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

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MECHANICAL SYSTEMS, CLASSICAL MODELS

592

To determine the smallest proper pulsation, we start from

000

123

1, 2, 1CCC===; it

results

22 22 22

123

2/, 3/, 2/CpCpCpωωω===. Because 2/1 2, 3/2 1.5==,

we make another iteration, starting from

(1) (1) (1)

123

2, 3, 2CCC===; we obtain thus

22 22 22

123

(7/2) / , 5 / , (7/2) /CpCpCpωω ω===. Comparing with the

previous results, we have

7 /2 3.50, 10/ 3 3.33==, so that, starting from

(2) (2) (2)

12 3

7, 10, 7CC C== =, we make a new iteration and get

22 22 22

123

12 / , 17 / , 12 /CpCpCpωωω===; in this case =12/7 1.714 ,

=17 /10 1.700 , and we can consider that the result is sufficiently exact. Applying the

formula (23.2.48'), we can write, e. g.,

()

=++

2123

CCCC

wherefrom

()

=++

17 12 17 12

;

we obtain

==17 /29 0.766ppω , hence the same result as that given by (23.2.38).

For the greatest pulsation, we start from

00 0

12 3

1, 1, 1CC C==−= and obtain

successively

(1) (1) (1)

22 22 22

12 3

3/, 4/, 3/CpC pCpωωω==−=, =

(2)

10 /Cpω ,

=− =

(2) (2)

22 22

14 / , 10 /CpCpωω,

(3) (3)

22 22

17 / , 24 /CpC pωω==−,

(3)

17 /Cpω ; observing that 17 / 5 3.40, 20/7 3.43==, we can stop to this

operation. Finally, it results that

==41/12 1.848ppω

, hence the name result as

that given by (23.2.38).

Various other methods of calculation have been considered, useful for particular

problems; we mention thus Holzer’s method, which allows writing the characteristic

equation in a general form in the case of a discrete mechanical system with several

masses and springs disposed in series.

23.2.1.11 Vibration of Discrete Mechanical Systems Subjected to Supplementary

Holonomic Constraints

Let be a holonomic and scleronomic mechanical system with s degrees of freedom,

for which the kinetic and the potential energy, expressed by means of the normal

co-ordinates

′′ ′

, ,...,

qq q, are given by (23.2.3'''). We suppose that a supplementary

holonomic and scleronomic constraint of the form

′′ ′

( , ,..., ) 0

fq q q

(23.2.50)

Stability and Vibrations

593

intervenes. Because we have to do with small oscillations around a stable position of

equilibrium, we can remain with the terms of first degree, in a power series expansion,

so that

′′ ′

+++= >

∑

11 22

... 0, 0

qq qαα α α ,

(23.2.50')

where

, 1,2,...,

jsα = , are known constants; the condition imposed to these

coefficients ensures us that they are not all zero, hence that the linear constraint relation

is effective.

The corresponding equations of Lagrange are of the form

′′

++==

0, 1,2,...,

jjj j

qq j sωλα ,

(23.2.51)

where

ω are the proper frequencies of the initial discrete mechanical system, while λ

is a non-determined multiplier of Lagrange. The

s normal co-ordinates and the

parameter

λ are determined by means of the s equations (23.2.51) and of the

constraint relation (23.2.50'). We search a particular solution of the form

′

==cos , cos

qa t tωλ μ ω;

(23.2.51')

replacing in (23.2.50') and (23.2.51), we obtain the relations

()

−+==

∑

0, 1,2,..., ,

jj j

ajs

ωω μα

wherefrom

−

∑

ωω

assuming that

≠ 0μ . This algebraic equation has − 1s roots for

, corresponding to

the

− 1s degrees of freedom of the new mechanical system (after the intervention of

the supplementary holonomic constraint); these roots are situated in the intervals

between the squares of the pulsations

22 2

, ,...,

ωω ω, ordered after their magnitude.

Assuming the existence of

m holonomic and scleronomic linear constraints of the form

(23.2.50'), we can state

Theorem 23.2.3 (Rayleigh). The intervention of m supplementary holonomic and

scleronomic constraints in a holonomic and scleronomic discrete mechanical system,

subjected to small motions around a stable position of equilibrium cannot bring down

the fundamental note, nor can carry it up over the value of the harmonic frequency of

order

+ 1m .

MECHANICAL SYSTEMS, CLASSICAL MODELS

594

23.2.1.12 Symmetry Properties of a Conservative Discrete Mechanical System

We assume, further, that

≤≤≤

...

ωω ω

A and B are the matrices (23.2.9), then the kinetic energy and the potential

energy, respectively, can be expressed in the form

(,), (,)

TA VB==



qq qq

(23.2.53)

where we have used the notation (23.2.10''). We will use now normal co-ordinates,

while in the expression of the kinetic energy we will replace the generalized velocities

by normal co-ordinates; we consider thus the quadratic forms

22 2

22 22 22

11 22

( , ) ... ,

(,) ... .

Aqqq

Bqqqωω ω

′′ ′ ′ ′

=+++

′′ ′ ′ ′

=+++

(23.2.53')

Let be the ratio

(,)

′′

(23.2.54)

It can be calculated for all the values of the normal co-ordinates, which are not

simultaneously zero; obviously,

> 0ρ .

Replacing all the pulsations by

ω (which is the smallest one) or by

ω (which is the

greatest one), we notice that

≤≤

ωρω;

(23.2.54')

as well, making

′′ ′

====

... 0

qq q or

−

′′ ′

=== =

12 1

... 0

qq q , we obtain

min , max

ρω ρω.

(23.2.54'')

If we fix the points

22 2

, ,...,

ωω ω on the real axis and if, at these points, we concentrate

the masses

22 2

1122

, ,...,

mqmq mq

′′ ′

== =, then we notice that ρ is the co-ordinate

of the barycentre corresponding to these masses, the relations (23.2.54') and (23.2.54'')

being thus justified.

Let us suppose now that one introduces the supplementary holonomic and

scleronomic constraint (23.2.50'), which will be denoted shortly

= 0α . One can

always determine values

′′

,qq, which – together with

′′ ′

====

... 0

qq q – do

satisfy the relation (23.2.50'). In this case,

′′

=≤

′′

22 22

11 22

ωω

ρω

;

Stability and Vibrations

595

as well

min for 0ρω α .

If we make vary the constraint relation (23.2.50'), then the above inequalities remain,

further, valid; in particular, for

′

0q ( =====

123

1, ... 0

ααα α) we have

22 22 22

22 33

22 2

...

qq q

ωω ω

′′ ′

+++

′′ ′

+++

wherefrom

′

min for 0qρω .

Taking into account the previous inequalities, we can state that

′

max min for 0qρω .

If one imposes

m holonomic and scleronomic linear constraints

0α = ,

0,..., 0

αα==, of the form (23.2.50'), then one can show that

====−

max min for 0, 1,2,..., , 1,2,..., 1

jmmsρω α ;

(23.2.55)

this is the property of maximinimum of the frequencies of the discrete mechanical

system. Analogously, together with the second formula (23.2.54''), one can show that

−

====−

min max for 0, 1,2,..., , 1,2,..., 1

jmmsρω α ;

(23.2.55')

this is the property of minimaximum of the frequencies of the same discrete mechanical

system. We have thus put in evidence the properties of extremum of the proper

frequencies of a conservative discrete mechanical system.

Let be now also a second conservative discrete mechanical system, specified by the

quadratic forms

′′

qq(, )A



and

′′

qq(, )B



with the principal frequencies

<<<

...

ωω ω

 

. In this case

max min for 0, 1,2,..., , 1,2,..., 1

jmmsρω α

====−



where we have denoted

(,)

′′



Let us suppose that the new system has a greater rigidity, with the same inertia,

hence that

MECHANICAL SYSTEMS, CLASSICAL MODELS

596

(,) (,),(,) (,)AABB

′′ ′′ ′′ ′′

=≥



qq qq qq qq ,

(23.2.56)

or that the new system has a smaller inertia, with the same rigidity, so that

(,) (,), (,) (,)AABB

′′ ′′ ′′ ′′

≤=



qq qq qq qq .

(23.2.56')

In both cases, we have

≤ρρ



for non-zero normal co-ordinates. In this case, both the

minima and maxima of these ratio are linked by the same inequality; in other words –

taking into account the properties of extremum found above – it results that

≤=, 1,2,...,

jmωω



(23.2.56'')

where we have a strict inequality, at least for one of the indices. We can thus state

Theorem 23.2.3' (Rayleigh). The frequencies of a holonomic and scleronomic discrete

mechanical system increase if its rigidity increases or if its inertia decreases.

This theorem, enounced in 1873, is another form of the Theorem 23.2.3 (the

supplementary constraints increase the rigidity of the mechanical system) (Rayleigh,

lord, 1899, 1900).

23.2.1.13 Free Damped Small Oscillations of a Discrete Mechanical System

We consider the case of oscillations with a damping in direct proportion to the

velocity (viscous damping); the matric equation of Lagrange is completed in the form

′

++=

 

Mx K x Kx ,

(23.2.57)

where

′′ ′

⎡

⎤

⎢

⎥

′′ ′

⎢

⎥

′

⎢

⎥

⎢

⎥

′′ ′

⎢

⎥

⎣

⎦

11 12 1

21 22 2

...

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

...

kk k

(23.2.57')

is the matrix of the damping coefficients.

In general,

, , , 1,2,...,

ij ji

kkijij n

′′

≠≠ = , the matrix K being asymmetric; if

ij ji

kkij

′′

=≠, then the matrix K is symmetric and one can introduce the quadratic

form

′

= xKx



(23.2.58)

so that the damping forces be given by

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597

∂

Φ=− =

∂

, 1,2,...,

x

(23.2.58')

The quadratic form

R is called dissipative energy (Rayleigh’s dissipative function).

We search solutions of the form

=xAe

tλ

, where A is a column matrix of

components

=, 1,2,...,

Ai n; introducing in the equation (23.2.57), we obtain

()

′

++ =

MKKA

λλ 0 .

(23.2.59)

We can write, in a developed form,

()

′

++ ==

∑

0, 1,2,...,

ij ij ij j

mkkAi nλλ ;

(23.2.59')

imposing the condition to have non-trivial solutions

=,1,2,...,

Aj m

, we get the

characteristic equation

′

++=

⎡⎤

⎣⎦

det 0

ij ij ij

mkkλλ

(23.2.59'')

For the damped vibrations, one can make a study analogue to that in Sect. 8.2.1.3, after

the position of the roots of this equation in the complex plane. As well, one can make

also a study of the self-sustained vibrations (called self-vibrations too), as in Sect.

8.2.1.4.

23.2.2 Small Forced Oscillations

If the small oscillations are influenced by the motion of perturbing forces, then they

become forced oscillations. We consider, in what follows, various types of such forces:

periodic forces, forces which do not depend explicitly on time, dissipative forces etc.

As well, we make distinction between undamped and damped small oscillations.

23.2.2.1 Action of Periodic Perturbing Forces on the Small Oscillations of a

Conservative Discrete Mechanical System

Let be a holonomic and scleronomic discrete mechanical system, acted upon – in

general – by viscous damped forces and by periodic perturbing forces. Lagrange’s

matric equation (23.2.57) will be completed in the form

′

++=

Mx K x Kx F cos tω

 

(23.2.60)

where

F is a column matrix, the elements of which are

, ,...,

FF F; the pulsation of

this perturbing force is

ω . The complete solution of the matric equation (23.2.60) is

formed from the solution of the homogeneous equation (a damped proper vibration)

over which is superposed a forced harmonic vibration (a particular solution of the

complete equation).

This latter vibration is obtained as a real part of the solution of the complex equation

MECHANICAL SYSTEMS, CLASSICAL MODELS

598

′

++=

Mz K z Kz Fe

tω

 

;

(23.2.61)

we search the particular solution of this equation in the form

=zA

tω

(23.2.61')

Calculating

i, ωω==−



zzz z, we obtain

()

′

−++=

MKKzF

tω

ωω ,

wherefrom

()

−

′

=− + +

zM KKF

tω

ωω .

(23.2.61'')

In the absence of the damping force (we take

′

K 0 ), we notice that the bracket is

real, while

Re e cos

ω= ; the forced vibration is given by

()

−

=− +xMKF

cos tωω.

(23.2.61''')

If the mechanical system has

s degrees of freedom, then we are in the space of

configurations

Λ ; the passing from the column vector q of the generalized

co-ordinates to the vector column

′

q or the normal co-ordinates is made by the formula

(23.2.23), written in the form

[

]

′

== ≠

qSqS S,,det0

s .

(23.2.23')

Analogously, we will try to pass from the column vector

Q of the generalized forces,

expressed by means of the generalized co-ordinates, to the corresponding column

vector

′

Q , expressed by means of the normal co-ordinates; to do this, let be the virtual

work

′′

δ=δ

ii ii

Qq Qq.

But

, 1,2,...,

iijj

qsqi s

′

δ=δ = , so that

′′′

δ=δ

iij j j j

Qs q Q q ; the virtual displacements

being independent (the constraints of the mechanical system are holonomic), it results

jiij

QQs

′

QSQ

;

(23.2.62)

one can thus write

()

−

′

QS Q

;

(23.2.62')

where the generalized forces have a contravariant transformation with respect to the

generalized co-ordinates.

Stability and Vibrations

599

S is an orthogonal matrix, then

−

=SS

() and the generalized forces are

transformed as the generalized co-ordinates.

Finally, we can write the system of Lagrange’s equations in normal co-ordinates in

the form

′′′

+= =

( ), 1,2,...,

jjj j

qqQtj sω ,

(23.2.63)

the general solution being given by

()

∗

′

=−+=( ) cos ( ), 1,2,...,

jjjjj

qt a t qt j sωϕ ,

(23.2.63')

where

∗

()

is a particular solution of the complete equation (23.2.63). Assuming that

′

()

Qt is a periodic function of time, which can be expanded into a Fourier series, we

choose a term of this series, so that

′

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

Qt A tj sω ,

(23.2.64)

of period

= 2/T πω; we obtain easily

∗

−

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

qt tj sω

ωω

(23.2.65)

Analogously, the expansion into a Fourier series

()

∞

′

=−=

∑

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

jjn jn

Qt A nt j sωϕ

(23.2.64')

leads to

()

∞

∗

=−=

−

∑

222

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

jjn

qt nt j s

ωϕ

ωω

(23.2.65')

nωω, while

≠ 0

, then takes place the phenomenon of resonance for the

respective normal co-ordinate.

=a , 1,2,...,

js, are the amplitude vectors of components =, , 1,2,...,

aij s,

then we can write

∗

=+qq q

(23.2.66)

where the free oscillations are given by

()

=−

∑

cos

jjj

Ctωϕ

(23.2.66')

MECHANICAL SYSTEMS, CLASSICAL MODELS

600

and the forced oscillations by

∗∗

∑

q .

(23.2.66'')

23.2.2.2 Small Oscillations of a Holonomic and Scleronomic Discrete Mechanical

System Subjected to the Action of Forces Which Do Not Depend Explicitly

on Time

If the holonomic and scleronomic discrete mechanical system is acted upon by forces

which depend only on the generalized co-ordinates and on the generalized velocities,

then Lagrange’s equations will be of the form

()

∂∂

⎛⎞

−= =

⎜⎟

∂∂

⎝⎠

12 12

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

Qqq qqq q j s

tq q

 



(23.2.67)

where the kinetic energy is a positive definite quadratic form (23.2.3'). If we expand the

generalized forces into a Maclaurin series around the origin, considered to be a stable

position of equilibrium, then we get

∂∂

⎛⎞ ⎛⎞

=+ + =

⎜⎟ ⎜⎟

∂∂

⎝⎠ ⎝⎠

, 1,2,...,

QQ q qj s



where we neglect the terms of higher order, because we assume that the oscillations are

small; we take

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

QQjs== = , the origin being a position of

equilibrium, and denote

∂∂ =−

(/)

iij

Qq b,

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

ji ij

Qq cij s∂∂ =− = , so

that

=− − =, 1,2,...,

jijiiji

Qbqcqj s .

(23.2.68)

The system of equations (23.2.67) takes the form

++==0, 1,2,...,

ij j ij j ij j

aq cq bq i s 

(23.2.67')

analogue to that in case of small damped free oscillations (see Sect. 23.2.1.13).

Introducing the matrices (23.2.9), the matrix

⎡

⎤

⎢

⎥

⎢

⎥

⎢

⎥

⎢

⎥

⎣

⎦

11 12 1

21 22 2

...

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

...

cc c

(23.2.9'')

and the column vectors (23.2.9'), we can write Lagrange’s system of equations in the

matric form

++=Aq Cq Bq 0

 

(23.2.67'')

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601

We search a solution of the form

=qae

tλ

(23.2.68)

where

a is a column amplitude vector, while λ is a scalar; introducing in the equation

(23.2.67''), we obtain the equation which determines the amplitude vector

()

++ =ACBa

λλ 0 ,

(23.2.69)

or, in a developed form,

()

++ ==

0, 1,2,...,

ij ij ij ij

acba i sλλ .

(23.2.69')

These equations have non-trivial solutions if and only if the determinant of the

coefficients vanishes

++=

⎡⎤

⎣⎦

ACB

det 0λλ ;

(23.2.70)

developing, we can write

++ ++ ++

22 2

11 11 11 12 12 12 1 1 1

22 2

21 21 21 22 22 22 2 2 2

22 2

111222

...

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

...

sss

ss ss ss

ssssss

acbacb acb

λλ λλ λλ

(23.2.70')

obtaining an algebraic equation of degree

2s in λ (the secular equation).

Let us suppose that all the roots

12 2

, ,...,

λλ λ of this equation are distinct. To each

root

λ will correspond non-zero amplitude a

, hence a particular solution e

a of

the equation (23.2.67'); the general solution is obtained in the form

∑

(23.2.68')

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

ksλ <= , then the position of equilibrium is asymptotically stable,

not only for the linearized system but also for the non-linear initial system.

In the case of a conservative discrete mechanical system we have

=C 0 , the

matrices

and

being positive definite. The secular equation +=AB

det[ ] 0λ

may be written also in the from

−BAdet[ ]μ , where μ is a positive real root; we

notice that, in this case

=±iλμ, so that the equation (23.2.70') has purely imaginary

roots in case of a conservative holonomic and scleronomic discrete mechanical system.

23.2.2.3 Small Oscillations of a Holonomic and Scleronomic Discrete Mechanical

System Subjected to the Action of Forces Which Depend Explicitly on Time

Unlike the case considered at the preceding subsection, we will suppose now that the

generalized forces depend explicitly on time. The system of equations (23.2.67') is

completed in the form