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
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.
