A problem on the stability of a spherical gyroscope

A

A PROBLEM ON THE STABILITY OF SPHERICAL GYROSCOPE (ODNA

USTOICAIVOSTI QIROSKOPA)

ZADACAA

08

Vo1.27,

No.6,

PMM

V.

1963,

V.

SHAROVOQO

pp.f005-1011

KREMENTULO

(MOSCOW) (Received

Under certain

assumptions

we consider scope. for

the

June

on the nature

stability

of

asymptotic the

stability.

case

of

method

The cases

rotation

of

the vertical

By the Liapunov-Chetaev

also

20,

with

of

1963)

the

viscous

rotation

gaseous

of

we obtain

sufficient

astatic

and heavy

variable

angular

shell,

a spherical

gyro-

conditions gyroscopes,

velocity,

and

are treated

in particular. The principal given

LlI.

in

scheme A steel

for

the motion

sphere

of

1 (Figure)

a spherical

is

placed

gyroscope

inside

was

a closed

spherical bowl tionary vertical wall

of

2 rotating with constant angular velocity around a staaxis 0~1. Gas supplied through special holes in the bowl forms a shell completely enveloping the sphere and

the

isolating

it

instant

the

thickness

(equal

the

sphere)

the

geometric

center

from the gaseous

to the

and remains center

the

gaseous

has

inside

axis of vertical

the

shell it

The

bowl.

thus

for

the

sphere

It

is

assumed

a homogeneous between all

shell

the

radii

subsequent coincides,

consequently,

is

time

that

at a certain

with

constant

of

the

of

the motion.

by hypotheses,

a stationary

bowl

and

with

point.

Thus, the

Since

the

viscosity,

the bowl in spinning carries along with it in its turn makes the sphere rotate. The sphere

which

be the

balanced

stability

the

difference

a cylindrical

0 will

of

becomes

groove

symmetry 0~ which 02,. The ellipsoid

point 1.

ing

of

of the bowl and,

gas has a specific

the

wall

shell

of

at the

3 for initial

creating instant

of inertia of the ellipsoid of rotation.

gyroscope. the established

The problem motion

1547

of

a definite coincides

sphere

with

consists the sphere

dynamic with

of

respect

the to

investigatas a rigid

1548

V. V.

body

in which

,pension the

the

center

0 and which

viscous

to the

gaseous

difference

of

is

gravity

acted

shell

Krementulo

coincides

between

the

angular

of

lar

the

the

velocity

instantaneous

velo-

of

the of

[21 prove

K is

of

the

us note

that

gaseous

A

here

the motion

is

the

choice

mentioned

problems

of

such

of

the

only

respect

action

velocity

of

gas is

to point

an induction

in balanced

value

for

considered

0.

Let

creation

The action

of

moment

maintaining

us write

the dynamic

the

a con-

equations

equations

of

A) qr = K (coy -

p),

-$+ qf-r-7’

I=

2 +

(A

-

C) pr=

q),

g

=- f)

K (coy” -

the

moment of

K (07’

-

equatorial of

respect under

moment of the

inertia

sphere,

corresponds

to

to the

p,

axes

investigation

P =

p==q=o,

ry

-

/7,-f

Euler

0

(1.2)

r),

inertia

with

stationary state system (1.2)

_t

and

the

-

Oz,

co-

with

motor,

motion.

sphere in the form of equations of Poisson

not

in connection

w =

a rotation

the

the

7

sphere,

C is

the

the projections

the sphere onto the principal direction cosines of the

Ox, Cb, and Oz, will

COIISL,

of

of

7 and r are

of the instantaneous angular velocity Q of axes of inertia Oxyz, y, y’ and y ‘* are the

which

a numerical

(C

Here A is

vertical

in hydroaero-

$- +

c -dr = rlt axial

of

analogous

our

moment M with to

of motion of the and the kinematic

A

certain

that

shell

rotating angular

the

valid.

M'1s analogous stant

of

the coefficient

solving

efficient

the

velo-

gas.

mechanics

Let

(1.1)

angular

d is the thickness and v is

7he results

that

in

o is the constant anguthe bowl, R is the radius

of

shell

viscosity

sus-

sphere,

of the sphere, gaseous

of

M originating

The

M = -K (!4 - w), K =8g2 city

the point

moment

and proportional

cities of the sphere and the bowl. moment M is taken in the form

where Q is

with

upon by a rotating

respectively.

be the particular

=

sphere

y’

zz 0,

and the

7” =z 1

bowl

The

solution

of

(1.X)

as a wliole

A problem

around

the

depending

on

vertical on p,

whose motion of equations

axis

q,

they (1.2)

0~~.

y,

r,

stability

the

y’

describe allows

of

a spherical

1549

gyroscope

The right-hand sides of equations (1.2)) and y”, show that the mechanical system

is essentially nonconservative. of only one known integral r2 +

7’2 +

TV2 =

The system

const

(1.4)

Let us determine the stability of motion (1.3) in the sense Liapunov with respect to the variables p, q, F, y, y’ and y”. The problem

rotation (1.3) with respect to body with a fixed oint in the Lagrange case, was solved, as is well known, by Chetaev [3 P by the method of constructing Liapunov functions in the form of a linear combination of the integrals of the equations of perturbed motion. In the given case p,

q,

F,

y,

of stability and y” for

of

y’

of vertical a heavy rigid

the number of integrals is not sufficient for the construction of a linear combination; however, we can try to construct a sign-definite combination from specially selected functions such that the conditions of

Liapunov’s Let

stability

us introduce

the projection and the

of

function

theorem

into the

consideration

angular

El

9 =

the kinetic

momentum

energy

G of the sphere

of

onto

the

sphere,

axis

Ozl,

Gz, = Apy + A qy’ + Cry”

+ Aq2 + CF), cll =

P =

satisfied.

0

T = +(Ap2

By denoting

are

the

perturbed

rl,

r=a+L

72 +

7’2 +

motion

7s2

(1.5)

corresponding 7

=

a,

7’

to =

(1.3)

P,

f’

by 1 +

=

6

(1.6)

we can write

F, = A (E” + q2) + CC”+ 2Co5 F, = A (b + qS) + CC6+ C (06 + 5) F, = a2 + p2 i Let

us make a linear

combination 2V =

It

is

not

difficult

2V = A (E” f

q2) -

to 2oA

see

F, that

(&I +

(1.7)

a2 + 26 of

20F,

functions +

2V will

q6) +

The derivative of function I’ taken turbed motion corresponding to motion

(1.7)

in

the

form

Co2F, be a quadratic

Co2 (a” + relative (1.3)

to

form

p”) +

C (< -

the

equations

oQ2 of

(1.8) per-

1.550

(l-9) will be sign-constant of a sign opposite to that of V. The function V, not being sign-definite in all of the variables <, q, <, a, p and 6, will be sign-definite in the variables <, q, a, p and 6 - o< when the condition C > A is satisfied, which indicates that in this case the axis Oz will

be the minor

of

the

ellipsoid

bility

with

stable y” are

with respect to p, q, y, y’ connected during the motion stability

follows

stability

respect

to y”

sequently, respect

the point

respect

all

The derivative general,

it

(1.3). is

is

sufficient

given

by the

asymptotic

with

my” follows

variables not

motion

in

c: = 0.

with

in equality variables

(1.6),

we obtain

it

is

seen

the modulus the

sphere

equals

does

motion not

motion the

function,

under for

therefore, of

with

study of

this

stability

is

theorem

t.hr

if

in detail

derivative

v’ is

between

motion

the

the nature

of

to

in accordance

(1.10)

= ---K/Q -aI2

proportional the

motions,

we <..n solutions except

from perturbations

(r-oy”)2l

state

derivative

to be asymptoti-

be demonstrated

difference

in of

the

asymptotic

we pass

and unperturbed

&n-

F.

be stable

stability

the perturbed

+

there

and y”.

us clarify

(1.9)

and

to

angular

the

square

velocities

of of

and consequently,

when Q = o.

or does

whether

the

vector

sphere

with

to

will

asymptotic

will

I.et of

respect

(1.3)

not wholly contain the corresponding to (1.3))

t_ (q--oy’)2

(1.2) shows that rotations around

except

perturbed

that

the

in perturbed

zero

System motions Y -

of

with y’

motion

(1.3)

If

the

to y and y’

sign-constancy

0 does motion

the original

V’ = --K[ (p-(0~)~ Hence

y,

the

the

itself.

of

the theorem on sta[41, motion (1.3) is

and from stability

[5,1 . On t,he basis

of

region

values

r,

infer

motion

(1.3’)

respect

criterion

establish that the region I’= of the equations of perturbed the

q,

cases

stated of

with to y”,

a sign-definite to

of

variables

stability

the unperturbed

stability

the

C > A, motion p,

possible

theorem

inertia

and r - my”. lhe variables y, y’ by relation (1.4), and therefore,

(1.3)

Such a sufficient

unperturbed

of

respect

condition

in certain for

stable.

solution

1’ is not

However,

cally

of

the

the

a portion

also and r -

under to

to

of

0. Thus, on the basis

respect

from the

to

axis

with

sphere

(1.3). contain

the sphere cannot the vertical with

Hence, the

corresponding for

the

entire

specified

to

accomplish established constant angular velpcity

question

whether

integral

curves

(1.3),

reduces

perturbations

the of

to the

region

the

question

can or cannot

V = 0

equations

of

of

accomplish

A problem

a continuous city

0.

fixed

sequence

In other

axoid

that

the moving bodies

us to

infer

axoid

gives that

achieving

asymptotic

it

is

sion

of

0.

that inertia

the of

Note.

condition

axis

the

C >

A motion

of

vertical

of

rotation

[61,

(1.3)

of

which

will

(1.3)

be the minor respect in

the

same time allows

be asymp-

of K. Consequently,

rotation

was obtained

velo-

that

The kinematics

question

the magnitude

with

at the

cone?

this

of

sphere

result

The results of

inertia

other

of

the

the

2v =

of

symmetry

Section

of

the

body with

assumptions

show that

of

to

of

the

for sphere,

axis

of

the

the point

of

suspen-

[71 by investigating

1 can be extended

rigid

body.

respect

to

Section

function

the

1 remain

satisfied,

in the

motion 2.

(1.3)

variables

of

us assume that dynamic

axis

of

The equations

$ + (C Ag+(A -

A

to

gyroscope. does

the

under not

center

of

It

axes is

Oxyz.

not

re-

The

difficult

to

motion

C) pr = -

(1.3).

stable

the

mgq,r

q,

r,

to

the

Hence,

under

with

equations

of

the unperturbed

conditions

of

the point the

(1.12).

sphere

suspension

is

located

0.

Let

on its

0.

sphere

K (coy +

y,

of

z,, from point

P-1)

p),

cg =

K (of

-

r)

K (07’ - q),

of a stationary solution not’ations, we investigate

respect to p, the functions

in-

Let us proceed to the question of the the action of moment (l.l), in which the

mgz,f +

A) qr =

when the

(1.12)

relative

gravity

of

(1.11)

Pz) + C(c - 06)~

p and 5 - 06

(1.9)

coincide of

a,

state

synnnetry Oz at a distance

when o = const, admit selves to the previous (1.3) with sideration

derivative

the sphere

gravity

the

C>B

be asymptotically

heavy of

we waive

principal

in force.

<, q,

A,

corresponding

will

The

stability center

has the

motion

its

if

B and C be the moments

(AEa + BqP) + Cw2(a2 +

C> perturbed

Let A,

V

At2 + BqZ - 20

Positive definite equal it ies

are

to

some angular

body move such line

a nondegenerate answer

with

rigid

1551

problem.

quirement of

straight

stability

An analogous

linear

to some fixed

the

gyroscope

rotations the

independently

sufficient

spherLcal

a

instantaneous

remains

under

stable

01

can or cannot

a negative

totically

ellipsoid

of

words,

degenerates

rigid

stabrlrty

the

on

y ’ and y”.

(1.3). the

By confining stability of

Let us introduce

into

ourstate con-

1552

Qr = A (5” + %

= A (Eo +

Functions F,

of

qB) +

q2) +

C66 +

Cl;” +

C (06

az and o3 coincide

Section

1.

It

is

easy

with

to

see

+

SCOT, + 6),

2mgz,6,

@a =

(2.2)

a2 + p2 t- h2 _t 26

the corresponding

that

the

functions

derivative

of

the

F, and functions

v, 2V, = taken form

relative (1.9).

to

the

a1 -

equations

The function 2vi

2&,

= A 6”

of

q2) +

+(Co2

motion

of

by virtue

of

asymptotic not

enter

directly

determine axis.

C >

the rate

( 2.4)

A inequalities

above,

motion

into

the damping

Inequalities

case

of

of

will

conditions

may hold (2.4)

are

in case

C < A they

reduce

3.

Ihe

velocity system sphere

case motion

u = o( t ). of

be the for

for

the

[31 when

(2.4)

conditions for the coefficient

motions

any relation

stability of

the h’ does if

the

gyroscope

between

A and C.

we In

to (2.5)

0 condition

We studied rotating

a sphere,

the

(2.3)

0

asymptotic

equivalent

C) 69 + mgz, <

(A -

established

to

has

(2.3)

20 <

the perturbed

to < but

function

As before,

(1.3).

of

by Chetaev

0,

C) to2 + mgz, <

what was stated

stability

(1.3),

(A& + A$ + ~56) + (a2 + fi2 + d2)

sign-definiteness

(A -

for

form

2w

mgz,)

resulting from, a more general form constructed A = - o and P = 0 (in the notations of [31). ‘Ihe conditions

mgz,) CD3

be a quadratic

Cc2 -

-

(Co2 -

perturbed

2V, will

+

+

0

(2.6)

above the stability around the vertical

of the with a

equal to the constant velocity o of the bowl. As shown by in the case o = o(t), the nonstationary motion of the (1.2))

p=q=o,

r =

does not occur. Motion (3. l), of special corrective devices

0

(t),

r = q-’ = 0,

nevertheless, which create

is possible an auxiliary

y” = 1

(3.1)

in the presence moment Mzk( t)

A problem

on the

stability

of

a spherical

gyroscope

1553

around axis 04. In every real gyroscopic system there are usually introduced corrective devices to ensure the realization of some designed motion; this justifies to a known degree the assumptions of the stated problem. If Mzk(t)

satisfy

the condition M*k (t) = Co.(t)

(3.2)

then the system of equations

A g Ag

(3.3)

+ (C - A) Q’ = K (W’ - p), + (A - C)

pF

=

K

admit of the particular we write the equations

(of

c -dr = R (a$

r) +

M*k (t)

solution (3.1). Retaining the adopted notations, of perturbed motion corresponding to (3.1)

df Az+(C-A)q(~+f)=K(

$

oa - 8,

A~+(A--C)((o+t)=K(oB-~),

c$ =K As the Liapunov

-

dt

q),

-

(08 -

function

2V = Fl -

let

Q, us

consider

20 (t) F, + Cd

=fw+~)-q(l+4

z

= E(1+6)--

d6 x

=w

the

-

(a+()

68

combination

(3.4) of

functions

(t) F, + p (t) Fda

by (1.7) and F, = 5. ‘be functions o(t), where F,, F, and F, are defined p(t) are assumed to be bounded, continuous, twice-differentiable on the infinite time interval t > ta; moreover 0-l (t)

a* >

>

0,

o (t) #

‘lbe function V which has been considered with variable coefficients

0 whent>to will

(3.5)

be a quadratic

form

3’ (t, E, 7, 5, a, B, 8) = A (E”+ q2)+ V + p) P -

-

20 (A@ + AT@ + C:8) + Co2 (a2 + fJ2 + d2)

‘Ihe derivative of be a quadratic form V’ = -

(3.6)

taken relative

to equations

(3.4)

(3.6) will

20 (Ea + $J + f6) + a2 (a2 + B” + WI

K IE” + q2 + G2 -

also -

-w~(A~u+A$+C~~)+C~~‘(U~+~~+~~)+&-~(O~-~)+ ++pX2

=

.- KY (t, 5, q, b, a, B, 6)

(3.7)

1554

V. V.

hrementulo

On the basis of Liapunov's theorem on asymptotic stability, the unperturbed motion (3.1) turns out to be asymptotically stable if the function (3.6) is positive-definite and function (3.7) negative-definite in the sense of Liapunov (the requirement of the presence of an infinitely small upper bound in form I’ is satisfied under the given assumptions on the nature of functions o(t) and u(t)). Thus, motion (3.1) is asymptotically stable if Sylvester's conditions are satisfied for the quadratic forms

(Al, A,, A, and A, are positive numbers as small as desired). Each of these forms can be represented as a sum of three forms in the variables (5, a), (n, 13)and (5, 6). The Sy1vester conditions for these two-variable forms are

In order to evaluate the basic results, let us analyze these inequalities when A, = h, = h, = A, = 0. Tn this case inequalities (3.9) give c I> ‘4,

p I-- 0

Inequalities (3.10) reduce to ‘M(A

-c)ow'-11120'2)o,

The first relation in

P'(w2-

(i1.11) (3.12)

,,;$‘j

(3.12) gives (3.13)

Let us now pass to the selection of function u(t) constrained by inequalities (3.11) and (3.12). ""Len

the

A problem

on

the

relation

in

(3.12)

second Taking

(3.11)

into

stability

is

account,

of a spherical

gyroscope

satisfied.

we get

the

following

bounds

(3.15)

inequalities By combining (3.13) and (3.15) we obtain ditions (3.9) and (3.10) reduce when A, = . . . A, = 0

If

the condition

then, the

as is

not

following

state

A, =

. . . = h, = 0 is eliminated to

sufficient

verify,

conditions

to which

from the hypothesis,

inequalities for

the

(3.16)

(3.9)

asymptotic

and (3.10) stability

EQ>$>-4Kc++e,<0,

El, -

(f)‘>o

positive numbers as small where Em, E, and .sJ are strictly Inequalitiei (3.17) define a certain class of asymptotically motions

(3.1).

special

kind

Conditions of

in accordance

(3.17)

function

with

(3.6)

(3.14).

functions

p satisfying

widen

the

of

class

corresponding

by considering

F, + 2hF, -

is

the aid

of

with

CohF, + pFd2 -

some function

conditions

for

Let us examine under (3.1) does occur in the with

p(t)

speaking,

motions

along

as desired. stable

from a consideration

(3.1).

(3.12)

which

This

aim can

a Liapunov

(3.6)

of

were defined we can find

inequalities

stable

(3.17)

can

function

form

2V* = where A = h(t)

generally

differential

asymptotically

be accomplished a more general

were obtained

when the coefficients

However,

other also

give of

(3.1):

C>A +

of

con-

(;)‘>o

O>f>-4Kq,

difficult

on o:

($ )‘>o

$
C>A,

1555

the

2V, =

conditions

A) F,

analogous

to

the

n(t).

what conditions an asymptotically stable motion case of heavy gyroscopes. ‘Ibis problem is solved

quadratic @,

satisfying

2C (o $ -

-

200~

form +

(Gz

-

mgz,) (&

+

p@)4”

(3.18)

where 0,, o2 and o3 are defined in accordance with (2.2) and 0, coincides with F,. As is not difficult to see, the derivative of (3.18) relative to the equations of perturbed motion coincides with (3.7). Along

with

(3.6)

in the

given

case

we obtain

a

1556

I’. 1’. Krrmentulo

The author thanks V.V. numiantsev interest in the work and for helpful

and S.A. Kharlamov for their remarks while it was in progress.

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4.

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7.

No.

with 4,

(On the

Appel' , P.,

Vol.

84.

Tcoret

gyro-

sfericheskogo bearing).

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(giroskopa)

axisymmetric Nauk

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pod-

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ical

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spherical

1955.

Chetaev, Izd.

Dynamics of an air-supported Vol. 32, No. 7, 1962.

na sfericheskoi body

(gyroscope)

CfkrSSR, NO. 1,

osesimmetrichnogo

pore

(Stability

of

on a spherical

tverdogo motion

support).

of

an Dokl.

1963.

Translated

by

N.H.C.