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Mech. Mac Theory

Vol. 22, No . 6 , pp. 563--568, 1987 0094-114X/87 3.0 0+ 0.00

Pr in t ed i n Gr ea t Br i t a in Pe r gam on Jour na l s L td

K I N E M A T I C S Y N T H E S I S A N D A N A L Y S I S O F T H E R A C K

A N D P I N I O N M E C H A N I S M F O R P L A N A R P A T H

G E N E R A T I O N A N D F U N C T I O N G E N E R A T I O N F O R

S I X P R E C I S I O N C O N D I T I O N S "

PETER DUSCH L

Technology Service Center, Eastern Michigan University, Ypsilanti, M I 48197, U.S.A.

STEVEN KRAM ER

Department of Mechanical Engineering, University of Toledo, Toledo, OH 43606, U.S.A.

Received

19

June

1986)

Ab stract- -T he rack and pinion mechanism is synthesized for generat ing both three prescribed path points

with input coo rdinat ion and three po si t ions of function generat ion. This mechanism has a number o f

advantages over the four bar f inkage. F irs t , since the ra ck is a lways tangent to the pinion , the transmiss ion

angle is always the same (optimum) value of 90 ° minus the pressure angle of the pinion. Second, with

both translat ion and rotat ion of the rack occurr ing, mult iple outputs are available . Other advantages

include the generat ion of monotonic functions for a wide var iety of motion and nonmononotomic

functions for the full range o f mo tion as well as nonlin ear amplified motions. In this work, the m echanism

is made to satis fy a num ber o f pract ical des ign requirements such as a completely rotatab le input crank

and others . The method of solut ion developed in this work ut i l izes the complex number method of

mechanism synthesis and the solut ion is program med on a VAX 11/785 comp uter and is being mad e

availa ble to interested readers.

I N T R O D U T I O N

T h e r a c k a n d g e a r m e c h a n i s m ( F i g . 1 ) h a s b e e n

s t u d i e d b y a n u m b e r o f r e s e a rc h e r s o v e r t h e

y e a r s [ I -6 ] . T h i s w o r k e x t e n d s t h e w o r k d o n e b y t h e

s e c o n d a u t h o r o n t h e s o l u t i o n o f t h e t h r e e p o i n t

f u n c t i o n g e n e ra t io n [ 3 ] a n d f o u r p o i n t p a t h g e n e r a t i o n

w i t h p r e s c r i b e d i n p u t t i m i n g [ 5 ] t o c o m b i n e p a t h

g e n e r a t i o n a n d f u n c t i o n g e n e r a t i o n i n t h e s a m e m e c h -

a n i s m t o a c h i e v e m u l t i p l e o u t p u t s .

T h e r a c k a n d g e a r m e c h a n i s m ( F i g . 2 ) i s c o m p o s e d

o f a n i n p u t c r a n k , Z 2 , w h o s e r o t a t i o n , ~ j , i s s p e c i f i e d

b y t h e m e c h a n i s m d e s i g n e r. T h e r a c k , Z ( , i s in

n o n s l i p c o n t a c t w i t h t h e p i n i o n s u c h t h a t i t r o t a t e s ~ j

a n d t r a n s l a t e s . T h e o f f se t , Z s , i s r i g i d l y c o n n e c t e d t o

Z ( a n d a l l o w s f o r g e n e r a l i t y b u t i n s o m e c a s e s m a y

b e o m i t t e d . T h e p i n i o n , w h o s e r a d i u s v e c t o r i s Z s ,

r o t a t e s a s t h e r a c k r o t a t e s a n d t r a n s la t e s . T h e v e c t o r

Z ~ d e f i n e s t h e t r a c e r p o i n t o f t h e p a t h g e n e r a t i n g

m e c h a n i s m a n d i s r i g i d l y a t t a c h e d t o t h e r a c k . T h e

f i x e d l i n k , Z j , c o n n e c t s t h e t w o f i x e d p i v o t s .

T h i s s p e c i a l i z ed m e c h a n i s m i s s i m i l a r t o t h e p r i s -

m a t i c m e c h a n i sm [ 7 , 8] i n t h a t t h e v e c t o r s Z s , Z ( a n d

Z 5 a l l r o t a t e w i t h t h e s a m e a n g l e , s i n c e Z ( i s a l w a y s

p e r p e n d i c u l a r t o Z s . T h e r a c k a n d p i n i o n m e c h a n i s m ,

h o w e v e r , p r o d u c e s a n a d d i t i o n a l o u t p u t , ~ , w h i c h i s

t h e r o t a t i o n o f t h e p i n i o n . T h e r a c k a n d p i n i o n

m e c h a n i s m h a s i n d u s t r i a l a p p l i c a t i o n s i n t h e p a c k -

t This paper is based on wo rk per formed at The Univers i ty

of T oledo, Ohio.

a g i n g i n d u s t r y a s w e l l a s t o y s a n d o t h e r l e i s u r e

e q u i p m e n t . T h e i n v e r s i o n o f t h e m e c h a n i s m , w h e r e

t h e p i n i o n i s t h e d r i v e r , c a n b e u s e d i n m e c h a n i c a l

a i r c r a f t c o n t r o l d e v i c e s , h o s p i t a l a n d l a b o r a t o r y

e q u i p m e n t , r a c k a n d p i n i o n a u t o m o t i v e s t e e r i n g

l i n k a g e s [ l l ] a n d s e v e r al t a s k s o n m a n u f a c t u r i n g

a s s e m b l y l i n e s .

M E T H O D O F SO L U T I O N

T h e v e c t o r r e p r e s e n t a t i o n o f th e r a c k a n d p i n i o n

m e c h a n i s m a l o n g w i t h c o o r d i n a t e a x e s a r e s h o w n i n

F i g . 2 i n t h e i n i t i a l d e s i g n p o s i t io n . T h e i n p u t c r a n k

r o t a t i o n s , ~ b j, a n d t h e p o s i t i o n v e c t o r s , 1 ~ , a r e k n o w n

b e c a u s e o f t h e p a t h g e n e r a t i o n s p e c i fi c a ti o n , w h e r e

j - - 1 , 2 a n d 3 . T h e m a g n i t u d e s o f a l l l i n k s a r e

u n k n o w n b u t t h e o r i e n t a t i o n s o f Z 3 a n d Z s r e l a t iv e

t o Z ( a r e k n o w n . S i n c e Z 3 i s r i g i d l y c o n n e c t e d t o Z ,

a t a r i g h t a n g l e a n d Z s i s p e r p e n d i c u l a r t o Z 4 d u e t o

t h e r a c k s t a n g e n c y t o t h e p i n i o n , t h e f o l l o w i n g

r e l a t i o n s h i p s a r e t r u e [ 3 ] :

Z 3 = Z4hse~n/2) = Z 4h3i (1)

a n d

Z s - - Z 4 h se 1 (- ~/2) ~ - Z 4 h 5 ( - i ) , ( 2 )

w h e r e t h e v a r i a b l e s h a n d h 5 ( s c a l a r u n k n o w n s a t t h i s

p o i n t ) a r e

h 3 ~ - I g s l

I z I

(3)

563

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564

PETER DUSCH Land STEVENKRAMER

Fig. 1. The rack and gear mechanism

a n d

IZ sI

h5 = [ 4----/ (4 )

A l t h o u g h h 3 a n d h 5 a r e r a t i o s o f m a g n i t u d e s , e i th e r

m a y b e n e g a t i v e t o i n d i c a t e a d i r e c t i o n o p p o s i t e t o

tha t a ssum ed in equa t ions (1 ) and (2 ) . Phys i ca l l y

s p e a k in g , t h e o f fs e t n e e d n o t b e p e r p e n d i c u l a r t o Z 4

bu t k inem a t i ca l l y speak ing t he re wi l l a lways be a

p e r p e n d i c u l a r v e c t o r , Z 3 , w h i c h c a n b e d r a w n . S i n ce

Z 3 and Z 6 a re r i g id ly conn ec t ed , t he fo l l owing

de f in i t i on i s adop ted fo r s im pl i c i t y :

Z36 = Z3 + Z6. (5)

S i n c e t h e m a g n i t u d e o f l i n k Z 4 d o e s n o t r e m a i n

cons t an t , a s t r e t ch r a t i o i s de f ined a s t he r a t i o o f t he

m a g n i t u d e o f Z 4 i n i ts j t h p o s i t i o n t o i ts i n i ti al

m a g n i t u d e s o t h a t

IZ4jl (6)

K j= 141

F o r p a t h g e n e r a t i o n , t h e l o o p c l o s u r e e q u a t i o n c a n

be wr i t t en f rom Fig . 2 fo r t he i n i t i a l pos i t i on a s :

Z 0 + Z 2 + Z ~ = R I . 7 )

Y

Fig. 2. The vector representation of the rack and gear

mechanism in its initial design position.

A d d i t i o n a l l o o p c l o s u r e e q u a t i o n s c a n b e w r i t t e n f o r

a n y ( j t h ) p o s i t i o n w h i c h r e p r e s e n t s t h e g e n e r a l

d i s p l a c e d p a t h p o s i t i o n . T h e y a r e

Z0 + Z~e~j + Z~ e ~j = Rj. (8)

sub t rac t i ng t he i n i t i a l pos i t i on f rom the gene ra l pos-

i t i on y i e lds t he fo l l owing l oop c losu re equa t ion :

Z2(eic~j -- 1) + Z~(e/~j -- 1)

= ( R I - R I ) f or j = 2 , 3 . (9)

F o r f u n c t i o n g e n e r a t i o n , a n e q u a t i o n c a n b e w r i t t e n

f rom Fig . 2 fo r t he i n i t i a l pos i t i on a s

Z2 + Z3 + Z4 + Z5 = Zt . (10)

A d d i t i o n a l f u n c t i o n g e n e r a t i o n e q u a t i o n s c a n b e

w r i t t e n f o r a n y ( j t h ) p o s i t i o n w h i c h r e p r e s e n t s t h e

g e n e r a l p o s i t i o n . T h e y a r e

Z 2 e i * j +

Z3eeej +

k j Z 4 e e e j

+ Z s e e j = Z l f or j = 2 , 3 . (11)

Sub t rac t i ng t he i n i t i a l pos i t i on f rom the gene ra l

p o s i t i o n a n d u s i n g e q u a t i o n s ( 1 ) a n d ( 2 ) y i e l d

Z2(e~ j - 1) + Z4[(h3 -

h s ) i e ° j -

1)

+ e ~ j k j - l ] = O

f o r j = 2 , 3 . (1 2)

Re fe r r i ng t o F ig . 3 , t he r e l a t i onsh ip be tween t he

p i n i o n a n d r a c k r o t a t i o n s c a n b e e x p r e s s ed u s i n g t h e

pr inc ip l e o f supe rpo s i t i on a s Re f . [3]

k - 1

~b = 7 + - - (13)

h5

E q u a t i o n ( 1 3 ) c a n b e w r i t te n i n t h e g e n e ra l f o r m f o r

s u b s e q u e n t p o s i t i o n s a s

~ j = y j + ( k j + l ) f or j = 2 , 3 . (14)

h5

A t ab l e[ 1 0 ] c a n b e c o n s t r u c t e d t o d e s c r ib e w h y t h e

n u m b e r o f p a t h a n d f u n c t i o n a l p o s i t i o n s f o r th i s

m e c h a n i s m c a n b e o b t a i n e d . T a b l e 1 i s s u c h a t a b l e

w h e r e t h e l e tt e rs p a n d f i n c o l u m n 1 r e f er t o

p a t h a n d f u n c t i o n r e s p ec t iv e l y, a t t h a t p o s i t i o n n u m -

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Kin em at i c sy nthes is an d an a l y s is o f t h e r ack an d p in io n mech an i sm 5 65

I I ~ q

@ -- @

t~5 : Y

Fig . 3 . The gear ro tat ion ¢ , i s found by super imposing the effects o f the rack 's ro tat ion an d t ranslat ion .

Tab l e 1 . D e t e rmin a t i o n o f th e m ax imu m n u m b er o f p o ssi b le p o s it i o ns o f

t h e mech an i sm

Nu m b er o f S cal ar S ca l a r F ree

posi t ions unkn ow ns equa t ions cho ices

p , f

12 ( Z o , Z I , Z 2 , Z 3 , Z 4 , Z s , Z ~ ) 4 8

2p 13 (same as abov e + Y2) 6 7

2 f 14 (same as

a b o v e

+ k2) 9 5

3p 15 (same as abov e + ~'3) 11 4

3 f 16 (same as above + k3) 14 2

b e r. T h e n u m b e r o f s c a la r u n k n o w n s a n d s c a la r

e q u a t i o n s a s a f u n c t i o n o f t he n u m b e r o f p r e s c r ib e d

p a t h a n d f u n c t i o n p o s i t i o n s a r e s h o w n t o b e s o lv a b l e

a t t h r e e p r e s c r i b e d p a t h p o s i t i o n s a n d t h r e e f u n c t i o n

p o s i t i o n s w i t h t w o a r b i t r a r y c h o i c e s . I t i s i m p o r t a n t

t o n o t e t h a t Z 3 a n d Z s a r e e a c h c o n s i d e r e d a s o n e

a d d i t i o n a l sc a l a r u n k n o w n b e c a u s e o f e q u a t i o n s

( 1 ) - ( 4 ) . S o i n r o w 1 , t h e s e v a r ia b l e s c o u l d h a v e b e e n

r e p l a c e d w i t h h 3 a n d h s .

T h e s o l u t i o n o f t h e m e c h a n i s m i s f o u n d b y f i r s t

t r a n s f o r m i n g e q u a t i o n ( 9) i n t o m a t r i x n o t a t i o n f o r

j = 2 an d 3 as fo l l o ws :

[ ( e ' O 2 - 1 ) ( e 2 - 1 ) 1 1 - 1

I - R . -

( e * 3 - l ) ( e O 3 - I J L Z ~ j = L R - R J ( 1 5 )

S i n c e t h e f r e c c h o i c e s a r c c h o s e n t o b e ~ 2 a n d ~ 3 , l i n k

v e c t o r s Z 2 a n d Z 3 ~ a r c t h e n e a s i l y d e t e r m i n e d t o b e

Z 2 = [ ( R 2 - R i ) ( e O 3 - 1 )

- - ( R 3 - R I ) ( e O 2 - I ) ] /

[ e ~ * 2 - l ) e o 3 - I )

- e ~ * 3 - I ) C 2 - I ) ]

a n d 1 6 )

Z s 6 = [ ( e ~ 2 l ) ( R s - R I )

- ( e * 3 - I ) ( R ~ - R 0 ] /

[ ( e ' # 2 - 1 ) ( e 3 - 1)

- ( ei #3 - 1 ) (e~2 - 1 ) ].

F o r t h e f u n c t i o n g e n e r a t io n p a r t o f t h e m e c h a n i s m ,

e q u a t i o n ( 1 4 ) i s s o l v e d f o r

kj:

k / = ~ j - - T j ) h s + l , f o r j 2 , 3 , ( 17 )

a n d t h i s r e s u l t i s t h e n s u b s t i t u t e d i n t o e q u a t i o n ( 1 2 )

t o y i e l d

Z4 {(C :2 - 1 ) (h3 -

hs)i +

[C:2(~b2 - )'2)h5

+ 1 ] - 1} - Z ~ ( e ' * 2 - 1 ) ( 1 8)

a n d

Z4 {(C :3 - l)(h 3 - h5)i + [e/~3(~//3 - )'3)h5

+ l] - l } = - Z 2 c ~ 3 - I ). 1 9 )

D i v i d i n g e q u a t i o n 1 8 ) b y e q u a t i o n 1 9 ) w i l l e l i m -

i n a t e Z 4 a n d Z 2 f r o m t h e a b o v e e q u a t i o n s a n d y i el d s

h 3 [ A ( C : 2 i - i ) - B ( i e ~ : 3 - i )]

+h 5 {A[e#2(~b2 - ) '2 ) - c#2 i + i ]

- - B [C :3 (~b3 - )73) C :3 i + i]}

= B ( e 3 - 1 ) - A ( e ~ 2 - 1 ) ,

w h e r e

A = ( c i * 3 -

I ) a n d B = c i* 2 I ) . 2 0 )

T h e s a m e r e s u l t s h o u l d a l s o h a v e b e e n o b t a i n e d i f

e q u a t i o n s 1 8 ) a n d 1 9 ) w e r e w r i t t e n i n m a t r i x

n o t a t i o n w h e r e t h e d e t e r m i n a n t w o u l d b e s e t e q u a l t o

z e r o [ 1 0 ]. B y d e f i n i n g

T = A ( e ~ 2 i - i ) - B ( e ~ 3 i - i ) ( 2 1 )

S = A [e~:2 ~b2 - ) '2) - e~ 2i + i]

- B[et t 3 (~3 - ) '3 ) - e~3i + i ] (22)

Q = B(e # 3 - 1 ) - A (e# 2 - 1 ) = - iT (2 3 )

a n d s u b s t i t u t i n g i n to e q u a t i o n ( 2 0 ) t h e f o l l o w i n g

s i m p l if i e d r e l a t i o n s h i p c a n b e w r i t te n :

h

T + h5 S = Q . (24)

S i nc e T , S a n d Q a r e c o m p l e x f u n c t io n s o f k n o w n

q u a n t i t i e s , h 3 a n d h5 c a n b e f o u n d b y s e p a r a t i o n o f

v a r i a b l e s a n d s o l v i n g t h e f o l l o w i n g t w o s c a l a r e q u a -

t i o n s :

Txh3 + Sxh5 = Q x

a n d

r y h 3 -] - S y h 5 = Q y 2 5 )

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566 PETER DUSCHL an d ST~WN K~M F.R

w h e r e t h e s u b s c r i p t s x a n d y r e f e r t o t h e r e a l a n d

i m a g i n a r y p a r t s , r e s p ec t iv e l y . T h e s o l u t i o n o f h 3 a n d

h s is

h a = Q . S y - Q y S ~ ) / T ~ S y - r y S x )

a n d

h 5 =

Q y T x - Q ~ T y ) / T ~ S y - T y S x ) .

(26)

L i n k v e c t o r Z 4 c a n b e f o u n d f r o m e q u a t i o n ( 18 )

- Z , e i ~ 2 - 1 )

Z4 = [(e~2 - 1)(h 3 - h s ) i + (e#2((~2 - y2)h5 + 1 ) - 1)]

(27)

T h e r e m a i n d e r o f t h e l i n k v e ct o r s c a n b e f o u n d b y

equa t ions ( 1) , ( 2 ) , ( 5 ) and ( 10) w her e

Z o = R i - Z ~ - Z 2 . 2 8 )

D E S I G N R E S T R I C T I O N S

O n c e t h e m e c h a n i s m h a s b e e n d e t e r m i n e d , i t i s

c h e c k e d f o r a d h e r e n c e t o p r a c t i c a l d e s i g n r e s tr i c t i o n s .

T h e s e r e s t r i c ti o n s i n c l u d e e l i m i n a t i o n o f e x t r a n e o u s

r o o t s a n d c e r t a i n g e o m e t r i e s w h i c h m a k e t h e m e c h -

a n i s m u n u s a b l e .

F i r s t , a ll m e c h a n i s m s t h a t y i e l d a t l e a s t o n e n e g a -

t i v e v a l u e f o r t h e s t r e t c h r a t i o a r e i n v a l i d s i n c e a

n e g a t i v e k j s ig n i f ie s a r e v e r s e d i r e c t i o n o f t h e r a c k

w h i c h i s p h y s i c a l l y i m p o s s i b l e . S e c o n d , t o i n s u r e t h e

i n p u t c r a n k c a n r o t a t e f u l l y

1221 + IZs I < 1211. (29)

T h i r d , t o i n s u r e t h a t t h e r a c k r e m a i n s i n c o n t a c t

I z 2 I z 3

Z I

Fig. 5. Rack interfering with gear when h3h ~ < O.

A N A L Y S I S

A n a l y z i n g t h e r a c k a n d p i n i o n m e c h a n i s m r e q m r e s

t h e d e t e r m i n a t i o n o f t h e s t re t c h r a t i o , k a n d r a c k

r o t a t i o n , y , f o r a n y a n d a l l m e c h a n i s m p o s i t io n s .

R e w r i t i n g e q u a t i o n ( 1 1 ) r e s u l t s i n t h e f o l l o w i n g

e q u a t i o n :

z

Z 1 - Z ~ e ' * J - Z W * J = e - ' ( 3 2 )

w h e r e n o s u b s c r i p t s a r e u s e d f o r t h e v a r i a b l e s k , tp

a n d y t o i n d i c a t e g e n e r a l i t y . I f t h e t e r m s i n p a r e n t h e -

s e s a r e d e n o t e d b y C a n d D , t h e n s e p a r a t i n g e q u a t i o n

( 32 ) i n t o r e a l a n d i m a g i n a r y p a r t s , s q u a r i n g b o t h

s i d e s o f t h e e q u a t i o n a n d a d d i n g y i e l d

C ~ q - k D x ) 2 q - C y q - k D y )2 = 1. (33)

T h e s u b s c r i p t x r e fe r s t o t h e r e a l p a r t a n d t h e

s u b s c r i p t y r e f e rs t o t h e i m a g i n a r y p a r t o f t h e c o m -

p l e x n u m b e r s . S e p a r a t i n g a n d c o m b i n i n g l i k e t e r m s

i n e q u a t i o n ( 3 3) y i e ld a q u a d r a t i c e q u a t i o n w h i c h ,

w h e n s o l v e d f o r k , y i e l d s

k =

- Cx Dx + C yD y) + x/ 4 C x D . + C yD y) 2 - 4 D 2 + Dy) C.2 2 + Cy2 _ l

(34)

w i t h t h e p i n i o n f o r t h e w o r s t p o s s i b l e c a s e , t h e

f o l l o w i n g m u s t b e t r u e :

IZ2l - t- IZ3l < IZsl + IZ ll i f h3h5 > 0. (30)

F o u r t h , t o i n s u r e t h a t t h e c r a n k d o e s n o t o v e r l a p t h e

p i n i o n , t h e f o l l o w i n g r e l a t i o n s h i p m u s t h o l d :

I Z2 1 l Z 3 1 l Z s I < I Z a l i f h 3 h 5 < 0.

3 1 )

F i g u r e s 4 a n d 5 i l l u s t r a t e t h e s e c o n d i t i o n s .

I f t h e a b o v e r e s t r i c ti o n s a r e m e t , t h e n c o m p l e t e

r o t a t i o n o f t h e i n p u t c r a n k i s i n s u r e d a n d b r a n c h i n g

c a n n o t o c c u r s i n c e h5 w i l l n o t c h a n g e s i g n .

Fig. 4. Rack leaving contact with gear when h3h 5 > 0 .

I t c a n b e s h o w n t h a t v e c t o r s C a n d D w i l l a l w a y s

b e n o r m a l t o e a c h o t h e r , h e n c e t h e i r d o t p r o d u c t w i l l

v a n i s h . T h i s m e a n s t h e t w o r o o t s o f k w i l l d i f f e r o n l y

i n s i g n a n d t h e a b o v e e q u a t i o n r e d u c e s t o

/ 1 I C l :

k = _ 2 IDI2 . (35)

S i n c e a n e g a t i v e v a l u e f o r k i s p h y s i c a l l y i m p o s -

s i b l e , t h e p o s i t i v e r o o t o f k i s u s e d f o r a n a l y s i s . T h e

r a c k r o t a t i o n c a n b e r e a d i l y d e t e r m i n e d f o r a n y

m e c h a n i s m p o s i t i o n f r o m e q u a t i o n ( 3 2 ) .

A l t h o u g h v e c t o r Z s r o t a t e s y b e c a u se i t is t a n g e n t

t o t h e r a c k , t h e p i n i o n w i l l r o t a t e a d i f fe r e n t a m o u n t

d u e t o t h e r a c k ' s t r a n s l a t i o n w i t h r e s p e c t t o t h e

p i n i o n . T h e e q u a t i o n d e t e r m i n i n g t h e p i n i o n r o t a t i o n

c a n b e e x p l a i n e d b y s u p e r p o s i t i o n a s s h o w n i n

Fig. 3 and is[3] :

k - I

= y + - - (3 6 )

h5

F i n a l l y , th e p o s i t i o n v e c t o r R c a n b e e s t a b l i s h e d f o r

a n y a n d a l l m e c h a n i s m p o s i t i o n s fr o m e q u a t i o n ( 8) .

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K i n e m a t i c s y n t h e s is a n d a n a ly s i s o f t h e r a c k a n d p i n i o n m e c h a n i s m

• R3

\

\ \ y

R

F i g . 6 . S y n t h e s i z e d m e c h a n i s m .

N U M E R I C A L E X A M P L E

. R a

567

T o t e s t t h e m e t h o d o f s o l u t i o n d e v e l o p e d i n t h i s

p a p e r , t h e f o l l o w i n g e x a m p l e w a s u s e d . T h e t h r e e

p a t h p r e c i s i o n p o s i t i o n s w e r e ( 3 . 0, 7 . 0 ), ( 7 .0 , 0 . 0) a n d

( 4 .0 , 6 .0 ) . T h e c o r r e s p o n d i n g i n p u t c r a n k r o t a t i o n s

w e r e 0 , 8 0 a n d 1 7 0 ° , r e s p e c t i v e l y . T h e f u n c t i o n t o b e

g e n e r a t e d w a s ~ = 4 0 s i n ~ w h e r e b o t h ~ a n d ~b a r e

i n d e g r e e s a n d t h e p r e c i s i o n p o i n t s a r e a l s o a t ~b = 0 ,

8 0 a n d 1 7 0 ° . R e s p a c i n g o f t h e p r e c i s i o n p o i n t s c a n b e

a c c o m p l i s h e d u s i n g s t a n d a r d t c c h n i q u es [ 4 , 1 0] b u t f o r

s i m p l i ci t y t h e fi rs t f u n c t i o n a n d p a t h p r e c i s i o n p o i n t s

a r e a t t h e i n i t i a l p o s i t i o n i n t h i s e x a m p l e .

T h e m e t h o d o f s o l u t io n d e s c r i b e d i n t h i s p a p e r w a s

p r o g r a m m e d o n a V A X 1 1 /7 8 5 c o m p u t e r . S i n c e 3'2

a n d 3'3 w e r e a r b i t r a r y c h o i c e s m a n y s o l u t i o n s w e r e

g e n e r a t e d . D u e t o t h e d e s i g n c o n s t r a i n t s a m u c h

l i m i te d n u m b e r o f s o l u t i o n s w e re e v e n t u a l l y f o u n d .

O n e s u c h m e c h a n i s m s o l u t i o n f o u n d w h e r e 3 '2 = 2 0 °

a n d 3'3 = 3 0 ° w a s

Z 0 = ( - 8 .4 8 5 , 0 . 4 4 2 ) )

Z I = ( - 9 . 7 9 1 , 3 . 9 2 5)

t

Z 2 = ( 0 .0 4 1 , - 3 . 3 9 0 )

Z 3 = - 7 . 1 6 1 , - 5 . 0 9 0 ) ( 3 7 )

Z 4 = ( - 6 . 7 5 4 , 9 . 5 0 2 )

Z 5 = ( 4 . 0 8 3 , 2 . 9 0 2 )

Z 6 = ( 1 8 .6 0 5 , 1 .0 3 8 ) .

T h e m e c h a n i s m i s s h o w n i n F ig . 6 a n d t h e a n a ly s i s

o f th i s m e c h a n i s m i s s h o w n i n T a b l e 2 . T h e a c c u r a c y

o f t h e f u n c t i o n g e n e r a t i o n c a n b e s e e n i n F ig . 7 w h e r e

t h e i d e a l c u r v e is s o l id a n d t h e g e n e r a t e d c u r v e i s

d a s h e d . T h e m a x i m u m s t r u c tu r a l e r ro r i n th e r a n g e

o f 0 - 1 8 0 ° w a s f o u n d t o b e 1 .6 ° .

T a b l e 2 . A n a l y s i s o f s y n t h e s iz e d m e c h a n i s m

~b ~ ~b K

0 . 0 0 0 . 0 0 0 . 0 0 1 . 0 0 0

10.00 2.44 7.76 1.040

20.00 4.96 14.87 1.074

30.00 7.53 21.23 1.103

40.00 10.11 26.75 1.125

50.00 12.67 31.38 1.140

60.00 15.19 35.05 1.149

70.00 17.65 37.74 1.151

80.00 20.00 39.39 1.145

90.00 22.23 40.00 1.133

100.00 24.29 39.55 1. 114

110.00 26.15 38.03 1.089

120.00 27.77 35.44 1.058

130.00 29.08 31.79 1.020

140.00 30.04 27.09 0.978

150.00 30.57 21.37 0.931

160.00 30.59 14.64 0.880

170.00 30.00 6.94 0.827

180.00 28.70 -- 1 .64 0.772

180.00 26.60 - 11.02 0.718

200.00 23.62 - 21.02 0.665

210 .00 19 .76 -3 1 . 32 0 .617

220 .00 15 .10 -4 1 . 48 0 .576

230.00 9.92 - 50.84 0.544

240.00 4.63 - 58.59 0.526

2 5 0 .0 0 - 0 .2 6 - - 6 3 .9 9 0 .5 2 2

2 6 0.00 - 4 . 3 1 - - 6 6 .5 2 0 .5 3 4

2 7 0.00 - - 7 .2 8 - 6 6 .1 1 0 .5 5 9

2 8 0.00 - - 9 .1 3 - 6 3 . 0 6 0 .5 9 6

2 9 0 .0 0 - 9 .9 6 - - 5 7 .8 6 0 .6 4 1

300.00 - 9 .94 -- 51.07 0.692

310 .00 - -9 .24 - -43 .20 0 .745

320.00 -- 8 .03 -- 34.66 0.800

3 3 0.00 - 6 .4 0 - - 2 5 .8 3 0 .8 5 4

340 .00 -4 .4 8 - 16 .96 0 .906

350.00 - 2 .32 -- 8 .29 0.955

360.00 0.00 0.00 1.000

P a t h v e c t o r

R e a l I m a g i n a r y

3.000 - 7.000

3 . 7 5 0 - 6 . 4 5 0

4.465 - 5 .776

5 .1 2 2 - 4 .9 9 1

5 . 7 0 3 - 4 . 1 0 9

6 . 1 9 2 - 3 . 1 4 9

6 . 5 7 7 - 2 . 1 2 9

6.848 - 1.071

7.00O 0.000

7.031 1.061

6.944 2.085

6.745 3.046

6.444 3.918

6 .056 4 .674

5.597 5.287

5.088 5.730

4.549 5.977

4.000 6.000

3.458 5.774

2.933 5.274

2.426 4.486

1.925 3.411

1.409 2.081

0.863 0.569

0.293 - 1.014

- 0 . 2 5 9 - 2 . 5 4 2

- 0 . 7 2 4 - 3 . 9 1 1

- 1.037 - 5.069

- 1 . 1 6 0 , 6 . 0 0 3

- 1.085 - 6 .726

- 0 . 8 2 7 - 7 . 2 5 5

- 0 . 4 1 1 - 7 . 6 0 6

0 . 1 3 4 - 7 . 7 9 1

0 . 7 7 6 - 7 . 8 1 8

i . 4 8 7 - 7 . 6 9 1

2 . 2 3 7 - 7 . 4 1 6

3.000 - 7.000

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568 PETER DUSCHL and STEvrN KRAMER

4 0

0 -

o= 3

e

I I I I I t I I ~

20 40 60 80 100 120 140 160 180

Cronk rotot ion q5 deg)

F i g . 7 . G e a r r o t a t i o n v s c r a n k r o t a t i o n .

CONCLUSION

The rack and pinion mechanism is a versatile

mechanism because it has two ma jor advantages over

the four bar mechanism. The first is that the trans-

mission angle is always at the same value o f 90 °

minus the pressure angle of the rack. Second, both

path and function generation are simultaneously

attained. In this paper the pinion rotation, which

represents this functional output, is the sum of the

rack rotatio n and the length of arc swept-out by the

pinion divided by its radius. The synthesis of this

single degree-of-frecdom rack and pinion mechanism

for multiple output (path and function generation)

makes it very valuable in machine and mechanism

design.

R E F E R E N E S

1. G. K. Kinzel and S. Chen, A general procedure for the

kinematic analysis of planar mechanisms with higher

pairs, A S M E M e c h . C o n f . Paper No. 84-DET-140

(1984).

2. D. Wilt and G. N. Sandor, Synthesis of a geared

four-link mechanism. J . Mech . 4, 291-302 (1969).

3. D. Gibson and S. Kramer, Kinematic des ign and

analysis of the Rack-and-ear mechanism for function

generation. Mech. Mach. Theory 19, (3) 369-375 (1984).

4. K. E. Hofmeister and S. N. Kramer, Kinematic syn-

thesis, Analysis, and optimization by precision point

respacing of the rack-and-gear function generating

mechanism. A S M E M e c h C on f. Paper 84-DET-134

(1984).

5. M. Claudio and S. Kramer, Synthesis and analysis of

the rack and gear mechanism for four point path

generation with prescribed input timing. A S M E J .

Mech. Transmiss . Automat. Design 108, (1) 10-14

(1986).

6. I. Meyer zur Capellen, Der einfache Zahnstangen-

Kurbeltrieb un Das entsprechende Bandgetriebe. Z.

Mach. Fert. Jahrg

2, 67-74 (1956).

7. J. E. Shigley and J. J. Uicker, Jr. Theory of Machines

and Mechanisms. McGraw-Hill, New York (1980).

8. Y. C. Tsai and A. H. Soni, Design of an inverted

slider-crank mechanism. 6th Applied Mechanisms Conf.

Denver. Paper No. 33 (1979).

9. G. N. Sandor, A general complex-number method of

plane kinematic synthesis with applications. Ph.D. Dis-

sertation, Columbia Univ., University Microfilms,

LCN59-2596 (1959).

10. G. N. Sandor and A. G. Erdman, Advanced Mechanism

Design: Analysis and Synthesis Vol. 2, pp. 180-184.

Prentice-Hall, Engelwood Cliffs (1984).

11. C. E. Zarak and M. A. Townsend, Optimal design of

rack-and-pinion steer ing linkages, A S M E J . M e c h .

Transmiss . Automat. Design.

105(2), 220-226 (1983).

K I N E M A T I S C H E S Y N T H E S E U N D A N A L Y S E DE S

Z A H N S T A N G E N - K U R B E L G E T R I E B E S A L S F U H R U N G S -

U N D U B E R T R A G U N G S G E T R I E B E F U R 6 G E N A U I G K E I T S L A G E N

Km zfamlg--E ine Komplexe-Zahlen-Methode wird angewandt fur den kinematischen Entwurf und zur

Analyse des funktionsgenerierenden Zalmstangen kurbeigetriebes mit Beziehung auf drei genau

voneinander getrenente Positionen. Das Zahnstrangen kurbeigctriebe ist ein nutzlicher ebener Mech-

anismus, der angewendet werden kann, urn viel¢ Funktionendes Viergliexi-SystemsZu generieren und der

zahlreiche zusatzliche Anwendungsmoglichkeiten hat. Wegen der zusatzlichen Komplikation des Za-

hnstangen kurbeigetriebes konnen sowohl monotone als auch nichtmonotone Funktionen, sowie nich-

tlineare, verstarkte Bewegungen erzeugt werden. Ein Hauptvorteil des getriebes besteht darin, dass der

Ubertragungswinkel immer bei seinem optimalen Wert verbleibt da die Zahnstange immer tangential zum

Zahnrad Liegt. Der Entwurf und die Analyse wurden fur den computer VAX 11/785 programmiert und

steben interessierten Lesern zur Verfugung.