Coupled frequencies of a hydroelastic system of an elastic two-dimensional sector-shell and frictionless liquid in zero gravity

Journal of Fhdds and Structures (1994) 8, 817-831

C O U P L E D F R E Q U E N C I E S OF A H Y D R O E L A S T I C SYSTEM OF A N ELASTIC T W O - D I M E N S I O N A L SECTOR-SHELL A N D FRICTIONLESS LIQUID IN ZERO G R A V I T Y H. F. BAUER

Institute fiir Raumfahrttechnik, Universitiit der Bundeswehr Miinchen, 8014 Neubiberg, F.R. Germany AND

K. KOMATSU

Structural Mechanics Division, National Aerospace Laboratory, Chofi~, Tokyo 182, Japan (Received 12 February 1993 and in revised form 25 May 1994) For a circular sector system consisting of an elastic shell and an incompressible frictionless liquid layer with a free surface, the coupled liquid-structure frequencies are determined and compared with the natural (uncoupled) frequencies of the shell and the liquid. The coupled frequencies always exhibit smaller values than those of liquid and shell alone. The deviation increases with decreasing thickness of the liquid layer. 1. INTRODUCTION THE ADVENTOF ORBITALSPACE FLIGHTSand orbital laboratories with longer time periods in a zero- or microgravity environment presents us with unique manufacturing and chemical engineering processing possibilities, which would hardly be possible on earth under the usual gravity conditions. Cylindrical configurations seem to be the basic geometry that will be used in space experiments. Since the structures should be light, they will exhibit elasticity and so interact with a liquid with a free surface and vice

versa. Various liquid configurations (rotating and nonrotating) have been investigated (Bauer 1982) for rigid structures under a zero-gravity condition. Two-dimensional results, i.e. investigations with no axial (z)-dependency may be found for quite a few configurations in Bauer (1987a). In circular cylindrical configurations, the liquid column exhibits Rayleigh instability in the axisymmetric mode m = 0, becoming unstable if the length of the column is equal or larger than the circumference of the column. This is valid for frictionless viscous liquid (Bauer 1984, 1986; Lamb 1945) and also for viscoelastic liquid (Rayleigh 1882). The hydroelastic problem has also been treated for cylindrical systems with a liquid layer on the outside or inside of an elastic cylindrical shell. This was performed for frictionless (Bauer 1987b) and viscous liquid (Bauer 1987c). In a z-independent circular cylindrical system, it was found that such a two-dimensional system does not exhibit any instabilities. This is different for circular sectorial systems. Here, instabilities may appear for certain vertex angles. Also, it is of importance to know the shift of frequencies if the system shows certain elastic behaviour. 0889-9746/94/080817 + 15 $08.00

~) 1994 Academic Press Limited

818

H. F. B A U E R AND K. KOMATSU

In addition, in many engineering disciplines, hydroelastic problems (Miles 1958) play an important role, since in most cases the structure is thin, exhibits high flexibility and reacts with the liquid (Yamaki et al. 1984). Whether one deals with large capacity oil storage tanks subjected to earthquakes or other wind disturbances, or one deals with propellant containers of airplanes, missiles, satellites or space stations, the knowledge of the coupled frequencies of the combined structure-liquid system is important to be known for design purposes. Numerous investigations, theoretical and experimental, have been performed in recent years, in which most of the emphasis has been centered on circular cylindrical up-right containers (Lakis & Pafdoussis 1971). The aim of the following investigation is, therefore, to study the interactions of the elasticity of the shell and the liquid with a free surface held in position only by surface tension effects. The side walls (radial walls) are considered rigid and the incompressible and frictionless liquid is allowed to either freely slip at the side walls or be anchored there by strong surface tension and/or a sharp-edged rim. Through the action of the free surface motion, the shell frequencies shall be influenced, thus yielding coupled shell frequencies: and by the motion of the elastic shell the free surface frequencies will be shifted to coupled liquid frequencies. For this reason, the natural (uncoupled) frequencies of the liquid with a free surface and those of the elastic shell alone will be determined first. The determination of the natural frequencies of the liquid in the circular sector geometry will allow the liquid to either freely slip or exhibit a stuck-edge condition. Then, in the second investigation the coupled liquid and shell frequencies will be determined, as well as the deviation of the coupled frequencies from the uncoupled frequencies. For the description of the shell motion, the Donneli shell equations (Leissa 1973) have been employed. 2. BASIC EQUATIONS An elastic sector shell (Figure 1) at r = b and a liquid layer of thickness of a - b , if disturbed, execute oscillations under the influence of the elastic structure and the free liquid surface motion. A sector of apex angle 2trc~, consisting of rigid walls at q~= 0 and 2tra, contains an incompressible and frictionless liquid. The liquid edge at the sector walls may either exhibit a freely slipping edge or an anchored (stuck) edge. The problem to be solved is the determination of the coupled frequencies of the hydroelastic structure-liquid system, which is assumed to be of axial (z-) independence. For small liquid velocity components and small elastic deformations, the governing equations may be linearized. With the liquid motion assumed to be irrotational, the continuity equation div v = 0 with v = uer + ve~, due to the representation of the liquid velocity as a gradient of a velocity potential 4,, i.e. v = grad 4,, leads to the Laplace equation V24, =0,

(1)

which has to be solved with the appropriate boundary conditions. At the rigid sidewalls, these are

104,

--- = 0 rOq~

at the sector walls q~= 0 and 2ira;

(2)

and at the free liquid surface: (i) the kinematic condition,

a~ --=--

at

a4, Or

at r =a,

(3)

SECTOR-SHELL HYDROELASTIC ZERO-GRAVITY OSCILLATIONS

Rigidw~ll A \.~" Z "

shell

~ .,"

~

,

,

.~(rp,t) "xx.~N___Freeliquid " . . ' x ' x ~ rface

~

•

(,/

./,

"

.

.

. .p~,,,--o

/ ~~ . ~,.- • . . b

819

"

,i

/ Rigid wall

a

r

(a) _ Free liquid

/"

.xq~. surface

,,,o,,/~~,,

Elastic

//~.".,

,( -

,,

':,

.

.

. D

.

'q,.

_

N.,).

. .

.

o

. ,q.

~,,,,,,,

,/

_

./

b

Rigid wall

a

r

(b)

Figure 1. Coordinatesystemand geometryof the hydroelasticsystemwith (a) freelyslippingcontact lines of the liquidand (b) anchorededge. and the d y n a m i c condition O2~ + ~ _ pa 2 04,

O~02

at r = a.

(4)

o" Ot

In these equations, ~'(q~, t) is the free liquid surface displacement, p the mass density of the liquid, and o- the free surface tension. At the shell, the condition

a~ a4,

-

-

=

_

0t

at r = b

_

0r

(5)

has to be satisfied. T h e values ~:(q~, t) and "O(~o,t) are the radial- and angular elastic d e f o r m a t i o n s of the sector shell, which has to satisfy the D o n n e l l shell equations

12b 2 taq~ 4

+ 2 ~__.~2+ sc + f + a - o + a~o

()b~ "~ i)2"g'] 0qo

0~ 2

E

8t z

p ( l -- v ) b 2 ~2T] = 0. E

Of2

-p

-~-t ,=,, D '

(6) (7)

820

H. F. BAUER AND K. KOMATSU

In these equations the Poisson ratio is denoted by V, the density of the shell by p, its thickness by h and its modulus of elasticity by E; also,/5 = Eh/(1 - V2). We may consider for the frictionless two edge conditions. (a) the freely slipping edges at the walls ~p = 0, 27ra; and (b) the anchored or stuck edge condition at ~p = 0, 2zra, which is due to surface tension effects expressed by =0

at ~ = 0 and 2Jra.

(8)

The boundary conditions of the shell are given by ~: = 0,

r / = 0,

0f 0~

=0

at ~ = 0 and 27ra.

(9)

3. M E T H O D OF S O L U T I O N The solution of the Laplace equation (1) satisfying the rigid wall boundary conditions (2) yields the expression

qb(r, ~p, t) = ~

[Amr ran'* + B,,r -'m2'*] cos[(ml2a)~o)]e '~'.

(10)

m=l

3.1. UNCOUPLED LIQUID MOTION WITH FREELY SLIPPING EDGES If the edges of the liquid at ~p= 0 and 2zra are freely slipping, we may combine the kinematic condition (3) with the dynamic condition (4) and obtain for the free liquid surface condition the expression

pa 2LOtp 20r

- ~ r J - 0t 2

at r = a

(11)

The contact angle is then assumed to be ½7r. In addition, we have to observe, that for a rigid shell, the boundary condition (b0___== 0 Or

at r = b

(12)

together with equation (10) yields

Bm = A,~b "'a,

(13)

and with equation (11) finally yields the uncoupled natural frequencies of the liquid: 2=

tom

o" r n r m 2

pa 32a L4a 2

-

] l - k 'm~ 1 1 + k m/'~'

(14)

where k = b/a < 1 is the diameter ratio. It may be noticed that the smaller the apex angle, 2zra, the higher the natural frequencies become. In addition, we find that the

821

SECTOR-SHELL HYDROELASTIC ZERO-GRAVITY OSCILLATIONS 60

50 m = 3,ct = 118

40 A"

X

\

X

30

%

m = 2 , or= 1/8

/

20 \

m=3, a=l14

10

m=2, ct=l/4 andm=l,a=l/8 m = l , a = ll4

00.0

'

0'.2

014

'

0:6

'

018

'

1.0

k = b/a

Figure 2. Uncoupled liquid frequencies for freely slipping contact lines.

thinner the liquid layer, the higher the frequencies (Figure 2). At a = ~ and a = 1, the liquid becomes unstable, since ~o~< 0 for a - > ½ and w2z = 0 for a = 1. For a modal number rn = 1, we notice that the liquid is therefore only stable as long as a < ½. For m = 2 and a = 1 we obtain instability for the liquid layer of apex angle 2zr and radial walls at ¢ = 0 and 2~r. 3.2. UNCOUPLED SHELL MOTION

If the liquid is not present, the right-side of equation (6) vanishes. The solution of the two homogeneous coupled differential equations (6) and (7) may be obtained with r / = Yea*e i'r,

= S e ' ~ ' P e i~'t,

(15)

leading to the characteristic systems [h 4 + 2A 2 + a 2 ] X

+ 12b2

h V = 0,

AX + [h 2 +/32]Y = 0

(16a) (16b)

which, together with a2=l+--

12b 2

12092fi(1- ~2)b4

h2

Eh 2

12b 2 --- 1 + - ~ - - (1 - Z ) ,

~2 = to2p( 1 -- v2)b2 ~ X > 0,

E yield the characteristic equation h2 ] j

+ --~-- (1

(16c)

This bicubic equation exhibits the solutions h 2, A2 and A2, and we obtain the solutions

H. F. BAUER AND K. KOMATSU

822

A~(j = 1, 2 . . . . . 6). With the modal relation (16b) the elastic deflections may be given as ~(~o, t) = ~ Xiea'*C "~',

rl(~p, t) =

_

XjAj , ]ea,¢ei., ,

With the boundary conditions of the shell ~: = 0,

a__{~= 0, 77 = 0 a~

we obtain the six algebraic equations 6 E Xj =0, j=l 6 E AjXj = O, j=~ ~

(18)

at ~ -- 0 and 2~rc~, t5 Z Xi e2'~"a' = 0' j=l 6 E A~X~e2"'"' = O, j=t

(19)

~a AjXj e 2rcc'A,

AJX~2=O,

~:l~+

,~,,~+~3~

:0,

the vanishing coefficient determinant of which represents the frequency equation of the sector shell; i.e., 1

1

1

1

1

1

e2rr-a ~

e2tr~A2

e2~r,~a~

eZm~a4

e2~r,.,~

e2~r,~.~.

A1 Ate 2re,A,

A2 A2e2~r'~a"

AI

A2

Ate 2~-a,

A3 A4 A5 A3e2n,~a~ A4e2m~a4 Age2,r,~,~ A3

A4

A6

A6e2tm,~,

A5

= 0.

(20)

A6

A2e2~-,~r A3e2n~,x~ A4e2n~a~ Ase2rr".~s A6e2~r,,a.

A~ + x A~ + x ,~ + x A4~ + x ,~ + x ,~ + x For given system parameters h, b and 15(1 - vZ)b2/E > 0, the magnitude of to*, may be obtained numerically. The two equations (16c) and (20) must be satisfied simultaneously. 3.3. COUPLED HYDROELASTIC SOLUTION WITH FREELY SLIPPING EDGES

Through the presence of sloshing liquid, the structural frequency will be changed; also, the sloshing frequency will differ, because of the elastically oscillating structure. We distinguish therefore coupled liquid and coupled shell frequencies. In this case, equations (6) and (7) have to be solved with the compatibility condition (5). The solution of equation (1) with boundary conditions (2) and free-surface condition (11) is given by pa 32ak4c~ 2 4)(r, ~, t) = .... ~ l A,,,e "°' r ""2~ + o" m rn 2

1) a'n'% -'"2~

m cos ~--~a~o. (21)

The right-hand side of equation (6) therefore reads ip°~b2 ~_, a,,,ei'~' a ' ' / ~ .... ~ ~

k ''/~

m ,p). pa32cl\4°t2 cos(~--~a + o- rn ( m 2 _ ) + w 2 j pa 3 2a \4c~ 2 1 .

.

.

.

(22)

SECTOR-SHELL HYDROELASTICZERO-GRAVITY OSCILLATIONS

823

The inhomogeneous solution of equation (6) is then given by =

t~'l= 1

C., cos

q~ ,

rI =

nl=l

D" sin

q~ ,

(23)

where C,. and D., are to be determined. Introducing these solutions into the differential equations (6) and (7) gives the following algebraic system for the determination of C,,, and D..: m" ~ m2 12b2 - Z ) ] + 12b 2 m D,,, C,,, [ (~--~-2)- - 2(~a2) + 1 + - - ~ (1 ~-~

• 1_4A -~mlot = t~ tztpwo u-

~)h 2 C

I

"b,,,,2-

m

(-)-

0v m m 2 ~_f] pa 32a ~ 1 w2, k"'~'+ o. m_(rn2 ) pa 32a \ 4a 2 1 +

[m 2

m(~-~ ) + [ 4ot2 - z ]O,,, = O:

(24a)

(24b)

and results for C,,, and Dm in the expressions

pa____~2a\4a_______~22_ 1) -¢'02 am/e~(m 2 ~0, { 12ipwb____~ 4 C,,, = ~)h 2 k"~'+ o. m ( m 2 _ l ) - - - i b , ~ , a ~ \ - - ~ 2 - Z ) , pa32a k4ot2 + m2J D,. =

Am ,.

a-,.,____~ (m] 12ipo~b4 k"/~'+ pa32 a (-~a z - 1 ) - J o" m /-'---~ m - "~ - "~z| b"Z2~'\2a/ 4h 2 pa--52---a~-~z-1) +w J _

(25a)

_

'

(258)

where ( m 2 ]2

m2

12b2

m2

Am=[\"~2012] --2(4--~2)+1---~-(1-Z)][--~z-Z] +

-{m]21262 \G/

(25c)

Therefore, the elastic deflection of the shell is

~(q~, t) = e i°"

Xje "~ +

m=l

C,. cos

q~

(26a)

and -e

|~-2

Lj= 1 Aj -- )(.

c

+

re=l-

D.,sin

~

.

(26b)

The compatibility condition (5) yields i¢o

Xje~"°+ ~

m=l

C~ cos ~a ~

m

=m:,Gam

aml°t I 1,m'° pa 3 2a \ 4a z

Then, with the shell boundary condition

~=n=0,

_=°~ o 3¢

at ~o= 0 and 2ga,

(27)

824

H. F. BAUER AND K. KOMATSU

we obtain 6

~

~'~ - A j X j

Ex + ....E l c,,,:o, /=, 6

~

E X~e2'~'a' +

....

( - 1)'C,,,

,

6

j=1 A2 -

= O,

~

=o, Z

-d~jXj

e 2 ~ , = O,

(28)

j=,

6

A / ~ : 0,

~ A~X/e 2'~'~' = 0.

j=l

]=1

These are six equations for the determination of the Xj as functions of C,,,. Expanding e A'~ into Fourier-cosine series (depending on the structure of the roots A/),

e A'*= ajo +

aim COS

~o

,

j = l , 2 . . . . . 6,

rrl= ]

and introducing these into equation (27) yields, by comparing coefficients, an infinite number of homogeneous algebraic equations, of which the vanishing coefficientdeterminant represents the frequency equation for the coupled frequencies of structure and liquid. Truncating this system to a determinant of finite order yields a good approximation for the lower frequencies.

3.4.

UNCOUPLED LIQUID MOTION WITH ANCHORED EDGES

If the edges of the liquid at ,p = 0 and 2Jra are anchored (stucked-edges), they have to satisfy, for the free liquid surface displacement at these locations, the condition =0

at ,p = 0 and 2zra.

(29)

The volume of the liquid is given by Vo= lraa 2, yielding a circular equilibrium surface. Therefore, with ~'(,p, t) = ei'°'~'(,p), the dynamic condition reads

d2--(~=ipa2t°[ ~ o r ,~,=, {Z"'a"2'~ + B,,,a -'"2'~} cos (m2a~o/+ ~ Po]' +

(30)

dip2

where Pc) is an additional term (Poei~') of the velocity potential satisfying the Laplace equation (1). The solution of this ordinary differential equation yields the expression

ipa2to[ ~(~)=mcos(p+Bsin,p+

tr

~ {m,,,am/2'~+Bma-m"2"} Po-,,~=l

FT_-~i L4a 2

{m

~]

cos~aa,p j

. (31)

J

The anchored condition gives two equations:

I- ~ ~ a,,,"2~,+ ~ ma-"r2~'tJ

- I ~ "~ t m

A - toi,,Z,__l

~m Y -----~

] /50 = 0

1

(32a)

and

-~,J=O, (32b)

SECTOR-SHELL

HYDROELASTIC

ZERO-GRAVITY

OSCILLATIONS

825

where

po=a~, , i , . - ~ , iPo

iA

~,~_~, iB.,

m

o5 -=

60

From the kinematic condition (3), with ~'(~o, t)

=

e i'°' ~ ~',~cos

( ~m_ ~ )

m=l

we obtain m

~, = _ m

2ao5 [ A , , a

(33)

"'~" - / L , a - " ' ~ ° ] .

Expanding cos ~o and sin ~o in Fourier-cosine series in the range 0 -< ~o<- 2Jra, i.e., sin(2Jra) + ~ ( - ~1 )o')-,1- ~ m sin(2~a)cos(~a S--cos ~ - - 2tra .... ' ffa [~-j2- 1] sin ~¢=

1 - cos(2zra) +,,~1 (-1)mc°s(2~a)-[--~--- lcos(~a ~o), 2~a = zva [~a_.- 1]

with expression (33), yields for the free surface displacement ~(~)=

A{sin(2zra) (-1) .... lsin(2zra) {m )} ( 27ra + .... ~ ~ " ~ TF m . . . . 1 c o s\t -2-a~ +B

1-cos( . . . 2~a o> + ~ (-1> . cos(2~a) m= l

+ O5 ~ ) _

~,,,a ....

= -

E

R'O~ ~

"p-o -

+ B,,,a

1

(m

)}

-- 1

,,.2.,}

()] m

,

_ ,,.z.~} ,,.a ''/2" - B,,.a

cos

D/

~o .

(34)

m = I

Comparison of coefficients renders (m + 1) equations: A sin(2n:a__) I- B 1 - cos(2zra) I- O5~)= 0 2~a 2~a

(35a)

and A ( _ 1),,,_ l sin(2tra)__w+B_ ( - 1 ) " ' c o s ( 2 z r a ) - 1 R'ot

o5 + A,,.a"'z"[~ (~24 - 1) -- O52]

R'o~

-/~,,.a .... a " [ ~ (~-~24- 1) + ~2] = 0

for m = 1, 2 . . . .

(35b)

826

H.F. BAUER

K. KOMATSU

AND

60 rn = 3,a

=

1/8

--~

-,,

50

40

-m--2-a-=-1/8. . . . . . .

.......

-

30

m = 3 , a = 1/4

20

m=l,a=l/8

m = 2 , a = 114 10

--'~-

m=l,a=ll4 I

I

O.0

012

0:4

'

0:6

'

0~'8

I'0

k = b/a Figure 3. U n c o u p l e d liquid frequencies for a n c h o r e d contact lines: - - -, a = ~; - - ,

a = ¼.

With the wall condition

a~

Or

0

at r

b

we obtain equation (13), i.e.,

nm a -m/2c, = k,~/,~ma,~a,,"

(36)

Equations (32), (35) and (36) represent (2m + 3) homogeneous equations in A, B, /5o, 7t,,,a ran" and Bma -mn~, the vanishing determinant of which is the frequency equation for the uncoupled stuck-edge natural frequency of the liquid. It may be noticed that, in the anchored edge case also, the layer a = ½ and a = 1 becomes unstable, since m2/4a 2 - 1 vanishes for m = 1, a = ½ or m =2, a = 1. Some uncoupled natural frequncies for a = -~, J are presented in Figure 3. 3.5.

COUPLED

HYDROELASTIC MOTION WITH

ANCHORED

EDGES

In the case of anchored (stuck) edges of the liquid, the Laplace equation (1) together with conditions (2), (3), (4) and with the stuck-edge ~"= 0 at q~ = 0, 2rra have to be solved, and yield equations (32a), (32b), (35a) and (35b). These are m + 3 equations for A, B, Po, A., and/~.,. With the velocity potential

{

(m)}

4fir, ~o, t) = e i~°' Po +

[Amr ma~' + B,,,r -m'2~] cos ~ ~o , m=l

the right-hand side of the shell equation (6) reads

12pb4i~o -~

.e

i,at( tPo+ ~ m=l

m [AmbmfZ~'+ Brab-mr2a]cos(G~p) }.

(37)

827

SECTOR-SHELL HYDROELASTIC ZERO-GRAVITY OSCILLATIONS

Thus, the inhomogeneous solution of shell equations (6) and (7) is 12pb4iw Po ~ (m) /~h2 12b2 + Cm COS ~£~ ~p m=I 1 +-~-(1-Z) and m=1

leading to 12b 2 / 12b 2 m +1 + - - ~ - ( 1 - Z ) j ~ + h---i-Z---~ D., 12pb 4it.o [Arab ''rz~" + B , . b - ' ~ ] , Dh 2 m

m 2

which yields

m2 ]}

1 ~.12pb4ko [Ambm,.2a + B b-mmq[ - C,,, = ~ [ b h 2 ,,, ~L4a2 1

X

(38a)

,

12pb4io~

(38b)

Then, the elastic deflection of the shell section is ~<¢, t ) = e/,O,f I2p b4

/DO

m

C,, cos(~__d~o) + ~ Xjea,,

~

12ba _ Z ) ] + [1 + --~-- (1 "=' and

(m)

rl(~o, t) = e i''

D,, sin ~ ~0

+

}

(39a)

J='

-xjxj j=~ (A~ - Z.) e*'*}.

(39b)

1

The compatibility condition (5) yields

{

[

]

m

12p b4i°a Po + ~ C,,, cos(~-~o) + ~ Sie a'~ ,,~=1 y=l ioJ /5h2 1 + ~12b 2 (1 - Z)

= ~

[

}

- B,,,b -''a'-'] cos

~o .

(40)

nl=l

Expanding the functions e a,~ into Fourier-cosine series and comparing coefficients yields an algebraic set of m equations. Therefore, the frequency equation for the stuck-edge liquid case is presented by the coefficient determinant of equations (32a), (32b), (35a), (35b) and those of equation (40). Truncating the determinant to a finite order gives the lower coupled frequencies of the hydroelastic system. 4. LIQUID INSIDE AN ELASTIC SHELL If we want to treat the system consisting of an elastic shell at r = b and liquid inside with a free liquid surface at r = a (a/b <- 1), we have to write on the right-hand side of

828

H. F. BAUER AND K. KOMATSU

equation (3) a minus sign. Accordingly, equations the related to equation (3), such as equations (11) and (14), must have a minus sign on their right-hand side. The determination of the natural and coupled frequencies, however, is similar. 5. N U M E R I C A L E V A L U A T I O N S AND CONCLUSIONS Some of the foregoing analytical results have been evaluated numerically. First of all, we determined the natural (uncoupled) frequencies of the freely slipping liquid and those of the liquid with anchored edges at ~o= 0 and 27ra. This was performed for an eighth and quarter sector system, with a = ~ and J, respectively. The lower natural liquid frequencies, tom/'k/(tr/pa3), are presented as functions of the diameter ratio k = b/a. The fundamental mode, m = 1, exhibits for a = J the lower natural frequency; with increasing diameter ratio, k, it has decreasing magnitude. This means that the natural frequencies decrease for thinner liquid layers. Figure 2 shows the curve of the natural (uncoupled) frequencies for freely slipping edges for a = ~ and ,~. It may be noticed that higher-mode frequencies, to.... decrease appreciably only for thinner liquid layers. In Figure 3 the natural (uncoupled) frequencies for anchored edges are shown, again as functions of the diameter ratio, k. They show similar behaviour with respect to the thickness of the liquid layer, but exhibit large values in comparison with the frequency of the liquid with freely slipping edges. The uncoupled frequencies of the elastic shell are presented in Figure 4, where for a = ¼ the first three natural frequencies are presented for a shell thickness ratio, h/a = 0.01 and the structural parameter, ~ ( 1 - v2)b2/E = 1 0 - 6 . The frequencies, f-It) t*F! 1 are presented as functions of the diameter ratio, k. With increasing k the natural elastic frequencies decrease strongly. In this figure the thickness ratio of the shell, h/a, is based on a. The liquid, however, was omitted. The radius a serves just as a reference 4OO

m=3 300 --2

"d

m=l

2oo

100

0-0

0"2

0'4

0"6

0"8

1"0

k = b/a F i g u r e 4. N a t u r a l f r e q u e n c i e s of shell; ot = ¼, tS(l - ¢)b2/E = 10 -~, h/a = 0.01.

829

SECTOR-SHELL HYDROELASTIC ZERO-GRAVITY OSCILLATIONS

.. 160

\

120

\,

""-."~S~e#

~ 80 Coupled shell

40

0

Uncoupled "~" " ~ ~ " L ~ L ~ q1' uid , Coupled l i q u i d ~ 0-1

0:3

0'5

0:7

0-9

k=b/a

Figure 5. Uncoupled and coupled frequencies for freely slipping edges; a = I, t S ( l h/a = 0.01, o ' / p a 3 = 100, p i p = 2.

~)b2/E

=

10 -~,

value to exhibit the increase of the shell radius, b, from zero to a (i.e., k = 1). This makes the comparison with the coupled frequencies more lucid. In Figure 5 we represent the fundamental natural liquid and shell frequency together with the coupled-liquid and the coupled-elastic-shell frequency, where freely slipping edges have been taken for the contact line of the liquid. The uncoupled liquid frequency(---) is decreasing with decreasing thickness of the liquid layer. So does the coupled liquid frequency (--) which, influenced by the elastic shell, exhibits a lower frequency than the liquid with a rigid wall at r = b. The coupled shell frequency ( - - -), being influenced by the oscillating liquid, exhibits much smaller magnitude than that of the elastic shell alone(..-), and it decreases considerably with decreasing layer thickness. The strong decrease of the coupled frequency is simply due to the effect of the added liquid mass. We notice that the deviation is larger for thicker liquid layers. In Figure 6 we see similar results for a liquid with anchored edges, where the uncoupled liquid frequency deviates much more from the coupled liquid frequency. The liquid frequency, influenced by the motion of the elastic shell, exhibits much lower magnitude for thinner layers. The uncoupled frequency of the shell is again larger than the coupled shell frequency, which again shows a drastic reduction of magnitude due to the added mass effect in the lower k-region. Comparing the results for freely slipping

\ 160

120

\ \ \

so

\-.~

Coupled shell

_..Z"_......

.....................

::_

40

00.I

0.3

0.5

0.7

0.9

k = b/a

Figure 6. Uncoupled and coupled frequencies for anchored edges for the same parameters as in Figure 5.

830

H. F. BAUER AND K. KOMATSU

edges (Figure 5) and stuck-edges (Figure 6) we notice a stronger coupling effect in the latter case. For higher modes the situation becomes more involved, since lower shell modes may come very close to higher liquid modes, which shows strong interaction between them; mode identification may then become quite difficult. 6. C O N C L U S I O N S The main conclusions of this study may be summarized as follows. (i) The natural liquid frequencies with anchored contact lines exhibit larger frequencies than those for a liquid with freely slipping contact lines. (ii) The natural liquid frequencies decrease for thinner liquid layers. This is more pronounced for higher k-values. (ii) The natural frequencies of the clamped elastic shell decrease considerably with increasing shell radius, b. (iv) The influence of the elastic shell oscillations upon the coupled liquid frequency results in decreased magnitude, which difference is more pronounced for thin liquid layers. (v) The effect of the liquid on the coupled shell frequency is to produce a strong decrease, which is mainly due to the effect of the liquid mass and to a smaller extent due to the sloshing of the liquid. (vi) The interaction effects are more pronounced for liquid with anchored contact lines. REFERENCES

BAUER, H. F. 1982 Coupled oscillations of a solidly rotating and non-rotating finite and infinite liquid bridge of immiscible liquids in zero-gravity. Acta Astronautica 9, 547-563. BAUER~ H. F. 1984 Natural damped frequencies of an infinitely long column of immiscible viscous liquids. Zeitschrift far Angewandte Mathematik und Mechanik 64, 485-490. BAUER, H. F. 1986 Free surface- and interface oscillation of an infinitely long viscoelastic liquid column. Acta Astronautica 13, 9-22. BAUER, H. F. 1987a Natural frequencies and stability of immiscible cylindrical z-independent liquid system. Applied Microgravity Technology 1, 11-26. BAUER, H. F. 1987b Coupled frequencies of a hydroelastic system consisting of an elastic shell and frictionless liquid. Journal of Sound and Vibration 113, 217-232. BAUER, H. F. 1987c Hydroelastic oscillation of a viscous infinitely long liquid column. Journal of Sound and Vibration 119, 249-265. LAKIS, A. A. & PA~'DOUSSlS,M. P. 1971 Free vibration of cylindrical shells partially filled with liquid. Journal of Sound and Vibration 19, 1-15. LAMB, H. 1945 Hydrodynamics. New York: Dover Publications. LEISSA, A. W. 1973 Vibration of Shells. NASA SP-288. MILES, J. W. 1958 On the sloshing of the liquid in a flexible tank. Journal of Applied Mechanics 25, 277-283. RAYLEIGH,LORD 1882 On the instability of cylindrical fluid surfaces. Philosophical Magazine 34, 177-180. YAMAKI, N. T., TANI, J. ~¢ YAMAJ1, T. 1984 Free vibration of a clamped-clamped circular cylindrical shell partially filled with liquid. Journal of Sound and Vibration 94, 531-550. APPENDIX: NOMENCLATURE a

b E h k

radius to the equilibrium position of the liquid surface radius of two-dimensional shell equal to E h / ( 1 - ~) modulus of elasticity shell thickness ratio of radii, b/a

SECTOR-SHELL HYDROELASTIC ZERO-GRAVITY OSCILLATIONS

in

circumferential mode number polar coordinates t time 2zu~ apex angle of sector system ~" radial free surface displacement Poisson ratio ~, 77 radial and circumferential shell deflections, respectively p density of liquid t5 density of shell o" liquid surface tension ~b velocity potential of liquid w,,, uncoupled natural frequencies of the liquid w* uncoupled natural frequencies of the elastic shell r, ~p

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