Nonlinear fuel sloshing in elastic cylindrical tank - steady state response and stability boundary determination

Chu, W. H.

English

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A THEORETICAL STUDY OF NONLINEAR F U E L  SLOSHING 
IN AN ELASTIC CIRCULAR TANK::: 
Wen Hwa Chu 
Senior  Resea rch  Engineer  
Department  of Mechanical Sc iences  
Southwest R e s e a r  ch  Ins i tut  e 
Investigations on breathing v ibra t ions  of a n  e l a s t i c  shel l  containing 
l iquid w e r e  m a d e  inRefs .  1, 2, and 3. In Ref. 3, it w a s  shown that  l i n e a r  
theory  and experiment  f o r  na tura l  f r equenc ie s  a r e  in good ag reemen t  if 
enough t e r m s  a r e  used  to r ep resen t  the  mode  shape,  
In Ref. 4, nonlinear  phenomena w e r e  r epor t ed .  The  ampli tude 
frequency cu rve  shows c h a r a c t e r i s t i c s  of softening. 
jump phenomena, m o r e  complex r e s p o n s e  of the she l l  resembl ing  b e a t s  
was  encountered.  In the m e a n  t ime,  the  l iquid exhibits l a r g e  ampli tude 
osci l la t ions with a mode  shape  different f r o m  l i n e a r  fuel  sloshing but  a t  
about the s a m e  low f requencies .  
phenomena,  , p re l imina ry  s tudies  w e r e  m a d e  t o  d e t e r m i n e  the approximate  
s teady-  s ta te  r e sponse  and the stabil i ty boundary,\  as follows. 
A. Formula t ion  of the P r o b l e m  
Aside  f r o m  the  
F o r  a be t t e r  understanding of th i s  
*- 
A p r e l i m i n a r y  investigation h a s  been  m a d e  of the  problem oftfuel 
s loshing in a tank consis t ing of an e l a s t i c  thin cy l indr ica l  shell , ;  with the 
following bas i c  assumpt ions :  
-.) 1 
r- 
I 
(1)  L inea r  shel l  theory; i. e . ,  d i sp l acemen t s  of t he  middle  su r face  
a r e  a s s u q e d  small compared  with the  th ickness  of the shel l ,  
and strarks a re  a s sumed  d i r ec t ly  propor t iona l  to  s t r e s s e s  
in the  s h d l .  
#! 
( 2 )  Poten t ia l  flow theory;  i. e . ,  the fluid is  incompress ib l e  and 
nonviscous,  and the f l  w is i r ro ta t iona l .  
Boundary conditions aye  expres sed  in te rms  up to  the th i rd  
o r d e r  in liquid a m p l i t u d ~ s , i ~ u t  include only t e r m s  of the  
f i rs t  o r d e r  in the  amplitT;@de’s of she l l  d i sp lacement .  
$( .ir +a. 
(3 )  
::-Prepared pa r t ly  under  Contract  No. NASr-94(03),  SwRI P r o j e c t s  No. 0 2 -  
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1329 and 02-1519. 
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? .. 
2 
Assumption (1 )  i s  based  on experimental  observation, of empty she l l  
vibrat ions,which indicates  v e r y  l i t t le of the nonlinear softening effect. 
Assumption ( 2 )  i s  made  fo r  the sake of mathemat ica l  t ractabi l i ty .  
Assumption ( 3 )  i s  m a d e  in o r d e r  to allow l a r g e  liquid ampli tudes while the 
ampli tudes of shel l  vibrat ions remain  relat ively sma l l .  
Since inclusion of the higher o r d e r  t e r m s  would have grea t ly  
complicated the boundary condition equations, i a t tention was l imi ted  to t e r m s  
of only the f i r s t  t h r e e  o r d e r s  in  fluid ampli tudes.  
poss ib le  to  cons ider  an infinite s e r i e s  in fluid ampli tudes,  the effort  requi red  
to  de r ive  corresponding boundary condition equations would be  prohibit ive,  
especial ly  s ince these  equations would s t i l l  be r e s t r i c t e d  to  the region of 
convergence of the infinite s e r i e s .  
Although i t  may be 
B. Methods of Solution 
Solutions to the c l a s s  of problems defined above may be  constructed 
in the  following manner .  
f r o m  a velocity potential  which sat isf ies  Lap lace ' s  equation throughout the 
volume occupied by the fluid. Exact solutions to the Laplace  equation for  
a c i r c u l a r  cylinder can be found by the method of separa t ion  of va r i ab le s .  
Solutions which approximately satisfy the she l l  equations and the boundary 
condition equations can then b e  constructed,  using the exact  solutions t o  
L a p l a c e ' s  equations obtained above, by the well-known method of Galerkin.  
Al te rna te  methods of solution include per turbat ion techniques,  in which each 
unknown i s  expanded in a power s e r i e s  of some s m a l l  p a r a m e t e r ,  and o ther  
approximate  methods (Ref .  5). Numer ica l  methods would be  l imi ted  t o  
t r ans i en t  solutions,  s ince  steady solutions,  which a r e  the p re sen t  objective,  
would be  obscured  by the accumulation of e r r o r s  a f t e r  many t i m e  s teps .  
However,  a seminumer ica l  method could be used  and s e e m s  to be  promis ing .  
By assumption ( 2 )  the  fluid mot ion ' i s  der ivable  
C ,  A Specific Example 
An upright cy l indr ica l  tank fi l led with a liquid t o  a depth equal to 
th ree - fou r ths  its length has  been considered. '  
tank-liquid sys tem a r e  extremely complicated,  and the re fo re  prec lude  
exhaustive theore t ica l  t rea tment .  However,  t he  need f o r  qual i ta t ive 
theore t ica l  r e su l t s  led to  d ra s t i c  s implif icat ions a s  r e g a r d s  a s s u m e d  
v ibra t ion  modes.  Thus,  even thgugh the f ree- l iqu id-  su r face  condition 
was  expres sed  a s  a n  infinite s e r i e s ,  the liquid modes  w e r e  a s s u m e d  to  
cons i s t  of one cos  (48)-mode,  one c o s  (88)-mode,  and two ax i symmet r i c  
modes ,  one of which is constant.  Likewise,  the she l l  was a s sumed  to 
v ib ra t e  in a single breathing mode. 
equation? r e s u l t s  f r o m  substituting these  a s sumed  modes  into the gene ra l  
The actual  v ibra t ions  of t h i s  
A k e t  of o rd ina ry  different ia l  
. I  . *  
, 
4 '  
3 
formulat ion.  1 The s teady s ta te  amplitudes can then be obtained approxi -  
mate ly  by the method of harmonic balance which is again l imi ted  to  a 
few t e r m s .  
The stabil i ty equations a r e  obtained by consider ing s m a l l  p e r t u r b a -  
t ions f r o m  the steady s t a t e  solution o r  by differentiation of the or ig ina l  
equation. 
proport ion to eXt which l eads  to an  eigenvalue problem fo r  de te rmining  X. 
The sign of the r e a l  p a r t s  indicates the stabil i ty:  
a r e  posi t ive the oscil lation i s  unstable.  
The s m a l l  o r  differential  d i s turbance  was  a s sumed  to b e  in 
i f  any of the  r e a l  p a r t s  
The rank  of the m a t r i x  can be reduced by the assumption tha t  the 
osci l la t ions a r e  slowly varying, an  assumption which can  f i r s t  be  used  to 
reduce  the o r d e r  of the differential  equations.  
p roblem,  with the assumption of slowly varying ampli tudes,  a 6 X 6 m a t r i x  
was obtained; without i t  a 16 X 16 m a t r i x  was  obtained. The problem was 
fu r the r  simplified to a 3 X 3 m a t r i x  fo r  checking purposes .  Th i s  3 X 3 
its eigenvalues a r e  s table  f o r  all t h r e e  b ranches  of i t s  response  curve .  
Thus ,  f o r  the p re sen t  
: m a t r i x  was  found to be unsat isfactory f o r  predict ing stabil i ty bounds, a s  
D. At tempts  to Determine  the Eigenvalues 
The following four  computer  p r o g r a m s  w e r e  avai lable  fo r  u s e  in  
computing the eigenvalues:  ; 
(1)  A Jacobi- l ike method fo r  r e a l  va r i ab le s  obtained f rom the  
Universi ty  of Roches te r ;  
( 2 )  The d i r e c t  and inverse  power method obtained f r o m  the 
Universi ty  of Texas;  
( 3 )  Determinat ion of roots  a f t e r  expansion into a polynomia1,based 
on SwRI p r o g r a m s ;  
(4) A complex code of the Jacobi- l ike method, a l s o  f r o m  the 
Univer si ty of Rochester  . 
Considerable  difficult ies w e r e  encountered with the f i r s t  two 
p r o g r a m s ,  as the magnitudes and s igns  of the r e a l  p a r t s  w e r e  often 
uncer ta in ,  varying significantly with the number  of i t e ra t ions  and with the 
convergence c r i t e r i a .  
3 x 3 mat r ix .  The Jacobi- l ike method (1)  fa i led using s e v e r a l  t r i a l  numbers  
of i t e r a t ions  and re f inements ,  although this  p r o g r a m  was  v e r y  successfu l  in  
prev ious  applications.  
The methods w e r e  f i r s t  t es ted  on the s implif ied 
After  severa l  t r i a l s  of both p r o g r a m s  ( 1) and ( 2 ) ,  
I ,  ,- 
4 
us ing  different  n u m b e r s  of i terat ions and different  convergence c r i t e r i o n s ,  
t he  power method ( 2 )  f inally yielded the c o r r e c t  a n s w e r ,  which w a s  a l s o  
obtained by the polynomial method ( 3 ) .  
Subsequent e f for t s  t o  apply the power method t o  the 6 x 6 matrix 
w e r e  again unsuccessful ,  s ince  the ( s m a l l )  r e a l  p a r t s  of the first eigenvalue 
continued to  f luctuate  in sign and magnitude with convergence c r i t e r i a ,  
and f u r t h e r  th i s  method fai led t o  converge  f o r  o the r  eigenvalues with t h e s e  
c r i t e r i a .  
The polynomial method ( 3 )  w a s  next t r i e d  on the 6 X 6 m a t r i x ,  but 
because  instabil i ty was  indicated on the s tab le  b ranch  of the r e sponse  
c u r v e  the r e s u l t s  a r e  somewhat  doubtful. 
to  the 16 X 16 matrix which somehow failed to  converge  and might  r e q u i r e  
ad jus tment  in the  ini t ia l  guesses .  A l so , the re  is s o m e  doubt,when using the  
polynomial method, as to  the accuracy  of the coeff ic ients  of the  polynomial 
which a r e  obtained f r o m  powers  of t he  matrix. 
Th i s  method w a s  a l s o  applied 
F o r  all the methods,  the difficulty could wel l  rest with the matrix 
being i l l-conditioned, o r  due t o  a p rec i s ion  problem in the computations.  
The eigenvalue problem can b e  wr i t ten  as XAx-Bx = 0. In a t tacking 
th i s  problem,  a n  a t tempt  was  made to  improve  the conditioning of matrices 
A and B by reducing the l a r g e s t  e lement  of each  column of A and B to  o r d e r  
unity. Th i s  was  done by absorbing different  f a c t o r s  into the components  of 
t he  eigenvectors .  Af te r  modification, the  Jacobi- l ike method (4) s e e m e d  
to  yield reasonable  answers .  However, f o r  the  real code, t h e r e  is no 
checking p r o g r a m  when the  eigenvalues a r e  complex. 
e igenvalues  of the 6 X 6 m a t r i x  indicated a v e r y  mi ld  instabil i ty every-  
where .  
ampl i tudes  of she l l  vibrat ion at resonance .  It is  doubtful that  d i scard ing  
the  assumpt ion  of slowly varying ampli tude,  as  d i scussed  in Section C,  
would lead  to  sa t i s fac tory  resu l t s .  
Unfortunately, 
Surprisingly,  th i s  instabil i ty r ema ined  mi ld  even f o r  qui te  l a r g e  
The  complex code of the Jacobi- l ike method w a s  u s e d  f o r  one c a s e ,  
but the  r e su l t  w a s  inaccura te  due to  l a r g e  off-diagonal t e r m s  in the 
eigenvalue matrix. 
equations was  then t r ied .  
computed f o r  a nondimensional  fo rce  equal to 0 . 0 0 2  at the  approximate  
brea th ing  na tu ra l  f requency.  
( o v e r  40 cyc le s )  f o r  a nondimensional f o r c e  equal t o  0 .0005 at a 
f requency  of 40 cps  above o r  below the  na tu ra l  f requency.  
of t h i s  approach  is that  the truncation e r r o r  would grow into significant 
va lues  if the motion is  unstable .  
Numer ica l  integration of six ampli tude governing 
The  r e su l t s  a r e  encouraging as  d ivergence  w a s  
Motion prac t ica l ly  maintained its shape 
The  philosophy 
Additional investigation of th i s  approach  
b 5 
wi l l  r e q u i r e  additional funds not avai lable  at p re sen t .  
made  avai lable ,  numer i ca l  integration of t he  s implif ied s tabi l i ty  equation 
could a l s o  b e  pe r fo rmed .  
If additional funds a r e  
Final ly ,  Routh 's  c r i t e r ion  (Ref.  6) and Hurwi t z ' s  c r i t e r i o n  (Ref.  7 )  
c a m e  to  the attention of the  wr i t e r .  
n e c e s s a r y  condition f o r  stabil i ty in that  the coefficients of the polynomial 
governing the eigenvalues m u s t  be of the  s a m e  sign. On the lower  s tab le  
b r a n c h  below resonance ,  the 3 X 3 and the  6 X 6 m a t r i x  all sa t i s fy  th i s  
condition, while the 16 X 16 matrix, without the a s sumpt ion  of slow vary ing  
ampli tude,  yields  a r e s u l t  of doubtful prec is ion .  
of the  Hurwitz c r i t e r i o n  with the method of n u m e r i c a l  integrat ion would b e  
rewarding  . 
In these  r e f e r e n c e s ,  t h e r e  is  a l s o  a 
It s e e m s  that  compar i sons  
1 .  . -  
m .  
c 
. 
6 
CONCLUSIONS 
The nonlinear liquid motion in an  e l a s t i c  c i r c u l a r  tank under  going 
breathing v ibra t ions  is a laborious task.  This  task  i s  amplif ied due to 
ce r t a in  "il l-conditions" of the mat r ix ,  the r e a l  p a r t  of the eigenvalues  of 
which de termines  the stabil i ty.  So far, the computer  p r o g r a m s  avai lable  
have fai led to  yield rel iable  eigenvalues.  
integrat ion h a s  been employed, but, unfortunately, additional funds wi l l  
be r equ i r ed  before  the stabil i ty boundary can be de te rmined .  
The method of numer i ca l  
In the mathemat ica l  analysis ,  the breathing problem i s  different 
in that  dynamic t e r m s  a r e  l a r g e r  due to  the high breathing frequency 
and that balance of the small constant t e r m s  led to apprec iab le  t ime  inde-  
pendent cos  (06)  and cos ( S O )  components of the f r ee  su r face  elevation; 
that  i s ,  they a r e  quali tatively in ag reemen t  with exper imenta l  observat ions.  
. 1 7 
REFERENCES 
1. Abramson,  H. N.,  Kana ,  D. D o ,  a n d  Lindholm, U. S . ,  "Breathing 
Vibrat ions of a C i rcu la r  Cylindrical  Shel l  Containing a n  Internal  
Liquid, Tech. Rept. No. 3 ,  Contrac t  No. NASw-l4b, SwRI 
P r o j e c t  No. 961-2, Februa ry  1962. See a l s o  Lindholm, U. S . ,  
Kana, D. D., and Abramson,  H. N., "Breathing Vibrat ions of a 
C i r c u l a r  Cyl indrical  Shell with a n  Internal  Liquid, ' I  J. Aerospace  
- S c i . ,  2, 9, pp. 1052-1059, Sep tember  1962. 
2. Chu, W. H.,  "Breathing Vibrations of a Pa r t i a l ly  F i l led  Cylindrical  
Tank - Linear  Theory, J. Appl, Mech. ,  30, 4, 532-536, December  
1963. 
-, 
3. Chu, W. H. ,  and  Gonzales,  R . ,  "Supplement to Breathing Vibrations 
of a Pa r t i a l ly  F i l led  Cylindrical  Tank - Linea r  Theory,  
Mech. ,  2, 4, pp. 722-723, December  1964. 
J. Appl. 
4. Kana, D. D. ,  Lindholm, U. S . ,  and  Abramson,  H. N., "An 
Exper imenta l  Study of Liquid Instabil i ty in a Vibrating E las t i c  Tank, I '  
Tech. Rept. No. 5, Contract  No. NASw-146, SwRI P r o j e c t  
No. 4-96 1-2, F e b r u a r y  1962. 
5. Sokolnikoff, I. S. ,  Mathematical  Theory  of Elast ic i ty ,  2nd Edition, 
McGraw-Hill Book Co.,  Inc. ,  New York, (1956). 
6. Von Mises ,  R . ,  Theory  of Flight,  McGraw-Hil l  Book, C o . ,  
New York, 1945. 
7. Uspensky, J. V . ,  Theory  of Equations,  McGraw-Hill  Book Co. ,  
New York, 1948, pp. 304-309. 


A theoretical study of nonlinear fuel sloshing in an elastic circular tank

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