Sloshing forces and moments on cylindrical tank - integration of nonlinear force equation

Rogge, T. R.

Weiss, H. J.

English

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I" 
N A S A  C O N T R A C T O R -  
R E P O R T  
A NONLINEAR ANALYSIS FOR 
SLOSHING FORCES A N D  MOMENTS 
ON A CYLINDRICAL TANK 
by H. J. Weiss and T. R. Rogge 
Prepared under Contract No. NAS 8-11193 by 
GENERAL DYNAMICS/ASTRONAUTICS 
San Diego, Calif. 
for  
N A T I O N A L   E R O N A U T I C S   A N D   S P A C E   A D M I N I S T R A T I O N  W A S H I N G T O N ,  D. C .  0 A P R I L  1 9 6 5  
https://ntrs.nasa.gov/search.jsp?R=19650013369 2020-03-16T23:40:44+00:00Z
NASA CR-221 
A NONLINEAR  ANALYSIS FOR SLOSHING FORCES 
AND MOMENTS ON A CYLINDRICAL TANK 
By H. J. Weiss and T. R. Rogge 
Distribution of this  report is provided  in  the  interest of 
information exchange. Responsibility for the contents 
resides i n  the  author  or  organization  that  prepared it. 
Prepared  under  Contract No. NAS 8 -11 193 by 
GENERAL DYNAMICS/ASTRONAUTICS 
San Diego, Calif. 
for 
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
For sale by the Clearinghouse for Federal Scientific and Technical Information 
Springfield, Virginia 22151 - Price $1.00 

A NONLINEAR  ANALYSIS  FOR SLOSHING FORCES 
AND MOMENTS ON A CYLINDRICAL  TANK 
By H. J. Weiss and T. R. Rogge 
ABSTRACT 
This  analysis of the  sloshing  forces and moments on a cylindrical 
tank w a s  prepared  for  use in conjunction  with NASA CR-220 entitled "An 
Approximate  Nonlinear  Analysis of the  Stability of Sloshing  Modes  Under 
Translational  and  Rotational  Excitation, " by T. R. Rogge and H. J .  Weiss. 
It is not intended  to be used as a separate  report. 
iii 
I 
SECTION 1 
DISCUSSION 
1.1 PRESSURE. For  the liquid tank. system,  the  pressure is given by (see Refer- 
ence 1, equations  2.38 and 3.14) 
where p is the ambient pressure. 
0 
The forces on the tank due to  the liquid a re  
where n. is the  unit exterior  normal to S, and S is the  surface of the  liquid in contact 
with  the  tank. On the  free  surface, p = po; thus,  there is no contribution  to  the force 
at  the  free  surface. 
1 
The  moments are given by 
where E.. is the  third  order  alternating  tensor. 
In Reference 1, the  potential is written as 
1Jk 
+ ( r y e ,  z, t ) = $ ( r , e ,  z, t ) - u  r c o s e - u  r s i n e + z r w  s i n e  
"ZW r c o s e  
1 2 1 
2 (4) 
1 
where’ # is approximated by 
1 - 
3 
q = E [ql (F, t) coso t  + x 1 (F, t) s i n w t  1 
2 - 
+ E 3  [ $ (F) + J ,  (F) cos 2ot + x2 (2) sin 2 0 t  
0 2 1 
+ E q (F) cos 3wt + x  (r) sin 3wt 
[ 3  3 I 
E indicates (r, e ,  z), and E is a small  parameter. 
The  perturbation  velocities a re  given  by 
u. = E C O S w t  
1 
w = “ O S W t  
E 
i h  
where  .h is the height of the  undisturbed  liquid  in  the tank. 
The  harmonic  functions # , X , X and X are  defined a s  (Reference 1, 
0 ’  $1, $2 1 2 3 
pp. 3-8 to 3-14). 
#o = constant 
cosh [Xl1 (z + h)] 
II, = [ fl (7) cos e + f (7) s ine  J~ hll 1 3 1 ( cosh X h 11 
OD cosh [X (z + h)] 
On 
2 coshAon h n=l 
cosh All (z + h)] 
[ 
1 
X = f2 (7) cos8  + f  (7) s h e ]  J1 (Xl1 r) coshX 1 4 11 
2 
cosh [Aon (z + h)] 
On 
x2 = 2 E (x ') coshX h On 0 On n=l 
T =  - 
solutions of J' (Amn a) = 0. The coefficients 
Reference 1, p. 3-11. 
Within the  order of the  approximations  used,  neither q3 nor X contribute  to  the 
pressure,  or  therefore  to the forces and moments. 
2 m  mn E 2'3 at, J are Bessel functions of the first kind of order   my and A are  the 
, 6 a re  defined as in A on' *2n' m 2n 
... 
3 
Substitute (4) into (1) 
0 
P - Po U 1 
= oz+- 3 
P 2 2 
" ( 9  2 a+ 
U 
+ - r  w sin e + w2 cos2 0) + 2 - ( r w  sin e - r w  cos e + - )  3 
2 2 az 1 2 2 
2 - 3 r  w w s i n e c o s 0 + u  w r s i n e - u w  r c o s 0 - z w u  
1 2  3 1  3 2  2 1  
+ z w u  - u  w r c o s e + z w w  r s i n e  
1 2  2 3   3 2  
azC, + u w r s i n e +  w - + Z W  w r c o s  e 
1 3  3 ae 3 1  
WJ + ;Z r cos e + G2 r s ine  - z rL  sin e + z L  r cos e - - 
1 1 2 a t  . (9) 
There  are two types of steady-state motion to  consider:  planar and nonplanar. For 
planar motion the  generalized  coordinates, f. (7) a r e  
1 
f = Y  
1 
f = f  = o  
2 4 J 
where Y is a parameter  that is independent of time and Q is defined as in  Reference 1 , 
(3.61). 
3 
I 
Thus, (see Reference 1, (3.41)) 
= AZn = BZn = o  n n 
On 
n 2 
DZn = -Y  Q Q2n 
2 
n "2n (Q2 c =  
2n 2 - 1) ' (11) 
2 
h "on c = -  (Q2 + 1) On 2 
$ = x  = o  
2 1  > 
where S2 and a are defined in Reference 1, p. A-12. Thus, from (7), (8), and 
( u ) ,  we obtain the  specific  forms of the  harmonic  functions. 
2n On 
2 2  cosh [ ion (z + h)] 
x = -  
2 (' 2 + 2 'On JO ( On ) coshX h h r  n=l On 
cosh IA (z + h)] 2n 
coshh  h 2n 
$ ( r ,  8 ,  Z ,  t )  = E  q1 COSWt + E 2'3 [Ji0 + x2 sin 2 w t  1 + 0 ( E )  (14) 
where 0 ( E )  includes  the  higher  harmonic  terms. Neglect terms of the  order 0 ( E )  and 
compute 
4 
. . .... . .._.... 
1/3 3 2/3 
$ = E  coswt+ 6 - ak sin 2 ~ t ,  k = r, e, z k ak 
(15.a) 
(15.b) 
cosh [All (z + h)] 
- = y  (- sine + Q COS e) J1 
ae ('11 '1 cosh$l  h (16.b) 
Y 2  (Q2 + 1) cosh [Aon (z + h)] 
- =  - a r  2 n=l 2 *On 'On Jb ('On '> cosh Aon h 
cosh [Aan (z + h)] 
coshin h 
cosh [Azll (z + h)] 
cosh\n  h 
(16.c) 
(17.a) 
(17. b) 
5 
a x2 
a z  2 
Y (Q- + 1) 2 2  sjnh [ A  (z + h)] On - = -  R A J . A r  On On 0 ( On ) cosh A h Ii=l On 
sinh [ A2n (z + h)] 
cosh A h 2n 
Disregarding  terms of 0 ( E  ) , then for k = r , 8 ,  z 
2 - 2 
*k = E 3  (2) cos ut  2 
2 
(Note in obtaining (18). the term 0 ( E  ) in J I  -- namely. E k 
cos ut sin 2 w t  -- introduces a harmonic  term of higher  order and is therefore 
neglected.) 
Substitute  equations  (15)  through  (18)  into (9) 
+ Q  cos e) J cosh hllh r 1 (hill) 
cosh [ All (z + h)] l2  
cosh A h 
11 
(17.c) 
sinh [Al1 ( z  + h) ]  
(COS e + Q sin e) J 
1 (hllr) cosh h h 
11 
6 
1 
3 
- cosh A (z + h) 
11 + E ywsin ut (cos 8 + Q sin 8) J 
1 (Illr) cosh h h 
11 
2 
3 
- cosh h (z + h) 
On 
cosh X h - 2 E wcos 2 w t  n = l  On 
cosh h (z + h) 
2n 
cosh h h 2n 1 
- r w E  sin w t  cos 8 - r u e  sin u t  sin 8 + - z r c  w h sin wt sin 8 
z r c  w 
h 
" sin w t  cos 8 
disregarding  terms of order  greater than 0 ( E )  and higher  harmonic  terms. 
1 . 2  FORCES. On the sides of the tanks, nl = cos 8, n = sin 8, n = 0 .  On the bottom 
of the tanks,  n = n  = 0,  n = -1. 
2 3 
1 2  3 
27r 0 P - P  
F = - p a  J ( O) cos 8 d8 dz, r = a  
0 -h 
1 P 
F = - p a  
2 
') sin 8 de dz, r = a  
0 -h 
F3 = P ( ) r dr  dB, z = -h 
(20.a) 
(20. b) 
(20. c) 
0 0 
7 
P - Po 
Note that because of the  trigonometric  functions  appearing in  and  because of the 
form of n. there are considerable  simplifications  in  evaluation of Fi. Thus 
P 
1 
21r 27r 
1 sin2 9 de = 1 cos2 9 de = T 
0 0 
2 2 3 ‘ 3  cos 8, sin 8, cos 8 sin 8, cos 8 sin 8, sin 8 cos 8, COS 8, sin 8, 
cos 2 8 cos 8, sin 2 8 cos 8, sin 2 8 sin 8, cos 2 8 sin 8 1 d 8 = 0 (21) 0 
and 1 
3 
- 
F = -  
- 
p a r y e  w 
1 
sin w t  J 
1 h l a )  Allh 
Y 
F = - a lr ” E Q w  sin w’t J1 (Alia) tanh Allh 
2 
All 
(22. a) 
(22.b) 
2 
2 1 3  2 Y T A1: (Q2 + 1) a 2 - a n a  h+--E  cos ut 
2 2 
cosh A h 
11 0 
2 
+ y2 T (Q2 + 1) /” J1 (hllr) dr 
cash h h 2 r 
11 0 
2 - 
3 n y  2 (Q2 + I )  a + E O C O S  2 u t  r Jo (honr) d r  ] (22.C) 
cosh A h On 
n = l  11 
0 
1.3 MOMENTS. The first moment is calculated from (3) as 
a 
" 
From the  geometry of the boundary, then, 
2 n  a 
P - Po 
M = - P  s J T  sin 8 r 2 dr dB 1 
0 0 
2 n  0 
P - Po 
- p a  sin 8 z dz  de 
0 -h 
r = a  
Si mi lar 1 y 
2 n  a 
P - Po 2 
cos 8 r dr de 
P 
2s 0 
i- Pa cos 8 z dz de 
0 -h 
(24. a) 
(24. b) 
r = a  
2a 0 
= a  J -[ (p - po) (cos 8 sin 8 - sin 8 cos 8 )  r de dz = o (24. c) 
0 
9 
Substitute (19) into (24a) and (24b) to obtain 
Y 
M = -  p Q  K C  Y wsin w t  
1 cosh A h 11 0 
3 
p a n e   Q y w s i n w t  J (hila) 11 - cosh A h 11 1 
All  cosh Allh L 
3 
M, = 0 
p a r e   y o s i n o t  J (Alla) 11 - cosh A h 11 I 
L L 
All cosh  hllh 
1 
3 
- a 
+ p K E  y w sin ut 
cosh A h 
11 0 
(25. a) 
(25 .b) 
REFERENCE 
1. T. R. Rogge and H. J. Weiss, An Approximate Nonlinear Analysis of the Stability 
of Sloshing Modes Under  Translational and Rotational  Excitation,  General Dy- 
namics/Astronautics Report GD IA-DDE64-059, November 1964. (NASA CR-220, 1965.) 
10 NASA-Langley, 1965 (33-221 


A nonlinear analysis for sloshing forces and moments on a cylindrical tank

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