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

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rototranslation are continuous functions) with respect to an inertial frame

′

of pole

′

; to find the form of the general theorems with respect to such a frame, we use the

results in Sect. 11.2.2. Thus, starting from the formulae (11.2.12), (11.2.12') and

observing, by applying a mean value theorem on intervals of continuity, that

′′ ′′

′′

′′ ′

−→+

⎡⎤

⎢⎥

⎣⎦

∫∫

FF0

() ()

lim ( )d ( )d

tt tt

because the complementary forces (the transportation and Coriolis forces),

corresponding to the centre of mass, vary continuously or have finite jumps (see the

formula (13.1.24') too), we can state that the Theorem 13.1.2 of the momentum

(formula (13.1.24)) and the Theorem 13.1.2' of motion of the mass centre (formula

(13.1.24'') take place also with respect to a non-inertial frame of reference. Applying the

relation (11.2.10') at the moments

′

and

′′

, subtracting the relations thus obtained one

of the other, multiplying by the mass

m , summing for all the particles of the discrete

mechanical system, passing to limit in the sense of the theory of distributions

(

′′ ′

−→+

00tt

) and taking into account the hypothesis iii) of the considered

mathematical model and that the rototranslation

{

}

′

v ω is continuous, we obtain

′

Δ=Δ

()().

(13.1.28)

As a matter of fact, applying the operator

Δ to the relation (11.2.11) we obtain the same

result, justifying thus the preceding affirmations (the resultants R and

R are invariant

to a change of pole).

Starting from the formula (11.2.18), corresponding to the theorem of moment of

momentum with respect to an inertial frame of reference, and observing that

[]

′′

′

′′ ′

−→+

′

×=

∫

lim ( ) ( ) d

tMttρ ,

we get Δ=+K

()

MM; hence, the jump (with respect to an inertial frame) of the

pseudomoment of momentum of a discrete mechanical system subjected to constraints,

with respect to an arbitrary pole, at a moment of discontinuity, is equal to the resultant

moment of the given and constraint external percussions which act upon the system,

with respect to the same pole, at that moment. Taking into account the relation

(11.2.17') and observing that

Δ=I0

(( ))

ω , it results that Δ=ΔKK

()()

; we

find thus again the formula (13.1.25'), written in the form (13.1.25). In general, starting

from (11.2.12') to (11.2.18''') and noting that

{

}

′′

′

′′ ′

−→+

⎡⎤

×+ =

⎣⎦

∑

∫

rF F 0

() ()

lim () ()d

ttt

we can state that the Theorem 13.1.3 of the moment of momentum (the formula

(13.1.25)) takes place with respect to a non-inertial frame of reference too. As a matter

of fact, starting from the relation (11.2.10') or applying the operator

Δ to the relations

13 Other Considerations on Dynamics of Mechanical Systems

143

MECHANICAL SYSTEMS, CLASSICAL MODELS

(11.2.16), (11.2.17'), taking into account that

= const





in the interval of percussion

and using the relation (11.2.24'), we may write

′

′′

Δ=Δ+×Δ

KKrH

00 0

()() ()

;

(13.1.28')

we find thus again the above affirmation, because

′

=+×

MMrR

and

′

=+×

MMrR

Moreover, from (11.2.16) one observes that this affirmation holds also in the case in

which the velocity of the pole

O has a jump for ≡OC, with the condition Δω = 0.

Finally, we can affirm that the Theorem 13.1.4 of torsor may be stated also with

respect to a non-inertial frame in continuous motion with respect to an inertial frame;

the fundamental equations of the mathematical model of the collision phenomenon do

not need a privileged frame of reference. Hence, in the collision phenomenon of a

discrete mechanical system, the jump relations are invariant to a change of frame of

reference or of pole.

If, in particular, we have

+=0RR , it results

Δ=H0

() ,

Δ=v0

(13.1.29)

′′ ′

HH,

′′ ′

(13.1.29')

so that the momentum of the mechanical system and the velocity of its mass centre

remain constant in the percussion interval. We find thus again the relation (1.1.26)

obtained by the collision of two homogeneous spheres of negligible dimensions, for

which the inertial character of the mass has been put in evidence (see Chap. 1, Sect.

1.1.6).

As well, if

+=0

MM , then we can write

Δ=K0

()

(13.1.30)

wherefrom

′′ ′

(13.1.30')

the moment of momentum of the mechanical system remaining constant in the interval

of percussion.

13.1.1.4 Case of a Discrete Mechanical System Subjected to Sudden Constraints

We introduce the notations

−+

()

3( 1)

()

3( 1)

= 1,2,...,in, = 1, 2, 3j ,

(13.1.31)

144

without summation with respect to i, where

()i

x and

()i

F are the components of the

position vectors

and of the given forces

, respectively; as in Chap. 3, Sect. 2.2.2,

one can pass from the geometric support Ω of the discrete mechanical system

S in the

space

E (formed by the geometric points

P ) to a representative point P (of co-

ordinates

ξ , k = 1,2,…,3n) in a representative space

 (of dimensional equation

1/2

M ). We assume that the mechanical system S is subjected to p bilateral geometric

constraints of the form (3.2.8''), expressed by

≡=

12 3

( ; ) ( , ,..., ; ) 0

lk l

tf tξξξξ, = 1,2,...,lp,

(13.1.32)

and by a unilateral geometric constraint

≥(;) 0

tξ .

(13.1.32')

Lagrange’s equations of motion (11.1.64), written for the mechanical system

S in the

space

E , take in the space

 the form

∂

=+ + =

∂∂

∑

kk l

ξφ λ λ

ξξ



, = 1,2,...,3kn,

(13.1.33)

where

λ , l = 1,2,…,p, and λ are the Lagrange multipliers. If we assume that the

constraint (13.1.32') is weak (

>(;) 0

tξ ), the motion of the mechanical system S

takes place as the respective constraint would not exist, after a law of the form

= ()

tξξ, k = 1,2,…,3n. Let

′

be the smallest positive root of the equation

=(();) 0

ttξ ; this is the moment at which the mechanical system reaches the

constraint, the corresponding co-ordinates and velocities being

′′

= ()

tξξ and

′′

= ()

tξξ



, respectively. In this case, the velocities thus found satisfy the conditions

′

∂

′

=+=

∂

∑



, = 1,2,...,lp,

(13.1.34)

′

∂

′

=+≥

∂

∑



(13.1.34')

If the velocities

′



satisfy the relation (13.1.34'), then the mechanical system S

leaves of new the constraint (strict inequality, sufficient condition) or remains on the

constraint (equality, necessary condition). But if the relation

′

<d/d 0

, takes

place, to be in concordance with the condition (13.1.34') we must assume that, in the

interval of collision, arise constraint percussive forces, which lead to constraint

percussions

()i

P and

()i

P , i = 1,2,…,n, respectively; there arise jumps of the

13 Other Considerations on Dynamics of Mechanical Systems

145

MECHANICAL SYSTEMS, CLASSICAL MODELS

velocities, which become

′′



, and a collision of the mechanical system S with the

constraint (13.1.32') takes thus place. If at the moment

′

the mechanical system leaves

the constraint, then one must see what happens for the following positive root of the

equation

=(();) 0

ttξ , till the exhaustion of all those roots or till one has a root for

which the system remains on the constraint, intervening the phenomenon of collision.

We assume that the three hypotheses in Sect. 13.1.1.1 which are mathematically

modelling this phenomenon hold further, and add a supplementary hypothesis:

iv) The collision phenomenon does not destroy the bilateral geometric constraints

(the constraints (13.1.32)) of the mechanical system.

At the end of the percussion interval (the moment

′′

) we have

′

′′

′

∂

′′

=+>

∂

∑



(13.1.35)

We suppose that, at a moment

[

]

,ttt

′′′

∈ , the velocities



are so that

′

∂

=+=

∂

∑



(13.1.35')

In the last two relations, we took into account the hypothesis iii). If the phenomenon of

collision is reduced to the first phase (the interval

[

]

′

,tt ), then the collision is a plastic

one; if the second phase (the interval

[

]

,tt

′′

) takes also place, then the collision has an

elastic-plastic character. We notice that the constraint percussions are given by

′

′′

()

grad

fλ ,

′

′′

()

grad

fλ ,

′

′′ ′′

()

grad

fλ ,

′

′′ ′′

()

grad

fλ ,

′

′′′

=+ =

PP P

() () ()

grad

iii

Rl Rl Rl

fλ ,

′

′′′

=+ =

PP P

() () ()

grad

iii

RR R

fλ ,

where the first and the second phase of the collision are specified by “prime” and by

“second”, respectively, the corresponding Lagrange multipliers being given by

′

∫

()d

ttλλ,

′′

∫

()d

ttλλ

′

∫

()d

ttλλ,

′′

∫

()d

ttλλ

′′′

lll

λλλ,

′′′

λλ λ.

Lagrange’s equations (13.1.33) lead thus to (we take into consideration the hypothesis

ii)

′′

∂

′′ ′

−= +

∂∂

∑

kk l

tt tt

ξξ λ λ

ξξ



(13.1.36)

146

′′

∂

′′ ′′ ′′

−= +

∂∂

∑

kk l

tt tt

ξξ λ λ

ξξ



(13.1.36')

′′

∂

′′ ′

−= +

∂∂

∑

kk l

tt tt

ξξ λ λ

ξξ



(13.1.36'')

for

k = 1,2,…,3n. Assuming that the collision is plastic, from the 3n + 1 equations

(13.1.35'), (13.1.36), to which we associate the

p equations

′

∂

=+=

∂

∑



, = 1,2,...,lp,

(13.1.37)

corresponding to hypothesis iv), we obtain the

3n + p + 1 unknowns



λ and λ. To

this goal, we multiply the equation (13.1.36) by

′

∂∂/

and by

′

∂∂/

respectively, and sum with respect to

k; taking into account (13.1.34), (13.1.35') and

(13.1.37), we obtain

[]

′

′′

+=−

∑

ff ff

λλ ,

[]

′′

∑

,,0

ff ffλλ ,

= 1,2,...,qp

(13.1.38)

with the notation

[][]

′′

∂

∂∂

∑

tt tt

ϕψ ψϕ

ξξ

(13.1.38')

It results

′

=−

′

=−

, = 1,2,...,lp,

(13.1.39)

where

Δ is the determinant of the system (13.1.38), (13.1.38') of p + 1 equations, while

Δ and

Δ are the normalized minors of the elements

[

]

f and

[

]

f of the first

line, respectively; replacing in the system (13.1.36), we find the unknown velocities



In the general case in which takes place the second phase of the collision

phenomenon, we use the equations (13.1.36), (13.1.36') or the equations (13.1.36''),

which represent, in fact, a consequence of the first system of equations; in this last case,

we dispose of

3n equations, to which we associate p equations

′

′′

′

∂

′′

=+=

∂

∑



, = 1,2,...,lp,

(13.1.40)

13 Other Considerations on Dynamics of Mechanical Systems

147

MECHANICAL SYSTEMS, CLASSICAL MODELS

corresponding to the hypothesis iv), for the

3n + p + 1 unknowns

′′



, k = 1,2,…,3n,

λ , l = 1,2,…,p, and λ. Using the method of calculation presented above, we obtain,

analogously,

′′ ′

⎛⎞

=−

⎜⎟

⎝⎠

tt tt

′′ ′

⎛⎞

=−

⎜⎟

⎝⎠

tt tt

= 1,2,...,lp

;

(13.1.41)

we notice that we do not know the total derivative

′′

d/d

, which contains the

unknown velocities

′′



, missing thus a last necessary equation to solve the problem. As

well, from (13.1.39), (13.1.41) it results

′′

, = 1,2,...,lp.

(13.1.41')

To solve the problem in the frame of the mathematical model considered above, we

assume that Lagrange’s multipliers corresponding to the unilateral constraint verify the

condition

′′ ′

= k

λλ

(13.1.42)

where

k is the coefficient of restitution introduced in Sect. 1.1.1. The first relations

(13.1.39) and (13.1.41') lead to

′′ ′

tt tt

;

(13.1.42')

it results

′′ ′

kλλ,

= 1,2,...,lp

(13.1.42'')

too for the bilateral constraints. We obtain, finally,

′′

∂

⎛⎞

′′ ′

−=+ +

⎜⎟

∂∂

⎝⎠

∑

(1 )

kk l

tt tt

kξξ λ λ

ξξ



(13.1.43)

the problem being solved as in the preceding case.

13.1.1.5 Carnot and Kelvin Theorems. Principle of Virtual Work. Conservation

Theorems

In what follows, we make a study of the variation of the kinetic energy

∑

T ξ



(13.1.44)

148

corresponding to the notations (13.1.31). Multiplying the equations (13.1.36) by



and

′



, respectively, and summing, we obtain

()

tt tt

Tffλλ

′′

−=− −

′

∑



()

tt tt

Tff

λλλ

′′

′

′′′′

−=− − +

′

∑



with

ξξ

′

∑



(13.1.45)

where we used the notations in Sect. 13.1.1.2, introducing the kinetic energy at the

beginning of the collision phenomenon and at the end of it and where we took into

account (13.1.34), (13.1.35') and (13.1.37). We assume that we have to do with

scleronomic constraints (in case of rheonomous constraints, the variation of the kinetic

energy depends explicitly on the variation of the constraints), so that

= 0



, = 0



l = 1,2,…,p; it results

T =

′

, ح

2d 2

λλ

′

′′ ′

−= =−

′

corresponding to the relations (13.1.39). From the above relations and by means of the

notations which have been introduced, we obtain, analogously,

2TT T

′′

+− =

′

where we have introduced also the kinetic energy of the lost velocities in the first phase

of the collision phenomenon

()

′′′

=−=

∑

022

TKξξ λ



, =

;

(13.1.46)

it is thus shown that the ratio

/ΔΔ is positive. Eliminating ح

′

, it results

′′

Δ+=

() 0TT ,

′′

Δ=−

()TTT

(13.1.47)

and we can state

Theorem 13.1.5 (Carnot, I). In the motion of a discrete mechanical system subjected to

holonomic and scleronomic constraints, the sum of the variation of the kinetic energy in

the first phase of the collision phenomenon, due to a sudden unilateral constraint, and

the kinetic energy of the lost velocities in the same interval of time, vanishes.

13 Other Considerations on Dynamics of Mechanical Systems

149

MECHANICAL SYSTEMS, CLASSICAL MODELS

One can see that, in the first phase of the collision phenomenon, the variation of the

kinetic energy is negative, corresponding thus to a loss of kinetic energy

′′

Δ=−Δ

() ()TT.

Starting from the equations (13.1.36'), multiplying by

′′



and



, respectively,

summing and taking into account (13.1.35'), (13.1.37) and (13.1.40), we can write,

analogously,

()

tt tt

Tffλλ

′′ ′′

−=− −

′′

∑



()

tt tt

Tff

λλ λ

′′ ′′

′′

′′ ′′ ′′ ′′

−=− − +

′′

∑



with

ξξ

′′

∑



(13.1.45')

where, using the notation in Sect. 13.1.1.2, we have introduced the kinetic energy at the

end of the second phase of the collision phenomenon. Assuming, further, that we

remain in the case of the scleronomic constraints, it results, taking into account the

relation (13.1.41'),

′′

, ح

2d 2

λλ

′′

′′ ′′ ′′

−= =

′′

We see thus that, in the frame of the hypotheses made above, we have

حح=

′′′

(13.1.45'')

so that we may write the relation

()

′′ ′

−=

∑

kk k

ξξ ξ

 

(13.1.45''')

too. Analogously, we have

2TT T

′′ ′′

+− =

′′

where we have introduced the kinetic energy of the lost velocities in the second phase of

the collision phenomenon

()

′′ ′′ ′′

=−=

∑

022

TKξξ λ



;

(13.1.48)

150

eliminating

′′

, it results

′′ ′′

Δ=

()TT,

′′ ′′

Δ=−

()TTT,

(13.1.49)

and we can state

Theorem 13.1.6 (Carnot, II). In the motion of a discrete mechanical system subjected

to holonomic and scleronomic constraints, due to a sudden unilateral constraint, the

variation of the kinetic energy in the second phase of the collision phenomenon is equal

to the kinetic energy of the lost velocities in the same interval of time.

Hence, in the second phase of the collision phenomenon, the variation of the kinetic

energy is positive, corresponding thus to an increase of kinetic energy

′′

()T .

From (13.1.46 to 13.1.49) it results

()

′′ ′ ′′ ′

−= −

22 2

TT Kλλ

too; taking into account (13.1.42), we can write

()

′′′

Δ=−− =− −

222 22

111

() 1 1

TkK K

λλ

wherefrom

()

′

Δ+− =

() 1 0TkT

(13.1.50)

()

′′

Δ+ − =

() 1 0

(13.1.50')

with

′′ ′

Δ=−

()TTT. We thus state:

Theorem 13.1.7 (Carnot, III). In the motion of a discrete mechanical system subjected

to holonomic and scleronomic constraints, the sum of the variation of the kinetic energy

in the collision interval, due to a sudden unilateral constraint, and the kinetic energy of

the lost velocities in the first phase of the collision phenomenon, multiplied by the

number

−

1 k , where k is the coefficient of restitution by collision, vanishes.

Theorem 13.1.7' (Carnot, III'). In the motion of a discrete mechanical system subjected

to holonomic and scleronomic constraints, the sum of the variation of the kinetic energy

in the collision interval, due to a sudden unilateral constraint, and the kinetic energy of

the lost velocities in the second phase of the collision phenomenon, multiplied by the

number

−

(1/ ) 1k , where k is the coefficient of restitution by collision, vanishes.

We can state thus that, in the collision interval, the variation of the kinetic energy is

negative, being put in evidence a loss of kinetic energy

Δ=−Δ

() ()TT.

Adding the relations (13.1.47) and (13.1.49) and observing that

′′′

Δ+Δ=

−Δ

()()

()

, we can write

13 Other Considerations on Dynamics of Mechanical Systems

151

MECHANICAL SYSTEMS, CLASSICAL MODELS

′′ ′

Δ=−

000

()TTT,

′′′

Δ=−

()TTT,

(13.1.51)

hence

′′′

TT; from (13.1.36 to 13.1.36'') it results, easily,

()

′′ ′ ′ ′′

−=+ − = + −

(1 ) 1

rr r r rr

ξξ ξξ ξξ

  

, = 1,2,..., 3rn,

(13.1.52)

so that the relations (13.1.51) lead to the same formula (13.1.23) as in the case of a

single particle; thus, the generalized Theorem 13.1.1 of Carnot is stated analogously for

a discrete mechanical system subjected to holonomic and scleronomic constraints. All

the considerations made in Sect. 13.1.1.2 hold further.

We notice that the relations (13.1.52) justify the relations (13.1.45''), (13.1.45''') too.

As in Sect. 1.1.3, we consider the relations (11.1.24), (11.1.25'), written for the time

interval

[

]

′′′

,tt ,

′′ ′

−<tt ε

, > 0ε arbitrary, which contains only one moment of

discontinuity

t ; passing to limit in the sense of the theory of distributions, we can

write (the sign “prime” at the sum indicates

≠ki)

[]

′′

′

′′ ′

−→+

⎧

⎫

′′

Δ+= ⋅ + + +

⎨

⎬

⎩⎭

∑∑

∫

vFRFR

( ) lim () () ' () () d

iii

ik ik

TT t t t tt,

[]

′′

′

′′ ′

−→+

⎧

⎫

′

Δ−= ⋅ + + +

⎨

⎬

⎩⎭

∑∑

∫

vFRFR

( ) lim ( ) ( ) ' ( ) ( ) d

iii

ik ik

TT t t t t t

where, for the sake of generality, we have introduced also the influence of the

constraint forces. We obtain thus

()

⎡

⎤

′′

Δ+= ⋅++ +

⎢

⎥

⎣

⎦

∑∑

vPP PP

() '

Ri ik Rik

(13.1.53)

()

⎡

⎤

′

Δ−= ⋅++ +

⎢

⎥

⎣

⎦

∑∑

vPP PP

() '

Ri ik Rik

(13.1.53')

so that we can state:

Theorem 13.1.8 (theorem of kinetic energy). The sum of the variation of the kinetic

energy of a discrete mechanical system subjected to constraints, at a moment of

discontinuity, and the kinetic energy of the lost velocities at the same moment is equal

to the sum of the scalar products of the percussions which act upon the particles by

their velocities after that moment of discontinuity.

Theorem 13.1.8' (analogue of the theorem of kinetic energy). The difference between

the variation of the kinetic energy of a discrete mechanical system subjected to

constraints, at a moment of discontinuity, and the kinetic energy of the lost velocities at

the same moment is equal to the sum of the scalar products of the percussions which act

upon the particles by their velocities before that moment of discontinuity.

()

⎡⎤

′′

⋅++ + =

⎢⎥

⎣⎦

∑∑

vPP PP 0

Ri ik Rik

(13.1.54)

152