Some Questions of the Dynamics of Self-Locking Mechanisms

7/21/2019 Some Questions of the Dynamics of Self-Locking Mechanisms

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Jnl. Mechanisms Volume 4, pp. 93-104 /Pergamon Press 1969 /Printed in Great Britain

S o m e Q u e s t io n s o f t h e D y n a m i c s o f

S e l f L o c k in g M e c h a n i s m s

P r o f . D r . T e c h n . V . L . V e i t z

P r o f . D r . T e c h n . N . I. K o l c h i n t

In g . A . M . M a r t y n e n k o

Received 17 June 1968

A b s t r a c t

The dynamical processes have been considered and l imi t ing con di t ions of the wo rk ing

regimes der ived for sel f - lock ing mechanisms w i th o ne degree of f reedom. I t has

been found that some restr ict ions of iner t ial parameters would have to be taken to

exc lude dyn am ic jamm ing. The dependence of f r ict ional characteristics as a fun ct ion

of the re lative veloc i ty of the l inks has been invest igated.

Zu sa m m en fas su ng - -E in ig e Fragen der Dynam ik yon se lbs tsper renden Getr ieben :

Prof. Dr. V. L. Veitz, Prof. Dr. N. I . Kolchin, Ing. A. M. Martynenko.

Die dynam ischen Prozesse s ind bet rachtet und G renzbedingung en abgele itet for

selbstsperrende Getr iebe m it einem Freiheitsgrad. Es ist festg este l l t dass einig e

Begrenzungen der Trgheitsparameter n f t i g sein wLirden zur Eliminierung der

dynam ischen Hemmung. Die Ab h~ ngig kei t der Reibung s-Charakter is t ik a ls

Fun kt ion der relat iven G es ch w ind igk ei t der Glieder ist untersu cht .

P e 3 1 o M e H e K o T O p b t e aonpocbt ~HHaMHK I4 CaMOTOpMO3~ttIP*XC~ M exa mi3 MOB : Hpoqb., ~.T.H ., BJI. BeRt L

H p o ~ . , ]I . T. H ., H . H . K o n q n H , 1 4i nK . . M . M a p r b i H c n z o .

PaccMarpHeamrca ]IrlUaMH~IeCKI Cnpoueccbt H ablao~xarca rpanHq ubte ycoaoaHapa6osa4x peze.a-Moa~ n

caMoropMoaa tuaxca Mexana3Moa c on no a c renen~m ceo6o gbt . Hma/ teno, qro u ezo rop uc

orpaun~eH la aHepu lam ubtx na pa Me Tp oa ~On ZHbl

6~,[rl, nptta,qTr~t~rlll

licrnloqeli;ia laa~mqeczaro

caMo3ar.nmfaealtHa 3aanc,Mo crb. Hc ca ca oe aa a na pa Me rp oe Tpc H,a zaz dpy~izu,n OTffOCaTenbaOil

c z o p o c r n 3 a c m ce .

LET u s f ir st co n s i d e r a d y n a m i c m o d e l o f a m e c h a n i s m h a v i n g o n e d e g r e e o f f r e e d o m , i n

w h i c h t h e k i n e m a t i c t r a in v a l u e a n d t h e m o m e n t s o f i n e r ti a a r e c o n s t a n t s , d e n o t e d a s i~. k + 1 ,

l k , l k + 1 T h e d i ff e r en c e s in s t r u c t u r e a n d d e s i g n o f m e c h a n i s m s a r e n o t c o n s i d e r e d h e r e . I t

is f u r t h e r a s s u m e d h e r e t h a t o n l y c e r ta i n m e m b e r s o f th e k i n e m a t i c c h a i n h a v e m a s s e s ,

n a m e l y : t h e l in k s t o w h i c h p o w e r i s p u t i n o r t a k e n o f f b y o u t s i d e s o u r ce s . W e s h a ll ca ll

t h e se l i n k s t h e i n p u t a n d t h e o u t p u t li n ks , c o r r e s p o n d i n g l y ; e x t e r n a l m o m e n t s M k a n d

M k + t w i l l b e a p p l i e d t o t h e s e l i n k s .

I n t h e s p e c i a l c a s e o f a s t e a d y u n i f o r m m o t i o n w h e r e C bk a n d tb k+ t a re z e r o , w e h a v e t h e

o b v i o u s e q u a l it ie s

A / k = M k + l , k ; M~ t = M k . k + t

* The Leningrad zav od-V TU Z.

t The Len ingrad Polytechnic Insti tute , Ch airm an of Machines and M echanisms.

Th e Machine-tool ]Designing Bureau, Len ingra d, USS R.

93


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9 4

A l e a k c h a r a c t e r i s t i c i n s u c h a r e g i m e i s t h e e f f i c i e n c y , w h i c h f o r p o s i t i v e d i r e c t i o n s o f t h e

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

f o r i n p u t l in k k

'eo + l '. 'V k + l _ X k , k + l .

? ] ~ . k + t ~

COkMk

i t . k + 1

f o r i n p u t l i n k

k + 1 '

0 9 k l ~ tk = _ _ X k + t , k

I ] k + l , k ~ - -

O-Dk+ 1 ~ / k + 1

i k + l , k

(1)

(2 )

F r o m ( 1) a n d ( 2 ), i t m a y b e s e en t h a t t h e d y n a m i c r a t i o x h a s t h e f o l lo w i n g f o r m s :

f o r t h e i n p u t l i n k

k

X k , k + l = - - i k , k + l q k , k + l ;

f o r t h e i n p u t l i n k k + 1

X k + l , k = - i k + l , k ? ~ k + l , t~ ;

X k + l , k = - i k + t , k q ~ . , t + t ;

X k , k + t ~ - - i k , k + l ? ] k + t l , k

3 )

(4)

I n a g e n e r a l c a s e f o r m e c h a n i s m s w i t h

ik,k t~ik l,k,

t he i ne qua l i t y r / k , + l : ~ g ] k + l , k

h o l d s . O u r e x p r e s s i o n s ( 3 ), (4 ) s o f a r h a v e r e la t e d t o s t e a d y m o t i o n s , b u t w e s h a ll b e a b l e to

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

M k l , k

a n d

M k . k + 1

i s t h e s a m e a s b e f o r e .

T h e r a t io o f p o w e r f r i c ti o n l o s se s a t t h e l in k s k a n d k + 1 f o r e q u a l p o w e r i n p u t s i s

d e n o t e d b y 6 , a n d t h u s

h e n c e

t ~ J ( f) 1 - - ~ ] k + l , k

t ~ V k + l _ _

N(~7)

1 - - r l k . k + t

q k + l , k = 1 - - 6 ( 1 - - q k , k + 1 )

5 )

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

e i t h e r t h e c o n d i t i o n ~ > 1 o r % , k ~> qk + ~ , k ho l ds t rue .

T h e t r a n s fe r o f f o r ce s i n a n y m e c h a n i s m o c c u r s t h r o u g h t h e c o n t a c t s in k i n e m a t i c p a i rs

( j o in t s ) in w h i c h n o r m a l a n d t a n g e n t i a l re a c t i o n s a r e c r e a t e d . T h e p o s s i b i l i ty o f m o t i o n i n

a n y g i v e n d ir e c ti o n o f t r a n s f e r o f fo r c e s m a y b e s o l v ed b y c o m p a r i n g t h e p r o j e c t io n s o f

f o r c es o r m o m e n t s in t h e d ir e c t io n o f m o t i o n .

T h e p h e n o m e n o n p r o h i b i t i n g a m o t i o n i n c e r ta i n d i re c ti o n s , n o t w i t h s t a n d i n g t h e

m a g n i t u d e o f t h e fo r c e s in t h e s a m e d ir e c t io n s , i s c a l le d s e l f - l o c k i n g . T h e r e a l m e c h a n i s m s i n

w h i c h s e l f- l o c k in g t a k e s p l a c e o n l y i n o n e d i r e c t io n a r e s o m e t i m e s c a l l e d

i r r e v e r s i b l e ,

t h o u g h

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

s e l f - l o c k i n g .

* f o r w h i c h t h e e f f ic i en c y o f t h e r e v e r s e dr o m ( 5 ) , l e t u s fi n d t h e v a l u e l / k . k + l = r h , k + l

m o t i o n i s e q u a l t o z e r o

* l = 1 - 6 - t ( 6 )

~]k . k +

-< * T h i s m e a n s t h a t t h e m e c h -

bv io us ly , w e sh ou ld ha v e qk, k + t ~< 0 i f /Tk,k + 1

~ t l k , k + t

a n i s m c a n n o t p e r f o r m p o s i ti v e w o r k t o o v e r c o m e th e r e s i st a n c e f o rc e s . I t is n e c es s a r y t o

a p p l y m o v i n g f o r ce s f r o m s o m e s o u r c e b o t h t o t h e l in k k a n d t h e l in k k + 1 t h a t m o t i o n


3/12

9 5

o ccu r , v iz . e i t h e r l i n k is an i n p u t . T h e p o w e r l o s t t o f r i c t io n i n t h i s c a s e i s eq u a l t o t h e s u m

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

N l I ~ = N k + t + N k .

T h e w o r k i n g r e g i m e d u r i n g w h i c h t h e r e i s n o s e l f- l o c k in g is c a l l e d t h e

pul l ing regime

U s i n g t h e s a m e n u m b e r i n g s y s t e m a s a b o v e f o r th e l i n k s, e x p r e s s i o n (3 ) is v a l id f o r a d y n a m i c

r a t i o i n t h e p u l l i n g r eg i m es . T h e w o r k i n g r eg i m e w h i l e t h e r e is s e l f - lo ck i n g , is c a l l ed a

disengaging regime I ts d i s t in g u i s h i n g c h a r a c t e r i s t i c is a d i s e n g a g i n g c o e f fi c i e n t e q u a l t o

t h e r a t i o o f th e i n p u t p o w e r s :

N k N I f}

~ L ~ + t . k - - - - - - - - 1 , 7 )

N k t [ V g l .

w h e r e N k is t h e i n p u t p o w e r o f t h e d i s e n g a g i n g l i n k .

W e c a n r e d u c e ( 7 ) t o

~lk 1, k - -

a n d h e n c e w r i t e :

t O k M k X k l . k

~ k + tM k + t

i k t , k

Xk 1, k = ik 1, k~ lk l , k

8 )

L a t e r w e s h a ll p r o d u c e e x p r e s s i o n s f o r t h e e f f i ci e n c y a n d t h e d i s e n g a g i n g c o e f f i c i e n t o f

s o m e s e l f - l o c k i n g m e c h a n i s m s .

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

e f f ic i e n c y a n d d i s e n g a g i n g c o e f f i c i e n ts a r e g i v e n b y t h e f o r m u l a e :

t an ) . . t - 4q . 2 . t a n ( p , - 2 ) . 1 + 0s . l

- - , F l2 t- -

t a n .

1-bl//s, 2 '

/ t 2 - - t a n ( 2 + p , ) 1 - - ~ .

w h e r e 2 i s t h e h e li x a n g l e o n t h e b a s e c y l i n d e r ( f o r a H i n d l e y w o r m g e a r 2 = 2 o , w h e r e 2 0 i s

t h e h e l ix ( le a d ) a n g l e in t h e m i d d l e o f th e h o u r g l a s s p i t c h s u r f a c e ) ; p , i s t h e r e d u c e d a n g l e o f

f r i c ti o n ; f o r e x a m p l e , P r o f a c y l in d r i c a l w o r m g e a r i s

p , = a r c t a n [ f x /1 + t a n 2 e c 0 s 2 2 ] ( 9 )

w h e r e f i s t h e c o e f f ic i e n t o f f r i c t io n f o r t h e c o n t a c t su r f a c e s ; ~ is t h e p r e s s u r e a n g l e i n t h e

ax ia l c ros s - s ec t io n of a th re ad (F ig . 1 ) ; ~Os,1, ~ks.2 a r e c o e f f ic i e n ts t o t a k e i n t o a c c o u n t t h e

f r i c ti o n l e a k a g e i n t h e b e a r i n g s o f t h e w o r m ' s a n d g e a r ' s s h a f ts , w h i c h a r e f u n c t i o n s o f t h e

d e s ig n p a r a m e t e r s . S e l f- lo c k in g m e c h a n i s m s o f t h e a b o v e - m e n t i o n e d t y p e h a v e p , > 1.

F o r a w o r m - g e a r d r i v e o f t h e t w i n w o r m t y p e ( F ig . 2 ) , i n w h i c h t h e r e is a n a n g l e

b e t w e e n t h e a x e s , w e h a v e f r o m [ 2 ] :

_ s in 2 1 . s in ( 2 2 + p , ) . _ s i n 2 2 . s i n ( p , - 2 I ) ( 1 0 )

q t 2 si n2 ~ 2 s i n ( , ; . l + p , ) ' # 2 t s i n 2 t s i n ( 2 2 _ p , ) ,

w h e r e ) .1 , 2 2 a r e t h e h e l i x a n g l e s o n t h e p i t c h c y l i n d e r s o f t h e w o r m s f o r

fl-----22--21;

p , = a r c t a n [ f s i n - t ~ ] is th e r e d u c e d a n g le o f f r ic t i o n ( h e re 7 is t h e a n g l e m a d e b y t h e

n o r m a l v e c t o r t o t h e s c r e w s u r fa c e a n d t h e r a d i u s v e c t o r o f t h e s a m e s u r f a c e a t a n y o n e

p o i n t ) .


4/12

96

\

k

F i g u r e 1

F i g u r e 2

S e v e ra l i m p o r t a n t f a c t s s h o u l d b e e m p h a s i z e d f o r t h e s a m e m e c h a n i s m s , n a m e l y : s e lf -

l o c k i n g o c c u r s w h e n p r > ) . t b u t w h e n 2 2~


5/12

7

F i gur e 4

A c c o r d i n g t o [4 ] f o r m u l a e m a y b e d e r i v e d f o r t h e tr i p l e sc r e w s e l f- l o c k in g m e c h a n i s m

( se e F ig . 4) , w h i c h t r a n s f o r m s r o t a t i o n i n t o a p r o g r e s s iv e t ra n s l a t i o n . T h e s e f o r m u l a e a r e

t an ) ~ t an 2 2 . t an ( 2 t + P t ) + A t an ( 2 2 + P 2 )

1 l l 2 - -

( I + , ~ ) t an ( 2 t + p t)t an (,~ .2 + p 2 ) t a n 2 t + A t a n 2 2

(1 + ~ ) ta n ( 2 t + p t ) t a n ( p 2 - 2 2) t a n 2 t + A t a n) .2

~t2t - (12 )

t a n 2 t t a n 2 , t a n ( 2 t + P l ) - A t a n ( p 2 - 2 2 )

w h e r e 2 t , 2 2 a re t h e h e li x a n g le s o f t h e i n n e r a n d o u t e r s c r e w p a i r s ; A = R / r i s t h e r a t i o o f

p i t c h r a d i i o f i n n e r a n d o u t e r p a i r s ; ~ = x / r s i n ( 2 t + P x ) i s a c o e f f ic i e n t o f f ri c t i o n a l l e a k a g e ;

P t , P 2 a r e t h e r e d u c e d a n g l e s o f f r ic t i o n i n t h e i n n e r a n d o u t e r p a i r s ; a n d x is t h e c o e f f ic i e n t

o f r o l l i n g f r ic t i o n o f t h e t h r u s t b e a r i n g s .

T h e t r a n s m i s si o n r a t i o m u s t t a k e i n t o a c c o u n t t h e t r a n s f o r m a t i o n o f t h e m o t i o n f r o m

r o t a t i o n i n t o p r o g re s s iv e m o t i o n , b y i n c l u d i n g th e r a d i u s R ; i t is d e t e r m i n e d b y t h e e x p r e s -

s ion

tan 2~ + A tan 22

i l2 = (13)

R tan ) .j tan 22

T h e s e l f - lo c k i n g c o n d i t i o n f o r t h e s a m e m e c h a n i s m i s ,02 t> ,~ .2 ; a n d d y n a m i c j a m m i n g

oc cu rs i f ( I /A ) tan (2 t + p t ) ~< tan (p2 - 22) .

A s e ri e s o f s e l f -l o c k in g m e c h a n i s m s w i t h h i g h e f f ic i e n c y i n t h e p u l l in g r e g i m e a n d h i g h

d i s e n g a g i n g c o e f fi c ie n t h a v e b e e n d e s c r i b e d b y N . S . M u n s t e r , B . S h . N o v o s c h i l o v , G . V .

Tsarev [5] .

W e t u r n n o w t o c o n s i d e r t h e p r o p e r t i e s o f s e l f -l o c k i n g m e c h a n i s m s i n d y n a m i c r e g i m e s .

H a v i n g s e e n th e s e m o d e l s o f s e l f- l o c k i n g m e c h a n i s m s , w e c a n w r i te t h e i r d i f fe r e n t i a l

e q u a t i o n s o f m o t i o n in th e f o r m :

lkt;J k

q- M g + 1 , ~ : = M k , ~ ( 14 )

]k

lt ~k+ l 1 Mk, k+ t ----M k + 1

T h e s e g i v e u s t h e r e l a t i o n s h i p b e t w e e n t h e m o m e n t s M k + t . k, M k . k+ 1 a n d t h e a c c e l e r a -

t ions cbk , cbk + z ; we der iv e the ex pre s s io ns

A l k , k I

= M k +

t l k - - i k k + t M k l k + t

; ( 15)

[ k - - i k t ,k X k t , k l k l

e~ = c bk - - M k - - X k + 1 . k M k ,

(16)

[ k - - i k l , k X k l ,k ~ k l


6/12

9 8

w h e r e Xk+ 1 .k i s t h e d y n a m i c r a t io , w h i c h t a k e s o n t h e v a l u e s

f . - j .

- - I k t , k ~ k , k l

Xk + t, k ~-'-'r.

i g + 1 , J A k + l , k

f o r M k , k + t < 0 ( p u l l i n g r e g i m e ) ;

f o r M k . k + t > 0 ( d i s e n g a g i n g r e g i m e ) .

( 1 7 )

S u c c e s s iv e a n a l y s i s o f th e g i v e n e x p r e s s i o n s p e r m i t s u s to f i n d t h e c o n d i t i o n s f o r

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

p a r a m e t e r s ( se e T a b l e 1).

T a b l e 1

Regimes

t2

Im

e-,

"gz

tall

i

e e , g t + l > O ,~ a'. ~ l < 0

I

M e > 0 ; M ~+ I > 0

h : , , ~ e > i e , ~:+1/eM,~+ Non-real izable

I I

M e > O ; M e ~ t < 0 , (: 'v /c , k : = - M J + I > 0 )

M e > i e L e q - t e , e l M c , e + l M t a < i x + t , k q - t e , e + l M e , e l

I I I

M e < 0 , (M e , = - - M e > 0 ) ;

N o n - r e a l i z a b l e

M e + l < 0 , ( M e. e . l = - M e + > 0 )

l eMe . e+l > ix+l. ele+lMc, ~:

I*

M e > 0 ; M ,~ +t > 0

L~ > i2 /c+t, t /J , l :+ 1 . e lk + I

IkMe+ t > &+ , e l f+ 1M e

M e > & + l, ,~:/.te+l M e+ t M e < ix+t, e/.*e+l,eM,~+

II*

M e < O , M e , e = - - M e > 0 ; M t+L > 0

No n-re a l i zab le l e> i2e +l . ,~ /.te+1. el,~:+

I I I *

M k < 0 , (M e . e = - - M e > 0 ) ;

N o n - r e a l i z a b l e

M e + l < 0 (M e . e+ l = - - M e i > 0 )

le > i2k+1, ea~+l , Je + l

I t+tMc. *:>i t . t+l l tMe, t+t

Me, e >ik +l . e / .te+l,

kMe,

k+l

L e t u s l o o k c l o s e l y a t t h e d i v i s o r s in (1 5 ), ( 16 ). O b v i o u s l y , w e h a v e f o r a p u l l i n g r e g i m e

i n a c c o r d a n c e w i t h ( 1 7)

[k - - i k 1 , kX k 1 , k[k I = I k i 2 - ~

, k ~ k ,k l ~ k l > 0 .


7/12

99

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

Ik - - i k tX ~ ,~ . 1 , k l k t = [ i - - i ~

1 . k k + t .

k / k I > 0

p r o v i d e d t h a t

lk > i~+ 1. k/tk+ 1. klk + t

~ 8 )

T h e i n e q u a l i t y (1 8 ) is a c o n d i t i o n f o r t h e a b s e n c e o f d y n a m i c j a m m i n g i n a s e l f -l o c k i n g

m e c h a n i s m . R e a l ly , t h e m o m e n t M k , k+ t a n d a c c e l e r a t i o n ek c o n v e r g e t o i n fi n it y a s I k t e n d s

to the va lue i~+ t .k~tk+l.klk+ 1 T h e s e c o n c l u s i o n s o f u n l i m i te d v a lu e s o f M k . k + 1 a n d ek

c o m e f r o m t h e in i ti a l a s s u m p t i o n a b o u t t h e in f i n it y o f t h e l in k ' s h a rd n e s s . A n a l y s i s o f

d y n a m i c p h e n o m e n a s h o w s f o r r e al m e c h a n i c a l sy s te m s t h a t a n a p e r io d i c in c r e as in g M k . k + 1

w ill b e re s t r ic t e d i f t h e l in k ' s h a r d n e s s i s l i m i te d . T h e m a x i m u m v a l u e o f m o m e n t M , . k + 1

d e p e n d s u p o n t h e l i n k ' s h a r d n e s s [ I ] .

I t is i m p o r t a n t f o r p r a c t i c a l n e e d s t o i n v e s t ig a t e t h e s t e a d y m u l t i h a r m o n i c m o t i o n o f a

m a c h i n e a g g r e g a t e c o n s i s ti n g o f a m o t o r d r iv e , a s e l f- l o c k in g m e c h a n i s m a n d a n o u t p u t

l i n k , t o w h i c h a n e x t e r n a l r e si s ti n g p e r i o d i c f o r c e ( o r m o m e n t ) i s a p p l i e d . S u c h a m a c h i n e

ag g r eg a t e m o d e l i s s h o w n i n F i g . 5 .

j~ ~ /

F i g u r e 5

T h e d y n a m i c c h a r a c t e r i s t i c o f t h e m o t o r a n d i ts d r i v e s h a ft h a s b e e n g i v e n b y [1 ] a s

~ l a + T a t M d - ( v T a ) - I S1 = 0 ,

w h e r e

S t = 1 - c o l / c o o )

O9

M e

T ~

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

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

is th e t o r q u e o f m o t o r d r i v e s h a f t ;

i s t h e t i m e c o n s t a n t o f th e d r i v e m o t o r , d e f i n in g t h e t r a n s i e n t

b e h a v i o u r i n t h e m o t o r d r i v e ;

i s a s h ap e f ac t o r o f t h e s t a t i c ch a r ac t e r i s t i c .

1 9 )

A d y n a m i c c h a r a c t e r i s t ic o f a m o t o r a n d d r i v e s h a f t s u c h a s is g i v e n b y (1 9 ) c a n b e

u t il iz e d f o r m a n y k i n d s o f e l e c t ri c d r i v e s a n d a l s o h y d r a u l i c d r iv e s . M e t h o d s t o d e t e r m i n e

t h e p a r am e t e r s v an d T d h av e b een g i v en i n [ 1] .


8/12

oo

T h c m e c h a n i c a l m o d e l o f t h is m o t o r d r i v e i s s h o w n i n F ig . 5 . I t h a s a d y n a m i c c h a r -

a c t e ri s ti c a s in ( 9 ) . a n d t h e p a r a m e t e r s o f t he m e c h a n i c a l m o d e l h a v e b e e n d e r i v e d t o b e

c a = ( v e a o T j ) -

t : /-Ca= ( r e ) o ) - i

T h e s y s te m o f d if f e re n t ia l e q u a t i o n s o f m o t i o n f o r t hi s m a c h i n e a g g r e g a t e c o n s i s ts o f

t h e eq u a t i o n s ( 1 4 ) an d ( 1 9 ) i n w h i ch

k = 1 ; M t = M a ( t ) ; M 2 = M c ( t ) .

H e r e t h e e x t e r n a l m o m e n t o f re s i st a n c e i s c o n s i d e r e d t o b e a p e r i o d i c f u n c t i o n o f t im e

w i t h p e r i o d T , v i z . M e( t ) = M c ( t + T ) . R ef e r r i n g ag a i n t o T ab l e 1 . w e f i n d t h a t t h e p u l l i n g

r eg i m e i s r e a l i z ed i f

1 2 M a ( t ) - i z , 211M c t) O , 2 0 )

an d t h e d i s en g ag i n g r eg i m e i f

l a M a ( t ) - i t . a l a M o ( t ) < 0 . ( 2 1 )

E x c l u d i n g M , 2 a n d , tl ,~ f r o m t h e e q u a t i o n s o f m o t i o n a n d m a k i n g u s e o f t h e d e p e n d e n c e

be tw ee n c.:)t an d ~b,_ , w e w r i t e t h i s s y s t em i n m a t r i x f o r m

,/* + C O = F ( t , t ) ) , (22)

w h e r e 4 ' i s a v e c t o r f u n c t i o n w i t h t h e t w o c o m p o n e n t s

4 l ( t ) = M a ( t ) ; ~ , _ ( t ) = s t ( t )

C i s a ( 2 x 2 ) m a t r i x w i t h e l em en t s

t = T ~- ~ - _ ( v T a ) - I

c , ~ , ~ c , , _ , - , { c h ~ = t * a ,o ) -~ { c } , , = 0 .

I * = 1 l - i , t x e t l 2 ;

a n d F ( t , ~ ) is t h e t w o - d i m e n s i o n a l v e c t o r fu n c t i o n w i t h c o m p o n e n t s

F t ( t ,

4 ' ) = 0 ,

k , ( t ,

4 ' )= r (~ , )M~( t ) ,

wh r

r ( 4 ) = - x , . . ~( l * o ) o ) - t

T h e w a t c h f u n c t i o n d e f i n i n g t h e w o r k i n g r e g i m e s i s

W ( 4 ) = I 2 4 1 ( t ) - i t , . 1 1 M c ( t )

an d . v 2 t i s g i v en b y t h e a l t e r n a t i v e s :

[ - - i 2 1 . ' 1 t 2 t

f o r W ( 4 ' ) > O

X2 = ~ i2 ~ 2t

[ 'O r W ( 4 ' ) , ~ O .

t (23)


9/12

1 1

W e n o w seek a so l u t i o n o f sy s t em (2 3 ) su p p o s i n g t h a t t h e m o m en t o f r e si s tan ce i s a

b o u n d e d , p i ecew i se co n t i n u o u s fu n c t i o n . L e t t h e s e t { t q . ;} b e t h e s e t o f t h e i n s t an t s i n t i m e

a t w h i c h t h e re o c c u r c h a n g e s i n t h e r e g im e s f o r a p e r io d i c m o v e m e n t o f t h e m a c h i n e

ag g reg a t e .

I t f o l l o w s t h a t

W [~ b( t q

; )] = 0 , an d t~ . ; can be expresse d by

w h e re

t q . ; = q T + t .

q = 0 , 1 , 2 . . . . ; ~ = 0 , I , 2 . . . . . : t - 1

(h e re :t is t h e n u m b er o f ch an g es o f w o rk i n g r eg i m es fo r h a l f t h e c lo sed t i m e i n t e rv a l O , T ) .

I f th e p e r i o d i c so l u t i o n o f th e n o n - l i n ea r d i f f e r en t i a l sy s t em (22 ) w e re k n o w n , t h en a f t e r

p u t t i n g i t in t o m a t r i x C an d v ec t o r fu n c t i o n s w e sh o u l d h av e a l in ea r sy s t em o f d i f f e r en t ia l

eq u a t i o n s w i t h p i ecew i se co n s t an t p e r i o d i c co e f f i c i en t s .

A g en e ra l so l u t i o n fo r su ch a d i f f e r en t ia l sy s t em can b e d e r i v ed b y m e t h o d s d e sc r i b ed i n

[ I ] t o t h e fo rm

( t ) = - t

~ ~ ( t ) q ; (24)

q O ~=l

w h ere q , ( i n d i ca t e th e ( r eg i m e o f t h e q p e r i o d an d

;= ; , / /* (t - tq , ; )q ' ;

( t ) q '

0

fo r

t ~[tq.

;,

tq,

; + ~), ~

o r

t ~ [ t q . ; , t q . ; + t ) .

H e r e ~ l * ( t ) q ' is a v ec t o r fu n c t i o n so l v i n g t h e d i f f e r en t i a l sy s tem (22 ) w i t h

C = C [ ( t q , ) ] = C q' , F ( ~ , t ) = F [ ~ b ( tq , ;) ; t ] = r q ' M c ( t )

fo r t e [0 , o o ) an d i n i t i a l d a t a ~ ' ~ .

(~ ,( t) i s t he pe r iod ic s o lu t io n o f equa t ion (22)).

T h e L a p l a c i a n m a p p i n g f u n c t i o n o f t h e f u n c t i o n

(t ) ~ ' ~

i s g i v en b y t h e fo rm u l a

(I)*( p)q ~= ( N q

; ) - l m * ( p ) ' t '

;

(25)

w h e r e N q ' ; is a ( 2 x 2 ) m a t r i x ;

m * ( p ) q ' :

is a v ec t o r fu n c t i o n . T h e e l em en t s o f t h e

N q ' '

m a t r ix a n d t h e c o m p o n e n t s o f

m * ( p ) q ~

a r e d e t e r m i n e d b y

( N q ' ; } l t = p + c t , ,

{ N q ' ; } 1 2 = c t 2 ; ( N q ' ; } 2 t = c ~ ' t ; ;

{ N q ' ; } 2 2 = p ;

, . q ,; ; L { M c ( t + t q . ; ) } ,

n~ (p ) q ' ~ k 1 o , m ' ~ ( p ) 'l ; = d / 2 o + r q,

where c~h; , r q ' ; a re t h e v a l u e s o f c2 t an d r fo r q , ( r eg im es r e sp ec t i v e l y ; L is a sy m b o l

o f L a p l a c i a n m a p p i n g .

R e v e r s i n g ( 25 ) a c c o r d i n g to R i e m a n n - M e l l i n f o r m u l a w e fi n d t h e c o m p o n e n t s o f v e c t o r

fun c t ion ~b*( t)q ; t o be as fo l lows:

I Fa+i~ ~,*[,,Vl..;_ ctznl~(p_~)q, eP,dp,

; = ._L_ , , , t_ w l _ _ _ _ _

~ ( t ) q ' 2 r t i J _ i ~

p 2 + c t t p - c t 2 c q ~

q , ; q , ~ * q , ;

. 1 I * + i ~ ( p + c t t ) m 2 ( p ) - - c z l m l ( p ) e Ptd p.

(26)


10/12

1 2

F r o m ( 2 6) a n d t a k i n g i n to a c c o u n t t h e e x p r e s s t o n s f o r m T I p ) + ( i = i . 2 ) . t h e s o l u t i o n

t/J(r)q ' :

m a y b e f o u n d in t h e fo r m

~ ( t ) q '" a l t ) q ' ; , ) q o ' ; + f ( t ) q ' ; .

127)

H e r e t h e m a t r i x e l e m e n t s

G (t ) q" ~

c a n b e r e c k o n e d b y m e a n s o f

I

i ~ t i ~ p e p . - ~ . . O d p "

{ o f t , . ; } , , = 2 - ~ , 3 ~ _ , ~ .

l l ( p ) q

~ i f C 1 ~ 2 e p ( t - t Z ' d P

G ( t ) q . ; } 1 2 = I , + i ~

. , - i ~ u fp ) q" ~

{ G( t ) q ' ; } 2 t = - ~- ~i~ i ~ _ i _ ~ u ~ , ~

P ~ t - ta , O d p ,

i " + '~ - p + c , , e m _ , . " O d e ( 2 8 )

{G(t)""}z2" - 2 x i ) ~ - i~ t-t(p)q--"

w h e r e u ( p ) q ' = p 2 + el IP -- Cl zC'~ 'l .

T h e c o m p o n e n t s o f t h e v e c t o r - f u n c t io n f ( t ) q

;

a r e

F q , . ~ f * a + i ~

J ' t ( t ) q ' - - - - + tq o ' )}ee - t , , , o d p

u ( P ) q ' . ; L { M ( t .

- - ~ l . J a _ i ~ C t 2

.'I,C,

/ ' ~ + i ~ ' . ,

f 2 ( t )q ' = r ' '' | P + C ~ L { M d t + t q , ) } e P " - '~ ' d p ( 2 9 )

2 n i J ~ _ i ~ u ( p ) q" "" "

A f t e r i n t e g r a t i n g ( 2 8 ) w e s h a l l h a v e

{ G ( t ) ' } , t = I - ~ s i n K q ' ~ ( t - t , , , - . ) cKq '( t - t q . z ) l e - ' t - t q'

C 1 2 - T l t - t q ,

{ G ( t) q ' } 1 2 = - -~ - ff T . e s i n K " ' ; ( t - t q , ; ) ,

q,~,

{ G ( t ) q ' ; } 2 t = c 2 't . e - ; ' " - t " ' s i n K q ' ( t - tq . ; ) ,

K q . .

{ G ( t ) q ~}2 2

= I ~ s i n K q ' ( t - tq . 0 + c o s K q ' ~ ( t - tq , r ) l t J - ' ( t -tq"

where?,=0 5ctt;

K q ' ; - q ,~ . ' 2

- 0 5 . v / - - 4 C l 2 C 2 t - - C t l .

H e r e w e s e e t h e p r a c t i c a l l y i m p o r t a n t c a s e w h e n 4 C t 2 C q 2 t ; + C f I < 0 t h a t c o r r e s p o n d s

t o t h e i n e q u a l i t y

T a ( T , . t ) - t > 0 . 2 5 ,

w h e r e 7 ~ = VOgoI* i s a m e c h a n i c a l t i m e c o n s t a n t f o r t h e m a c h i n e a g g r e g a t e .


11/12

1 3

I n (2 9 ) t h e v a l u e s o f t h e v e c t o r f u n c t i o n f ( t ) ~ ' a r e d e p e n d e n t o n t h e s p e ci fi c f o r m o f t h e

m o m e n t o f re s i st a n c e M ' = ( t) . T h e v a l u e s o f t q . ; a t w h i c h t h e c h a n g i n g o f r eg i m e s o c c u r s ,

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

4 t t ) = i t

: [ ~ - L M c t ) .

30 )

A s i t w as s h o w n i n [ l ], t h e co n d i t i o n s f o r ex i s t en ce o f a p e r i o d i c s o l u t i o n o f t h e d i f -

f e r e n t ia l e q u a t i o n s o f m o t i o n a r e g iv e n b y

4 o = 4 ~ ' . ( q = 1 , 2 . . . . )

( 3 1 )

L e t u s n o t e t h a t

G t o , ; + O , ; = G o . ; , f t o . + l ) , ; = f . ; ,

(32)

c o n s i d e r i n g t h a t

4 00 , ; = 4 ( t 0 , ~ - 0 ) ; - 1

(33)

F r o m ( 3 2 ) , ( 3 3 ) a n d ( 2 7 ) w e c a n o b t a i n a r e c u r r e n c e f o r m u l a

40. ~+ t = Go, ~4o ; + f0 , ; . (34)

T h i s e q u a t i o n ( 34 ) f o r = 0 , 1, 2 . . . . ~ - l a n d t h e c o n d i t i o n o f e x i st e n c e o f p e r i o d i c

s o l u t i o n s ( 3 1 ) c a n b e u s e d t o g e t h e r t o g e t 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 v e c t o r 4 0 f o r t h e

p e r i o d i c s o l u t i o n .

w h e r e

( H - I ) 4 o = - P , ( 35 )

k . l 2 -1

H = I - [ O o . 1 - [ C o . ; B _ = I P = y .

k = z- I i= ~ -1 k = 0

an d 1 is a u n i t m a t r i x .

I t c a n b e p r o v e d [1 ] t h a t t h i s p e r i o d i c s o l u t i o n is e q u a l t o t h e p e r i o d i c s o l u t i o n o f th e

d i f f e r en t i a l s y s t em ( 2 2 ) , i f t h e i r i n i t i a l d a t a a r e eq u a l t o e a ch o t h e r .

A n i t e ra t i v e a l g o r i t h m h a s b e e n u s e d f o r s e a rc h i n g o u t a p e r i o d i c s o l u t i o n ; i t c o n s i s ts

o f t h e f o l l o w i n g :

I . W e c h o o s e s o m e p e r i o d i c f u n c t i o n 4 ( ~ (t ) ( f o r e x a m p l e , a s o l u t i o n o f t h e d i f f e re n t i a l

s y s t em o f eq u a t i o n s ( 2 2 ) f o r I * = I = co n s t . ) an d t h u s f i n d t h e s e t { t~ . ~ [0 ]} f o r e q u a t i o n s

( 2 2 ) a f t e r l i n ea r i z a t i o n .

2 . W e u s e t h is a p p r o x i m a t e s e t i n th e p r o p e r d i f fe r e n t i a l s y s t e m o f e q u a t i o n s t o f i n d it s

per io d ic so l u t io n 4(1) ( t ) an d sea rch o u t a new se t { tq, ~ [1]} .

3 . W e co n t i n u e t o i t e r a t e a cc o r d i n g t o t h e s t ep s 1 , 2 u n t i l { tq , ; [ m - 1 ]} w i ll b e eq u a l t o

{ t~ , ;[ m ]} w i t h t h e ex a c t i t u d e d em an d ed .

I t c a n b e s h o w n t h a t t h i s i te r a t iv e a l g o r i t h m w i ll i n d e e d s e a r c h o u t t h e s e t o f f u n c t i o n s

{ 4 (k )( t) } w h i c h c o n v e r g e a s k ~ o o t o t h e f u n c t i o n ~ k(t), t h e p e r i o d i c s o l u t i o n o f th e e q u a t i o n s

( 2 2) . T h e p e r i o d i c s o l u t i o n 4 ( t ) e x i st s i f t h e i t e r a t iv e a l g o r i t h m c a l c u l a t i o n c a n b e m a d e .


12/12

1 4

Q u e s t i o n s c o n c e r n i n g t h e u n i q u e n e s s o r t h e s o l u t i o n a n d t h e e x i s te n c e o f s u b h a r m o n i c

s o l u t i o n s a r e s o lv e d b y in v e s t i g a ti n g t h e v a l u e o f t h e d e t e r m i n a n t ( H - I ) . w h e r e

H = [ im H [ k ] .

I f w e g e t d e t ( H - I ) # 0 , t i~ e s y s t e m o f d i f fe r e n ti a l e q u a t i o n s ( 2 2 ) h a s a u n i q u e p e r i o d i c

s o l u ti o n . N e c e s s a r y c o n d i t io n f o r t he e x is te n c e o f s u b h a r m o n i c s o l u ti o n s is d e t ( H - l ) = 0 .

S u f f ic i en t c o n d i t i o n s f o r t h e ex i s te n c e o f s u b h a r m o n i c s o l u t i o n s c a n b e f o u n d i f t h e e ig e n -

v a l u e s o f m a t r i x H a r e i n v e s t ig a t e d . R e s e a r c h i n to t h e s e e ig e n v a l u e s o f H p e r m i t s u s t o

e s t a b l i s h w h e t h e r t h e s o l u t i o n s a r e s t e a d y - s t a t e o r n o t .

T h e r e s u lt s o b t a i n e d h e r e c a n b e u ti li z ed t o i n v e s t i g a t e t h e d y n a m i c p r o c e s s e s in m a c h i n e

a g g r e g a t e s w i th m o r e c o m p l e x f o r m s o f s e l f -l o c k i n g m e c h a n i s m s , s u c h a s , f o r e x a m p l e , t h o s e

w i t h e l as t ic l i n ks , o r w i t h a d y n a m i c r a t i o d e p e n d i n g o n t h e v e l o c i t y o f th e c o u p l i n g .

R e f e r e n c e s

[I] VErrz V. L.

Dinamika 34ash inykh Agrega tov

Izdat. M ashinostroenie, M osc ow (1969).

[2] POPPER B. and PESSE,~ D. W. The twinwo rm dr ive --a self-locking worm -gear transmission o f high

efficiency.

T r a n s . A S M E ,

Aug ust 1 960, pp. 191-199.

[3] PAINLEV~P. and PRAYDTLL.

Leons stir le Frottement.

Paris (1895).

[4] TURPAEVA. I.

Samotorm ozyashch i sya M echan i smi , p . 110 .

Izdat. Mashinostroenie, Moscow (1966).

[5]

Teoriya Mechanismov i l~Iashin,

V oI. 39, lzdat. FA N, Tashk ent (1967).

[6] KOLCmS

N. I . K . Voprosy d inamik i samotorm ozyashch i k sya S i s t em.

Trudy, Leningrad Polytechnic

Institute, No . 25 4, Leningrad (1965).

N o t a t i o n

CO k

E k = ( ~ ) k

i k

k + I = ( 2 ) k / ( O g + |

~ [ k . k + l

M k

V k.k + I = M k . k + I / M k + I . k

lk

qk . ~+,

/ l k , k 1

A n g u l a r v e l o c i ty o f t h e l in k i d e n ti fi e d b y s u b s c r i p t k .

A n g u l a r a c ce l e r a ti o n o f l in k k .

K i n e m a t i c tr a n s m i s s i o n r a t i o o f m e c h a n i s m .

M o m e n t a c ti n g o n th e l in k k d u e t o t he m e c h a n i s m ( an

i n t e r n a l m o m e n t ) .

E x t e r n a l m o m e n t a c t i n g u p o n t h e l in k w i t h s u b s c r i p t k .

T h e d y n a m i c r a ti o o f m o m e n t s .

M o m e n t o f i n e r t ia o f t h e li n k w i t h s u b s c r i p t k .

E f f ic ie n c y o f t h e p a r t o f t h e m e c h a n i s m i n w h i c h t r a n s f e r o f

m o m e n t s is in t h e d i r e c t io n f r o m k ' t o k + 1

D i s e n g a g i n g c o e f f ic i en t o f a s e l f -l o c k i n g m e c h a n i s m .

Some Questions of the Dynamics of Self-Locking Mechanisms

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