axisymmetric vibrations of a shell with liquid
TRANSCRIPT
SOVIET APPLIKI) MECHANICS 23
AXISYMMETRIC VIBRATIONS OF A SHELL WITH LIQUID
E. T. Gr igo r ' ev
P r ik l adnaya Mekhanika, Vol. 2, No. 4, pp. 39-49, 1966
The vibrat ion problem of a cy l indr ica l shell with l iquid was solved in [3] in its l inear formulat ion. The non-smal lness of vibrat ion of the
l iquid was t aken into account ear l ie r only for its vibrations within a cav i ty in a r igid body [2]. In this paper we present the solution of the nonl inear problem of vibrations of a cy l indr ica l shel l with liquid, when
vibrat ions of the l iquid surface occur with la rge ampl i tudes . The solu-
t ion is ob ta ined in the fo~m of an inf in i te system of nonlinear differen-
t i a l equations, t /e r ta in features of vibrat ion observed in carrying our
the tests are exp la ined theore t i ca l ly .
We cons ider the problem of de t e rmin ing the c h a r a c - t e r i s t i c s of a x i s y m m e t r i e v ib ra t ions of a th in -wal led c i r c u l a r cy l indr i ca l shel l with a shallow spher ica l bottom, when this shel l is pa r t i a l ly f i l led with an ideal liquid. The acce l e ra t i on g of the field of m a s s forces and the externa l forces are a s sumed to be d i rec ted along the axis of the shell . The mot ion of the l iquid is a s sumed to be non- ro ta t iona l ; the field of m a s s forces is taken to be potential .
The o r ig in of the cy l ind r i ca l sys t em of coord ina tes (a, R, x) is located at the level of the lower ex t remi ty on the axis of the cy l indr i ca l shell ; the x axis is d i - rec ted upwards along the axis of the shell .
The a x i s y m m e t r i c v ib ra t ions of the shell with l iquid a re de t e rmined by the following funct ions which did not depend on the polar angle ce (t is t ime): u(x, t ) , w(x,t) a r e the axial and n o r m a l d i sp l acemen t s of the medium sur face of the cy l ind r i ca l shel l ; w0(R , t) a re the n o r m a l d i sp l aeemen t s of the median sur face of the bottom; 4"(x, R, t ) is the potent ia l of d i sp lacemen t s which e h a r a c - t e r i z e s the mot ion of the l iquid:
The d i sp l acemen t s u, w and w 0 of the shel l a re a s - sumed to be smal l . We denote the d i sp l acemen t of the su r face pa r t i c l e s of the l iquid at the shel l wall for the i - th na tu ra l mode by ri( t) , and a s s u m e that the a m - pli tude of v ib ra t ion of the l iquid sur face is l a rge only for one na tu ra l mode; this mode is condi t ional ly denoted by the index "1". Such a mode can be any one of the na tu ra l modes , i . e . it need not n e c e s s a r i l y co r r e spond to the lowest f requency. Such an a s sumpt ion is j u s t i - fied by the c i r c u m s t a n c e that at a r e sonance a con- s i de rab l e p r edominance of d i sp l acemen t s of one of the modes upon the r e s t takes place. We a s s u m e that the n o n - s m a l l n e s s of ampl i tudes of d i sp l acemen t s r l is such that the ampl i tude r 2 has the o rde r of s m a l l n e s s of the ampl i tudes of the r e m a i n i n g d i sp l acemen t s . The ve loc i t i es and a c c e l e r a t i o n s a re a s s u m e d to be of the same o rde r of s m a l l n e s s as the c o r r e s p o n d i n g d i s - p l acemen t s . Only such t e r m s whose o rde r of s m a l l - hess is not h igher than r~ a re t aken into account in the
equat ions , The c a m b e r of the sphe r i ca l bottom is a s s u m e d to
be sma l l in c o m p a r i s o n with the depth h of the l iquid and the rad ius R0 of the med ian su r face of the e y l i n - d r i ea l shell . In this case the sphe r i ca l bottom can be rep laced by an equiva len t p lane d iaphragm.
The potential 4, of d i sp lacements is defined as a harmonic function sa t is fying the boundary condit ions
0r 0--R = w(x, t) for R = Ro;
0r O--f( : u(O. t} + w.(R. t) for x = 0 ; (1)
v 2 Or-- v + U + y + X(t) = 0 on a,
where U is the potential of the mass forces ; v is the veloci ty of l iquid pa r t i c l es ; X (t) is a c e r t a i n funct ion of t ime; a is the d i s turbed free sur face of the liquid.
The las t one of the above condi t ions follows from Cauchy 's in tegra l and f rom the faet that the p r e s s u r e on the free sur face of the l iquid is zero.
We define the potential of d i sp l acemen t s as the following sum of ha rmonic funct ions:
�9 = Oj (x, R, 0 + O.2 (x, R, t) + Os (x, R, t). (Z)
Here 4,1 is given by the deformat ions of the cy l ind r i ca l shell ; 4'2 is g iven by the deformat ions of the bottom; 4"z is given by the wave mot ion of the l iquid.
In [3] the following fo rmu la s have been obtained
for 4"1 and 4"2:
R ~ - - 2x~ ~ w (x O~ 2hRo ~ , t) dx +
knit2 Io (~,R)([ikRo) S ~- ? j cosl~x w(~, t) cos I}~d~:
lr=t 0
2x r = xu (0, t) q- -.-2 ~ wo (R, t) RdR -
tr 2 o
_,~"~2chk,,(x--h) Jo(k.R) wo(R ' RdR,
r~=l 0
(3)
where J0, I0 and I 1 a re B e s s e l ' s funct ions of reaI and i ma g i na r y a rgumen t s ;
kn ~. 1~, = T ' k = - -
i n a re the roots of the equat ion J~(~) = 0. The ha rmon ic funct ion 4"3 is defined by the boundary
condi t ions
0r 0 r s = 0 for R = R o ; Ox - 0 f o r x = 0 ;
24 PRIKLADNAYA MEKHANIKA
Here n is a n o r m a l to the d i s t u r b e d su r f ace of the l iquid; u n is the n o r m a l component of the ve loc i ty of l iquid p a r t i c I e s .
The boundary condi t ions for the potent ia l of v e l o c - i t i e s 0~53/0t , specified on the non-disturbed free sur- face of the liquid, are found by the method proposed by G. S. Narimanov [2].
The equation of the disturbed �9 surface of the liquid is represented in the form
x = h + ~ r, (t) U~RJ, (R), (4) i=1
where f i (R) a r e the e x p r e s s i o n s of the na tu ra l modes of v ib ra t ions .
The funct ions f i(R) fo rm a comple t e s y s t e m of func- t ions which a r e or thogonal and no rmed within a c i r c I e of r ad iu s R0:
1 do (kiR) f f~(R) = VaRo Jo (~) ; 2~,) [~RdR = 1. (5)
0
The n o r m a l component of ve loc i ty of l iquid p a r - t i c l e s on the su r f ace ~ is e x p r e s s e d by the f o r m u l a
co
" - ( d t , V y r , r , (V~Ro) f, \ a~ ) �9 i : 1
and for the ve loc i ty potent ia l we obta in the e x p r e s s i o n
OO 3 �9 . -Or = 2 aiAi~ +a' 2 aiAu + al 2 aiAi' "-~ a'a~'A' (6)
i : 1 i : 1 i : 2
where ai( t ) = v~R0ri( t) ; Ai0, A~i, A i [ a n d A a r e ha rmon ic funct ions in the vo lume occupied by the l iquid; they s a t i s fy the boundary condi t ions given on the wet ted s u r f a c e ~ of the she l l and on the n o n - d i s t u r b e d f r ee s u r f a c e (for x = h)
-an-aA~~ i ~OA'~ : aA~,~ ~= aA ~ o&O_ax j~ = ~ - : O; x=h = l i ( R ) ,
OAl~ x = a dfi OAm O~Am 1 aA~o o~ : = ~ - ~ R + Z , - ~ m - + g L o ~ '
OAil i d~t OAi~ a=Aio l . dam ax ix=~ = Z ~ - ~ - + f, ~ + ~- :, o R '
OA !~ d[] OA n O~An 1 . OAn ~ - =~ = ~ - O R - + l, ~ + -k- 5 " - ~ +
-'-t, t dR) + g , -d~+2R. ,dR �9
As the r e s u l t of so lv ing the boundary value p r o b l e m s we obta in
A ch kix 2cth k~h Nis~ ch knx . . . . . [~ (R), A~i = 2 k. ~ in (R), gO �9 ki sh k~h I/"~R~ sh
(7)
A = 4clh k,h NilraNml n cih k.,h + M. sh-~-~h [" (~). n = l \ m = l ]
Here the fol lowing notat ion has been used for the con- s tant s:
Nun = Jo(gi)Jo(~i)Jo(~n) ~i J' ([i z)J ' (giz) J o ( ~ n z ) z d z - -
, ] - - ~l S Jo (~i z) Jo (~i z) Jo (~,,z) zdz ; (8)
0
3 [2J~' (~,z) - s~ (~,z)l ao (~,z) ]o(~.z) zdz (9) M. = j~ (~,) So (L) o
Owing to the orthogonality of the functions ji(R) relative to the constant, the third one of the boundary conditions (1) can be written in the form
2a t-~fi--+U-l-gv2otl(R)RdR=O. (10)
The potent ia l of m a s s f o r c e s is r e p r e s e n t e d in the fo rm
Then
Odp U =gos (11)
co
O (01 + ( P ~ ) U [~ = ~ ~x x=h + g 2 a ] .
i=l
F o r the quant i ty vX/2 we obta in
, I i .2[(OA,o V (OA oV] 2 - v ' = 2 t g r a d o { ) = 2 - a ' [ k Ox ) -kl-3-R- ) I +
�9 aAio aA1o aA~o'~ + a ' 2 a ' ( aa~ ~ ax ~- oR o R ) +
"2 [ a A l o OAn OAxo OAu ) + %a, ~ ~ Ox +OR OR +
(z2)
�9 J O_Alo as (% + %) OA~o a, (% + %) I + al t Ox axat + oR- OROt . (13)
The so lu t ion is sought for by the Bubnov-Ga le rk in method�9 The d i s p l a c e m e n t s of the she l l a r e p r e s e n t e d in the fo rm of e x p a n s i o n s
u = ~ U i (x) qi (t); W = ~ w k (X) q~k (t); i = l k = l
% = ~, %= (R) qom (t), r r l = ]
(14)
w h e r e {uj(x)}, {Wk(X)}, and{w0m(R)} a r e a p p r o p r i a t e l y chosen c o m p l e t e s y s t e m s of or thogonal and n o r m e d funct ions which s a t i s f y the g iven bounda ry condi t ions .
Subs t i tu t ing the e x p r e s s i o n s (2), (5), (12) and (13) into the f o r m u l a (10), t ak ing into account the e x p a n - s ion (14) and c a r r y i n g out the in t eg ra t ion , we obta in
SOVIET A P P L I E D MECHANICS 25
the v ib ra t ion equat ion of the l iquid in the shel l
h = l m = l
+ ~ o,,h,~,+ B,,,~,h: + (15)
r ~ 2 i= )
+ . ~ % , A 4 , + ~' "o.o~,q;.. = 0 , s = , , 2 . . . .
where ~ is the dens i ty of the l iquid. We w r i t e the equat ion of the d i s t u r b e d mot ion of the
she l l , us ing the f l e x u r a l eng inee r ing t h e o r y of s h e l l s
[11,
L . u + L,,2w 1 - - v ~ [ ~ O~u O~u = EF [ - - O~ - - ~ (x) ~o OF
P~)
~ - RdR --
0
(16) - ] --a~ (x) 2r~ ~ p (0, R, t) RdR + P (x, t) ,
0
~ ] Lz, u + L,~w = - - O o F ~ + 2~tRoa o (x)p (x, R o, t) ,
~_ . / o ~ 0 O~u ~ - 2 ,w (o, ~, O. Lo%= - - -~oOo~-~ - + Ot~ ,=o)
Here u, E and P0 a r e P o i s s o n ' s r a t i o , modulus of e l a s t i c i t y and the dens i ty of the m a t e r i a l of the she l l ; F and ~ a r e the a r e a and t h i cknes s of the c y l i n d r i c a l she l l ; 60 and P0 a r e the t h i c k n e s s of the bot tom and i ts m a s s ; p(x, R, t) is the p r e s s u r e of the l iquid; P(x, t) is the e x t e r n a l fo rce ; e~(x), a0(x) a r e funct ions s a - t i s fy ing the condi t ions
%(x)=O for x > 0 , lim [o 1(x) dx= 1,
% ( x ) = 0 for x > h , % ( x ) = l f o r 0 ~ x < : h .
F o r the d i f f e r e n t i a l o p e r a t o r s we use the e x p r e s - s ions [11
0 2 v 3 La-- - - 1 6 2 0 4 Lu = ~i- , LI~ = L2~ = Ro Ox' ~ + 12 Ox 4 '
#E6: (02 l 0 ) ' _E6 o L ~ ~ + R - O R + 2 , , /~l '
whe re R1 is the r a d i u s of the s p h e r e of the bot tom. The value of the t h r u s t p(x, R , t ) , ac t ing on the she l l
f rom the s ide of the l iqu id and e n t e r i n g into equa t ions (16), is d e t e r m i n e d f rom C a u c h y ' s i n t e g r a l
p (x, R, t) = -- . ~ t ~ + U + g + X ( t ) . (17)
We a s s u m e that the pe r t inen t f requency of v i b r a - t ions of the shel l with a l iquid f i l l e r is not ve ry low. Then in the fo rmula (17) we can igaore the h y d r o s t a t i c component - 0 U owing to i t s s m a l l n e s s in c o m p a r i s o n with the hyd rodynamic t e r m s , i . e . we can a s s u m e that
Ul~=o = U I~=R~ = 0. (18)
A d m i s s i b i l i t y of this is e a s i l y e s t a b l i s h e d by nu- m e r i c a l ca l cu la t ions .
When wave mot ions of the l iquid a r e absen t (at = 0), f rom the condit ion
[ p (h, R, t) RdR = O, 0
e x p r e s s i n g the absence of e x t e r n a l f o r c e s ac t ing on the f r ee s u r f a c e of the l iquid, we d e t e r m i n e the func- t ion X(t)
Ro
2 ~ [ 02(q)l+dp2) ( q b l + ~ ) ] RdR. (19) x (t) = - R=~o ot~ + g o dx ~=h
0
Subst i tu t ing (2), (18), (13), and (19) into Eq. (17), we find the e x p r e s s i o n s p(0, R , t ) and p(x, R0, t ) in t e r m s of the r e q u i r e d funct ions u, w , w 0, a i and t h e i r d e r i v - a t ives . Then the equat ions of the s y s t e m (16) wil l con - t a in only t h e s e unknowns. The c o r r e s p o n d i n g equa t ions of the Bubnov-Ga le rk in a s s u m e the fo rm
(aO~" + ~oqi) + ~ (~,*q'~k + u +
~ ~ "q" + D a2a + + Oim Om I i 1 1
m = l
+ D2ia ~ + ~ Dzi,aiq'=~ + ~ D,ima,qo, n = F, (0, ~ i m==l
i ~ 1 , 2 . . . . .
k = l j=l m-~-I r = l
+ ,~ e.,~>, + ,~ e~,o~a; + e~,a:o~ + ej;o, + r = l r = 2
r = l j ~ l k ~ l rrt== i
t = l , 2 . . . . .
+ +,2 + t ~ l r = 2
n = I, 2 . . . . (20)
The s y s t e m of Eqs. (15) and (20), s u p p l e m e n t e d by the bounda ry and in i t ia l cond i t ions for the funct ions u,
26 PRIKLADNAYA MEKHANIKA
w, w0, and as , fully d e t e r m i n e s the mot ion of the she l l with l iquid. All coef f ic ien t s of the equat ions of th is s y s t e m a r e cons tan t quant i t i es , whi l s t the equat ions t h e m s e l v e s a r e nonl inear . When the non l inear t e r m s a r e neg lec ted , the s y s t e m of the r e s u l t i n g equat ions y i e lds the s y s t e m obta ined e a r l i e r by V. P. Shmakov [3].
Equat ions (15) and (20) can be. s imp l i f i ed c o n s i d e r - ably. The e x p r e s s i o n s of c e r t a i n coef f ic ien t s ( inc lud- ing those of the non l inear t e r m s ) c o n t a i n s h k ih in t h e i r denomina to r s and, s ince ~min = 3.8317, t he se coe f f i - c ien ts a r e s m a l l , p rov ided the depth h of the l iquid i s not sma l l ; for example , if h > R 0. Because of th is we can take
XOs m = B 7 s ] = B 8 s k = Bgsrn = Dii . =
= E6t / = ETI k = Est m -- H/n " = 0 .
The subsequent s i m p l i f i c a t i o n of the equat ions of m o t i o n is connected with the t r a n s i t i o n f r o m the in - f ini te s y s t e m to the f ini te s y s t e m by tak ing into a c - count only a f in i te number of t e r m s in the expans ions (4) and (14). In inves t iga t ing the s t eady s t a t e v i b r a - t ions in a c o m p a r a t i v e l y n a r r o w f requency range of the exc i t ing fo rce , we can c o n s i d e r only one of the p o s s i b l e modes of v ib ra t ion , the ampl i tude of which is t aken to be l a r g e .
The equat ions of v ib r a t i ons of the she l l with l iquid a s s u m e the fo rm
] k m
i = 1 , 2 . . . . .
S "" ' c + X (~'#'q~ + Yltqi) + Z A's + • + k ] m
w-' E,,ata,'" '--Ea/a]a~+E4,a2at+Esfi~=O,, t 1,2 = . . . . ;
m i k
n = l , 2 . . . . ;
B ,~I + ~ea, + E "~q'~ + ~2;;,", + ~: i< + ~,~ + ~;0, = o.
where a l l the sums conta in a f ini te n u m b e r of t e r m s , and the number of equa t ions i s a l so f ini te .
The fo l lowing f o r m u l a s have been ob ta ined for the coe f f i c i en t s of Eq. (21):
%i = OoF6zi + (M + ~Xo) ui (0) u/(0), l
EF f u~(x) u i(x) dx, ff ~/ 1 - - v "~
o
h
o
-i EF u~ (x) w'~ (x) dx,
f)
R,)
~'o~,,, = (M + l~J R~ ~ ~ (0) Wo. , (R) RdR,
~t~ = r + 2h ~ (R~~ + 4hz -- 4x2) wt (x) dx w~ (x) dx -1- 0 o
1 lo O.Ro) + 'lRoO n I~ O.Ro) wt (x) cos E:dx ~ (x) cos ~.xdx +
n...~ 1 0 0
h h
+ 3h ,} wt(x) dx (h 2 - 3 x 2) w k(x) dx, 0 0
EF [ 1 . 8 , : ..e,~(x). ] c,, = v ~ [~o,k + ~ 0j t o , , x , - - ~ - ax ] .
A,~ = ~ (x - - h) wt (x) dx Wo~ (R) R d R - -
[ 0 0
11
2 ]/'~RoQ S tot (x) ch klxdx,
O
h
E u ---- 2~RoQ ~ A . (x, R o) to, (x) dx, 0
E~ = 2~RoO ~ A (x, R o) w, (x) dx. 0
h
E4t = 2~Ro~ 2A .+ Ox Ox ~=~ wt 0
h
11
4~ho 1%,~ --- 2~Oo6o6nm + ' ~ o S Won(R) RdR y wo.,(R)RdR +
0 0
r &
J Jo (g~) (R) RdR, ~ t 0 0
~ I( ) ] Conm ~ Won(R) 0 1 0 6 (I O~Y + ~- ~ Wo,,, (R) RdR + 0
QRo 2Nut ~- 2z~E6o o B1 = B2 = Q ,
B3 -- Ul~oo~ 1 a + 4 Nu,nN,~ 1 ,
!
B 4 == B 2 + V.~RoJao (~) l J30 (~,z) + J~ (~,z) Jo (~,z)] zdz, 0
SOVIET A P P L I E D MECHANICS 27
B s = 2 B ~ 2 ~ 7 " aR~oao %)
1 t " Jo + (g~z) J~ (g,z) J~ (~z) zdz
5 ~
Here M is the mass of the liquid contained in the shell; I is the length of the cylindrical shell; 5ii = I, 6ij = 0 f o r i ~ j ; we h a v e t a k e n c th k ih = 1 and h a v e a s s u m e d the fo l l owing n o r m cond i t i ons of the func t ions :
0 0 0
The solution of the system of Eq. (21), for partic- ular boundary conditions and parameters of the shell,
can be carried out on a digital computer. To analyze
the character of vibrations, we use the method of suc-
cessive approximations. At the same time we take into account the following results of the tests: at re- sonances the liquid in the shell vibrates with a fre-
quency which is an integer number of times lower than
the excitation frequency. Let us consider the most characteristic and often
e n c o u n t e r e d c a s e , when the f r e q u e n c y of t he e x t e r n a l e x c i t i n g f o r c e is t w i c e t he f r e q u e n c y w of v i b r a t i o n of the l i qu id , w h e r e w is c l o s e to t he n a t u r a l f r e q u e n c y
of v i b r a t i o n s of the l iqu id . We r e p r e s e n t the e x t e r n a l f o r c e s in the f o r m
and t a k e
F~ = F~o cos 2(o/
al = a0 sin ~t, (23)
as the f i r s t a p p r o x i m a t i o n , w h e r e a 0 i s a c e r t a i n c o n -
s t an t . Substituting (23) into all the equations of the system
(21), with the exception of the last one, and taking a 0
as the unknown, we obtain a system of linear equa-
tions for determining the first approximation of the
functions q~, qwk and q0m. Since
"" 2 ~ { 1 1 ala , = ~ a # ~ f f - - ~- cos 2~t
"" ~ ( 3 -~I sin 3o~t) a,a~ = -- #oo~ \ 4 sin a~t - - P
�9 1 a~a, = a~co' (-4- sin ot + l s i n 3cot) ,
�9 ~ ~[1 12. cos2~)t) a~ = a,o~ ~-~ +
t he s o l u t i o n can be r e p r e s e n t e d in t h e f o r m
q, : bt~ -i- %/sin ~ot + g,i cos 2o,! -i- z~i sin 3o)/;
q~k = b,~/, + x2k sin (,Jr -l- Y2u cos 2oJt -{- z2u sin 3cot;
q,., .= b:~ -i- x:~,~ ~i ~ a)! + gain cos 2o~t -l- za,~ sin 3o,t. (24)
The constants bni, Xni, Yni and Zni are determined by solving, independently of one another, the systems
of linear algebraic nonhomogeneous equations. Sub-
stituting (23) and (24) in the last equation of the sys- tem (21) and neglecting the constant terms and the harmonics with frequencies 2w and 3w, we obtain the the equation for determining a 0. The determinant of the
system of equations for Xnk
(~ii -- %~) -.- (~'i, -- ~'H ~2) "- �9 (-- ~0H r176 �9 ' "I I
(~ . - - ~.,o;-) . . . (% - - ~h,o;) . . . ( - - Al,~)~) . . . / (25)
/ (-- ~'o,Y) �9 �9 �9 ( - - a , ,o;) . . . (co, , - - ~oH<O ~-) . . . [
!
coincides with the characteristic determinant of the
system (21), when a I - 0. If w is not a natural fre-
quency of vibrations of the shell with liquid when a wave motion of the liquid is absent (this corresponds to the assumption about the smallness of displace- ments of the shell), the determinant (25) is non-zero,
and the values Xnk are finite. The relationship between
X2k and a 0 has the form
x2k = ao% + ao313 k,
w h e r e ce k and fik a r e d e t e r m i n e d only by the c o e f f i - c i e n t s of Eqs . (21) and the f r e q u e n c y co.
T h e n f o r a 0 we ob ta in the e x p r e s s i o n
a o = ~ ( l u u y,__ ) " ~o B , - - t 3 , - - z_J ~k•
k
In t h i s c a s e the a m p l i t u d e a 0 w i l l be a f in i t e q u a n t -
i ty fo r a l l f r e q u e n c i e s . T h e s e c o n d a p p r o x i m a t i o n fo r a 1 is found by s o l v -
ing the l a s t one of Eq. (21), in wh ich the f i r s t a p - p r o x i m a t i o n s h a v e b e e n s u b s t i t u t e d fo r qwk, wi th t he f i r s t a p p r o x i m a t i o n fo r a 1 s u b s t i t u t e d into the n o n -
l i n e a r t e r m s . It h a s the f o r m
al =bo +aosin~ot +yocos2cot +zosin3oJt. (26)
T h e a p p e a r a n c e of the c o n s t a n t t e r m s bni in the solution (24) is physically explained by the fact that
the nonlinear action on the shell, from the side of the
liquid, contains a constant component in addition to
the oscillatory terms. This leads to a variation in the
average level of the liquid, about which the vibration
take place. The constant components are not large and, in a number of cases, can be neglected without
any harm. The solutions thus obtained fairly well charac-
terize the vibration features of a shell with liquid. In cases when the frequency of the external disturbing force is close to a frequency which is a nmltiple of the frequency of vibrations of the liquid, we observe vibrations with several multiple frequencies.
The distribution of the amplitude of vibrations of the shell with liquid depends on the value of the de- terminant (25). It is of i n t e r e s t to note the c a s e when
28 PRIKLADNAYA MEKHANIKA
this de t e rminan t becomes ze ro and the resonance of the shel l occurs at the f requency which is half the f r e - quency of exci ta t ion (general ly , at the f requency which is an in teger number of t imes the f requency of e x c i - tation).
A she l l with the fol lowing d imens ions was ca lcu la ted by the fo rmulas de r ived above: R 0 = 300, 6 = 1, 50 = = 0.5, 1 = 1360 mm; it was f i l l ed with wate r up to the l eve l h = 1000 mm. In the ca lcu la t ions it was a s sum ed that the v ib ra t ions of the l iquid in the th i rd natura l mode with the f requency 2.9 cps (43 = 10.173) were not smal l . In the t e s t s the range of f r equenc i e s of the long- i tudinal exc i ta t ion was inves t iga ted c lose to 5.8 cps. The r e s u l t s of the t es t s coincided with the data of ca lcula t ion: a r e sonance took place at the f requency 5.8 cps. Here v ib ra t ions with the exci ta t ion f requency w e r e p redominan t for the shel l ; for the l iquid v i b r a - t ions at half this f requency (2.9 cps) w e r e predominant . The su r face of the l iquid v ib ra ted in the th i rd na tura l
mode. Analogously we can obtain the m o r e g e n e r a l i z e d
nonl inear v ib ra t ion equat ions of a cy l indr ica l shel l
with a liquid filler under longitudinal excitation, which take into account the possibility of the shell and liquid vibrating in non-axisymmetric modes that are periodic functions of the polar angle ~.
REFERENCES
I. V. Z. Vlasov, The General Theory of Shells and its Application in Engineering [in Russian], GITTL,
1949. 2. G. S. Narimanov, "On the motion of a ves-
sel partially filled with a liquid; consideration of non-
smallness of the motion of the latter," PMM, vol. 21,
no. 4, 1957.
3. V. P. Shmakov, "On the equations of axisym-
metric vibrations of a cylindrical shell with a liquid filler," Izv. AN SSSR, Mekhanika i mashinostroenie, no. l , 1964.
3 March 1965