Teodorescu P.P. Mechanical Systems, Volume III: Analytical Mechanics

Подождите немного. Документ загружается.

MECHANICAL SYSTEMS, CLASSICAL MODELS

542

,0,const

λλλ=+ > =

(23.1.55)

(,,..., )

Wx x x being a function identically zero or of constant sign, and if V is not

of constant and opposite to

W sign, then the unperturbed motion is instable.

We can put together the two theorems of Lyapunov in

Theorem 23.1.16 (Chetaev). If, for the system of non-linear differential equations

(23.1.29) (of the perturbed motion), we can build up a function

( , ,..., )

Vx x x so

that, in an arbitrary small neighbourhood of the origin, there exists a domain

0V >

(on the frontier of which we have

0V = ), for which – according to the system – the

derivative

′

be positive definite on this domain, then the unperturbed motion is

instable.

23.1.2.3 Rheonomous Mechanical Systems

In the case of rheonomous discrete mechanical systems the temporal variable appears

explicitly, the non-linear differential equations being of the form

( , ,..., ; ), 1,2,...,

Xxx xt i n

;

(23.1.56)

they define a non-autonomous system of differential equations.

In the following we say that a function

( , ,..., ; )

Vx x x t is positive definite if, in

the spherical domain

S definite by

xhtt

≤>

∑

(23.1.57)

we have

12 12

( , ,..., ; ) ( , ,..., ), (0,0,...,0; ) 0

Vx x x t Wx x x V t≥=,

(23.1.57')

the function

(,,..., )

Wx x x being positive definite in the domain

S . Analogously, a

function

( , ,..., ; )

Vx x x t is negative definite if, in the domain

S , we have

12 12

( , ,..., ; ) ( , ,..., ), (0,0,...,0; ) 0

Vx x x t Wx x x V t≤− = .

(23.1.57'')

The total derivative with respect to time of the function

( , ,..., ; )

Vx x x t is

calculated in the form

VVV

tx t

∂∂

∑

(23.1.57''')

In connection with the system of differential equations (23.1.56), we assume that the

Stability and Vibrations

543

conditions

(0,0,...,0; ) 0, , 1,2,...,

Xtttin≡> = ,

(23.1.56')

hold; taking into account (23.1.56'), we can state that the system (23.1.56) is verified by

the solution

( ) 0, 1,2,...,

xt i n== , which is the unperturbed solution, called

sometimes the trivial solution.

Let

( , ,..., ; )

Vx x x t be a positive definite function and l be the inferior limit of

the function

(,,..., )

Wx x x in the spherical domain S

, 0 hε<<; it results

( , ,..., ; )

Vx x x t l≥ ,

(23.1.58)

for

tt≥ . Taking into account the continuity, we can determine a quantity δ so that

00 0 0

( , ,..., ) for , 1,2,...,

Vx x x l x i nδ<<=,

(23.1.58')

according to the second relation (23.1.57''). Let us suppose that the functions

00 0

( ; , ,..., )

xftxx x

represent the solution corresponding to the initial conditions

( ) ,| | , 1,2,...,

iii

xt x x i nδ=<=. Let us suppose that, for >

tt, takes place a

relation of the form

≥

∑

f ε

and let be

′

tt the moment for which takes place the equality (the moment

′

exists,

because the solutions

can be as close as possible to the trivial solution for which

∑

f ). In this case, according to the relation (23.1.58), we can have

′′ ′′

≥

((), (),..., ();)

Vf t f t f t t l; in the hypothesis in which the derivative d/dVt is

non-positive, the preceding inequality is contradictory to (32.1.58'), because

V could

not increase in the interval of time from

t to

′

. Hence,

∑

f ε

the trivial solution being thus stable.

We can state

Theorem 23.1.17 (Lyapunov; first theorem). If, for the system of non-linear differential

equations (23.1.56), which verifies the conditions (23.1.56'), as well as the conditions of

existence and uniqueness of the solution, relative to the initial conditions (at the

moment

t )

, 1,2,...,

xi n= , in the spherical domain

S , defined by (23.1.57), one

can set up a function

( , ,..., ; )

Vx x x t of definite sign, so that – according to the

system – one obtains a derivative

d/dVt, which does not have the sign of

in the

domain

S , then the unperturbed motion is stable.

MECHANICAL SYSTEMS, CLASSICAL MODELS

544

Analogously, one can state

Theorem 23.2.18 (Lyapunov; second theorem). If, for the system of non-linear

differential equations (23.1.56), which verify the conditions (23.1.56'), as well as the

conditions of existence and uniqueness of the solution, relative to the initial conditions

(at the moment

t )

, 1,2,...,

xi n= , in the spherical domain

S , defined by

(23.1.57), we can set up a function

( , ,..., ; )

Vx x x t positive definite, with the

superior limit arbitrarily small, so that – according to the system – to obtain a

derivative

d/dVt negative definite, then the unperturbed motion (the trivial solution)

is asymptotically stable.

23.1.2.4 Stability in First Approximation

Considering the autonomous system of equations (23.1.29), we get the system of the

first approximation (23.1.31), by an expansion into a Maclaurin series (23.1.30). The

problem is put to see in what conditions the stability in first approximation, considered

in Sect. 23.1.1.2, is sufficient for the system (23.1.29) too; a study in this direction has

been made be Lyapunov too.

Let

( , ,..., ; )

Vx x x t be a form of degree m , for which

′

=VVλ

(23.1.59)

where

′

is the derivative calculated according to the system (23.1.31).

= 1m , then the linear form can be written

∑

,const

ii i

Vxαα

(23.1.60)

Replacing in (23.1.59) and taking into account the system (23.1.31), we get

, 1,2,...,

ij i j

ajnαλα

∑

;

(23.1.60')

to can obtain the non-zero coefficients

α , we determine the parameter λ by means of

the characteristic equation

=() 0D λ , given by (23.1.34).

> 1m , then we denote by

, ,...,

αα α the coefficients of the form V ;

proceeding as above, it results a system of linear algebraic equations of the form

, 1,2,...,

ij i j

AjNαλα

∑

(23.1.61)

where

are linear combinations of coefficients

a ; this system has non-zero

solutions if and only if

λ is a root of the characteristic equation

Stability and Vibrations

545

11 12

21 22

...

() 0

... ... ... ...

...

NN NN

AA A

−

(23.1.61')

Lyapunov showed that all the roots of this equation are given by

=+++

11 22

...

mm mλλ λ λ, (23.1.62)

where

, ,...,

λλ λ are the roots of the characteristic equation =() 0D λ , while

are non-negative integers linked by the relation

+++ =

...

mm m m

. (23.1.62')

We will determine a form

V of degree m , which does satisfy the equation

′

=VW

(23.1.63)

the derivative being calculated according to the linear system (23.1.31), while

is a

form of degree

m too. If

, ,...,

αα α are the coefficients of the form V and

, ,...,

ββ β are the coefficients of the form W , then the relation (23.1.63) leads to

the system of linear algebraic equations

, 1,2,...,

ij i j

AjNαβ

∑

(23.1.63')

where

A are the coefficients which intervene in the characteristic equation (23.1.61').

The determinant of the system is

=det[ ] (0)

AD . Hence, if no one of the roots λ

given by (23.1.62) is zero, then – for any form

W – there exists a form V and only

one which satisfies the equation (23.1.63); the coefficients

=, 1,2,...,

iNα , are

obtained from (23.1.63'), because

≠det[ ] 0

A .

We can thus state

Lemma 23.1.1 If all the roots of the characteristic equation associated to the linear

system (23.1.31) have the real part negative, then – for any form

(,,..., )

Wx x x of

definite sign – there exists a unique form

( , ,..., )

Vx x x of the same degree as W ,

which satisfies the condition (23.1.63) and which is of definite sign, opposite to that

W .

Let be now the quadratic form

V , defined by the equation

11 1

nn n

ij j i

ij i

ax x

== =

∂

=−

∂

∑

∑∑

;

MECHANICAL SYSTEMS, CLASSICAL MODELS

546

on the basis of Lemma 23.1.1, this quadratic form exists and is positive definite. Taking

into account this equation, the equation (23.1.29) and the Maclaurin series (23.1.30), we

can express the derivative

′

in the form

∂

⎛⎞

′

=−+

⎜⎟

∂

⎝⎠

∑

VxR

;

because in the second term of the above bracket intervene powers at least of the third

degree in the variables

x , the derivative is negative definite and we may state

Theorem 23.1.19 (Lyapunov; first theorem). If all the roots of the characteristic

equation which correspond to the system of first approximation (23.1.31) have the real

part negative, then the unperturbed motion of the autonomous system of differential

equations (23.1.29) is asymptotically stable (for any terms

R of higher order).

We denote

=A []

A and = []

δδ and consider the equation

(

)

det 0, 0

μα

⎡⎤

−+ = >

⎣⎦

A δ ,

(23.1.63'')

the roots of which are linked to the roots of the characteristic equation

=() 0D λ by

the relations

=− =, 1,2,...,

μλ .

(23.1.63''')

We can choose the number

α so that the equation (23.1.63'') have, as the characteristic

equation

=() 0D λ , at least a root with positive real part, the number

+++

11 21

...

mm mμμ μ

being non-zero; we deduce from here that there exists a

function

V of the same degree as W and not of a constant opposite sign, so that

(

)

⎡⎤

+++− ++ =

⎣⎦

∑ 11 22

... ... 0

ii i in

ax ax a x ax

for any form

W of definite sign. But – according to Euler’s theorem – we have

∂∂=

∑

(/)

Vx mV, so that we obtain, finally

′

=+ >,0VVWαα

(23.1.64)

We can thus state

Lemma 23.1.2 If, between the roots of the characteristic equation associated to the

linear system (23.1.31) there exists at least one root with a positive real part, then – for

any given form

W , of definite sign and of degree m – there exists always a function

V of the same degree and a positive number α , so that the relation (23.1.64) does

take place, the function

V being not of constant sign, opposite to that of W .

Stability and Vibrations

547

Let be now the quadratic form

V , defined by the equation

11 1

nn n

ij j i

ij i

axVx

αα

== =

∂

=+ >

∂

∑∑ ∑

;

on the basis of the Lemma 23.1.2, this quadratic form exists and there are points at

which it is positive. Taking into account this equation, the equation (23.1.29) and the

expansion into a Maclaurin series (23.1.30), we can express the derivative

′

in the

form

∂

⎛⎞

′

=+ +

⎜⎟

∂

⎝⎠

∑

VV x R

;

in a sufficiently small neighbourhood of the origin, the sign of

′

is specified by the

sum

++++

22 2

...

Vx x xα , which is positive. In these conditions, the theorems of

instability of Lyapunov (see Sect. 23.1.1.7) allow to state

Theorem 23.1.20 (Lyapunov; second theorem). If there exists at least a root of the

characteristic equation corresponding to the system of the first approximation

(23.1.31), with a positive real part, then the unperturbed action of the autonomous

system of differential equations (23.1.29) is instable (for any terms

R of higher order).

To be more specific, Lyapunov showed that the terms

R can be neglected if

()

<+++ >=

1/2

22 2

12 2

( , ,..., ) ... , 0, const

Rxx x Mx x x M

μ .

(23.1.65)

If the characteristic equation considered above has not roots with a positive real part,

but has purely imaginary roots (with zero real parts), then one cannot decide on the

stability without taking into consideration the terms

=,1,2,...,

Ri n, of higher order;

the respective cases are critical ones. Studies in this direction have begun in the last part

of nineteenth century by H. Poincaré; they are contained in a paper on the system of

differential equations

=− + =− +

2 112 1 212

(, ), (, )

x Xxx x Xxx

λλ

(23.1.66)

and have been continued by Lyapunov and other researchers.

23.1.2.5 Lyapunov’s Function

Practically, to apply the theorems stated above, one must effectively build up

functions

V of Lyapunov; there does not exists a general method in this direction, but

one can give some indications. We consider only autonomous systems of differential

equations.

If the equations (23.1.29) of the perturbed motion admit a first integral

( , ,..., ; )

Fx x x t , then we can take

=−

12 12

( , ,..., ) ( , ,..., ) (0,0,...,0)

Vxxx FxxxF ,

(23.1.67)

MECHANICAL SYSTEMS, CLASSICAL MODELS

548

if the difference in the second member is a function of definite sign; the derivative

′

is – obviously – identically zero.

If one can determine several first integrals

( , ,..., ), 1,2,..,

Fxx x j m= , for the

equations (23.1.29), so that for no one of them the difference

−

( , ,..., ) (0,0,...,0)

Fxx x F

is a function of definite sign, then – after Chetaev –

one can search the function

V in the form

[]

=−

∑12 12

( , ,..., ) ( , ,..., ) (0,0,...,0)

jj j

Vx x x F x x x Fλ

(23.1.68)

or in the more general form

[]

{

}

12 12

( , ,..., ) ( , ,..., ) (0,0,...,0)

( , ,..., ) (0,0,...,0) ,

jj j

Vxxx FxxxF

Fxx x F

=−

+−

⎡⎤

⎣⎦

∑

(23.1.68')

the constants

=, , 1,2,...,

jmλμ , being determined so that the function V be of

definite sign. Also in these cases, the derivative

′

is identically zero.

In the case of mechanical systems, the first integrals can be obtained with the aid of

the universal theorems.

We have seen in Sects. 23.1.1.5 and 23.1.1.6 that, in case of linear systems, the

stability problem can be directly studied; in this case, the functions

V and

′

will be

quadratic forms in the co-ordinates

, ,...,

xx x. In a matric formulation, let be the

autonomous system of differential equations

(23.1.69)

where we have introduced the matrices

⎡⎤

⎡

⎤

⎢⎥

⎢

⎥

⎢⎥

⎢

⎥

⎢⎥

⎢

⎥

⎢⎥

⎢

⎥

⎢⎥

⎣

⎦

⎣⎦

11 12 1

21 22 2

...

... ... ... ...

...

nnn

aa a

(23.1.69')

We choose Lyapunov’s function in the form

V = xxα ,

(23.1.70)

where the matrix

Stability and Vibrations

549

11 12 1

21 22 2

...

... ... ... ...

...

αα α

⎡

⎤

⎢

⎥

⎢

⎥

⎢

⎥

⎢

⎥

⎣

⎦

(23.1.70')

is symmetric. The total derivative with respect to time is

′



xxxxαα

;

taking into account the matric equation (23.1.69), we can write

()

TT T T T

′

=+= +



xa x x ax x a axαα αα

too. We put the condition that this quadratic form be of the form

−+++

22 2

( ... )

xx x

, hence that

′

=−xx;

in this case, the matrix α will be determined by the matric equation

+=−aaαα δ.

(23.1.71)

If the equation (23.1.71) admits a positive definite solution for the symmetric matrix

then the solution

=x 0

is asymptotically stable.

Let us consider now Hamilton’s canonical equations (19.1.14) in the space

Γ , the

unknown functions being the generalized co-ordinates

q and the generalized momenta

, 1,2,...,

pj s= , for a conservative, holonomic, scleronomic, discrete mechanical

system; Hamilton’s function

==−=

12 1 2

( , ,..., , , ,..., )

HHqq qpp p TUh,

= consth , is a first integral of this system of differential equations. The kinetic energy

T is a positive definite form; if the potential energy −U is a positive definite form too;

then

=HE (E is the mechanical energy) is – as well –a positive definite form in a

domain situated in the neighbourhood of the origin

(0,0,...,0,0,0,...,0).

The condition that the potential energy

−U

be a positive definite form in the

neighbourhood of the origin (we have

= 0U at the origin) is equivalent to the

condition that

−U have a minimum at this point, obtaining thus again the

Dirichlet-Lagrange theorem (see Sect. 23.1.1.3); in exchange, if the potential energy

−U has a maximum, then we find again Lyapunov’s Theorem 23.1.7 of instability.

Let us choose as Lyapunov’s function the expression (we use the summation

convention of dummy indices)

=−

VHqp,

(23.1.72)

MECHANICAL SYSTEMS, CLASSICAL MODELS

550

Hamilton’s function being given by

=−= −

( , ,..., )

ij i j

HTU pp Uqq qβ ;

(23.1.73)

we express then

in the form

=++

12 1 2

( , ,..., , , ,..., )

ij ij

Hpp qqqqqpppβαθ ,

(23.1.73')

where

ij ij

αβ are constants, while the function θ contains terms of higher order

than

. Assuming that

−U

has not a minimum at the origin, it results that, for small

values of the generalized momenta, there exists a neighbourhood of the origin for which

< 0H , i.e. the neighbourhood for which 0U−<. Let us denote by D the domain for

which

< 0H and > 0

qp . We have > 0U in D , while on its frontier we have

= 0U , because = 0H or = 0

qp ; we notice that the origin is a point of this

frontier.

We can write

()

12 1 2

, ,..., , , ,..., ,

ij ij

jj jj jj j j

ij ij

VHqpHqpqp Hp q

Hpp qq qqqppp

βα θ

∂∂

⎛⎞

′′

=− +=− −

⎜⎟

∂∂

⎝⎠

=− − +





where we took into account Hamilton’s equations (19.1.14), the fact that

H is a first

integral of these equations and the expression (23.1.73'); we denoted by



the set of

terms of degree greater than

2 .

Taking into account that in the domain

D we have < 0U , the principal part of the

kinetic energy being positive definite, it results that the principal part of the potential

energy must be negative to have

′

> 0

V ; the function



does not have any influence

on this result. The potential energy cannot have a minimum (

′

V is a function positive

definite), obtaining thus again Chetaev’s Theorem 23.1.8 of instability.

23.1.2.6 Systems of Cyclic Co-ordinates. Routh’s Theorem

Let us consider a natural, holonomic and scleronomic discrete mechanical system,

for which the first

r generalized co-ordinates which intervene in Routh’s autonomous

system of equations (19.1.31), (19.1.31') are cyclic co-ordinates, so that

/ 0, 1,2,...,Hq r

α∂∂= = ; in this case, = 0p

 , resulting the first integrals

, 1,2,...,pc r

αα

α== . Routh’s function is of the form (19.1.30); we can state that

we have – as well –

∂∂=/0q

L , hence / 0, 1,2,...,Rq r

α∂∂= = , too. The

equations (19.1.31) lead – in this case – to the same result as the Routh–Helmholtz

theorem (see Sect. 18.2.3.7). There remain to be integrated the equations (19.1.31').

which are of Lagrange type.

Stability and Vibrations

551

We call stationary motion that motion in which all the non-cyclic co-ordinates

, 1, 2,...,

qjr r s=+ + , have constant values (equal to that at the initial moment);

without any restriction on the generality of this motion, we can suppose that, for some

values of the constants

, the system (19.1.31') admits the particular solution

====

... 0

qq q,

(23.1.74)

to which corresponds the unperturbed motion (for all the co-ordinates

, ,...,

qq q).

The initial generalized velocities which correspond to the non-cyclic co-ordinates are –

as well – zero, while the cyclic generalized velocities are constant. Let us consider now

some initial perturbations of this system, without changing the constants

As in the general case of Lagrange’s equations, we multiply each equation by the

generalized velocity

, 1, 2,...,

qjr r s=+ + , sum and obtain a first integral of Jacobi

type

∂

−=

∂

∑

qRh



(23.1.75)

where

= consth

is the energy constant; let be the function

∂

=−−

∂

∑

VqRh



(23.1.76)

where

h corresponds to the solution (23.1.74). If

h is an extreme value (minimal or

maximal), then the function

V will be of definite sign (positive or negative).

Calculating the total derivative with respect to time and taking into account the first

integral (23.1.75), we obtain

=d/d 0Vt ; according to the Theorem 23.1.17, we can

state

Theorem 23.1.21 (Routh). The stationary (unperturbed) motion of a natural holonomic

and scleronomic discrete mechanical system, relative to the cyclic co-ordinates is stable

if the mechanical energy of this system, after exclusion of the cyclic velocities, admits

an extremum relative to the perturbation of the constants

This result has been obtained by Routh in 1877 (Routh, E.J., 1898).

We must notice that the stability with respect to Routh’s variables

, , 1, 2,...,

qq j r r s=+ + , and , 1,2,...,pr

α = , does not also mean stability with

respect to the cyclic co-ordinates

, 1,2,...,qr

α =

23.1.2.7 Limit Cycles

Let be the autonomous system of differential equations with one degree of freedom

(23.1.18), considered in Sect. 23.1.1.4; in certain conditions, this system admits periodic

solutions to which correspond closed curves, called limit cycles. The denomination

results from the behaviour of some solutions of the system.