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

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where

M is the mass of the element, while

′

is the velocity of its centre of mass

C , the same for all the points of the element. If the element is in motion of finite

rotation about an axis which passes through its mass centre, we can write

TIX

′

(17.3.11)

where

I is the moment of inertia of the element with respect to the central axis of

rotation,

being the corresponding rotation angular velocity. In case of a general

motion of the element, the formulae (14.1.29'), (14.1.35) allow to write

()

jjj

TMv

′′

=+⋅Iω ω ,

(17.3.12)

where

I is the corresponding central tensor of inertia. If the motion is a plane-parallel

one, as – in general – in case of an engine, it results

TMv IX

′′

(17.3.12')

where

I is the moment of inertia with respect to the axis of rotation (axis normal to

the fixed plane and fixed with respect to the considered element) which passes through

the mass centre

C . Finally, we can assume that the kinetic energy of the engine (of the

mechanical system

S ) with respect to the frame

′

is given by

TMv IX

′′

¦¦

(17.3.13)

The linear and angular velocities of the elements vary during a cycle of running of

the engine; hence, the kinetic energy of the motion has a cyclic variation during its

running too. Corresponding to the Theorem 11.1.10'', the variation of the kinetic energy

in a finite interval of time is equal to the work of the given external forces which act

upon the engine, in that interval of time. If, in an interval of time, the work of the

external forces is positive (negative), then the kinetic energy increases (decreases); if

this work vanishes, then the kinetic energy is conserved. In the latter case, the increases

of kinetic energy of some elements take place on account of the decreases of kinetic

energy of other elements, because the kinetic energy of each element has a cyclic

variation.

17.3.1.3 Reduced Mechanical Quantities

Because an engine has only one degree of mobility (degree of freedom), it is sufficient

to know the law of motion of a single element of it (which is, usually, the initial element

of the engine); one arrives thus to the idea of replacing the study of the engine as a

MECHANICAL SYSTEMS, CLASSICAL MODELS

524

mechanical system by the study of a fictitious element, having some of the properties of

the engine, necessary to the search one wishes to make.

Thus, we can replace the mass of the engine by a conventional one

red

M (reduced

mass), situated at a part of the engine called reduction centre; as well, we can introduce

a reduction element (even an element of the engine), having a reduced moment of

inertia

red

I . The reduction of the masses of the engine to a reduction centre or to a

reduction element is made so that their kinetic energy be, at any moment, equal to the

kinetic energy of the engine; obviously, the law of motion of this conventional

mechanical system

(of translation velocity

′

v or rotation angular velocity about its

mass centre

ω ) is the same to that corresponding to the situation in which the

respective system is a part of the considered engine. Equating the kinetic energies, we

obtain

()

red

MM I

′

§·

¨¸

′′

©¹

¦¦

(17.3.14)

the reduced mass of the engine being thus a non-negative quantity (it vanishes if the

engine is not in motion), which has a periodical variation, function of the linear and

angular velocities of its elements. Analogously, it results

()

red

IM I

′

§·

¨¸

©¹

¦¦

;

(17.3.15)

this quantity has the same properties as

red

M . We notice that a relation of the form

red red

IMl=

(17.3.16)

takes place, where

/lvX

′

= is the length of the reduction element. If we denote by

the rotation angle of the reduction element, then it results

red red

()IIK= ; as well,

red

()TTI

′′

= , wherefrom ()TTK

′′

= , so that we can draw both diagrams.

Analogously, we can reduce the given external forces and their moments with

respect to a pole. For this, we calculate the power produced by the forces and the

moments which act upon the element of engine in the form

PFv MX

′

¦¦

(17.3.17)

where

F and

represent the resultant and the moment resultant, respectively,

which act upon the element of index

j at its mass centre,

′

and

ω being the

corresponding linear and angular velocities, respectively (along the directions of the

corresponding forces and moments, respectively). Equating to the power

red red

redC

PFvMX

′

== , we get

17 Dynamics of Systems of Rigid Solids

525

red

FF M

′

′′

¦¦

(17.3.18)

as well as

red

MF M

′

¦¦

(17.3.19)

The obvious relation

red

redC

MFl=

(17.3.20)

takes place. The reduced force and the reduced moment are variable quantities. In case

of an element of reduction

red red

()K=FF and

red red

()

K=MM , we can draw the

corresponding diagrams.

F and

M are the counter-balance force and moment, respectively, components

of the counter-balance torsor, applied on the same reduction element, then we can write

red

=−FF

red

=−MM.

(17.3.21)

17.3.2 Applications

In what follows, we present some interesting applications of the theoretical results

obtained above. We consider thus the problem of the equilibration of the mobile masses

and the problem of work of engines (including their settlement).

17.3.2.1 Equilibration of the Mobile Masses

In case of engines which have some elements deficiently assembled (e.g., a runner

deficiently centred), which are not perfectly homogeneous or which are performed with

some tolerance, appear non-equilibrated forces of inertia, as well as their non-

equilibrated moments, yielding supplementary dynamic efforts in the kinematic

couples. We say that these elements are out-of-balance; the elimination of the

corresponding injuries is obtained by the balance of the inertia forces, hence by the

balance of the mobile masses.

Let be an element of mass

M, which is uniformly rotating with an angular velocity ω

about an axis

%, which does not pass through the centre of mass C. Thus arises a radial

(centrifugal) non-equilibrated force of inertia

F , applied at C and normal to the axis

%, as well as a non-balanced moment of this force

, normal to the same axis and

applied at a point of it (if the motion would be varied, then there would appear also a

tangential force of inertia, contained in the plane of rotation of the centre

C too). We

choose the axis

% as

′′

-axis, the

Ox x

′′′

-plane passing through

()

,,0C SS

′′

(

OC S

′′

= ). The centrifugal force is given by (the formulae (14.2.1))

MECHANICAL SYSTEMS, CLASSICAL MODELS

526

FMXS= ,

0F = ,

(17.3.22)

the moment of the forces of inertia being specified by (the formulae (14.2.1'))

123

MIX=

231

MIX=−

0M =

(17.3.22')

To equilibrate this element, we must have

=F0 (hence,

0SS==

, the mass

centre

C being thus on the rotation axis %) and

=M0 (hence,

23 31

0II==, the

rotation axis

% being a principal axis of inertia for the point of application of the

moment

). Hence, for its equilibration, the element must rotate about a principal

axis of its central ellipsoid of inertia.

≠F0

=M0

, then the torsor of the forces of inertia is reduced to a centrifugal

unbalanced force, the support of which passes through the mass centre

C, eccentrically

situated with respect to the rotation axis

%; there results a static out-of-balance (which

is put in evidence even at rest), which can be eliminated by an operation of static

balance. If

=F0,

≠M0, then the mass centre C is on the axis of rotation %, the

element being balanced at rest; during the motion of rotation appears a moment of

deviation of the axis of rotation (unbalanced moment of the forces of inertia), leading to

a dynamic out-of-balance, which can be eliminated by a dynamic balance. In general, if

≠F0,

≠M0, then takes place a general (complete) out-of-balance, which can be

eliminated by an operation of general (complete) balance. These operations are realized

by eliminating some masses (from those parts of the element which are heavier) or by

adding some supplementary masses (to those parts of the element which are lighter),

placed conveniently, so that the quantities

F and

M do vanish.

In case of masses in rotation, the forces of inertia are decomposed in components

contained in two parallel planes (equilibration planes); the resultants of these forces in

the two planes can be annulled by the application of a supplementary mass in each of

them, along the direction of the corresponding force. Obviously, this balance can be

realized in various modes, because the equilibration planes can be chosen arbitrarily, as

well as the position and magnitude of the supplementary masses.

In case of a mechanism, the relations (17.3.2) lead to the components of the torsor of

inertia forces of the form

Rmx

′

=−



kl k l

Mmxx

′

′′

=−∈



, 1, 2, 3j = ,

(17.3.23)

where the sum (which can be an integral too) is referring to an arbitrary point of the

mechanism. If the mechanism is plane (a plane-parallel motion), then the equations

(17.3.23) read

Rmx

′

=−



Rmx

′

=−



0R = ,

Mmxx

′

′′



Mmxx

′

′′

=−



()

12 21

Mmxxxx

′

′′ ′′

=− −

 

(17.3.24)

17 Dynamics of Systems of Rigid Solids

527

Because ()xx

′′

= ,

1, 2B =

, where

()tKK= is the angle of rotation of the

leading element, it results

′′

′





, 1, 2B = ,

so that

Rmm

′′

§·

=− +

¨¸

©¹

¦¦



, 1, 2B = ,

(17.3.25)

Mmx mxKK

′

′′

¦¦



MmxmxKK

′

′′

§·

′′

=− +

¨¸

©¹

¦¦



(17.3.25')

12 12

21 21

dd dd

xx xx

Mmxmx mxmxKK

′

′′ ′′

§·

′′ ′′

=−+−

¨¸

©¹

¦¦ ¦¦



(17.3.25'')

We notice that the moment

′

is contained in the moment of the given and

constraint forces, so that only the relations (17.3.25), (17.3.25') remain to be

considered. To balance the forces of inertia, the quantities

and

′

, 1, 2B = ,

must also vanish for any



and



; to do this, the relations

′

1, 2B =

(17.3.26)

are sufficient. Indeed, in this case we have

′

, 1, 2B =

too. Taking into account

mx M

′′

, 1, 2B = , where M is the mass of the

mechanism, the first relations (17.3.26) lead to

const

′

= , 1, 2B = . As well,

observing that

Imxx

′′

=−

, the last relations (17.3.26) allow to state that

constI

= , 1, 2B = . Hence, to equilibrate the resultant force

R , the masses of the

mechanism must be distributed so that their mass centre be fixed during the motion. To

balance the resultant moment

′

M , it is necessary that the

′′

-axis be a principal

axis of inertia at the point

′

MECHANICAL SYSTEMS, CLASSICAL MODELS

528

case of an articulated quadrangle, the centre of mass will have a fixed position on the

straight line joining the two fixed articulations if two masses of balance are introduced.

17.3.2.2 Work of Engines

In the work of an engine, we mark out – in general – three phases: the phase of starting

(of duration

t ), the phase of regime (of duration

tkt= , assuming the existence of k

cycles of motion, of duration

t ) and the phase of stopping (of duration

t ); the total

duration of motion of the engine will be, obviously,

Tt t t=++. The diagram

()tXX= of variation of the angular velocity of the leading element (the tachogram of

the machine) will be represented in Fig. 17.23. In the phase of regime, the mean angular

velocity

()

max

min

XX X=+ is put in evidence.

The kinetic energy of the engine is given by the relation (17.3.13) or, using the

reduced mass or the reduced moment of inertia, by one of the relations

red red

TMv IX

′′

(17.3.27)

The work effected by all the external forces which act upon the elements of the

engine in the whole duration of running of it is of the form

WW W

′′ ′

=−

, where

′

is the motive work, while

′

is the resistant work. Applying the theorem of

kinetic energy, we may write

dd d

TW W

′′ ′

=−.

(17.3.28)

Hence, the kinetic energy of the engine increases or decreases as

′′

< ; if dd

′′

= , then the kinetic energy passes through an extremum.

The theorem of kinetic energy in finite form leads to (between the limits

t and t for

the time)

() ()

22 2 2

00 0

red red

TT I M v v W WXX

′′ ′ ′ ′ ′

−= − = − = −.

(17.3.28')

The expressions at the left represent the work

of the forces of inertia, which is

positive if the angular velocity increases (

XX> ) and negative otherwise (

XX< );

as well, the difference

′′

− is the work of the reduced moment (at the reduction

element) or of the reduced force (at the reduction centre).

17 Dynamics of Systems of Rigid Solids

Often, the moment of the inertia forces is neglected, this one being sufficiently small,

remaining to be fulfilled the condition concerning the centre of mass. For instance, in

529

Fig. 17.23 Diagram of variation of the angular velocity of the leading element

(the tachogram of the machine)

The phase of starting is characterized by

0X = and

XX= , where

X is the

angular velocity of regime (of mean value

X ); it results

′′

> . To diminish the

starting time

t , the no-load start of the engine, by uncoupling the useful load, is

recommended. In the phase of regime, the angular velocity has a cyclic variation

(

XX= ) and we have

′′

= (the motive work during a cycle is consumed by

the respective resistant work). In the stopping phase, the angular velocity decreases

from

X to zero; we have

′′

< . Practically, the motive source is uncoupled

(

′

= ), and, to reduce the time of stopping

t , the work

′

is increased, by

braking.

The resistant work can be expressed, in general, in the form

rupg

WWWW

′′′′

=+±,

(17.3.29)

where

′

is the useful work consumed by the engine to overcome the technological

resistances,

′

is the passive (lost) work (by friction, resistance of the medium etc.),

while

′

is the work of the gravity forces (one takes the sign + or the sign − as the

gravity centre of the elements in motion moves downwards or upwards). Taking into

account (17.3.28'), we get

mupg

WWWWW

′′′′′

=+±±,

(17.3.30)

putting thus in evidence the equilibrium of the motive work and of the other works

consumed or produced during its running (

′

is the inertial work). Hence, it results

ddddd

mupg

WWWWW

′′′′′

=+±±; dividing by dt, we obtain

mupg

PPPPP

′′′′′

=+±±,

(17.3.30')

where

′

is the power due to the motive forces,

′

is the useful power,

′

is the

positive (lost) power,

′

is the gravitational power (the power necessary to balance the

MECHANICAL SYSTEMS, CLASSICAL MODELS

530

forces of weight if the centre of gravity of the elements moves up or the power

produced by these forces if the centre of gravity moves down), while

′

is the inertial

power (the power necessary to increase the kinetic energy of the engine or the power

obtained by decreasing the kinetic energy). The equation (17.3.30') represents the

equation of energetic balance of the engine, specifying the power necessary to the

engine and allowing an estimation of it from an economical point of view.

Taking into account Chap. 6, Sect. 1.1.3, we will use the notion of the mechanical

efficiency

01I≤<, given by (6.1.17), as well as that of coefficient of loss

WWK

′′

; it results 1IK=−. As the coefficient of loss is smaller, so the

mechanical efficiency is greater (the work is better used by the engine); if the engine is

in no-load moving (

′′

= ), then we have 0I = . In case of self-braking of the

engine (

′′

> ), the mechanical efficiency would become negative (the engine

cannot work in regime).

Let be

n mechanisms or engines

M , 1,2,...,jn= , grouped together in series. We

assume that

′

is the work of the setting by means of the mechanism

M and that

′

is the useful work of it by means of the mechanism

M . We may write

WWI

′′

221

/WWI

′′

,…,

WWI

−

′′

; we obtain thus

1, 1 2

...

IIII= ,

(17.3.31)

with

WWI

′′

= . Hence, the mechanical efficiency of a setting composed by

mechanisms (or engines) coupled in series is equal to the product of the mechanical

efficiencies of the mechanisms (or engines) which compose this setting. The mechanical

efficiency of the setting is smaller than each of the component mechanical efficiencies;

this mechanical efficiency is as smaller as the setting is more complex. As well, the

formula (17.3.31) puts in evidence the impossibility of existence of a perpetuum mobile

(which works without loss of energy, having the mechanical efficiency

I = ).

17.3.2.3 Adjustment of the Work of Engines

The angular velocity

ω of the leading element of the engine varies periodically or non-

periodically during the motion of rotation. As it is seen in Fig. 17.23, the periodic

variations take place between

min

X and

max

X ; from the equation

red

redC

IMX =



results that for the extreme values we have

red

M = . The non-periodic variations,

which can appear because of various causes (coupling or non-coupling of

supplementary charges, accidental charges, random deficiencies etc.), must be limited

too. The problem of adjustment of the work of engines is thus put (maintaining the

angular velocity

X between some limits or even constant). From the relation

red

dd/

MtIX = one observes that the variations of the angular velocity are as

smaller as

redC

M and their duration are smaller or as

red

I is greater.

The increasing of the reduced moment of inertia

red

I can be obtained with the aid of

a fly wheel (a great and heavy wheel) fixed on the element in motion of rotation; hence,

17 Dynamics of Systems of Rigid Solids

531

red redm

II I=+, where

redm

I is the reduced moment of inertia without fly wheel,

while

I is the moment of inertia of the fly wheel. As a matter of fact, the fly wheel is

an accumulator of kinetic energy. Indeed, during an energetic cycle of an engine, the fly

wheel accumulates kinetic energy in excess, when the angular velocity is increasing,

handing it back when the angular velocity decreases; thus, the angular velocity

ω is

close to the mean velocity (of regime). But the fly wheel cannot impede an engine to

pass from a state of regime to another one; it makes only easier the passing (which takes

thus place less sudden) and is not efficacious in case of non-periodical variations of the

velocity.

To maintain the mean angular velocity

X one uses particular mechanisms, called

regulators (which act, especially, on the admission of the motive energy). Hence, the

regulator establishes and maintains the value of the mean velocity of the engine, while

the fly wheel diminishes the periodic variations of the velocity around this value.

The regulator is a mechanism which controls automatically the work of the motive

force, decreasing it when the angular velocity is increasing and increasing it when the

angular velocity decreases, so that the equilibrium between the motive couple and the

resistant one is re-established. The regulators can be centrifugal (influenced by the

centrifugal force), of inertia (influenced by the tangential inertia forces) etc. We

mention also the power controllers (e.g., for lifting pumps). Between the centrifugal

regulators, one can mention: the Watt regulator (which has the advantage to be stable)

and the Rankine regulator (which is isochronous, but has the disadvantage to be labile)

etc.

MECHANICAL SYSTEMS, CLASSICAL MODELS

532

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