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Coupled frequencies of a hydroelastic system, consisting of an elastic shell and frictionless liquid
Journal of Sound and Vibration (1987) 113(2), 217-232
COUPLED FREQUENCIES OF A HYDROELASTIC SYSTEM, CONSISTING OF AN ELASTIC SHELL AND FRICTIONLESS LIQUID H. F. BAUER lnstitut flir Raun~ahrttechnik, Unit,ersitiit der Bundeswehr, D-8041 Neubiberg, German), ( Receired 30 December 1985)
For a frictionless liquid layer around a cylindrical shell and a frictionless liquid inside a shell in a zero-gravity condition the hydroelastic interaction of the shell- and liquidmotion is treated. The coupled frequencies for the two-dimensional and three-dimensional cases are presented for various elasticity parameters and liquid layer thicknesses. There are strong interactions in both cases. I. INTRODUCTION The availability of extended manned space flights in an earth-orbiting space laboratory makes unique experiments possible. Such experiments cannot be performed under the action of earth gravity, since a liquid with a free surface will exhibit completely ditterent equilibrium positions. One of these experiments is the behavior of an annular liquid column outside or inside a cylindrical shell, where the liquid layer under the action of gravity would bulge out to a geometric configuration not representing the annular layer form, as it does under zero-gravity, where the liquid is just held together by surface tension. It offers, due to the lack of gravity, a cylindrical liquid form and a number of advantages for exploratory experiments. The natural frequencies of such a liquid layer, thick or extremely thin, should be known, in order to avoid during the experiments certain frequency ranges appearing in a space laboratory known as the so-called g-jitter. Such vibrations of the laboratory appear during the operation o f machines on board, motion of the crew or control maneuvers, etc. For a rigid center shell surrounded by a frictionless liquid layer or a rigid cylindrical container partially filled with a frictionless liquid the natural frequencies have been determined [1, 2]. The condition of a rigid shell, however, may be violated in such a way that the shell-structure may be elastic and perform vibrations. Such vibrations would also affect the motion of the free liquid surface and couple with it, in which case the shell frequencies as well as the free liquid surface frequencies would in the coupled form shift to other magnitudes. Of interest, therefore, is the behavior of the coupled liquid-structure system as a function of the thickness of the liquid layer, its surface tension, the wall thickness, the density ratio of structure and liquid and the elasticity o f the shell. Lord Rayleigh [3] treated the three-dimensional case of a cylindrical air column in an infinite medium and found that the axisymmetric cylindrical configuration becomes unstable and exhibits its most pronounced instability at an axial wave length A.. = 12.96 a. Lamb [4] performed similar investigations, in which mainly the simple liquid column was treated. In addition the remarkable work of Plateau [5], who performed a large number ofuseful simulation experiments, should be mentioned. No work, however, has been performed yet upon the interaction of an elastic structure with a liquid in a zero-gravity environment. In what follows an analysis is presented for 217 0022-460X/87/050217+ 16 $03.00/0 (~) 1987 Academic Press Inc. (London) Limited
218
mr.
F. BAUER
the determination of the coupled frequencies of a two-dimensional and three-dimensional liquid-structure-system; i.e., a system exhibiting no z-dependency and with z-dependence respectively. The main interest here is in the behavior of the coupled liquid-shell system, and the influences of the tension parameter T / p a 3, the thickness of the liquid layer (k = b / a is the diameter ratio of the shell and the liquid surface location), the density ratio of shell and liquid fi/p, the thickness of the shell h and the elasticity parameter f i ( I - f i 2 ) a 2 / E are investigated (a list of symbols is given in the Appendix).
2. BASIC E Q U A T I O N S
Under the influence of an elastic structure a liquid in a micro-gravity environment will execute oscillations, if disturbed (see Figure l). The problem to b~ solved is the determination of the coupled frequencies of the elastic structure and the liquid with a free surface. The liquid is considered frictionless and incompressible. For small liquid velocity components and small elastic deflections the governing equations may be linearized. With the liquid motion assumed to be irrotational the continuity equation leads due to the representation of the velocity as a gradient of a velocity potential tb, i.e., v = g r a d ~ b,
(1)
V2@=O,
(2)
to the Laplace equation
which has to be solved with the appropriate boundary conditions. If the free surface is at r = a outside an elastic inner cylinder at r = b, b < a, the free surface conditions are u = a~o/Ot = O@/ar
at r = a,
the kinematic condition,
(3)
and
-p a@lat+(Tola2)[~o+a2~o/a~2+ a" a2~'o/aZ"-] = 0 at r = a,
(4)
this being the dynamic surface condition. Combining equations (3) and (4) yields after differentiating equation (4) with respect to time, -a2,1'/at 2 + ( To/pa")[ u + ,'~:u/O~o2 + a: F u / a z 2] = 0
Freehquidsurface ~
T
o
at r = a.
(5)
TosurfacetensJon ....
tens,onSUrface ^
/ "r/.~ (
FreeI,quid E~ShC shell
Figure I. Geometry and co.ordinates of system.
IIYDROELASTIC
SYSTEM
COUPLED
219
FREQUENCIES
The normal elastic deflection of the structure is denoted by ~(~, z) in the r-direction, and "0(~, z) and ~'(~,, z) in the ~- and z-directions. The boundary condition at the inner cylindrical structure is ,~/at = ~/ar
at r = b.
(6)
A cylindrical shell of length I has to satisfy the shell equations I ~b ~.~..~+at/+ h2 -4 _ (I - ~2). 2 a2~: t]tP b2 az a~p ' ~ + i - ~ v ~ : + P - - f f - - b ~ - ~ = P - ~ - , o b ~ atr=b,
~(l+O)b ~2~r+0~+I(I a z O~ b2~2~+1
0r
2
(7)
21}2"0+'~2r/ (I-~2) 2a2r/- ~') b -~z2 -~-~2 - fi - - E
~2~.. ! . . . . ~2~7 l,)O,~za ~
+
(8)
-- b Ot2 - 0 '
~b~: _(1-~2).2~2~ ^ -~z-pTo ~t2=u
atr
=
b.
(9)
These are the DonnelI-Mushtari equations, which easily could be changed to other circular cylindrical shell theories by employing additional terms multiplied with the small value h2/12b 2. Here the operator V2 = b 2 ~2/~z2 + +-~2/;~2. For a cylindrical shell of infinite length without axial ( z - ) dependency the equations of motion of the elastic shell yield, upon using the concept of plane strain, the equations h2 [ ,~2~ ,.}4~.-]+~:+~__~qtS(l_~2)b2~2,~ ~b 2 12b 2 2 o~ ~ ~ t 2 - P - ~ t - "~ a t r = b , (10) ~'~+ t32r/ fi(1-p2)b2a2"r/-0 9}~ ~ 2 E r]t2
atr=b.
(11)
In these equations h is the thickness of the shell, P the Poisson ratio, E the elasticity modulus and /~ = Ell/(l - p2), while t5 is the density of the shell material and p that of the liquid. As may be noticed, these equations are coupled with the liquid motion by the term p ~ r and the right-hand sides ofequations (7) and (10) as well as by the boundary conditions (6). If the liquid is in an elastic container r = a and has its free surface at r = b, then the free surface condition reads
a2el2_,gt~-~[ "F+ u + b 2~2u
~2u1
atr=b
(12)
while equation (6) has to be satisfied at r = a. In the shell equations b has to be replaced by a and the right-hand sides of equations (7) and (10) show a minus sign ( - ( p a 2 / D)OdP/Ot). 3. METHOD OF SOLUTION To determine the frequency equation for the coupled frequencies of the systems comprising the elastic structure and frictionless liquid equations (2), (5), (7), (8), (9) for the three-dimensional case, or equations (2), (5), (10) and (1 !) (with ~ / ~ z - 0 ) for the two-dimensional case, have to be solved with the appropriate boundary conditions. 3.1.
FRICTIONLESS
LIQUID
DEPENDENCY
(TWO.DIMENSIONAL)
AROUND
ELASTIC CENTER
SHELL
WITIIOUT
AXIAL
In this case equation (2) yields the velocity potential q 3 ( r , % t ) = ~ [ A , , r " + B m r - " ] e i"'~+''. rn=l
(13)
220
H.F. BALIER
With the boundary condition (5) one obtains, with ~/az = 0,
[s2+(Tom/pa3)(m2-1)]A,~a"+[s2-(Tom/pa3)(m2-1)]B,o/a"=O,
(14)
and with ~(~p, t) = X,, e imr247
7/(~, t) = Y,,,e i'r §
(15)
from the shell equations (10) and (11) the expressions
{(h2/12b2)[m'-2m2+ 1]+ 1 +,5(1 - ~Z)b2s2/E}aX,,k+im Y,,ak = (pb3s/D)[Ab"'+ Bb-m],
(16)
im Xma -{m2 + iS(1 - ~'2)b2s2/E} Yma = 0.
(17)
The boundary condition ensuring that the motion of the elastic wall is equal to that of the liquid at the wall (equation (6)) gives the expression
sX,. = mA,,b "-I - mB.,/ b "+l.
(18)
Equations (14)-(18) are four homogeneous algebraic equations, and hence the vanishing of the coefficient determinant provides the frequency equation for the determination of the coupled frequencies:
II~,,ll =0,
j,!=1,2,3,4.
(19)
The elements ~Sslare given in terms of k = b/a, ~2 = ( Tom/pa3)(m 2 _ 1),/3 = ,5(! - V2)b~/E, and a = D/pb 3 by (see Figures 3 and 4 of section 4)
,5,2=s2-d, 2,
~5,1 = s2+ a32,
823=-ot{(m2-1)2(h2/12a2k2)+l+fls2}k, 833 = ira,
,521=sk",
813= ~ 4 = 0 ,
834 = -{m2 + fls2}, c543= -sk,
822=s/k",
824 = - i m a k ,
841
=
mkra,
831=832=0,
~42 =
--m/kin,
c544= O.
(20)
For an elastic cylinder without a liquid layer, one has only the elements ~23, ~524, ~33 and $34, and hence obtains the frequencies given by Reismann and Pawlik [6]; i.e. (see Figures 3 and 4 of section 4), S2 =
-
~
2,5(!
_E t32)b2/n ! [ z + I q- i~(I/'1 112 - 2
- -
1) 2
(m2 + 1)2- 2 ( m 2 - 1)3 .--T~,z + ~ ( m 2 12b 144b
1)4
"
(21)
If the structure is rigid, ~ and ~7 vanish and one has only the elements 8tl, ~t2, ~21 and ~22. The determinant results then in the frequency equation for a frietionless liquid: i.e. [1,2] (see Figure 2 of section 4), s 2 = - Tom(m 2 - 1)( 1 - k2")/pa3( i + k s'').
(22)
HYDROELASTIC SYSTEM COUPLED FREQUENCIES
221
The complete determinant (19) gives a bicubic equation of the form with X ~ (h~/12a2)(m 2- 1)2,
s6fl2[ k( l - k2") + m P- h ( l + k2") } pa + s4fl{ (l + k2")[kfl~2 + mfi h (l + m2 + x)] +(l -k2")m[kn, + fltb2 ~ h ] l +s2 (! +k2m) flk ~52+m~
p
X m~+(l-k2"')m
I+m~+X)
pa
+ nt2c52 -P h x ( 1 -/,:2,,,) = 0.
(23)
pa
3.2. FRICTtONLESS LIQUID INSIDE ELASTICCONTAINER WITHOUT AXIAL DEPENDENCY (TWO-mMENSIONAL) For a frictionless liquid inside an elastic cylinder of radius a and a free liquid surface at r = b one obtains, with the boundary condition (12), the shell equations (10) and (1 I) at r=a and the wall condition (6) at r=a with equations (13) and (15), the equations
km[s 2- (T,m/pa3k3)(m 2- I)]Ama '~ + ( I / k m ) [ s 2 + (T,m/0a3k3)(m 2- I)] B,,/a m = 0, (24) (pa2/ O)s[ Ama m+ (Bin~am)] +{(h2/12a2)(m4-2m2+ l ) + 1 + ~ ( I - r,2)a2s2/E}X,~ +ira Y,~ = 0, (25) imX,,-[m2+~(l-P2)a2s2/E]Y,~=O, mA,,a"-m(B,,/a")-saX,,=O. (26,27) The equations (24)-(27) are four homogeneous algebraic equations, ofwhich the vanishing determinant is the frequency equation of the coupled frequencies. It is Here
II, ,ll =0, j, I = 1,2,3,4. the elements are, with a52 -~ T,m(m 2- 1)/pa~k 3, 6 =- l)/pa 3, and/3 -= ,6( ! - f,2)a2/E, g,, = [ s 2 - ~ 2 ] k '' , g.=(1/k")[s2+&2], g,3 = g,., = 0, gn=s=g22, = a{i + = in,a, = =0.
g~3 = ira,
g3+ = - [ m 2+/3s2],
g41 = m,
g.~2= - m ,
g43 = -s,
g.~4= 0.
(28)
Without a liquid layer equation (21) is again obtained, if b is replaced by a. For a rigid structure, one has only the elements g,,, 6,2, 62, and g22, which yield from the determinant [ 1, 2] (see Figure 5 of section 4)
s 2 = - T,m(m 2- 1)( 1 - k2")/pa3k3( 1 + k2").
(29)
The determinant with the elements (28) gives the bicubic equation (see Figures 5 and 6 of section 4)
s6fl2{ ( l - k2") + m fi- h ( l + k2") } pa + s4fi{ (l + k2"')[fit~2 + m~ h (l + ,~ + m2)] +(1-k~")n,[m + flt~2 P- h]} pa p L
pa
+ m3a3t ~ h)?(I - k t'') = O,
in which
pa ,~ -= (h2/12a2)(m 2-
I) 2.
.j
pa (30)
222
H.F. BAUER
3.3. F R I C T I O N L E S S L I Q U I D A R O U N D E L A S T I C I N F I N I T E C Y L I N D R I C A L S I I E L L When an elastic infinitely long cylindrical shell is considered the liquid satisfies the Laplace equation (2) and is represented by the potential
/nrrr\ 9(,,~,z)_._,.oo~ L """i-7-)+~~
/nrrr\']
rA
_ ~
....'"
(~')
Satisfying the remaining boundary conditions at r = a (i.e., equations (5) and (6)) and at r = b (i.e., equations (7), (8) and (9)), with the elastic deflections
n=l m-O
n-I
m=O
(32) n=l m=O
yields the following system of homogeneous algebraic equations:
AL~ xA,t-7-d)+p~,(l/o)(m=~ (,/~)="'== , .7. ,.,,,.,, nrr,/nrr\ A T l,,,~/--~a) +
/nrrk) apsl,,~7-~a
(,/~ ')~~
o,
nz" ,/nrr \
BTK,,t~ak ) -sX h2 [
(34)
=0,
nrr XD +BpsK'.(~7-~k ) --gr{' +]~-~[m
2 n2"n'2b2"]2
+ ;-~--J
+
fi(l-~2)b2s 2}
imD Y- imnrrb/ D\ Z , ~y)=o. imX-Y 7
(33)
(3s)
{"" m2+~(1-
~)4 ~ ( l -
X-
)b2s~ ------~-(1 mnrrb fn=~r2b2
+
+ ~)Z = 0,
(36)
- E--z)b2s2} = 0.
(37)
The coupled frequencies are obtained from the vanishing coefficient determinant II~,~.ll =o, where the elements a,~. are given by .
In,<+gro~
n. (.,'+
a,,=s I,,t'~-d} \pa'] (i/a)\ .
~ ~
/,,=\+(ro~
n,~ /
,
n'~ ~
\
"'#
\ , I,l<
(I/a) 2 l)l"klT-a )' ,/,,=~
K4,,7-a) ~ , ' / ( , - ~ t ''+(,/.)' ' ) ~ ' k ~ ) ' n~ , I . n ~ \
~,,=-
1+ ~
~
"~ K' [ k ,I,*~
~ ,-Ta/'
n,'+ (t-i~o)~-] ~
o.,=~176
~"=-~'
~(1- r
~,,=o.=o,
HYDROELASTIC SYSTEM COUPLED FREQUENCIES
C~'41 ~ ~'42 ~
/~34
im( h / a )(~ / p ) /5(1 - ~,2)b2/E'
0,
a4, = in/,
ct35 =
i,nn~rk( h / a )(~ / p ) (I/a)~(l - ~2)b2/E'
[ 2 H2n2k2 9 /5(1 - p2)b2s2" ~ o~44 = - - t m + ~ ( 1 - ~)-t~J'
mn~rk
a45=
223
2(I/a) (1+~')"
inrtr, k aSI -----a'52---- 0'
l/a'
a53
~fn2~2k 2
nnl,'rk a,,=---2(i/a)(l+~),
a 5 5 = - t ~ * : ( l - P 2 ) bm: s2: + ~
} "-~-'(1-~ 2) .
(38)
The determinant gives the coupled frequencies for the modes (m, n) as functions of the various parameters f i ( l - ~,2)b2/ E, To/pa 3, h/a, l/a, k = b/a, and fi/p (see Figure 8 of section 4). For a shell with liquid the lower-right determinant of third order gives the frequency equation i.e.: 118,~11=0, i, K =3, 4, 5. This yields the hi-cubic equation with, ), =/~(1 - ~3)b2/E, 2 4[m2,~ 2 ( !!1
n'77'
4
4
I| 77
I12 1
4
2
111
2/j2
n%r 2 \ : 1 ~T
2
11111277 2 -
+TS t-~-(l-P)+~(l-P)2--m+~(I--P)
+rl+
h2 I n 2+ 112772\2-1rnl2
(lib)2u
1121"1"2
kl_
+":"
~ ~
.,,+ +q
,
=0
(39) For a rigid shell one obtains the natural frequency of the liquid from the upper-left determinant of second order. It is [1] (see Figure 7 of section 4)
r, I n77\ , I k ,m-\ s2
Tomr [ ,
= ,~t'"
,i%.:
, I k n=\ , I n = \ ]
.'~[l"t~-~)K"~,77~)-I"[,)7~)K"W~;J
+ (~1,~)2 'Jr
,' 11,.,.\ , / k 1"-" L''t,l-7-~)K' q"~-d )
: '1"'\ '~ t k"-l: ~
(40)
K"[l-7~)J/"=\]"
3.4. FRICTIONI.ESS L I Q U I D INSIDE AN INFINIJ'E ELASTIC C O N T A I N E R
In this case a must be replaced by b, and To by -Ti, while in equation (10) the sign on the right-hand side is altered. This yields a determinant Ilflm.II =o, of which the elements are (see Figure 8)
/,,~ k)~_[\pa3 T,~] k2(I/a)\nI24 ,i,, C --"':: fl,,=s l..tl~a (i/a)2
f12, : ~ 7 ~ l , . \ l / a ] ,
fl,2 =/-7-daK,-t/-7-~a) ,
l)~o,~kG),'": ' ~
f12.,= -s,
f12+= 0,
224
H.F.
~3,,=sire (mr)/Ta'
/3,,=0,
(h/a)(/5/p)
[nrr' /332= sK,,~)7-a),
/5(1-02)a 2 2
[
im( h/ a)(/5/ p ) /334= tS(1 - ~2)a2/E' /3., = fl42 = 0.
BAUER
!12 [ 2
inrrO( h/ a )(/5/ p) /355 = (i/a)/5(l - ~2)a2/E'
f n2"n"2 fl4a=-/~(1-
/34s=im.
~)+ m2-1
mmr(1 + ~) /345 =
2(l/a)
n27r2 ]2"1
/5(1-: ff2)a2 2] ~s
/'
inTrf' '
/35~ =/352 = O,
/353 = ( t / a ) '
[" n2~'2 /5(1--/52)a2 2InI2""7 ] /355 = - L ( / - - ~ 4 ~ s -e-~-tl ~)_.
/354 = - (i/a)mnrr(1 + 9),
(41)
From this equation the coupled frequencies s may be determined for the modes (m, n) and for the various parameters k = b/a, l/a,/5/p, h/a,/5(1 - 92)a2/E and Tdpa 3. It may be noticed that these results also satisfy the problem of a finite shell of length l with a freely supported or shear diaphragm end condition at z = 0 and /. This means that a circular cylindrical shell of infinite length vibrating in a mode of which the half-length of the wave in the axial direction is I has the same frequency as a shell of that finite length I with these boundary conditions. If at z = 0, ! a bottom and a top were included, the above solution for the liquid could not satisfy the condition of vanishing normal velocity. For a rigid container the natural frequencies of the liquid layer inside the container may be obtained from the upper-left determinant of second order. It yields, with s = ito,,,, the expression 2
nz_kk
to...-(1/a ~ [
r,
n~\
1
-m 2
n2rr2k2~[ , [nTrk'~ , [ nTr'~ , [ nTr\ ,2/a 2 / / i . k ~ ) ~ k ~ ) - , ~ k ~ ) ~ k ~ ) j /
, [n~-\
I' / m r \
-
[nz-
\'1
J
, [nrrk\] /
(42)
which is represented in Figure 7 of section 4.
4. NUMERICAL EVALUATIONS AND CONCLUSIONS The hydroelastic (coupled) frequencies have been evaluated, in the first instance, for various stiffness parameters /3 (/3) and given thickness ratio h/a=O.O1, density ratio /5/p = 2 (shell density to |iquid density), and modes number m - - 2 as a function of the diameter ratio k = b/a. Small k-values represent an elastic shell v)ith a thick liquid layer around it, while the magnitude k ~ l (<1) represents a very thin liquid layer. The uncoupled frequencies are given as ( . . . . ) curves, where the value s~ denotes the uncoupled liquid layer frequency and s# and s o denote the uncoupled shell frequencies in the radial (~:)- and angular (r/) directions. The mode m = 2 describes an elliptic free surface shape during the oscillation: i.e., a q-dependency of cos 2~. In Figure 2 the uncoupled liquid frequency to2 is given for the tension parameter To/pa s = 100 (s-:) and /3 = 0 (i.e., E --~oo, a rigid shell) as a function of the diameter ratio of shell toundisturbed D
H Y D R O E L A S T I C SYSTEM C O U P L E D F R E Q U E N C I E S
225
2,50 . . . .
~ .
o
\.
\
20 0
"\
\ kSt
\
150
100
\
\
\ \
50
O0 O0
,
I
02
,
I
,
04
I
06
,
I
08
10
k = blo
Figure2. Uncoupled frequency of frictionless liquid around rigid center shell (equation (22)) (twodimensional), m = 2, To/pa ~= 100 (s-~),/3 = 0 (s2).
liquid surface k = b/a. The natural frequency of the liquid in the mode m = 2 decreases with decreasing liquid thickness. 'In Figures 3(a)-(c) the coupled and uncoupled liquid and shell frequencies are presented for the mode m = 2, the tension parameter T o / p a 3 = 1 0 0 ( s - 2 ) , the density ratio ~/p = 2, the shell thickness ratio h/a = 0.01 and various stiffness parameters/3 = (,6(1 - ~ ) a ~ / E ) (s:). The uncoupled frequencies are given by ( - . - ) curves, and the coupled frequencies by (- - -) curves. Figure 3(a) shows the result for a very elastic shell, for which/3 = 10 -3 (s2). It can be seen that the coupled liquid frequency is strongly influenced by the coupling with the shell. This is more pronounced for thin liquid layers, as the coupled liquid frequency increases strongly with increasing k = b/a. The uncoupled shell frequencies s~ and s o are also presented. While %, the natural frequency of the shell in the angular direction is of large magnitude, the uncoupled frequency (sr in the radial direction has a small value for this particular case. The coupled structural frequencies sr and % show in the radial direction (s~) a further decrease in magnitude, with the difference between the uncoupled and coupled values decreasing with decreasing liquid layer thickness. The coupled structural frequency % in the angular direction is on the whole smaller than the uncoupled one. For a further increased stiffness, say/3 = 10 -6 (s 7') the results are shown in Figure 3(b), where the lower frequency range is presented. A further increase of stiffness, i.e., an increase of the modulus of elasticity E, has the results shown in Figure 3(c), where only the liquid and lower structural frequencies are shown. This is a very stiff shell, for which the coupled liquid frequency is hardly affected by the motion of the shell. The liquid, however, through its mass, affects the structural frequency s~ in the radial direction. It can be seen that the coupled structural frequency s~ approaches the uncoupled value for vanishing liquid layer thickness.
226
H.F.
BAUER
8oo[
80 0
I
/ "/--
$I
!(~)
64 0
640]-
//
"- 4 8 0
/
/
/
/
I
I I I
iL
st// u.
32"0
,/
/
320
=-:'-== ~
,
02
~
~ 04
~
l
//
\
/z-s, \
I
/
O6
~ . . . . . -~..~.
16 olN.
O0
/
,~----~'-"~..s L
O0
I I I I
i
\.
"
08
00I O0
0
/
~
\ 02
04
0.6
08
10
k = b/a
300 0
t
(c)
\
I
240 0
\
\
\\
"\
,? ,/
\
1800
/ \
\
\ \
\ /
u: 1 2 0 0
/
/
/
I I
60'0
,~_SL
O0 0"0
~
I
0"2
i
I
0 4
i
I
06
~
I
0 8
~ 10
k = b/o
F i g u r e 3 . U n c o u p l e d a n d c o u p l e d frequencies of liquid layer a r o u n d elastic shell for m o d e m = 2 (Iwod i m e n s i o n a l . T, J p a ~ = 100 (s-"), ~ / p = 2, h / a = 0.01. (a) fl = l 0 -3 (s2); (b) fl = l 0 -6 (s"); (c) fl = l 0 -9 (s2). - - -, U n c o u p l c d ; - - -, c o u p l e d .
I I Y I ) R O E L A S T I C SYSTEM C O U P L E D
227
FREQUENCIES
For a frictionless liquid inside an elastic container (without axial dependence) the results for the coupled and uncoupled frequencies are given in Figure 4. Figures 4(a) and (b) are for cases of the mode m = 2 with the tension parameter T,/pa 3 = 100 (s-2), the density ratio ~/p = 2, the thickness ratio h/a = 0.01, and various stiffness parameters ~ = ~ ( l - f J ) a 2 / E . For a very elastic shell / 3 = 1 0 -3 (s 2) the coupled ( - - - ) and the uncoupled ( . . . . ) frequencies are shown in Figure 4(a). The uncoupled liquid frequency sl ( . . . . . ) (equation (29)) shows a decreasing magnitude with decreasing liquid layer thickness. The coupled liquid frequency decreases first for increasing k-values and then increases again for smaller liquid layer thickness. The coupled radial shell frequency is very small (se=0.1) and not indicated in this figure. The structural frequency s, in the angular direction first remains nearly constant for decreasing liquid layer and then decreases, through the influence of the oscillating liquid surface, to smaller values, and finally increases again for thin liquid layers. For a further increase of the stiffness fl = 10 - 6 (5 - 2 ) the results are shown in Figure 4(b). For the case of an infinitely long elastic shell (three-dimensional case) the natural frequencies of the frictionless liquid are presented in Figure 5 for various modes (m, n = 1) for the tension parameter To/pa 3= 100 (s -2) and the length/radius ratio 1/a = 2 as a function o f t h e diameter ratio k = b/a. These are the frequencies for a liquid layer around a rigid central cylinder. The case n = 1 is shown in Figure 5(a) for the axisymmetric case m = 0 and for m = 1, 2, 3 and 4. With increasing k-value, i.e., decreasing thickness of the layer, the frequency decreases towards zero. Similar results may be obtained for n = 2, etc. The uncoupled liquid frequencies are also shown in Figure 5(b) but as a function of the length ratio I/a for the tension parameter To/pa 3 = 100 (s-2), diameter ratio k = b/a = and n = 1. It may be noticed that with increasing length ratio the natural frequencies decrease, with a rapid decrease within the region 0 < l / a < 2. The axisymmetric motion m = 0 becomes unstable for l / a -- nrr and for l / a > n,'r the root has a real part. iO0 0
250 0
(a)
(b)
~!~
80
/"
150 0
" / I
!
\ \
400
l
L / \ \ . \
g
l l l
200 0
\\
600
l
\
100 0
\s,
I
x
/
\ k.
\
k.
20 0
\.
\
, "~
O0 O0
\ ; ..~. '-1 . . . . 02
s{=O.1
- 4 - - - ~--..-~ . . . . . 04
06
s(
L_ . . . . . 08
" \ . "~"~...~. ~ ~ . . . J st
50 0
x.
/
0
0 OL-. O0
02
04
06
08
k = b/n
Figure4. Uncoupled and coupled frequencies of liquid in an elastic container for m o d e m = 2 (twodimensional. Parameter values and key as Figure 3 except ( a ) / 3 = 10 -3 (s2); (b) ~ = I0 -t' (s:).
228
H.F. BAUER
100 0
I
m
J (a)
80 0
?\
\
\
600
]
400
I[Z]
200
1--
I I I
I
O0 O0
04
02
-J \
i ~ ~=/ ~-..,,. 3"'--~. 12 J~
2L~t_ /~
O6
X\
\
\4i
F-"--'-.-' 08
k= bla
150 0 (b)
120 0
90 0
i\'x,~ m = 4
600
30 0
I \'.., O0 O0
--?--i
"
I 20
~"
40
60
80
100
L/a
F i g u r e 5. U n c o u p l e d f r e q u e n c y o f f r i c t i o n l e s s l i q u i d a r o u n d rigid c e n t e r shell ( t h r e e - d i m e n s i o n a l ) ( e q u a t i o n (40)). To/pa 3 = 100 (s-2), n = I. (a) I/a = 2; (b) k = 0.5. , Real part; - - -, imaginary part.
The coupled frequencies of the system of an elastic shell surrounded by an inviscid liquid layer are presented in Figures 6(a) and (b) for tile elasticity parameter fl = ~(1 - ~ Z ) a 2 / E =0.01 and in Figures 6(c) and (d) for 0.1 (s2), the density ratio of shell to liquid density #/p = 2, length ratio l/a = 2, Poisson ratio ~ = 0.3 and wall thickness ratio h/a =0.01, as a function of the diameter ratio k =b/a. The tension parameter To/pa 3 was chosen to be 100 (s-~). Three coupled structural and one coupled liquid frequencies are obtained for each mode (m,,t). It may be noticed that the largest frequency (in comparison with the results given for the uncoupled liquid frequencies in Figures 6) is the coupled liquid frequency (m = 0, n = 1) (Figure 6(a)). This frequency is least coupled with the structure for small k, i.e., when the liquid layer thickness is relatively large. This coupled liquid frequency is marked by o~. For a thin liquid layer the coupled liquid frequeficy ~o, does not have small values as for a rigid structure, but rather large values. This is the case for a larger elasticity parameter fl = ,,5(I - ~2)a2/E, while for values below,
tlYDROELASTIC 50O
SYSTEM COUPLED
(o)
(b)
I
l
I
I I /-
I l--
400
I
I
30 0
/
SJ
,~,.&., #'
100
W~
..... ----::
20 O
"~
229
FREQUENCIES
..........
#
I
oo S
..
'-:SL2 --- L : - -
-
- -!
OOC t~ 125 C (c)
I ! ,-I
100 C
('J)
I
! ~ I I
J
750 9
WL S 9
50( 25( O0 (
. . . . . 0
02
04
06 k
08
1000
02
=9 ~=",-.7-.: 04
06
08
10
k
Figure 6. C o u p l e d frequencies of liquid and shell for liquid layer a r o u n d an elastic shell (three-dimensional) (elasticity p a r a m e t e r / 3 = ,(3( ! - p 2 ) a 2 / E = 0 . 0 1 , 0 ' I (sZ)). T o / p a ~ = 100 (s-2); ~ / p 2 , f, = 0-3, I / a = 2-0, h / a = O. I, m = 0 . ( a ) / 3 = 0.01, n = I; (b) /3 = 0.01, n = 2 ; (c)/3 = 0 - 1 0 , n = I; ( d ) / 3 =0"10, n = 2 .
say, 10 -7 the coupled frequency of the liquid approaches zero as k approaches unity. Comparison of the two-dimensional case, which has been evaluated numerically more extensively confirms this statement too (see Figures 3(b) and (c)). In Figures 6 the elasticity parameter was chosen to be/3 = 0.01 (s:). If/3 approaches zero, i.e., E -->oo, the structure becomes more stiff and yields for/3 = 0 the case of a rigid cylindrical shell. The lowest frequency is the shell frequency ~ in the radial direction, which experiences, of course, the largest coupling effect, due to the additional liquid mass around the shell. This radial frequency we is considerably reduced from the value of the uncoupled structural frequency in this direction. It shows with increasing k-values, i.e., decreasing layer thickness, a decrease in magnitude. The coupled frequency in the angular direction ~ is denoted by w,7 and shows an increase in magnitude with a decreasing liquid layer thicknesss. The largest coupled structural frequency ~o~ in the axial direction z also increases with increasing k. For higher modes, the coupled liquid frequency wl remains nearly constant over a rather wide range of small and moderate k-values. Only for thinner liquid layers does the liquid frequency increase rapidly, as does the coupled structural frequency o.,r in the radial direction for thick liquid layers. The coupled frequencies of the liquid structure system for decreased stiffness of the shell, i.e., smaller modulus of elasticity, for which /3 is 10 times as large, namely /3 =0.1 (s 2) are presented for the same fixed values of the other parameters in Figures 6(c) and (d). The trends of the coupled frequencies wl, w~, o% and oJ~ are very similar to those for the case/3 = 0.01 (s2), except for the numerical values. The influence of elasticity is rather more pronounced, since the shell is less stiff and this gives larger structural frequencies (0,7 and Lo~.The coupled liquid frequencies o.,~show, for thinner liquid layers, a slightly reduced frequency compared to those of the stiffer shell (/3 = 0.01 (s2)). The coupled radial structural frequency o.,~shows,
H.F.BAUER
230
in comparison with that for the stiffer shell (/3 =0.01 (s2)) a decrease in magnitude, particularly for thicker liquid layers, i.e., smaller k-values. It also may be noticed, that the influence of the stiffness parameter/3 is quite small upon higher modes of the liquid layer. This is, of course, due to the fact that for higher modes the liquid waves penetrate less deeply below the free liquid surface. The important effect found in this investigation of a liquid layer around an elastic shell is that the coupled radial structural frequency w, is drastically reduced and that the coupled liquid frequency is drastically increased for small thicknesses of the liquid layers. For the case (three.dimensional) of a frictionless liquid inside a rigid container the natural frequencies may be obtained from reference [ 1] (see equation (42)). The frequencies are presented in Figures 7(a) and 7(b). The frequencies for the case n = 1 are shown in Figure 7(a), for m = 0, 1, 2, 3 and 4, the aspect ratio l/a = 2 and the tension parameter To/pa 3= 100 (s-2), as functions of the diameter ratio k = b/a. It may be noticed that the axisymmetric mode m = 0 is unstable for k < 2 / r r , as indicated by an imaginary value
/ \
44000
~L~ /
300 0
I
I
2000
%'
,
\
i
.
x
9000
0"0 0 0
~,.
~._
0"2
0.4
' ~
06
0.8
6.0
80
1-0
k = b/a
~ 500
0
(b)
I.t.
400-0
300 0
2000
_~~
100C
O-C O0
20
4-0
10-0
t/a
Figure 7. Uncoupled frequency of frictionless liquid layer inside a circular cylinder (three-dimensional). n = i, To/pa ~= 100 (s-Z). (a) I/a =2; (b) k =0.5. - - - , Real part; - - -, imaginary part.
HYDROELASTIC SYSTEM COUPLED FREQUENCIES
231
(. . . . ) for tOo~. Figure 7(b) shows the natural frequencies to,,, for a fixed diameter ratio k = 0 . 5 as functions o f the aspect ratio i/a. T h e natural frequency decreases rapidly with increasing l/a and exhibits, for m = 0 , instability for 1/a > n,-rk, i.e., I/a > n~r/2, for the d i a m e t e r ratio k = 0.5. The coupled frequencies o f the system o f an a n n u l a r liquid layer inside an elastic cylindrical shell are presented in Figure 8 ( a ) - ( d ) for the elasticity p a r a m e t e r s fl = 0.01 (s 2) and 0.1 (s2), respectively. The other parameters, such as the ratio of the density o f the structure to that o f the liquid fi/p = 2, Poisson's ratio ~ = 0.3, length ratio 1/a = 2, wall thickness ratio h / a = 0.01, and the tension p a r a m e t e r Tdpa 3 = 100 (s-2), are the s a m e as in the previous case. They are presented as functions of the d i a m e t e r ratio k = b/a, which c o r r e s p o n d s to a layer thickness of (a - b). For small k-values the liquid layer is large, while for k close to unity the thickness is small. Again one can notice that the coupled liquid frequency wl is the largest o f the four hydroelastic frequencies for each m o d e (nl, n). The axisymmetric case nl = 0 has been omitted, since the liquid is at a ratio 1/a = 2 already unstable (as m a y be seen from the results for m = 0 in Figure 7). It m a y be seen that the coupled liquid frequency to~ is drastically changing to large magnitudes for thin liquid layers (as has been observed in the previous case). A strong change in the coupled radial structural frequency to~ to smaller values m a y also be noticed especially in the small k-range, i.e., for thick liquid layers, where the inertial effect o f the liquid is p r o n o u n c e d . The numerical results m a y be seen in Figures 8(a) and (b) for some m o d e c o m b i n a t i o n s (m, n). One can notice again that for higher m o d e s the coupled liquid frequencies (o~ show less interaction effects. For a more elastic shell, i.e., less stiff shell, o f f l = 0.1 (s 2) all coupled structural frequencies decrease in magnitude, while the c o u p l e d liquid frequency (o~ shows hardly any visible change. More numerical results m a y be f o u n d in reference [7]. 80 0
~
(o)
1
/
640,
I
1 ~
/ "~%%
32.0
t.gl. ' S IS
~ --
......
\
I I
i!.__
--""
"~ 16.0
~
oJ~ O'O - -
80-0
:111\ %
48"0
o
(b)
160
Ic)
,
lI (d) t
64-0
\
128
Y !
t
I
48-0
~
96
j
0.11.
~-- ~-. . . . . . . . .
.,
64 32
16.0 r
o.o'
i
jI
%
J
32"0
I
oJ
oJ : _
:-
0 0
. . . . . . . . . . . .
0-2
0-4
"7--7-~--:~---:
k
0 6
0'8
'
10
C
0 0
. . . . . . . . . . . . .
0-2
04
k
06
0 8
10
Figure 8. Coupled frequencies of liquid and shell for liquid layers inside an elastic shell (three-dimensional)
fl =~(l-~2)aZ/E = 0.01.0.1. TJpa 3= 100 (s-:). ,6/,o=2-00. ~ =0"3, I/a =2, h/a =0.01, m = I. (a) fl =0.01, n = 1; (b) fl =0-01, n =2; (c) fl =0.10, n = I; (d) fl = 0.10, n =2.
232
H.F. BAUER REFERENCES
1. H. F. BAUER 1982 Acta Astronautica 9, 547-563. Coupled oscillations of a solidly rotating liquid bridge. 2. H. F. BAUER 1981 Forsch. Ing.-Wes. 47, 190-198. Freie Schwingungen nichtmischbarer Fliissigkeiten im rotierenden Kreiszylinder unter Beri]cksichtigung der Oberfl~ichenspannungen. 3. LORD RAYLEIGll 1982 Philosophical Magazine 34, 177-180. On the instability of cylindrical fluid surfaces. 4. H. LAMB 1945 Hydrodynarnics. New York: Dover Publications. See pp. 471-473. 5. J. A. F. PLATEAU Smithsonion Institute Annual Report. 1863, 207-285; 1864, 286-369; 1865, 411-435; 1866, 255-289. Washington: Government Printing Office. Experimental and theoretical researches on the figures of equilibrium of a liquid mass withdrawn from the action of gravity. 6. H. REISMANN and P. S. PAWLIK 1968 Journal of Applied Mechanics 35, 297-305. Plane-strain dynamic response of a cylindrical shell: a comparison study of three different shell theories. 7. H. F. BAUER 1985 Forschungsbericht der Unicersitiit der Bundeswehr Miinchen LRT-WE-9-FB- 12. Coupled frequencies of a frictionless hydroelastic system. APPENDIX: LIST O F SYMBOLS b E h I,,,, K,, k I n! n
P r,~,z S t
T ii, ~, W
4, P
to
radius of container or radius of the free surface of the liquid layer around center shell radius of free liquid surface inside container or radius of inner shell for liquid layer case Young's modulus of elasticity wall thickness of cylindrical shell modified 13essel functions of ruth order and first and second kinds, respectively b/a, diameter ratio (-<1) length in axial direction angular modal number axial modal number liquid pressure cylindrical co-ordinates = o-+ i~, complex frequency time free liquid surface tension liquid velocity component elasticity parameters liquid velocity potential liquid density shell density Poisson ratio coupled frequency and abbreviation &2 .~ Tm(m 2_ 1)/pa 3 shell displacement in radial, angular and axial directions, respectively
COUPLED FREQUENCIES OF A HYDROELASTIC SYSTEM, CONSISTING OF AN ELASTIC SHELL AND FRICTIONLESS LIQUID H. F. BAUER lnstitut flir Raun~ahrttechnik, Unit,ersitiit der Bundeswehr, D-8041 Neubiberg, German), ( Receired 30 December 1985)
For a frictionless liquid layer around a cylindrical shell and a frictionless liquid inside a shell in a zero-gravity condition the hydroelastic interaction of the shell- and liquidmotion is treated. The coupled frequencies for the two-dimensional and three-dimensional cases are presented for various elasticity parameters and liquid layer thicknesses. There are strong interactions in both cases. I. INTRODUCTION The availability of extended manned space flights in an earth-orbiting space laboratory makes unique experiments possible. Such experiments cannot be performed under the action of earth gravity, since a liquid with a free surface will exhibit completely ditterent equilibrium positions. One of these experiments is the behavior of an annular liquid column outside or inside a cylindrical shell, where the liquid layer under the action of gravity would bulge out to a geometric configuration not representing the annular layer form, as it does under zero-gravity, where the liquid is just held together by surface tension. It offers, due to the lack of gravity, a cylindrical liquid form and a number of advantages for exploratory experiments. The natural frequencies of such a liquid layer, thick or extremely thin, should be known, in order to avoid during the experiments certain frequency ranges appearing in a space laboratory known as the so-called g-jitter. Such vibrations of the laboratory appear during the operation o f machines on board, motion of the crew or control maneuvers, etc. For a rigid center shell surrounded by a frictionless liquid layer or a rigid cylindrical container partially filled with a frictionless liquid the natural frequencies have been determined [1, 2]. The condition of a rigid shell, however, may be violated in such a way that the shell-structure may be elastic and perform vibrations. Such vibrations would also affect the motion of the free liquid surface and couple with it, in which case the shell frequencies as well as the free liquid surface frequencies would in the coupled form shift to other magnitudes. Of interest, therefore, is the behavior of the coupled liquid-structure system as a function of the thickness of the liquid layer, its surface tension, the wall thickness, the density ratio of structure and liquid and the elasticity o f the shell. Lord Rayleigh [3] treated the three-dimensional case of a cylindrical air column in an infinite medium and found that the axisymmetric cylindrical configuration becomes unstable and exhibits its most pronounced instability at an axial wave length A.. = 12.96 a. Lamb [4] performed similar investigations, in which mainly the simple liquid column was treated. In addition the remarkable work of Plateau [5], who performed a large number ofuseful simulation experiments, should be mentioned. No work, however, has been performed yet upon the interaction of an elastic structure with a liquid in a zero-gravity environment. In what follows an analysis is presented for 217 0022-460X/87/050217+ 16 $03.00/0 (~) 1987 Academic Press Inc. (London) Limited
218
mr.
F. BAUER
the determination of the coupled frequencies of a two-dimensional and three-dimensional liquid-structure-system; i.e., a system exhibiting no z-dependency and with z-dependence respectively. The main interest here is in the behavior of the coupled liquid-shell system, and the influences of the tension parameter T / p a 3, the thickness of the liquid layer (k = b / a is the diameter ratio of the shell and the liquid surface location), the density ratio of shell and liquid fi/p, the thickness of the shell h and the elasticity parameter f i ( I - f i 2 ) a 2 / E are investigated (a list of symbols is given in the Appendix).
2. BASIC E Q U A T I O N S
Under the influence of an elastic structure a liquid in a micro-gravity environment will execute oscillations, if disturbed (see Figure l). The problem to b~ solved is the determination of the coupled frequencies of the elastic structure and the liquid with a free surface. The liquid is considered frictionless and incompressible. For small liquid velocity components and small elastic deflections the governing equations may be linearized. With the liquid motion assumed to be irrotational the continuity equation leads due to the representation of the velocity as a gradient of a velocity potential tb, i.e., v = g r a d ~ b,
(1)
V2@=O,
(2)
to the Laplace equation
which has to be solved with the appropriate boundary conditions. If the free surface is at r = a outside an elastic inner cylinder at r = b, b < a, the free surface conditions are u = a~o/Ot = O@/ar
at r = a,
the kinematic condition,
(3)
and
-p a@lat+(Tola2)[~o+a2~o/a~2+ a" a2~'o/aZ"-] = 0 at r = a,
(4)
this being the dynamic surface condition. Combining equations (3) and (4) yields after differentiating equation (4) with respect to time, -a2,1'/at 2 + ( To/pa")[ u + ,'~:u/O~o2 + a: F u / a z 2] = 0
Freehquidsurface ~
T
o
at r = a.
(5)
TosurfacetensJon ....
tens,onSUrface ^
/ "r/.~ (
FreeI,quid E~ShC shell
Figure I. Geometry and co.ordinates of system.
IIYDROELASTIC
SYSTEM
COUPLED
219
FREQUENCIES
The normal elastic deflection of the structure is denoted by ~(~, z) in the r-direction, and "0(~, z) and ~'(~,, z) in the ~- and z-directions. The boundary condition at the inner cylindrical structure is ,~/at = ~/ar
at r = b.
(6)
A cylindrical shell of length I has to satisfy the shell equations I ~b ~.~..~+at/+ h2 -4 _ (I - ~2). 2 a2~: t]tP b2 az a~p ' ~ + i - ~ v ~ : + P - - f f - - b ~ - ~ = P - ~ - , o b ~ atr=b,
~(l+O)b ~2~r+0~+I(I a z O~ b2~2~+1
0r
2
(7)
21}2"0+'~2r/ (I-~2) 2a2r/- ~') b -~z2 -~-~2 - fi - - E
~2~.. ! . . . . ~2~7 l,)O,~za ~
+
(8)
-- b Ot2 - 0 '
~b~: _(1-~2).2~2~ ^ -~z-pTo ~t2=u
atr
=
b.
(9)
These are the DonnelI-Mushtari equations, which easily could be changed to other circular cylindrical shell theories by employing additional terms multiplied with the small value h2/12b 2. Here the operator V2 = b 2 ~2/~z2 + +-~2/;~2. For a cylindrical shell of infinite length without axial ( z - ) dependency the equations of motion of the elastic shell yield, upon using the concept of plane strain, the equations h2 [ ,~2~ ,.}4~.-]+~:+~__~qtS(l_~2)b2~2,~ ~b 2 12b 2 2 o~ ~ ~ t 2 - P - ~ t - "~ a t r = b , (10) ~'~+ t32r/ fi(1-p2)b2a2"r/-0 9}~ ~ 2 E r]t2
atr=b.
(11)
In these equations h is the thickness of the shell, P the Poisson ratio, E the elasticity modulus and /~ = Ell/(l - p2), while t5 is the density of the shell material and p that of the liquid. As may be noticed, these equations are coupled with the liquid motion by the term p ~ r and the right-hand sides ofequations (7) and (10) as well as by the boundary conditions (6). If the liquid is in an elastic container r = a and has its free surface at r = b, then the free surface condition reads
a2el2_,gt~-~[ "F+ u + b 2~2u
~2u1
atr=b
(12)
while equation (6) has to be satisfied at r = a. In the shell equations b has to be replaced by a and the right-hand sides of equations (7) and (10) show a minus sign ( - ( p a 2 / D)OdP/Ot). 3. METHOD OF SOLUTION To determine the frequency equation for the coupled frequencies of the systems comprising the elastic structure and frictionless liquid equations (2), (5), (7), (8), (9) for the three-dimensional case, or equations (2), (5), (10) and (1 !) (with ~ / ~ z - 0 ) for the two-dimensional case, have to be solved with the appropriate boundary conditions. 3.1.
FRICTIONLESS
LIQUID
DEPENDENCY
(TWO.DIMENSIONAL)
AROUND
ELASTIC CENTER
SHELL
WITIIOUT
AXIAL
In this case equation (2) yields the velocity potential q 3 ( r , % t ) = ~ [ A , , r " + B m r - " ] e i"'~+''. rn=l
(13)
220
H.F. BALIER
With the boundary condition (5) one obtains, with ~/az = 0,
[s2+(Tom/pa3)(m2-1)]A,~a"+[s2-(Tom/pa3)(m2-1)]B,o/a"=O,
(14)
and with ~(~p, t) = X,, e imr247
7/(~, t) = Y,,,e i'r §
(15)
from the shell equations (10) and (11) the expressions
{(h2/12b2)[m'-2m2+ 1]+ 1 +,5(1 - ~Z)b2s2/E}aX,,k+im Y,,ak = (pb3s/D)[Ab"'+ Bb-m],
(16)
im Xma -{m2 + iS(1 - ~'2)b2s2/E} Yma = 0.
(17)
The boundary condition ensuring that the motion of the elastic wall is equal to that of the liquid at the wall (equation (6)) gives the expression
sX,. = mA,,b "-I - mB.,/ b "+l.
(18)
Equations (14)-(18) are four homogeneous algebraic equations, and hence the vanishing of the coefficient determinant provides the frequency equation for the determination of the coupled frequencies:
II~,,ll =0,
j,!=1,2,3,4.
(19)
The elements ~Sslare given in terms of k = b/a, ~2 = ( Tom/pa3)(m 2 _ 1),/3 = ,5(! - V2)b~/E, and a = D/pb 3 by (see Figures 3 and 4 of section 4)
,5,2=s2-d, 2,
~5,1 = s2+ a32,
823=-ot{(m2-1)2(h2/12a2k2)+l+fls2}k, 833 = ira,
,521=sk",
813= ~ 4 = 0 ,
834 = -{m2 + fls2}, c543= -sk,
822=s/k",
824 = - i m a k ,
841
=
mkra,
831=832=0,
~42 =
--m/kin,
c544= O.
(20)
For an elastic cylinder without a liquid layer, one has only the elements ~23, ~524, ~33 and $34, and hence obtains the frequencies given by Reismann and Pawlik [6]; i.e. (see Figures 3 and 4 of section 4), S2 =
-
~
2,5(!
_E t32)b2/n ! [ z + I q- i~(I/'1 112 - 2
- -
1) 2
(m2 + 1)2- 2 ( m 2 - 1)3 .--T~,z + ~ ( m 2 12b 144b
1)4
"
(21)
If the structure is rigid, ~ and ~7 vanish and one has only the elements 8tl, ~t2, ~21 and ~22. The determinant results then in the frequency equation for a frietionless liquid: i.e. [1,2] (see Figure 2 of section 4), s 2 = - Tom(m 2 - 1)( 1 - k2")/pa3( i + k s'').
(22)
HYDROELASTIC SYSTEM COUPLED FREQUENCIES
221
The complete determinant (19) gives a bicubic equation of the form with X ~ (h~/12a2)(m 2- 1)2,
s6fl2[ k( l - k2") + m P- h ( l + k2") } pa + s4fl{ (l + k2")[kfl~2 + mfi h (l + m2 + x)] +(l -k2")m[kn, + fltb2 ~ h ] l +s2 (! +k2m) flk ~52+m~
p
X m~+(l-k2"')m
I+m~+X)
pa
+ nt2c52 -P h x ( 1 -/,:2,,,) = 0.
(23)
pa
3.2. FRICTtONLESS LIQUID INSIDE ELASTICCONTAINER WITHOUT AXIAL DEPENDENCY (TWO-mMENSIONAL) For a frictionless liquid inside an elastic cylinder of radius a and a free liquid surface at r = b one obtains, with the boundary condition (12), the shell equations (10) and (1 I) at r=a and the wall condition (6) at r=a with equations (13) and (15), the equations
km[s 2- (T,m/pa3k3)(m 2- I)]Ama '~ + ( I / k m ) [ s 2 + (T,m/0a3k3)(m 2- I)] B,,/a m = 0, (24) (pa2/ O)s[ Ama m+ (Bin~am)] +{(h2/12a2)(m4-2m2+ l ) + 1 + ~ ( I - r,2)a2s2/E}X,~ +ira Y,~ = 0, (25) imX,,-[m2+~(l-P2)a2s2/E]Y,~=O, mA,,a"-m(B,,/a")-saX,,=O. (26,27) The equations (24)-(27) are four homogeneous algebraic equations, ofwhich the vanishing determinant is the frequency equation of the coupled frequencies. It is Here
II, ,ll =0, j, I = 1,2,3,4. the elements are, with a52 -~ T,m(m 2- 1)/pa~k 3, 6 =- l)/pa 3, and/3 -= ,6( ! - f,2)a2/E, g,, = [ s 2 - ~ 2 ] k '' , g.=(1/k")[s2+&2], g,3 = g,., = 0, gn=s=g22, = a{i + = in,a, = =0.
g~3 = ira,
g3+ = - [ m 2+/3s2],
g41 = m,
g.~2= - m ,
g43 = -s,
g.~4= 0.
(28)
Without a liquid layer equation (21) is again obtained, if b is replaced by a. For a rigid structure, one has only the elements g,,, 6,2, 62, and g22, which yield from the determinant [ 1, 2] (see Figure 5 of section 4)
s 2 = - T,m(m 2- 1)( 1 - k2")/pa3k3( 1 + k2").
(29)
The determinant with the elements (28) gives the bicubic equation (see Figures 5 and 6 of section 4)
s6fl2{ ( l - k2") + m fi- h ( l + k2") } pa + s4fi{ (l + k2"')[fit~2 + m~ h (l + ,~ + m2)] +(1-k~")n,[m + flt~2 P- h]} pa p L
pa
+ m3a3t ~ h)?(I - k t'') = O,
in which
pa ,~ -= (h2/12a2)(m 2-
I) 2.
.j
pa (30)
222
H.F. BAUER
3.3. F R I C T I O N L E S S L I Q U I D A R O U N D E L A S T I C I N F I N I T E C Y L I N D R I C A L S I I E L L When an elastic infinitely long cylindrical shell is considered the liquid satisfies the Laplace equation (2) and is represented by the potential
/nrrr\ 9(,,~,z)_._,.oo~ L """i-7-)+~~
/nrrr\']
rA
_ ~
....'"
(~')
Satisfying the remaining boundary conditions at r = a (i.e., equations (5) and (6)) and at r = b (i.e., equations (7), (8) and (9)), with the elastic deflections
n=l m-O
n-I
m=O
(32) n=l m=O
yields the following system of homogeneous algebraic equations:
AL~ xA,t-7-d)+p~,(l/o)(m=~ (,/~)="'== , .7. ,.,,,.,, nrr,/nrr\ A T l,,,~/--~a) +
/nrrk) apsl,,~7-~a
(,/~ ')~~
o,
nz" ,/nrr \
BTK,,t~ak ) -sX h2 [
(34)
=0,
nrr XD +BpsK'.(~7-~k ) --gr{' +]~-~[m
2 n2"n'2b2"]2
+ ;-~--J
+
fi(l-~2)b2s 2}
imD Y- imnrrb/ D\ Z , ~y)=o. imX-Y 7
(33)
(3s)
{"" m2+~(1-
~)4 ~ ( l -
X-
)b2s~ ------~-(1 mnrrb fn=~r2b2
+
+ ~)Z = 0,
(36)
- E--z)b2s2} = 0.
(37)
The coupled frequencies are obtained from the vanishing coefficient determinant II~,~.ll =o, where the elements a,~. are given by .
In,<+gro~
n. (.,'+
a,,=s I,,t'~-d} \pa'] (i/a)\ .
~ ~
/,,=\+(ro~
n,~ /
,
n'~ ~
\
"'#
\ , I,l<
(I/a) 2 l)l"klT-a )' ,/,,=~
K4,,7-a) ~ , ' / ( , - ~ t ''+(,/.)' ' ) ~ ' k ~ ) ' n~ , I . n ~ \
~,,=-
1+ ~
~
"~ K' [ k ,I,*~
~ ,-Ta/'
n,'+ (t-i~o)~-] ~
o.,=~176
~"=-~'
~(1- r
~,,=o.=o,
HYDROELASTIC SYSTEM COUPLED FREQUENCIES
C~'41 ~ ~'42 ~
/~34
im( h / a )(~ / p ) /5(1 - ~,2)b2/E'
0,
a4, = in/,
ct35 =
i,nn~rk( h / a )(~ / p ) (I/a)~(l - ~2)b2/E'
[ 2 H2n2k2 9 /5(1 - p2)b2s2" ~ o~44 = - - t m + ~ ( 1 - ~)-t~J'
mn~rk
a45=
223
2(I/a) (1+~')"
inrtr, k aSI -----a'52---- 0'
l/a'
a53
~fn2~2k 2
nnl,'rk a,,=---2(i/a)(l+~),
a 5 5 = - t ~ * : ( l - P 2 ) bm: s2: + ~
} "-~-'(1-~ 2) .
(38)
The determinant gives the coupled frequencies for the modes (m, n) as functions of the various parameters f i ( l - ~,2)b2/ E, To/pa 3, h/a, l/a, k = b/a, and fi/p (see Figure 8 of section 4). For a shell with liquid the lower-right determinant of third order gives the frequency equation i.e.: 118,~11=0, i, K =3, 4, 5. This yields the hi-cubic equation with, ), =/~(1 - ~3)b2/E, 2 4[m2,~ 2 ( !!1
n'77'
4
4
I| 77
I12 1
4
2
111
2/j2
n%r 2 \ : 1 ~T
2
11111277 2 -
+TS t-~-(l-P)+~(l-P)2--m+~(I--P)
+rl+
h2 I n 2+ 112772\2-1rnl2
(lib)2u
1121"1"2
kl_
+":"
~ ~
.,,+ +q
,
=0
(39) For a rigid shell one obtains the natural frequency of the liquid from the upper-left determinant of second order. It is [1] (see Figure 7 of section 4)
r, I n77\ , I k ,m-\ s2
Tomr [ ,
= ,~t'"
,i%.:
, I k n=\ , I n = \ ]
.'~[l"t~-~)K"~,77~)-I"[,)7~)K"W~;J
+ (~1,~)2 'Jr
,' 11,.,.\ , / k 1"-" L''t,l-7-~)K' q"~-d )
: '1"'\ '~ t k"-l: ~
(40)
K"[l-7~)J/"=\]"
3.4. FRICTIONI.ESS L I Q U I D INSIDE AN INFINIJ'E ELASTIC C O N T A I N E R
In this case a must be replaced by b, and To by -Ti, while in equation (10) the sign on the right-hand side is altered. This yields a determinant Ilflm.II =o, of which the elements are (see Figure 8)
/,,~ k)~_[\pa3 T,~] k2(I/a)\nI24 ,i,, C --"':: fl,,=s l..tl~a (i/a)2
f12, : ~ 7 ~ l , . \ l / a ] ,
fl,2 =/-7-daK,-t/-7-~a) ,
l)~o,~kG),'": ' ~
f12.,= -s,
f12+= 0,
224
H.F.
~3,,=sire (mr)/Ta'
/3,,=0,
(h/a)(/5/p)
[nrr' /332= sK,,~)7-a),
/5(1-02)a 2 2
[
im( h/ a)(/5/ p ) /334= tS(1 - ~2)a2/E' /3., = fl42 = 0.
BAUER
!12 [ 2
inrrO( h/ a )(/5/ p) /355 = (i/a)/5(l - ~2)a2/E'
f n2"n"2 fl4a=-/~(1-
/34s=im.
~)+ m2-1
mmr(1 + ~) /345 =
2(l/a)
n27r2 ]2"1
/5(1-: ff2)a2 2] ~s
/'
inTrf' '
/35~ =/352 = O,
/353 = ( t / a ) '
[" n2~'2 /5(1--/52)a2 2InI2""7 ] /355 = - L ( / - - ~ 4 ~ s -e-~-tl ~)_.
/354 = - (i/a)mnrr(1 + 9),
(41)
From this equation the coupled frequencies s may be determined for the modes (m, n) and for the various parameters k = b/a, l/a,/5/p, h/a,/5(1 - 92)a2/E and Tdpa 3. It may be noticed that these results also satisfy the problem of a finite shell of length l with a freely supported or shear diaphragm end condition at z = 0 and /. This means that a circular cylindrical shell of infinite length vibrating in a mode of which the half-length of the wave in the axial direction is I has the same frequency as a shell of that finite length I with these boundary conditions. If at z = 0, ! a bottom and a top were included, the above solution for the liquid could not satisfy the condition of vanishing normal velocity. For a rigid container the natural frequencies of the liquid layer inside the container may be obtained from the upper-left determinant of second order. It yields, with s = ito,,,, the expression 2
nz_kk
to...-(1/a ~ [
r,
n~\
1
-m 2
n2rr2k2~[ , [nTrk'~ , [ nTr'~ , [ nTr\ ,2/a 2 / / i . k ~ ) ~ k ~ ) - , ~ k ~ ) ~ k ~ ) j /
, [n~-\
I' / m r \
-
[nz-
\'1
J
, [nrrk\] /
(42)
which is represented in Figure 7 of section 4.
4. NUMERICAL EVALUATIONS AND CONCLUSIONS The hydroelastic (coupled) frequencies have been evaluated, in the first instance, for various stiffness parameters /3 (/3) and given thickness ratio h/a=O.O1, density ratio /5/p = 2 (shell density to |iquid density), and modes number m - - 2 as a function of the diameter ratio k = b/a. Small k-values represent an elastic shell v)ith a thick liquid layer around it, while the magnitude k ~ l (<1) represents a very thin liquid layer. The uncoupled frequencies are given as ( . . . . ) curves, where the value s~ denotes the uncoupled liquid layer frequency and s# and s o denote the uncoupled shell frequencies in the radial (~:)- and angular (r/) directions. The mode m = 2 describes an elliptic free surface shape during the oscillation: i.e., a q-dependency of cos 2~. In Figure 2 the uncoupled liquid frequency to2 is given for the tension parameter To/pa s = 100 (s-:) and /3 = 0 (i.e., E --~oo, a rigid shell) as a function of the diameter ratio of shell toundisturbed D
H Y D R O E L A S T I C SYSTEM C O U P L E D F R E Q U E N C I E S
225
2,50 . . . .
~ .
o
\.
\
20 0
"\
\ kSt
\
150
100
\
\
\ \
50
O0 O0
,
I
02
,
I
,
04
I
06
,
I
08
10
k = blo
Figure2. Uncoupled frequency of frictionless liquid around rigid center shell (equation (22)) (twodimensional), m = 2, To/pa ~= 100 (s-~),/3 = 0 (s2).
liquid surface k = b/a. The natural frequency of the liquid in the mode m = 2 decreases with decreasing liquid thickness. 'In Figures 3(a)-(c) the coupled and uncoupled liquid and shell frequencies are presented for the mode m = 2, the tension parameter T o / p a 3 = 1 0 0 ( s - 2 ) , the density ratio ~/p = 2, the shell thickness ratio h/a = 0.01 and various stiffness parameters/3 = (,6(1 - ~ ) a ~ / E ) (s:). The uncoupled frequencies are given by ( - . - ) curves, and the coupled frequencies by (- - -) curves. Figure 3(a) shows the result for a very elastic shell, for which/3 = 10 -3 (s2). It can be seen that the coupled liquid frequency is strongly influenced by the coupling with the shell. This is more pronounced for thin liquid layers, as the coupled liquid frequency increases strongly with increasing k = b/a. The uncoupled shell frequencies s~ and s o are also presented. While %, the natural frequency of the shell in the angular direction is of large magnitude, the uncoupled frequency (sr in the radial direction has a small value for this particular case. The coupled structural frequencies sr and % show in the radial direction (s~) a further decrease in magnitude, with the difference between the uncoupled and coupled values decreasing with decreasing liquid layer thickness. The coupled structural frequency % in the angular direction is on the whole smaller than the uncoupled one. For a further increased stiffness, say/3 = 10 -6 (s 7') the results are shown in Figure 3(b), where the lower frequency range is presented. A further increase of stiffness, i.e., an increase of the modulus of elasticity E, has the results shown in Figure 3(c), where only the liquid and lower structural frequencies are shown. This is a very stiff shell, for which the coupled liquid frequency is hardly affected by the motion of the shell. The liquid, however, through its mass, affects the structural frequency s~ in the radial direction. It can be seen that the coupled structural frequency s~ approaches the uncoupled value for vanishing liquid layer thickness.
226
H.F.
BAUER
8oo[
80 0
I
/ "/--
$I
!(~)
64 0
640]-
//
"- 4 8 0
/
/
/
/
I
I I I
iL
st// u.
32"0
,/
/
320
=-:'-== ~
,
02
~
~ 04
~
l
//
\
/z-s, \
I
/
O6
~ . . . . . -~..~.
16 olN.
O0
/
,~----~'-"~..s L
O0
I I I I
i
\.
"
08
00I O0
0
/
~
\ 02
04
0.6
08
10
k = b/a
300 0
t
(c)
\
I
240 0
\
\
\\
"\
,? ,/
\
1800
/ \
\
\ \
\ /
u: 1 2 0 0
/
/
/
I I
60'0
,~_SL
O0 0"0
~
I
0"2
i
I
0 4
i
I
06
~
I
0 8
~ 10
k = b/o
F i g u r e 3 . U n c o u p l e d a n d c o u p l e d frequencies of liquid layer a r o u n d elastic shell for m o d e m = 2 (Iwod i m e n s i o n a l . T, J p a ~ = 100 (s-"), ~ / p = 2, h / a = 0.01. (a) fl = l 0 -3 (s2); (b) fl = l 0 -6 (s"); (c) fl = l 0 -9 (s2). - - -, U n c o u p l c d ; - - -, c o u p l e d .
I I Y I ) R O E L A S T I C SYSTEM C O U P L E D
227
FREQUENCIES
For a frictionless liquid inside an elastic container (without axial dependence) the results for the coupled and uncoupled frequencies are given in Figure 4. Figures 4(a) and (b) are for cases of the mode m = 2 with the tension parameter T,/pa 3 = 100 (s-2), the density ratio ~/p = 2, the thickness ratio h/a = 0.01, and various stiffness parameters ~ = ~ ( l - f J ) a 2 / E . For a very elastic shell / 3 = 1 0 -3 (s 2) the coupled ( - - - ) and the uncoupled ( . . . . ) frequencies are shown in Figure 4(a). The uncoupled liquid frequency sl ( . . . . . ) (equation (29)) shows a decreasing magnitude with decreasing liquid layer thickness. The coupled liquid frequency decreases first for increasing k-values and then increases again for smaller liquid layer thickness. The coupled radial shell frequency is very small (se=0.1) and not indicated in this figure. The structural frequency s, in the angular direction first remains nearly constant for decreasing liquid layer and then decreases, through the influence of the oscillating liquid surface, to smaller values, and finally increases again for thin liquid layers. For a further increase of the stiffness fl = 10 - 6 (5 - 2 ) the results are shown in Figure 4(b). For the case of an infinitely long elastic shell (three-dimensional case) the natural frequencies of the frictionless liquid are presented in Figure 5 for various modes (m, n = 1) for the tension parameter To/pa 3= 100 (s -2) and the length/radius ratio 1/a = 2 as a function o f t h e diameter ratio k = b/a. These are the frequencies for a liquid layer around a rigid central cylinder. The case n = 1 is shown in Figure 5(a) for the axisymmetric case m = 0 and for m = 1, 2, 3 and 4. With increasing k-value, i.e., decreasing thickness of the layer, the frequency decreases towards zero. Similar results may be obtained for n = 2, etc. The uncoupled liquid frequencies are also shown in Figure 5(b) but as a function of the length ratio I/a for the tension parameter To/pa 3 = 100 (s-2), diameter ratio k = b/a = and n = 1. It may be noticed that with increasing length ratio the natural frequencies decrease, with a rapid decrease within the region 0 < l / a < 2. The axisymmetric motion m = 0 becomes unstable for l / a -- nrr and for l / a > n,'r the root has a real part. iO0 0
250 0
(a)
(b)
~!~
80
/"
150 0
" / I
!
\ \
400
l
L / \ \ . \
g
l l l
200 0
\\
600
l
\
100 0
\s,
I
x
/
\ k.
\
k.
20 0
\.
\
, "~
O0 O0
\ ; ..~. '-1 . . . . 02
s{=O.1
- 4 - - - ~--..-~ . . . . . 04
06
s(
L_ . . . . . 08
" \ . "~"~...~. ~ ~ . . . J st
50 0
x.
/
0
0 OL-. O0
02
04
06
08
k = b/n
Figure4. Uncoupled and coupled frequencies of liquid in an elastic container for m o d e m = 2 (twodimensional. Parameter values and key as Figure 3 except ( a ) / 3 = 10 -3 (s2); (b) ~ = I0 -t' (s:).
228
H.F. BAUER
100 0
I
m
J (a)
80 0
?\
\
\
600
]
400
I[Z]
200
1--
I I I
I
O0 O0
04
02
-J \
i ~ ~=/ ~-..,,. 3"'--~. 12 J~
2L~t_ /~
O6
X\
\
\4i
F-"--'-.-' 08
k= bla
150 0 (b)
120 0
90 0
i\'x,~ m = 4
600
30 0
I \'.., O0 O0
--?--i
"
I 20
~"
40
60
80
100
L/a
F i g u r e 5. U n c o u p l e d f r e q u e n c y o f f r i c t i o n l e s s l i q u i d a r o u n d rigid c e n t e r shell ( t h r e e - d i m e n s i o n a l ) ( e q u a t i o n (40)). To/pa 3 = 100 (s-2), n = I. (a) I/a = 2; (b) k = 0.5. , Real part; - - -, imaginary part.
The coupled frequencies of the system of an elastic shell surrounded by an inviscid liquid layer are presented in Figures 6(a) and (b) for tile elasticity parameter fl = ~(1 - ~ Z ) a 2 / E =0.01 and in Figures 6(c) and (d) for 0.1 (s2), the density ratio of shell to liquid density #/p = 2, length ratio l/a = 2, Poisson ratio ~ = 0.3 and wall thickness ratio h/a =0.01, as a function of the diameter ratio k =b/a. The tension parameter To/pa 3 was chosen to be 100 (s-~). Three coupled structural and one coupled liquid frequencies are obtained for each mode (m,,t). It may be noticed that the largest frequency (in comparison with the results given for the uncoupled liquid frequencies in Figures 6) is the coupled liquid frequency (m = 0, n = 1) (Figure 6(a)). This frequency is least coupled with the structure for small k, i.e., when the liquid layer thickness is relatively large. This coupled liquid frequency is marked by o~. For a thin liquid layer the coupled liquid frequeficy ~o, does not have small values as for a rigid structure, but rather large values. This is the case for a larger elasticity parameter fl = ,,5(I - ~2)a2/E, while for values below,
tlYDROELASTIC 50O
SYSTEM COUPLED
(o)
(b)
I
l
I
I I /-
I l--
400
I
I
30 0
/
SJ
,~,.&., #'
100
W~
..... ----::
20 O
"~
229
FREQUENCIES
..........
#
I
oo S
..
'-:SL2 --- L : - -
-
- -!
OOC t~ 125 C (c)
I ! ,-I
100 C
('J)
I
! ~ I I
J
750 9
WL S 9
50( 25( O0 (
. . . . . 0
02
04
06 k
08
1000
02
=9 ~=",-.7-.: 04
06
08
10
k
Figure 6. C o u p l e d frequencies of liquid and shell for liquid layer a r o u n d an elastic shell (three-dimensional) (elasticity p a r a m e t e r / 3 = ,(3( ! - p 2 ) a 2 / E = 0 . 0 1 , 0 ' I (sZ)). T o / p a ~ = 100 (s-2); ~ / p 2 , f, = 0-3, I / a = 2-0, h / a = O. I, m = 0 . ( a ) / 3 = 0.01, n = I; (b) /3 = 0.01, n = 2 ; (c)/3 = 0 - 1 0 , n = I; ( d ) / 3 =0"10, n = 2 .
say, 10 -7 the coupled frequency of the liquid approaches zero as k approaches unity. Comparison of the two-dimensional case, which has been evaluated numerically more extensively confirms this statement too (see Figures 3(b) and (c)). In Figures 6 the elasticity parameter was chosen to be/3 = 0.01 (s:). If/3 approaches zero, i.e., E -->oo, the structure becomes more stiff and yields for/3 = 0 the case of a rigid cylindrical shell. The lowest frequency is the shell frequency ~ in the radial direction, which experiences, of course, the largest coupling effect, due to the additional liquid mass around the shell. This radial frequency we is considerably reduced from the value of the uncoupled structural frequency in this direction. It shows with increasing k-values, i.e., decreasing layer thickness, a decrease in magnitude. The coupled frequency in the angular direction ~ is denoted by w,7 and shows an increase in magnitude with a decreasing liquid layer thicknesss. The largest coupled structural frequency ~o~ in the axial direction z also increases with increasing k. For higher modes, the coupled liquid frequency wl remains nearly constant over a rather wide range of small and moderate k-values. Only for thinner liquid layers does the liquid frequency increase rapidly, as does the coupled structural frequency o.,r in the radial direction for thick liquid layers. The coupled frequencies of the liquid structure system for decreased stiffness of the shell, i.e., smaller modulus of elasticity, for which /3 is 10 times as large, namely /3 =0.1 (s 2) are presented for the same fixed values of the other parameters in Figures 6(c) and (d). The trends of the coupled frequencies wl, w~, o% and oJ~ are very similar to those for the case/3 = 0.01 (s2), except for the numerical values. The influence of elasticity is rather more pronounced, since the shell is less stiff and this gives larger structural frequencies (0,7 and Lo~.The coupled liquid frequencies o.,~show, for thinner liquid layers, a slightly reduced frequency compared to those of the stiffer shell (/3 = 0.01 (s2)). The coupled radial structural frequency o.,~shows,
H.F.BAUER
230
in comparison with that for the stiffer shell (/3 =0.01 (s2)) a decrease in magnitude, particularly for thicker liquid layers, i.e., smaller k-values. It also may be noticed, that the influence of the stiffness parameter/3 is quite small upon higher modes of the liquid layer. This is, of course, due to the fact that for higher modes the liquid waves penetrate less deeply below the free liquid surface. The important effect found in this investigation of a liquid layer around an elastic shell is that the coupled radial structural frequency w, is drastically reduced and that the coupled liquid frequency is drastically increased for small thicknesses of the liquid layers. For the case (three.dimensional) of a frictionless liquid inside a rigid container the natural frequencies may be obtained from reference [ 1] (see equation (42)). The frequencies are presented in Figures 7(a) and 7(b). The frequencies for the case n = 1 are shown in Figure 7(a), for m = 0, 1, 2, 3 and 4, the aspect ratio l/a = 2 and the tension parameter To/pa 3= 100 (s-2), as functions of the diameter ratio k = b/a. It may be noticed that the axisymmetric mode m = 0 is unstable for k < 2 / r r , as indicated by an imaginary value
/ \
44000
~L~ /
300 0
I
I
2000
%'
,
\
i
.
x
9000
0"0 0 0
~,.
~._
0"2
0.4
' ~
06
0.8
6.0
80
1-0
k = b/a
~ 500
0
(b)
I.t.
400-0
300 0
2000
_~~
100C
O-C O0
20
4-0
10-0
t/a
Figure 7. Uncoupled frequency of frictionless liquid layer inside a circular cylinder (three-dimensional). n = i, To/pa ~= 100 (s-Z). (a) I/a =2; (b) k =0.5. - - - , Real part; - - -, imaginary part.
HYDROELASTIC SYSTEM COUPLED FREQUENCIES
231
(. . . . ) for tOo~. Figure 7(b) shows the natural frequencies to,,, for a fixed diameter ratio k = 0 . 5 as functions o f the aspect ratio i/a. T h e natural frequency decreases rapidly with increasing l/a and exhibits, for m = 0 , instability for 1/a > n,-rk, i.e., I/a > n~r/2, for the d i a m e t e r ratio k = 0.5. The coupled frequencies o f the system o f an a n n u l a r liquid layer inside an elastic cylindrical shell are presented in Figure 8 ( a ) - ( d ) for the elasticity p a r a m e t e r s fl = 0.01 (s 2) and 0.1 (s2), respectively. The other parameters, such as the ratio of the density o f the structure to that o f the liquid fi/p = 2, Poisson's ratio ~ = 0.3, length ratio 1/a = 2, wall thickness ratio h / a = 0.01, and the tension p a r a m e t e r Tdpa 3 = 100 (s-2), are the s a m e as in the previous case. They are presented as functions of the d i a m e t e r ratio k = b/a, which c o r r e s p o n d s to a layer thickness of (a - b). For small k-values the liquid layer is large, while for k close to unity the thickness is small. Again one can notice that the coupled liquid frequency wl is the largest o f the four hydroelastic frequencies for each m o d e (nl, n). The axisymmetric case nl = 0 has been omitted, since the liquid is at a ratio 1/a = 2 already unstable (as m a y be seen from the results for m = 0 in Figure 7). It m a y be seen that the coupled liquid frequency to~ is drastically changing to large magnitudes for thin liquid layers (as has been observed in the previous case). A strong change in the coupled radial structural frequency to~ to smaller values m a y also be noticed especially in the small k-range, i.e., for thick liquid layers, where the inertial effect o f the liquid is p r o n o u n c e d . The numerical results m a y be seen in Figures 8(a) and (b) for some m o d e c o m b i n a t i o n s (m, n). One can notice again that for higher m o d e s the coupled liquid frequencies (o~ show less interaction effects. For a more elastic shell, i.e., less stiff shell, o f f l = 0.1 (s 2) all coupled structural frequencies decrease in magnitude, while the c o u p l e d liquid frequency (o~ shows hardly any visible change. More numerical results m a y be f o u n d in reference [7]. 80 0
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Figure 8. Coupled frequencies of liquid and shell for liquid layers inside an elastic shell (three-dimensional)
fl =~(l-~2)aZ/E = 0.01.0.1. TJpa 3= 100 (s-:). ,6/,o=2-00. ~ =0"3, I/a =2, h/a =0.01, m = I. (a) fl =0.01, n = 1; (b) fl =0-01, n =2; (c) fl =0.10, n = I; (d) fl = 0.10, n =2.
232
H.F. BAUER REFERENCES
1. H. F. BAUER 1982 Acta Astronautica 9, 547-563. Coupled oscillations of a solidly rotating liquid bridge. 2. H. F. BAUER 1981 Forsch. Ing.-Wes. 47, 190-198. Freie Schwingungen nichtmischbarer Fliissigkeiten im rotierenden Kreiszylinder unter Beri]cksichtigung der Oberfl~ichenspannungen. 3. LORD RAYLEIGll 1982 Philosophical Magazine 34, 177-180. On the instability of cylindrical fluid surfaces. 4. H. LAMB 1945 Hydrodynarnics. New York: Dover Publications. See pp. 471-473. 5. J. A. F. PLATEAU Smithsonion Institute Annual Report. 1863, 207-285; 1864, 286-369; 1865, 411-435; 1866, 255-289. Washington: Government Printing Office. Experimental and theoretical researches on the figures of equilibrium of a liquid mass withdrawn from the action of gravity. 6. H. REISMANN and P. S. PAWLIK 1968 Journal of Applied Mechanics 35, 297-305. Plane-strain dynamic response of a cylindrical shell: a comparison study of three different shell theories. 7. H. F. BAUER 1985 Forschungsbericht der Unicersitiit der Bundeswehr Miinchen LRT-WE-9-FB- 12. Coupled frequencies of a frictionless hydroelastic system. APPENDIX: LIST O F SYMBOLS b E h I,,,, K,, k I n! n
P r,~,z S t
T ii, ~, W
4, P
to
radius of container or radius of the free surface of the liquid layer around center shell radius of free liquid surface inside container or radius of inner shell for liquid layer case Young's modulus of elasticity wall thickness of cylindrical shell modified 13essel functions of ruth order and first and second kinds, respectively b/a, diameter ratio (-<1) length in axial direction angular modal number axial modal number liquid pressure cylindrical co-ordinates = o-+ i~, complex frequency time free liquid surface tension liquid velocity component elasticity parameters liquid velocity potential liquid density shell density Poisson ratio coupled frequency and abbreviation &2 .~ Tm(m 2_ 1)/pa 3 shell displacement in radial, angular and axial directions, respectively