The effects which moderately large deflections have on the frequency spectrum of rotating rings and cylinders are considered. To develop the requisite solution, a variationally constrained version of the Lindstedt-Poincare procedure is employed. Based on the solution developed, in addition to considering the effects of displacement induced nonlinearity, the role of Coriolis forces is also given special consideration

Padovan, J.

English

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CORIOLIS EFFECTS ON NONLINEAR OSCILLATIONS OF 
ROTATING CYLINDERS AND RINGS 
Joseph Padovan 
University of Akron 
SUMMARY 
The effects which moderately large deflections have on the frequency 
spectrum of  rotating rings and cylinders are considered. To develop the 
requisite solution, a variationally constrained version of the Lindstedt- 
Poincare procedure is  employed. 
tion t o  considering the effects of displacement induced nonlinearity, the 
role of Coriolis forces i s  also given special consideration. 
Based on the solution developed, in addi- 
INTRODUCTION 
Numerous engineering applications ( t i res ,  turbines, satel l i tes ,  etc.) 
contain ro tor  systems which are essentially rings or shells of revolution 
ro t a t ing  about  their axes. 
design, a thorough dynamic analysis is necessary. 
papers have been published which deal w i t h  the free vibration properties 
of such systems. 
as can be seen from the excellent surveys by references 1 and 2. The effects 
of rotation, i n  particular Coriolis forces, have been discussed by references 
3 t o  7. With the exception of references 6 and 7 which treated small dynamic 
deformations superposed on large s ta t ic  deformations, the previous investiga- 
t ions  incorporating Coriolis acceleration forces have been limited t o  linear 
shell theories. 
satel l i tes ,  and turbines are flexible enough t o  undergo significant deflections 
i n  the form of moderately large rotat ions.  
I t  i s  the purpose of this paper t o  consider the effects which such 
moderately large rotations have on the frequency spectrum of r o t a t i n g  struc- 
tures. In particular, the analysis presented will consider the free 
vibration characteristics of rotating rings and cylinders wherein the deflec- 
tions involve moderately large rotations.  Since the analytical model used 
t o  characterize the stated problem involves nonlinear par t ia l  differential 
equations, a modified version of the renormalized perturbation procedure i s  
employed t o  evaluate the overall solution. This modification was undertaken 
since the usual renormalized procedure is  unwieldy for systems of equations 
involving a multitude of frequency eigenvalue branches and secondly yields 
steady s ta te  results which are irregular for the linearized case. The modi- 
fication employed involves prescribing the system energy i n  advance; hence, 
a hierarchy of energy states i s  obtained from which the strained parameter 
can be evaluated. 
regular, and thus, the proper limiting behavior is  obtained for the linearized 
Obviously, i n  order t o  properly influence their 
In this regard, numerous 
Most such work has centered on stationary configurations, 
T h i s  is  a shortcoming since numerous rotor systems, t i res ,  
The resulting solution employing this procedure i s  
409 
https://ntrs.nasa.gov/search.jsp?R=19770003326 2020-03-22T12:11:48+00:00Z
case. 
nonl inear i ty ,  special  emphasis i s  centered on determining the e f f e c t s  o f  
C o r i o l i s  forces i n  the range o f  deformations marked by moderately l a rge  rota-  
t ions.  Hence the effects on the backward and forward t r a v e l i n g  waves w i l l  be 
evaluated. 
Based on the solut ion,  i n  add i t i on  t o  considering the global e f f e c t s  of 
GOVERN I NG EQUATIONS 
Since the nonl inear o s c i l l a t i o n s  o f  ro ta t i ng ,  e l a s t i c a l l y  supported r i n g s  
and i n f i n i t e  cy1 inders undergoing def lect ions i nvo l v ing  moderately l a rge  rota-  
t i o n s  are considered herein, the governing displacement equations of motion 
employed t o  model the s tated problem are def ined by ( re fs .  2,4,6, and 7) 
(1 1 3 F ~ A  W,tW,ee+f  cos (me) cos (wt)+ph(W,tt-2fiV,t-a2W) = 0 
2 
A ( V  ,ee+W , e + ~ W  ,,W ,ee)-p h( V , tt+2~W, t- f i2V)  = 0 
2 
where 
Eh E1 
R4 R2 
, A = -  A = -  ( 3 )  
such t h a t  E = Wm/R and 0, t, ( ),e, ( ):t, W, V, W m y  E, I ,  h, R, p ,  P, K, w, and 
a respect ive ly  represent c i rcumferent ia l  space, time, space and t ime d i f -  
f e ren t i a t i on ,  r a d i a l  and c i rcumferent ia l  s h e l l  displacements, maximum r a d i a l  
displacement, Young's modulus, moment o f  i n e r t i a ,  s h e l l  thickness, radius and 
density, i n t e r n a l  pressure, foundation e l a s t i c i t y ,  e x c i t i n g  frequency, and 
l a s t l y ,  the r o t a t i o n a l  speed of t he  shel l .  Due t o  the inherent nature o f  the 
c i rcumferent ia l  coordinate space and the f a c t  t h a t  the steady s t a t e  response 
i s  being sought, i t  fo l lows t h a t  W and V are pe r iod i c  i n  both space and time. 
To round ou t  the r e q u i s i t e  f i e l d  equations, the fo l l ow ing  po ten t i a l  energy 
funct ional  i s  associated w i t h  equations (1 ) and ( 2 )  , namely 
T 2n 
Y = J J I A  1 W,t,+A 2 (V,g+2V,eW+W2)+~A 2 (VyeW,:+WW,i) + 
0 0  
* 2  1 P F ~ A  W,;+(K+?r)W2+2f cos (me) cos (wt)W-ph[a2(R +W) +W,t + 
fi2V2+V,;+2a( R*+W)V, t-2nW, tV] Idedt  
where T = - ' and R* = R/Wm. 2nw 
2 L 
(4) 
4 10 
SOLUT I ON 
As noted, earl ier ,  the standard renormal ized perturbation procedure has 
This difficulty is circumvented by pre- 
the twofold difficulty of yielding irregular results as e 0  and secondly, i s  
unwieldy when more than one equation of motion involving several frequency 
eigenvalue branches i s  considered. 
scribing the systems potential energy i n  advance such t h a t  ( W ;  V )  = (W(e,t ,f ,  
m,y); V(e,t,f,m,r)). Once the solution i s  obtained, the role o f  y and W are 
reversed t o  t h a t  employed i n  the traditional version of the renormalized pro- 
cedure. 
hence W ,  V , and w are expanded in the following perturbation series 
To ini t ia te  the solution, w is treated as  the strained parameter; 
co 
i <w; v; W> = c <wi ;  v i ;  W . > E  
1 i =O 
(5) 
such tha t  time is stretched so that T = u t .  
In order t o  obtain the zeroth order equations, E i s  set  to zero; this 
yi el ds 
+ ( A  +K$)W~+A V +ph(w6Wo,tt - 
2 0 , e  
A W  0,eeee 
2 w o n V o  -n2Wo)+f cos (m6) cos ( T )  = 0 
ST 
+W ) = ph(w2V + 2 ~  QW -Q2Vo) A2("o,eo o,e 0 0 , T T  0 0,T 
* 2 
2f cos (me) cos (T)Wo-ph[a2(R +Wo) + +o2V2+w2V2 + 
WOWO,r 0 0 0 , T  
* 
2 w o n ( R  +W 0 ) V  0,T - ~ W ~ Q W ~ , . ,  V 3 1ded-C (8) 
whereas w i t h  time, the potential energy space i s  stretched so t h a t  r = y/Q. 
Since the steady state solut ion i s  sought,  
where W 
t o  the Following matrix set of ordinary differential equations, namely 
are time dependent. Employing equations (9), (6), and (7 )  reduce 
w2[B ]Y +W [B ]Y +[B ]Y +f COS ( T )  = 0 (1 0) o i m  ,mo,-c.c o 2111 ,mop 3111 ,mo , , 
such tha t  
411 
Noting that [B2m] i s  skew symmetric while [B,,] and  [B,,] are purely sym- 
metric, the steady s ta te  form of Y i s  given by ,mo 
Y = Z cos ( T )  + Z sin ( T )  (1 2) ,mo ,mc -ms 
where Z and Z satisfy the matrix equation ,mc ,rns 
Noting that the pencil of equation (13) yields the characteristic equation 
of equation ( lo) ,  equation (12) becomes unbounded for wo equal t o  the natural 
frequency eigenvalues of the linear case. The properties of such eigenvalues 
can be ascertained by developing the appropriate Rayleigh quotient. 
possible by inserting.y dmejT intolequation (10) t o  yield a complex second 
order regular polynomi2y matrix problem. The inner product of this  expression 
and i$, yields a bilinear form from which the following modified version of 
Raylgigh’s quotient is  obtained, namely 
This is 
As can be seen from equation (14) ,  Coriolis forces cause a twofold bifurcation 
i n  the number of eigenvalue branches. 
relationship betweenr and q-,, Wo, and Vg must be evaluated by inserting equa- 
tions (9)  and (12) i n t o  equation (8); this yields 
Following the previous comments, the 
[A m4+A +K+- P - p h ( ~ ~ + w $ n ~ ) ]  ( W ~ c o + W ~ s o + W & o + W ~ s o )  + 
1 2 R  
[A m2-p h (  a2+m2wi)] ( V ~ c o + V ~ s o + V ~ c o + V ~ s o ) + 2 m A  ( VscoWcco + 
2 2 
-v w -v w )-zPhw,Qm(Wcco v - w v  cso cso cco + vssowcso cco sco cso sso 
wscovsso sso sco cco cso cso cco sco sso -w v -v w +v w -v w + 
I 
denote coefficients of the Wo and Vo solution, namely where Wcco,.. , V  sso 
412 
bJo ;Vo) = (Wcco ;Vcco)cos (me )cos (T)+. . . +( wss0 ;vss0) s i n  (me )sin (T ) 
As can be seen from equations (13). and (15), four potential energy resonances 
are initiated for uo-O(u,,,f) wherein %f are the frequency eigenvalues of the 
linear problem. 
derivative of equations (1), ( Z ) ,  and (4 )  with respect t o  E and then setting E 
t o  zero. This yields 
Hence equation (5) is  regular for E -t 0 (the linear case). 
The f i r s t  order set of field equations can be obtained by taking the f i r s t  
wowlw; ,r+W;Wo, TW 1, T +Q2VOV1+wOw~V~,+ + 
* * 
0 0 , T  1,T 9T yT 0 1 0,'C 
w2V V +fiwlR Vo +61wOR V, +Qw W V + 
"0 wov 1 , 'i:+Qw lWOV0, '1: - QWOWl , T vo-Qwowl , TV 1 - 
Noting the form of the inhomogeneities appearing in equations (16) and ( 1 7 ) ,  i t  
follows t h a t  W1 and VI can be taken i n  the form 
where the coefficients W , i Y .  . . are directly obtained upon inserting equation 
(19) into equations (16) and (17). Furthermore, employing equation (19) in 
conjunction w i t h  the f i r s t  order potential energy constraint, equation (18) 
the following functional relationship i s  obtained for w1 , namely 
413 
where D is the determinant of the pencil of equation (13). 
u1 is  bounded and positive definite. T h i s  follows since 1/D(2%,0),. ..'etc. 
remain bounded for WuO~fO,~). Therefore, unlike the zeroth order set, W1 and 
V1 remain bounded for Wq,. 
In order t o  obtain the second order field equations, equations (1), (2), and 
(4)  are differentiated twice and then E is  set t o  zero. T h i s  operation yields 
Hence fo r  uo-O((,,+,,f), 
28 28 
0 = J J 
{A1W:,ee+'A1Wo,eew2,ee+A1(Vl,e +w1)2 + 0 0  
2A2(Vo,e+Wo)(V23e+W2) + A2(2(V0,e +w o )w o,e w 1,e + 
+W )W2 ) +  - A  1 W4 + ( K + E ) ( W : + ~ W ~ W ~ )  P + 
%,e 1 0 3 e  4 2 o,e 
2f  cos (me) cos ( T )  W2-ph[2n2(R +Wo)W2+a2W: + * 
n2(2VoV2+V;)+w;(2V2 v +v2 )+4w w v v + 
2nw2(R +Wo)Vo,T+2~woW1V1 ,T+2~wlW1VoyT+2~woW2,T v -  
3T 0,T 1 , T  1 0 1 , T  OYT 
0 2,T 1 ¶T 
* * 
Pwow2V~,T+2nwo(R +W )V +2nwl(R +Wo)V + 
* 
(continued) 
414 
v -2Qw w v1-2Qw1w1,Tv0 - 
2Q~0w0,Tv2-2Q~0w2,T 0 1 0,T 
2Qw2w0,T V 0 ]}ded~ (23) 
As in the zeroth and first order cases, noting the inhomogeneities of 
equations (21) and (22), W2 and V2 take the form, namely 
(w2; v,) = (wc2; vc2) cos (2-c) + ... 
+ (wss2; vss2 ) sin (3me) cos (3-c) (24) 
Employing equations (24), (21) ,  and (22), it can be shown that the following 
proportionalities exist, that is 
Hence W and V2 become unbounded for w -O(u,,,f). 
functional , namely equation (23). 
yields the following proportionalities for 012, that is 
The requisite form of w2 can be 
After extensive manipulations, this operation 
obtaine i by inserting equation (24) in?o the second order potential energy 
Thus for wo"O(wmf) , w2-0(l/D2(w0,m)) where, since D2(wmf,m) is singular, w 2  
is itself unbounded and negative definite. 
themselves unbounded at such values of w0. 
Additionally W2 and V2 are 
DISCUSSION 
Stopping the solution at this point, W, V, and w are given by 
(w; v; w)- (wo;  vo; wo) + (wl; VI; wl) E + 
(w2; v,; @,iE2 + 0(~3) (27) 
Due to the procedure employed, it follows that W and V are regular in E, 
including E E 0. This result is,in contrast to standard renormalized 
perturbation procedures which do not yield zeroth order solutions exhibit- 
ing the proper unbounded behavior for w on the order of the linear system 
frequencies. 
The softening behavior of the ring or infinite cylinder can be directly 
obtained by considering the fundamental relationship between w and r. 
Before doing this, the nature of the w0 dependency of w must be ascertained. 
In particular, for wo-O(wmf), 
4 15 
0 - W  + &0(1) * €20 ( )+ 0(&3) 
0 D 2 ( w o , m )  
where since ~2 is negative def ini te  and unbounded, w i s  i t s e l f  negative 
def ini te  and unbounded. 
values of the pencil of equation (13). Note a s  0 i s  set t o  zero, the t w o  pairs 
of eigenvalue branches merge back to  the two frequency branches of the station- 
ary state, and hence, the tradit ional frequencies a re  obtained. 
Such unboundedness occurs a t  each of the eigen- 
Eliminating o from equations (28) and (15), i t  follows tha t  since w is 
unbounded and negative def in i te  for  wo-O(q,,f), the overall steady s t a t e  harmonic 
behavior of the r i n g  or in f in i t e  cylinder i s  of the softening type. 
as .u is  raised o r  lowered, the usual softening type jump phenomenon is 
encountered. 
Hence, 
In the context of the foregoing, the resu l t s  can be summarized by the 
following remarks: 
(1 ) Coriolis forces induce bifurcations i n  the frequency spectrum; 
( 2 )  Such bifurcations extend i n t o  the range of deflections marked by 
moderately large rotations;  
(3) All branches exhibit  a softening type behavior; this applies t o  the 
branches associated w i t h  forward as we1 1 as backward traveling waves; 
(4) Driving frequencies i n  the neighborhood o f  the l inear  system 
frequency may induce jump phenomena; 
(5) S e t t i n g  Q-tO yields the resu l t s  for  stationary rings and cylinders. 
1. 
2. 
3. 
4. 
5. 
6. 
7. 
416 
REFERENCES 
Bert, C.W. and Egle, D.M.,  Journal of Spacecraft and Rockets, Vol. 6, 
Leissa, A.W. , "Vibration of Shells," NASA SP-288, 1973. 
Srinivasan, A.V. and Lauterbach, G.F . ,  Journal of  Engineering for  
Penzes, L .E .  and Kraus, H., AIAA J r . ,  Vol. 10, 1972, pp. 1309-1313. 
Padovan, J . ,  Journal of Sound and Vibration, Vol. 31, No. 4, 1973, 
Padovan, J . ,  Computers and Structures, Vol. 5, 1975, pp. 145-154. 
Padovan, J . ,  International Journal of Sol i d s  and Structures, Vol . 11 , 
No. 12, Bec. 1969, pp. 1345-1361. 
Industry, Vol . 93, 1971 , pp. 1229-1231. 
pp. 469-482. 
1975, pp. 1367-1380. 


Coriolis effects on nonlinear oscillations of rotating cylinders and rings

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