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

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MECHANICAL SYSTEMS, CLASSICAL MODELS

same pole by the angular velocity vector of the non-inertial frame

R with respect to

the inertial frame

′

R ), the moment of momentum of the mass centre, translated at the

pole

O, at which is considered to be concentrated the whole mass of the system S, with

respect to the pole

′

, in the frame

′

R , and the moment of momentum of the pole O,

translated at the centre of mass at which is assumed to be concentrated the whole mass

of the system

S, calculated with respect to the pole O, in the same inertial frame

′

R .

Differentiating the relation (11.2.16) with respect to time in the fixed frame, we can

write

[]

()

OCOC CO O O

′

′′′′ ′ ′ ′

=+ ×+×+×+××+×

vvra vv v aωρ ρ

;

observing that

′

=+×MMrR

′

=+×MMrR

for the moments of the given and constraint external forces and taking into account

(11.2.14), (11.2.15), V. Vâlcovici showed that the theorem of moment of momentum

becomes

()

OOO

′

×+=+

aMMρ ,

(11.2.18)

where

() ()

ddd

OO O O

OOOO

ttt t t

∂∂

=+ =+×+ ++×

∂∂



KK K I

IKIIωω ωωωω

(11.2.18')

Observing that

() ()2()()()

iiiiii iiii

×× +×× = ⋅ −⋅ −⋅rvvrrv vrrvωω ωωω

()()

() () () () () () () () () ()

ii ii ii ii ii

kkjkk

jj j

ll k k ll k

xx xx xx xx xx

δωδω

∂

=−− = −

∂



multiplying by

m , summing for all the particles of the discrete mechanical system S

and introducing the tensor of inertia, we find the remarkable relation

() ()

ii i i i i

∂

×× + ×× =

∂

∑∑

rv vrωωω

(11.2.18'')

corresponding to the relation (11.2.17).

Starting from the equations of motion (11.2.13), (11.2.13'), we effect a vector

product at the left by

r and sum for all the particles of the system S ; we get thus

()

() ()

∂

=++ × +

∂

∑

MM rF F

(11.2.18''')

We notice that

() ( ) ( )

ii i

OO O

′

+× +× = × ×

∑



II a rr

ωω ω ρ ω

()

111

()

nnn

ii i ii i

iii

===

′

+ × × × + × =− ×

∑∑∑

rr ra rFωω ,

because

[

]

[

]

()()() ()

ii iii i

××× =−⋅ ×=×××rr rrr rωω ωω ωω,

as well as (we use the relation of definition (3.1.81))

() ()

1 1

n n

ii i j i j

Okjkllmnk

i i

m mxv

ω ω

= =

∂

×+ =× ×+ =∈∈

∂∂

∑ ∑

Krvi iωωω

()

() () () () () ()

1 1

n n

ii ii ii

pp mn

ijjmjnij

kjk knkmk

i i

mxx xx mxv

ωδ δδδδω

= =

∂

+−=−

∂

∑∑

()()

() () () () () () () () () ()

ii i i ii ii ii

pp pp

ijijj

kjk k

jj j

kk k

m xv vx xv mxv xvωδ ωω

+ −−=−

∑∑

[]

()

111

2()()2 ()

nnn

iii i i ii i i

iii

===

=⋅−⋅=××=−×

∑∑∑

rv r v r v r Fωω ω ;

one can thus state that the relations (11.2.18) and (11.2.18''') are equivalent.

One can easily see that the theorem of moment of momentum (11.2.18) reads

()

OOO

′

×+=+

aMMρ

(11.2.18

)

if we have

const

JJJJG

KIω

with respect to the inertial frame

′

R .

In the particular case in which the frame

R does not rotate ( = 0ω , hence

=KK), it results the formula

() ()

OO C O

′

′′′ ′

=+× +×KKr v v

(11.2.19)

obtained by V. Vâlcovici as a particular case of the Theorem 11.2.7; the theorem of

moment of momentum becomes

()

OOO

∂

′

×+=+

∂

aMMρ

(11.2.19')

and we can state

11 Dynamics of Discrete Mechanical Systems

MECHANICAL SYSTEMS, CLASSICAL MODELS

Theorem 11.2.8 (V. Vâlcovici). The theorem of moment of momentum of a free discrete

mechanical system maintains its form with respect to a non-inertial frame of reference

which does not rotate (it moves with the axes parallel to themselves) if, from the

moment of the given and constraint external forces, we subtract the dynamic moment

(with respect to an inertial frame) of the pole of the non-inertial considered frame,

translated at the centre of mass of the system, at which is assumed that is concentrated

the whole mass of the system, with respect to this pole.

One can thus see that the theorem of moment of momentum of a free or constraint

discrete mechanical system maintains its form with respect to a non-inertial frame of

reference which does not rotate if and only if

′

×=a0

, hence 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 thus inertial too;

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

= 0

, hence

OC≡

, Theorem 11.2.4);

(iii) the support of the acceleration

′

of the pole of the non-inertial frame, with

respect to an inertial frame, passes always through the centre of mass

C. As a matter of

fact, the case (ii) is a particular case of (iii).

We say that a non-inertial frame

R is a frame of Koenig type if = 0ω but its pole

is not the mass centre; in this case, the frame

R is just such a frame. As it was shown

by V. Vâlcovici, to can write a formula of Koenig type

()

OO O

′

′′′

=+×KKr v

(11.2.20)

it is necessary and sufficient that (we notice

COC

′′

=+vvv

)

() ()

OC O

′′

×+×=rv v0

(11.2.20')

In the particular case in which the pole of the non-inertial frame coincides with the

centre of mass of the system (

OC≡ , = 0

), we obtain a formula of Koenig type

()

′

′′′

=+×KK vρ

=+KKK,

(11.2.21)

and the theorem of moment of momentum reads

;

(11.2.21')

if we have

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

()

′

′′′

=+×KK v

∂

(11.2.22)

Taking into account the results in Sect. 11.1.2.3, we notice that we may write

()

OOC

′

′′′ ′

=×KKr v−

; the formula (11.2.16) takes thus the form

()

OO O

′′

=+×KK vρ ,

(11.2.23)

the moment of momentum

′

being calculated with respect to the inertial frame

′

R .

Analogously, we can write

() () ()

OC CO C C C

MM M

′′′ ′′ ′ ′ ′

=+× −× =+×KK v r v K v

ρρ

with respect to the inertial frame, as well as

()

OC C

M=+×KK v

, with respect to a

non-inertial frame. Taking into account (3.1.116'), (11.2.14) and (11.2.17') and

observing that

()

M=× ×I S

, we can write the relation (11.2.23) in the

form

()

CCC CC

′

==+=+KKKKKI

S ω ,

(11.2.23')

stating

Theorem 11.2.9 (C. Iacob). The moment of momentum of a discrete mechanical system

with respect to the centre of mass, in an inertial frame of reference

′

R , is equal to the

moment of momentum of this system with respect to the centre of mass, calculated in a

non-inertial frame

R of Koenig type (the sum of the moment of momentum of the

system with respect to the centre of mass, in an arbitrary non-inertial frame

R, with

the same pole as

R , and the contracted product of the tensor of inertia, with respect

to the same pole, by the angular velocity vector of the frame

R, with respect to the

frame

′

R ).

The privileged rôle of the centre of mass is thus put into evidence. If

= 0ω , the

arbitrary non-inertial frame

R being in motion of translation with respect to the inertial

frame, then we get

′

=KK

;

(11.2.23'')

this result corresponds to the relation (11.2.19) of V. Vâlcovici. If, in particular, the

pole of the non-inertial frame coincides with the centre of mass, then we can write the

relation

()C

′

(11.2.23''')

which corresponds to the Theorem 11.2.1 of Koenig.

If in the formula (11.1.23) we take

QC≡ , then the theorem of moment of

momentum reads

()()

ddd

dddd

CCC

tttt

′

==+ =+

KKK

IMM

S ω ,

(11.2.24)

where we took into account (11.2.23'); we mention that the derivatives are calculated

with respect to the inertial frame

′

R . If = 0ω , then it results

∂

(11.2.24')

11 Dynamics of Discrete Mechanical Systems

MECHANICAL SYSTEMS, CLASSICAL MODELS

the derivative being calculated with respect to the non-inertial frame

R with the pole at

O, while if, in particular, OC≡ , then we obtain

()C

∂

(11.2.24'')

the non-inertial frame being with the pole at the centre of mass; this last result

corresponds to the Theorem 11.2.4 of Koenig.

Introducing the dynamic moment (11.1.67) and taking into account (11.1.23) and

(11.1.67'), we get

()

OOC

′

′′′

=+×

Dvv

(11.2.25)

Using the relation (11.2.23), we find again the theorem of dynamic moment in the

inertial frame

′

R , with respect to an arbitrary pole O, in the form

OOO

′

DMM.

(11.2.25')

Taking into account the relations (11.2.18), (11.2.18'), we can write this theorem in the

non-inertial frame of reference

R too; we obtain

() ( )

OOO OOO

′

+× + +× = +DKI aMMωωρ

(11.2.25'')

= 0ω , then it results

()

OOOO

′

+× = +DaMMρ ,

(11.2.25''')

corresponding to the Theorem 11.2.8. If we have

OC≡ too, then we can write

()C

=+DMM,

(11.2.26)

which corresponds to the relation (11.2.24''); starting from the relation (11.2.24'), it can

be easily verified that

CCC

=+DMM,

(11.2.26')

with respect to an arbitrary non-inertial frame with the pole at

O. From (11.2.11') to

(11.2.25') it results also a theorem of dynamic torsor in an inertial frame with respect to

an arbitrary pole

{

}

{

}

{

}

ii i

OOO

′

τ=τ+τAFR

(11.2.27)

We notice that, starting from the theorem of momentum, one can obtain conservation

theorems (hence, first integrals) only with respect to the inertial frame

′

R . Thus, if

RR 0+=

, then from (11.2.15) it results that

′

=a0

, wherefrom

′

=vC

′′

+CC

= , ,const

′

JJJJG

, with respect to the frame

′

R .

In what concerns the theorem of moment of momentum with respect to the pole of a

non-inertial frame of reference

R, one can make some interesting considerations. Thus,

′

×=a0

, hence if the support of the acceleration of the pole O with respect to an

inertial frame passes through the centre of mass of the discrete mechanical system

(we eliminate the trivial case in which the pole

O has a uniform and rectilinear motion

with respect to a frame

′

R , hence the case in which the frame R is inertial too, as

well as the case in which

OC≡

), and if the sum

+=MM 0, then

const

JJJJG

the moment of momentum

K (calculated with respect to the non-inertial frame R )

being conserved in time with respect to the inertial frame

′

R ; one obtains thus a

vector first integral. If we have

= 0ω too (the non-inertial frame R does not rotate

with respect to the inertial frame

′

R ), then const

JJJJG

(the moment of momentum

with respect to the pole

O, in the frame R , is reduced to the moment of momentum

with respect to the same pole, in the frame

R ) with respect to the non-inertial frame

If in the relation (11.2.24) we make

+=MM 0, then it results const

′

JJJJG

the moment of momentum

′

(calculated with respect to the non-inertial frame R )

being conserved in time with respect to the inertial frame

′

R ; we notice that this result

cannot be obtained from the above one, making

OC≡ , because it takes place with

respect to an arbitrary non-inertial frame

R. If we have = 0ω too, then

const

JJJJG

with respect to the non-inertial frame R, and if, in particular, OC≡ , then

()

const

JJJJG

with respect to a non-inertial frame with the pole at the centre of mass

(this last first integral can be obtained from the Theorem 11.2.4 of Koenig).

As a conclusion, one can obtain a vector first integral (equivalent to three scalar first

integrals) with respect to a non-inertial frame too, but which is not independent of that

which can be obtained with respect to an inertial frame.

11.2.2.2 Kinetic Energy and Work with Respect to an Arbitrary Non-inertial

Frame of Reference. Comoment of Two Torsors

Squaring the relation (11.2.10'), we can write

22 2

()2()()

ii iiii

OOO

′′′

++× = + ⋅ +×+ +×

vv r v vv r v rωωω.

Multiplying by

m and summing for all the particles of the discrete mechanical system

S, it results (we take into account (11.2.14))

11 Dynamics of Discrete Mechanical Systems

MECHANICAL SYSTEMS, CLASSICAL MODELS

(,,)

OOC O OOC

TT Mv M M T Mv M

′′′ ′ ′′′

=+ + ⋅ + =− + ⋅vv v vvωρ ,

(11.2.28)

where we have introduced a quantity of the nature of a kinetic energy

()

22d

ii i i

Tm m

=+×=

∑∑

vrω

(11.2.28')

which, as in the case of the moment of momentum, does not represent the kinetic

energy of the discrete mechanical system

S with respect to the non-inertial frame;

indeed, the derivative of the position vector

r , in the non-inertial frame R, is taken

with respect to the inertial frame

′

R . This quantity, which is – in fact – the kinetic

energy of the discrete mechanical system

S with respect to the non-inertial frame R ,

becomes

()

OO O

TT T T=+ ⋅ + =+⋅ +KK Kωω,

(11.2.28'')

where

Tmv

∑

(11.2.28''')

is the kinetic energy of the discrete mechanical system

S with respect to the non-

inertial frame

R, while

()

222

11 22 33

111

()

222

Ojkk

TIIIIωω ω ω ω=⋅ = = + +Iωω ;

(11.2.29)

we have put thus in evidence also the representation with respect to the principal axes

of inertia, that is a quantity of the nature of a kinetic energy, which we call

pseudokinetic energy, of the discrete mechanical system

S with respect to the non-

inertial frame

R having the pole at O. We state

Theorem 11.2.10 (V. Vâlcovici). The kinetic energy of a discrete mechanical system

with respect to a given inertial frame of reference

′

R is equal to the sum of the kinetic

energy of this system with respect to an arbitrary non-inertial frame

R , which does

not rotate with respect to the frame

′

R (the sum of the kinetic energy of the system

with respect to a non-inertial frame

R with the same pole O as the frame R , the

scalar product of the angular velocity vector by the moment of momentum of the system

with respect to the same pole, in the same frame

R, and the semi-scalar product of the

angular velocity vector by the contracted product of the tensor of inertia with respect to

the pole

O by the angular velocity vector) and the scalar product of the velocity of the

pole

O with respect to the frame

′

R by the momentum of the system with respect to

the same frame

′

R , from which the kinetic energy of the pole O, in the frame

′

R ,

where the whole mass of the discrete mechanical system is considered to be

concentrated, is subtracted.

()tω is along the direction Δ of unit vector n at the moment t, then we can write

11 1

() ()

22 2

ωω⋅=⋅ =InInωω ,

(11.2.29')

where we have taken into consideration (3.1.82'),

()tωω= being the magnitude of the

angular velocity of instantaneous rotation; the axial moment of inertia

plays thus

the rôle of a mass for the instantaneous motion of rotation.

Taking into account (11.1.16), we find also the remarkable solution

()

11 1

()

22 2

OCOOOC

TT M M

′

′′ ′′′′

=+ ⋅ + + ⋅ −× + ⋅KK vv r vvωω

(11.2.30)

The elementary work of the given and constraint external and internal forces which

act upon the discrete mechanical system

S is given by

int

111

dd d d ( )d '( )d

nnn

ii i i

RR ikik

iik

WW W W

===

′′ ′ ′ ′ ′

++ + = +⋅+ +⋅

∑∑∑

FR r F R r

111 1

() '( )d()d

nnn n

ii ii i

ik ik O

iik i

=== =

⎡⎤

′

=++ +⋅++⋅

⎢⎥

⎣⎦

∑∑∑ ∑

FR F R r FR r

11 1 1

'( ) d '( ) d

nn n n

iiii

ik ik ik ik

ik i k

== = =

⎡

⎤

++⋅+×⋅+++

⎢

⎥

⎣

⎦

∑∑ ∑ ∑

FR r rFR FRω ,

where we have used the formula (11.2.10'); introducing the torsor of the given and

constraint external forces at the pole of the non-inertial frame

(

{}

{

}

OOO

τ+= +FR RM RM, the torsor of the given and constraint internal

forces being equal to zero) and taking into account the properties of the triple scalar

product, we read

int int

dd d d dd d d

RR RR

WWWW WWWW

′′ ′ ′

++ + =++ +

()d( )d

OOO

′

++ ⋅ + + ⋅RR r M M ω ,

(11.2.31)

finding thus the relation between the elementary work of the given and constraint forces

in the two frames: inertial and non-inertial. For the power of the given and constraint

forces we have

int int

() ( )

RR RR OOO

PPPP PPPP

′′ ′ ′ ′

++ + =++ + ++⋅+ + ⋅RRv M M ω ,

(11.2.31')

so that we can state

11 Dynamics of Discrete Mechanical Systems

MECHANICAL SYSTEMS, CLASSICAL MODELS

Theorem 11.2.11 The power of the given and constraint external and internal forces

which act upon a discrete mechanical system, with respect to an inertial frame of

reference, is equal to the sum of the power of the same forces with respect to an

arbitrary non-inertial frame and the power of the given and constraint external forces

applied at the pole of the latter frame with respect to the inertial frame.

Let

{}

{

}

, 1,2,...,

in≡=VV and

{}

{

}

, 1,2,...,

′′

≡=VV be two systems of

(bound or sliding) vectors of torsors, at the pole

{}

{

}

τ≡VRM and

{

}

{

}

′′′

τ≡VRM

, respectively. We call torsor product (or comoment) of the two

torsors the scalar quantity

{}

{

}

()

′′′

ττ =⋅+⋅VV RMRM

(11.2.32)

where

O is an arbitrary pole. Observing that the resultants R and

′

are invariant by a

change of pole and that

′

=+×

JJJG

MM R

′

′′′′

=+×

JJJG

MM R

, we can

write

()

′′

′′ ′ ′′′

⋅+⋅=⋅+ +⋅

JJJG

RM R M RM R R R M

()

′′ ′ ′

+=⋅+⋅

JJJG

RRRMRM

;

hence, the comoment of two torsors does not depend on the pole with respect to which

it is calculated, the definition being thus consistent.

Introducing the kinematic torsor at the pole

{

}

′

≡

′

vT ω ,

(11.2.33)

for which

′

′′′

=+ ×

JJJG

vv ω

, we can write the relation (11.2.31') in the form

{

}

{

}

()

int int

RR RR

PP P P PP P P

′′ ′ ′

++ + =++ + + τ +τ

′

FRT

(11.2.31'')

where we have put in evidence the torsor of the given and constraint external forces.

Applying the theorem of kinetic energy with respect to an inertial frame of reference

and starting from (11.2.28), (11.2.28''), (11.2.31'), we can write

int

OO O C OC R R

MMM PPPP

′

′′ ′′ ′′ ′ ′ ′ ′

= − ⋅+ ⋅+ ⋅=++ +va av va

int

() ( )

RR OOO

PP P P

′

=+ + + + + ⋅ + + ⋅RRv M M ω ;

taking into account (11.2.14), (11.2.15) and (11.2.17), we get

()

int

,, ( ) ( )

OOC R ROO

MMPPPP

′′

++⋅=+++++⋅aav MMω

ω ,

(11.2.34)

where

tt t t

∂∂

∂

⎛⎞

=+⋅+⋅ +⋅

⎜⎟

∂∂∂

⎝⎠



ωω ωω,

(11.2.34')

because

() ()

1111

2222

ij j i ij j i

IIωω ωω⋅+⋅= +



IIωωωω

() ()

()

ij ji i j i j

II Iωω ωω=+ = =⋅ =⋅



IIωωωω.

A scalar product of the equation (11.2.18), (11.2.18') by

ω leads to

int

()

OOCRR

MPPPP

∂

⎛⎞

′

+⋅ − ⋅ + ⋅ = + + +

⎜⎟

∂∂

⎝⎠



Kavωωω

(11.2.34'')

where we have used the equation (11.2.34), (11.2.34').

The theorem of kinetic energy for the system of equations of motion (11.2.13) has

the form

() ()

int

PP P P

∂

=++++⋅+⋅

∂

∑∑

Fv Fv;

(11.2.34''')

taking into account (11.2.13') and (11.2.18'') and observing that the last power vanishes,

we find again the relation (11.2.34'').

As V. Vâlcovici has shown, the theorem of kinetic energy with respect to an inertial

frame of reference maintains its form with respect to a non-inertial frame if and only if

()

OO C

∂

⎛⎞

′

⋅+⋅ =⋅

⎜⎟

∂

⎝⎠



Ka vωωω

(11.2.35)

If we have

= 0ω , hence TT= , then the non-inertial frame R does not rotate and

we obtain

() ()

OO C OO C

T T Mv M T Mv M

′′′ ′′′

=+ +⋅ =− +⋅vv vv

(11.2.36)

so that the theorem of kinetic energy takes the form

int

()

OC R R

MPPPP

∂

′

⋅+=+++

∂

(11.2.36')

From here (or from (11.2.35)) we deduce that the theorem of kinetic energy of a free

or constraint discrete mechanical system maintains its form with respect to a non-

inertial frame which does not rotate (

= 0ω ) if ()0

OC O C O

′′′′

⋅=⋅ − =av a v v

, hence

11 Dynamics of Discrete Mechanical Systems