a torsional vibrations problem
TRANSCRIPT
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