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INVESTIGATION OF THE STABILITY OF Bffi 
VERTICAL POSinON OF A GYROSCOPi: OF 
VARIABLE MASS 


^ DTIC 

I^ELECTEf^ 

^ AUG 02 19941 I 


- USSR - 


by V.S, Novoselov 


LIBRARY COPY 

APR 21 1960 


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INVESTIGATION OF THE STABILITY OF THE VERTICAL 
POSITION OF A GYROSCOPE OF VARIABLE MASS 


/ Following la a translation oF an article 
written by V* 3. Novoselov in Vestnlk Lenin- 
p:pad3kop:o Universlteta (Hem Id of Leningrad 
University, Ho, 5er Mat, Mekh 1 Astron, 
No. 4, 1959, pages 121-129. j 


This article Is devoted to a purely theoretical study 
of the stability of the vertical positinn of a gyroscope of 
a variable mass, the particles of which may possess an inter¬ 
nal motion. Technical features are not considered, Basic 
results of this article were presented in the author's the- 
sls / 

1 


1. Let UB obtain the "apex” equation CzJ of a gy¬ 
roscope oi’ variable mass. By this term we understand a body 
of variable mass that has kinetic symmetry and a stationary 
point. Ne shall also assume that a process of ejection but 
no accretion of particles might be taking place. The direc¬ 
tions of the principal axes of the body are assumed to be fi¬ 
xed. 

The formula of the variation of the kinetic moment of 
a body of a rarlable mass witn respect to a stationary point 
has the forraj 

( 1 , 1 ) 

for the general case, allowing for the internal movement of 
particles / 3_/. _ 

In this formula Aqj-^Crk is the aforementioned 

kinetic moment expressed in the principal axes with the ortho- 
nomeal vectors i, jf* A and C are the moments of inertia; 
p, q, r the projections of the angular velocity. Further: 

D/Dt is the symbol of the derivative, with respect to time, 
which is computed on the as_sumptlon that all point masses of 
the rigid body are fixed. L and are the moments of the 
acting forces and of forces resulting from the variability of 



Codes 

y i/or 





/ 



the total mass. 

Let u3 aae v for the velocity of the,end-point of the 
rector & which corresponds to the gyroscope's apex. We find{ 

+ ( 1 . 2 ) 

Consequently, the kinetic moment shall be written in the 
foxrm: 

J^A(kxv)-^Cri. ( 1 , 3 ) 

on the basis of the formula (1*3) the equation (1,1) 
assumes the forms 

A{%xv)^CrS-^CfV^»t + K . (1,4) 

The moments of the active forces shall be composed of 
the moment of the force of gravity and the moments of the re¬ 
sistances to the gyroscope's motion, which will include the 
damping forces of the ejected particles. On neglecting fric¬ 
tion in the suspension and using formula (1,2) we may assume: 

—(••'>) 

Here h is the distance of the gyroscope's center of 
gravity from the point of suspension. The quantity h will 
oe greater than zero, equal to zero, or smaller than zero, 
depending oh whether the center of gravity is above the sta¬ 
tionary point, coincides with it, or is below it. V) is a po¬ 
sitive function of time, and vp a positive constant. (The 
quantities v^ and vg stand for resistance and damping). For 
a rapid rotation the resistance V 2 r must be replaced by the 
experimentally determined function f(r) of the resistance. 

It is not too difficult to realize a construction of 
a gyroscope, for which the relative kinetic moment of parti¬ 
cles moving in the gyroscope will be equal.to zero. In the 
case of such a construction the moment M / "bj would be 
/written in the form 

( 1 . 6 ) 

Here V is the loss per second of the relative kineticfo¬ 
ment of the particles across the gyroscope's surface; 
is the loss per second of the transferable kinetic moment of 
the particles through the same surface. 

Assuming the process of the ejection of particles to 
be symmetric we shall have: 

where K is a known function of time. And we shall have further: 


... 2 - 






ol -^OiXi) + C>i: 


(1.7) 

M.H) 


The syn?'^''! &nd.^, in the formula (1,8) are the 

known oositlv ttlona of time, corraapondln^^ to the mo¬ 

ments Gi iner . of loss of mass ^er second, Vie shall con¬ 
sider gyroscopes which Ay ^ 1^1/ ^ I Cf • The func¬ 
tions of time K, A'-y as well as A', C, h, ra and , 

can be determined experimentally upon the Btand._^ 

Takia^ the scalar and vector products of k and the 
equation (1,4), and teklrxf, Into consideration formulas (1,5) 
— (l»'"j)j obtain 

^ 4*^1^^ 

4 V, A)v (1.10) 

The equation (1,9) shall be called ''equation of the a- 
pex" as used by Yu, A, Krutkov, 

2, Equation (1,9) has the Solution 




(1.11) 


For a very rapid rotation of the gyroscope, when the 
resistance may no longer be considered linear, the equation 
of rotation about the instrument's own axis can be integrated 
approximately. 

It is obvious that equation (1,10) has a particular 
solution __ _ 

v-ssO. (1,11-9 

where a is the unit vector of the vertical axis, 

vie shall study the stability of the vertical position 
(1,12), For this we shall have equations by the first appro¬ 
ximation. 

Let k = a + R, we will consider the fixed system of 
the coordinates (x, y, z), with origin o at the end-point of 
the vector a. The axes ox, oy shall be horizontal, while oz 
is directed vertically downwards, 

V/e shall have the formulas 


v^R'y 





A?(l (1,10) 

On projecting, the equation (1,10) onto tne axes ox 


and oy by the first approximation with respect to quantities 
X and z and their derivatives, we obtain: 


* -T-*—Py-f fljf. j 

y^pjf -iy+ «y. I 


(l.l-V 


A A Cr ,• 

wiiece . P i Y *• Ih +''i + ai'e some bounded functions of 

time. 

i'^or the projection z ws have the formula 

z * 1 — V\ - 


Therefore, it follows from stability with respect to 
X and y and their derivatives, and of their derivative func¬ 
tions, that motion in the z direction is also stable. 


2 


Multiplying the first equation of the system (1,13) 
by X and the second by y we obta^, on adding, 

i V'-.~.2t(i* + y*)-i(x»+y’), (2.1) 

where we introduce the notation 

V^->+y‘-a(x’-ry*) (2.2) 

Let the center of gravity be located below the sta- 
tlnnarv^ point; then^<0 and consequently «<0. The function 
V shall be definite positive, since^*, /A/ and ^>0* Let 
OC no’w be a mon^otonlcaliy Increasing function. In that ease 
the derivative -^Kis definite negative, as follows from for¬ 
mula (2,1). For sufficiently small initial conditions, this 
derivative will remain negative definite if computed using 
the non-linear equations of the gyroscopic motion, because 
terms of the second and of higher orders cannot affect the 
sign in tne ri<i:ht member of the relation (2,1). Hence, by 
the iundamental theorem of A. 'A. Lyapunov's second method 
l_ 4_J7 vje conclude that tne vertical position of the gyros¬ 
cope of variable mass is asymptotically stable in the case 
considered. 

Let us suppose that the center of gravity coincides 
with the stationary point, i,e. Using the same theorem 

of Lyapunpv we shall prove asymptotic stability with respect 
to X and y. Since these latter are small for small initial 
conditions, we can use for their evaluation the equations 


4 





oftha firct approximation (1,13), wlilch In this case admit" 

tna l:iX.o J-C.1 

jS)^ (2.3) 

Denoting by the symbol the averaging we find for 
the case when oL^C from the equations (1,13) 

A — ^ ~ V* ( j : _ -“?•(>- Vo). 

y— y« «• p* (jf ~ x^) - f* (y — y^). 

Therefore^ for small velocities, the coordinates will 
suffer negligibly from the initial values, 

2. If the process of variation in mass occurs only 
in a finite time interval tJ a-nd should the resistance of 
the air to the rotation of the gyroscope about its own axis 
be negligible, the coefficients factors of the system (1,13) 
will be constant for t > T, 

The characteristic equation shall be written as: 

X-fr-l-2|>^ 4-2»-I* ?•) i* - «**0. 

In order to satisfy Hurwitz's conditions It is both 
sufficient and necessary'that In our case < 0, Therefore, 
if at the instant t the quantity h Is negative,, the vertical 
position of the gyroscope shall be stable. The nature of the 
gyroscope'^ motion can be examined by the method of B, 7, 
Bulgakov / 5_J7, 

Let the coefficients In the equations of the system 
(1,13) approach finite limits as t infinity. Assuming 
vg = 0, we obtain a boundary system with constant coefficients. 
If the limit _i3 less than zero, then by the theorem of K, 

P, Persidskiy vertical position of the gyroscope is 

also stable In this case, 

3 

1, Should internal motion of the particles in the gy¬ 
roscope be absent, as for example in the case of 

we shall have a purely superficial burning away, 

v'e shall neglect the air resistance. Then as a conse¬ 
quence of formulas (1,11) and (1,13) we shall find: 


5 







(3.1) 


^ rf/, 

f- I 

> • pi + ay. I 


(3.2) 


2* As a preliminary we will prove a theorem refer¬ 
ring to the approximate solution of a system of linear dif¬ 
ferential eauatlona with variable coefficients. 

Let us suppose we have a linear system: 


- <i« 1.2,. ., (.1.S) 

/-« 


where are continuous functions of time, and where /a^j/ 
< a for'O <1 t < infinity. V/e shall consider an arbitrary 
time interval ^0, T_7 subdivide It by points 

X — X ® parts so that <, 4 ^ — ■= 'C. 
X write the approximating system 


with the notation: 


dt 

• 2 


All 


a,y(0) 

npH 

0<r</,. 

aQitf) 

• 

npM 

4 

* 


4 

) npH 

• 


(3.4) 


(3.'}) 


The theorem consists in the following: for any arbl- 
trarilv small £. > 0 and an arbitrarily large T 0 we can 
find a" sufficiently small ■t:> 0, so that for -//ir 
0 ^ t ^ T we shall have: 

The continuous solution of the system {3»^) with the 
Initial conditions yj,© will be unique, Now we shall use the 
method of successive approximations for computing yi, taking 
XjL as the zero zpproxlmatlon. By virtue of the system (3»^) 
we have: 

yi^yiv-irJ^o'uyjdf 


6 






Thus we arrive at the following result: 

y, ^ X (y?* ~ *')’ 

AkI 

The functions a^j are continuous, therefc^e they shall 
he uniforialy continuous In the closed interval Gon- 

seouently, a value of /C can be found for which 

This gives the estimate (:5,6). It must be mentioned 
that this estimate shall be uniform relative to T as varies. 

3. Let us examine the stability of the zero solution 
of the system (3,2), This system Is linear and the theorem 
Just proven applies. 

Let us consider the approximating system; 


where fi<'and /i*are constants obtained by the approximation 
rule oi' (3#5)« Settins? + (i» t ^ t ) , we reduce the 
ayaten (3,7) to a single equation 

S A- /f'u — a'tt - ‘t 


.-or the jth interval we have 


Let us set 


*“yf +-a;«y*0. 


Uj^rVjt 


1%^ 


♦ 


( 3 . 1 ", 


which yields the equation 

for the variable Wj, 

r'ormulas (3,10) and (3,11) show that the motion of a 
point having the coordinates X| and can be represented as 
a set of relative motion in accordance with equation (3,11) 
and translatory motion of uniform rotation with an angular 
velocity /3j/A. 

V7e shall note that the stability of the sequence of 
functions u^, Implies the stability of the zero solution 

of system (3,7), since by formula (3,10) we have 


A solution in the variables u and d must be continu¬ 
ous, hence by virtue of (3,12) we have the following transi¬ 
tion conditions from the interval j to the Interval J + 1: 




(3 1?. 





where is the modulus of the complex number uj ; la 

th‘" aa xj u' velocity of relative motion or u derivative 
wliu ri.iocr t to -iniG w:.oi’e ars^ument in the complex number 
; anu hj Si I — (l ^, 

Let us suppose that- tat . Denoting by the 

quantity Indicated on the interval, we obtain tne fol¬ 
lowing obvious integrals of tne equation (3,11); 


//^-consl, ^3 

_- ‘^onsl. ( 3 . 15 , 

where is the relative velocity on the j^h in¬ 

terval. 3y virtue of formulas (3,13) - (3,15) the following 
relation will hold; 

I 

Let us suppose 4r^ Is decreasing (not necessarllv ra¬ 
pidly), Then by virtue of the formulas (5,13) - (3,16) 

-f — b] -h ^ p*(t) 4* 


13 , 17 ) 


The expression in square brackets In (3,17) equals: 


In it the index indicates the averaging In the Indicated 
ib-^ interval a(; and fi; are derivatives computed at the ins¬ 


tant 


Sinceis decreasing we can expect that for suffi¬ 
ciently high relative velocities and a sufficient momentum 
K, «<0. Consequently, for any value of the indices 

1 and j we shall have 


3rom the formulas (3,17) - (3,18) we obtain: 

//,>! < 4 2| Si + A«A^( HeCr+'f). 


( 3 . 19 > 


in wiaich we use the notations; 


9 


W w |^ll9j *♦* y ‘V(r) arsupj 




Juppose the initial conditions are sufricleritl 7 small 


.hen lattin-^ 






MNiTl 


('•i) 




we obtain tne inequality 


K ^ t-+ 'i^"P 5 )^ t ti\»> 

The estimate (.5,20) is independent or 1', consequently 
it is valid lor the interval 0 4t '2 <- Ini’lnlty. 

ihc case v;here./3 and Increase is also of interest. 
(The increase does not have to be fast). Cn account of the 
Intewrai (3,1-^), the expression 

-tc 

^ Gj p ^ conM. Oj ■» c(>n«ii ri. 211 

will ai'5o bo an Intej^ral of the equation (2,7) on tne 
time interval. 

Due to the formulas (3,13) - (3,l6) and (3,21), we ha 
ve 



where v:e introduce the notation 



llnca b2 is an Increasia'? function, we have 

*V ^ piT" ■*" (P/,i ““ 

ij’urther on we will have 



10 








Let us suppose that the relationship (I < is sa¬ 
tisfied for a given number q. For sufflolently high rela¬ 
tive velocities of the moving particles, is considerably 
larger than \dj > Consequently, a sufficiently large number 
P can be found with the property that 

- —J— 

jrom the above, for a sufficiently small X, and 

fci <» 

the inequality 



will be satisfied. 

-/e have proved that in the two cases discussed the 
stability of the vertical position of a gyroscope of varia¬ 
ble mass is valid by the first approximation. And we ob- 
talnel, in addition, the estimates for the changes of the 
variables. 




BIBLIOGRAPHY 


1* Novoselov, V* S, Certain Problems of the Hechanics of 
Variable Hasses, Author's Abstracts Candidate's 
Dissertation. Leningrad State University imeni A, A. 
Zhdanov, 1952* 

2, Krutkov, Yu, A. On the Equations of the "Tops” Apex, I. 

News of the Acad Soi USSR, Ser. Math* and Hatural 
Sciences, 1932, pages 489-501, 

3, Novoselov, V. S. Certain Problems of the Mechanics of 

Variable Masses in Considering the Internal Motion 
of Particles, I. Herald of Leningrad State Univer¬ 
sity, No. 19, 1956. 

4, Lyapunov, A. M, General Problem of the Stability of Mo¬ 

tion. Moscow State Publishing House of Technical and 
Theoretical Literature, 1950, 

5, Bulgakov, b. V. Vibrations. Moscow, Gostekhlzdat, 1954. 

6, Persidskiy, K, P. On the Characteristic Numbers of Dif¬ 

ferential Equations, News of the Acad, Sci, KazSSR, 
Ser. Math, and Mech., No. 42, vyp 1, 1947* 


Article received by the edi¬ 
tor on 6 February 1958, 


2023 


END 


12 





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