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

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MECHANICAL SYSTEMS, CLASSICAL MODELS

602

++= =( ), 1,2,...,

ij j ij j ij j i

aq cq bq Q t i s  ;

(23.2.71)

in a matric form, we can write

++=Aq Cq Bq Q

 

(23.2.71')

The general solution is given by

∗∗

=+= +

∑

qq q a q

(23.2.71'')

where

is the general solution of the homogeneous equation, while

∗

q is a particular

solution of the complete equation. We assume that the origin (

====

... 0

qq q )

represents an asymptotically stable position of equilibrium, so that

Re 0, 1,2,...,2

ksλ <= ; hence, →q

0 for →∞t , so that, for a sufficient long

time

t , one can neglect q

with respect to

∗

q . In what follows we make a study of the

particular solution

∗

q .

We assume that only one component of the generalized force

Q is non-zero, let be

( ) 0, ( ) 0, 2,3,...,

Qt Qt j s≠==; because the system is linear, we can obtain a

general result by superposition of effects. Choosing the non-zero component of the

generalized force in the form

() e

Qt A

, the system (23.2.71) will be given by

++=

++==

111

0, 2,3,..., .

jjj

ij j ij j ij j

aq cq bq A

aq cq bq i s

 

We search a solution of the form

e , 1,2,...,

qa j s

the amplitude

being given by the system of algebraic equations

++=

⎡⎤

⎣⎦

++==

⎡⎤

⎣⎦

111

(i ) (i ) ,

(i ) (i ) 0, 2, 3,..., ;

jjj

ij ij ij j

acbaA

acbai s

ωω

(23.2.72)

the solution of this system is

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

aW Aj sω ,

(23.2.72')

where

(i )

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

(i )

Wjs

(23.2.72'')

Stability and Vibrations

603

is a function in the form of a rational fraction with real coefficients, depending on

iω ;

the hodograph of this function in the complex plane (or even the function itself) is

called the characteristic frequency or the characteristic amplitude-phase.

Let us write the characteristic frequency in the form

i()

(i ) ( )e , 1,2,...,

WR j s

ψω

ωω ,

(23.2.72''')

where

() 0

R ω

is the characteristic amplitude, while

()

ψω

is the characteristic

phase; finally, it results

i[ ( )]

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

qt R A j s

ωψ ω

(23.2.73)

If we choose now the non-zero component of the generalized force in the form

() cosQt A tω , we get, analogously,

()

=+=

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

qt R A t j sωωψω .

(23.2.73')

We remark that, passing from the generalized force

()Qt to the generalized

co-ordinate

()

, we amplify the amplitude by the characteristic amplitude and shift

the phase by the characteristic phase

()

ψω for = 1,2,...,

s . We can thus state that

()

R ω increases or decreases the amplitude of the perturbing force; the filter principle

is based on this observation.

23.2.2.4 Action of Small Dissipative Forces Upon a Conservative, Holonomic and

Scleronomic Discrete Mechanical System

Let us suppose that the matrices =B []

b and =C []

c which intervene in the

expression (23.2.68) of the generalized forces are symmetric and positive definite.

Introducing the potential energy and Rayleigh’s dissipative function (see Sect. 23.2.1.13

too), we can write

∂∂

=− − = = =

∂∂

, 1,2,..., , ,

jijijijij

QjsVbqqRcqq





(23.2.74)

The dissipative forces do not change the stability of the position of equilibrium;

moreover, in some cases, this position can be asymptotically stable.

To integrate the equation of motion (23.2.67''), we search a solution of the form

(23.2.68); the amplitudes

=,1,2,...,

aj s, will be given by the algebraic system

(23.2.69'). Multiplying by the amplitudes

a , complex conjugate to

a , we obtain

++=

ij i j ij i j ij i j

aaa caa baaλλ ,

(23.2.75)

or, in compact notation,

(, ) (, ) (, ) 0ACBλ ++=aa aa aa ,

(23.2.75')

MECHANICAL SYSTEMS, CLASSICAL MODELS

604

where we have used the bilinear form (23.2.10); we notice that

( , ) 0, ( , ) 0, ( , ) 0ABC>>>aa aa aa , so that <Re 0λ . Let i,λγδ=+

iλγδ=− be two complex conjugate roots of the secular equation (23.2.70); we

notice that these roots verify the equation (23.2.75') too. Denoting

i, i=+ =−auvauv, where

and

are real column vectors, we can write

(, ) (, ) (, )

2Re 0,

(, ) (, ) (, )

CCC

AAA

BBB

AAA

λλλ

=+=− =− <

== = >

aa uu vv

(23.2.75'')

where we have used the property (iv) in Sect. 23.2.1.2. Hence, to two complex conjugate

roots

λ and λ correspond two complex conjugate oscillations ae

tλ

and ae

tλ

. Taking

two complex conjugate constants

=+(1/ 2)( i)C αβ, =−(1/2)( i)C αβ, we may

write

()()

[]

+= − −+aa uv vueee cos sin

ttt

CC t t

λλγ

αβ δ αβ δ.

(23.2.75''')

If to the real roots

λ and

′

λ correspond the column vectors a and

′

a , respectively,

we obtain, analogously, the equation

(, ) (, ) (, ) 0ACBλ

′′′

++=

aa aa aa ,

(23.2.76)

wherefrom

(, ) (, )

λλ λλ

′′

+=− =

′′

aa aa

(23.2.76')

Passing to normal co-ordinates, we can express the kinetic and the potential energy,

respectively, in the form (23.2.3'''); Rayleigh’s dissipative function will be given by

===

′′′

∑∑∑

111

sss

ii ijij

iij

Rq qqββ,

(23.2.77)

where we assume that the coefficients

0, , , , 1,2,...,

iij

ijij sββ>≠=, are small

(the product and the squares of these coefficients are negligible). We obtain Lagrange’s

equations

′′

++ ==

∑

0, 1,2,...,

iii ijj

qq q i sββ   ;

(23.2.78)

searching solutions of the form

e , 1,2,...,

qis

′

, we get the system of linear

algebraic equations

Stability and Vibrations

605

()

++ + =≠=

∑

0, , 1,2,...,

iii ijj

iji sλβλωαλβα .

(23.2.78')

To have non-trivial solutions, one must equate to zero the determinant of the

coefficients; it results the characteristic equation

11 12 1

21 2 2 2

...

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

...

λβλω βλ βλ

βλ λ βλ ω βλ

βλ βλ λ βλ ω

(23.2.79)

Developing the determinant and neglecting the product of coefficients

iij

ββ, as it has

been mentioned above, we obtain

()

++ =

∏

λβλω

(23.2.79')

so that we can write, with a good approximation,

≅− ± =i , 1,2,...,

λω

(23.2.79'')

Let us consider

=− +

111

(1/2) iλβω and let us calculate the amplitudes

, 1,2,...,

isα = . Replacing in the last − 1s equations (23.2.78'), we can write

122 3

21 2 1 23 2

11 1 1

2133

31 32 3 1 3

1111

i ... 0,

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

ωωα α

βββ β β

ωα α α

αωωα

ββ ββ β

αωαα

−

⎛⎞

+−+ + ++ =

⎜⎟

⎝⎠

−

⎛⎞

++−+ ++=

⎜⎟

⎝⎠

23 1

12 3 1

11 11

... i 0,

ss s

αα ωω

ββ β ββ

αα ωα

−

⎛⎞

++++−+ =

⎜⎟

⎝⎠

obtaining a system of

− 1s equations for the ratios

/ , 1,2,...,

jsαα = ; neglecting

the terms of higher order in

β and

β , we obtain

== =

−

i , , 2,3,...,

βω

εε

ωω

(23.2.80)

, 2,3,...,

jsε = , being real small quantities, of the same order of magnitude. There

results the normal co-ordinates (we assume that

− i

=Ae

α )

MECHANICAL SYSTEMS, CLASSICAL MODELS

606

−−

−−+

′

(/2)i( )

(/2)i( /2)

() Ae e ,

( ) Ae e , 2, 3,..., .

qt j s

βωα

βωαπ

(23.2.80')

Taking into account

(1/2) iλβω=− − too, we obtain, by a linear combination,

()

(

)

(/2)

() Ae cos ,

( ) Ae cos , 2, 3,..., .

qt t

qt t j s

ωα

εωα

−

′

=−

′

=−+=

(23.2.80'')

We get analogous expressions for the other principal oscillations.

We notice thus that the dissipative forces do not modify the frequencies of the

conservative discrete mechanical system; the oscillations induced by these forces tend

to zero for

→∞t . In the principal oscillation of order j , all the co-ordinates are small

with respect to the co-ordinate of order

and differ in phase from it with a quarter of a

period (

=/2 2 /4ππ).

23.2.3 Non-linear Vibrations

The non-linear vibrations appear in the case of many phenomena of mechanical nature,

their mathematical modelling leading to non-linear differential equations or systems of

differential equations. In general, a differential equation of these vibrations is of the form

+=(,,) 0xfxxt 

(23.2.81)

and is obtained from Newton’s equation, which links the elastic forces to the

displacements. In particular, one can consider autonomous vibrations, which lead to

equations of the form

+=(,) 0xfxx 

(23.2.81')

or to non-autonomous vibrations of the form

+=(,) 0xfxt 

(23.2.81'')

or of the form

+=(,) 0xfxt .

(23.2.81''')

It has been shown in Sect. 8.2.2.13, in some particular cases, how one can obtain

such equations.

As well, it has been put in evidence an exact method to determine the period of

non-damped free non-linear vibrations, establishing the formula (8.2.86).

An important class of non-linear equations is that with small non-linearities (which

are put in evidence by a small parameter) of the form (8.2.88), (8.2.88''), and these

equations will be called quasilinear.

Stability and Vibrations

607

Besides the topological methods of calculation, as S.P. Timoshenko’s method, the

Ostrogradskiĭ-Lyapunov method, the least squares method and the method of the

harmonic balance. One has considered also the method of the equivalent linearization

for a quasilinear equation of the form

++ =

(,) 0xxfxxωε 

and for Van der Pol’s equation

()

+− − = >

10,0xx xxλλ  ;

we mention also the method of perturbations, initiated by Poincaré and used by Krylov

and Bogoliubov (Krylov, A.N., 1958; Bogoliubov, N. and Mitropolsky, Y., 1961).

We mention also the graphic methods of computation, between them the isoclinal

method and the delta method.

In what follows, we present the method of the variation of constants too; one makes,

as well, some considerations concerning the parametric vibrations and the non-linear

self-vibrations.

23.2.3.1 Parametric Vibrations

The parametric vibrations are vibrations with variable characteristics, hence

vibrations of a discrete mechanical system, the parameters (mass, frequency,

dimensions, elastic coefficient, damping coefficient etc.) of which are functions of time.

Such an equation is, e. g., the equation (8.2.82') of the mathematical pendulum of

variable length

= ()llt.

In the absence of a perturbing force, a mathematical model of such a mechanical

phenomenon leads to a differential equation of the form

++=() () 0xtxtxαβ  ,

which – by a certain substitution – leads to the equation

+= =−−

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

utu t t t t

γγβαα  ,

studied in Sect. 8.2.2.12.

In the following, we consider the equation

[

]

++ = =() 0, , constuabftu ab  ,

(23.2.82)

where

()ft is a periodic function of time ( =+() ( )ft ft T). If =() cos2ft tω , then

one obtains Mathieu’s equation (see Sect. 8.2.2.13 too) for a variation of the rigidity in

time.

The general solution of the equation (23.2.82) is of the form

MECHANICAL SYSTEMS, CLASSICAL MODELS

608

11 22

() () ()ut Cu t Cu t ,

(23.2.82')

where

()ut and

()ut are two linear independent particular solutions of this equation,

while

C and

C are two arbitrary integration constants. Obviously, we can write

+= ++ +

11 22

() () ()ut T Cu t T Cu t T

too, where

()ut T and +

()ut T are also particular solutions, because ()ft is a

periodic function. Hence,

+= +

1111122

2211222

( ) () (),

ut T aut aut

where

()ut and

()ut, as well as ()ut , are functions which – in general – are not

periodic ones.

Let us suppose that

+=()()ut T sut ,

(23.2.82'')

hence, that, after a period, the amplitude of the action is amplified by

s ; let be

> 0s

After

n periods, the amplitude is amplified by

s . If > 1s , then the amplitude

increases indefinitely, the motion being instable, while if

< 1s , then the amplitude

tends to zero, the motion being stable. Replacing in the formula (23.2.82''), we may

write

[

]

[

]

[

]

+= ++ +

1 1 2 2 1 11 1 12 2 2 21 1 22 2

() () () () () ()sCut Cut C aut aut C aut aut ;

this relation is independent on time if

() ()

−+ = +−=

11 1 21 2 12 1 22 2

0, 0asCaC aC asC .

The constants

C and

C are non-trivial if the determinant of the coefficients vanishes

−

11 21

12 22

asa

aas

(23.2.83)

hence for

s given by

()

−+ + − =

11 22 11 22 12 21

0saasaaaa ;

(23.2.83')

||1s or >

||1s , then the motion is instable, while if ≤

||1s and ≤

||1s ,

then the motion is stable. The difficulty of calculation consists in the searching of the

particular solutions

()ut and

()ut.

Stability and Vibrations

609

23.2.3.2 Method of the Variation of Constants

Let be a discrete mechanical system subjected to non-damped free non-linear

oscillations, which lead to the non-linear equation

++ =(,) 0xx fxxε  ,

(23.2.84)

where

ε is a small parameter, the function (, )fxx containing both linear and

non-linear terms in

x and x . Denoting =xy , one obtains the equivalent system

==−−,(,)xyy x fxyε ,

(23.2.84')

the associated linear system of which is (we make

= 0ε )

==−,xyy x,

(23.2.85)

with the solutions

=+ =−+

12 12

cos sin , sin cosxC tC ty C tC t.

(23.2.85')

In the method of variation of constants, we write

=+ =−+

12 12

()cos ()sin, ()sin ()cosxCttCttyCttCtt;

(23.2.85'')

introducing in the non-linear system (23.2.84'), one obtains

()

1212

()cos ()sin 0,

()sin ()cos

( )cos ( )sin , ( )sin ( )cos .

Ct t Ct t

fCt t Ct tCt t Ct tε

=− + − +



Hence, it results

()

11 21 2

21212

() ()cos ()sin, ()sin ()cos sin,

() ()cos ()sin, ()sin ()cos cos;

Ct fCttCttCttCttt

Ct fCt t Ct tCt t Ct t t

=+−+

=− + − +



(23.2.86)

by integration, we get the exact solution. In general, it is very difficult to integrate this

system of equations for an arbitrary function

f . In an approximate calculation, we

assume that the functions

()Ct and

()Ct are constant during a period 2π , so that

they can be approximate by their mean values on that period; we can write

()

11212

21212

() ()cos ()sin ()sin ()cos sin d ,

() ()cos ()sin ()sin ()cos cos d .

Ct fCt t Ct t Ct t Ct t tt

=+−+

=− + − +

∫



(23.2.86')

MECHANICAL SYSTEMS, CLASSICAL MODELS

610

where

C and

C are considered to be constant in the integrand. Calculating the

integrals, it results a system of the form

() ()

11122212

() (), (), () (), ()Ct FCtCt Ct FCtCtεε



(23.2.86'')

where

and

are of new considered as functions of time. Integrating the latter

system and replacing in (23.2.85), we get the solution of the non-linear equation

(23.2.84).

Let us consider, in particular, the equation (

(,)fxx x )

++ =

0xx xε ;

(23.2.87)

the formulae (23.2.86') lead to

()

=+ =+

=− + =− +

∫

112 212

212 112

() cos sin sin d ,

( ) cos sin cos d ,

Ct C t C t t t CC C

εε



We notice that

=−

12 21

//CC CC



, wherefrom +==

222

constCC a , a being an

amplitude. Eliminating

C and

C , respectively, we can write

+=+==

2224

11 22

0, 0,

CCCC a

ωωω

 

wherefrom

=+ =−+

( ) cos sin , ( ) sin cosCt A t B tCt A t B tωω ωω,

the integration constants

A and

being determined by the initial conditions. In this

case

=+++

==−+ +++ +

( ) cos(1 ) sin(1 ) ,

() () (1 ) sin(1 ) (1 ) cos(1 ),

xt A t B t

yt xt A t B t

ωω

ωωω ω

where

(23.2.87')

Putting the initial conditions

(0) , (0) 0xax== , it results the solution of the equation

and the period, respectively,

=+ =−+ +=

( ) cos(1 ) , ( ) (1 ) sin(1 ) ,

xt a txt a tT

ωωω

 .

(23.2.87'')

Stability and Vibrations

611

23.2.3.3 Non-linear Self-vibrations

In the vibratory motions considered till now one has introduced resistance forces

with a sense opposite to that of the velocity; as a matter of fact, the rôle of these forces

is that of damping the free motion or of limiting the increasing of the amplitudes of the

forced vibrations. In some cases, one can introduce forces with the same sense as that of

the velocity, which amplifies the amplitudes of the vibrations; these forces correspond

to a negative friction and lead to self-vibrations.

Unlike the case in which we have to do with perturbing forces, external to the

discrete mechanical system, which depend explicitly on time and which lead, e.g., to the

phenomenon of resonance, the negative friction depends – in general – only on the

velocity, being internal to the mechanical system.

The self-vibrations can be produced, e.g., by a dry friction. Other self-vibrations can

appear in case of the rotation of an axletree in a bearing, in case of the combined action

of the resistance by driving and the lifting power on an airplane profile or in case of the

fuel injection valves of Diesel motors.

In the above mentioned cases, we are led to differential equations of the form

′′

−+= >0, , , 0mx k x kx m k k 

which have been studied in Sect. 8.2.2.7. The corresponding motions are called

self-sustained motions too.

But it can happen that the coefficient

′

be not constant, depending on x or on x ;

in this case the vibrations are non-linear. Let thus be the equation of Van der Pol type

()

−− +=

0mx c c x x kx 

(23.2.88)

and the equation of Rayleigh type

()

−− +=

0mx k k x x kx  .

(23.2.88')

In case of the first equation, the coefficient

−−

()ccx is negative for small values of

the elongation (

/xcc), obtaining self-vibrations; if the elongations are great

(

/xcc), then this coefficient is positive, appearing the phenomenon of damping

of the vibrations. One can make analogous considerations for the equation of

Van der Pol type, after the magnitude of the velocity (

/xkk or >

/xkk ).

As a matter of fact, the equations (23.2.88) and (23.2.88') are equivalent; indeed, if

we differentiate the equation (23.2.88') with respect to time and if we denote

kc,

3kc and =xy , then we find again the equation (23.2.88).

Starting from the equation (23.2.88) and making the change of variable

/kmtτ = , we can write

ddd d

;

ddd

xx kx

ttm

xx kx

ttm

ττ

== =



