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3.10. Analytical mechanics as an “ideal” theory 111
whose geometrical interpretation is Zhukovsky’s lever. It is interesting that
the kinematical method allows to reduce the problem of an equilibrium of
a rigid “lever”;
• the cooperative use of principle of virtual displacements and D’Alembert’s
principle (common dynamics equation), which allows to rise the equations
of kinetostatics (3.74) to the higher “energetic” level,
n
(Fi+Φi+Ri)δri=0. (3.78)
i=1
• The use of common dynamics equation written in generalized coordinates
instead of this equation in “usual” coordinates. Using the ratios
∂v
∂ ˙q
and
which determine the velocityV
∂˙r
i
=
∂ ˙q
j
d
(
dt
∂q
of any point of a system in linear function
i
∂r
i
j
i
j
=
)=
∂r
∂q
∂ ˙q
∂˙r
i
,
j
i
,
j
of a generalized velocity, we obtain
k
(Qj−
j=1
dt
∂T
d
(
∂ ˙q
j
)+
∂T
) δri=0. (3.79)
∂q
j
• The reverse “descent“ to the first level (level of the differential equations
of a system motion), but this “descent“is to the next “coil” of a cognition
spiral, namely – to the writing equation in Lagrange’s form. Since equality
(3.76) is satisfied for arbitrary δq
only in case if all expressions in round
i
brackets equal zero, we obtain
d
∂T
(
(
dt
∂ ˙q
j
) −
∂T
) δri= Qj(j =1, 2, ...k) . (3.80)
∂q
j
System information operator of a null action in dynamics of constrained
system
System information operator of a null action was introduced in the first
chapter for statics and kinematics. Now we consider the its usage for dynamics.
112 Chapter 3
So, because of the introduction of generalized coordinates (without changing the
mechanical state of a system) we can minimize the number of coordinates in
the differential equations of motion. The generalized velocities and generalized
forces are also the operators of a null action by the similar reasonings.
The introduction of imaginary virtual displacements (without changing the
mechanical state of a system) permits to obtain the equilibrium condition of a
system in terms of work, i. e. in “energetic” level.
The notion of ideal constraints can be regarded as the information operator
of a null action too, since the work of these constraints equals zero for any
virtual displacement of a system, i. e. they don’t change the sum of works of
forces acting on a system.
The introduction of D’Alembert’s inertia forces, and their imaginary actions
on the system points without change of the system mechanical state allow to
write the differential equations of motion of a system as statics equations.
Common dynamics equation also can be regarded as the example of the
information operator of null action. Acting on a system imagined D’Alembert’s
inertia forces and imagined virtual displacements of a system don’t change the
mechanical state of a system, but using these notions we can write the differential
equations of motion in form of statics equation, on an “energetic” base.
Finally, Lagrange’s equation is also one of examples of the information
operator of null action. It does not change the mechanical state of a system but
permits to formalize writing up differential equations of motion of any systems
even very complex. This becomes possible because the generalized inertia forces
are expressed in terms of a kinetic energy.
Thus, the system information operator of a null action is the universal tool
for system organization of “ideal” theories in mechanics.
3.11. System classification of forces
Classification of forces by A. J. Ishlinsky
The notion of a force and the classification of forces cause much discussion
in mechanics. The motives of such discussion are analyzed in [8].The author
of this work suggests the following classification of forces: physical forces and
inertia forces.
Physical forces are:
• long-range forces (distant forces) F — gravity, electric, magnet forces;
• contact forces N –reactions of constraints, surface forces;
3.11. System classification of forces 113
• resistance forces K — forces which a body exerts on other bodies; they are
connected with the property of inertia according to which a body resists
being accelerated; these forces were called inertia forces by Newton; A.J.
Ishlinsky suggested to call them Kepler’s forces.
Inertia forces are [8]:
• a) Euler’s forces (P, Q) — inertia forces of transport ( P ) and Coriolis
inertia forces (Q); they are considered to be applied to particles or elements
of continued bodies; ones in the state of a relative rest they are referred to
D’Alembert’.
• b) D’Alembert’s forces — the product of mass of a particle (or element of a
continued body) and the negative acceleration vector of this particle’s.
In this classification the physical forces are forces of interaction between
two bodies. The bodies can come into contact ( N ) or interact being some
distance apart (F ). Therefore they obey the information operator of null action
in its application for the equilibrium of two interacting bodies. The physical
sense holds whether we consider a force applied immediately to the first body
(F,N) or we consider the first body and a force exerted on it by the second
body (K).
The physical sense of interaction losses if one of bodies becomes infinitely
distant; then the force acting on the first body becomes a fiction (P,Q,D).
Classification of forces by the basic mechanics operator
The classification of forces by the basic mechanics operator is more con-
venient for system mechanics than the classification by A. J. Ishlinsky, whereas
it has the physical sense of the latter classification.Let’s apply information operator (with physical, dialectical, mathematical components)for the analysis of
our notion of a force (see Fig. 3.42). As a result we obtain three types of
forces. Physical forces including all physical forces by A. J. Ishlinsky except
for reactions of constraints. System forces, i.e. forces determined, in addition
to interaction, by some hypotheses (for example the axiom about the actions of
constraints). Reactions of constraints are referred to the second type, i.e. to
system forces. The third type is mathematical forces, or informational forces,
having no physical sense.
Now let’s use the operator which represents every notion as a “positive” or
“negative” one, and also “extends” one notion to the “area” of another notion.
As a result we obtain four types of forces: “positive” forces (actual physical
and system ones); “negative” forces, opposite to the first ones (pure informational forces); “positive – negative” forces deal with no actual physical process
114 Chapter 3
INFORMATION
OPERATOR
SYSTEM COMPONENT
MATHEMATICAL COMPONENT
PHYSICAL COMPONENT
+-+ -
- +
NOTION OF FORCE
● Σ ⊕
=0; ≠ 0
Figure 3.42
(physical-informational forces); “negative – positive” forces concerned with first
an information and second a physical sense (informational — physical) forces.
At last we use the basic operator of mechanics (•;;;0;=0)to obtain single forces (•), force systems (), resultant force (), forces equal to zero (0),
forces not equal to zero (=0).
Thus, we obtain the following classification of forces (see Fig. 3.43).
All forces are divided into four groups (see Fig. 3.43):
• physical;
• physical – informational;
• informational;
• informational – physical.
There are two types of forces inside these four groups: single (physical and
physical-informational); force system (physical-informational, informationalphysical).
Besides, physical and physical-informational forces are divided into pure
physical and system forces, i.e. reactions of constraints.
The forces of three groups (physical, physical — informational, informa-
tional — physical) don’t equal zero (=0), but the forces of the fourth group
3.12. Classification of “physical” bodies 115
+ PHYSICAL
PHYSICAL
SYSTEM
≠0; •
– INFORMATIONAL
=0; Σ;
+ - PHYSICAL - INFORMATIONAL
PHYSICAL
SYSTEM
Σ; ; ≠0
- + INFORMATIONAL - PHYSICAL
Fig. 3.59
System classification of forces
• ; ≠0
Figure 3.43
(informational) are always zero. All the forces appeared in a system on the base
of information operator of null action (the sum of internal forces,resultant force
equal to zero, etc.) are referred to the fourth group.
This classification doesn’t deny the classification by A. J. Ishlinsky [8], but,
in addition, it incorporates the operators of system mechanics.
3.12. Classification of “physical” bodies
In classical mechanics the integrational mechanics is submitted by special
system operators (general operator of information, information operator of null
action, system operator of stability and others), compact of Newton‘s laws, compact of Lagrange‘s mechanics, a system method of setting up the equations if
motion.
The classical course begins with concept of physical bodies, the integration
mechanics comes to this concept at the final stage. The inversion is obvious.
The integrational mechanics in comparison with classical mechanics considers the much greater circle of problems, therefore it is necessary to introduce
a special compact of “physical” bodies. The compact of “physical” bodies consists of three groups of bodies (physical group of bodies, mathematical group of
bodies). Each group is divided into subgroups, as a result we have six subgroups
116 Chapter 3
of “physical” bodies: physical and mathematical group of bodies; physical and
system group of bodies; mathematical and physical group of bodies; mathematical and system group of bodies; system and physical group; system and
mathematical group of bodies(Fig. 3.44).
In this paragraph we shall consider physical and mathematical bodies as
well as physical and system bodies.
The physical and mathematical group includes bodies of two mutually opposite physical properties (rigid body and deformable body). Besides, the bodies
can be integrated (a big body) ir differential (a small body). Physical and system group of bodies represents synthesis of physical and mathematical bodies
with the use of ideas of the system approach (rigid body plus plastic deformations at shock loading; rigid body plus elastic deformation at a place of contact;
deformable body plus the principle of removing of constraints etc.).
The introduction of a small (differential) rigid body in classification allows
to remove the paradox. A particle free only to translate not to rotate is attributed
to mathematical and system bodies.
The complex of physical and system bodies incorporates four groups of
bodies, two groups concern to rigid bodies but with the small additive if plastic
body (the Newton‘s stereo - mechanical theory of impact in which the factor
of restitution is less than unit always because of plastic deformation). The second variant is when the whole body is rigid but there is an elastic zone in the
place of contact. The two groups concern to physical and system bodies which
are designated by(+ −). The factor β concerns to these groups. It shows applicability of basic notions and laws of classical mechanics. For appropriate
factor β Newton‘s mechanics can be applied also for deformable bodies (+ −)
of small dimension.Strength of materials (bodies “− +”) uses physical and system bodies where separate ideas of the theory of elasticity are applied, but with
essential simplification of technical problems due to use of the principle of removing of constraints. The Sirs‘ synthesized theory was earlier mentioned. In
this theory the whole body is deformable and wave process is taken into account,
but at a place of contact wave process is neglected and Hertz‘ contact theory is
used.
Mathematical modelling, system and linearized theories in classification of
“physical” bodies
The mathematical and physical group of bodies is intended for carry of
complex problem of vibration, statics, stability, impact (a model of equivalent
rod in helical springs applies the ready mathematical theory of straight beams
for helical springs). This group of bodies essential uses unity of mathematical
3.12. Classification of “physical” bodies 117
Figure 3.44. Compact of “physical ”bodies
118 Chapter 3
solution (for example, wave equation) for various physical problems. Fig. 3.44
show how the model of equivalent bar for helical springs is formed.The height
of a rod (beam) and spring is identical, (longitudinal,displacement, transverse in
to planes) of a rod is equal to that of a string, weights of a rod and a spring
are equal. But physically these models are not equivalent,this leads to additional
paradoxes fir springs in relation to the well - known theory of straight beams.
Very much frequently for the analysis of parametrical vibrations a given problem
is simplified in order to use a ready mathematical formulas (Matye‘s equation
and its graphical solution as Ains – Strett diagram).
Mathematical and system group of bodies is submitted by a mass-point
particle, weightless spring, flexible inextensible weightless thread (stiffness on
a stretching is infinitely large, cross-sectional stiffness is zero) and other similar
bodies.
This group of bodies is used for for setting up equations if motion.The usual
method is that all mass of rigid body is concentrated in the center of gravity, and
elastic bodies possess only stiffness not mass.This group includes inextensible
thin threads for which the line of action of force and velocity vector trace the
form of this thread.
The system and physical group of bodies, on the contrary with first four
groups where there were individual bodies, represents systems of bodies having
united physical process. For example, not elastic impact of a weight on a rod,
system of bodies in classical mechanics.
For system and mathematical group of bodies the diagram contact force
versus compression of two bodies(Hertz‘ contact theory) is shown is shown in
Fig. 3.44. In this case we deal with two bodies from the group of physical and
system bodies as we have system of bodies.
The system and mathematical group of bodies includes system of bodies
such that the exact solution of a problem (impact of two bodies under Hertz‘
contact theory) can be replaced by the system of bodies of other physical nature(linearized theory of impact).This group of bodies allows to create effective
engineering techniques. In a brief complex on impact we considered linearized
theory of impact, but to get the simple and effective theory of impact it is needed
also knowledge of the most exact theory(Hertz‘ theory in this case). Two different are incorporated in linearized theory of impact. A classical example of a
system and nonlinear problem.
Chapter 4
EXAMPLES OF PROBLEM ANALYSIS
4.1. Kinematics of mass – point particle. Analogies
The main goal of solving sample problems is to give the students methods
of practical usage of theoretical notions. Since the method of solving problem
being under consideration is usually predetermined by the subject of discussion, the analysis of problem is often formalized that does not assist for deep
understanding.
The better results can be evidently achieved by solving problem on the
base of the individual accounts, with using support information and dialectical
techniques.
Students study some notions of classical mechanics at school, in the institute course of physics, and in course of classical mechanics. The points of view
and degree of thinking and understanding are different in all three cases. Therefore it is advisable to base on the knowledge achieved, to place them in order,
and to extend them instead of its ignoring. For example, instead beginning with
principle of conservation of linear momentum student can deduce this principle
as a himself result of solving certain problem.
There is a lot of support information (such as, for example, vertical or
horizontal plane, number of reaction forces, absence of external forces, absence
of acceleration of the body being at rest etc.) in the text of the problems and in
the figures. The only attention is needed to see this information. Besides, known
basic mechanics laws (even only Newton’s laws) can be considered as support
information.
The basic dialectical techniques which help to organize the solving process
are listed in [2]. Some of them are: dividing one complex problem into a number
of simpler ones, inversions, representation of information as a sum of the basic
(primary) and secondary ones, using analogies.
The attempt of problem analysis in accordance with these ideas is made in
the present chapter. Besides given solutions all the problems have ones based
on successive use of classical mechanics principles.
120 Chapter 4
Sample problem 1. Determine equations of motion of a point lying on the
rim of the car’s wheel, if the car moves at the speed of 20m/s along the ox –
axis, and the wheel has radius R =1m. Let the initial position of the point
on the axis be zero, and the wheel rolls without slipping (see Fig. 4.1) [9,
problem 10.13].
Solution. The car travels a rectilinear path,
y
therefore the problem is a two – dimensional one,
and solution should be of x = x(t),y= y(t).Weuse
the principle of dividing one complex problem into a
number of simple ones. So, we can consider the motion of point M on the wheel’s rim as a composition
of the motion of transport with the mass center and
x
the relative motion with respect to the same center.
M
C
R
O
(t)=xC(t)+x
x
Figure 4.1
M
y
(t)=yC(t)+y
M
For the constant speed of the center of mass (V
=20m/c) and a zero initial
C
M/C
M/C
(t),
(t).
coordinate:
x
=20t, yC= R.
C
The relative motion of point M can be regarded as a circular motion at a
constant speed. In its turn, this type of motion can be presented as a harmonic
one in projections onto the coordinate axes.
Such a harmonic motion along the ox – axis is illustrated in Fig. 4.2.
x
2
1
3
4
It is the sinusoidal curve of x
2
1
Figure 4.2
3
t
4
T
= −R sin ωt (where R — amplitude
M/C
of “oscillations”, ω — angular speed of point’s motion along the rim) which
satisfies initial conditions of the relative motion. There is negative sign in the
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