Response of structural elements to random excitation.

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Response of structural elements to random excitation.
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ix, 94, 1 leaves : ill. ; 28 cm.
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Trubert, Marc Roger Pierre, 1927-
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Vibration   ( lcsh )
Mechanics, Applied   ( lcsh )
Structural frames   ( lcsh )
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non-fiction   ( marcgt )

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Thesis--University of Florida.
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Bibliography: leaves 91-93.
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By Marc Roger Pierre Trubert.
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Manuscript copy.
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Vita.

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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 Two-Point 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 CONTENTS--Continued

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, Coil-Magnet and Stand . .. 25

5. Location of Acceleration Measurement and
Forces on the Model .. . 26

6. Flow Chart for Loading and Measurement 27

7. Coil-Magnet System . 30

8. Calibration of Coil-Magnet System by Static Test 33

9. Verification of Coil-Magnet 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 FIGURES--Continued


Figure

18. Cross-Correlation of Acceleration and Force for
Determination of Transfer Function H'( .

19. Cross-Correlation 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 Two-Point
Loading (Point 1, Case 1) . .

25. Cross-Correlation of Forces at Points 1 and 2,
Series I . . .

26. Autocorrelation of Acceleration at Point 0 for
Two-Point 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 FIGURES--Continued


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 Cross-Correlations of Forces on the
Experimental Accelerations . 82

38. Computed Acceleration, Correlated and
Uncorrelated ., ..... 83

39. Cross-Correlation 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

Cross-correlation 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 space-time 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 space-time response, UL(Fi ) of a

linear structure, S subjected to a lateral space-time 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 well-known harmonic analysis for the

one-dimensional 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

space-time 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
space-time 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 cross-correlation 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)
T--O 0 I T Z _T


where T is a time delay. The cross-power 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
ST-wo -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 t-a ) e-letdd ,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


S-r I+T-J,
0 [ =-T- ,


, e l d ,h
][+0"




)=, L
:,e de 6,
= --T-


lirm
T-o- T


-[


(17)


e *d


(18)

e d c ds. 8d a









The curled bracket is simply the definition of the cross-power 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 cross-power spec-

trum for the response ) at two points r and r one

needs to know the cross-power 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 well-known case of a one-dimensional 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 cross-power 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 two-dimensional

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 cross-spectra,


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 high-speed 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

cross-power spectra of the load at n points



square complex matrix1 of order m of the

cross-power 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 B-A' G +B' Gp A
(29)
+j (A ,A+B'GB)









8. Particular Case of Two-Point 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 cross-correlation
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 (ea-B0)









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

input-output cross-correlation of a discrete load p applied to a linear

system with the response U It is known that the measurement of the

cross-correlation of the input of a one-dimensional 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 cross-power 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 right-hand 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 two-point 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 cross-power 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',



I-4







Co


C'

02










weighs 90 grams and was clamped on a 90-1b 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 two-channel 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 15-ohm 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 50-ohm 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)
U



@*
N
0)

U

a


3-0


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 6-gram crystal-type 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 Coil-Magnet 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. Coil-Magnet 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. 009-in. 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 Coil-Magnet 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 degree-of-freedom 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

















Coil-magnet no. Z _
k = 245 grams/amp
k = 0.111 lb/amp


Coil-magnet no. 1
k = 180 grams/amp
k = 0. 0815 lb/amp


0.6
Amps


Figure 8. Calibration of Coil-Magnet 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 half-oscillations. 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/m-sec-2, 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 coil-magnet no. 2


:omputed acceleration
for coil-magnet no. 1


* Measured values


0.6
Amps


0.8


Figure 9. Verification of Coil-Magnet 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 Two-Point 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 500-cps 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 multiplier-DC 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 Cross-Correlation and Cross-Power Spectral Density

Direct experimental determination of the cross-correlation

and the cross-power 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 cross-power 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 cross-correlation, which can be done numerically

(Sec. 5). The amount of computation becomes very large when the

cross-correlation curve is oscillating and approaching zero at a slow

rate. Only high-speed 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 cross-power

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 cross-power 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 cross-correlation. If the cross-correlation 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 cross-correlation 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 cross-correlation and true cross-correlation 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 cross-correlation 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 cross-correlation functions were first measured for 20-sec 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 4-sec loop could be adopted as a good rep-

resentative sample of the random signals, and that (52) would give a

good estimate of the cross-correlation 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(L-T) 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

low-frequency 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 large-scale 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(t-2) 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 cross-power 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
C-1 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 cross-power 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 large-scale 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 cross-correlation technique explained in Section 9, Chapter I.

Autocorrelation and cross-correlation 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 cross-power

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
P-o
o






v.4



0
u








0
r4
o k
E 4






















0
4)
4-1
"4



Q) )0
E ^








0

-l


4oa







r-

4)

kZ


O 4 m O I C O















S-4

0

o
4-A


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 two-point 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
E-4
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

10-4 sec


Curves rms

1-11 0. 216 Ib

1-22 0. 298 Ib

1-12 0. 254 lb


1-11


1-12-
Real


Imaginary


I..


500


o000
Frequency, cps


Power Spectral Densities of Forces, Case 1


Power


1-22


-2-


-4-


1500


i


lo-


Figure 27.










Power spectral density
100 10-4 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
to-4 1 2
10 sec



Curves rms

Sz--22z 2-11 0.214 lb

2-22 0.311 lb

2-12 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 10-4 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

10-4 e
10-4 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


3-11


3-12


Real' --- S


/

3 /



/





3-12

Imaginary


500


1000
Frequency, cps


Figure 31. Power Spectral Densities of Forces, Case 3


to


8s-


(6 -


Curves rms


3-11 0.217 lb


3-22 0.305 lb


3-12 0. 258 lb


1500


I
-~---3-2
I
!^32
|
|
i
|







0
f 1 2


Power spectral density

loo- 10-4 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 /, ** ^ ^ ^


\~


-- .,-.


/ m--4-12

/ / Imaginary


500


tooo
Frequency, cps


1500


Figure 33. Power Spectral Densities of Forces, Case 4


0
-= 1 2Z


4-11


Curves rms


4-11 0. 220 lb


4-22 0. 297 lb


4-12 0.256 lb


r


-2 [







0
-Al I z E


Power spectral density
00 10-4 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
10-4 sec
10 sec


o -


5-11


0
-4 1 2




Curves rms

5-11 0. 214 lb

5-22 0. 297 Ib

5-12 0.252 lb


5-22


..... 5-12 .........
J/ / 2Imaginary


5-12
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- 10-4 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 cross-correlation. 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 cross-power

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 cross-power 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 1-12 is not zero. Nevertheless, the correlation is strong,

as indicated by the real part 1-12. 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 cross-correlation between them was lost, contrary

to case 1 where the cross-correlation 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 cross-correlation.

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 cross-correlation 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- 10-4 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 Cross-Correlations 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

cross-correlations 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 cross-correlation 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
O-4
14
14


B












Power spectral density
00o 10-4 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 10-4 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
I-nlIiE


Power spectral density
100- 10-4 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 10-4 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 two-point random load-

ing with reasonable accuracy.

Also, it has been found that the cross-correlation 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 cross-correlation effect.

These experiments have shown that it is perfectly feasible to

determine the response of a structure to a random space-time discrete

input by a semiexperimental method, at least for a two-point 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 cross-power 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.




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