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

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MECHANICAL SYSTEMS, CLASSICAL MODELS

562

holds; hence, the zeros of the trinomial in

λ must be real and distinct. It results the

condition

4IJRlω

(23.1.91')

and the constant

λ must be contained between the two zeros. As a matter of fact, the

inequality (23.1.91') represents a condition for

ω . In this case, the motion of rotation

of the projectile is stable, having a strict minimum for

at ====0ααββ



 .

23.1.3.3 Circular Motion in a Central Field

Let us consider a particle of mass

m acted upon by a force of intensity <() 0Fr ,

where

r is the distance to a fixed point O (a centre of attraction); obviously, this is a

central force (see Sects. 8.1.1.1, 18.3.2.2 and 19.2.3.1).

The trajectory of the particle being plane, we can write the kinetic energy and the

potential energy in polar co-ordinates

(,)r θ in the form

()

=+−=−

∫

222

,() ()d

Tmrr Ur Frrθ



;

(23.1.92)

it results Hamilton’s function

()

=−= + − = +

222 2 2

111

()

222

HTU mr r Ur p p



 ,

(23.1.92')

where we have introduced the generalized momenta

∂∂

== == =+

∂

pmrpmrTU



L ,

corresponding to the generalized co-ordinates

(,)r θ . We obtain thus the canonical

equations

== =+ =

,, (),0

rpFrp

mr mr



 

(23.1.93)

where

θ is a cyclic co-ordinate; the solutions

==− + = =−

()

,,0,()

rR t p p RmRFR

θθ

(23.1.93')

corresponding to a uniform circular motion, which is unperturbed, are verified.

Let be the perturbed motion

Stability and Vibrations

563

()

,().

rR t

pprmRFR

εθ θ ε

ηη

=+ =− ++

==− +

(23.1.93'')

Replacing in the equations (23.1.93), we obtain the perturbed equations

()

−+

== −−

−+

⎡⎤

⎣⎦

=++=

112

() ()

()

,0,

RmRFR FR

mmR

RmRFR

εε

ηεη



(23.1.94)

corresponding to Hamilton’s function

()

⎧⎫

−+

⎡⎤

⎪⎪

⎣⎦

=+ −+−−

⎨⎬

⎪⎪

⎩⎭

112

()

RmRFR

HUR

mmR

ηεη

(23.1.94')

We notice that the potential energy

−+

()UR ε does not admit a local minimum for

0ε

, because it does not contain the co-ordinate

ε , so that an eventual minimum

of this function for

0ε could take place for any

ε and could no more be a local

minimum; hence, the motion is instable in Lyapunov’s sense.

We notice, from (23.1.94), that

constηη , where

η can be taken

arbitrarily small, hence

(, )HHεη . Neglecting the term

(1/2 )m η ,

corresponding to the kinetic energy, we consider the function

()

−+

⎡⎤

⎣⎦

=−+−−

112

()

RmRFR

VUR

εεη

;

(23.1.95)

we have

()

−+

⎡⎤

⎣⎦

=− − + =

()

RmRFR

(23.1.95')

for the position of equilibrium. The condition of minimum is

()

−+

⎡⎤

⎣⎦

′

=−+>

3()

RmRFR

(23.1.95'')

Eliminating the rectangular brackets between the last two relations, we can write

()()()

′

−+−+ +>

11 1

30FR R F Rεε ε;

(23.1.96)

MECHANICAL SYSTEMS, CLASSICAL MODELS

564

→

0ε , then this condition becomes

() ()

′

30FR RF R

(23.1.96')

In the important particular case in which

,() , 0, ,, const

kks

UFr ksks

==−>∈=] ,

(23.1.97)

we have

() ( 1)/

Fr kss r

′

, and the condition (23.1.96') gives

2s <

; (23.1.97')

in this case, the motion of the particle is orbitally stable in Poincaré’s sense (e.g., in

case of the Keplerian motion for which

1s = ).

We will consider now the more general case of motion on a sphere, having a circular

motion on the equator (the radius

R and the angle ϕ in the equatorial plane – the

longitude) and on the meridian (the radius

R and the angle θ , measured on the

meridian from the equator towards the pole – the latitude). We denote (unlike the

preceding case, we introduce in these expansions also the mass

m of the particle –

besides the constant

k – for subsequent simplifications)

,() ,, 0, ,,, const

km kms

UFr kmskms

==−>∈=] ,

(23.1.98)

for a central force of attraction; the equation of motion admit a solution of the form (see

Sect. 8.1.1.1)

,,() 0rR FR mRϕϕ ϕ≡≡ + =  .

(23.1.98')

We will make a study of the stability of the motion with respect to all variables

,,,rrθθ



 and ϕ .

The mechanical system is conservative, while

ϕ is a cyclic co-ordinate

(/ / 0)HTϕϕ∂∂=∂∂=, so that we obtain the first integral

()

222222

cos ,

ETU mr r r mh

mr ma

θθϕ

θϕ

=−= + + − =

∂







(23.1.99)

where

,constha= .

The perturbed motion will be characterized by the variables

123

,,ηηη in the form

1212 3

,,,,rR rεθ ε ηθ ηϕ ϕ η=+ = = = = +



;

(23.1.100)

Stability and Vibrations

565

the corresponding first integral will be

()

()( )

()

()( )

11 12

132

21 32

cos ,

FR a

ηεη

εϕη ε

=++

++ + − =

=+ + =



(23.1.99')

but no one of them is of definite sign with respect to the variables

12 12 3

,,,,εεηηη.

Hence, we search a Lyapunov function in the form

[

]

11 22 2 2

(0) (0) (0)VFF FF F Fλμ=− + − + −

⎡

⎤

⎣

⎦

(23.1.101)

trying to determine the constants

λ and μ so that this function be of definite sign; we

also use the relation

= ,

(23.1.98'')

obtained from (23.1.98), (23.1.98'). Expanding the non-polynomial terms into a

Maclaurin series and neglecting the terms of a degree higher then

2 , we can write

()

22 22222

0000

11 1 3 1 2

2222

123 13

2222

000

22 1 3 1 2 13

22 32 4 222

000

22 1 3 1

422 42 3

23 13

(0) 4 2

(0) 2 2 ,

(0) 4 2 6

28;

FF R R s R

FF R R R R

FF R R R

RRR

ϕε ϕη ϕε ϕε

ηηηϕεη

ϕε η ϕε ϕε εη

ϕε ϕη ϕε

ϕε η ϕεη

−= + −−

++ + +

−= ++− +

−= + +

−++









we replace in (23.1.101) and equate to zero the linear terms (the origin must be a point

of extremum), which leads to the condition

()

21Rλϕ μ=− + ,

(23.1.102)

remaining only the determination of the parameter

μ . We can express now Lyapunov’s

function in the form

VVV=+, where

() ()

=−− + ++

=++

2223 222

11133

222 2 22

2212

424 1,

VRs R RR

VR R

μϕεμϕεημη

ϕε η η





We notice that quadratic form

V is positive definite with respect to the variables

212

,,εηη. To have also the quadratic form

V positive definite with respect to the

MECHANICAL SYSTEMS, CLASSICAL MODELS

566

variables

12 12 3

,,,,εεηηη (it is sufficient to be positive definite with respect to

ε and

η ), we use Sylvester’s criterion; hence, it is necessary and sufficient to have

()

223

322

422

42, 0

Rs R

RRR

μϕμϕ

μϕ μ

−−

>+ >





which leads to the conditions

()

RsR

μμ

−

(23.1.103)

In the particular case in which

1s = (Keplerian motion), both inequalities are

satisfied if

3/Rμ > ; we can thus state that the Keplerian motion is stable with

respect to any perturbations, on a circular orbit.

23.2. Vibrations of Mechanical Systems

In many domains of physics, chemistry, biology appear oscillatory phenomena; the

mechanical oscillations can be also vibrations. From a mathematical point of view, the

development of the knowledge in this direction has led to a development of the

qualitative theory of differential equations; indeed, together with vibrations appear

problems of stability of motion. We mention that the respective phenomena are

encountered everywhere in techniques and technology.

The basic notions of the theory of oscillations can be found in Christian Huygens’s

fundamental work “Horologium oscillatorium sive de motu pendulorum ad horologia

adaptato demonstrationes geometricae”, dedicated to the French king Louis XIV in

1673, and in his famous “Undulatory theory of light”, presented at the Academy of

Sciences in Paris, in 1678. The knowledges in this direction are enriched by Leonhard

Euler in “Mechanica sive motus analyticae exposita” in 1736, by J.-L. Lagrange, in

1788, in “Mécanique analytique”, the first systematic presentation of this new method

of calculation, by E. J. Routh in his “Dynamics of particles”, in 1898, by

A. M. Lyapunov in his famous doctor thesis, in 1892, and by others; the actual

possibilities of calculation allow, besides a linear study of vibrations, a non-linear one

too (Huygens, Cr., 1920; Lagrange, J.-L., 1788; Lyapunov, A.M., 1949; Routh, E.J.,

1898).

In what follows, we consider undamped and damped, free and forced small oscillations

about a stable position of equilibrium, non-linear vibrations and limit cycles; the results thus

obtained are followed by applications with theoretical and practical character.

23.2.1 Small Free Oscillations About a Stable Position of Equilibrium

Many motions of discrete mechanical systems with one or several degrees of

freedom are periodical; in such a motion, the geometric parameters which determine the

position of a mechanical system are periodic function of time. The oscillatory motion is

a motion with a single degree of freedom, in which the geometric parameter which

determines the position of the discrete mechanical system changes periodically the

sense of its variation. An important rôle is played by the small oscillations about a

Stability and Vibrations

567

stable position of equilibrium; the case of a heavy particle constrained to stay on a fixed

surface has been considered in Sect. 7.2.3.3. The oscillatory motion is periodical if the

constitutive oscillations of the motion and the parameters (the geometric parameter

which determines the position of the mechanical system, the velocity and the

acceleration) are repeated with the same detail, in the same order after a certain interval

of time

T , called period. The simplest periodic oscillation is the harmonic oscillation –

a periodic motion in which the periodicity is expressed by the sinus or by the cosines of

a linear function of time, e. g.,

()

cosxa tωϕ=−,

(23.2.1)

where

x is the elongation, a is the amplitude, ϕ is the phase shift, ω is the pulsation

(circular frequency),

2/T πω= is the period and 1/ /2fTωπ== is the

frequency. Thus, in Sects. 23.2.1 and 23.2.2, we have considered the oscillations of the

discrete mechanical systems with two degrees or with one degree of freedom,

respectively.

After establishing the equations of motion, we put in evidence the proper forms of

vibration, we pass to normal co-ordinates and give some methods to determine proper

pulsations; we mention, as well, the introduction of the influence of a supplementary

constraint. The results thus obtained are applied to the study of small oscillations with

one degree or two degrees of freedom, as in the case of two harmonic oscillations. The

small oscillations of damped free oscillations will be considered too.

The theory of small oscillations with which we deal corresponds to the cases of

motion of the discrete mechanical systems which remain close to a stable position of

equilibrium or – more general – to a state of motion in permanent regime.

23.2.1.1 Equations of Motion

Let us consider a conservative holonomic and scleronomic discrete mechanical

system with

s degrees of freedom. We assume that the potential energy

(,,..., )

VVqq q= has an isolated minimum at the position of equilibrium

, 1,2,...,

qqj s== ; because the generalized forces are given by

, 1,2,...,

Qjs

∂

(23.2.2)

these ones will vanish at the respective point, the position of equilibrium being thus a

stable one, Without any loss of generality, we assume that

0, 1,2,...,

qj s== ; as

well, because the potential energy

is determined neglecting an arbitrary constant, we

can take this minimum equal to zero (

(0, 0,...,0) 0VV==), so that in the

neighbourhood of the origin we have

0V > .

Expanding

V in a power series at the origin, we can write (we notice that

(0, 0,...,0) 0VV== and 0V > , so that the linear terms must disappear)

,const

ij ijij ij ji

Vqqbqqbb

∂

⎛⎞

====

⎜⎟

∂∂

⎝⎠

(23.2.3)

MECHANICAL SYSTEMS, CLASSICAL MODELS

568

the sum being made – obviously – from

1 to s in the hypothesis of small motions, we

neglect the terms of higher order. It results that the quadratic form

V is positive

definite. It can happen that, by neglecting some terms of higher order, the minimum at

the origin does no more be strict (e. g.,

222444

12 1 2

( ... ) ( ... )

Vaq q q bq q q= + ++ + + ++ , ,0ab> ); we will not take into

account such special cases.

The kinetic energy is given by

()

, , ,...,

ij i j ij ij

Tgqqggqqq== ;

expanding the coefficients

g into a power series and neglecting all the terms which

contain generalized co-ordinates, because we have to do with small oscillations, we

admit that

const, , 1,2,...,

ij ij

ga ij s== = . In this case

ij i j ij ji

Taqqaa==

(23.2.3')

the kinetic energy being, as well, a quadratic positive definite form, assuming that the

origin is not a singular point.

If, at the initial moment, the position of the discrete mechanical system is sufficiently

close to the position of stable equilibrium and if the initial velocities are sufficiently

small in absolute value, then – during the motion – both the deviations from the

position of equilibrium and the generalized velocities will be small in absolute value.

One can thus justify the above linearization of the problem.

Lagrange’s function

TV=−L leads to Lagrange’s equations in the form

0, 1,2,...,

ij j ij j

aq bq i s+==

(23.2.4)

corresponding to the small motions of a discrete mechanical system, subjected to

holonomic and scleronomic constraints and to conservative forces, around a stable

position of equilibrium; we will try to determine a motion described by particular

solutions of the form

() cos( )

qt a tωϕ=−

(23.2.5)

which – as a matter of fact – are harmonic oscillations.

Introducing in the system of equations (23.2.4) and using the notation

λω= ,

(23.2.6)

we obtain the homogeneous system of linear algebraic equations

()

0, 1,2,...,

ij ij j

baa i sλ−==,

(23.2.7)

which has non-trivial solutions for values given by the secular equation

Stability and Vibrations

569

[]

11 11 12 12 1 1

21 21 22 22 2 2

1122

...

det 0

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

...

ij ij

ss ss

ssss

baba ba

λλ λ

−− −

−= =

−− −

(23.2.8)

Developing this determinant, we obtain a polynomial equation of degree

s , which is

the frequencies equation, with

s roots

0, 1,2,...,

ksλω=> = (the link between

the frequency (circular frequency)

and the pulsation

ω is given by

fωπ=

);

we find, for any root

λ , a solution of the homogeneous system (23.2.7), which

corresponds to the amplitudes of the harmonic oscillations. We can thus state that there

are at the most

s distinct frequencies. If 0

λ < , then the corresponding normal mode

is not oscillatory (the generalized co-ordinates

q increase or decrease exponentially in

time). The general solution will be a linear superposition of such particular solution as it

will be shown in the following subsection.

Let us introduce the matrices

11 12 1

21 22 2

...

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

...

bb b

aa a

bb b

aa a

bb b

==AB

(23.2.9)

and the column vectors

⎡

⎤

⎡⎤

⎢

⎥

⎢⎥

⎢

⎥

⎢⎥

⎢

⎥

⎢⎥

⎢

⎥

⎢⎥

⎢

⎥

⎢⎥

⎣⎦

⎣

⎦



(23.2.9')

We say that the symmetric matrices

A and B are positive definite because the

corresponding quadratic forms are positive definite. In this case, the kinetic and the

potential energy can be written in the form

TV==



qAq qBq

(23.2.3'')

being led to the kinetic potential

TV=−L

(23.2.3''')

and to Lagrange’s equations

MECHANICAL SYSTEMS, CLASSICAL MODELS

570

+=0



Aq Bq ;

(23.2.4')

the normal mode of motion is given by

() cos( )ttωϕ=−qa .

(23.2.5')

Replacing in (23.2.4'), we obtain the matric equation

()

λ−=0BAa ,

(23.2.7')

which has non-zero solutions if and only if

[

]

det λ−=0BA ;

(23.2.8')

we obtain thus the amplitude vectors

23.2.1.2 Secular Equations

To can find the nature of the roots of the secular equation, we will put in evidence

some properties of the quadratic forms with real coefficients (we assume that

, , 1,2,...,

ij ji

aaij s==).

We notice first of all that to any quadratic form

ij i j

auu corresponds the bilinear

form

(,)

ij i j

auv=Auv ,

(23.2.10)

where

u and v are the column vectors

⎡

⎤⎡⎤

⎢

⎥⎢⎥

⎢

⎥⎢⎥

⎢

⎥⎢⎥

⎢

⎥⎢⎥

⎢

⎥⎢⎥

⎣

⎦⎣⎦

uv;

(23.2.10')

in this case, the quadratic form is written in the form

(,)

ij i j

auu=Auu ;

(23.2.10'')

Following properties can be easily verified:

(i)

12 1 2

(,)(,)(,)+= +Au uv Auv Auv;

(ii)

(,) (,)λλ=Auv Auv, λ scalar;

(iii)

(,) (,)=Auv Avu;

(iv)

(,) (,) ( , ),=+Auu Avv Aww with i, i=+ =−uv wuvw is a real number;

(v) if

(,)Auu is a positive definite quadratic form and ≠ 0u is an arbitrary

Stability and Vibrations

571

complex vector, then

(,) 0>Auu ;

(vi) if

λ is a root amplitude of the secular equation det[ ] 0λ−=BA and a is the

corresponding amplitude vector (

, λ=≠0Ba Aa a

), then for any vector v one

has

(, ) (, )λ=Bav Aav .

One can now easily show that the relation

(, ) 0

′

=Aaa

(23.2.11)

takes place for two amplitude vectors

a and

′

a , which correspond to two different

roots

λ and

, λλ λ

′′

≠

, of the secular equation.

Let

λ be a complex root of the secular equation (23.2.8') with λλ≠ and let a be a

corresponding complex vector;

λ is, in this case, also a root of this equation, to which

corresponds the amplitude vector

a . Because λλ≠ , it results the equality (23.2.11),

in contradiction to the property (v). One can thus state that the secular equation

(23.2.8') has only real roots.

One can choose a real amplitude vector

a to a real roots λ . Using the property (vi),

putting

=va and observing that (,)Aaa is a positive definite quadratic form, we may

write

(,)

λ =

Baa

Aaa

;

but

(,)Baa is a positive definite quadratic form too, so that 0λ > . Taking into

account (23.2.6), we can state that the secular equation (23.2.8') has

s positive roots

λ , to which correspond the positive pulsation

ωλ= and the real amplitude

vectors

, 1,2,...,

js=a .

Assuming firstly that all the roots of the characteristic equation are real roots, to each

λ corresponds

()

cos , 1,2,...,

jj j j

tj sωϕ=−=qa ,

(23.2.12)

where the amplitude vectors must satisfy the linear system

()

λ−=0BAa .

(23.2.7'')

Because the system (23.2.4') is linear, any linear combination of the solutions

(23.2.12) is a solution of this system; hence

()

cos

jjj

Ctωϕ

=−

∑

qa ,

(23.2.12')

where

, 1,2,...,

Cj s= , are arbitrary constants, is a solution of this system.

Let be