NASA Technical Memorandum 102 134

.. .

AVSCOM Technical Report 89-C-004

Topology of Modified Helical Gears

F.L. Litvin and J . Zhang University of Illinois at Chicago Chicago, Illinois

R. F. Handschuh Propulsion Directorate U. S. Army Aviation Research and Technology Activity-A VSCOM Lewis Research Center Cleveland, Ohio

and

J.J . Coy Lewis Research Center Cleveland, Ohio

Prepared for the 5th International Power Transmission and Gearing Conference sponsored by the American Society of Mechanical Engineers Chicago, Illinois, April 24-27, 1989

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\ {NASS-Tff-102134) TOPOLOGY OP MODIFIED 189- 280 15 H B I C A L GEARS @&SA. ~ L e w i s Besearch Center) 21 p CSCL 131

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https://ntrs.nasa.gov/search.jsp?R=19890018644 2018-04-22T17:04:31+00:00Z

TOPOLOGY OF M O D I F I E D HELICAL GEARS

I .

' .

F .L . L i t v i n and J . Zhang U n i v e r s i t y o f I l l i n o i s a t Ch icago

Depar tment o f Mechan ica l E n g i n e e r i n g Ch icago, I l l i n o i s 60680

R . F . Handschuh P r o p u l s i o n D i r e c t o r a t e

U . S . Army A v i a t i o n Research and Techno logy A c t i v i t y - AVSCOM Lewis Research Cen te r C l e v e l a n d , O h i o 4 4 1 3 5

and

J . J . Coy N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o n

Lewis Research Cen te r C l e v e l a n d , G h i o 44115

SUMMARY

The t o p o l o g y of s e v e r a l t ypes of m o d i f i e d s u r f a c e s o f h e l i c a l gear; a r e p roposed. The m o d i f i e d sur faces a l l o w a b s o r p t i o n o f l i n e a r or a l m o s t l i n e a r

=I- f u n c t i o n o f t ransmis ; ion errs-s caused by gear r n i s a l i g n m e n t , and improvement o f t h e c o n t a c t o f gear t o o t h s u r f a c e s . P r i n c i p l e s and c o r r e s p o n d i n g programs

l o p e d . The r e s u l t s o f t h i s i n v e s t i g a t i o n a r e i l l u s t r a t e d w i t h n u m e r i c a l examples.

Ln 0 d- W I f o r computer a i d e d s i m u l a t i o n o f meshing and c o n t a c t o f gears have been deve-

1 . INTRODUCTION

T r a d i t i o n a l methods f o r g e n e r a t i o n of i n v o l u t e h e l i c a l gears w i t h p a r - a l l e l axes p r o v i d e deve loped r u l e d t o o t h s u r f a c e s f o r t h e gear i z e t h ( f i g . 1 ) . The tooth surfaces c o n t a c t each o t h e r a t e v e r y i n s t a n t a l o n g a l i n e , L , t h a t i s t h e t a n g e n t t o t h e h e l i x on t h e base c y l i n d e r . The s u r f a c e no rma ls a l o n g L do n o t change t h e i r o r i e n t a t i o n . The d i s a d v a n t a g e o f r e g u l a r h e l i c a l gears i s t h a t t h e y a r e very s e n s i t i v e t o m i s a l i g n m e n t s such as t h e c r o s s i n g or i n t e r - s e c t i o n o f gear axes . The m i s a l i g n e d gears t r a n s f o r m r o t a t i o n w i t h a l i n e a r f u n c t i o n or t r a n s m i s s i o n errors and t h e b e a r i n g c o n t a c t i s s h i f t e d t o t h e edge o f t e e t h . The f r e q u e n c y o f t r a n s m i s s i o n er rors c o i n c i d e s w i t h t h e f r e q u e n c y of t h e change of t e e t h b e i n g i n mesh. The a c t u a l c o n t a c t r a t i o ( t h e average number o f t e e t h b e i n g i n mesh a t e v e r y i n s t a n t ) i s close t o one and i s f a r from t h e expec ted v a l u e .

These a r e t h e reasons why we have t o r e c o n s i d e r t h e c a n o n i c a l i d e a s on i n v o l u t e h e l i c a l gears and m o d i f y t h e i r t o o t h s u r f a c e s . Crowning t h e gear su r - f a c e i s needed t o negate t h e e f f e c t s o f t r a n s m i s s i o n e r r o r s and t h e s h i f t o f c o n t a c t between t h e gear t o o t h s u r f a c e s . D e v i a t i o n s o f s c r e w i n v o l u t e gear t o o t h s u r f a c e s t o p r o v i d e a new t o p o l o g y t h a t can reduce t h e gear s e n s i t i v i t y t o m i s a l i g n m e n t w i l l be deve loped. T h e o r e t i c a l l y t h e m o d i f i e d t o o t h s u r f a c e s w i l l be i n c o n t a c t a t e v e r y i n s t a n t a t a p o i n t i n s t e a d o f a l i n e . A c t u a l l y , due t o t h e t r a n s m i t t e d l o a d t h e c o n t a c t w i l l be spread o v e r an e l l i p t i c a l a r e a

whose d imens ions may be c o n t r o l l e d . Methods f o r gear t o o t h s u r f a c e g e n e r a t i o n t h a t p r o v i d e t h e d e s i r a b l e s u r f a c e d e v i a t i o n a r e proposed. For economica l r e a - sons o n l y t h e p i n i o n t o o t h su r face i s m o d i f i e d w h i l e t h e gear t o o t h su r face i s k e p t as a r e g u l a r screw i n v o l u t e s u r f a c e .

2 . SIMULATION OF M E S H I N G

The i n v e s t i g a t i o n o f i n f l u e n c e of gear m i s a l i g n m e n t r e q u i r e s a n u m e r i c a l s o l u t i o n for t h e s i m u l a t i o n o f meshing and c o n t a c t o f gear t o o t h s u r f a c e s . The b a s i c i d e a s o f t h i s method ( r e f . 2 ) a r e as fol lows:

( 1 ) The meshing o f gear t o o t h s u r f a c e s i s c o n s i d e r e d i n a f i x e d coor- d i n a t e system, Sf . U s u a l l y , t h e genera ted gear t o o t h s u r f a c e s may be r e p r e - sen ted i n a t h r e e p a r a m e t r i c form w i t h an a d d i t i o n a l r e l a t i o n between these parameters - Gaussian c o o r d i n a t e s . Such a form i s t h e r e s u l t of r e p r e s e n t a - t i o n of t h e gear t o o t h s u r f a c e as an enve lope o f t h e f a m i l y o f t h e tool su r - face ( t h e g e n e r a t i n g s u r f a c e ) and two of t h e t h r e e Gaussian c o o r d i n a t e s a r e i n h e r i t e d from t h e tool s u r f a c e .

The c o n t i n u o u s tangency o f gear t o o t h s u r f a c e s i s r e p r e s e n t e d by t h e f o l l o w i n g e q u a t i o n s

- r ( l ) ( u l ,e, '9, A ~ ) = r ( 2 . 1 )

( 2 . 2 )

( 2 . 3 )

( 2 . 4 )

Here: U i and 6 i a r e t h e tool su r face c u r v i l i n e a r c o o r d i n a t e s , + j i s t h e pa ramete r o f m o t i o n i n t h e p r o c e s s of g e n e r a t i o n o f t h e gear t o o t h s u r f a c e , 8 i 1 s t h e a n g l e o f r o t a t i o n o f t h e gear b e i n g i n mesh w i t h t h e m a t i n g g e a r .

Equa t ions ( 2 . 1 ) t o ( 2 . 4 ) p r o v i d e t h a t t h e p o s i t i o n v e c t o r s r(I) and r(2) - - and s u r f a c e u n i t normals

s u r f a c e s i n c o n t a c t ( f i g . 2 ) . V e c t o r e q u a t i o n s ( 2 . 1 ) and ( 2 . 2 ) y i e l d f i v e

independent e q u a t i o n s and t h e t o t a l e q u a t i o n system i s

and n ( 2 ) a r e equa l f o r t h e gear tooth .., -

An i n s t a n t a n e o u s p o i n t c o n t a c t i n s t e a d o f a l i n e c o n t a c t i s gua ran teed i f t h e Jacob ian d i f f e r s from z e r o , i . e . i f

( 2 . 6 )

2

i f t h e i n e q u a l i t y e q u a t i o n ( 2 . 6 ) i s obse rved , t h e n t h e sys tem o f e q u a t i o n ( 2 . 5 ) may be s o l v e d i n t h e ne ighborhood of t h e c o n t a c t p o i n t b y f u n c t i o n s

( 2 . 7 )

These f u n c t i o n s o f c l a s s t h e f i r s t o r d e r ) . F u n c t i o n s ( 2 . 7 ) and e q u a t i o n s ( 2 . 5 ) p r o v i d e t h e i n f o r m a t i o n o n t h e t r a n s m i s s i o n e r r o r s ( d e v i a t i o n o f $2($1) from t h e p r e s c r i b e d l i n e a r f u n c t i o n ) and t h e p a t h of t h e c o n t a c t p o i n t o v e r t h e gear t o o t h s u r f a c e .

C1 ( a t l e a s t th.y have c o n t i n u o u s d e r i v a t i v e s o f

For t h e case when t h e gear tooth s u r f a c e i s a r e g u l a r screw i n v o l u t e s u r f a c e , i t may be d i r e c t l y r e p r e s e n t e d i n a two-pa ramet r i c form and t h e number o f e q u a t i o n s i n sys tem ( 2 . 5 ) may be reduced t o s i x .

3 . SIMULATION OF CONTACT

Due t o t h e e l a s t i c approach of t h e gear t o o t h s u r f a c e s t h e i r c o n t a c t i s spread o v e r an e l l i p t i c a l a r e a . I t i s assumed t h a t t h e magn i tude o f t h e e l a s - t i c approach i s known from e x p e r i m e n t s or may be p r e d i c t e d . Knowing i n a d d i - t i o n t h e p r i n c i p l e c g r v a t u r e s arid d i r e c t i o n s for two c o n t a c t i n g su r faces a t t h e i r p o i n t o f c o n t a c t we may d e t e r m i n e t h e d imens ions and o r i e n t a t i o n o f t h e c o n t a c t e l l i p s e ( r e f . 2 ) .

The d e t e r m i n a t i o n of p r i n c i p a l c u r v a t u r e s and d i r e c t i o n s f o r a s u r f a c e r e p r e s e n t e d i n a t h r e e - p a r a m e t r i c form i s a c o m p l i c a t e d c o m p u t a t i o n a l p rob - lem. A s u b s t a n t i a l s i m p l i f i c a t i o n o f t h i s p rob lem may be a c h i e v e d u s i n g t h e r e l a t i o n s between p r i n c i p l e c u r v a