a torsional vibrations problem

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A Torsional Vibrations Problem R. T. GRAY S. W. McELHENNY MEMBER AIEE TORSIONAL studies of vibration simple me- 6-CYLINDER AUXILIARY GENERATOR Q Ka chanical systems are usually handled by methods which, while not difficult, often re- quire the expenditure of con- siderable time when done by hand. T h e use of a digital computer shortens the time considerably. One method fre- quently used is the Holzer table calculations resulting in the determination of the natu- ral frequencies of the system. Another method is a matrix solution of the differential equations of the system. While these methods produce good results for simpler systems, the introduction of nonlinear driving functions; discontinuities in the form of gear backlash; and system components, such as hydraulically connected vibration dampers, often require simplifying assumptions which affect the validity of the results. The proposed method overcomes some of these difficulties and predicts the actual operation of the system more accurately. Using the analog computer, nonlinearities and discontinuities in the system function, as well as in the driving function, can be accounted for. The system studied includes a 6-cylinder diesel engine with its torsional damper and two direct-current genera- tors. The main generator is direct-driven by the engine, while the auxiliary generator is driven through a set of spur gears from the main generator. A simplified sche- matic diagram of the system is shown in Fig. 1. The system is broken into discrete lumps and the standard practice for analyzing reciprocating engines is used. The rotating and reciprocating parts are lumped to form average equivalent moments of inertia. The actual system is distributed, of course, and not lumped. A lumped mass torsional spring analogy of a simple mechanical system is shown in Fig. 2, and is used as the basis for the analog computer simulation. The analog computer was used to simulate the system shown in Fig. 1, including the driving torque pulses of the diesel engine, the torsional damper, and the gear branch with backlash. The derivations of analog com- puter parameters for all these simulations are included in the report. During the simulation, the actual system from which test data were obtained was available. A C o x recorder was used to take recordings of the damper out- put, main generator shaft output, and auxiliary gener- ator output (points A, B, and C in Fig. 1). T h e C o x recorder was simulated, and readings were also taken at INDICATES BACKLASH ALL GEAR RATIOS HAVE BEEN TAKEN INTO ACCOUNT ON J AND K. ENGINE FIRING ORDER, 1,5,3,6,2,4 BORE = 9 " STROKE = 10.5" Fig. 1. Schematic of equivalent system. Fig. 2. Schematic of typical system. Ti, Tt, and Ta are the externally applied driving or load torques; Ji, J2, and Ja are the moments of Inertia of the lumped masses; , (a)2, a n d 0 )3 are the angular velocities of the masses; Oi, 2, and Ga ore the angular displacements of the masses with respect to an arbitrary datum; and Ko, Kt, K2, and K3 are the torsional spring constants of the connecting links between the various lumped masses. the same locations of the simulation. Comparisons of test and computer data are included in the report. There appears to be good correspondence between test and computer data of both magnitude and frequency. An interesting observation was that the order and amplitude of vibration are affected by the amount of backlash between gear teeth. Only qualitative results have been obtained; but they indicate that, as backlash is increased, the vibration amplitude increases and the vibration frequency changes u p t o a point. These quali- tative results agree with prior analytical work done by W. A. Tuplin in England. Quantitative relationships for this are not established in this report. Results indi- cate that this method produces better results for a complex system than the Holzer calculation because nonlinearities and discontinuities can be included. Digest of paper 58-502, "An Anal Vibrations Problem," recommended Committee and approved by the AI Computer Study of a Torsional >y the AIEE Land Transportation EE Technical Operations Depart- ment for presentation at the AIEE-ASME Railroad Conference, Cleveland, Ohio, April 9-10, 1958. Scheduled for publication in AIEE Applications and Industry, 1958. R. T. Gray and S. W. McElhenny are with the General Electric Com- pany, Erie, Pa. DECEMBER 1958 Gray, McElhenny—Torsional Vibrations Problem 1109

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Page 1: A torsional vibrations problem

A Torsional Vibrations Problem R . T . G R A Y S. W . M c E L H E N N Y

MEMBER AIEE

Τ T O R S I O N A L s tud ies of

v i b r a t i o n s i m p l e me-

6 - C Y L I N D E R

A U X I L I A R Y G E N E R A T O R

Q Ka

c h a n i c a l systems a r e u s u a l l y h a n d l e d by m e t h o d s w h i c h , w h i l e n o t difficult, o f ten re­q u i r e t h e e x p e n d i t u r e of con­s i d e r a b l e t i m e w h e n d o n e by h a n d . T h e use of a d i g i t a l c o m p u t e r s h o r t e n s t h e t i m e cons ide r ab ly . O n e m e t h o d fre­q u e n t l y u s e d is t h e H o l z e r t ab l e c a l c u l a t i o n s r e s u l t i n g i n t h e d e t e r m i n a t i o n of t h e n a t u ­ra l f r equenc i e s of t h e sys tem. A n o t h e r m e t h o d is a m a t r i x s o l u t i o n of t h e d i f fe ren t i a l e q u a t i o n s of t h e system.

W h i l e these m e t h o d s p r o d u c e g o o d r e su l t s for s i m p l e r systems, t h e i n t r o d u c t i o n of n o n l i n e a r d r i v i n g f u n c t i o n s ; d i s c o n t i n u i t i e s i n t h e f o r m of g e a r b a c k l a s h ; a n d sys tem c o m p o n e n t s , such as h y d r a u l i c a l l y c o n n e c t e d v i b r a t i o n d a m p e r s , o f ten r e q u i r e s imp l i fy ing a s s u m p t i o n s w h i c h affect t h e va l i d i t y of t h e resu l t s . T h e p r o p o s e d m e t h o d ove rcomes s o m e of these difficulties a n d p r e d i c t s t h e a c t u a l o p e r a t i o n of t h e system m o r e a c c u r a t e l y . U s i n g t h e a n a l o g c o m p u t e r , n o n l i n e a r i t i e s a n d d i s c o n t i n u i t i e s i n t h e system f u n c t i o n , as we l l as i n t h e d r i v i n g f u n c t i o n , can be a c c o u n t e d for.

T h e system s t u d i e d i n c l u d e s a 6-cyl inder d iese l e n g i n e w i t h its t o r s i o n a l d a m p e r a n d t w o d i r e c t - c u r r e n t g e n e r a ­tors . T h e m a i n g e n e r a t o r is d i r e c t - d r i v e n by t h e e n g i n e , w h i l e t h e a u x i l i a r y g e n e r a t o r is d r i v e n t h r o u g h a set of s p u r gears f rom t h e m a i n g e n e r a t o r . A s impl i f i ed sche­m a t i c d i a g r a m of t h e sys tem is s h o w n in F ig . 1 . T h e system is b r o k e n i n t o d i sc re te l u m p s a n d t h e s t a n d a r d p r a c t i c e for a n a l y z i n g r e c i p r o c a t i n g e n g i n e s is u sed . T h e r o t a t i n g a n d r e c i p r o c a t i n g p a r t s a r e l u m p e d t o fo rm a v e r a g e e q u i v a l e n t m o m e n t s of i n e r t i a . T h e a c t u a l system is d i s t r i b u t e d , of cou r se , a n d n o t l u m p e d .

A l u m p e d mass t o r s i o n a l s p r i n g a n a l o g y of a s i m p l e m e c h a n i c a l sys tem is s h o w n i n F ig . 2, a n d is u s e d as t h e basis for t h e a n a l o g c o m p u t e r s i m u l a t i o n .

T h e a n a l o g c o m p u t e r was u s e d t o s i m u l a t e t h e sys tem s h o w n in F ig . 1 , i n c l u d i n g t h e d r i v i n g t o r q u e pu l s e s of t h e diesel e n g i n e , t h e t o r s i o n a l d a m p e r , a n d t h e g e a r b r a n c h w i t h b a c k l a s h . T h e d e r i v a t i o n s of a n a l o g com­p u t e r p a r a m e t e r s for a l l these s i m u l a t i o n s a r e i n c l u d e d in t h e r e p o r t . D u r i n g t h e s i m u l a t i o n , t h e a c t u a l sys tem f rom w h i c h test d a t a w e r e o b t a i n e d was a v a i l a b l e . A C o x r e c o r d e r was u s e d t o t a k e r e c o r d i n g s of t h e d a m p e r o u t ­p u t , m a i n g e n e r a t o r shaf t o u t p u t , a n d a u x i l i a r y gene r ­a t o r o u t p u t ( p o i n t s A , B , a n d C i n F ig . 1 ) . T h e C o x r e c o r d e r was s i m u l a t e d , a n d r e a d i n g s w e r e a l so t a k e n a t

• INDICATES B A C K L A S H

A L L G E A R RATIOS HAVE B E E N TAKEN INTO A C C O U N T ON J A N D K. E N G I N E F I R I N G ORDER, 1,5,3,6,2,4 BORE = 9 " S T R O K E = 10.5"

Fig. 1 . Schematic of equivalent system.

Fig. 2. Schematic of α typical system. Ti, Tt, a n d Ta are the externally applied driving or load torques; Ji, J2, a n d Ja are the moments of Inertia of the lumped masses; ωι, (a)2, a n d 0)3 are the angular velocities of the masses; Oi, Θ2, a n d Ga ore the angular displacements of the masses with respect to an arbitrary datum; a n d Ko, Kt, K2, a n d K3 are the torsional spring constants of the connecting links between the various lumped masses.

t h e s a m e l o c a t i o n s of t h e s i m u l a t i o n . C o m p a r i s o n s of tes t a n d c o m p u t e r d a t a a r e i n c l u d e d i n t h e r e p o r t . T h e r e a p p e a r s t o b e g o o d c o r r e s p o n d e n c e b e t w e e n test a n d c o m p u t e r d a t a of b o t h m a g n i t u d e a n d f r equency .

A n i n t e r e s t i n g o b s e r v a t i o n was t h a t t h e o r d e r a n d a m p l i t u d e of v i b r a t i o n a r e affected b y t h e a m o u n t of b a c k l a s h b e t w e e n g e a r t e e t h . O n l y q u a l i t a t i v e resu l t s h a v e b e e n o b t a i n e d ; b u t t h e y i n d i c a t e t h a t , as b a c k l a s h is i nc r ea sed , t h e v i b r a t i o n a m p l i t u d e increases a n d t h e v i b r a t i o n f r e q u e n c y c h a n g e s u p t o a p o i n t . T h e s e q u a l i ­t a t i v e r e su l t s a g r e e w i t h p r i o r a n a l y t i c a l w o r k d o n e by W . A . T u p l i n i n E n g l a n d . Q u a n t i t a t i v e r e l a t i o n s h i p s for t h i s a r e n o t e s t a b l i s h e d i n th i s r e p o r t . R e s u l t s i n d i ­c a t e t h a t t h i s m e t h o d p r o d u c e s b e t t e r r e su l t s for a c o m p l e x sys tem t h a n t h e H o l z e r c a l c u l a t i o n b e c a u s e n o n l i n e a r i t i e s a n d d i s c o n t i n u i t i e s c a n b e i n c l u d e d .

Digest of paper 58-502, "An Anal Vibrations Problem," recommended Committee and approved by the AI

Computer Study of a Torsional >y the AIEE Land Transportation EE Technical Operations Depart­

ment for presentation at the AIEE-ASME Railroad Conference, Cleveland, Ohio, April 9-10, 1958. Scheduled for publication in AIEE Applications and Industry, 1958.

R. T. Gray and S. W. McElhenny are with the General Electric Com­pany, Erie, Pa.

D E C E M B E R 1 9 5 8 Gray, McElhenny—Torsional Vibrations Problem 1 1 0 9