T.K. Caughey- Whirling of a Heavy Chain

8/3/2019 T.K. Caughey- Whirling of a Heavy Chain

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REPRINTED FROM THE

F TH E T H I R D U.S. N A T I O N A L C O N G R E S S O F

HELD ATBROWNUNIVERSITYPROVIDENCE, RHODE ISIANDJUNE 11-14, 1958

E D I T O R I A L C O M M I T T E E

R. M. Haythornthwaite,Chbirman cmdEditor of the Proceedings

Lloyd H. bonnellM. M. FrochtJ. N. Goodier4irthuri KantrowitzHans W. LiepmannPaul M. NoghdiN. M. NewmarkWalter RambergHunter RouseW. R. SearsJ. J. StokerEdward Wenk, Jr.

P U B LI SH E D O N B E H AL F OF T H E C O N G R E S S B YI C A N S OC I E T Y OF M E C H A N I C A L E N G I N E E R S

29 WEST 39th STREET NEW YORK 18, NEW YORK


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WHIRLING O F A H E A V Y CHAIN

of a heavy chain. It i s shown tha t for a given mode of whirl ing, tr ip le valued T . K . Cuughe-yare obtained for speeds of rotation above the l inear cr i t ic al speed for thate. I t is also shown that two of these solutions are stable while the third i s California Institute of Technology

is problem i s of some antiquity, the linea rized solu-D. Bernoulli (1700-1782)

L. Euler (1707-1783). A simple experiment with as sufficient to show that the linearized solu-

lem. Almost two hundred year s pas se d before a moreble solution wa s obtaine d which included the non-

In 1955 Kolodner [ l I a t New York Uni-

s a continuous function of the s peed of

a heavy chain. It willr sp eed s of rotation above the linear

tic al spe ed for that mode. It will further be shown

Formulation

ut a vertical ax is with an angular velocity o, thet from the

s of rotation, the lower end being free.is to determise the possible configura-

ch a chain wilI tak e in st eady rotation. By con-in comparison with t he other force s acting on

Equations of Motion

Denote by:S = the arclength measured from the free end of the

chain.Z = (x(S,t), y(S,t), z(S,t)j -t he position vector of a

point in the chain.

T(S,t) = the tensiong= o,o, -g) - he acceleration of gravity vector.Consider the motion of the chain in a set of rectangu-

lar coordinate ax es rotating about the z axis with anangular velocity a.

The equations of motion are then

With the end conditions

Writing out the three components of (1)

The geometric condition (2) becomes

Steady State SolutionsFor a stead y state of rotation, the velocities and

accel erati ons in the rotating coordinate syst em must bezero.Hence


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Equations (41, (51, an d (6) therefore reduce to:I "

(8)Before proceedi in^ to the solution of (12) and ( 1 3 ) ,-consider the lineariz ed problem given by assuming xd- T(s) g) - p o 2 y ( 9 ) y small. In this ca se i t can easi ly be shown that thedS solution of ( 1 2 ) and ( 1 3 ) is:

Integrating ( 10 ) and using the boundary condition a t S = 0 y = O J provided o i f o

Substituting ( 11 ) into ( 8 ) and ( 9 ) and using ( 7 )

where

a i a r e theroo ts of Jo(ai) O (and

i s the critic al speed for the i th mode in free whirlingFrom ( 15 ) it will be observed that a s o- i ,A i- ; thus, in the vicinity of the crit ica l spe ed asingle term represents the summation with considerab

accuracy. In the ca se of the non-linear problem it sex reasonable to e xpress the solution in the form:

where the A;s are to be chosen to sat isf y the differ-ential equation

I Thu s the problem of solv ing the non-linear differentiaequation i s reduced t o the problem of solving an infinFTGURE I. FORCED WHIRLINGOF HEAVY CHAIN. s e t of non-linear algebrai c equations. In @;enera1hi


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prob lem is s c a r c e l y l e s s d i f f i cu l t t h a n t h e s o l ut i o n oft h e d i f f e r e n t i a l eq u a t io n ; h o w e v e r , t h e l i n e a r i z e d s o l u -i o n s u g g e s t s t h a t i n t h e v i c i n i t y of a c r i t i c a l s p e e d a

s i n g l e t e r m i n t h e s u m m a t io n w i l l g i v e a r e a s o n a b l e a p -o t h e s o l u t i o n .

T h u s , t a k e1

x Y. + Ai J o ( c L ~ ( S / L ) ~ ) I n t e g r a t i n g t h e f i r s t t e r m b y p a r t s :(19 )

i s c o n d i t i o n i m p o s e s a n u p p e r b o un d t o Ai i n ( 1 9 ) ,2

a. 1( 2 0 ) ( 2 4 )

S i n c e t h e m a xi m u m v a l u e o f J L ( x ) / x o c c u rs a t x = 0.

T h i s r e s t r i c t io n o n A i s a c o n s e q u e n c e o f t h e a p-(19) a n d h a s n o t h i ng t o d o w it h t h e T h e f i r s t te rm v a n i s h e s s i n c e lo = 0 , b y t h e c h o i c e

( 1 8 ) p e r s e . B e a r i ng t h i s r e s t ri c t io n of T h u s (24 can b e w r i t t e n i n t h e f o rmm i nd , s u b s t i t u t e (19) i n t o ( 1 8 ) . T h u s

w2 .- A i + 2 = L F i ( A i ) ( 2 5 )g / Lw h e r eo2- [ A i J o ( ~ i ( ~ / ~ ) ~ ) + c I=O ( 2 2 ) - 2 rg r . = - ( 2 6 )u i J , ( a i )

o e v a l u a te A , mul t ip ly ( 2 2 ) b y Jo(a i -JVEJn d i n t e- ' J L 1 : ( a i f l r ) d sf r om 0 t o L . T h u s

a n d


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Consider Fi Ai) defined by (27):1) Fi(AJ is an odd function of A i

i . . , FAA2 = -FJ-A 32) Fi(Ai) is a monotonically increasing function of1 A I provided A i < A i*3) The slope of the function Fi(Ai) is:f 7

i s a rnonot0nical1~ ncreasin g function of 1 A i 1 . Thefunction Fi(Ai) has its smallest slope at A i = 0

FTGURE 2. GRAPHICAL SOLUTION OF EQUATION (25 ) .

From Fig. 2 i t i s possib le to construct an amplitude/rotational velocity curve su ch a s that in Fig. 3. Theanalysis breaks down beyond an amplitude A* given by(21) sin ce F i(Ai) is complex beyond this point.

W .2JFi (Ai ) ( -2 Stability of Steady State Solutions- (28b)aAi ~ ~ - 0 In order to establish the range of o within which the

approximate solution (19) is valid and als o to answerthe question of which of the thre e so lutions a, b, or c

Graphical Solution of (25) will occur in steady rotation, i t i s nece ssary to studythe stabil i ty of the stead y st at e solutions.Equation (25) lends itself to simple gaphical interpre-

tation, IfA, = [m2/(g/LII (A 0.WZ 0;2) If- -, then three intersections are possi-g / L g / Lble, one i s positive while two are negative.

Give sm all perturbations t, q , 4 o x , y , z n (41, (5),(6) and (7); thes e will c aus e a small perturbation 7 n thtension T .

Substituting

x = x , + cy = o - t q into (41, (51, (6) and (7) the followiz = z , + [ equations are obtained:T = T , + T


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i A i l

Substituting into (29) and (30)

;. ,((A2-02)[--2wXq) (35)0 indicates steady state solutions.[, q, [ nd 7 and the time dependence may

assumed to be exponential. Thus

Substituting for To rom (11) and using (71, (35) and (36)(33) become


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Subj ect to the boundary condition tha t

Solution of (37 ) and (38)The method of solution is to assume 6 ( S ) and 11 ( S ) to

have the fonn

where the ai)s are the roots of Ja ( a J = 0 .Substitution of ( 40 ) into ( 37 ) and ( 38 ) and the use of

the orthogonality properties of the Be ss el functions willlead to a Hill's determinant for the determination of A .In general the evalua tion of such a determinant i s diffi-cult, and this particular case, almost impossible. Sincethe solutions x , an d To are only approximate it is hardlyworth th e additio nal labor to tr y to compute any but thesimples t approximation to the Hill's determinant.

Thus, to the same order of approximation a s the stea dystate solution take

Substituting (41) nt o ( 3 7 )and ( 38 ) , two simultaneousequations in a j an d bj may be obtained by expanding theleft hand side of ( 3 7 ) and ( 38 ) in a ser ies of Bes se lfunctions and equating the coefficients of theJo ( a ja)erms. Thus

Integrating by parts, Gii may be put into the form


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Frequency Equation Expanding (31, the condition becomes( 4 2 ) and ( 4 3 ) either a = bi = 0 ori

I f w is restricted to the range < o w i + l anddL 0 is the condition forstability for JL


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that solutions a an d b are st able while solution c i sunstable.

Fig. 4 shows a plot of the unstable regions for the i thmode, for 6 small. It will be observed that the approxi-mate solution x = A i ,(aim)ecomes unstable atthe end points of the int erval mi -1 < o < m i + . Withinthis interval there are two sta ble solutions, a largeamplitude solution a and a small amplitude solution b,and an unstable solution c .

Experimental Qbservations

Experiments with a chain rotated by a variable speedmotor show that:I) For a given mode, there are two stable solutions for

rotational sp ee ds above the linear critical spee d fortha t mode.

2) The small amplitude solutions become unstable in thevicinity of the linear critical speeds, giving ri se tomode conversion.

3) Fo r i > 1, the la rge amplitude so luti ons become un-stable for high enough rotational speeds.4) In the lower modes particularly, amplitudes are ob-served which ar e in exc es s of A:, the maximum forapproximate theory.

5) Subharmonic modes of rotation are observed in whicthe fixed end of the chain an integra l numof rotations in the time that the free end takes tomake one complete revolution.

Observations 1) and 2) are in agreement with the approximate theory of this paper and cannot be explainedby the linear ized theory. Observat ions 3), 4) and 5 ) cnot be explained by the approximate theory and will require a much refined analysis.

1.Heavy Rot at ing String-'4 Nonlinea r EigenvalueProblem, by Ignace I. Kolodner, Communications onPure & Applied Mathematics, Vol. VIII, pp. 395-40(1955).

T.K. Caughey- Whirling of a Heavy Chain

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