Acoustic Characteristics of an Expansion Chamber With Constant Mass Flow and Steady Temperature Gradient (Theory and Numerical Simulation).pdf

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Yang-Hann Kim

Associate Professor.

Jae Woong Choi

Research Assistant.

School of Mechanical and

Material Engineering,

Korea Institute of Technology,

400 Kusung-dong, Yusung-gu, Taejon-shi,

302-338, Korea

Byung Duk Lim

Research Associate.

Acoustics and Vibration Laboratory,

Korea Standards Research Institute,

Taejon-shi,

302-340, Korea

Acoustic Characteristics of an

Expansion Chamber With Constant

Mass Flow and Steady

Temperature Gradient (Theory and

Numerical Simulation)

The governing equation of acoustic w ave propagation in a circular expansion cham

ber with me an flow and tempe rature gradient is

solved.

T h e

circular

chambe r is

divided into N se gments and the flow spee d and tempe rature are assumed to be

constant in each segment. T h e solution is obtained in recursive form by applying

the

matching condition on

the

boundary of adjacent e lements. T h e solution

is

verified

by comparing it with the experimental results. The results demonstrate that the

present

theory

can well predict the

transmission loss

of an

expansion chamber which

h as offset, a twisting angle , mean flow, and temperature gradient.

Introduction

An understanding of acoustic wave propagation through an

expansion chamber is essential in order to design the silencer

system of autom obiles, blowers, and other machines. The main

difficulty of this subject is that the governing wave equation

is not linear because of the nonuniform mass flow of gas as

well as the temperature gradient along the chamber.

Munjal and Prasad [1] applied the perturbation method to

solve the wave equation which represents the acoustic char

acteristics of a uniform pipe with mean flow as well as linear

temperature g radient. The solution shows an unrealistic offset

of the transmission loss curve. Peat [2] derived the transfer

matrix considering variable m ean gas density and the velocity

gradient along uniform pipe, but this result also has the offset

which cannot be explained in a physical sense. El-Sharkawy

and Nayfeh [3] successfully analyzed the acoustic character

istics of an expansion chamber by theory and experiment, but

the effects of mean flow and tem perature v ariations along the

chamber on the propagation of acoustic wave were not con

sidered. Due to the complexity of the wave equation, which

represents sound wave through an expansion chamber with

mass flow and tem perature g radient, M unjal [4] tried to solve

the equation by numerical technique. The drawback is that

many meshes are required to account for higher order modes.

Apa rt from this approach , Ih and Lee [5] succeeded in getting

theoretical results which included higher order modes of a

circular expansion chamber whose inlet and outlet ports have

offset and a twisting angle. The temperature effect was not

considered in their study. Davies [6] discussed the effect of

sudden expansion and contraction on acoustic wave propa

gation in terms of

acoustic

impedance and the reflection factor.

Viscothermal loss on the pipe wall was considered by the mod

ification of the wave number.

In this study, the general equation for acoustic wave prop

agation through an expansion chamber, which includes the

effect of m ean flow and steady tempera ture gradient

is

derived.

The numerical solution of the equation is also introduced.

Results are compared with the results of experiments which

were performed by the authors [9]. The numerical solution

was obtained by dividing the expansion chamber into N cy

lindrical elements and applying the requirement of pressure

and velocity continuity between the bound ary of adjacent ele

ments. It can be assumed that each element had homogeneous

mass flow and temperature; that is, the governing equation

for each element has uniform mass flow and convective terms.

By applying the boundary conditions, the solution was ob

tained in recursive forms.

Theoretical Formulation and Solution Method

A three-dimensional acoustic wave equation for the isen-

tropic process can be easily derived from momentum, conti

nuity, and state equations:

V« - V P + V • ( K • V ) K -

±/±_dP\

dtypc

2

dt)

Contribu ted by the Design Engineering Division for publicatio n in the JOURNAL

O F

VIBRATION AND ACOU STICS. Manuscript received November 1989.

PC

2

(VP)>V = 0

(1)

The first term represents the variation of the body force.

4 6 0 / V o l . 1 12 , O C T O B E R 1 9 9 0

Tra ns a c t ions o f the ASM E

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The second and fou rth term s express the change of energy due

to the change of velocity and temperature in space. The third

term is the time rate of change of energy within the system.

The general solution of this equation has not been found yet.

O ne of the reasons is that the acoustic velocity and density of

gas are not constant, varying with respect to a spatial coor

dinate. For example, temperature (7), density (p), acoustic

velocity (c), and mean flow velocity

(W

0

)

of a simple expansion

chamber which has a l inear temperature gradient (Fig. 1) can

be expressed as follows:

T

_

T

~

T

_

T

=

Ti Tl

T

x

+ T

2

'

T(z) =T,

p z) =p

m

/

1-T,

(H

(2)

c(z) = 20 .05 yJT{z)

W(z)

=

W

m

- ( H

But if there is no temperature variation, then the density and

acoustic velocity are constants . Therefore, equation (1) can be

simplified as

V

2

P - -5

D

Z

oP



The superscripts + and - represent the wave going to the

right and the wave going to the left, respectively. The n rep

resents the mode number in the radial direction and

m

is the

mode number in the circumferential direction. E quation (6)

ca n

be a

solution

of

equa t ion

(4) for a

circular expansion

chamber when there is uniform tempera ture .

The boundary condi t ions

of

equation

(4) for the

circular

expansion chamber (Fig. 1) can be decomposed as

90

dz

dz

z=o oz

z= i

dr

-0 {la)

z = l

= VJ

2

(r,8),

30

dz

= 0,

z=o

dr

= 0 (lb)

where f

t

(r, 0) = Heav iside unit step function

1 at the inlet/ou tlet pipe at

r

= 5,

0 other locations

C

It

can be

assumed that

the

wall

of the

cylinder

is

acoustically

rigid, as is expressed in the third equations of equations (7a)

and (76). Applying the rigid boundar y con dition in the r di

rection

to

equa t ion

(6), the

following equations

can be ob

tained.

^ ( X

n m

) = 0

(8)

The wave number in the z direction can be obtained by

substituting equations (8) and (6) into equation (4):

kf

nm

=

k[ M =F

V l - ( l - M

2

) ( f c

r a m

/ f c )

2

/ ( l - M

2

) ] (9)

where M = W/c: Mach number

k = w/c : wave number

It

can be

also assumed th at there

is no

energy loss across

the

wall. If there is the viscothermal effect, the wave num ber will

be complex as Davies [5] suggested. If boundary condi t ions,

equation (7a), are applied on each of the elements whose tem

pera ture is uniform (Fig. 1), then the following matching con

ditions

can be

obta ined.

Pq-\\l-l

a

-Pq\l-l.

dz

I->a

dz

l-l

a

- d6

P=jup4>

+ pW-^-

dz

(10)

( ID

(12)

U s ing the second equation of equation (la), the relation

between R%

m

and R^

m

can be found. Then the matching con

ditions which are expressed by equa t ions (10) and (11), lead

to

the

following recursive equa tions.

K-qnm ~ ^qnm^qnm

**-qnm qnmK-(q—\)nm

(13)

The coefficients

S

qnm

and

W

qnm

are

shown

in

Appendix

A.

With these relations, we can obtain the velocity potential for

the first element as

n m \

a

/

(14)

To apply the first equation

of

equation (7a)

to

equation (13),

we must expand f

x

(r, 6) in te rms of the Fourier-Bessel series

[7],

Tha t is

j \rf(r,6)J

n

(\„

m

r

Acosnddrdd

V \mj

(\m)

( -\

CO S/20 (15)

where

n =

0 ,1 ,2 ,

. . .

m = 0 ,1 ,2 , . . .; n - 0 and m = 0 should not be as

signed to zero simultaneously

_ ( 1 n = 0

7

" ~ ( 2

n*0

Hence, th e velocity potentials

for

the first

and

the last elem ents

are

(

a

i

a

> (

e

i

k+

izM

z

+ s

l00

e >

k

'iz0O

z

)

4>i=JVi

HzOO

+

Sinak]

100

ft

l«00

2

ln

ai/a)J_„ \

nm

-A h \

nm

?A

" ' (K„

m

+

S

lznm

ky

znm

)\\-

^-\Jl(\

nm

)\

n

(16)

(e>* I**™* + S

lnm

e >

k

w»>>

z

)J,

V~J

V

\

^iz00

+

ino fc

00«-lz00

« ( K m ~

a

)cosndl

*& (e

/k

y^

:

+

S

y00

e

/k

y^o

z

)

+EE

n m

2

7

„ (a

1

/ a ) / _„

(\

nm

b

A J

{

(\„

m

^ W

ynm

...W

2

,

\ K\znm "i Inm^lznm )

(-£) •

(

e

i

k

'yznnfi + s

ynm

e

ik+

yz'»»

z

)J „ ( „

m

- J cosnd

\

«/ J

( n )

where « = 0 ,1 ,2 , . . .

m = 0 , l , 2 , . . .; « = 0 and m = 0 should not be as

signed to zero simultaneously

The coefficients

S and W

are shown

in

Appendix

A.

The mean pressure acting on the inlet and outlet ports of

the expansion chamber

can be

readily obtain ed

by

substituting

equations

(16) and (17)

into e qua tion (12). This gives

P

x

and

P

y

. Integrating these with respect to the area of the inlet and

outlet ports leads

to

5

»

=

i f I

P

rdrdd=U

i

Z

i

. - ^ I l ^ y * * - ^ A

(18)

£/] ( = 7ra V{) is the volume flow rate of the inlet p ort.

By using Graf's a dditional theorem [8] to get the integration,

Zi

and Z

3

can be obtained . T he details are shown in Appendix

A .

Similarly,

we

have

to

apply

the

bou nda ry condition which

is shown in equa t ion (lb). After similar derivation, the mean

pressures

of the

inlet

and

outlet ports

can be

obtained.

1

Pl2 - 7

TTOT

— j

\P

l

rdrde=U

2

Z

2

l

'

Tfl? j ] '

(19)

P

y

rdrd6=U

2

Z^

U

2

( = ir a

2

V

2

) is the volume flow rate of the outlet port. P

it

P

y

,

Z

2

and Z

4

are shown in Appendix B.

Finally, the mean pressures of the inlet and outlet ports of

the expansion chamber

can be

obtained from equations

(18)

and (19).

4 6 2 / V o l . 112, OCTOBER 1990

Transac t ions

of the

ASME

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2.6kHz

=L3ZJ=

,w

vvW

a .a 4.a

s i

f r e que nc y CkHz J

CdBJ

SB

L

1 1

=H.

—

ZGktiz

nf \ f \f If it l i / \ l / / J A A* l\

...

1 1

l_

r

IM^

3 . 2

f requency

S.4

CkHzJ

( a ) D = 1 5 0 m m , L = 3 0 0 m m , C e n t er

to

C e n t e r

(

- :

E x p e r i m e n t a l r e s u l t . • • •

:

P l a ne w a ve t he n r v

1

( c) D = 1 5 0 m m , L = 4 5 0 m m , C e n t e r

to

C e n t e r

( - : E x p e r i m e n t a l r e s u l t , • • • : P l a ne w a ve t he o r y )

JVxWI

J

[Mb]

(b) D=150mm, L=300mm, Cente r to Center

( Present theory; 1 e l e m e n t , n= 2 , m~7)

[dB]

60

JvL :O TYYYliiklU

C E E

s

J

1 3 3.2 4.B

frequency

(d) D~150mm, L—450mm, Cente r to Center

(Present theory; 1 element, n~ 2, m— 7)

BA

Fig . 2 TL of circular simple expansion chamber w ith the same diameter

but different lengths (Effect of lengths on the TL of circular simple

expansion chamber with the same d iameter)

P

l

=P

n

+ P

a

=U

l

Z

i

+

U

2

Z

2

P

2

=

P

21

+ P

22

=U

l

Z

3

+ U

2

Z

4

Rewriting equation

(20) in

matrix form leads

to

where T

n

••

T,,=

T

22

—

M

M

n -

* ]

- ^ « - 0

• 2Jll

2

= 0

l

]

=

T

u

[T

21

z

3

Z\Z

4

z,

1

z

3

-

z

4

z

3

T

l2

T

22

(20)

(21)

These are the same results as those obtained by Munja l and

Prasad

[1] for a

straight pipe with mean flow. This result

confirms that the derived solution is correct.

The transmission loss

of an

expansion cham ber with

ane-

choic termination can be readily obtained from equation (21)

by simply regarding that there is no reflective wave at the outlet

port

of the

chamber . Tha t

is

I Tn + T

i2

/Z

m

+ T

2l

Z

02

+

T

22

1

Z ,

r L

=

201og

w h e r e

Z

0

i=PiCi/(ircr\)

Z

02

= p

y

c/(iraj)

(23)

+ Z

2

Tjj(i

= 1,2

and,/ ' =1 , 2 ) are often called four-pole para met ers.

When there is mean flow along a uniform pipe , the four-pole

parameters in equation (21) for plane wave are

T

n

=

e -

jklM/ii

-

M

\os(kl/(l

- M

2

) )

T

u

=jpc/(Tra

1

)e-J

klM/

U-

M

\m(lcl/(

1 - M

2

)) (22)

T

2i

=jira

2

/(pc)e-J

klM/

^-

M

\m(kl/(l

- M

2

))

T

22

= e -

JklMni

-

M

\os(kl/(\-M

2

))

C ompar i son B e tw e e n N ume r ic a l

and

E xpe r ime nta l

Re-

sults

To verify the solution which can account for the effect of

mean flow, temperature gradient, and geometrical parameters

(length, diameter, offset distance, and twisting angle) on the

acoustic wave propagation in a circular expansion chamber,

comparisons between the numerical solutions and the exper

imental results [9] are performed. For the numerical simula

t ion , a simple expansion chamber depicted in Fig. 1 was u sed.

Transmission Loss Without Me a n Flow). Figure 2 shows

the transmission loss curves for expansion chambers whose

lengths are 300 mm and 45 0 mm and whose diameters are the

same (300 mm). Figure 2(a) demonstrates that the experimental

result and the result ob tained by the plane wave theory agree

well in the low frequency range . Transmission loss of the ex-

Journal of Vibration and Acou stics

OCTOBER 1990, Vol. 112 / 463




Exper imenta l resul t

: P resent theory( l e lement , n=2, m=7)

Fig . 3 TL of circular simple expansion chamb er (D = 300 mm , L = 450

mm, center to center)

1 . 6 3 . 2

frequency

: Exper imenta l resul t

4 . a 6 . 4

CkHi l


Fig .

4 Effect of offset of Inlet/ou tlet port to TL (D = 150 mm , L = 300

mm; twisting ang le = 0°, offset to offset (50 mm each))

Exper imenta l resul t

—.—.— : Present theory fl elemen t, n= 5, m= 7)

(a) Twisting ang le= 120°, O ffset to O ffset(50mm each)

: Exper imenta l resul t


(b) Twisting an gle= 180°, Offset to O ffset(50mm each)

Fig .

5 Effect of twisting ang le of inlet/outlet port to TL (D = 150 mm,

L = 300 mm)

pansion chamber based on the plane wave theory can be ex

pressed by

7Z,= 101og

I(

1

+

R )

sin

2

W

in which

a

is the area ratio between the expansion chamber

and the inlet-outlet ports, that is (D/df, I is the length of the

expansion chamber, and k is the wave number.

For the high frequency range, in this case over 2.6KHz,

where higher order modes start to participate in the transmis

sion loss, the plane w ave theory is no longer valid as Fig. 2(a)

demonstrates. Figure 2(b) shows the comparison between the

results of the present theory and the plane wave theory. This

confirms th at the present theo ry predicts the transmission loss

exactly the same as the plane wave theory does in the low

frequency range.

For the high frequency range, the results obtained by the

present theory do not coincide with the experimental results

[Figs.

2(a) and 2(b)]. The sharp peak above 3.2KHz is pro

foundly evident in Fig. 2(b), but no t in Fig. 2(a). This is because

the resolution of Fig. 2(a), which is about 25Hz, is much lower

than that of Fig. 2(b). This can be readily verified by carefully

observing each hump and bump of Figs. 2(a) and 2(b). The

trend is quite similar but not the magnitude. Comparing Figs.

2(c) and 2(d) also reveals similar results. These conclude tha t

the developed theory can predict the transmission loss of an

expansion chamber whose inlet and outlet ports do not have

offset and a twisting angle. Figure 3 also confirms the ability

to predict the characteristics of

wave

propagation

in

the circular

expansion chamber.

The effect of offset between inlet and outlet ports of an

expansion chamber on the transmission loss is well demon

strated in Fig. 4. Comparing Fig. 4 with Figs. 2(a) and

2(b),

we can see that the occurrence of the modes in the radial

direction essentially m akes a different transmission curve. The

first radial mode is observed at

1.33KHz

which satisfies X

0

i =

1,84.

The presence of a twisting angle between the inlet and outlet

ports controls the transm ission loss in a high frequency region.

This can be understood by Fig. 4 and Fig. 5.

Transmission Loss in the Presence of Mean Flow and Tem

perature G radient. Figure 6 shows the transmission loss of an

expansion chamber whose diameter and length are 150 mm

and 450 mm, respectively. In this case, the flow velocity is 50

m/s which corresponds to M ach number 0.14. Com paring Figs.

2(c) and

2(d)

with Fig. 6 does not reveal the effect of mean

flow on the transmission loss. The presence of mean flow

modifies the wave number inversely propo rtional to

(1

- M

2

)

as can be seen in equation (22). Mach number 0.14, which is

the case in Fig. 6, only modifies a wave number of 2 percen t.

464 / Vol. 112, O CTO BER 1990

Transact ions of the ASME




"L

_r

mm

3 -t.S

frequency [kH

: Experimental result(Mean flow velocity—50m/s)

: Present theory(Mean flow velocity=50m/s,

1 e lement , n=2, m~7)

Fig . 6 Effect of mean flow to TL of circular simple expansion cham ber

with concentric inlet and outlet port (D = 150 mm , L = 450 mm, center

to center)

Fig . 7 Effect of mean flow on TL of simple expansion chamber withou t

temperature gradient (Tr = 0; D = 150 mm , L = 300 mm; 6 elem ents,

n = 5, m = 7)

This is the reason why

Figs.

2(c) and 2(d), and Fig. 6 are almost

identical.

The effect of mean flow is demonstrated better in Fig. 7

than Fig. 6. In the low frequency range, the frequency shift

due to the presence of mean flow can be observed easily. In

the high frequency range where higher order modes control

the transmission loss, the mean flow modifies the curve of

transmission loss in a more complicated way than it does in

the low frequency range [equation (22)].

The effect of temperature gradient on transmission loss is

shown in Fig. 8. As the temperature gradient increases, so do

the amplitude of transmission loss and the cutoff frequency.

Acoustic impedance is inversely proportional to \pT. There

fore,

the acoustic impedance of the outlet port is higher than

that of the inlet port. This introduces a higher transmission

loss in magnitude.

It is very difficult and expensive to obtain a high mach

number and meaningful temperature gradient in the experi

ment. Also, the measurement of transmission loss of an ex

pansion chamber in the presence of a high Mach number and

temperature gradient has many difficulties [9]. Noise due to

the turbulence of the flow must be properly handled, and

special sensors which can be operated in hot gas must be de

signed or obtained. The numerical solution does not have these

limitations.

The number of elements which were required for conver

gence was 6 to 9 in most cases. The number of modes for

convergence was always less than 10. In most cases, it was 8,

as can be found in all of the figures.

frequency Dti9s]

Fig . 8 Effect of temperature gradient on TL of simple expansion cham

ber without mean flow (M = 0; D = 150 mm, L = 300 mm; 6 elements,

n = 5, m = 7)

Conclusions

The solution of the acoustic wave equation which governs

the acoustic wave propagation in a circular expansion chamber

in the presence of mean flow and temperature gradient was

derived in recursive form. The solution was verified by nu

merical simulation. The num erical results of transmission loss

were compared with the experimental results. We found that

the suggested numerical solution well predicted the transmis

sion loss of an expansion chamber. The number of elements

and number of modes in radial and circumferential directions,

which are required for convergence, are sufficiently small

enough to extend this scheme to more general cases; i.e., an

elliptic expansion chamber, pulsating gas flow, etc.

References

1 Munja l , M. L. , and Prasad, M. G. , 1986, "O n P lane Wave Propaga t ion

in a Uniform Pipe in the Presence of a Mean Flow and a Tem perature Gra dien t, ' '

Journal oflheAcoust. Soc. Am.,

Vol. 80, No . 5, pp . 1501-1506, Nov .

2 P eat, K. S., 1988, "T he Transfer M atrix of a U niform Duct with a Linear

Tempera ture Gradient ," Journal of Sound and Vibration, Vol. 123, No . 1, pp.

43-53 .

3 El-sharkawy, A. I . , and Nayfes, Ali H., 1978, "Effect of an Expansion

Chamber on the Propaga t ion of Sound in Circula r Duc t , ' '

Journal oftheAcoust.

Soc. Am., Vol . 63, No. 3 , pp. 667-674, Mar .

4 Munja l , M .L . , 1987, "A S imple Numer ica l Method for Three Dimensiona l

Analysis of Simple Expansion Chamber Mufflers of Rectangular as well as

Circular Cross Section with a Stationary Medium," Journal of Sound and

Vibration,

Vol. 116, No. 1, pp. 71-88.

5 Ih , J eong-Gu on, and Lee , Byung-Ho, 1985, "Analysis of Higher O rder

Mode Effects in the Circular Expansion Chamber with Mean Flow," Journal

oftheAcoust. Soc. Am., Vol. 77, No . 4, pp . 1377-1388, April.

6 Davies , P . O . A. L. , 1988, "Prac t ica l F low Duct Acoust ics ," Journal of

Sound and Vibration,

Vol. 124, No . 1, pp . 91-1 15.

7 Hildebrand, F. B., 1976, Advance d Calculus for Application, Prentice Inc.,

C ha p .

5.

8 Watson, 1966,

A Treat ise on the Th eory of Bessel Functions,

Cambr idge

U niv. Press , Chap. 6 .

9 Kim, Y.-H., Lim, B. D., Kim, S. H., and Kwak, Y. K., 1989, "Experimental

Study on Acoustic Wave Propagation in Circular Expansion Chamber with Mean

Flows," ASME Paper 89-WA/NCA-7, a lso presented a t the Winte r Annua l

Meeting, San Francisco, California. December 10-15, 1989.

A P P E N D I X A

3

(q-\)nm'

w =

Tr

qnm

Xq \

\Xq22 — X

qn

X

q

2\

.e

ik

\q~\)zmn^ + - • • + / (? -1) )

Xq2lQ(q- \)nm ~ X

q

\

2

k(

q

- \)

znm

X„ \ \X

a

Y> — X„i

2

X,

^q\\

A

qTL q\2

A

ql\

• e ' ' '

r

( ? - l ) « « m ( ' l

+

- -

. +

' ( 8 - 1 ) )

wnere t i

qn m

=

o

qnm

K

qnm

^ynm

e/(*;

Journal of V ibrat ion and Acoust ics

OCTOBER 1990, Vol . 1 12 /46 5


http://xq22/http://xq22/http://xq22/



Qi

nm

=p

q

^^W

q

k^

nm

)

Xqn

=e/*5w«('i+ - • •

+

' (» -

1

)

)

G ,

+

„

m

+ S,

nm

e * ? W ' i + - • • + ' < « - i ) > g -

n f f l

^ 2 .

^ w ^ ^ - • •

+ / ( < ?

-

i ) )

+s ,

n m

^„

m

* , = y * 2

(a

2

/a)

2

kyzOO + SyaAyzOO

( /

-

z )

) . /

n

(x „

m

0cosn0

Z i = -

Pi

«i/«)

2

•UE

^ 0 2 2

" - { g - l ^ n m

1 7 w

2y„(.a

2

/a)J_„(

X

nm

-^\

7, (X„

m

-^ j

' ^yznm "* ^ynm^yznm) \* ~~

-v

2 I «

">"

«

T « l W z OO +SlOO^ljOO

[co l+S

100

)

( e

i

' K -

l

' -

l

+ S . / j « " -

l

) / ,

"\

X

"

r a

a)

cosn9

+ î(*i

+

zoo + Sioo*iloo)]

4 7 „ A „ ( x „

m

^ 7 ^ X ^

"

m

Knm + S\

nm

kunm)\A~ r̂)

fyKn,)H

[u(l + S

lnm

) + W

l

(kt

z

nm +

nm

where n = 0,1,2, . . .

m = 0,1,2, . . .; n = 0 and m = 0 should not be assigned

to zero simultaneously

- 1 ) 0 0

TTtfJ

C ^jOO

+

ÔO^VzOO

[co(e

/

*W '

)

+ S

100

e'* i*

)o(

'

)

)

- ̂ "»

l

+Sy„

m

e >

k

yz">»

1

) + J¥y(k+

,„

m

e >

k

yz»»i

l

+

S

ynm

ky

Z

„

m

e

ik

yznmi)}

where n = 0,1,2, . . .

/« = 0,1,2, . . .;« = 0 and w = 0 should not be assigned


z

4 = L + /̂f̂ - [«(1 + S>oo) -*W*x> + W£ oo ) l

7T0£ lAj/tfX) +Z>y00

K

yz00

-EE-

A P P E N D I X

B

«>./•-.(»•.;) 4 * )

( -£) •

V z n m ^ynm'^yznm.

- 90i

P , =76)00 +p*P,

—

[Cl)(l +Sy„

m

) — Wy(ky

Z

„

m

+ Sy„

m

ky

Z

„

m

)]

Py=jWp(j>y + pWy

dz

^ynm)

rr

yK^ywrn'

LJ

ynm'

y

yznm

where « = 0,1,2, . . .

m

=

0,l,2,

. .

.; « = 0 and m

=

0 should not be assigned


y"K

2

(a

2

/a)

2

W

m

. ..W,

K

yzWS ~r 'JyOoKyzM

( J

°° ( ^ I J O O C - ^

+

^ ^ I Z O O C - - ^ )

X?2 lQ( ?+ l)nm + X?llk(

9

+

1)znm

X

9

n ^ ? 2 2

—

XtfuXî

,&

k

( ? + l ) z n m ( ' - ' l (?)>

« m

2

7

„(«

2

/a) (/_„ (\

nm

^ 7, (̂ X„

m

̂ W

lnm

...W,

y~\)nm

W

1 m

\ Kyznm ~i~ ^ynm^yznm )

( - w ^

K

X

g 2 2 Q ( g +

l)«m ~

X

g l 2 k (

+

g

+ j)

znm

+ ( , _ / , _ . ,, ,,,

tgl" (q+iyznm

1

-

1

1 ' ( 9 -1 ) )

X

« l 1

X

?2 2 ~

X

9 l 2

X

9

2 1

466/Vol. 112, OCTOBER 1990

Transactions of the ASME




Kqnm ^qnm^-qnm

X

qn

=-^

k

ig

+

Dznm(l-n lg)Q-

q +

\)nm

^ \ 7

Iznm

3

lnm

Iznm k i

znm

)l

H-lznm

Xql\~

k^fii^vmii-'i

— - / , ) +

S

qn m

k-

znm

e

k

~^^-n

- -lq)

< l l —

~lq)

••

• "

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