ANALYSIS OF THE MOTION OF A GYRO-THEODOLITE.pdf

5/19/2018 ANALYSIS OF THE MOTION OF A GYRO-THEODOLITE.pdf

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

(Engl i sh Edi t ion, Vol .8 , No.9, Sep 1987)

P u b l is h e d b y S U T ,

Shangha i , China

A N A L Y S I S O F T H E M O T I O N O F A G Y R O - T H E O D O L I T E

W a n g H o n g - l a n ( 7 : ~ ~ . )

(Shandong Min ing Institute, Tai an, Shangdong)

(Received June 5, 1986; Com municated by Zhu Zao -xuan)

A b s t r a c t

W ith the metho d of analytical mechanics, this pape r studies the motions o f a gyro-

theodolite under the action o f (l ) the torque o f gra vity only, (2) the torque applied by the

band suspension, (3) the torque o f the band suspension with air damping considered, t h e

equations o f mo tion a re then established and their solutions are foun d. Furthermore,

analysis o f the law o f mo tion an d the behaviour o f gyro-theodolite during the orientation is

made.

G y r o - t h e o d o l i t e is a m a i n i n s t ru m e n t f o r o r i e n t a ti o n i n m i n e s u rv e y i ng . A n a n a l y si s a b o u t t h e

dyn am ics o f o r i en t a t i on i n Ref s . [5 ], [6 ], [10] a r e no t qu i t e r i gorou s , o r even invo lve mi s t akes . W i th

t h e m e t h o d o f a n a ly t ic a l m e c h a n i c s , t h is p a p e r s t u d ie s t h e m o t i o n s o f g y r o - th e o d o l i te u n d e r t h e

a c t i o n o f (1 ) th e t o r q u e o f g r a v i t y o n ly , ( 2) th e t o r q u e a p p l i e d b y t h e b a n d s u s p e n s io n , ( 3 ) th e t o r q u e

o f th e b a n d s u s p e n s i o n w i t h a ir d a m p i n g c o n s i d e re d , t h e e q u a t i o n s o f m o t i o n a r e th e n e s t a b l is h e d

a n d t h e ir s o l u t io n s a r e fo u n d . F u r t h e r m o r e , a n a l y s is o f l a w o f m o t i o n a n d t h e b e h a v i o u r o f g y r o -

t h e o d o l i t e d u r i n g t h e o r i e n t a t io n i s m a d e . T h e r e s u lt s a n a l y s e d i n th i s p a p e r m a y b e u s e f u l t o c o r r e c t

t h e e r r o r s in s o m e l it e ra t u re , t o i m p r o v e t h e c o m p u t a t i o n a l m e t h o d a b o u t t h e o r i e n t a ti o n o f a g y r o -

t h e o d o l it e a n d t o m a k e t h e c o m p u t a t i o n m o r e a c c u r a te . M o r e o v e r , t h e y m a y b e v a lu a b l e f o r

a n a l y s in g t h e s t ru c t u r e a n d t h e q u a l i ty o f a g y r o - t h e o d o l i t e .

I A M e c h a n i c a l M o d e l a n d K i n e m a t i c a l A n a l y s i s

A g y r o - t h e o d o l i te i s c o m p o s e d o f tw o p a r ts , a g y r o s c o p e u s e d a s t h e s e n si ti v e e l e m e n t a n d a

t h e o d o l i t e w h i c h o r i e n t a t e d b y t h e s e n si ti v e e l em e n t . I t c a n b e s im p l i fi e d a s a m e c h a n i c a l m o d e l a s

fo l l ows : a ro to r o f t he gyrosc op e i s i ns ta l l ed i n a ro to r cas ing , on w hich . the ro to r ax i s and a

suspen ding po l e is f ixed . Th e cas ing is f r ee ly susp end ed u nde r t he b ox o f t he i ns t rumen t wi th a

f le x ib l e m e t a l b a n d t h r o u g h t h e .p o l e . W e d e n o t e t h e d i s t an c e o f t h e c e n t re o f g r a v i t y o f t h e c as i n g

f r o m t h e s u s p e n s i o n p o i n t O b y a , a n d a s s u m e t hi s in s t r u m e n t t o b e f ix e d t o t h e s u r fa c e o f th e e a r t h

at la t i tude ~v, for ope rat ing .

W e c h o o s e t h r e e c o o r d i n a t e s y s t e m s d e s c ri b e d b e l o w ( F i g . 1, F i g . 2 ):

1 . a t r ans l a to ry coord ina t e sys t em O~ r /~ , w i th t he suspens ion po in t O as t he o r ig in and

t r ans l a t i ng i n an i ne r t i a l r e f e rence f r ame;

2 . a d i r ec ti ng coo rd in a t e sys t em O~ 0.r/,~ , , f i xed on the ea r th , w i th po in t O as t he o r ig in , an d

r o t a t in g w i t h a n g u l a r v e l o c it y n o o f t h e e a r t h s r o t a t i o n i n t h e t r a n s l a t o r y c o o r d i n a t e s y s t e m ;

3 . a R e s a l s c o o r d i n a t e s y s t e m O x y z , f i xed on the cas ing , w i th po in t O as t he o r ig in , i t s ax i s

O y

h a v i n g t h e s a m e d i re c t i o n a s th e a n g u l a r m o m e n t u m H o f t h e r o t o r r o t a t i o n . W i t h t h i s s y s t e m , i t

889


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8 90 W a n g H o n g - l a n

is poss ib i e t o sepa ra t e t he ro t a t i on o f t he ro to r ab ou t i ts ax i s, so t ha t w e can de vote ou r a t t en t ion t o

s tudy the m ot ion o f t he ro to r ax i s i n t he d i rec t i ng coord ina t e sys t em O~Qr/0~0 f ixed on th e ea r th .

T h i s m o t i o n c a n b e i n d i ca t e d b y th e r u ie o f t h e c h a n g e o f th e R e s a l s a n gl e s a , f l , w i t h w h ic h t h e

Resa i s sys t em tu rns abo ut t he d i r ec t ing coord ina t e sys t em. W e cons ide r t ha t t he t rue nor th Of t0 i n

d i r ec t i ng sys t em i s t he d i r ec t i on which the ro to r ax i s shou ld be o r i en t a t ed .

T h e p o s i t io n o f t h e R e s a l s s y s t e m

O x y z

s o b t a i n e d b y t u r n i n g t h e d ir e c ti n g c o o r d i n a t e s y s te m

O~0r/0~0in such a way:

d

O ~ o r l o G ) - - ~ ~ ( O x ~ o )

(OxnCo) ~p -> ( O x y z )

Fig. 1

~ ~

J - \ / \

Fig. 2

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

COSQr

= . t = s i n =

0

s l

, r ~ = 0 c o s B

s i z ~ f i

COS12

9 ~ 0

- -s iac 0

c o s a 0 )

1

0 ~

- - s i n f l )

cosfl

- s i a a

os f l

.C0 C C O S

s i n f l

s i n a s i a f l

c o s a s i n f l

C O S f l

The angu la r v e loc i ty wi th which the R esa l s sys t em ro t a t e s i n t he d i r ec t ing sy s t em i s :

o = a L , + f l x

- - /~ X q ~ s i n f l y + ~ c o s B z

1.1)


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A n a l y s i s o f t h e M o t i o n o f a G y r o - T h e o d o l i t e 891

Th e an gu la r ve ]oc i ty o f the d i rec t ing sys tem in th e t r an s !a to ry sys tem i s :

= c o , c o s ~ ( s i n a x + c o s a e o sf l y - - c o s as i n fl z )

+ co~sincp(sinfl y + eo sfl z )

Thus the angu la r ve loc i ty o f the Resa l ' s sys tem in the t r ans la to ry sys tem i s :

P. =.o.0 + (o

= ( ~ e o s r + ,g ) x

+ (co , cos~ 0cosa cosB + co~s i.- .r + ds in f l ) y

+ ( - - co.co s~ pc os as in fl + c_o~sinq0eos,8+ d eos f l ) Z

S u p p o s i n g t h e a n g u l a r v e l o c i ty o f t h e r o t o r a b o u t i ts a x i s is

the ro to r in the t r ans !a to ry sys tem i s :

~ =~+~y

( 1 . 2 )

( . 3 )

q; , t hen the angu la r ve loc ity o f

1.~)

I I . M o t i o n u n d e r t h e A c r o f t,h e T o r q u e o f G r a v i t y O n l y

Ta k ing ang le s /~ , f l , ~ a s gene ra l i z ed coord in_~tes , and sup pos ing tha t the ca s ing ' s

p r i n c ip a l m o m e n t o f in e r ti a l a r e A , B , C , abou t ax i s x , y , z , a n d t h e r o t o r ' s P r i n c i p a l m o m e n t s o f

iner t ia l J~ , Jg , J . about ~xes

x , y , z ,

t hen the k ine t i c ene rgy o f th is sys tem i s :

T 1 A ~ , , 1 - - , 1 - - , a - l a " = 1 9 ~ 1 ,

+ Jo c + + e ,

Z ~ . . . . g L '

= 1 ( A -} - r =) ( c o ~ e o s ~ s i n a + / ~ ) z

2

+' ~ r + . l , ) ( --co~eosopeosa, s in f l + o)osinqgcosfl-k , d cos fl) z

l e

+ -~ -B ( c o o e O S me o .a e o s B + c o ~s ia q ~s in fl + d s in 8 )

c 1

+ - ~ L j ( co~eo , ~ e o s a c o s f l + c o,s inq~s in f l + a s in f l -r . ~ z

N e x t w e d e t e r m i n e t h e g e n e r a l iz e d f o r c e s o f t h i s s y s te m .

t h e t o t a l t o r q u e a c t i n g o n t h e g y r o s c o p e i s:

m = m , x r n g y m ~ z

( v . . )

I n t h e t r a n s l a t o r y s y s t e m , s u p p o s e

a s t h e g e n e r a l i z ed c o o r d i n a t e s h a v e a c e r t a i n v i r t u a l d i sp l a c e m e n t , t h e n t h e v i r t u a l w o r k is

3 .A_= (ra , x - t- rr~y - I - re , z ) . [3f ix + (s in f ld }a + ~r q- cosf l~3az ]

= ( m v s i n f l + m ~ co s0 g)

3a + m , c3 l + m~3r


4/

8 92 W a n g H o n g - l a n

so tha t

Q,,=ra~sinfl+ra~cost~ Qp=ra,, Q,=-.mw ( 2 . 2 )

Subs t i t u t i ng t he k ine ti c ene rgy and genera li zed fo rces i n to t he L agrange ' s equa t ion o f t he

s e c o n d k i n d , w e o b t a i n t h e d y n a m i c a l e q u a t i o n o f t h e g y r o s c o p e . T o t hi s e n d , s o m e a s s u m p t i o n s

s h o u l d b e m a d e :

(1) t he two ang les . a , f l w i th whic h the ro to r ax i s de f l ec ts f rom the d i r ec t ing coord ina t e

system are smal l quan t i t ies , and ~ i, f f are a l so smal l ;

(2 ) the angula r m om entu m H of the ro to r i s su f f ic i en t ly l a rgy ;

.(3 ) when the gyro scop e is runn ing s t ead i ly , t he d r iv ing to rq ue o f the gyro m oto r i s ba l anced by

the t o rque du e t o f r ic t ion a t t he b ea r ings o f the ro to r , so ne i the r o f t he t o rque s i s t aken in to a cco unt

in the fol lowing.

N o w w e s t u d y th e c a s e th a t t h e m o m e n t o f g r a v it y is th e o n l y e x te r n a l m o m e n t a c t i n g o n t h e

sys t em, i,e . we can t em pora r i l y i gnore t he ac t i on o f t he mom ent o f the ba nd suspen s ion and the a i r

d a m p i n g .

W e have coord ina t es o f t he cen t r e o f g rav i ty i n t he Resa l ' s sys t em: xq = 0, ya- -- -0 , z a = - - a ,

grav i ty fo rce : P = - -P t , o= - - P ( s i n f l y + c o s f l z )

m o m e n t o f e x t er n a l I o rc e s: M ,,xt = M v = O G x P = - -Pas in f l x ( 2 . 3 )

g en era liz ed fo rc es : Q , = 0 , Q p = - P a s i n f l , 9 ..

We no t i ce t ha t

OT/Or ,

an d -Q~-----0, an d ~0 is a cycl ic co ord ina te.

I t i s poss ib l e t o e s t ab li sh t he dynam ica l equ a t ions wi tho u t t he cyc l ic coo rd ina t e b y us ing the

Ro uth ' s equa t ion . Fo r t h is purpose , i t is necessa ry t o fi nd t he in t egra l o f cyc l i c m om en tum :

# T

P ~= #~ = J ,( co.cosqocosacosB

+ WoSin~0sinfl+ 5 s in f l r = H

H

= - ~ , - - (w , cosq~cosacosfl+ c o . s i n g s i n f l + a s in /~ )

T h e R o u t h ' s f u n c t i o n i s

R = T - - P , ~

1

= ~ - (. ,4 + 1 . ) (c o, c o s ~p sin a + / ~ 2

1

+- -~ ( C q- J , ) ( - c0. co s (pcosa s i n i l + co,s in q~cos~6+ 6 c os f l ) 2

2

+ T B ( w 0 c o s ~ p c o s a c o s f l + c o+ sin q~ sin fl+ r

+ H ( c o e c o s ~ c o s a c o s , 6 + co s incp s in f l+c? s in f l ) 1 H 2

2 lw ( 2 . 4 )

W e s u b s t i tu t e t h e R o u t h ' s f u n c t i o n a n d g e n e ra li z ed f o r ce s in . to t h e R o u t h ' s e q u a t i o n :

d [ 8 R ~ O R 3

= ]

t


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A n a l y s i s o f t h e M o t i o n o f a G y r o - T h e o d o l i t e 8 93

t o o b t a i n d y n a m i c a l e q u a t i o n s f r ee f r o m c y c li c c o o r d i n a t e , i f H is l a rg e e n o u g h , a ll t h e t e r m s i n t h e

l e f t- h a n d s id e o f t h e e q u a t i o n n o t i n c l u d in g H c a n b e n e g l ec t e d , a s t h e y a r e m u c h s m a l l er t h a n t h o s e

i n c lu d i n g H . T h e s im p l i fi e d d y n a m i c e q u a t i o n is t h e n

H co ,cosq0s ina + /~ )

=

0

2 .

6

)

-- H - COoeOsq~eosasinfl + co, sinq ge osfl + ti cos f l ) = - P a s i n f l J

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

f l x H = M , , , , ,

x y z

Q x H = S 3 , . O , . 0=

0 H 0

= - -

~ =H O H

=M.x M,y M=z

w e h a v e :

H D , = M =

- - H O = = M , } 2.7)

S u b s t i tu t i n g t h e a n g u l a r v e l o c i ty i n d i c a t e d b y E q . 1 .3 ) a n d t h e m o m e n t o f e x t e r n a l f o r c e s

i n d i c a t e d b y E q . 2 .3 ) i n t o t h e a b o v e e q u a t i o n , w e c a n a l so o b t a i n E q . 2 .6 ). T h i s s e t o f e q u a t i o n s

d i sc r ib e s t h e m a i n f e a t u r e o f t h e m o t i o n o f t h e r o t o r a x i s - - a p r e c es s io n , w i th n u t a t i o n n e g l ec t ed .

Because o f a ~ 0 , f l~ - ,0 , Eq . 2 .6 ) c an be s impl i fi ed to :

/ ~ + c o, c o s ~ 9a = 0

Oe OS~

w h e r e

z

a P

h = co, cosq9 + ~ ) c o , cos~0

coo s inq~

f f ~ = C O o C O s ~ + a P / H

e l i m i n a t in g ~ , ~ , w e h a v e

a + h = a = O

+h= Pfh= } 2 .8 )

whose so lu t ion i s

a = c c o s h t + e )

f l - - - - -~*- - bs in ~- t - e ) }

2 . 9 )

w h e r e


6/

8 94 W a n g H o n g - l a n

T h i s s o l u t i o n c a n b e d e s c r i b e d w i t h t h e g r a p h i n F ig . 3 : w k e n v i e w e d i n t h e d i r e c t i o n f r o m t h e

s o u t h t o t h e n o r t h , a p o i n t o f t h e r o t o r a x i s t ra c e s o u t a n e l l ip s e w i t h i t s c e n t r e a t ( 0 , i f " ) . T h i s

s o l u t i o n i n d i c a t e s th e u n d a m p e d o s c i l la t i o n o f t h e r o t o r a x is . T h e r o t o r a x is o s c il la t e s w i t h a

d e f l ec t i o n a n g le a ( ~ ) a b o u t t he m e r i d i a n , b u t o n l y t h e t w o e x t r e u m , p o s i t io n s o f t h e a = c a n d

a = - c a r e n e e d e d t o d e t e r m, . ' n s t h e t r u e me r i d i a n .

/ / ' + b

F i g . 3

S u b s t i t u t i n g E q . ( 2 . 9 ) b a c k i n t o E q . ( 2 . 7 ) , w e h a v e :

b c o o e o s ~

~ - = k = J

w h e r e j i~ u s u a l l y v e r y s m a l l , w h i c h m e a n s t h a t t h e s h a p e o f th e a b o v e e l ! i p ~ i s v e r y fia t~ a n d t h e

c h a n g e o f a n g l e f l is d i f f i c u lt t o o b s e r v e .

T h e c i r c u l a r f r e q u e n c y o f t h is o s c i l l a t io n i s

9 ~ p

- ~

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

9 2 ~

_ / H

-, = -- - r - - - = z~ r . d . . . . . ( 2 1 O)

, o , e o s ~ P o

-

W e h a v e a n a l y s e d t h e m o t i o n o f t h e r o t o r a x i s in t h e i .n .e rtia l c o o r d i n a t e s y s t e m , n e x t w e s h a l l

a n a l y s e it in t h e n o n - i n e r t i a l c o o r d i n a t e s y s t e m w i t h t h e e a r t h a s r e f e re n c e b o d y . T h e e o n v e c t e d

m o t i o n i s a r o t a t i o n w i t h t h e a n g u l a r v e l o c i ty o ~, o f t h e e a r t h ' s r o t a t i o n . N o w l e t u s i n t r o d u c e t h e

i n e rt ia l g y r o s c o p e t o r q u e M s ,. ! t is c o n v e n i e n t t o r e s ol v e e% i n t o t h r e e c o m p o n e n t s :

a , = ~ , s i n g

o

C % ~ a ) ~ c o s ~ ps i s a X

0 --'- . ) s e o ~ 0 c o f , ~

O n

a n d t o r e s o lv e c o r r e s p o n d i n g l y t h e in er ti al y r o s c o p e t o r q u c M . i n t o t h r e e c o m p o n e n t s :

M , ~ = M , , 2 M s ~

+ M e ,

M~2=H X e ~ = H r ~ = s i n g ~ c o s f l x .

M~s=H x o~ = v - H o ~ c o s 4 ~ . n a z

9

M ~. =H .r - - H r o , c o s g~ c . os a s in r x


7/

A n a l y s i s o f t h e M o t i o n o f a G y r o - T h e o d o l i t e 8 95

s o w e h a v e

i l f l t H

w ~ = M a z M a s M v 4

= H c o ~ x

- - H c o , c o s ~ o s i n a

( 2 : 1 1 )

T h i s i n e r t ia l g y r o s c o p e t o r q u e s h o u l d b e t h o u g h t t o b e r e a l l y a c t i n g o n t h e r o t o r a x i s. T h e t o t a l

m o m e n t o f e x t e rn a l f o rc e s is :

M ~ , = M , + M r

= H c a ~ s i ~ q ~ c o s f l - - H c o , c o s q a c o s a s i n f l - - P a s i n f l ) x

- - H c o , c o s ~ s i n a z

( 2 . 1 2 )

S u b s t i t u t i n g t h i s t o t a l m o m e n t o f e x t e r n a l f o r c e s a n d t h e a n g u l a r v e l o c i t y ~ o f t h e R e s a l ' s

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

Hoe,,.---M .

w e c a r , o b t a i n t h e s a m e s e t o f dy na ~n ~c al e q u a t i o n s o f g y r o s c o p e a s E q . ( 2. 6) .

T h e in e r t i a l m o m en t ~ a3----- -H c0 o co sq~ s in~ zz m ak es th e ro to r ax i s to o sc i l l a t e in a v e~ ica l -

p l a n e a n d a ia g ie f i t o c h a n g e , w h e n p o i n t G o f t h e c e n t r e o f g r a v i t y i s d e f l e c t i e g f r o m t h e v e r t ic a l

l in e w h i c h p a s s es t h r o u g h p o i n t O , tb_ ere ~ u s t b e a m o m e n t ~ o f g r a v i ty , w ~ c h m a k e s t h e r o t o r

ax i s to p recess to th e r,ae~ d ian in th e h o r / z o n m l . In R efs . [5] , [6] , [ I0 ] e t c . , t h e in e r t i a l m o m en t ~ as

w a s k n o w n a s t h e d i re c t m g m o ~ n e n t ' , it w a s sa i d t o b e t h e m o m e n t w h i c h m a k e s L he r o t o r a x is

m o v i n g t o t h e m e r i d i a n , b u t t h i s is n o t e x a c t, t h u s c a u s i n g s o m e m i s t k e s a n d c o n f u s i o n s .

iiT~. M o t i o n u n d e r ~ h e A c~ i~ xa o ~ t h e ' ~ o r q u e o f t h e G r a - r a n d o f t h e B a n d

S u s p e n s i o n

S u p p o s e t h e r i g i d tw i s z in g c o e f f i c i e n t o f t h e b a n d s u s p e n s i o n i s D B, t h e t o r q u e o f t h e b a n d

s u s p e n s i o n i s d i r e c tl y p r o p o r t i o n a l t o t h e a n g l e o f t o r s i o n ,

T h e e x t e r na l m o m e n t i s

M D = r - D B ~ Z

~ i e,, ~ M r M ~ = - - P a s i n f l x - - D a a z

S u b s t i t u t i n g i t i n t o t h e p r e c e s s i o n E q . ( 2 . 7 ) , w e o b t a i n

H (c o ,e os ~p sin a+ /~) ----- -

D p a

- - H - - c o , c o s r p c o s a s i a f l + COosinq~eosfl+Seosfl)---- - P a s i n f l

w h i c h m a y b e s i m p l if ie d a s

3 . z )

/ ] H ~ , c o ~

D~a=O

v B m 0

fOeCOSlj

( 3 . 2 )


8/

8 96 W a n g H o n g - l a n

where k , / 3 a r e t he same as were d i f ined . Eq . (3.2 ) can be chang ed in to

~ + H c o oe os q~ + D B ~ = 0

k~

i ~ = 0

Oe O~

El iminat ing cL /~ , we hav e:

/~ + Hco.cosep+ DB M fl--

H c o . e o s ~ p + D B

Mfl*

- ~ e e o s ~ p ' - - H o ~ , e o s ~ p 9

~ + H c o . e o s ~ o + D B , kZf l O

H c o o e O S ~ p

Let t i ng

k l H co , c o ~ + D 8

= "

Hco,cos~ p

D B a P

w e h a v e

~ + k l p - - - - k l f f~

~ + k | a = O } ( 3 .3 )

and i t s so lu t i on i s

a - ~ c i c o s k t t - t - e 1 )

f l = l ~ - - b l s i n k t t- t - el

3 . 4 )

w h e r e

b l - - h t

I t i s ob v iou s t ha t t he ro to r ax i s has s ti ll a pe r iod i c osc i l l a ti on t r ac ing o u t an e l l ipse , bu t t he re i s

som e chang e in t he e ll ipse ' s pa ra me te r s a nd in t he pe r io d o f osc i ll a ti on . The long - to - shor t ax i s r a t i o

of t he e l li pse is now

J l ~ b , - - -- c o . c o s ~

c , k ,

t he c i r cu l a r f r eque ncy i s

D jL aP L co, co ( a P ~ - l J s )

4

and the pe r iod i s

I V .

9 = _ 9 / b /

T -

- k , - x , q o ~ .c o ~ (a P + D m ) ( 3 . 5 )

M o t i o n u n d e r t h e A e t i p n o f t h e T o r q u e o f t h e G r a v i t y a n d o f t h e a n d

S u s p e n s i o n w i t h A i r D a m p i n g C o n s id e r e d


9/

A n a l y si s o f t h e M o t i o n o f a G y r o - T h e o d o l i t e 8 9 7

S u p p o s e t h e c o e f f ic i e n t o f a i r r e s i st a n c e is n, t h e m o m e n t o f r e s is t a n c e is d i r e c t ly p r o p o r t i o n a l

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

= - n B x -

n~sin

y - - n 3 r e o s fl z

N e g l e c t i n g t h e s e c o n d o r d e r s m a l l q u a n t it i e s, w e m a y s i m p l if y it t o

M x = - . a z - n B x

T h e m o m e n t o f e x te r n a l fo r c e s is

M,~,,,2 = M p + M D + M ~

= - - P a s i n f l - - n ~ ) x + - D s a - - n a ) z

S u b s t i t u t i n g i t i n t o t h e p r e c e s s i o n E q . ( 2 . 7 ) , w e o b t a i n

H cg, eosq~s ina + /3 ) = -- Dn a -- n~

Y

- H -co ~ + COosinqgeosfl + + dt eos f l ) = - P a s i n f l - n~

H / ~ + m ~ + ( H ~ o e O S ~ p +

Ds)a---- 0 T

H a - n B - H c o , eo scp+ P a ) f l + H c o,s in q~ = O ~

{ = - - - H - B - (c o . e o s a

9 n q~+ ' ~ - - ) f l + c g , s i m p = 0

II DB

e + - m + .eo s = 0

T h e s o l u t i o n f o r E q . ( 4 . 4 ) i s

4 . i )

4 . 2 )

( 4 . 3 )

4 . 4 )

4 . 5 )

( 4 . 6 )

~ 4 . 7 )

n ~ D s

/~ ---- - - ~ r - - ( co . co s q~+ - - ~ ) a

4 . 8 )

T h e s o l u t i o n f o r E q . ( 4 . 6 ) i s

n D

S u b s t i tu t i n g t h e m i n t o E q . ( 4 .7 ), w e h a v e

n* n O s

r , i P a D s \ P a D s 7

L ~ , eo s ~ ~ , . ~ o s

~1,-=~

+--~ +- - -a r - . l~=o

S i n c e H i s v e r y l a rg e , a n d n , D a a r e q u i t e s m a l l , i t m a y b e s i m p l i f i e d a s

~ _ 2 n + c o , c o s ~ ( w , c o s q ~ + + ~ --~ ---~ a =

t - - - ~ - c o , c o s ~ = - - - ~ 0 ,


10

8 9 8 W a n g H o n g - l an

i f w e l e t

w e h a v e

n Pa

/J = - ~ c ~ , e o s ~ o , k | = a ~ , c o s < p (c o.e os ~0+ - - ~ + - - ~ - )

t ~ + 2 / ~ + k | a -- - -0 ( 4 . 9 )

W e o n l y c o n s i d e r t h e c a se u n d e r t h e a ir d a m p i n g , i.e . t h e C as e o f s m a l l d a m p i n g . D u e t o # ( ( k ~ , t h e

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

a = c 2e x p [ - - # t ] s i n ( d k | - -# 2 t + e2) ( 4 . I 0 )

T h i s s o l u t i o n in d i c at e s t h a t t h e a m p l i t u t e o f t h e d a m p e d o s c i ll a ti o n o f t h e r o t o r a x is i s a t t e n u a t e d

a c c o r d i n t o t o a g e o m e t r i c a l p r o g r e s s i o n w i t h t i m e . I t s m o t i o n i s s h o w n i n F i g . 4 . T h e p e r i o d o f

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

T 2 z

B e c a u s e o f t h e a ir d a m p i n g , th e p e r i o d o f o s c i l la t i o n b e c o m e s lo n g e r , b u t d u e to /z r k s , t h e p e r i o d

c h a n g e s o n l y w e a k l y .

F r o m E q . ( 4.1 0 ), w e h a v e

c~ - - c z e x p [ - - # t ] [ d k | - - ~ e o s ( , , / ~ t + e2 )

--

i n ( , , / k - - ~ ~ - -~ f + e ~ ) ] ( 4 . 1 2 )

S u b s t i t u t i n g E q ( 4 . 8 ) i n t o E q . ( 4 . 5 ) , w e o b t a i n

. o+ - ~ - - ) a - - ( co . c os ~p + - - - - ~ ) f l + c o~ n~p= O

n 2 . n

w h i c h m a y b e s i m p l i fi e d a s

+ - - ~ ~ . c o s ~ . a - - ( c ~ .c o s ~ + ~ ) f l + c o . s i m p =P a

S u b s t i t u t i n g E q . ( 4 . 1 0 ) a n d E q . ( 4 . 1 2 ) i n t o t h i s e q u a t i o n , w e h a v e

c 2 e x p [ - - # t ] ~ /'~ 2 - - ff ~ c o s ( , v / h l - # 2 t + e 2 ) + c o, s i n ~

P a

w h o s e s o l u t i o n i s

w h e r e

f l = f l * + b2 e x p [ -- /J t ] c o s ( ~ / k | - -# 2 t + e2 )

c o o s i n ~ c 2 ~ / | _ 2

fl*= co.eosq~+Pa/H b = co,cos~+ Pa /H

( 4 . 1 4 )

T h e s o l u t io n s f o r a n g l e s a a n d f l e x p r e ss t h e r u le o f t h e m o t i o n o f t h e r o t o r a x is u n d e r t h e

a c t i o n o f d a m p i n g . I t c a n b e d e s c r ib e d b y t h e g r a p h i n F i g. 4 : W h e n v i e w e d i n t h e d i r e c t i o n f r o m t h e

s o u t h t o t h e n o r t h , a p o i n t o n t h e r o t o r a x is t ra c e s o u t a s p i ra l p a t h w i t h t h e p o i n t ( 0 , f i x ) a s i ts


11

Analysis of the Motion of a Gyro-Theodolite 899

centre, as shown in Fig. 5.

References [6], [10] etc. Said that The friction in the bearings of the sensitive element makes

the amplitude gradually attenuated in azimuth and in altitude, the degree ofat ten uat iond epends on

the moment of friction force, the characteristics of oscillation depends on whether the moment of

friction forces is constant or not. I f the value and the direction of the moment o f friction force are

constant, the weakly attenuated oscillation of the rotor axis Possesses the followinf Property: the

ratio of any amplitude to the succeeding one, i.e. attenuate coefficientf, is a constant

f ~ _ a t ~ t l~ ~ . . . a n _ l c o n s t .

a ~ a 3 a n

c~ ~. a C~exp[ ~t]

~ i / , j \ , / ~ x , d /

I k l Z ~ l / 1 1 3 d 'Z " ~ - - - - - '~

J,, a---- --c2exp[--pt]

--.c2. / [ _ T

.I

Fig. 4 Fig. 5

From the structure o f gyroscope, we know that fiJction exists only in the bearings between the

rotor and the rotor axis. When this instrument is operating normally, the rotor rota tes steadily, the

angular velocity of rotor is constant, the friction moment in the bearings is balanced with the driving

momen t of the gyro motor and furthermore we know that the friction moment vector is always in

the direction of the rotor axis, therefore, the friction moment does not influence the oscillation of the

rotor axis. Besides, there is the act ion of the torque supplied by the band suspension~ but from the

above derivation, we know that when the rotor axis oscillates periodically around an ellipse, the

torque of the band suspension only makes the parameters o f the ellipse and the period of oscillation

to change, hut does not make the amplitude to attenuate. The main cause of amplitude attenuation

is the damping o f air which causes the amplitude to a ttenua te according to the exponential law, as

shown in Fig. 4. The ratios o f amplitude are

f____ a~ a~ a._ 1

= . ---- . . . - =- exp [/~T ]

0~2 ~S t/ n

We have analysed thirty groups of data determined by the gyro-theodolite of model

J6-J60

and

found that the main feature are coincides with the property analysed above in this paper.

e f e r e n c e s

[ l ] Cheng Bin,

Analytical Dynamics

Peking University Publishing House. (in press)

[ 2 ] Zhu Zhao-xuan, Zhou Qi-zhao and Yin Jin-shcng, Theoretical Mechanics Peking University

Publishing House 0982), 217-255. (in Chinese)

[ 3 ] Nikolai , Ye.L.

The Theory about Gyroscope The

Science Publishing House (1956), 84-93.

(Chinese Version)


12

9O0

[4

W a n g H o n g - l a n , ..

X i a o S h a n g - b i n a n d D o n g Q i n - q u a n ,

Gyro M echanics The

P e o p l e s E d u c a t i o n P u b l i sh i n g

H o u s e (1 98 0 ), 5 9 - 6 1 . ( in C h i n e s e )

[ 5 ] T h e g y r o - t h e o d o li t e g r o u p , T a n g s h a n I n s t i tu t e o f t h e C o a l M i n e A c a d e m y ,

The Basic Theory

Construction and Orientation about Gyro-Theodo lite Th e C o a l I n d u s t r y P u b l i s h i n g H o u s e

(1 9 8 2 ) , 7 7 - 8 4 . ( in Ch in e s e )

[ 6 ] T h e t e a c h i n g a n d r e s e a r ch s ec t i o n a b o u t s u r v e y i n g , C h i n a M i n i n g I n s t i tu t e ,

The Mine

Surveying

T h e C o a l I n d u s t r y P u b l i s h i n g H o u s e ( 1 9 7 9 ) , 1 5 9 - 1 8 0 . ( i n C h i n e s e )

[ 7 ] L i Q in g -y u e ,

Engineering Surveying

T h e S u r v e y a n d D r a w i n g P u b l i s h i n g H o u s e , ( 1 9 8 4 ) ,

2 4 5 - 2 7 2 . ( i n C h i n e s e )

[ 8 ] L i u Y a n - b o , E n g i n e e r i n g S u r v e y i n g , T h e M e t a l l u r g ic a l I n d u s t r y P u b l i s h i n g H o u s e ( 1 98 4 ),

2 4 8 - 2 7 2 . ( i n C h i n e s e )

[ 9 ] W a n g H o n g - l a n , A d y n a m i c a l t h e o r y o f th e o r i e n t a t io n o f a g y r o - t h e o d o li t e , M ine Surveying

3 (1 9 8 5 ) , 4 3 - 4 6 . ( in Ch in e s e )

[ 1 0 ] K a z a k o v s k i i , D . A . ,

The Mine Surveying

M o s c o w P u b l i sh i n g H o u s e (1 95 9), 4 6 1 - 4 7 0 . ( in

R u s s i a n )

[ 1 1] A r n o l d N . N . a n d L . M a u n d e r ,

Gyrodynamics

A c a d e m i c P r es s, N e w Y o r k a n d L o n d o n

(1961).

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