RESPONSE OF A CYLINDRICAL SHELL TO RANDOM ACOUSTIC EXCITATION

by Daniel D. Kana

INTERIM REPORT Contract No. NAS8-21479

Control No. DCN 1-9-53-20039 (1F) SwRI Project No. 02-2396

Prepared for

National Aeronautics and Space Administration George C. Marshall Space Flight Center

Huntsville, Alabama

15 July 1969

SOUTHWES HOUSTON SAN ANTONIO

https://ntrs.nasa.gov/search.jsp?R=19700003973 2020-01-26T04:29:34+00:00Z

S E A R C H I N S T I T U T E 8510, 8500 Culebra Road

San Antonio, Texas 78228

YLINDRICAL SHELL us XCITATI

bY Daniel D. Kana

INTERIM REPORT Contract No. NAS8-2 1479

Control No. DCN 1-9-53-20039 (1F) SwRI Project No. 02-2396

Prepared for

National Aeronautics and Space Administration

George C. Marshall Space Flight Center

Huntsville, Alabama

Approved:

H. Norman Abramson, Director Department of Mechanical Sciences

ABSTRACT

Response and equivalent force spectra have been investigated

f o r random acoustic excitation of a cylindrical shell within a frequency

band of relatively low modal density.

resul ts a re compared f o r single point t ransfer functions, acoustic

mobility functions, response and equivalent force power spectral den-

s i t i es , and coherence functions. In general , it is found that a purely

theoret ical prediction of response based on l inear random process theory

is severe ly l imited because of the inability of current ly available expres-

sions for t r ans fe r functions to account for various deviations which r e -

sult principally f r o m imperfections and eccentricit ies in the cylinder.

However, good agreement is achieved between measured response and

that calculated with measured t r ans fe r functions. It i s further indi-

cated that a ra ther c o a r s e d iscre te representation of a continuously

distributed excitation is possible.

Theoretical and experimental

.. 11

T A B L E O F CONTENTS

G E N E R A L NOTATION

LIST O F ILLUSTRATIONS

INTRODUCTION

ANALYTICAL EXPRESSIONS

R e s p o n s e t o Mul t ip l e D i s c r e t e Exc i t a t ion

E q u i v a l e n t F o r c e Spectra

E X P E R I M E N T A L PROCEDURES AND RESULTS

M e a s u r e m e n t of E x c i t a t i o n Field

H a r m o n i c E x c i t a t i o n and R e s p o n s e

R a n d o m E x c i t a t i o n and R e s p o n s e

ACKNOWLEDGMENTS

R E F E R E N C E S

A P P E N D I X

P a g e

iv

V

1

8

8

13

1 9

3 3

34

35

iii

G E N E R A L NOTATION

a

Be

f

h

ne

RC

P S

w

wmn

shel l radius

equivalent f i l ter bandwidth for spectral density analysis

analysis frequency, Hz

thickness of shell wall

shell length

s ta t is t ical degrees of f reedom for spec t ra l density analysis

averaging t ime constant for spec t ra l density analysis

shel l mass density

excitation c i rcu lar frequency

natural frequency of m, n-th mode

iv

LIST OF ILLUSTRATIONS

1.

2.

3 .

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

C oo r dinate Sys tem

Apparatus for Measuring Acoustic Field

Cross-Spectral Densities of Acoustic Field

Spatial Distribution of Acoustic Field

Apparatus for Point Excitation of Cylinder

T rans fe r Function Between Y 1 and x = (0 , 0 )

T rans fe r Function Between Y 1 and x = (4, 0)

Acoustic Mobility Function for Y 1

Trans fe r Function Between Y 2 and x = (0,O)

Trans fe r Function Between Y2 and x = (4,O)

Acoustic Mobility Function fo r Y2

Single Point Random Response for Y 1

Acoustic Random Response for Y1

Acoustic Random Response for Y2

Theoret ical Equivalent F o r c e Spectra for Y1 and Y2

Variation of Acoustic Response a t Y 1 with Excitation Mesh Size

Coherence Function for Random Excitation

V

INTRODUCTION

Dynamic loading on launch and space vehicle s t ruc tures i s comprised

to a g rea t extent of spatially-distributed random acoustic energy which is

generated by various sources within the vehicle environment. At launch,

high-level engine noise i s reflected f r o m the ground up onto the s t ructure,

while during flight aerodynamic turbulence a s well a s engine noise excite

s t ruc tura l response. This response i s important f rom the point of view of

i ts influence on internal components and systems in addition to that of

s t ruc tura l integrity itself. Since cylindrical shells a r e a typical component

i n cur rent s t ruc tura l designs, it i s particularly important that their response

to such distributed loads be understood.

A general analytical approach to determine the response of elastic

1 s t ruc tures to distributed random excitation has been given by Robson a s

well a s others. The essence of this approach involves f i r s t the determina-

tion of theoret ical s t ruc tura l admittances or t ransfer functions between

response a t some appropriate point and harmonic excitation a t a. single

point. These functions can conveniently be expressed a s se r i e s expansions

of the normal modes of the system. Then, by means of generalized h a r -

monic analysis and superposition propert ies of l inear random process

theory, expressions a r e obtained which relate statist ical propert ies of the

response to the t r ans fe r functions and s tat is t ical propert ies of the excita-

tion over the aggregate of points i n the a r e a over which a distributed load

acts . This approach has been applied to the case of cer tain types of random

2

response of a cylindrical shel l by Nemat-Nasser2, and m o r e recently by

Hwang It is ve ry evident f r o m these analyses that the par t icular fo rm

of damping mechanism which is assumed has a profound influence on the

response, as it does in any force vibration problem. It is fur ther apparent

that no modal distortions, and thereby deviations in response distributions,

such as m a y be caused by eccentricit ies in the cylinder, can be predicted by

3

such a theory.

In view of the fac t that previous investigations4 of dynamic shell

behavior under harmonic excitation have uncovered various deviations of

response f r o m that predicted by normal modal theories , an experimental

p r o g r a m was conducted to determine the validity of such theories as applied

t o the case of random excitation as well as to investigate several concepts

important to the design of environmental t e s t s fo r internal systems which

may be attached to the shell.

F o r convenience, first a discussion of appropriate analytical relationships

between excitation and response a r e given for the cylindrical shell. These

are presented i n a f o r m for synthesizing a continuously distributed load into

a gridwork of multiple d iscre te loads, so that information on the required

mesh density fo r such a procedure can be obtained.

given for replacing the distributed load by an equivalent concentrated random

load acting at the response point.

importance in the design of force spec t ra for environmental testing of

internal systems.

The resul ts of this work a r e presented herein.

Then, expressions a r e

Both of these concepts a r e of eminent

3

A NA L Y T IC A L EXPR .ESSIONS

Response to Multiple Discrete Excitation

It will be of convenience to decompose a continuously distributed

random acoustical load into a n aggregate of multiple discrete loads. F o r

such a representat ion i n a l inear ly elastic s t ruc ture , the following matr ix

relationship can be written’ between excitation and response:

4-

Syy(f) = (Hxy) [SX,] px;) (1)

where

S ( f ) = displacement power spectral density of the response a t y YY

(Hxy) = row vector of displacement admittance at y to force

applied at x

[S,,] = square mat r ix of c ross -spec t ra l densities of excitation

forces a t various d iscre te points

[H;!) column vector of complex conjugates of elements of ( ~ ~ ~ 1 .

The use of any t r ans fe r function in Eq. (1) i s valid; h e r e , however,

=

displacement is chosen mere ly for convenience. F o r a purely theoretical

application of this equation analytical expressions must be utilized for the

admittance functions.

shell with coordinate sys tem as indicated in F igure 1, the following dis-

placement admittance between a lateral excitation a t x and displacement

a t y can be obtained:

1 Analogous to Sec. (4. 7 ) of Robson for a cylindrical

4

R

Circle In Tangent Plane

237

ystern

5

where

- - d a m n 5

and P 2Tr

Mmn = pshaJ j” 0 0

wmn ( z , 0 ) dzd0

F o r a simply-supported cylinder

m r z P - cos ne wmn(z,O) = s i n ( 3 )

so that

Mmn = psh (F) It should be noted that a s t ruc tura l type of damping has been speci-

fied, and that the response pat tern is assumed to follow the point x of load

application s o that

m r z x wmn (zX,ex) = s in -

I m r z

wmn (z ,e -ex) = s i n J c o s n (ey-ex) Y Y I

The latter behavior occurs in a perfectly symmetr ical cylinder.

Equivalent F o r c e Spectra

In o rde r t o determine the power spec t ra l density of a single excita-

tion fo rce acting at point y in such a way a s to replace the effects of the

distributed or multiple d iscre te load, and to allow fo r the additional re-

action effects of a n internal sys t em package, it is f i r s t necessary to

develop seve ra l admittance relationships. A ssuming steady-state harmonic

6

excitation and response, we introduce the following notation:

Fy(f) = amplitude of applied excitation at y

Gy(f)

Wy(f)

= amplitude of reaction force of internal sys tem at y

displacement of shell at y when internal sys tem is

attached

displacement of shell a t y due to force F ( f ) when

=

WyF(f) = Y

internal sys tem is detached

displacement of shel l at y due to the action of G ( f ) WyG(f) = Y

alone

Hal ( f ) = driving point admittance of internal sys tem

H ( f ) = driving point admittance a t point y of shell only with YY

internal sys tem detached

The complex admittance functions may be defined a s having zero

phase angle when displacement and force ac t in the same direction.

for the cylinder and attached internal sys tem we can wri te

Thus,

Wy(f) = WyF(f) f WyG(f) (5)

and

Gy(f) = W y ( f ) /Ha 1 ( f ) (6c)

Upon substitution of Eqs. (6 ) into (5) and solving for Gy(f), t he re results

7

These express ions a r e valid f o r steady s ta te harmonic excitation.

F o r the c a s e of random excitation, by means of generalized spectral

analysis5, Eqs. (6a) and (7) become

It can be seen f r o m Eqs. (8a) and (1) that SF(f) i s an equivalent concen-

t ra ted fo rce spec t r a which ac ts at y in such a way as to produce the same

response S ( f ) of the shell alone a s that of a distributed load. Fu r the r , YY

SG(f) is the fo rce spec t r a t o which the internal sys t em alone must be sub-

jected in o rde r t o duplicate the environment it must withstand a s a resul t

of the action of the distributed load on the shell.

8

EXPERIMENTAL PROCEDURES AND RESULTS

Measurement of Excitation Field

In o rde r to employ Eq. (1) f o r prediction of shell response it is

necessa ry to determine the elements of the mat r ix [Sxx] for a given dis-

tr ibuted load. That i s , the propert ies of the acoustic excitation field must

be measured. F o r this purpose, a microphone was mounted in a simulated

cylindrical section and placed in the same position relative to the acoustical

speaker as that to be used when exciting the actual cylinder.

cylindrical section and the cylinder had the same radius.

Both the

A photograph of

this p a r t of the apparatus i s shown in Figure 2.

tion could be moved vertically and swung horizontally, while the speaker

The microphone and sec-

was dr iven by a Gaussian noise generator.

Thus, the speaker output was measured over its ent i re effective

field, and recorded on analog tape. The subsequent data was then processed

by means of analog spec t ra l analysis equipment in order t o determine its

s ta t is t ical propert ies .

su re at the speaker centerline and other off-center positions a r e shown in

Figure 3 for a l imited frequency band.

the imaginary p a r t of the ordinarily complex functions was essentially zero.

F r o m such data it was determined that the excitation field was of purely

convective fo rm, was essentially symmetr ica l , and that the elements of

Samples of the c ross -PSD between the acoustic p r e s -

It is implied f r o m the figure that

[SXXI w e r e

9

s eld

0 h

m N @A bl

N I - V c Q) 3 U aJ L U

11

where

kx = e

and rx i s the radial distance off the speaker axis in a plane tangent to the

cylinder as shown in Figure 1. Fu r the r , Soo(f) i s the force PSD of the

excitation at the point of intersection of the speaker centerline with the tank

wall. Equation (10) i s an empir ical relationship which has been f i t to the

experimental data a s shown in Figure 4.

excitation field i s discussed on pp. 77-81 of Robson . F o r this case ,

This type of directly correlated

1

Eq, (1) reduces to

where

is defined as an acoustical mobility function,

and Soo(f) i s the p r e s s u r e power spectral density measured a t the inter-

A is the gridwork mesh s i ze ,

P

section of the centerline of the speaker with the shell.

F o r l a t e r calculations, it was determined f rom Figure 3 that at

v ) = (0,O) we have q

P 2 Soo(f) = 3. 66 x p s i /Hz

a s an average value throughout the frequency range of 100 to 180 Hz.

1.04

0.9

0.8

X x 0.7 -

CI .- v) S

0.6 - E .c, u a

0.5 v) cn 0 L 0

Z 0.4 N

m L

.- - E

2 0.3

0.2

0.1

1 2 4 5 6 7 0

0

Off-Centerline Position rx, i

13

Fur the r , for single point excitation to be described l a t e r , the same taped

signal was utilized to drive a point electrodynamic coil a t a value of

Soo(f) = 1.26 x lb2/Hz

Harmonic Excitation and Response

Experiments with harmonic excitation of a cylinder were conducted

both with single-point and distributed acoustic excitation in order to mea-

s u r e t r ans fe r (or admittance) functions and to observe the qualitative be-

havior of the system. The apparatus for single-point excitation is shown

in Figure 5.

8 gm. which is attached to the cylinder. Thus, m a s s loading effects were

The electrodynamic shaker has a moving element of only

negligible in the frequency range of interest .

t ransducer was used for measuring displacement (not shown in the photo-

graph). The cylinder is attached to the end sk i r t s p r imar i ly f o r ease in

handling, and the combination is bolted to a solid closed base a s shown,

A noncontacting Bentley

but is open at the top.

The cylinder shown in F igure 5 i s the same a s that used in a p r e -

vious study , except that no bulkheads were incorporated for the present

experiments e

4

Prope r t i e s of this cylinder along with some experimentally

measured natural frequencies are given in Table I.

qmn w e r e determined by measuring the t ransfer function for harmonic

single-point excitation a t the various natural frequencies and observing

Damping factors

the displacement with the following pa rame te r s :

ex = 0 0 ey = 1 8 0 0

zx = 15 in. z Y = 12. 5 in.

14

.- u X w + S

L 0 LL

m a 2

15

T A B L E I

P R O P E R T I E S AND NATURAL FREQUENCIES O F TEST CYLINDER

a = 12.42 in. p s = 2.59 x lb s e c / in

h = 0. 020 in B = 15. 0 in.

2 4

M O D E n - m -

F R E Q U E N C Y DA MPING F A C T OR Hz - qmn

5

6

7

8

9

10

11

12

13

214. 7

176. 1

144.5

132. 8

137.2

150.4

168. 0

193.9

224. 8

0. 00668

0 . 00922

0. 01870

0. 00977

0. 00386

0. 01380

0. 01080

0. 00531

0. 00884

2 1 0

2 11

2 1 2

2 13

237. 0

241. 0

244. 0

259. 0

16

The measu red f o r c e and displacements were then substituted into Eq. (2)

s o that the various qmn could be calculated. The summations in Eq. (2)

w e r e neglected for this purpose.

well separa ted modes and low damping.

the damping factors a r e dependent on frequency, a resul t which is not

predicted by s imple s t ruc tura l damping theory.

This simplification is valid only for

It is obvious f rom Table I that

T rans fe r functions w e r e measured over a frequency band which

included seve ra l of the lower natural frequencies and the results were

compared with numerical resul ts computed f rom Eq. (2) including the

damping fac tors given i n Table I.

the series expression.

The lowest nine modes were used for

A comparison of resul ts for one response point ( Y l ) and two dif-

ferent excitation points is given in F igures 6 and 7.

respectively to the real and imaginary par t s of the t ransfer function.

It is obvious f rom the two figures that the modal pat tern does indeed

shift re la t ive to the space fixed coordinate sys t em as the excitation point

is moved around the Fircumference of the cylinder.

pancies between theoret ical and experimental resul ts further indicate

that some distortion of modal pat terns a l so occurs. Nonalignment of

peaks in the theoretical and experimental Quad p a r t of the t ransfer

functions indicates that slightly different resonant frequencies can be ob-

tained by observing response at different points.

be achieved only when the point of observation for experimental response

C o and Quad re fer

However, discre-

Pe r fec t alignment can

O. 04

0

-0.04

- 0.08

I I

zy = 12.5 inches, eY = 204"

zx = 15.0 inches,8, = 0" -

Theory Co E X ~ . Co - Theory Quad E X ~ . Quad

--- .--..-- --

-0.12 I 130 132 134 136 138 140 142 144

Frequency, Hz 2375

Figure 6. Transfer Function Between Y 1 and x = ( 0'0 1

0.08

0.04

0

-0.04

I I

- 0.08

I I

z y = 12.5 inches,Qy = 204" z x = 15.0 inches, Ox = 18.45'-

--- Theory Co

- -- -. Theory Quad -- Exp. Quad

- E X ~ . CO

130 13 2 134 136 138 140 142 144 Frequency, Hz 2376

-0.12

Fiaure 7. Transfer Function Between Y 1 and x = ( 4.0 1

19

i s the s a m e as that f o r which damping fac tors a r e calculated.

of behavior resu l t s f r o m eccentricit ies in the cylinder.

This type

Distortion in modal response is even m o r e apparent in Figure 8

where a comparison in resul ts i s given f o r the acoustical mobility func-

t ion between Y 1 and harmonic acoustic excitation. Theoretical values

a r e calculated f r o m Eqs. (12) , (2), and ( l o ) , along with a mesh s ize of

A = 1 /4 in.

all subsequent resul ts .

indicates that distortion in response pat terns a r e even more prevalent

for distributed excitation. Fur ther , evidence of split modes i s present ,

and will be even m o r e apparent in subsequent resul ts .

Except where stated otherwise, this mesh s ize was used for

The discrepancy in resul ts evident in Figure 8

6

A s imi l a r comparison between theoretical and experimental t rans -

fer and acoustic mobility functions is given in Figures 9 - 11 for an addi-

tional observation point (Y2).

previously descr ibed for ( Y l ) .

The general behavior is s imi la r to that

Random Excitation and Response

Responses t o single-point and acoustic random excitation were

determined a t the s a m e two observation points which were previously

described.

descr ibed in an ea r l i e r section were used to drive the cylinder for both

types of excitation while response data were simultaneously recorded on

analog tape.

by analog analysis equipment.

The s a m e taped analog random signal whose propert ies were

Subsequent processing of the taped data was accomplished

A speed fac tor increase of 32 was utilized

20

1.0

0.8

0.6

0.4

0.2

0

-0.2

- 0.4

-0.6

-0.8

- 1.0

c z y = 12.5 inches,By = 204" zxo= 15.0 inches ,exo= 0"

Theory Co . E X ~ . Co Theory Quad E X ~ . Quad

--- ---- --

I

130 132 134 136 138 140 142 144

Frequency, Hz 2377

Fiaure 8. Acoustic Mobility Function For Y 1

21

I I

T- 2 8 0 0

I 0 B

0 0 0 1 . I

22

0 B 0

I

h x h l N

2 3

/-

==-" \

.--- / /

i C

0 0 0 I I I I

24

in processing response data in o rde r t o allow an effective Be = 0. 312 Hz

with a 10 Hz filter. Results a r e presented in Figures 12 and 13 respec-

tively.

Eq. (11) i n which CY’

Theoretical resu l t s for single-point excitation a r e based on

is replaced by a single theoretical t ransfer function XY

Hxy between the response and excitation points, and Soo(f) P is replaced

by SO0(f), the input force PSD for this case. Theoretical resul ts for

acoustic excitation a r e based direct ly on Eq. ( l l ) , including the appro-

pr ia te theoret ical expressions which make up CY’ as given by Eq. (12). XY

Average values for Soo(f) P and Soo(f) were taken as given in a previous

section.

originated by Trube r t . That is , the response values a r e calculated by

Semi-experimental values of response are based on a procedure

7

means of Eq. ( l l ) , except that experimentally measured , ra ther than

theoret ical , t r ans fe r functions o r acoustic mobility functions a r e used a s

appropriate. F o r point Y 1 , these experimental functions have been given

in F igures 6 and 8.

It is obvious f rom Figures 12 and 1 3 that considerable discrepancy

exists between purely theoretical and experimental results. This is not

surpr i s ing in view of s imi l a r discrepancies encountered for the t ransfer

functions.

shift in peak frequencies, quite good comparison is achieved between semi-

experimental and experimental resul ts . This indicates that the basic l inear

However, it i s equally obvious that except for some very slight

random process theory is applicable, s o long a s good representation of the

t r ans fe r and acoustic mobility functions can be achieved.

25

0 130 132 134 136 138 140 142 144

Frequency, Hz 2381

2.0

1.8

1.6 r- f

53

2

*- - 1.2

5: fn 1.0 E

a - 0.8 e x a, 0.6

x 1.4

\

% .c V I=

- n Y- - .- v) c a,

+ V

v)

L

Q

0.4

0.2

Figure 12. Single Point Random Response For Y 1

2 6

130 13 2 134 136 138 140 142 144 Frequency, Hz 2226

Figure 13. Acoustic Random Response For Y 1

27

In comparing Figures 12 and 13 it can be seen that a partially

spli t mode occurs for the acoustical input between 136 and 138 Hz and

similar peak responses occur for slightly different frequencies for the

two different kinds of excitation. Such split resonances a r e even more

apparent i n F igure 14 where additional response resul ts a r e given for

observation point Y2.

Equivalent force spec t ra for both points Y 1 and Y2 a r e presented

in Figure 15 fo r purely theoretical data only.

Eqs. (8a) and (2).

purely experimental values in the right side of Eq. (8a). However, the

given resul ts a r e sufficient to indicate the complexity of force spectral

density which would be required to s imulate the environment.

plex force spec t ra are difficult to achieve with present-day electro-

dynamic shaker equalization equipment.

The resul ts a r e based on

More accurate values would be obtained by using

Such com-

The influence of mesh s i ze on the theoret ical resul ts i s indicated

in F igure 16 for one typical frequency and observation point Y1 . The

resul ts are normalized to that fo r a mesh s ize of A = 1 /4 in . , which r e -

qu i res 2304 mesh points.

length for the dominant mode of response present , while A/rmax i s

mesh s i z e to maximum rx, which f r o m Figure 4 was taken a s 6 inches.

It is ve ry interesting that equally valid resul ts can be achieved with a

mesh s i ze as coa r se as A = 2 inches which requires only 144 mesh points.

The dependence of this t r end on frequency and dominant modal pat tern

The rat io A / L is that of mesh s ize to wave

remains to be determined.

28

0 v;

0 3 "0 11

I 0 ti

a

__to__

I .

n

/---

H4

\ c

C

1 0 m

0 c\i

0 -

29

0

I I I

Y 1 v-7

zy = 12.5 inches, 8, = 204"

zy = 15.0 inches, 8y = 222" Y2 - ---

zxo = 15.0 inches, 8x0 = 0"

J

- 130 134 138 142 146 150

Frequency, Hz 2 383

Figure 15. Theoretical Equivalent Force Spectra For Y 1 And Y 2

30

1.6

1.4

1.2 a

2 a

m \

1.0 v,

A

v) S Q)

.c, .- n e 0.8 - .c, V Q) a m L Q)

0.6 e x

& 0.4

N

tu .- - E

z

0.2

= 12.5 inches, By = 204"

zxo = 15.0 inches, 8 x 0 = 0"

m = l , n - 9

AIL N rmax

zY

f = 137.2 Hz, rmax/a -0.483

--- --e-

0 0 0.1 0.2 0.3 0.4 0.5 0.6

Mesh - Wavelength Parameter, A/ L 2384 or Mesh -Field Size Parameter, A/ rmax

Figure 16. Variation Of Acoustic Response At Y 1 With Excitation Mesh Size

31

A final correlat ion of purely experimental data i s presented in

Figure 17 , where the ordinary coherence functions5 have been calculated

f rom

for single point excitation and

for acoustic excitation. In these expressions S (f) and So P (f) is the OY Y

c ros s - spec t r a l density measured between the response a t y and respec-

tively the force and p r e s s u r e a t rx = 0.

value nea r unity for a perfectly l inear sys tem, and will be l e s s than unity

These functions should have a

otherwise. However, they a r e highly sensitive to small deviations, and

often a value of y > 0. 6 i s a fair indication of linearity. It should be

mentioned that for the acoustical input, an ordinary coherence value near

2 XY -

unity is not necessar i ly an indication of l inearity. However, a s shown in

the Appendix, it does provide such a n indication in the present case where

the excitation field is of the special convective form.

Figure 17 it can be concluded that excellent l inear i ty is indicated for

single point random excitation, while somewhat diminished l inearity is

In general , f rom

suffered fo r acoustic random excitation.

to ag ree with the data previously presented.

These resul ts appear generally

32

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

0 - Avg. Y 1 S. P. --

Avg. Y 2 Acoustic Avg.Y l ---

Acoustic .~

-

6 0

0

C 0

= 12.5 inches, 8y, = 204" = 15.0 inches, €Iy2= 222"

zY I

zY2 zxo = 15.0 inches, Oxo = 0"

0.0 Y 1 Single Point o o o Y 1 Acoustic o o o Y 2 Acoustic

60 100 140 180 220 260 300

Frequency, Hz 2385

Figure 17. Coherence Function For Random Excitation

33

A CKNOWLEDGMENTS

The author wishes to express his sincere appreciation t o several

of his colleagues f o r help throughout the conduct of this program.

mention should be given t o Mr, Thomas Dunham and Mr. Dennis Scheidt

fo r aiding with the experiments , t o Dr. Wen-Hwa Chu f o r theoret ical

d i scuss ions , and t o Mr. Robert Gonzales fo r digital computer programing.

Special

34

REFERENCES

1. Robson, J. D . , AN INTRODUCTION TO RANDOM VIBRATION, Edinburgh University Press, Edinburgh, Elsevier Publishing Company, New York, 1964.

2. Nemat:Nasser, S . , "On the Response of Shallow Thin Shells to Random Exci ta t ions," AIAA Journal , 5, 7, pp. 1327-1331, July, 1968.

-

3. Hwang, Chintsun, "Random Acoustic Response of a Cylindrical Shell , e las t ic i ty Specialist Conference, New Orleans, La. , pp. 112-120, A p r i l , 1969.

Proceedings of A I A A Structural Dynamics and Aero-

4. Kana, D. D. and Gormley, J. F . , "Longitudinal Vibration of a Model Space Vehicle Propel lant Tank, 'I Jour . Spacecraft and Rockets , - 4, 12, pp. 1585-1591, Dec., 1967.

5. Bendat, J. S . , and P ie r so l , A . G. , MEASUREMENT AND ANALYSIS O F RANDOM DATA, John Wiley & Sons, Inc. , New York (1966).

6. Tobias , S. A. , "A Theory of Imperfection for the Vibration of Elas t ic Bodies of Revolution, I ! Engineering, Vol. 172, pp. 409- 411, 1951.

7. T r u b e r t , M a r c R . P . , "Response of Elast ic Structures to Statist i- cal ly Corre la ted Multiple Random Excitations, ' I Jour . Acous. SOC. Amer., 35, 7, pp. 1009-1022, July 1963.

3 5

A P P E N D I X

Ordinary Coherence Function for Perfect ly Correlated Excitation Field

F r o m Reference 5:

@xy(f)J = [ S a l {Hxyl

Le t point x = (0 , 0 ) be the excitation point for the f i r s t element of the

column ma t r i ces in this expression. Then, in view of Eq. (9b), the

f i r s t e lement of the left-hand mat r ix i s

N N

Combining this resul t along with Eqs. (11) and ( 1 2 ) , it is found that

The validity of this expression is a joint indication of the l inearity of the

sys tem a s well a s an indication of the validity of Eq. (9b) for representing

the experimental excitation field.