Teodorescu P.P. Mechanical Systems, Classical Models Volume II: Mechanics of Discrete and Continuous Systems

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MECHANICAL SYSTEMS, CLASSICAL MODELS

in one of the following cases: (i) the pole of the non-inertial frame has a uniform and

rectilinear motion with respect to an inertial frame, the movable frame being inertial

too; (ii) the non-inertial frame is a Koenig frame (

=v0

and

const=

JJJJG

, hence one

can have

OC≡

, Theorem 11.2.5); (iii) the projection of the velocity of the centre of

mass with respect to an inertial frame, on the direction of the acceleration of the pole of

a non-inertial frame is equal to the projection of the velocity of this pole on the same

direction, as it was shown by O. Bonnet. As a matter of fact, the case (ii) is a particular

case of the case (iii).

As well, from (11.2.28) we obtain a result of O. Bonnet, in conformity to which a

non-inertial frame of reference which is not rotating is a frame of Koenig type for the

kinetic energy if

()0

OC O C O

′′′′

⋅=⋅ − =vv v v v

, hence if the projection of the

velocity of the mass centre with respect to an inertial frame on the direction of the

velocity of the pole of a non-inertial frame is equal to the magnitude of the velocity of

that pole.

If the pole of the non-inertial frame of reference coincides with the mass centre of

the discrete mechanical system (

OC≡

, hence

= 0

), then we obtain a formula of

Koenig type

() 2

TT Mv

′′

(11.2.37)

and the theorem of kinetic energy reads

()

() () ()

()

int

()

CCC

PPPP

=+++++⋅MMω

;

(11.2.37')

if we have

= 0ω too, then the non-inertial frame R is a Koenig frame, so that

() 2

TT Mv

′′

=+ ,

()

() () ()

()

int

CCC

PPPP

∂

=+++

∂

(11.2.38)

From (11.2.28'') we notice that the first formula (11.2.38) of Koenig can take place also

for

≠ 0ω if

() ()

CC CC

⋅−⋅ =⋅+⋅ =KIKIωωωωωω;

(11.2.39)

we can also write

ω =− K

(11.2.39')

where

I is the moment of inertia of the discrete mechanical system with respect to the

instantaneous axis of rotation of the non-inertial frame of reference with the pole at the

centre of mass. We find thus again a result of V. Vâlcovici, according to which the first

formula (11.2.38) of Koenig takes place only and only if the angular velocity vector of

the non-inertial frame with respect to the inertial one is directed in an opposite sense to

that of the projection of the moment of momentum of the discrete mechanical system on

the instantaneous axis of rotation, having a magnitude equal to the double of the

magnitude of this projection divided by the moment of inertia of the system with

respect to the instantaneous axis.

As well, using the condition (11.2.35), we notice that the second formula (11.2.38)

of Koenig takes place for

0ω ≠ too if

∂

⎛⎞

⋅=⋅

⎜⎟

∂

⎝⎠



Kωωω

(11.2.40)

Equating to zero the relative motion with respect to the non-inertial frame of

reference, we find – from (11.2.28), (11.2.28'') – the transportation kinetic energy

(tr )TO

′

of the discrete mechanical system S, by the non-inertial frame R, with

respect to the inertial frame

′

R (we make

=v0 in the formula (11.2.10'), so that

(tr )

′′

=+×vvrω ). To calculate the transportation kinetic energy (tr )TO

′

the discrete mechanical system

S by the inertial frame

′

R , with respect to the non-

inertial frame

R, we introduce the kinematic torsor

{

}

′′

≡− vωT ; observing that

OO O O

′

∂

′′ ′ ′

=+×=−+×=−+×

∂∂

vr rvrωωω

we have

(tr ) ( ) ( ) (tr )

iiii

′

′′

=+−×+=−−×=−vv rrvrvωω

Thus, C. Iacob showed that

(tr ) (tr ) ( , , ) ( )

OO O

TOTO Mv M

′′′′

==+ +⋅vIω

ωω.

(11.2.41)

With this result, the relations (11.2.28), (11.2.28'') may be written also in the form

(tr )

OC O

TTTOM

′′ ′

=+ + ⋅ +⋅vv Kω ,

(11.2.42)

(tr )

OC O

TTTOM

′′ ′′′

=− + ⋅ +⋅vv Kω ,

(11.2.42')

where we have invert the rôle of the frames

′

R and R, taking into account the

previous results and the relation

()

OO C

′

′′′ ′

=+×KKr v

11 Dynamics of Discrete Mechanical Systems

Thus, as it has been stated by V. Vâlcovici, the theorem of kinetic energy remains

invariant in its form with respect to a non-inertial frame of reference with the pole at the

mass centre if and only if the scalar product of the angular acceleration vector of the non-

inertial frame with respect to an inertial one by the moment of momentum of the

discrete mechanical system, relative to the non-inertial frame, is equal to the half of

the scalar product of the angular velocity vector of the same frame with respect to the

inertial one by the contracted product of the derivative with respect to time of the tensor

of inertia with respect to the centre of mass of the system, in the non-inertial frame, by

the considered angular velocity vector.

MECHANICAL SYSTEMS, CLASSICAL MODELS

Summing the relations (11.2.42), (11.2.42'), observing that

M=Hv,

′′

=Hv,

introducing the kinematic torsor (11.2.33) and the kinetic torsors

{

}

τ=HK ,

{

}

′′′

τ=HK

, we get

()

ττ

′′

=+ +

′

Let be also the kinematic torsor

{

}

′

=−

′

vωT , which corresponds to the motion of

the inertial frame of reference with respect to the non-inertial one; taking into account

the relation between

′

v and

′

and the relation ()

′′

=+×KKr v, we

obtain the reciprocity relation

()

ττ

′′

−=−

′

(11.2.43)

due to C. Iacob.

In particular, if

= 0ω , then we have

(tr ) (tr )

TOTO Mv

′′′

==.

(11.2.41')

From (11.2.36') we notice that for

= 0ω and 0

′

⋅=av

, hence for the conditions

found by O. Bonnet in case of scleronomic constraints (for which

int

PP==

assuming that the given internal forces derive from a simple or from a generalized

potential, we can write a theorem of mechanical energy of a discrete mechanical

system, free or with scleronomic constraints, with respect to a non-inertial frame which

does not rotate with respect to an inertial one, in the form

∂

, ETV=+,

(11.2.44)

where

V is the potential energy of the system with respect to the pole O of the movable

frame. If

0P = , in a certain interval of time, we get constE = for this interval,

hence a conservation theorem of mechanical energy (a scalar first integral), which is not

independent from that which could be obtained with respect to an inertial frame.

11.2.2.3 Problem of n Particles

We will consider now the problem of

n particles with respect to a non-inertial frame of

reference

R, which does not rotate with respect to an inertial frame

′

R , but which is

not a Koenig frame (see Sects. 11.1.2.8 and 11.2.1.3 too). If the origin

O of the non-

inertial frame coincides with one of the particles, e.g., with the particle

P , of position

vector

′

, with respect to the frame

′

R , we can write

′′

=+rrr

, 2,3,...,in= ,

=r0. The equations of relative motion with respect to the frame R are

()

∑



rFF

()i

′

=−



Fr,

2,3,...,in=

;

noting that

3333

11 1 2

'' '

nn n n

ik k i ik

ik k k k

kk k k

ik ik

fm fm fm fm

rrr

== = =

−

==− =−+

−

∑∑ ∑ ∑

rrr r

333

22 2 2

nn n n

kkk

kk k k

fm fm fm

== = =

−

==− =

−

∑∑ ∑ ∑

rrrr

fm m

⎛⎞

⎜⎟

⎝⎠

∑

we can write these equations in the form (the operator

∇ is considered with respect to

the non-inertial frame)

()

iiii

mm R

++ =



r ∇ ,

Rf m

⋅

⎛⎞

=−

⎜⎟

⎝⎠

∑

2,3,..,in=

(11.2.45)

In the particular case

2n = , it results 0

R = (we have only 2i = ); the equation

(11.2.45) is reduced to an equation of the form (11.1.51), hence to the equation of

motion corresponding to the problem of two particles. Thus, the influence of the other

particles upon the motion of the particle

P is given by the perturbing function

R .

These equations are used in the study of the motion of the planets with respect to the

Sun, in the frame of the solar system. In this case, the Sun is considered to be the

particle

P (the centre of mass of the Sun is chosen as origin O of the frame R ),

having the mass

m , which does not intervene in

R ; we may thus state that the values

of the perturbing function are very small.

Another possibility to study the problem, due to Jacobi, is based on the introduction

for the discrete mechanical subsystems

{

}

, ,...,

PP P≡S , 1,2,..., 1in=−, of

1n − non-inertial frames

R , with the poles at the mass centres

C of position

vectors

ijj

′′

∑

rρ ,

∑

, 1,2,..., 1in=−,

(11.2.46)

with respect to the inertial frame

′

R , and which do not rotate about the latter frame;

the motion of the particle

P is thus considered with respect to the inertial frame

′

R ,

while the motion of the particle

is considered with respect to the non-inertial

frame

R , having the position vector

1i +

r with respect to that frame. Observing that

′′

=−rr

, it results

11 Dynamics of Discrete Mechanical Systems

MECHANICAL SYSTEMS, CLASSICAL MODELS

ii jj

MM M

′′

=−

∑

rr r;

(11.2.46')

taking into account (11.1.48'), we can write the equation of motion with respect to the

frame

R in the form

MUU

′′

=−

∑



r ∇∇

, 1,2,..., 1in=−,

(11.2.47)

the operator

()

′′ ′

=∂∂i∇ being taken with respect to the inertial frame

′

R . By

means of the relation (11.2.46') and using the notation (11.2.46), we can express the

operators

′

∇

, taken with respect to the frame

′

R , by the operators

1i +

∇

, taken with

respect to the frames

R ; we obtain thus

−

′

=−

∑

∇∇,

111

iii

−

+++

′

=−

∑

∇∇ ∇,

1,2,..., 2in=−

′

=∇∇

(11.2.46'')

Replacing in (11.2.47) and observing that

121

iiin n

jj i

jjjkj ki

−−

==== =

′

=− =−

∑∑∑∑ ∑

∇∇ ∇ ∇

(11.2.46''')

we can write the equation of motion (11.2.47) of the particle

with respect to the

frame

R in the form

11 1ii i

++ +



r ∇ , 1,2,..., 1in=−,

(11.2.47')

where we have introduced the notation

111

(/ )

iii

mMMm

+++

= and where

()()

′′

−= −rr rr

. To can calculate the last form

of the potential, we start

from the relation (11.2.46), which leads to

11 22 1 2 1 1

() ( )

ii ii i i i i

MM M M

−− −− − − − −

′′

−=−− −rr r r

ρρ

11 1 2 1

()

ii i i i

mMM

−− − − −

′′

==−rr

wherefrom

−

−−

−

′′

−=−rr r r

;

(11.2.48)

writing these relations for successive values of the index

i and summing for all the

indices, we obtain the relation

ij ij

−

′′

−=−+

∑

rr rr r,

(11.2.48')

useful for the goal had in view.

Introducing the canonical co-ordinates (the generalized co-ordinates

()i

qx=

and

the generalized momenta

()i

kkk

pmx mq==, 3( 1)ki j=−+, 1,2,..., 1in=−,

1, 2, 3j =

), we can write the equations of motion (11.2.47') in the canonical form

∂

 ,

∂

=−

∂

 , 1,2,...,3( 1)kn=−,

(11.2.49)

where we have introduced Hamilton’s function

3( 1) 3( 1)

11 1

kkk

HpUpqU

−−

=−=−

∑∑



(11.2.49')

These equations are particularly useful for the study of the motion of satellites if

these ones are influenced by only one celestial body, excepting that one with respect to

which one considers the motion (e.g., the study of the motion of the Moon, taking into

account the influence of the Sun and of the Earth).

11 Dynamics of Discrete Mechanical Systems

Chapter 12

Dynamics of Continuous Mechanical Systems

The motion of a continuous mechanical system with respect to an inertial (Galilean)

frame of reference is studied in this chapter, considering free and constraint systems;

one passes then to the case of a non-inertial (non-Galilean) frame of reference. In

particular, some one-dimensional mechanical systems are dealt with.

12.1 General Considerations

A continuous mechanical system (continuous medium, continuous body, continuous

material)

S represents the mathematical model of a body which is entirely immersed in

a domain

D of the space

E ; this domain is the geometric support Ω of the considered

system. After some notions with introductory character and after presenting the general

principles which allow to set up the mathematical model of such a mechanical system,

one passes to the corresponding general theorems and to the conservation ones.

12.1.1 Introductory Notions. General Principles

To pass from a discrete mechanical system to a continuous one represents – in fact – to

pass from a mechanical system with a finite number of degrees of freedom to a

mechanical system with an infinite number of degrees of freedom. The mathematical

model used in the first case must be completed and Newton’s principles must be

consequently adapted.

12.1.1.1 Introductory Notions

As it was shown in Chap. 1, Sect. 1.1.8, a continuous mechanical system constitutes a

mathematical model formed by a domain

D to which is associated a mass; the

quantities in connection with such a system are, in general, represented by continuous

functions. In computations, we will consider a point

P of a continuous mechanical

system

S ; but that one is not a particle in the sense considered till now, having not a

finite mass (however, we will use also the denomination of particle of the continuous

mechanical system). The topology of the Euclidean domain

D is the topology of the

mechanical system

S too, while the distance between two points of that system is the

Euclidean distance between their positions at the same moment t; these positions will be

considered, in general, with respect to an inertial frame of reference. If the distances

between all pairs of points of the continuous mechanical system

S remain invariable in

P.P. Teodorescu, Mechanical Systems, Classical Models,

time, then we have to do with a rigid solid; otherwise, this mechanical system is

deformable (deformable continuous medium). In Chap. 1, Sect. 1.1.10 has been made a

classification of such media, i.e.: deformable solids and fluids (liquids, gases, plasma);

as well, the properties of elasticity, plasticity and viscosity of those mechanical systems

have been put in evidence.

Under the action of external charges (which can be: concentrated or distributed

forces, other concentrated charges (applied on the external surface of the body, hence

on the frontier of the domain in which is immersed the body or on its interior), volumic

(or massic) forces or moments, forces of inertia, charges produced by a thermic or by an

electromagnetic field, charges produced by radioactive radiations, deformations

provoked by various causes, imposed displacements etc.), the particles (infinitesimal

elements) which form a solid body change (eventually, in time) the position (with

respect to an inertial frame of reference, considered fixed) which they had before the

action of those charges. If, after a translation and a rotation, all the particles of the body,

subjected to the action of charges, have the same mutual positions as before the

application of those charges, then we say that we have to do with a rigid solid motion;

otherwise, the body is subjected to a deformation. The totality of the deformations of a

particle of the body forms the state of deformation at a point (the point is the geometric

support of the considered particle). The totality of the states of deformation

corresponding to all points (particles) of the solid body constitutes the state of

deformation of the body. Together with the notion of deformation, the notion of

displacement is put in evidence too. The totality of the displacements corresponding to

all the points of the solid body constitutes the state of displacement of the body.

Corresponding to what was specified before, the bodies which allow only

displacements of rigid body are called rigid solids; the other solid bodies are

deformable solids. Due to deformations, the (static or dynamic) equilibrium of the

constraint forces which act between the particles of the body does no more hold, so that

supplementary internal forces arise; the totality of those internal forces (called efforts, if

they act upon an arbitrary section of the body, or stresses, if they correspond to the

efforts acting on a unit area), which correspond to a particle, form the state of stress at a

point (geometric support of the considered particle). The totality of the states of stress

corresponding to all the points (particles) of the solid body forms the state of stress of

the body.

In case of a fluid (which changes much its form under the action of the external

causes), the deformation at a point is replaced by the velocity of deformation at that

point, while the displacement of a point of the mechanical system is replaced by its

velocity; thus, we have to do with a state of velocity of deformation and with a state of

velocity, respectively.

The mathematical model of continuous deformable media (solids or fluids) must be

completed by a constitutive law (of theoretical and experimental nature), which

represents a relation between the state of deformation (of velocity of deformation) and

the state of stress of the respective continuous medium. The results of theoretical nature

(the geometrical-kinematical and mechanical aspects) mentioned above, valid for an

arbitrary continuous deformable mechanical system, are thus specified for a certain

continuous deformable medium.

MECHANICAL SYSTEMS, CLASSICAL MODELS

In general, the motion of a particle

P of the continuous mechanical system S is

given by an equation of the form

()

;t=rrP ,

(12.1.1)

which specifies thus the position of that particle at any moment

[

]

,ttt∈=T , with

respect to an inertial frame of reference, considered fixed. The velocity and the

acceleration of the particle are thus defined by the relations

()

;



vr r

()

;

===



avr rP .

(12.1.1')

The totality of motions of each particle defines the motion of the continuous mechanical

system

S. In this case, to the continuous deformable medium corresponds a domain

()t=DD , by the mapping

()

;tr S , that is the geometric support Ω of the medium at

a moment t. This domain represents the configuration of the continuous mechanical

system

S at the respective moment; hence, the motion of the mechanical system S is a

succession of configurations. The configuration in which the particles

P of the

continuous mechanical system

S are identified with their positions (the locations

occupied by them, specified by the position vectors

r ) is called reference

configuration. In case of a deformable solid, this configuration is called the non-

deformed configuration (state), while in case of a fluid it will be the initial

configuration (state)

D , corresponding to a moment

tt= ; the configuration at an

arbitrary moment t, specified by the point P of position vector r, is called actual

configuration (state). Corresponding to Chap. 1, Sect. 1.1.9, the motion of the particle

P is described by the equation

()

;t=rrr ,

()

000

123

,,;

xxxxxt=

1,2, 3i =

(12.1.2)

defined on

×DT. The vector r travels through the configuration D if the vector

travels through the configuration

D ; we may write

()

;t=DDD ,

(12.1.2')

defining thus a mapping of the space

E in itself. The velocity and the acceleration at a

certain moment

t are given by (the position vector

r is considered to be constant)

()

;



vr rr

()

;

===



avr rr

(12.1.2'')

The co-ordinates

x , 1,2, 3i = , are called material co-ordinates (identifying the

particle

P by the point

P of position vector

r , one follows the motion of that one in

12 Dynamics of Continuous Mechanical Systems

MECHANICAL SYSTEMS, CLASSICAL MODELS

time), corresponding to a material (Lagrangian) description of the motion; they are

called Lagrangian co-ordinates too (although they have been introduced by Euler) or

reference co-ordinates (corresponding to a reference configuration). In this case,

x ,

1,2, 3i =

, are the unknown functions of the problem. If the displacements are great

(e.g., in case of a fluid), it is convenient to choose as independent variables the co-

ordinates

x ,

1,2, 3i =

, which will be called spatial co-ordinates (of the place P in

space, through which passes a particle

P at the moment t), corresponding to a spatial

(Eulerian) description of the motion; they are called Eulerian co-ordinates too (in fact,

they have been introduced by d’Alembert). In this case,

123

(,, ;)

vxxxt,

1,2, 3i =

are the unknown functions of the problem. We assume that the Jacobian

J, defined by

the relation (1.1.75'), is non-zero in

D (eventually, excepting some singular points,

lines or surfaces), so that – from the relation between the elements of volume

(

ddVJV= ) – we deduce that the geometric support Ω cannot vanish or become

infinite; to the axiom of continuity (the particles preserve their individuality) correspond

thus the indestructibility and the impenetrability of the matter, respectively. From a

mathematical point of view, the mapping (12.1.2) is a bijection between

D and D.

The theorem of implicit functions allows to write

()

;t=rrr,

()

123

,,;

xxxxxt= , 1, 2,3i = ,

(12.1.2''')

×DT; these functions are univocally determinate (in particular,

()

0 000

123

,,;

xxxxxt= , 1, 2, 3i = ) if (necessary and sufficient condition), at least in a

neighbourhood of the considered point

P , the functions (12.1.2) are of class

()

C D

(as a matter of fact, in this case also the functions (12.1.2''') are of class

()C D ). If the

functions (12.1.2) are of class

()

C D

, then the points which are initially

neighbouring remain neighbouring also at the moment

t; in this case, the functions

(12.1.2''') are of a class

()C D

too.

In a material description (convenient in case of deformable solids) we can express

the equation (12.1.2) in the form

(;)t=+rr ur

(12.1.3)

×DT, where u is the displacement vector of the particle P from the non-

deformed configuration

D to the deformed configuration D (Fig. 12.1); the velocity

and the acceleration of that particle will be expressed in the form

∂

== =

∂



vr u

∂

== =

∂



ar u

(12.1.3')

the vector

r being constant with respect to time. In general, in case of a (scalar or

vector) field

(;)tΦ r or

(;)trΨ , respectively, defined on S, the derivative with