transpose of H
square complex matrix1 of order n of the
crosspower spectra of the load at n points
square complex matrix1 of order m of the
crosspower spectra of the response at tn
point s.
iThis matrix is Hermitian.
We have
,H = H (as
S a
F finally,
u = H H(26)
One can also separate real and imaginary parts of the ma
trices and write
i,. = Fu + j G,
(27)
H = Fp A +j
H = A +jB
H = A'jB'
where the prime denotes the transpose of a matrix. Then
S[A j B[F, +j Gp [A + B] (28)
which gives
A, A'FA + B' F BA' G +B' Gp A
(29)
+j (A ,A+B'GB)
8. Particular Case of TwoPoint Loading
For the experimental investigation we shall consider the case
of correlated loadings at two points, 1 and 2, on a structure and the
response at one point, 0, on this same structure. In this case s is
a Zx2 matrix, H is a column matrix and A reduces to one real term.
This means that the imaginary part of (29) must vanish.
We have the following:
A : B Bz
(30)
F;' F "0 G,
F = G =
F1 Fp G O
Then the power spectrum of the response has the form
u F) = F() T(W) + F'(2) T (2)
(31)
+F (w) T,.(() + Gp 4) T, ()
where the Ts have been referred to as transfer coefficients and are
given by
T,(c) = T;, = (A:) (B:)'. H12
,()= T = (A +(B)a = H:
TT=T.= 2 (A: A*+B:B:) (32)
Ta, () = ,, (A: B, A B,)
The first two terms in (31) represent the uncorrelated part of the ran
dom loading and the last two terms the effect of the crosscorrelation
of the same loading.
r r
If the modulus Ms and the phase angle e are used in the
representation of the complex transfer functions, we have
H = A:+jB = M (cos6 +jsin:n')
(33)
H A: + B M (cos 6 + j in )
and
T, (M:)
(34)
S 2 M M; cos(8 ) ()
T,= 2 M M; sin (eaB0)
9. Determination of the Transfer Function
Before turning to the experimental aspect of this work, we
shall consider the important conclusion that can be drawn from the
inputoutput crosscorrelation of a discrete load p applied to a linear
system with the response U It is known that the measurement of the
crosscorrelation of the input of a onedimensional electrical system
with its output leads to a formulation that can be used to determine the
transfer function [11, 13]. This result can be extended to the case of
a multidimensional mechanical structure [6]. If a structure is loaded
at one single discrete point by a random load,
p(=fb) (: (r) S (( (35)
and the response LL = LU (r', ) is measured at another point r then
the structure can be considered as the usual "black box" of electrical
engineering and we have
UP) = H (c) sPs() (36)
In (36) p (sc) is the crosspower spectral density between the re
sponse UL at point F and the load p at point S and sf's (w) is the
power spectral density of the load at point ? The first term is
complex, but the last one is real. Written in terms of real and imagi
nary parts, we have for the transfer function H;(u):
H,(O) Fu() + (37)
It should be pointed out that when the transfer function is
needed for the corrputation of the response, as in the case of a con
tinuous distributed load, the discrete transfer function HN(w) has to
be determined for a large number of locations S in order to have
an adequate substitute for the continuous function H (S,~ ). Al
though it requires numerous measurements and numerical transforma
tions, the righthand side of (37) can be determined experimentally as
shown later.
CHAPTER II
EXPERIMENTAL INVESTIGATION
1. Scope of the Study
The scope of the experimental investigation will be as follows.
First we desire to check the validity of formula (19) for the prediction
of the response of a structure subjected to a random loading. We will
consider only the simple case of a random loading applied simultane
ously at two discrete points of the structure and the response at a third
point. Therefore, equation (31) will be substituted for equation (19),
thereby replacing the integration by a summation. An experimental
determination of the transfer function will be made according to equa
tion (37), using a random loading at a single point. Then experiments
with the twopoint loading will be conducted and the response meas
ured. Finally, a comparison will be made between the computed re
sponse and the measured response.
Another interesting aspect that will be investigated here is
the effect of the correlation of simultaneous loadings on the response.
Most existing work to be found in the literature has been concerned
with input loadings restricted to the hypothetical uncorrelated white
noise. In this case the imaginary part of the crosspower spectra
vanishes, and equations (19) and (31) are greatly simplified. Actual
random loads having various correlations leading to input power spec
tra that retain the imaginary part will be investigated.
It has been found in the course of this investigation that the
conducting of the experiments and the processing of data require the
setting of numerous electronic and electromechanical components. It
was possible to use standard electronic equipment for most of the
components, or at least to use this equipment with only minor modi
fications. On the other hand, several electrical circuits and electro
mechanical devices were specially built. Matching the different pieces
of equipment and keeping them running simultaneously was sometimes
a delicate problem. Nevertheless, the experiments proved to be fea
sible. Figure 2 shows a picture of the experimental arrangement.
2. Experimental Apparatus
2. 1 The Model
The model tested was a cantilever beam, 7 in. long, 1 in.
wide and 1/2 in. thick, specially constructed in order to have a large
damping characteristic. The model was made of a 1/4 in. thick plexi
glass core, on both sides of which a 1/8 in. layer of a plastic base
damping material was cemented, as shown in Figure 3. The beam
TI k
i 4
p.'
1+x
23
I
C',
I4
Co
C'
02
weighs 90 grams and was clamped on a 901b stand that provides a
virtually rigid immovable reference as shown in Figure 4. Figure 5
shows the location of the applied forces at points 1 and 2, situated re
spectively at 1 1/2 in. and 4 1/2 in. from the free end of the beam.
The response was monitored at point 0, situated 1/2 in. from the free
end.
2. 2 Loading and Measurement System
Figure 6 is a block diagram of the loading and the measure
ments on the beam. The twochannel tape recorder no. 1 provides an
input to the beam of two random signals of various correlations. These
signals were obtained in the course of previous work from pressure
measurements at the boundary of an air jet [15]. The tape was played
back at a speed of 15 ips, which is half the recording speed, in order
to meet the frequency range of power amplifier no. 2.
The signal from channel no. 1 was fed into one power ampli
fier, and that from channel no. 2 into a separate power amplifier.
Amplifier no. 1 was able to deliver 1 amp rms into a 15ohm resis
tance for a frequency band ranging from 10 cps to 5000 cps. Amplifier
no. 2 was able to deliver 1.5 amps rms into a 50ohm resistance for a
frequency band ranging from 10 cps to 1500 cps. These amplifiers
were standard power amplifiers for sinusoidal waves and had no pro
vision for random wave equalization.
4,
T,
4,
*1
bO
Cd
4
' 0
Cd
bo
f
tj~ I
.4
26
4
4)
0
4)
~1 14
N 0:
No 4
LI
14 4
0C
4)
N1
Fr '3
N 
, U
P10 r0
oC"4
ER 4
0
J E
E fi
o5
o (Q
u
4
4)
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30
k
'0
34
U
40
04
The output circuit of each amplifier was composed of a load
resistance R a coil C the resistance of which was 4 ohms, and
a measuring resistance 'L of Z. 5 ohms, all coupled in series. The
load resistance R was added in order to reduce the feedback effect
from the beam to the amplifier. This procedure realized a partial
equalization [1].
Force measurement. It is shown in Section 2. 3, equation
(38), that the force acting on a coil is proportional to the current, t ,
flowing along the wire of the coil. Let t, represent the current in the
output circuit of amplifier no. 1. All the components of this circuit
are in series; therefore, the same current t, flows through the coil
and the resistance L, The voltage V, across the resistance tL, is
proportional to the current t, (Ohm's law, V'= L). Consequently, the
voltage 'V can be calibrated to represent the force P,(t) applied at
point 1. Also, the voltage V,, across 'L, can represent the force p (t)
at point 2. The voltages I, and V2 were recorded simultaneously on
tape recorder no. 2, using FM recording at a speed of 30 ips. This
tape recorder had seven channels, but only two channels were equipped
with an FM recording device.
Measurement of the re sponse. The monitored response was
the acceleration. The sensor was a 6gram crystaltype accelerome
ter with a frequency range of 20 to 80, 000 cps and a sensitivity of
0. 114 g/mV when connected across the preamplifier. After amplifi
cation the acceleration signal was recorded on tape recorder no. 2
simultaneously with the force signal no. 1. Two successive runs
were necessary to record simultaneously the two force signals of the
previous paragraph and then the acceleration signal and the force sig
nal no. 1.
2. 3 The CoilMagnet System
The two concentrated forces P (L) and p(t) were applied to
the beam by means of a specially built electrodynamic device. The
basic principle of this device is the one used in any commercial shaker.
According to Laplace's law, an electric current itk)flowing through a
helicoidal wire, the coil, placed into a radial magnetic field 6 cre
ates a resultant force F(i) acting along the axis of the coil (Fig. 7).
This force F(k) is given by
F = G = k I (38)
where
F = F(t)
= induction of the magnetic field
= length of the wire
i = i(t) = current
k = e = constant
Ft
1'1

4F 
N
Annular
magnet
Figure 7. CoilMagnet System
The main difference between the device built here and the
commercial shaker is in the attachment of the coil and its weight. In
the commercial shaker the coil is suspended permanently to a frame
by a flexible fixture. The weight of the entire moving part is usually
substantial and when the part is attached to the structure, it introduces
additional mass and rigidity that would have been prohibitive in the
case of the relatively light beam used here. For this reason, ex
tremely light coils weighing only 4 grams each were built by winding
two layers of a 0. 009in. copper wire on a light cylinder of paper, 1 in.
in diameter. The wire was cemented to the paper with a commercial
epoxy, and a conical cap was attached to the cylinder to allow its at
tachment to the structure. Once fixed to the beam, the coils were an
integral part of it (Fig. 3).
A radial magnetic field produced by a permanent annular
magnet and its casing was placed around each coil. The gap in which
the coil was moving was approximately 1/16 in. Each magnet was
rigidly fixed on the stand and there was no mechanical connection be
tween the coil and the magnet. The beam itself procured the proper
centering of the coil playing the role of the flexible fixture of the
shaker. Therefore, the disturbance was reduced to the weight of the
coils.
Z.4 Calibration of the CoilMagnet System
The constant 1 of formula (38) was determined directly by
a static test. The coil was attached to the end of a very flexible can
tilever beam and was moving into the magnet. The equilibrium of the
beam was noted. Dead weights were placed on the beam vertically
above the coil, and the equilibrium was restored by adjusting a direct
current through the coil. Figure 8 indicates the results of these
experiments.
In order to check the validity of this calibration for the case
when the current through the coil was not DC, but was rapidly varying
with time, an impedance test was performed. A stiff cantilever beam
with a large damping coefficient was built and driven through the coil
by a sinewave at the resonance frequency (j, = 500 cps) of the funda
mental flexion mode. It is known that near the resonance one can con
sider a structure as a one degreeoffreedom system. The relation
between the complex driving force N and the complex response X at
the point of application of the force is given by
[(R oR + ) + j b (]% = T3 (39)
where i is the generalized mass, IT the generalized rigidity and b
the generalized damping constant. At the phase resonance frequency
,A) we have
Coilmagnet no. Z _
k = 245 grams/amp
k = 0.111 lb/amp
Coilmagnet no. 1
k = 180 grams/amp
k = 0. 0815 lb/amp
0.6
Amps
Figure 8. Calibration of CoilMagnet System by Static Test
Grams
zoo
Sca = (40)
If j is the complex acceleration and IX the reduced damping coeffi
cient, we have
a t (41)
2 PJ C
Therefore, equation (39) becomes (shifting from complex to real
representation)
J= 2~ ~ ((42)
where W and A are either the maximum value or the rms value of
the force and the acceleration. Formula (42), together with (38), was
used to compute the acceleration 4 once the force J the damping
coefficient (o and the generalized mass e were measured. This com
puted value of the acceleration J. was then compared to the measured
value by means of an accelerometer.
The damping coefficient C( was determined by a transient at
the natural frequency :
t = 0.734 n log (43)
where im is one maximum amplitude of the transient and mn the
maximum amplitude after n halfoscillations. For the calibration
beam used here we obtained 0( = 0. 017.
The generalized mass A was determined by the method of
variation of frequency [16] resulting from small additional masses on
the vibrating structure. If a small additional mass A P is added to
the structure at the point of measurement of the response, the struc
ture has then a new resonance frequency f+ Af, From equation
(40) we have
A + A (44)
or
A (45)
2
The frequency shift 6h was measured by a beating process with the
original natural frequency f before the additional masses were added.
The experiment gave j = 456/g in grams/msec2, where g is the
acceleration of gravity. Finally, the relation between the acceleration
and the current through the coil for the calibration beam is
V
.4 = k_ (46)
9 15.4
where
O4
S= the acceleration in number of g's
i = the current in amps
k = the constant given in Figure 8.
Figure 9 gives a comparison between the computed values
according to (46) and the measured values of the acceleration. It can
be seen that the constant 1 measured by the static calibration gives
a satisfactory value for dynamic loading, at least for the frequency
tested, i.e., 500 cps. It was assumed that the agreement was also
acceptable for the whole range of frequencies of the tested model, i.e.,
from 100 to 1500 cps.
3. Testing Procedure
3. 1 Relative Sign of Forces and Acceleration
Measuring the force by the voltage 'V across the resistance '
(Fig. 6) does not provide the sense of the applied force. This sense
depends upon the relative sense of the induction 'A and the current i
in the coil, and not upon the resistance 'L It is important to measure
the force and the corresponding acceleration with the proper rela
tive sign. This sign was determined by sending through the coil a
sinewave at a frequency near the fundamental natural frequency of the
beam. Force and acceleration have to be in phase below the natural
frequency and out of phase above.
Acceleration in g's
15 
10o 
Computed acceleration
for coilmagnet no. 2
:omputed acceleration
for coilmagnet no. 1
* Measured values
0.6
Amps
0.8
Figure 9. Verification of CoilMagnet Calibration
3. 2 Experimental Determination of the Transfer Function
Only one force at a time was applied in this series of experi
ments. Tape recorder no. 1 was played back and only channel no. 2
fed into amplifier no. 2, giving a random force at point 2 of the beam.
Force and acceleration signals were recorded simultaneously on tape
recorder no. 2 for 15 sec.
Amplifier no. 2 was then disconnected from coil no. 2 and
connected to coil no. 1. The whole procedure was repeated, force and
acceleration signals being recorded for points 1 and 0 on tape recorder
no. 2. The setting of the attenuation for the acceleration amplifier and
the force attenuator Aa (Fig. 6) were carefully noted for subsequent
determination of the rms values. These two tests provided enough
data for the computation of the transfer function corresponding to
points 1, 2 and 0.
3. 3 TwoPoint Loading Tests
In this series of tests both channels of tape recorder no. 1
and both amplifiers were used in order to deliver simultaneously two
correlated forces, j,() and p,(t) at two points, I and 2, of the beam.
The desired data to be recorded were the two forces p,() and P, (k)
and the acceleration C4() taken at point 0. As stated above in Section
2. 2, two runs were necessary to record these three signals. Figures
10 and 11 show a picture of a sample of the random forces and the
Figure 10. Typical Force Signal (Points 1 and 2)
Figure 11. Typical Signals of Acceleration (Point 0)
and Force (Point 1)
acceleration displayed on an oscilloscope. Attenuations for the ac
celeration and the forces were carefully noted as above for the deter
mination of the rms values.
3.4 Calibration of the Electronic Chains
The calibration of the electrical network for measurement of
the two forces was done by employing a 500cps sinewave rather than
tape recorder no. 1. The gain of the amplifiers was adjusted to have
3 volts rms across each resistance l, and Ott These two voltages
V and Vh were measured with a standard vacuum tube AC volt
meter and recorded on tape recorder no. 2 with a known attenuation.
They were equivalent to the following forces on the coils and could be
interpreted as forces:
V, = 0. 216 kg on coil no. 1
VC = 0. 290 kg on coil no. 2.
Likewise, the calibration of the electric network for measurement of
acceleration was accomplished by means of a sinewave at 500 cps. An
rms voltage of 61 mV was substituted for the output of the accelerome
ter. The output Vco of the voltage amplifier was recorded on tape re
corder no. 2. According to the calibration of the accelerometer, the
voltage Vc, recorded on the tape is equivalent to an acceleration of
6.95 g's.
A comparison of the random signals ,(t) p() and Q(t) re
corded on the tape with the calibration signals V Vc and iro pro
vides the rms value for these random signals at playback. It must be
noted here that the rms of the random signals cannot be measured by
the standard AC voltmeter. 1 A true squaring device is necessary.
This was done by using the combination multiplierDC voltmeter de
scribed in the following section.
Eight different correlated loads divided into two series were
tested. The correlation of these loads is shown in Figures 25 and 39
in Chapter III. The case of uncorrelated load was also investigated
(Fig. 35, Chap. III).
4. Data Reduction
4. 1 CrossCorrelation and CrossPower Spectral Density
Direct experimental determination of the crosscorrelation
and the crosspower spectral density are both possible. The cross
correlation has the simpler experimental determination. For an er
godic process, the latter determination requires a delay line, a mul
tiplier and an average. If the crosspower spectral density is also
needed, as it is for this case, it can be obtained by taking the Fourier
A standard AC voltmeter is a rectifying device, the calibra
tion of which depends upon the wave shape. It is calibrated only for a
sinewave.
transform of the crosscorrelation, which can be done numerically
(Sec. 5). The amount of computation becomes very large when the
crosscorrelation curve is oscillating and approaching zero at a slow
rate. Only highspeed computing machines can then handle the prob
lem. Electronic systems have also been devised for this purpose
[13, 17].
The direct experimental determination of the crosspower
spectral density is also possible [18]. The method uses a spectrum
analyzer as a basic instrument. A spectrum analyzer can measure
only real power and can process only a single signal at a time, ex
cluding any cross measurement between two signals as required in
the crosspower spectral density. This difficulty has been overcome
by using adding and subtracting devices together with a delay line [18].
This method seems to be somewhat more difficult than the direct de
termination of the crosscorrelation. If the crosscorrelation is
needed, the inverse Fourier transform can be taken numerically, but
the numerical integration is in no way quicker than the one in the pre
vious method. The direct determination of the crosscorrelation will
be performed here.
4. Z Normalization
All the correlation curves and the spectral density curves
reported in this study have been normalized. This procedure permits
a better comparison between the different curves. The normalization
is such that all the autocorrelation functions are equal to unity for a
zero time delay: viz.,
^(o) (= I(47)
This means practically that each signal has been divided by its rms
value, since
) = 'lin I L) Xcr(t+o)dt = [r(t (48)
T
where the bar denotes mean value. Therefore, the relation between
normalized crosscorrelation and true crosscorrelation is
Tx (k)(49)
V[]2 V[^)y(^
where [ l ) is dimensionless.
For the power spectra we have
=TW) (50)
l N [Y(t)
where has the dimension of time.
LY N
4. 3 Length of the Sample
The computation of the crosscorrelation was done according
to its definition, equation (8), and is repeated here in its simplest
notation:
+T
S() = lim(t) y(t) dt (51)
1xY T T0 g
/T
This definition requires the length of the records 'X(t) and y(L) to be
infinite. For practical consideration, only truncated samples are
available. Therefore, formula (51) is replaced by
+T
x(t,t) Jy() dt (52)
where T is the total time length of the sample. Choosing the proper
time T is a delicate problem. It was assumed here that if the time T
was much larger than the period of the smallest frequency component
of the truncated record, then formula (52) was a good substitute for
(51). It was also kept in mind that the time delay T should not exceed
a reasonable percentage of T which has been suggested to be about
5 per cent [19].
Different times T were tried for the determination of (52).
The crosscorrelation functions were first measured for 20sec sam
ples, playing back and rewinding the tape for each chosen value of T .
Then subsamples of 8, 4 and 2 sec were cut from the original samples.
With such shorter samples it was possible to make loops that could be
played back continuously on the tape recorder. Very little variation
was found in the computation of TP () for the different loop lengths.
The time delay t never exceeded T = 20 millisec. The ratio T/T is
then 0.5 per cent, which is much lower than the value suggested [19].
It was then concluded that a 4sec loop could be adopted as a good rep
resentative sample of the random signals, and that (52) would give a
good estimate of the crosscorrelation function.
4.4 Correlator
A simple analog computer was assembled for the computation
of (52). The sequence of operation is
1. Introduce a time delay in one of the signals ? or Y
2. Perform a multiplication
3. Take a time average of the product.
Figure 12 shows a schematic of the computing system. The
two signals 2(t) and y(t) are picked up simultaneously from the play
back of the FM tape recorder no. 2. One of the signals, depending
upon the sign given to T is fed into an audio delay line that provides
a given time delay, variable by steps of 0. 1 millisec from 0 to 2 milli
sec. Then the signals are amplified and the multiplication is per
formed by a commercial multiplier,
m(L) = X(LT) yCL) (53)
Figure 13 shows the square of ?A(t) and Figure 14 shows the product
%m(L).
The third operation is nothing more than determining the
mean value of the product signal Im(t)~ Electrically this means the
measure of the DC component of nM(t), which can be done by a DC
voltmeter. However, the product operation X(a) by y(t) introduces
lowfrequency components in n(t) although they did not exist in
1The product signal ,m(t) can be considered as having the
following Fourier series expansion:
m (t) =Mo + C t in (e th +
where M, is the mean value measured by the DC voltmeter.
'4
0
4.J
0)
0
C)
11'
Figure 13. Typical Squaring of Random Signal
Figure 14. Typical Product of Random Signals
'X and y and makes the meter oscillate. To smooth out the reading,
a resistance capacitor box was built, the effect of which was to elimi
nate the ripple of the DC component by shorting out all AC components.
This box was placed between the output of the multiplier and the meter.
The loading of the capacitors of the box required about 20 sec.
4.5 Procedure
The use of a loop greatly reduced the processing time by
eliminating the rewinding time of the tape recorder and having the RC
box almost continuously loaded. According to formula (49), three
averages
(X ) y(t) [_()r)]2 [yCt)]9 (54)
are necessary to obtain the normalized correlation. The procedure is
to fix a value of t (negative T being introduced by delaying the second
signal y(k) ), then perform successively the three averages (54). The
actual measurement of the two mean square values and, consequently,
the division in (49) were avoided by adjusting the gain of each amplifier
in such a manner that the meter would read unity
[X(t~)] = [y(f)]M2= (55)
in performing the mean square operation. The reading of the mean
value of the product tvk) gives directly the normalized correlation
(Tmi The method was even rendered more rapid by using
two multipliers simultaneously, one for the product and the other for
the mean square. The procedure was repeated for each value of T ,
and the correlation curves plotted point by point on largescale paper.
Constructing a correlation curve required an average of 100 meter
readings.
elay lihue
Figure 15. Recopying the Delayed Signal on the Loop
The delay line used in Figure 12 has a range of only 2 milli
sec, but the correlation curves reported are extended to 20 millisec
in some cases. This long time delay was done by copying the 2
millisec delayed signal X(t2) and the undelayed signal y(t) from two
channels of the loop to two other channels on the same loop (Fig. 15).
When played back, these two new signals have an original time shift of
2 millisec that comes in addition to the value of t set on the delay line.
This procedure can be repeated at will to produce time delay up to the
desired value. It should be pointed out that this was possible only with
an FM recording technique, since this technique erases whatever has
been recorded on the track before a new signal is to be recorded.
5. Numerical Computation of the
Power Spectral Density
All the power spectral densities (and crosspower spectral
densities) reported in this experimental study have been obtained by
taking the Fourier transform of the autocorrelation (and cross
correlation) curves. Let us rewrite (10) in the discrete notation;
omitting Z and y for simplicity, we have
o00
]() cprs (z) e'd t (56)
We now separate real and imaginary parts in (56) and use the symme
try property
tp(r) = jTC r (57)
Using the notations of (13) for the real and imaginary part of s() ,
we have
cp T) P (CT)
2T, r. r
C1 L*
Figure 16. Fourier Transform Integration
F's() "() + (r) cos ru dt (58)
0
G"'() = O IT sr TCT) 5i. Sill dr (59)
The integration was performed numerically by approximating
rs so
the curve (C(t) + T (z) by n horizontal segments of straight lines
of length AT (Fig. 16). The interval of integration was also split into
n equal subintervals:
rs ro
F (W) = Co) Cos t + ...T + (ra o Z dt +...
Then the cosine was integrated into each interval. The sarr
dure was followed for (59). Finally we have
(60)
ie proce
F LsitITf A r ( iP + [, s t+ sk]o(TOICos (61)
:F L. I
G 6 WiTsi n {L,6 ^ [0ce t s (T()]S] s P t1 j
L11 I
(62)
where it is recalled that the indices '( and y have been omitted for
simplicity. These formulas derived for the crosspower spectral
densities are evidently valid for the ordinary power spectral density
rs
for which (62) will be identically zero. The numerical values of Tj (T)
were read from the largescale correlation curves, and the interval
AT was chosen as 0. 08 millisec. The numerical computation was
done on an IBM 650 computer.
CHAPTER m
INTERPRETATION OF THE RESULTS
1. Transfer Functions
a g
The transfer functions H,(f) and Ha(f) have been determined
by the crosscorrelation technique explained in Section 9, Chapter I.
Autocorrelation and crosscorrelation functions were measured ac
cording to the procedure of Section 4. 4, Chapter II, and are shown in
Figures 17, 18 and 19. Then the power spectral and crosspower
spectral densities, (.) () *, (2) ( were
computed numerically according to (61) and (62). Finally, the trans
fer functions H (f) and H,(f) were obtained by using formula (37) and
are represented in Figures 20 and 21. The modulus and phase angle
representation has been chosen for ease of plotting. Normalized
values of the modulus are plotted in Figure 20. It can easily be shown
by using (36) and (50) that the actual value of the transfer function H (
t" hi
is related to the normalized value H (4 n by
[H(()= [ ] i (62)
where F is the rms value of the acceleration 4(i), and P is the rms
4
0
f4
0
*4
'44
0
Q)
OD
Po
o
v.4
0
u
0
r4
o k
E 4
0
4)
41
"4
Q) )0
E ^
0
l
4oa
r
4)
kZ
O 4 m O I C O
S4
0
o
4A
o.,
0o0
u I
010
0 N
*a a
0
4)
ok oo
Ii
E 4
0
0
I
4
o
S I 1 14
*E 0
O0
61
I 44
0
o V0
\ ,"
\o ~ ^" K
\i t 44
\ 3 0
\ 3 0
\. 6 "5
^ > "4
^Ja 4)
____  * ^^ 14
^ * *o i
., _ 0 u 0
 ^^ C
  * K *
 ~ ^ *
>0
1 I
4
,u
0 0
.. 0"
I' a
.I1
44 f'44
o
a 0
o H
0
4,
0
0
IU
a 0
,
0
H '
'414^ I I
t '.1 0
Normalized modulus
Curves r/P rms
[H,] 10.60 g/lb
[H:l 12. 95 g/lb
QH: ]
'\
ii
I i
I I
I
,
I.. S
, '
Soo
1ooo
Frequency, cps
Figure 20.
Modulus of Transfer Functions H (f) and H;(f)
1500
0
n1 2
Phase angle, radians
2
1 
0I
1
V 
500
1000
1500
Frequency, cps
Figure 21. Phase Angle of Transfer Functions H:(f)and H(T)
I
8
:J 1
value of the applied force p(k). The ratio F/p is indicated
in Figure 20.
The modulus IHJ and the angle 0, for point 1 do not ex
hibit very much variation with the frequency except at the two ends of
the spectrum. But the modulus IH for point 2 has a sharp maximum
for a frequency of about 500 cps, which corresponds to the natural
frequency of the second mode of the beam. The angle Ga also has a
rapid variation for that same frequency. The values obtained for the
frequencies above 1200 cps must be considered with caution since the
input force spectra P and F and the cross spectra and
C all have small amplitudes in this region. Therefore, equation
uap
(37) is close to the indeterminate form 0/0 which evidently gives a
poor accuracy. Nevertheless, a tendency for a second peak at a fre
quency in the neighborhood of 1400 cps can be detected. The transfer
coefficients, T a T I corresponding to the Hhave
been computed according to (32), and are represented in Figures 22
and 23. We note that in these figures actual values of the T"s are re
ported rather than the normalized values.
2. Loading of Series I
Figure 24 represents a typical autocorrelation of one force
for a twopoint loading. Figure 25 represents the various cross
correlations of the input forces of series I, applied simultaneously at
61
Transfer coefficients
8 102 gZ/lb2 1 2
I
1 I
I I
I I
i I
I
I
5
I I
I i
6
T
I I
(( t
i
II
t
i
S / I
/
\ ,
Frequency, cps
Figure ZZ. Transfer Coefficients T1 and
,'
500 100 1SO
Frqeny cp
Fiur 22 TaseCoficint I, and I i
Transfer coefficients
3 102 g2/lb2
z
0
1 2
= I
' ,
I
I
I *
I .
I,
I
I
500 tOOO
Frequency, cps
Figure 23. Transfer Coefficients T and T
la #I
I1
\I
W4
0
oo
40
cd
0
0
u 0
4
4
0
a
'4
4,
o 0
0
E4
4,
'4r
E 4
0
0h 8
2d
C.,'
4
4)v
'.4
0 I* ,, 0
0 0 e0 .. .O
64
r 4
0
c)
uJ
u
tn
.SU
bo
to.4
01
ID O
0~
t a
4)) '.
4* L
TjH
i," I 4
'4
44 S
C~o ) 9 ~ ~ ~ro0
t'4
65
~4
cc
4) 04 ) ) *l 0
UUUU 0(
H
j I I $
I .. .4
I .4 '4.
I, 0 0
/e O
U
'I~ '4.
*I /
*~ /
I 0
r 4
.5: 'h 0
r$4
o$4
k ~ ;;~~~0
~51rU
O 0
Uo q S4
'.
points 1 and 2. These forces correspond to actual pairs of air pres
sures measured at different locations downstream from an air jet [18].
The effect of these different loadings on the acceleration of the model
is shown by the autocorrelation curves in Figure 26. A reasonably
rapid decay of these curves has been obtained because of the large
damping coefficient originally introduced into the model. The Fourier
transform of all these curves leads to the corresponding cross spectra.
Each of the figures, 27 through 36, represents the normalized
spectra and cross spectra for the input loadings and the corresponding
accelerations. Actual values of these spectra and cross spectra are
obtained according to formula (50). The spectra and the cross spectra
of the input forces exhibit occasional fluctuations in terms of the fre
quency which did not exist in the input pressures from the tape before
amplification (Fig. 5). These fluctuations are believed to be due to the
feedback effect of the beam, through the coil, on the output of the power
amplifiers, nos. I and 2. It must be noted here that since the coils C,
and C. are each in series with the measuring resistances t, and
ta the voltages V, and V across I., and 'L give the real
forces acting on the beam, i.e., the intended forces minus the feed
back forces. Therefore, the forces applied at points 1 and 2 are not
exactly the same as the pressures jC6) and ?() from the tape, but
the departure of w() and p.(t)from J(t) and () is small enough to
spectral density
104 sec
Curves rms
111 0. 216 Ib
122 0. 298 Ib
112 0. 254 lb
111
112
Real
Imaginary
I..
500
o000
Frequency, cps
Power Spectral Densities of Forces, Case 1
Power
122
2
4
1500
i
lo
Figure 27.
Power spectral density
100 104 sec
io0
Frequency, cps
Figure 28. Power Spectral Densities of Acceleration, Case 1
Curves rms
 Measured 3. 90 g
* Computed 3. 80 g
500
1000
1500
n 1 z E~
69
Power spectral density 0
to4 1 2
10 sec
Curves rms
Sz22z 211 0.214 lb
222 0.311 lb
212 0. 258 lb
I
/ Imaginary
0r
I
0  " \....  ... '" *^'" ^.
r i
2\ /1 1
"Real
4
500 tooo 1500
Frequency, cps
Power Spectral Densities of Forces, Case 2
Figure 29.
Power spectral density
S 104 sec
100 
0
1 1 Z
10 
* **
* O0
500
Frequency, cps
1000
1500
Figure 30. Power Spectral Densities of Acceleration, Case 2
Curves rms
 Measured 4. 20 g
* Computed 4.45 g
~
Power spectral density
104 e
104 sec
i
i", r
r I ,
\r
I
I
I
1
I
I
r
t
I
I
I
r
t
I
r
I
I
r
I
r
'
N
41
0
1 1 2
311
312
Real'  S
/
3 /
/
312
Imaginary
500
1000
Frequency, cps
Figure 31. Power Spectral Densities of Forces, Case 3
to
8s
(6 
Curves rms
311 0.217 lb
322 0.305 lb
312 0. 258 lb
1500
I
~32
I
!^32


i

0
f 1 2
Power spectral density
loo 104 sec
I I __ I
500
1500
0ooo
Frequency, cps
Figure 32. Power Spectral Densities of Acceleration, Case 3
Curve s rm s
 Measured 4. 15 g
SComputed 4.45 g
Power spectral density
104 sec ", '
I
.0 I
i I
t I
I I
I (
t I
4
t
1
I
I
1
I
I
I
I
r
r
Ir
I
R eal / '". ,
/* e l /, ** ^ ^ ^
\~
 .,.
/ m412
/ / Imaginary
500
tooo
Frequency, cps
1500
Figure 33. Power Spectral Densities of Forces, Case 4
0
= 1 2Z
411
Curves rms
411 0. 220 lb
422 0. 297 lb
412 0.256 lb
r
2 [
0
Al I z E
Power spectral density
00 104 sec
Curves rms
 Measured
* Computed
4. 10 g
4.30 g
*
*
500
tooo
Frequency, cps
* 7 *
1500
Power Spectral Densities of Acceleration, Case 4
10o
I_
Figure 34.
Power spectral density
104 sec
10 sec
o 
511
0
4 1 2
Curves rms
511 0. 214 lb
522 0. 297 Ib
512 0.252 lb
522
..... 512 .........
J/ / 2Imaginary
512
Real
2?
500
1000
Frequency, cps
Figure 35. Power Spectral Densities of Uncorrelated Forces, Case 5
1500
I
0
Hi 2z
Power spectral density
1o0 104 sec
*
*
500 1.000
500 1000
Frequency, cps
Figure 36. Power Spectral Densities of Acceleration, Case 5
Curves rms
 Measured 4. 20 g
*Computed 4. 60 g
1500
produce only a small change in the intended crosscorrelation. Never
theless, since we measure the actual forces applied at points 1 and 2,
we can forget about the feedback effect. The correlations of Figure 24
and also of Figure 39 represent the real loading, which is very close
to the intended one.
Figure 27 represents the power spectra and crosspower
spectrum of the loading denoted as case 1. In this case the same pres
sure signal JCt) was fed simultaneously into the two power amplifiers,
nos. 1 and 2 (Fig. 6). This procedure gives the maximum cross
correlation between the two forces P(() and p(t), i.e., a zero imagi
nary part in the crosspower spectrum. However, due to the phase
shift between the two amplifiers (they were of different construction)
and the feedback effect, it can be seen from Figure 27 that the imagi
nary part 112 is not zero. Nevertheless, the correlation is strong,
as indicated by the real part 112. The spectrum of the measured ac
celeration at point 0 is represented in Figure 28 by the solid line. The
plot in this figure, as well as for the subsequent spectra of the accel
eration for other loadings, has been made on semilog paper because of
the sharp variations of the spectrum in the neighborhood of 500 cps.
The dots on Figure 28 represent the computed spectrum according to
formula (31), using the transfer functions determined experimentally
above. The computation of (31) requires the knowledge of the rms
values of all the quantities involved besides the normalized spectra
and transfer functions. Indeed, the response depends not only upon
the nature of the two forces P() and p(t) but also upon their rela
tive strength, and upon the size and stiffness of the structure. The
rms values of the forces P [)a= = V ]'and P:=
are indicated in Figure 27. Formula (50) was used to obtain the actual
spectra of the input forces.
00
Once the spectrum of the acceleration 4 (f) had been com
puted by (31), then the computed rms value could be determined by
integrating the spectrum over the whole range of frequency, viz.,
r= L[aC)] 7 =$( df (63)
The computed rms value J is indicated in Figure 28, and the dots
represent normalized values of the computed spectrum.
The agreement between the experimental curve and the com
puted values is not, in general, excellent, but is nevertheless accept
able if one considers the number of successive experimental measure
ments and numerical computations involved between the test runs and
the curves in Figure 28. Running several identical tests rather than
only one of each case as reported here, would certainly show the mag
nitude of the dispersion and improve the agreement by averaging over
the number of runs. The computed rms value and the rms value de
termined experimentally differ by only 3 per cent which is good but is
considered as an exception, since this difference ranged between 5 and
10 per cent for the other cases of loading.
The general shape of the power spectrum is reasonably well
predicted, at least for the values of the frequency below 1000 cps.
Above 1000 cps the agreement is poor as can be expected, since the
transfer function is itself not determined with great accuracy in this
region.
Figures 29 through 36 represent different cases of correlated
loadings that were tested in series I and the corresponding measured
and computed spectra of the acceleration. It can be seen that all out
put spectra have a maximum at about 500 cps, but the shape of the
curves varies from one case to the other, showing the effect of the
various loadings. Analogous to the situation in case 1, the general
shape of each curve is reasonably well predicted except for the fre
quencies above 1200 cps. The computed points fall off the measured
spectra with about the same discrepancy as for case 1. We also note
that these computed values appear to be, for most cases, shifted to
the right by a small amount. No explanation other than a general im
precision in the determination of the transfer function has been found
for this shift. It is remarkable to note that the rms values for all
these cases remain relatively close together. The following table con
tains all rms values measured and computed for all cases of series I
and series II.
RMS VALUES OF THE ACCELERATION AT POINT 0 FOR LOADS
AT POINTS 1 AND 2 FOR SERIES I AND II
Difference r
Series Case Computed Measured above
Series Cases above
rms in g's rms in g's measured
1 3.8 3.9 3 15.3
2 4.45 4.2 +7 16.3
I 3 4.45 4.15 +7 16.1
4 4.3 4.1 +5 15.1
5 4.6 4.2 +10 16.6
6 5.0 4.6 +9 18.2
7 4.6 4.2 +10 16.0
8 4.55 4.8 6 19.0
9 4.35 4.15 +5 16.8
3. Effect of Correlation
Special attention is now given to cases 1 and 5. Case 5 has a
completely uncorrelated loading as opposed to case 1, which is com
pletely correlated. In both cases the pilot signals fk) used to drive
the two amplifiers, nos. 1 and 2, were selected from pressure rec
ords made previously at the same location of the air jet [15] so that
they all have the same autocorrelation function. In case 1 the forces
p,(t) and P () were obtained by feeding the same record (t) simul
taneously into the power amplifiers, nos. 1 and 2. In case 5 two dif
ferent records P(t) and !t) taken at two different times were con
sidered so that the crosscorrelation between them was lost, contrary
to case 1 where the crosscorrelation was maximum since only one
single record was used. This idealistic situation was only partially
altered by the amplifiers and the feedback effect as shown in Figures
27 and 35.
Output acceleration spectra of cases 1 and 5 are shown in
Figure 37 in an attempt to point out the effect of the crosscorrelation.
The distribution of frequencies is different for the two cases. There
is more power distributed in the higher frequency range for the un
correlated load than for the correlated case.
This difference has been explained by analyzing the contribu
tion of each term in the computation of the response in case 1. Corre
lated and uncorrelated parts were isolated in the computation as stated
in Section 8, Chapter I. These two terms are represented in Figure 38.
It can be seen that above about 700 cps the correlated part is of the
same order of magnitude as the uncorrelated part. This shows clearly
that the crosscorrelation between the input forces has a real impor
tance and that neglecting it would lead to an erroneous spectrum.
Moreover, above 700 cps the total spectrum is the difference of two
0
Power spectral density
loo 104 sec
10 
I'
L?
\\
I
I
Case 1 
(correlated forces)
AI
L Case 5 (uncorrelated forces)
I
t
1
500
1.000
Frequency, cps
o500
Figure 37. Effect of CrossCorrelations of Forces on the
Experimental Accelerations
Curves rms
 Correlated 3. 90 g
 Uncorrelated 4. 20 g
Power spectral density
4
50o 10 sec
0 
Total 
to
0
A 1 2
Case 1

!
t I
S 'Correlated
I /
I
I I
/
i ,
Soo
tooo
Frequency, cps
Figure 38. Computed Accelerations, Correlated and Uncorrelated
Curves rms
 Uncorrelated 3. 80 g
 Correlated 3. 80 g
.**** Total 3. 80 g
10
15oo
large numbers, which explains, at least partially, the larger discrep
ancy between measured and computed values in this region.
4. Series II Loading
In this series a more hypothetical loading was investigated.
The two coils were hooked up in such a way that for a direct current
through the coils, one coil would push while the other would pull. Then
a single random signal ?(t) was sent through amplifier no. 1 and the
same random signal, but delayed by a time T J(t't) was sent to
the second amplifier. Four cases of time delay, T = 0, T = 0. 5 mil
lisec, Z = 1 millisec, T = 2 millisec, were tried, leading to the
crosscorrelations indicated in Figure 39. The rms values F and
P, of the input forces p(t) and P.(t) are also indicated in Figure 39,
and for simplifying the computation, the autocorrelation was assumed
to be the same as for case 1.
The resulting acceleration spectra were measured and com
puted with the same technique as above. The results are shown in
Figures 40 through 43. The discrepancies between measured curves
and computed values are of the same order as for the previous series.
However, the curves indicate a definite effect of the time delay t on
the shape of the spectra. This series shows that not only the maxi
mum of the crosscorrelation is important, but also that its time posi
tion on the time scale of the correlation function plays a determinant
I I I I
a 1 o 0 4. m g
I I I
a
uM
4)
0)
0
*l
k
*E M
#1
4
0
o
U)
U4)
"4
E o
R,
o .
4
0
0
U
a'
U

cL4
t ________ _______
E 
l J
I I I I
0
*I
O4
14
14
B
Power spectral density
00o 104 sec
10 
* 0
* 0
*.
* I I I
Soo Iooo
Frequency, cps
Figure 40. Power Spectral Densities of Acceleration, Case 6
15oo
Curves rms
 Measured 4.60 g
SComputed 5. 00 g
* *
0 *
0
41
0
4 1 1
Power spectral density
00 104 sec
*
* *
0 *
500 iooo 1500
Frequency, cps
Figure 41. Power Spectral Densities of Acceleration, Case 7
Curves rms
 Measured 4. 20 g
* Computed 4. 60 g
0. I
InlIiE
Power spectral density
100 104 sec
O* *
**
**
* *
500
1000
Frequency, cps
Figure 42. Power Spectral Densities of Acceleration, Case 8
Curves rms
 Measured 4. 80 g
SComputed 4.55 g
1500
0
f 1 1 g
Power spectral density
ioo 104 sec
10 
.*
0 0
i 
* 0
o00
1000
Curves rms
 Measured 4. 15 g
SComputed 4. 35 g
Frequency, cps
Figure 43. Power Spectral Densities of Acceleration, Case 9
1500
role. It is noteworthy to mention here that similar kinds of experi
ments were previously conducted on a string [3].
5. Conclusion
In this experimental investigation it has been shown for sev
eral loading configurations that once the transfer function of a canti
lever beam has been determined, at least for a finite frequency band,
then it is possible to predict the response to a twopoint random load
ing with reasonable accuracy.
Also, it has been found that the crosscorrelation of the load
ing plays a definite role in the determination of this response, and
should not be ignored when the spectral distribution is of interest.
However, the root mean square value of the response has been found
to be relatively insensitive to the crosscorrelation effect.
These experiments have shown that it is perfectly feasible to
determine the response of a structure to a random spacetime discrete
input by a semiexperimental method, at least for a twopoint loading.
The extension to the case of a large number of loading points does not
present any mathematical difficulties, but the amount of data process
ing and numerical computation would become rather large. In this re
spect an automatic correlator or even an electronic system for a direct
determination of the crosspower spectrum would be desirable.
LIST OF REFERENCES
1. S. H. Crandall. Random Vibration. The Technology Press,
Cambridge, Mass., 1958.
2. V. L. Lebedev. Random Processes in Electrical and Mechani
cal Systems. Trans. from Russian by the National Science
Foundation and the National Aeronautics and Space Administra
tion, 1958.
3. R. H. Lyon. "Response of Strings to Random Noise Fields."
Journal of the Acoustical Society of America, Vol. 28, 1956.
4. A. C. Eringen. "Response of Beams and Plates to Random
Loads." Journal of Applied Mechanics, Transactions of the
ASME, Vol. 79, 1957.
5. J. C. Samuels and A. C. Eringen. "Response of a Simply Sup
ported Timoshenko Beam to a Purely Random Gaussian Process."
Journal of Applied Mechanics, Transactions of the ASME, Vol.
80, 1958.
6. F. P. Beer. "On the Response of Linear Systems to Time
Dependent, Multidimensional Loadings." Journal of Applied
Mechanics, Transactions of the ASME, Vol. 83, March, 1961.
7. W. A. Nash. "Response of an Elastic Plate to a Distributed
Random Pressure Characterized by a Separable Cross
Correlation." Technical Note No. 1, Contract No. AF49(638)
328, University of Florida, Gainesville, September, 1961.
8. S. H. Crandall and A. Yildiz. "Random Vibrations of Beams."
Contract No. 49(638)566, Massachusetts Institute of Technology,
Cambridge, April, 1961.
9. S. H. Crandall. "Dynamic Response of System with Structural
Damping." Contract No. 49(638)564, AFOSR 1561, Massachu
setts Institute of Technology, Cambridge, October, 1961.