Us 2476269

8/11/2019 Us 2476269

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1

I I S

AGENT

f e m

m aw-MM* Wu9476925@

R .

B . BLACKMAN

MECHANICAL

DIFFERENTIATOR FOR

SMOOTHING

T A R G E T

TRACKING DAT A

F i l e d

A l l g .

2 0 , 1 9 4 5

ammi ;

. F u ñ y i 9 , 1 9 4 9 .

_

/M/E/VTOR

@ i B . BLACK/WIV

V ,

8/11/2019 Us 2476269


2 9 4 7 7 6 , 2 6 @

â@MC/WAN

R . B .

BLACKMAN

MECHANICAL DIFFERENTIATOR FOR SMOOTHING

TARGET TRACKING DATA

, 1945

J u l y

ï g ,

i l e d A u g . 2 O

w

S

8/11/2019 Us 2476269


P a t e n t e d J u l y

1 9 , 1 9 4 9

2 , 4 7 6 , 2 6 9

UNITED STATES

PATENT

OFFICE

2 , 4 7 6 , 2 6 9

MECHANICAL

DIFFERENTIATOR

FOR

SMOOTHIN

G

TARGET

TRACKING

DATA

Ralph B .

Blackman,

C r a n f o r d , N . J . , a s s i g n e r t o

B e l l

Telephone L a b o r a t o r i e s , I n c o r p o r a t e d , New

York, N. Y . ,

a

corporation

o f New o r k

A p p l i c a t i o n

A u g u s t 2 0 ,

1 9 4 5 , S e r i a l N o . 6 1 1 , 6 3 9

( C l .

2 3 S - 6 1 . 5 )

Claims.

1

This

i n v e n t i o n

r e l a t e s t o

a n

i mpr ove me nt

i n

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

p a r t i c u l a r l y u s e f u l

i n

a mechanical system f o r

i n d i c a t i n g

a n angular

v e l o c i t y .

The

g e n e r a l

o b j e c t

o f

t h e

i n v e n t i o n

i s t o p r o

v i d e improved

means

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

v e l o c i t y

o f

a

s h a f t o f

which t h e v a r y i n g angular

p o s i t i o n a n d t h e angular

v e l o c i t y

r e p r e s e n t

r e

s p e c t i v e l y

t h e varying value o f an observed q u a n

t i t y and t h e time

r a t e

o f

change t h e r e o f .

The o b s e r v e d

q u a n t i t y

m a y be t h e h o r i z o n t a l

displacement

o f

a n

a i r p l a n e

i n

i i i g h t

o f

which

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

by

an o b

s e r v e r o n

the

groun d. T h e o b s e r v e r ’ s

instruments

may

themselves

be i i a w l e s s but h i s use o f t h e m

i s

u s u a l l y

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

overrun

n i n g or u nd er r u nn i ng

a n d

when a shaft

i s

ro

t a t e d t o f o l l o w t h e v a l u e o f t h e o b s e r v e d q u a n t i t y ,

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

t h i s

s h a f t d o e s n o t change

unif ormly with time even

f

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

i s

s o

c h a n g i n g .

I t

thus

becomes n e c e s s a r y

t o

p r o v i d e

m e a n s

f o r

measuring

t h e s h a f t

angular

v e l o c i t y

and smoothing

o u t

t h e

r a n d o m

v a r i a t i o n s

t h e r e i n .

A u r t h e r

o b j e c t

o f

t h e i n v e n t i o n i s t h e r e f o r e

t o

provide i m p r oved m e a n s

f o r

obtaining

a

smoothed

average o f t h e

angular v e l o c i t y

o f the

r o t a t i n g

s h a f t by e l i m i n a t i o n o f

r a n d o m e r r o r s

o f

o b s e r v a t i o n .

I t may be shown b y mathematical a n a l y s i s

t h a t

i f

a

s e r i e s

o f o b s e r v a t i o n s

o f

instantaneous

an gular v e l o c i t y are averaged over a

convenient

t i m e

i n t e r v a l ,

t h e a n g u l a r v e l o c i t y

b e i n g

found

constant

f p e r f e c t l y o b s e r v e d ,

t h i s

average should

be

obtained

by

p a r a b o l i c weighting

o f

the

s u c c e s

s i v e a n g u l a r v e l o c i t i e s ; t h a t

i s , t h e l e a s t

w e i g h t

should be

g i v e n

t o

the

v e l o c i t i e s

observed

a t

the

b e g i n n i n g a n d a t

t h e

e n d , and

t h e

maximum

weight

t o

t h o s e

a t

the c e n t e r ,

o f

the i n t e r v a l .

T h e

average v e l o c i t y

s o

obtained w i l l be i d e n t i c a l

with

that d e r i v e d

f r o m

the s l o p e o f

a

s t r a i g h t

l i n e

ñtted by a l e a s t squares adjustment

t o

the

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

time

a n d

the r a n d o m e r r o r s a l l u d e d t o

w i l l

be

f i l t e r e d o u t .

A c c o r d i n g l y ,

i t

i s

another o b j e c t

o f

t h e

inven

t i o n t o

p r o v i d e means

f o r

o b t a i n i n g with a p p r o x i

mately p a r a b o l i c

weighting over

a

convenient

time

i n t e r v a l

the

average value over

that

i n t e r v a l

o f t h e angular v e l o c i t y o f a

r o t a t i n g

s h a f t .

While

t h e

apparatus o f t h e

i n v e n t i o n

may be

m a d e

s u f û c i e n t l y

complex

t o

approach

a s

c l o s e l y

a s d e s i r ed t h e i d e a l

p a r a b o l i c

Weighting

o f i n s t a n

t a n e o u s a n g u l a r v e l o c i t i e s , t h e

i n v e n t i o n

w i l l b e

s e t f o r t h with r e f e r e n c e t o two embodiments

which approximate t h e

i d e a l

a n d

c o n s t i t u t e

i m

provements

o v e r the mecha n isms k n o w n

t o

the

p r i o r

a r t .

The e l e c t r i c a l a n a l o g u e o f t h e mechanical

p r o b l e m

i s f o u n d

where t h e v a r y i n g o b s e r v e d

10

1 5

20

25

30

35

40

45

50

6 0

2

q u a n t i t y

i s r e p r e s e n t e d by a v a r y i n g

v o l t a g e

which i s d i f f e r e n t i a t e d with r e s p e c t t o time

a s ,

f o r

e x a m p l e ,

by t h e d i f f e r e n t i a t i n g a m p l i f i e r d i s

c l o s e d b y E . L .

N o rt o n

i n

United

S t a t e s

Patent

2 , 3 2 4 , 7 9 7 , J u l y 2 0 , 1 9 4 3 . I t i s k n o w n t h a t a n am

p l i ñ e r

having a feedback

path i n c l u d i n g an

i n t e

g r a t i n g c i r c u i t

p r o v i d e s an

o u t p u t

v o l t a g e

which

i s t h e t i m e d e r i v a t i v e o f t h e i n p u t

v o l t a g e .

The

p r e s e n t i n v e n t i o n a p p l i e s t h i s p r i n c i p l e t o

me ch

a n i c a l d i f f e r e n t i a t o r s , s o

t h a t

another o b j e c t o f

t h e i n v e n t i o n

i s

t o p r o v i d e an

improved

me chani

c a l t r a n s m i s s i o n

system

o f t h e t y p e

having be

tween i n p u t a n d output a

mechanical feedback

path c o m p r i s i n g mechanical i n t e g r a t i n g means.

The i n v e n t i o n w i l l

be

understood f r o m

t h e

f o l

lowing d e s c r i p t i o n read with r e f e r e n c e t o

t h e a c

companying

d r a w i n g s

i n

w h i c h :

F i g .

1 p r e s e n t s

c u r v e s

e x h i b i t i n g c e r t a i n

weighting f u n c t i o n s , i d e a l a s

w e l l a s p r a c t i c a b l e ;

F i g .

2 shows a simple mechanical system

ac

c o r d i n g t o t h e p r i o r

a r t f o r i n d i c a t i n g

t h e angu

l a r v e l o c i t y o f

a

r o t a t i n g s h a f t ;

F i g .

3 shows

s c h e m a t i c a l l y

one

embod i me nt

o f

t h e

p r e s e n t

i n v e n t i o n ;

and

F i g .

4 s i m i l a r l y

shows a n o t h e r embodiment o f

t h e

i n v e n t i o n .

In a l l f i g u r e s , l i k e numerals

and

l e t t e r s d e n o t e

l i k e elements.

I t

w i l l

be

convenient

t o

use the n o t a t i o n

o f

t h e o p e r a t i o n a l

c a l c u l u s t o

d e s i g n a t e t h e

v a r i o u s

s h a f t

r o t a t i o n s i n v o l v e d

i n t h e apparatus shown

i n

F i g s .

2 t o 4 ,

i n c l u s i v e .

In t h i s

n o t a t i o n

p f

( t ) i n d i c a t e s t h e

t i m e d e r i v a t i v e ,

w h i l e

1

i f

i )

i n d i c a t e s t h e t i m e i n t e g r a l o f t h e f u n c t i o n f ( t ) .

No

o r e than the r e s u l t s o f mathematical analy

s i s w i l l be s t a t e d h e r e , f o r the reason that

a math

ematical

d i s c u s s i o n i s

unnecessary

t o a n u nd er

s t a n d i n g

o f t h e i n v e n t i o n .

The

a n g u l a r p o s i t i o n

o f a

s h a f t w i l l

be

d e s i g n a t e d by 0 , t h e angular

v e l o c i t y by

0 w i t h s u b s c r i p t s where

a p p r o p r i a t e .

Th e

mechanical

t r a n s m i s s i o n c i r c u i t

beyond

t h e

r o t a t i n g s h a f t

o f

which

t h e a n g u l a r v e l o c i t y

i s t o

be i n d i c a t e d ,

i s analogous

t o

a n e l e c t r i c a l

t r a n s m i s s i o n c i r c u i t i n which t h e output c u r r e n t

e q u a l s

i n p u t

v o l t a g e m u l t i p l i e d b y

t h e s t e a d y

s t a t e t r a n s f e r a d m i t t a n c e ,

which

may e chosen

b y c i r c u i t

d e s i g n

t o r e s u l t i n

an output c u r r e n t

p r o p o r t i o n a l

t o

t h e

t i m e

d e r i v a t i v e o f

t h e i n p u t

v o l t a g e . The i n d i c i a l admittance o f

t h e

c i r c u i t

i s

t h e

w e i g h t i n g

f u n c t i o n

a c c o r d i n g t o

which

i s

a v e r a g e d

t h e

o u t p u t c u r r e n t

t o r e p r e s e n t t h e

b e s t

t i m e d e r i v a t i v e o f t h e i n p u t

v o l t a g e ,

which

may

e varying

a t a u ni fo r m r a t e p l u s the

super

p o s i t i o n o f r a n d o m i r r e g u l a r i t i e s .

T o s o l v e

t h e

problem o f mechanical d i i î e r e n t i a

t i o n ,

one

s p e c i e s

t h e d e s i r e d

mechanical

i n d i c i a l

8/11/2019 Us 2476269


2 , 4 7 6 , 2 6 9

3

a d m i t t a n c e , t h a t

i s ,

a mechanical w e i g h t i n g func

t i o n ; d e r i v e s

f r o m

t h i s t h e mechanical t r a n s f e r

a d m i t t a n c e ; and f i n a l l y

d e s i g n s

a

mechanical

system p o s s e s s i n g

a p p r o x i m a t e l y t h e

t r a n s f e r

admittance d e s i r e d . In both the e l e c t r i c a l a n d

the

mechanical

c a s e ,

the

i d e a l

s o l u t i o n

i s

f o r e

g o ne f o r p r a c t i c a l reasons.

R e f e r r i n g

n ow

t o

F i g . 1 , c u r v e s

A ,

B , C a n d D

a r e

w e i g h t i n g

f u n c t i o n s

p l o t t e d

with weight

f a c

t o r a s o r d i n a t e a g a i n s t time a s a b s c i s s a , the

u n i t

o f time b e i n g


t o

t h e time c o n s t a n t

o f t h e

averaging mechanical network. This

time

c o n s t a n t i s

d e t e r m i n e d ,

a s l a t e r e x p l a i n e d , b y

the

g e a r r a t i o s

and

m otor s p e e d s i n F i g s . 2 , 3

a n d

.

In F i g .

1 , c u r v e A

s

t h e e x p o n e n t i a l weighting

f u n c t i o n g i v e n by t h e s i m p l e d i f f e r e n t i a t o r o f F i g .

2 . Curve

B

s t h e i d e a l p a r a b o l i c W e i g h t i n g which

i s zero a t time z e r o a n d u n i t y . Curve C s o ne

a p p r o x i m a t i o n t o

c u r v e B ,

p r a c t i c a l l y

r e a l i z e d

by

t h e p r e s e n t i n v e n t i o n i n t h e embod i me nt shown

i n

F i g . 3

a n d

curve

D

s

a

b e t t e r

approximation

than curve

C which

i s

r e a l i z e d

by the

system

shown i n F i g .

4 .

The

improvements

p r o v i d e d

by t h e p r e s e n t i n

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

F i g . 2 , which shows

a s i m p l e system

o f apparatus

whereby t h e a n g u l a r v e l o c i t y

o f

a r o t a t i n g s h a f t

may

e measured. Shaft I 0 ,

t u r n i n g

i n

the

s e n s e

i n d i c a t e d

by the a r r o w ,

o c c u p i e s

momentarily

an

angular

p o s i t i o n

0 1 which v a r i e s with time a t

the

r a t e 0 1

which

i s

t o be determined.

Thr ough

d i f f e r e n t i a l gear I I , s h a f t I 0 d r i v e s s h a f t I 2 o n

which i s c a r r i e d p o i n t e r I 3 , t h e a n g u l a r

p o s i t i o n

o f which i s

read

o n

d i a l I 4 .

From s h a f t I 2

through b e v e l

g e a r s

I 5

i s

d r i v e n s h a f t I 6 c a r r y

i n g p i n i o n

I ‘ I

which moves

rack

I 8

t o

t h e

r i g h t ,

when

s h a f t I 6 turns i n the i n d i c a t e d

d i r e c t i o n .

A onstant speed mot o r

CSMI,

o f any

k n o w n

d e s i g n ,

d r i v e s a s i n d i c a t e d d i s c 2 0 ,

which

through

c o n t a c t i n g s p h e r i c a l b a l l s

2 | t u r n s r o l l e r 2 2 about

a n a x i s

a t r i g h t

a n g l e s t o the

a x i s

o f m o t o r

CSMI a t a

r o t a t i o n a l speed

determined b y

the

r a d i a l l o c a t i o n o f b a l l s 2 | o n d i s c 2 D .

The

x i s

o f r o l l e r 2 2

i s

prolonged

a s

s h a f t 2 3 ,

terminated i n

spur gear 2 4 meshing

with

r i n g gear 2 5 o f

d i f

f e r e n t i a l I I . B a l l s 2 | , which may be o f any de

s i r e d

even

number, two

b e i n g shown

i n

F i g .

2 ,

are b y a n y s u i t a b l e

m e a n s constrained b y

the

motion

o f rack

I 8

t o m ove

r a d i a l l y

with

r e s p e c t

t o d i s c 2 0 which r e s u l t s i n a n angular

speed

o f

r o l l e r

2 2 p r o p o r t i o n a l t o

t h e

a n g u l a r

p o s i t i o n

o f s h a f t

I 6 .

Th e d i s c

a nd

r o l l e r system j u s t de

scribed may

be such a s

s h ow n

i n United

States

Patent

1 . 3 1 7 . 9 1 5 ,

October

7 , 1 9 1 9

t o

H. C . Ford o r

i n United S t a t e s

Patent

2 , 0 0 2 , 5 8 5 , May 8 , 1 9 3 5

t o J . J .

Rothwell e t a l . I t w i l l

a t

o n c e

b e

recog

n i z e d t h a t t h e angular p o s i t i o n

o f

s h a f t

2 3

changes i n a n y

time

i n t e r v a l

b y an a m o u n t pro

p o r t i o n a l t o t h a t i n t e r v a l m u l t i p l i e d by

t h e

r a d i a l

s h i f t o f

b a l l s

2 | f r o m the c e n t e r

o f d i s c 2 0 ;

that

i s ,

t o

t h e

r o t a t i o n o f

s h a f t

I 5 ,

which

i s

t h e

s a m e

as

that

o f shaft

I 2 i f gears

I 5

are

o f

unity

r a t i o . When

s h a f t s

I 2 and I 6 through p i n i o n

I 1

a n d rack I 8

a r e

s e t t o

c e n t e r

b a l l s 2 l i o n d i s c

2 0 , p o i n t e r

I 3

i s

a d j u s t e d o n

s h a f t I 2 t o

read

zero

o n

s c a l e

I 4 .

Now, i f

t h e

angular p o s i t i o n 9 2 o f s h a f t I 2

r e p r e s e n t s

an angular v e l o c i t y , t h e r o t a t i o n o f

s h a f t 2 3 d u r i n g any time i n t e r v a l i s p r o p o r t i o n a l

t o

the

time i n t e g r a l o f 6 2 over

that

i n t e r v a l . In

t e r m s

o f

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

p o s i t i o n

0 3

o f

s h a f t 2 3 i s

1

-0

p

1 5

25

3 0

35

4 0

60

7 0

By s u i t a b l e c h o i c e o f t h e g e a r r a t i o between

g e a r s

2 4 a n d 2 5 a f r a c t i o n

o f t h i s i n t e g r a l

may

b e impressed

a s

a

r o t a t i o n

o f

r i n g

g e a r

2 5 j u s t

s u f f i c i e n t

t o

o f f s e t t h e r o t a

t i o n o f

s h a f t

I 0 ,

whereupon s h a f t

I comes t o

r e s t

a t a n angular departure 0 2

f r o m

the zero

s e t t i n g above d e f i n e d . When h i s

c o r n e s

about

n o further r o t a t i o n o f

shaft

I 2 takes p l a c e .

0 2

read

o n s c a l e I 4 i s

then

the measu r e o f the a n g u

l a r v e l o c i t y o f

s h a f t I 0 : 01=p0i=6a

The elements f r o m

g e a r s

I 5 through r i n g

gear

2 5 c o n s t i t ut e a n i n t e g r a t i n g mechanical feedback

path

between

o u t p u t

s h a f t I 2 and i n p u t s h a f t I 0 .

The

r e g u l a t e d

power

s u p p l y 3 0 , o f

any

k n o w n

f o r m ,

s u p p l i e s

power

t o d r i v e

mot or I 9 , t h e s p e e d

o f

which m a y

b e

a d j u s t e d

by v a r y i n g r e s i s t o r

3 | . The i n t e g r a t o r above d e s c r i b e d i s g e n e r a l l y

d e s i g n a t e d

a s 9 0

i n c l u d e d

i n

dashed

r e c t a n g l e

I ,

F i e . 2 .

Whenever

a sudden change

i s

brought i n

the

angular

v e l o c i t y

o f

s h a f t I 0 a c e r t a i n

time i n

t e r v a l must

e l a p s e b e f o r e s h a f t I 2

t u r n s

t o

the

corresponding

new

steady reading

o n s c a l e I

4 .

The readjustment

o f s h a f t I 2

i s a t f i r s t r a p i d a n d

then

m o r e

s l o w .

A s may b e

shown

b y a n a n

a l y s i s h e r e o m i t t e d , t h i s

r e a d j u s t m ‘ e n t o b e y s

t h e

law o f c u r v e A f F i g .

1 ,

t h e

e x p o n e n t i a l

weighting

f u n c t i o n

which

a t t a c h e s g r e a t e s t w e i g h t t o

t h e

e a r l i e s t

i n f o r m a t i o n .

F i g ,

2

shows a l s o how

the

angular

v e l o c i t y

02=p01

may

e shown

l e c t r i c a l l y

as w e l l a s me

c h a n i c a l l y . Shaft

I 2

i s p r o l o n g e d

t o

c a r r y brush

3 2

t r a v e r s i n g

l i n e a r

p o t e n t i o m e t e r

3 3

s u p p l i e d

w i t h v o l t a g e f rom b a t t e r y 3 4 , which

may

e i n

cluded i n power

supply 3 0 . Variable

r e s i s t a n c e

3 6 may be a d j u s t e d t o c o n t r o l

t h e

s c a l e

f a c t o r

a t

which

v o l t r n e t e r 3 5 , connected between b rush

3 2 a n d t h e

grounded

end o f potentiometer

3 3 ,

i n

d i c a t e s

a

v o l t a g e p r o p o r t i o n a l t o 0 2 .

A change i n v e l o c i t y o f s h a f t

I 0 o c c u r s

when

t h a t

s h a f t i s f i r s t conn ected with

t h e

apparatus

with which a n observer

i s

tracking a n a i r p l a n e ,

f o r

example,

a n d o t h e r v e l o c i t y changes o c c u r

when

shaft I l l

i s advanced

o r

retarded b y the

observer in correcting for overrun o r

u n d e r r u n

i n h i s o b s e r v a t i o n s .

In

such t r a c k i n g , t h e e a r l i e r

c o r r e c t i o n s a r e

u s u a l l y

l a r g e r

than

t h e l a t e r ,

a n d

the

e a r l i e r o f

t h e

v e l o c i t y

v a r i a t i o n s

av

eraged b y t h e system o f F i g . 1

a r e

g i v e n undue

w e i g h t by


w e i g h t i n g . I t i s

t h e r e

f o r e d e s i r a b l e t o p r o v i d e

a

weighting f u n c t i o n

approximating

the

i d e a l

curve

B f F i g . 1 .

This

i s

done by a n

e l a b o r a t i o n

o f

the

simple

s y s t e m , i n

which

again a r e used the elements

i n c l u d e d

i n dashed

r e c t a n g l e

I o f F i g . 2 .

To

d e r i v e

a

p h y s i c a l l y r e a l i z a b l e

appr o x i ma

t i o n

t o

t h e

p a r a b o l a

o f F i g . 1 ,

c o n s i d e r

t h e i d e a l

i n d i c i a l

admittance

A o < t > and

t h e

correspond

t r a n s f e r

admittance

Y o m ) , where

p i s

t h e d i f

f e r e n t i a l

o p e r a t o r

d

d t

a n d

p“

s

t h e

nth

time

d e r i v a t i v e

ï

d i

I t

may e shown t h a t

d e s i r a b l y

A o ( t ) =6t(1t) f o r t between 0 a n d 1

=0 o r t

<0

or>

1 ,

( 1 )

I 5

where t h e t h e o r e t i c a l r e s p o n s e time o f t h e

me

8/11/2019 Us 2476269


2 , 4 7 6 , 2 6 9

c h a n i c a l

c i r c u i t

i s u n i t y . A o ( t ) i s d r awn a s curve

B f F i g .

1 .

The c o r r e s p o n d i n g t r a n s f e r admittance m a y

b e

shown t o b e

g i v e n

by

T he

s p e c i ñ c

e mb o d i me nts shown i n F i g s . 3 a n d

4 c o n s t i t u t e

a p p r o x i m a t i o n s

i n v o l v i n g

t h e

r i g h t

h a n d

m e m b e r o f Equation 2 t o t h e second a n d

t o the fourth power o f p , r e s p e c t i v e l y .

Y 1 ( 1 U ) :ggü 1

P

3 2 2 2 +

1 0

- l - 1 0

which

may

e

written

( 3 )

2 9 .

Y1(P)=

_3 _ _

( 4 )

` 1

-

l

p < p + 5 + â v 5 ) ( p + 5 1 2 J

5 )

where

z ` \ / 1

The c o r r e s p o n d i n g i n d i c i a l a d m i t t a n c e

A 1 ( t )

may

e shown

t o be

s a

40 2-

.

N/ Et

: J n / E e sm ’ à

T h i s i s t h e w e i g h t i n g

f u n c t i o n shown

i n

F i g 1

as c u r ve C.

As another

a p p r o x i m a t i o n ,

l e t

then

1

18 3

_

l

G1=ìña b

_£1 ¿Ddo

t h a t

Y u p )

: i 1 2

p+70

( 6 )

p

4 1 3 + 4 _ 5 _ 0

2 + 2 1 6 0

4 2 0 0

p 4 1 p

41

p 4 1

which may e w r i t t e n

( 7 )

Y2 7 1 ) =

P

4 0 p + 7 0

4 1 ( p 4 - 4 . 2 6 ) @ +3.36 + 1 3 5 7 ) ( p + 3 . 3 c ~ i 3 . 5 7 )

l e a d i n g t o t h e i n d i c i a l a d m i t t a n c e

A z ( t ) ,

p l o t t e d

i n F i g . l as curve

D:

S t a t e d

i n

i t s

most

g e n e r a l f o r m , t h e

r e l a t i o n

s h i p o f

t h e

a n g u l a r d i s p l a c e m e n t 0 2 o f o u t p u t

s h a f t

I 2

t o

0 1 , t h e a n g u l a r d i s p l a c e m e n t o f s h a f t

l l l ,

F i g .

2 , i s 02=Y(p) 0 1 and t h i s i s

a n a l o g o u s

t o

t h e

e l e c t r i c a l c i r c u i t formula I = Y ( p) E , i n which

I i s the

output

current o f

a n

e l e c t r i c a l

network

o f t r a n s f e r admittance Y(p)

a nd

E s the e l e c

tromotive f o r c e impressed a c r o s s the network

i n p u t . By

s u i t a b l e

c h o i c e o f

Y W ) ,

t h e

o u t p u t

1 0

1 5

20

25

30

3 5

40

45

50

5 5

6 0

65

7 0

7 5

6

c u r r e n t may e

m a d e

p r o p o r t i o n a l t o e i t h e r t h e

time

i n t e g r a l o r t h e

time

d e r i v a t i v e o f

t h e

i n

put v o l t a g e ,

a n d

the s a m e

may

be c l o n e i n a

mechanical s y s t e m .

T he

p r e s e n t i n v e n t i o n con

t e m p l a t e s

t h e l a t t e r

n a m e d

r e l a t i o n s h i p between

i n p u t

and

o u t p u t

s h a f t

a n g u l a r

d i s p l a c e m e n t s .

In

t h e s i m p l e d i i f e r e n t i a t o r shown

i n F i g . 2 ,

a n d

the

angular displacement

o f

s h a f t l 2 r e p

r e s e n t s

the

angular v e l o c i t y o f s h a f t

I 0 , ex

p o n e n t i a l l y ave r a g e d .

In

any

c a s e , where

0 2 ( t ) = Y ( p ) . 0 ( t ) , i t may

be

shown t h a t t h e r e s p o n s e

0 2 ( t ) a t time

t i s

g i v e n

b y

t h e e qu a t i o n

z .

c e )

= f _ œ

A c - n 1 o ) d n < 9 1

Where

) 1

i s

time counted

f r o m

the

p a s t

f r o m

the

same

p o i n t

o f r e f e r e n c e

a s t ,

a n d

0100 i s

the

a n g u l a r

v e l o c i t y

(measured c o n t i n u o u s l y a t s u c

c e s s i v e v a l u e s

o f

A )

o f the

s h a f t having t h e a n g u

l a r displacement mt). The age

a t

time t

o f

the me as u re me nt made

a t

time

A

i s written

a s

1 = t > `

a n d

z

.

f _ œ A c - m a m d r

b e c o m e s

L 1 ( 1 ) @ c - f w f ,

where

A ( f ) i s

t h e

w e i g h t f a c t o r

a p p l i e d

t o t h e

data o f age r .

I f

such a

weighting function

as

Ah)

c a n

b e

p r o v i d e d ,

then

0 2 ( t ) = 1

where

_

ó i

i s t h e Weighted average o f

@ . 1 ( Ä )

i s

t h e t r a n s f e r admittance d e t e r m i n i n g t h e f i n a l

v a l u e

o f

6 2 .

Y<p>

i s

t o

be

mechanized.

I t c a n

be

f u r t h e r shown t h a t

t o

be

p r a c t i c a l l y

mechanized

Y(p)

must

be

o f t h e

f o r m

where M(p) a nd

N(p)

a r e

polynomials

i n p

with

c o n s t a n t

r e a l c o e f f i c i e n t s . We h a l l c o n s i d e r

t h e

mechanization o f the c a s e s where N MO)

i s

e i t h e r

one

o r two d e g r e e s higher i n p

than

Mw).

To

p u t

t h e o p e r a t i o n a l e q u a t i o n

i n t o

a

f o r m

s u i t a b l e f o r

mechanization,

i t i s

car

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

s t e p s a s

f o l l o w s :

w h e r e a=0

i f

N ( p ) l i s o n l y o n e f d e g r e e h i g h e r

i n

8/11/2019 Us 2476269


» 7

p

than

Mo».

' r ï f m i s a t most o f

t h e

same de

gree a s Mm). Then

N 0 1 )

P M Q J )

i s expanded i n t o

p a r t i a l

f r a c t i o n s o f which one

i s o f

the

f o r m

N o w ,

l

P

a n d

the

o t h e r s a r e each o f

the f o r m

1

1 1 + @

P

out

i n

common

o a l l t h e s e

t e r m s ,

we have

( 1 0 )

1 1

0 2 : . » 1 1 1 7 - 1 -

1 0 1 - g l a i -

Y »

( p ) l Y . . ( p ) + e t e 1 o 2 ]

I n Equation

1 0 t h e

Y ( p ) ’ s

a r e

each

o f

t h e

f o r m

The

terms

Ym(p),

Y u m ) ,

e t c . , a r e

a l l

o f

the

f o r m

PM@

p )

a n d

s o introduce n o

new roblems

i n the design

now o be

proposed.

When

i = 0

a nd B 1 = 1 a n d t h e

terms i n Ym(p),

e t c . , a r e o m i t t e d , Equation

1 0

r e d u c e s t o t h e

f o r m

@ F m - § 0 2

( 1 0 )

which

i s t h e c a s e o f

the

system

o f F i g . 2 .

The

system

o f

F i g . 3 c o r r e s p o n d s

t o v a l u e s

o f A1 a n d

B1 both

g r e a t e r than

z e r o , s t i l l o m i t t i n g

Ym(p),

e t c . Adding Ym(p) l e a d s t o t h e system

o f

F i g . 4 .

E a c h o f

the

systems o f F i g s . 3 a n d 4 e s t a b l i s h e s

a weighting

function approximating

the i d e a l

p a r a b o l a , c u r v e

B

o f F i g . 1 , and o b v i o u s l y b y

p r o g r e s s i v e l y

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

c o m p l e x i t y

o f

t h e

m e c h a n i s m

the

a p p r o x i m a t i o n

c a n

b e

carried as

far as d e s i r e d .

Mechanization

o f such

a

factor as

P - I - b

o b v i o u s l y i n v o l v e s

the g e n e r a t i o n

o f

0 2 a s the

i n

t e g r a l

o f

6 1 4 - 1 2 0 2 ,

t h a t

i s ,

and

i n

t h e

s y s t e m o f

F i g . 3 s h a f t I 2 , i n s t e a d o f

b e i n g

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

d i f f e r e n t i a l

I I

i s t h e r e t o connected

through

a n i n t e r v e n i n g me

c h a n i c a l

i n t e g r a t o r

o f

t h e

g e n e r a l

fo r m i n c l u d e d

i n dashed r e c t a n g l e I o f

F i g .

2 .

10

1 5

20

25

30

35

40

45

5 0

55

6 0

6 5

70

7 5

R e f e r r i n g

now t o

F i g .

3 ,

i n p u t s h a f t

I 0 and

output s h a f t

I 2 c o n t i n u e

t o be

connected by t h e

f e e d b a c k p a t h

( d a s h e d

r e c t a n g l e I ) e n t e r i n g d i f

f e r e n t i a l gear I I . The

d i r e c t

connection f r o m


gear

I l

t o

output

s h a f t I 2 i s

here i n

t e r r u p t e d t o i n c l u d e, i n dashed

r e c t a n g l e

2 ,

t h e

i n t e r v e n i n g

i n t e g r a t o r

mentioned

i n

t h e

p r e c e d

i n g p a r a g r a p h . O p p o s i t e s h a f t

I l l ,

o u t p u t s h a f t

4 0 l e a d s

t o d i f f e r e n t i a l

gear

4 I

a n d

therethrough

t o s h a f t 4 2

terminating

i n b e v e l g e a r s 4 3 .

Through

g e a r s 4 3 ,

s h a f t 4 2

d r i v e s

s h a f t 4 4

c a r r y

i n g

p i n i o n 4 5

engaged by r a c k 4 6 .

Rack 4 6

c on

t r o l s ,

a s d o e s

rack I 8 ,

an i n t e g r a t i n g

element

g e n e r a l l y

d e s i g n a t e d

a s 5 0

a n d i d e n t i c a l l y

l i k e

elements

CSMI

o 2 2 o f F i g . 2 . Output s h a f t 5 I

o f

i n t e g r a t o r

5 0 d r i v e s , through b e v e l

g e a r s

5 2

and

5 3 , ñnal

o u t p u t s h a f t

I 2 . The last-named

s h a f t

i s provided with

b e v e l g e a r s

I 5 a n d c a r r i e s

p o i n t e r I 3

a s

i n F i g . 2 . The feedback

path

b e

tween

s h a f t s I 2 a nd I 0

i s

i d e n t i c a l

with that i n

F i g .

2 .

On

h a f t

I 2 ,

a t

t h e

end

o p p o s i t e p o i n t e r

I 3 ,

i s

c a r r i e d

spur gear 5 4 meshing with r i n g - gear

5 5 o f


gear 4 I .

The angular d i s p l a c e

me nt 0 2 o f

s h a f t I 2

i s here

r e l a t e d

t o

t h e

angular

d i s p l a c e m e n t 6 4

o f

s h a f t 4 0 by t h e

r e l a t i o n

1

( 9 a - m 0 4

S i n c e 0 4 i s , a s i n

F i g .

2 f o r 6 2 , e q u a l t o

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

s h a f t

I 2 i n

F i g .

3 i s

1

a

9 2 _ p + b 0 ‘ - P 0 2 )

Exami nat i o n o f the d i ag r am o f F i g . 3 shows that

while the

i n i t i a l

e f f e c t o f the r o t a t i o n 0 4 i s t o

b r i n g about

a


angular

v e l o c i t y o f

s h a f t 5 I ,

t h e i n c r e a s i n - g displacement H z o f s h a f t

I 2 p r o g r e s s i v e l y c a n c e l s

t h i s

e f f e c t s o that

i n t e

g r a t o r

5 U c e a s e s

t o

o p e r a t e s i m u l t a n e o u s l y with

t h e

co mi ng

t o r e s t o f

s h a f t

I 2 ,

p r o v i d e d

t h e

a n g u

l a r

v e l o c i t y 0 1 i s

s t r i c t l y c o n s t a n t .

When h a f t

I 2

i n

F i g .

3 c omes t o r e s t , i t

may

be shown

t h a t

i t s angular p o s i t i o n 0 2 i s propor

t i o n a l

t o

t h e

angular v e l o c i t y 0 1 o f s h a f t I 0

weighted s u b s t a n t i a l l y i n accordance with c u r v e

C , F i g .

1 . I t

w i l l

be o b s e r v e d t h a t angular p o s i

t i o n

feedback has been introduced

f r o m

shaft

I 2 t o r i n g gear 5 5 ,

i n

a d d i t i o n t o t h e feedback

me

c h a n i c a l

i n t e g r a t i n g

path

f r o m

s h a f t I 2 t o r i n g

gear 2 5 .

The

added p a t h ,

i n c l o s e d i n dashed

o u t l i n e 2 , i s

a n

improvement

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

p r e s

ent i n v e n t i o n ; a n a d d i t i o n a l

i m p r oveme nt

i s

shown i n F i g .

4 .

Without d e t a i l i n g t h e n e c e s s a r y

mathematical

a n a l y s i s , i t may be s t a t e d t h a t the appr o x i ma

t i o n

t o the p a r a b o l i c Weighting f u n c t i o n , curve

B f F i g .

l , may e

p r o g r e s s i v e l y improved

b y

a d d i ng

a n y n u m b e r

o f

admittance paths i n par

a l l e l with

the

path between

g e a r s I 5 a n d

i n t e

g r a t o r 9 0 o f F i g . 2 . E a c h o f t h e s e added ad

mittance paths

r e q u i r e s

the

i n t r o d u c t i o n

o f a

d i f f e r e n t i a l gear t o e n a b l e i t s output t o j o i n that

f r o m

g e a r s

I 5

t o

be operated

o n

b y

i n t e g r a t o r

9 0 .

I t w i l l s u i i i c e here t o

i l l u s t r a t e

the

a d d i t i o n

o f

o ne

such admittance.

Referring

now o F i g . 4 , a system providing a

w e i g h t i n g f u n c t i o n s u b s t a n t i a l l y t h a t shown by

c u r v e D , F i g . 1 , the system o f F i g . 3

i s

e l a b o r a t e d

b y adding

a

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

i n c l u d e d

i n dashed

r e c t a n g l e

3 . Bevel

g e a r s

6 0 transmit t h e mot i o n

o f s h a f t

I 2

t o a u x i l i a r y

s h a f t

6

I ,

from which

b e v e l

8/11/2019 Us 2476269


ö t i i t i i i i ‘ i

N U U W É

2 , 4 2 8 , 2 6 6

g e a r s 6 2 r e d i r e c t t h e motion t o e n t e r d i f f e r e n t i a l

g e a r

6 3 .

The output

s h a f t

6 4

o f gear 6 3

c a r r i e s

b e v e l

g e a r s

6 5 whereby i s d r i v e n

rack

6 8 c o n t r o l

l i n g

t h e i n t e g r a t o r

g e n e r a l l y

d e s i g n a t e d

a s

1 0 .

o f

l i k e fo r m

t o

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

t o r s

i n

F i g s .

2

and

.

Thr ough

b e v e l g e a r s

‘ I 2

a n d

1 3 , and r i n g g e a r I 4 t h e o u t p u t

o f

i n t e g r a t o r I 0

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

6 3

a m ot i on propor

t i o n a l

t o

t h e

time i n t e g r a l o f

t h e motion

o f

s h a f t

G 4 a n d i n a

sense adapted t o check

the

m o t i o n

o f

t h e

l a s t

n a m e d s h a f t .

T o ad d the mot i o n

o f

s h a f t 6 4 t o that o f s h a f t

I 6 , d i f f e r e n t i a l 1 5 i s i n t r o d u c e d

between

g e a r s I 5

and p i n i o n

I l

a n d

rack I 8 i s m o v e d f r o m a b o v e ,

a s i n

F i g . 2 ,

t o

below

p i n i o n

I I . Shaft

6 4

i s pro

l o n g e d t o

d r i v e

through b e v e l

g e a r s

1 6 ,

r i n g

g e a r

I l

o f

d i f f e r e n t i a l I 5 i n

t h e

s e n s e

t o

add t h e

m ot i on o f s h a f t 6 4

t o that

o f s h a f t I 6 ,

the

l a t t e r

d r i v e n through

d i f f e r e n t i a l I 5 by s h a f t I 6 .

The

c om b ined m otion o f

s h a f t s 6 4

a n d

I 6 i s then

i n t e g r a t e d

by

i n t e g r a t o r

9 D ,

a s p r e v i o u s l y

d e

s c r i b e d i n

connection with F i g . 2 ,

t o c a n c e l

pro

g r e s s i v e l y t h e motion o f t h e output s h a f t o f d i f

f e r e n t i a l I l .

In

F i g s .

2 , 3 a n d 4 , f u l l l i n e arrows

i n d i c a t e

t h e

ñnal

d i r e c t i o n s

o f

movement

o f

t h e

r a c k s i n

i n

t e g r a t o r s

9 0 ,

5 0

and l 0 , when

s h a f t

I 2 i s moving

f r o m z e r o t o i t s

f i n a l

r e s t

p o s i t i o n ,

a s indicated

b y the

dashed arrows.

The

r a c k s

o f i n t e g r a t o r s 5 0

a n d I 0 i n i t i a l l y

m o v e a s

shown b y the

dashed arrows a n d t h e n ,

a s

s h a f t

I 2

approaches i t s

ñnal p o s i t i o n , r e t u r n

( f u l l

l i n e

a r r o w s )

t o

c e n t e r t h e t r a n s m i t t i n g b a l l s

o n

t h e i r d r i v i n g d i s c s

i f b 1 i s

c o n s t a n t .

The rack

o f i n t e g r a t o r

S 0

moves o n l y a s

shown b y

the f u l l

arrow,

c o m i n g

t o

a

f i n a l p o s i t i o n a s

shaft

I 2

does

s o .

I t w i l l

be noted that the d i r e c t i o n s o f r o t a

t i o n

o f t h e d i s c s d r i v e n by c o n s t a n t speed motors

in

a l l

three integrators

are

the

same.

I t i s t o

b e

un derstood that

t h i s

i s only i l l u s t r a t i v e , a n d ` » t h a t

the mot o r

r o t a t i o n s

may

e

conveniently chosen

i f

regard

i s h a d t o

preserve

the

sense o f

the

shaft

motions a s i n d i c a t e d . Further, a

l i t t l e

consid

e r a t i o n o f

F i g .

4 w i l l show that a s

f o r

i n t e g r a t o r

5 0 i n

F i g .

3 ,

t h e co mi ng

t o r e s t o f

s h a f t I 2 w i l l

b e accompanied by

t h e

c e s s a t i o n o f i n t e g r a t i n g

o p e r a t i o n by b o t h

i n t e g r a t o r s

5 0

and I I I

i f

t h e

m ot i on

0 1 o f s h a f t

I U i s

s t r i c t l y un i f or m with t i m e .

Only i n t e g r a t o r

9 0

w i l l i n t h a t e v e n t c o n t i n u e t o

o p e r a t e

t o m a k e 0 2 = p ü i = 0 1 . S i n c e t h e c o n s t a n t

s p e e d

motors r un

c o n t i n u o u s l y ,

a l l

t r a n s m i t t i n g

b a l l s

c o m e t o c e n t e r when s h a f t

I 0

i s a t r e s t .

The

time

c o n s t a n t o f t h e system i s determined

b y

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

i n t e g r a t o r

d i me n

s i o n s ,

t o g e t h e r w i t h t h e

s p e e d s

o f t h e c o n s t a n t

s p e e d motors CSMI, CSM2

and

CSM3, a n d

may

b e c h o s e n a t p l e a s u r e

by a d j u s t i n g t h e s e

s p e e d s .

I t i s : t o be understood

t h a t

a l l

t h e c o n s t a n t

s p e e d

motors m a y

b e

s u p p l i e d ,

by means shown

o n l y

f o r

CSMI, f r o m power

s u p p l y

3 0 ,

F i g . 2 .

S i n c e

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

i t i s deemed unnecessary herein t o p r e s c r i b e gear

r a t i o s

a n d dimensions

o f p a r t s ; t h e s e may be

chosen a s

seems

convenient t o

the

d e s i g n e r .

A d j a c e n t t h e important

s h a f t s

i n F i g s . 2 , 3 and

4 a r e

i l l u s t r a t e d

t h e i r

r e s p e c t i v e

angular

d i s p l a c e

ments

a s

0 i ,

0 2 ,

e t c . I t may

e p r o f i t a b l e t o

tabu

l a t e

t h e

r e l a t i o n s

o f

t h e s e d i s p l a c e m e n t s

i n t h e

s e v e r a l f i g u r e s ,

s t a t i n g

t h e

c o r r e s p o n d i n g

W e i g h t

i n g f u n c t i o n s .

In F i g .

2 :

a

_

P

.

0 2 - 9 1 - 5 - 9 2 _m 1

w e i g h t i n g

f u n c t i o n

e x p o n e n t i a l ,

c u r v e A

f F i g . 1 .

i 6

I n F i g . 3 2

1

9 = _ _ _ _ . . , 9

’

n + 1 »

‘

:___-___P 0

p2+pb+a ‘

l o w e i g h t i n g

f u n c t i o n

a p p r o x i m a t e l y p a r a b o l i c ,

Equation , c u r v e

C

f F i g . 1 .

In

F i g . 4 :

l

@ g m t

a l

20

2 5 ‘ 0 1 1

w e i g h t i n g f u n c t i o n

a p p r o x i m a t e l y

p a r a b o l i c ,

Equation , curve

D f

F i g .

1 .

In t h e

a b o v e

e x p r e s s i o n s

f o r 0 2 , t h e c o n s t a n t s

a , b

a n d c

r e f e r o f course t o the gear r a t i o s c o n

cerned

i n d r i v i n g t h e r i n g

g e a r s .

O b v i o u s m o d i f i c a t i o n s

i n

t h e g r a d u a t i o n

o f

s c a l e

I 4

and

i n

t h e b a t t e r y c o n n e c t i o n

t o

p o t e n t i o m e t e r

3 3 can be m a d e t o e n a b l e the

system

t o

i n d i c a t e

angular v e l o c i t i e s o f

s h a f t

I I ) i n

t h e

o p p o s i t e

sense to that shown n the

drawin gs .

C a l i b r a t i o n o f

t h e system i s

m a d e b y

i m p o s i n g ,

f r o m

a n y s u i t a b l e source

a

known

constant

a n g u l a r

v e l o c i t y

on

s h a f t

I 0 , t h e t r a n s m i t t i n g

b a l l s

o f

t h e

s e v e r a l


9 0 , 5 D a n d I 0 b e i n g

5 c e n t e r e d o n t h e i r r e s p e c t i v e d r i v i n g d i s c s which

a r e i n r o t a t i o n a t constant

s p e e d .

T he

f i n a l

p o s i

t i o n o f p o i n t e r I 3 ,

a n d

t h e time taken t o reach

t h a t

p o s i t i o n , a r e n o t e d . T he d i s c

speeds

may

e

changed i f t h i s

time

i s t o o

l o n g

o r t o o s h o r t ,

a n d

p r o v i s i o n , not shown, may e m a d e t o gear u p o r

d ow n

the

m o v e me nt o f

p o i n t e r I 3 .

I t

i s

obvious

fr o m

a

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

o f

F i g .

2 ,

f o r

e x a m p l e , t h a t

t h e

h i g h e r

t h e

s p e e d o f d i s c 2 0 ,

t h e

l e s s

t h e

d i s

p l a c e m e n t o f

s h a f t

I

2

f o r a , g i v e n a n g u l a r v e l o c i t y

5 o f

s h a f t I 0 . I f the response time s t o be shortened

a n d

the s c a l e f a c t o r o n s c a l e I 4 i s t o be kept the

same, p o i n t e r I 3 must be geared u p i n the s a m e

r a t i o

a s

t h e d i s c speed i s i n c r e a s e d .

The c h o i c e a m o n g t h e s y s t e m s o f F i g s . 2 , 3

a n d

4

60 i s

t o

be

based o n the

character

o f

the

m ot i on

0 1

o f

s h a f t

I 0 . I f t h i s m ot i on i s uni for m with t i m e ,

t h e

simple

system o f

F i g . 2 w i l l

s e r v e .

I f 0 1

i s

a

m ot i on un i f or m with time

p l u s

r a n d o m i n c r e a s e s

a n d

d e c r e a s e s

i n

v e l o c i t y , t h e system

o f

F i g . 4

5 5 i s p r e f e r r e d

t o

t h a t

o f F i g .

3

u n l e s s

t h e a d d i t i o n a l

me ch a n i s m i s unwelcome. T he apparatus o f F i g .

4 ,

r e a l i z i n g

c u r v e

D

f

F i g .

1 ,

m o r e

c l o s e l y

f o l l o w s

the

parabola B, a n d thus provides a m o r e a c c u

r a t e

r e s u l t

than

d o e s t h e apparatus

o f

F i g .

3

70

which

c o r r e s p o n d s t o c u r v e C .

Moreover,

s i n c e

curve D eaches zero sooner than curve

C ,

the

error due t o the curve

area

b e y o n d

t = 1 i s

l e s s .

I t

w i l l

be u n d e r s t o o d

t h a t

t h e s p e c i f i c i n t e g r a t

i n g

e l e m e n t s shown i n i n t e g r a t o r s 9 0 , 5 0 and 1 0

7 5 a r e i l l u s t r a t i v e o n l y , t h e r e b e i n g numerous me~

30

35

4 0

8/11/2019 Us 2476269


2 , 4 % , 2 6 9

c h a n i c a l s p e e d changing s y s t e m s

which m a y

b e

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

shown,

t o produce

t h e same

r e s u l t .

In F i g .

2 ,

f o r

e x a m p l e ,

a l l

t h a t i s r e q u i r e d

i s

a means o r

conferring

o n

shaft 23 a n a ngula r

v e l o c i t y p r o p o r t i o n a l

t o

t h e

a n g u l a r

d i s p l a c e m e n t

o f

shaft

I 2 f r o m

,

zero

p o s i t i o n ,

a n d

a n y

means

o r

s o

doing may

e p l a c e

t h a t

shown without

d e p a r t

i n g f r o m the p r e s e n t i n v e n t i o n .

For a

s h a f t

I l l turning

with

a g r a d u a l l y

varying

angular v e l o c i t y , with

o r

without r a n d o m i r r e g u

l a r i t i e s , t h e i n v e n t i o n p r o v i d e s ,

i n

t h e embodi

ments

shown

i n F i g s .

3

a n d

4 , an


a v e r a g e d

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

over a

time

i n t e r v a l o f

s e l e c t a b l e

l e n g t h .

The

s p e e d o f t h e

t i m i n g

d i s c s

i n

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

be s e t

f o r

a response time

s e v e r a l

times the period

o f

t h e p r i n c i p a l i r r e g u l a r i t i e s . The system

o f

t h e

i n v e n t i o n

then p r o v i d e s

a

run n i n g average

a n g u l a r v e l o c i t y

o f t h e

i n p u t s h a f t ,

smoothing o u t

r a n d o m

f l u c t u a t i o n s .

I

am ware

that

m e c h a n i c a l

d i f f e r e n t i a t i n g

a n d

s m o o t h i n g

n e t w o r k s ,

c o m b i n i n g


g e a r s

a n d mechanical

i n t e g r a t o r s ,

are k n ow n . I - I o w

e v e r , the d e s i g n herein d i s c l o s e d a n d based u p o n

the

derivation o f E qua t i o n

10 i s

believed t o i n - ~

c l u d e advantages not heretofore r e a l i z e d i n such

mechanisms. These advantages a r e :

1 . The

d e s i g n

i s independent o f t h e n u m b e r o f

r e a l o r conjugate, o r b o th r e a l

a n d conjugate,

complex p a i r s o f e x p o n e n t i a l

f u n c t i o n s

o f t i m e

which

are c o m b i n e d

t o

obtain a

s a t i s f a c t o r y

w e i g h t i n g f u n c t i o n .

2 .

T he n u m b e r

o f p r e c i s i o n

p a r t s

i s

a l s o i n d i .

pendent o f

t h e n u m b e r

o f


f u n c t i o n s .

Since i f t h e angular v e l o c i t y o f s h a f t I 0

i s

ab

s o l u t e l y

c o n s t a n t

a l l


e x c e p t t h a t

g e n e r a l l y

designated

a s 9 0

e v e n t u a l l y

return t o

a

n e u t r a l p o s i t i o n , o n l y i n t e g r a t o r 9 0

needs

t o

b e

o f

high

p r e c i s i o n .

In a d d i t i o n , a l l g e a r s may e o f

low

p r e c i s i o n ,

e x c e p t ,

a s i n F i g . 4 , g e a r s I 5 ,

d i f f e r e n t i a l 1 5 , p i n i o n

I 1

and

r a c k

I 8 , and

p r e c i s e

s p e e d r e g u l a t i o n i s r e q u i r e d o n l y

f o r

motor

CSMI.

What s claimed

s :

1 .

Mechanis m

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

angular

v e

l o c i t y

o f

a f i r s t

r o t a t i n g

s h a f t comprising a second

s h a f t , m e a n s

including

a f i r s t d i f f e r e n t i a l gearing

f o r

d r i v i n g t h e second s h a f t

f r o m

t h e

f i r s t

s h a f t

a n d i n the o p p o s i t e sense o f

r o t a t i o n ,

a

t h i r d s h a f t ,

m e a n s including a sec on d


gearing

f o r

driving the t h i r d shaft f r o m the

second

shaft

i n

the

sense

o f

rotation

o f

the

f i r s t

s h a f t ,

a fourth

s h a f t ,

me a ns i n c l u d i n g a

mecha ni ca l

i n t e g r a t o r

f o r d r i v i n g

t h e f o u r t h

s h a f t f r o m t h e t h i r d s h a f t

i n the sense

o f

rotation o f

the second

shaft a t a n



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

p l a c e m e n t

of the

third

shaft,

means for intro

ducing

f r o m the fourth s h a f t t o the t h i r d s h a f t

a n g u l a r

d i s p l a c e m e n t

f e e d b a c k


t o

a n d i n the s e n s e

o f

the angular di splacement

o f

the fourth s h a f t , m e a n s including a m e c h a n i c a l

i n t e g r a t o r connected between

t h e

fourth s h a f t

a nd t h e ñ r s t d i l f e r e n t i a l g e a r i n g f o r adding t o t h e

angular

v e l o c i t y o f

the second

s h a f t

a n o p p o s i t e



t o

t h e

a n g u l a r d i s

placement o f

the

fourth s h a f t , a n d

m e a n s

f o r i n

d i c a t i n g t h e last-named a n g u l a r

d i s p l a c e m e n t .

2 . A

mechanical system

c o n t r o l l e d

b y

the

ro

tation o f

a f i r s t

shaft c o m p r i s i n g a

sec on d

shaft

d r i v e n

through d i f f e r e n t i a l

g e a r i n g

f r o m the

ñ r s t

s h a f t ,

a t h i r d s h a f t driven through


gearing f r o m the second ` s h a f t , a fourth s h a f t , a

mechanical path i n c l u d i n g i n t e g r a t i n g

means b e

I I )

1 A

2 0

65

N

r

12

tween

t h e t h i r d s h a f t a n d t h e f o u r t h s h a f t , means

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

f r o m

the fou rth s h a f t

t o

t h e se co nd- name d


gearing a n d a

mecha ni ca l

feedback

path

including

i n t e g r a t i n g m e a n s

between

the

fou rth shaft a n d

t h e

first-named


g e a r i n g ,

whereby

r o

t a t i o n o f the f i r s t shaft occasions a n an gular d i s

placement o f

the fou rth

s h a f t


t o the


o f

t h e

f i r s t

s h a f t .

3 . Me c h a n i s m

f o r

i n d i c a t i n g the angular

ve

l o c i t y

o f a

f i r s t

r o t a t i n g

shaft comprising

a

second

s h a f t , me a ns including

a

ñ r s t d i f f e r e n t i a l

gearing

f o r d r i v i n g t h e

second

s h a f t f r o m the f i r s t s h a f t

a n d i n the o p p o s i t e

s e n s e

o f r o t a t i o n , a t h i r d s h a f t ,

m e a n s including

a

second d i f f e r e n t i a l gearing

f o r

d r i v i n g

t h e t h i r d s h a f t f r o m t h e second

s h a f t

i n

the sense o f

r o t a t i o n

o f the

f i r s t

s h a f t , a fourth

s h a f t , me a ns i n c l u d i n g

a mechanical

i n t e g r a t o r

f o r d r i v i n g the

fourth

s h a f t f r o m

the t h i r d

s h a f t

i n

the sense o f rotation o f the sec on d shaft a t

a n

angular

v e l o c i t y


t o

t h e

angular

d i s

placement o f

the

t h i r d s h a f t , m e a n s f o r introduc

i n g f r o m the

fourth shaft

t o the

t h i r d shaft

a n g u

l a r displacement feedback


t o

a n d i n

t h e

s e n s e o f the


o f the

fourth

s h a f t , a f i f t h

s h a f t , m e a n s

including

a

t h i r d d i f f e r e n t i a l g e a r i n g f o r d r i v i n g t h e ñfth

s h a f t

f r o m the

fourth

s h a f t

i n the o p p o s i t e

s e n s e

o f

r o t a t i o n , means including a

mechanical

i n

t e g r a t o r

connected

between the f i f t h s h a f t a n d

t h e t h i r d d i f f e r e n t i a l gearing f o r adding

t o

the

m o ti o n

o f the f i f t h

shaft

a n angular

v e l o c i t y

p r o p o r t i o n a l a n d opposed

t o

t h e

a n g u l a r d i s p l a c e

m e n t

o f

the f i f t h s h a f t , a s i x t h s h a f t , m e a n s

i n

c l u d i n g a fourth


gearing f o r impart

ing

t o

the

s i x t h

shaft

a n



t o

t h e

s u m o f

the

angular d i s p l a c e

ments o f

the

fourth

a n d f i f t h s h a f t s , m e a n s

i n

cluding

a

mechanical

i n t e g r a t o r connected

be

tween

the sixth shaft

and

the f i r s t


g e a r i n g f o r adding

t o

the

angular v e l o c i t y o f

t h e

second s h a f t a n o p p o s i t e angular v e l o c i t y

pro

p o r t i o n a l t o

t h e

angular displacement o f

t h e

s i x t h s h a f t , a n d me a ns

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

angular

displacement

o f t h e f o u r t h

s h a f t .

4 .

A echanical system a s i n c laim 2 i n c l u d i n g ,

i n

a d d i t i o n ,

a

f i f t h s h a f t d r i v e n

through

d i f f e r e n

t i a l

gearing

f r o m

the fourth

s h a f t , a mechanical

feedback path

i n c l u d i n g i n t e g r a t i n g

means

b e

tween the

f i f t h

s h a f t a n d the last-named d i f f e r e n

t i a l gearing

a n d m e a n s

f o r a d d i n g

the m o t i o n

o f

the

f i f t h

shaft t o

that

o f

the fourth s h a f t i n the

feedback path between

the fourth

shaft a n d

the

f i r s t - n a m e d d i f f e r e n t i a l

g e a r i n g .

RALPH

B.

BLACKMAN.

REFERENCES CITED

The

following references are

o f record i n

the

f i l e o f t h i s p a t e n t :

UNITED

STATES PATENTS

N u m b e r

Name Date

2 , 071, 4 2 4

P a p e l l o ___________ __ Feb .

28,

19 3 7

2,089,878

Corbin ___________

_

Aug. 1 0 ,

1 9 3 7

2 , 136 , 213 Hodgman

________

_ Nov. ,

1938

2 , 206 ,875

Chaffee ___________

_

Ju ly 9 , 19 4 0

2 , 2 48, 072 Fry _______________

__ Jul y

8 , 1 9 4 1

2,377,898 Myers

___________

_

June 1 2 , 1 9 4 5

2 , 4 2 6 , 584 Baker ____________ __ SeptA 2 , 19 4 7

2 , 4 3 3 , 0 0 6 Weiss ____________

_ Dec .

23,

19 4 7

2 , 4 4 2 , 792

White

____________

_

June ,

1948

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