Estimate of effect of the vertical component of inertial forces of translational motion on the...

ESTIMATE OF EFFECT OF THE VERTICAL COMPONENT

OF INERTIAL FORCES OF TRANSLATIONAL MOTION ON

THE STABILITY OF A GYROCOMPASS

A . A . M a r t y n y u k a n d A . P . Y a n t s h e v s k i [ UDC 531.383

w C o n s t r u c t i o n o f L y a p u n o v F u n c t i o n s f o r N o n s t a t i o n a r y

L i n e a r S y s t e m s [2]

We c o n s i d e r the s y s t e m of d i f f e r en t i a l equa t ions

n

dxs dt ~ Ps~ (t)xh = 0 (s = l, 2 . . . . . n), (1.1)

h = l

whose coe f f i c i en t s Psk(t) a r e def ined and con t inuous func t ions of t for e v e r y pos i t ive va lue tEJc~ = [0, c~).

We take an a u x i l i a r y func t ion v(t, x) in the f o r m of the quadra t i c f o r m

v(t,x)= ~ bsr(t)XsX r, (1.2) s , r ~ [

having the fo l lowing p r o p e r t i e s : 1) v(t, x) = 0 only for x i = x 2 = . . . = Xn = 0; 2) the coe f f i c i en t s b s r ( t ) a re def ined for t _--e_ 0 con t inuous ly d f f fe ren t i ab le (or p [ecewise con t inuous ly d i f fe ren t i ab le ) on J0 = [0, + ~) such that the func t ions d b s r / d t have a f in i te n u m b e r of f i r s t - o r d e r d i s c o n t i n u i t i e s .

The total d e r i v a t i v e of the quad ra t i c f o r m v(t, x) by v i r t u e of s y s t e m (1.1) has the f o r m

n

d_~v =: V d~xsx,, (1.3) dt .a

s , r = l

where

, , z

dt (&ibi" q- b#P~i) (s, r = 1,2 . . . . . n).

i=1

Let the se t M not con ta in a l l the t r a j e c t o r i e s of s y s t e m (1.1) b e s i d e s the point x = 0. F r o m the B a r b a s h i n - - K r a s o v s k i i t h e o r e m it is known that if the set M is d e s c r i b e d by the equa t ion v2(x) = 0, then for th i s it is su f f ic ien t that v2(x) # 0 on M. The equat ion dv /d t on the set M is p o s s i b l e for XsX r # 0 only for d s r = 0 (s, r = 1, 2 . . . . . n). Hence, the condi t ion dv /d t = 0 will be s a t i s f i ed if the coe f f i c i en t s bs r ( t ) of the f o r m (1.2) sa t i s fy the s y s t e m of

n

db~, d ~ - - ~ (&ibi r + b/~o ;) = 0 % r = 1,2 . . . . . n). (1.4)

He re the re ex is t [(n 2 - - n) /2] + n = C2n+1 unknown func t ions bs r ( t ) = b rs ( t ) (r, s = 1, 2 . . . . . n).

Along with the s y s t e m (1.4) we c o n s i d e r the s y s t e m

Ins t i tu te of M a t h e m a t i c s , P r i k l a d n a y a Mekhanika, Vol. 10, 15, 1974.

Academy of Sc i ences of the U k r a i n i a n SSR, Kiev . T r a n s l a t e d f r o m No. 12, pp. 66-70, D e c e m b e r , 1974. Or ig ina l a r t i c l e s u b m i t t e d March

�9 76 Plenum Publishing Corporation, 22 7 West 17th Street, New York, N. Y. 10011. No part of this publication may be reproduced, t stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $15.0~_~

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dbsr -- s Z~ (P,ibi, q- bi,P, ;) (s, r = 1,2 . . . . . n), dt

1=1 where e is an aux i l i a ry p a r a m e t e r .

We seek the solut ion of s y s t e m (1.5) in the f o r m of the s e r i e s

b~- (t) %0) 8b(O __ ~.(.) . . . . + sr + . . . + 8 o ~ + . . . .

a f t e r a s s u m i n g fo r the funct ions bsr(U)(t) to be d e t e r m i n e d that

b~7 ) ( 0 ) = 0 ( v > l ; s , r = l , 2 . . . . . n).

The m a t r i x 0asr(~ n d e t e r m i n e s the initial app rox ima t ion of the quadra t ic f o r m (1.2).

A s s u m i n g

b(0) 6~ .= /0 s=/=r; s r [ 1 S ~ r

(1.5)

(1.6)

(1.7)

(1 .s)

and taking account of (1.7), in o r d e r to f ind the funct ions bsr(U)(t) we obtain a sequence of r e c u r s i o n s y s t e m s of d i f fe ren t ia l equat ions ,

.n

db~) ~ (o b (~'-1) -k b (~'-1)- " dt - - Z . ~ " s i J r ~; Pit) ( v ~ l ; s , r = l , 2 . . . . . n). (1.9) 1=1

F r o m this s y s t e m of d i f fe ren t ia l equa t ions the funct ions bsr(U)(t) a r e d e t e r m i n e d by q u a d r a t u r e s

b~7 > (t) = t . .~,(~-~) ~.~,vir q- b~7-~ dr. (1.10)

We now p rove the c o n v e r g e n c e of expans ions (1.6). Let B(t) be a pos i t ive function, no l e s s than the m a x i m u m of the modut i of the coef f i c ien t s Psk(t) fo r all t(: (0, T), i . e . ,

and, f u r t h e r m o r e ,

max{ p~, (t)! ..<B(t) for all t > l 0 (1.11)

t

~ l B ( x ) I d x ~< (z < to. (1.12) t ,

F r o m e x p r e s s i o n s (1.10) and (1.11) by induct ion we have

t

I bi~ ) (t) l ~ - - ~ / J B (~) dr . (1 .13) 0

By v i r t ue of (1.12) f r o m e x p r e s s i o n (1.13) it fo l lows that the s e r i e s (1.6) is m a j o r i z e d by the s e r i e s of the funet ion

- 6 - k! k=O

and, hence, c o n v e r g e s un i fo rmly and abso lu te ly fo r all t~j0 (t o >_ 0) and e -< 1.

Thus , the fol lowing a s s e r t i o n is va l id .

THEOREM 1. Let : a) the quadra t i c f o r m v( t , x ) = "~ bsrx~xr be such that its coef f i c ien t s bsr( t ) s , r ~ |

sa t i s fy the s y s t e m of d i f fe ren t ia l equat ions (1.10); b) v(t, x) = 0 only for x 1 = x 2 = . . . = Xn; c) the t r a c e of the m a t r i x B(t) = {bsr(t)h n be bounded fo r all t _> 0; d) the Sy lves t e r condi t ions

bl, (t) b12 (t) bn (t) ~ 8, > O; b~ (t) b~ (t) ~ 8~ > 0 . . . . . an > 8, > 0

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be satisf ied. Then the quadratic form (1.2) with coefficients defined by Eqs. (1.10) is a Lyapunov function.

We note that the question of the stabili ty of the zeroth solution of sys tem (1.1) is solved by the following theorem.

THEOREM 2. If the conditions a-b of Theorem 1 are satisfied, and the set M for sys tem (1.1) does not contain all the t r a j ec to r i e s besides x = 0, then the unperturbed motion of sys tem (1.1) is stable.

Taking into account that s e r i e s (1.6) converges absolutely and uniformly for all t _> 0, for the solu- tion of the question of the stability of the zeroth solution of sys tem (1.1) we allow the use of a finite number of t e rms of se r i e s (1.6) and the corresponding conditions of Theorems 1 and 2. Determination of the posi - tive quadratic form

n )

yields an express ion in t e rms of the cons t ra in ts for the eigenvalues of the matr ix

N

~,vsr I"

It is known that the quadratic fo rm (1.15) will be posit ive-definite if in the chain of inequalities for its e igenvalues

~,L (B (NI) -,< ~2 (B INI) 4 . . . 4 M (B (N~)

the following inequality is satisfied:

If the quantity Xn(B(N)) has an upper bound for t ~ .% small higher l imit .

)~, (B {NI) > cr > 0 (cr - - const).

then fo rm (1.15) allows an infinitesimally

w I n v e s t i g a t i o n o f S t a b i l i t y o f a S p a t i a l

G y r o s c o p i c - H o r i z o n C o m p a s s

In o rder to derive the equations of motion of the gyroscopic element of a gyroscopic compass it is usual to neglect the ver t ical component of the inertial forces of the t ranslat ional motion, considering them to be considerably less than the force of gravity. Moreover , as is shown by an analysis, this force can considerably change the nature of the motion of the gyroscopic element of the instrument. The par t icular case of the motion of the object on which the instrument is mounted (true circulation) was considered in [3]. We show that the ver t ical component of the inertial forces in the case of a spatial gyroscopic -hor izon com- pass general ly causes loss of stability in the motion of the gyroscopic element of the instrument for an a rb i t r a ry nature of the motion of the object near the ea r th ' s surface. The deviation in the shape of the ear th f rom a sphere is neglected.

The equations of the per turbed motion of a gyroscopic element of the gyroscopic-hor izon compass have the fo rm [1]

- - PIV'~ - - m l V ~ + IF~ = - - ~22B~ sin o;

2B6 sin o J~ + ~ ~Y; Y q n d R =

~2t5;

- - y = ~2mlVa. (2.1)

Here o~ is the angle of deviation of the gyroscopic element of the compass in the azimuth; ~ is the angle of elevation of the nor thern d iameter of the gyrosphere over the plane of the geocentr ic horizon; T is the angle of rotation of the gyroscopic element with respec t to the axis of the total kinetic moment; 6 is the angle of deviation of the natural axes of rotat ion of the ro to r s of the gyroscopes f rom a cer ta in regulated value a of the propagation angle of the gyroscopes , for which the conditions of total compensation of forced ball ist ic deviations of the compass are satisfied; m is the mass of the gyroscopic element; l is the meta- centr ic height of the compass ; B is the kinetic moment of each of the two gyroscopes; F is the gravitational

at t ract ive force; tl is the radius of the earth, assumed to be a sphere; ~2 is the project ion of the absolute

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angular veloci ty of the Darboux t r ihedron on the geocentr ic ver t i ca l .

Below, it is convenient to conver t in Eqs . (2.1) to the new var iab les

Vcr 2B sin c R"~ = xl; ~ = x2; ? = x3; m l R ~ 6 = x 4 (2.2)

(u is the Schuler f requency) .

In the var iab les (2.2) the equations of the pe r tu rbed motion of the gyroscopic e lement of a gyro- scopic-hor izon compass are wri t ten in the fo rm

x l = vx2 + P2r x2 = - - v x l + g~x~;

;~ = - - m , - - v x 4 ; x , = - - a x e + ( ~ " v~ ~ (2.3) \ ~/~ ] x~,

where f~ and V a re known functions of time, defined by the equations

~ = U sin q~ +---~- tgcp + ~z" (~z* = v~v ) RU cos r + v~ " ;

(2.4) V 2 = (RU cos q~ + v~) ~ + v~,

where U is the angular veloci ty of the daily rotat ion of the earth; and ~ is the geographic lati tude.

(1.6), Applying the method of w to sys tem (2.3) and r e s t r i c t i ng ourse lves to the f i r s t two t e r m s in se r i e s

we obtain for the Lyapunov function the equation

t +S v ( t , x ) = x l t 2 + X s + X 4 + V~(~)dTx3x4. (2.5)

0

On the bas is of T h e o r e m 2 it is easy to find that the zeroth solution of sys tem (2.3) will be stable for all t ~ J 0 if

t

0

Substituting in inequality (2.6) the value of V(t) f rom (2.4:), we obtain

t

_ Ucos~p ve(~)]d'~ < v. (2.7) S [(U cos r + ~2~) + 2 -------ff--- 0

Here v2(t) is the square of the absolute value of the re l a t ive velocity of the vessel , vE(t) is its e a s t e rn component.

Since v 2 (t) .. U c~cpv~ << v, (UcosqD)~+ ---~-I<.~,v; 2

and the maneuver is c a r r i e d out for an interval JT of finite length, inequality (2.6) or (2.7) is prac t ica l ly not violated.

Thus, neglect of the ve r t i ca l component of the inert ial force of the t ransla t ional motion for a finite interval (of the o rde r of 10-30 min) does not cause instabil i t ies in the motion of the gyrocompass .

L I T E R A T U R E C I T E D

A. Yu. Ishlinskii, Mechanics of Gyroscopic Systems [in Russian], Izd-vo Akad. Nauk SSSR, Moscow (1963). A. A. Martynyuk, nIterational fo rmula for construct ion of Lyapunov functions, " Ukrainsk. Matem. Zh. , No.2 (1973). S. P. Sosnitskii , ~Investigation of stabil i ty of a gyroscop ic -hor izon c o m p a s s , , Izv. Akad. Nauk SSSR, Mekhan. Tve rd . Tela, No.3 (1971).

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3.

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