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2.16. A system way to derive Coriolis acceleration 41
using know technique [1]
d−→V
a
=
M
a
=
M
a
d
M
=
(−→V
+ ω ×ρ +−→Vr)=
dt
d−→V
dt
e
= aA+ ε × ρ + ω × [ω ×ρ]; aM= a
M
A
dt
dρ
dρ
=
dt
A
+−→ω × ρ + ω ×[−→ω × ρ]+−→ω ×V
+−→ω × ρ;
dt
dρ
= ω ×−→V
dt
+ ε × ρ +−→ω × [−→ω × ρ]+ar+ aC;
= a
ω ×
A
d−→V
a
+
dt
d−→V
dt
d−→V
r
=
dt
+−→ω × [−→ω × ρ];
r
d−→ω
× ρ + ω ×
dt
r
+−→ω ×−→V
+
r
d−→V
e
M
dt
dρ
dt
;
r
r
+−→ω ×−→V
+ ar+ ac.
r
d−→V
+
;
dt
=
r
(2.33)
When we derived the acceleration of a point in compound motion, the
principle of inversion has been applied. We have got initially basic components
of the equation and only then all values of the equation are obtained. Whether
has Coriolis acceleration physical sense? Coriolis acceleration is informational
acceleration sense it appears if the initial information is specified for subsystems
not for the system as a whole. Coriolis acceleration gains physical sense only
if a system of reference could be chosen such that the absolute acceleration
with respect to it coincides with Coriolis acceleration. But this is impossible
since Coriolis acceleration cannot exist without normal acceleration. Therefore
Coriolis acceleration should be considered as informational acceleration as well
as physical one (informational and physical acceleration).
r
w
r
e
a
C
The null-vector for Coriolis acceleration
Let’s construct the null-vector for Coriolis
acceleration (see Fig. 2.18): a thin disc rotates about axis O
point B has linear velocityV
with angular velocity we,
1O2
in the plane of
e
the disc, a conventional hinge is at point O
which permits two simultaneous opposite rotations with the same angular velocity w
axes OO
and O2O. There are two null-vectors
1
e
about
of information — for Coriolis acceleration, for
angular velocity about a fixed axis and an instantaneous axis — in Fig. 2.18.
r
a
r
-w
e
Figure 2.18. Null-vector of
Coriolis acceleration
C
V
r
r
42 Chapter 2
2.17. The analogy principle and compound rotational motions
of a rigid body
It is noted in textbook [1] that there exists a complete of concurrent forces
in statics and the reduction of a system of instantaneous angular velocities of
bodies. In system mechanics an analogy is a technique of the system component
of the information operator, but the analogy as an information object has three
components: mathematical, physical, system.The notion of a complete analogy
is incorrect in system mechanics. The analogy between the reduction of a system
of concurrent forces in statics and the reduction of the instantaneous velocities of
bodies is only mathematical analogy. The cylinder rotates (see Fig. 2.19) about
axis OZ
motion is a rotation about axis OZ
and about axis OZ2simultaneously, the axes interest, hence the total
1
with total angular velocity
3
Ω= w
+ w2. (2.34)
1
r
w
r
w
2
r
3
w
1
Figure 2.19. Compound motion of a cylinder about two intersecting axes
We obtain the pair of rotations from the null-vector of an instantaneous
angular velocity, if half — axes OO
Fig. 2.20). Now OO
and O1O2are the axes.
1
and O1O2are separated for distance d (see
1
The pair of the instantaneous rotations of a rigid body is equivalent to the
instantaneous translation with the velocity which equals the vector moment of
the pair of angular velocities. In [1] it is written as
V =M( w
; −w1). (2.35)
1
In Fig. 2.21 the plane of the instantaneous angular velocity vectors of w
and −w1realizes the fact of simultaneous body rotations with equal and opposite
angular velocities about two parallel axes. Therefore all points of this plane
1
2.17. The analogy principle and compound rotational motions of a rigid body 43
r
w
1
M(; )ww
-
11
-w
r
1
r
w
1
d
Figure 2.20. Pair of rotation
w
1
Figure 2.21. Rotation of a
body out two parallel axes
have no rotation,but the plane possesses the instantaneous translation with linear
velocityV determined by (2.20).
The superposition principle (sign “S” in the basic operator of mechanics)is
used for the reduction of an arbitrary set of rigid body motions to a simple motion: a set of only number of simultaneous rotations and translations of a rigid
body can be reduced to two simultaneous motions (rotation and translation [1]).
There is also the analog in statics: a force and a couple as an indivisible information.
Chapter 3
DYNAMICS
3.1. Newton’s laws systematization
Let’s consider a mass-point particle in the abstract condition of its isolation
i. e. in its infinite away from other bodies. This may be in two particle states.
The first one is rest. The second is a straight line motion with constant velocity,
called the inertial motion. These two possible states of a particle isolated are
postulated by Newton’s first law of inertia.
When a particle is placed in space near other bodies acting on it immediately or by means of a field, the velocity of the particle will change in a general
case. The cause of this change is the mechanical action on the particle by ambient bodies. It is common to determine a force as a measure of this internal
action on the particle. It is needed to mention that the inertial motion of the
particle is also possible when the applied forces are balanced. In this connection
it is possible to give the dynamical treatment of the statics axiom about the two
forces equilibrium: two forces applied to a rigid body are balanced in case if
they do not change the body velocity or do not accelerate it. The quantitative
connection between the force applied to the particle and the change of its motion
is given by Newton’s second law, or the basic dynamic equation
ma =F. (3.1)
The change of motion is proportional to the force applied and takes place in
the direction of the force action [1]. Thus, the force and the acceleration caused
by it are always collinear.
It is incorrect to consider equation (3.1) as a definition of force because in
mechanics the action of a force causes not only a body acceleration but also its
deformation without any acceleration.
Newton’s third law states that if one particle exerts on another particle, then
the second particle exerts a force on the first one the force equal in magnitude,
opposite and collinear with the first force. This law is often called the law of
action and reaction. However, Newton’s third law has a more deep physical content that mere equality of forces of action and reaction. Let’s write equation (3.1)
3.1. Newton’s laws systematization 45
for each of the two interacting particles:
dt
d
(m
)=F2. (3.2)
2v2
d
(m
1v1
dt
By Newton’s third law,F
d
(m
+ m2v2)=0, whence m1v1+ m2v2=const. (3.3)
1v1
dt
)=F1and
= −F2. Now let’s add equations (1.2) termwise:
1
Newton’s law can be formulated as a law of conservation of the sum of
interacting particles linear momenta when the external forces are absent [1]
Newton deduced the principle of linear momentum conservation from his own
third law. Unlike he, Descartes regarded the “movement conservation principle”
as a basic dynamics law. We differentiate equation (3.2) with respect to time
using the technique of inversion. Then
m
+ m2a2=0.
1a1
Rewriting this equation with the second term in the right part
= −m2a
m
1a1
2
and taking into account that −m2a2= −F2and that by Newton’s third
law −F
particle:
IfF
=F1, we finally obtain the expression of Newton’s law for the first
2
=F1.
m
1a1
=0,thenm1v1=constor, in particular case, v1=0, i. e. we obtain
1
Newton’s first law. However, it would be incorrect to regard the first law as
a particular case of the second law, since equation (3.1) acquires the proper
physical treatment only if the inertia law is satisfied.
Let’s show that simple problems can not be solved by classical mechanics
methods within a rigid body model. We shall consider the central impact of two
balls as an illustration. Since there are no internal forces acting on the system
of two balls, the linear momentum of the system just after impact should be the
same as one just before impact (see Fig. 3.1):
where v
1,v2
collision, v
collision.
m
+ m2v2= m1v
1v1
are the projections of the balls velocities onto the x-axis before the
,v
the projections of the balls velocities onto the x-axis after the
1
2
1
+ m2v
, (3.4)
2
46 Chapter 3
y
V
m
1
P
1
1
Figure 3.1
m
2
There are the projections of the velocities v
1
P
2
and v
V
2
x
in equation (3.4),
2
therefore the problem becomes indeterminate.
To solve this problem Newton introduced the additional information about
the coefficient of restitution:
K =
v
v1− v
1
− v
2
.
2
Although the notion of coefficient of restitution K was introduced by Newton in
the terms of kinematics, this coefficient really depends on the elastic properties
of the colliding bodies and is determined experimentally.
This example shows that as early as in XVII century Newton used the
techniques of system approach (how we call them now), and in property, the
introduction of an additional information.
Thus for the systematic summation of Newton’s laws we used mainly the
two typical procedures of system approach: the inversion and the introduction
of an additional information.
3.2. Informational compact of Newton’s vector dynamics
Let’s introduce a system operator which can be represented by the following
set of mathematical operators and symbols:
d
dt
0 =0
•
. (3.5)
3.2. Informational compact of Newton’s vector dynamics 47
The two mutually opposite mathematical operators of differentiation and
integration are placed in the first line of the compact. The second line symbolizes
that that some expression (value) is equal to zero, or it isn’t. The third line
symbolizes the transfer from one particle to the system of particles using the
mathematical operator or summation, and then to the particle (point) being a
mass center of the mechanical system.
We can represent the laws of Newton’s vector dynamics in the compressed
form (3.6) using mass m and position vector r of a particle as a basic infor-
mation, and also using the system operator (3.5). Expression (3.6) is called the
information compact of Newton’s dynamics.
−→ d/dt
r
mrm˙rm
0 r ×m˙rr × m¨r
←−
˙
r
¨
r
¨
. (3.6)
r
The first line of compact (3.6) is formed by a basic information r differ-
entiated and integrated by the mathematical operators
the first line of system operator (3.5).The use of the mathematical operator
d
,which is placed in
dt
d
dt
for r gives velocity vector˙r and acceleration vector¨r. Inversely, the use of
the mathematical operatorfor¨r gives velocity vector˙r and r. Thus, the first
line of the compact is formed by the kinematics characteristics of a motion r, v,
and a.
The multiplication of elements of the first line by a scalar coefficient ”m”
(mass) gives the elements of the second line: mr — a statical moment of a
particle about the same center, m˙r — a particle momentum vector, m¨r — a vector
which is equal in magnitude and direction to a force acting on a particle.The
elements of the third line are obtained by vector multiplication of the elements
of the second line by r : r × mr =0,r × m˙r — a moment of momentum
about the same center, r × m¨r = m
(F ) — a vector moment of a force about
0
the same center.
Let’s consider the laws of Newton’s dynamics compressed by compact
(3.6). Let’s apply the first and the second lines of the system operator (3.5)
for the second and the third lines of the compact. Really, we get the expression
48 Chapter 3
d
m˙r = m¨r. Then we can rewrite it in the form
dt
integral form mv
− mv0=
1
expression of the principle of linear momentum : the change in the linear mo-
t
2
Fdt. The latter equation is the mathematical
t
1
mentum of a particle during some time interval equals the impulse of a resultant
force acting on the particle during this time interval. If t
is called the finite impulse of a instantaneous force, or the impact. Using the
operator of summation, we obtainQ =m
of a system of particles, andR =
forces acting on a system. Then
=m
and Mvc=mkvk, we obtain Mvc=R — the mathematical
krk
dQ
=R, and taking into account that Mrc=
dt
v — a principal momentum vector
k
F — a principal vector of the external
d
(mv)=F , and in the
dt
= t1,then
2
t
t
1
2
Fdt
form of the principle of the motion of a mass center for the collection of particles: the center of mass of a mechanical system moves as though all the mass
were concentrated in a single fictitious particle at the center of mass and the
principal vector of the external forces acted on that fictitious particle. Thus, the
third line of the system operator (3.5) is realized.
In the particular case of zero principal vector of external forces we obtain
the linear momentum conservation principle and the principle of conservation
of motion of a mass center: mv
− mv0=0and ac=0,orvc=const.
1
The similar operations can be fulfilled also for the third line of compact. In-
deed,
d
(r × m˙r)=r ×F ,or
dt
d
(r ×mv)= m0(F ) gives the mathematical
dt
expression of the moment of momentum principle for single particle: the time
rate of change of the moment of momentum about the same center equals the
moment of a force acting on the particle, about the same center. Using the op-
erator of summation we obtainL
moment of momentum for the collection of particles, andM
the principal moment of external forces acting on a system. Then
=(rk× mkvk) — the principal vector
0
=m0(Fk) —
0
dL
0
=M
dt
is the mathematical equivalent of the moment of momentum principle for a
mechanical system.
If the principal vector of external forces equals zero, then we obtain the
dL
moment of momentum conservation principle for a system:
0
=0,i.e.L0=
dt
=const.
Thus, compact (3.6) and system operator (3.5) help to represent the basis
notions and laws of Newton’s dynamics in instructive compressed form.
The symbol “ =0
of the second line of system operator (3.5) corresponds
0
3.3. The basic information compact of dynamics problems 49
to the laws of change (f.ex. motion of a mass center principle), but the sym-
bol “0
corresponds to the laws of conservation of the analogous entities.
3.3. The basic information compact of dynamics problems
Description of the compact
The aim of the compact is to present the course of classical mechanics in a
compressed form.
For an engineer the most interesting things are:
• mechanics of a rigid body;
• mechanics of a deformable body;
• engineering mechanics (one part of information is taken from rigid-body
mechanics, another - from mechanics of deformable body);
• integrating mechanics in terms of which both rigid body mechanics and
deformable body mechanics can be considered.
The basic information compact of dynamics problems contains:
• determination of the limits of dynamics methods applicability;
• methods of composition of the differential equations of a motion;
• analysis of the differential equations of a motion;
• methods of the differential equations solving;
• information block of the particular dynamics problems.
Criterion of appliance of classical mechanics basic notions
The first statement of the basic compact: the coefficient of applicability of
the classical mechanics basic theorems is β =
2θ
,whereT
— the time of
imp
T
imp
impact, θ — the time of running a deformation wave.The base of an information
is an experiment. If β>3·· 8, then we use the methods of Newton’s classical
mechanics, but T
and θ are determined from a real process and connected
imp
with the notions of deformable body mechanics. Waves run on the body many
times, stresses increase, and the wave process itself does not present a danger
(see Fig. 3.2).
50 Chapter 3
ε
t
2θ
T
y
Figure 3.2
Methods of composition of equations of motion
There are two types of equations in classical mechanics: equations of motion, equations of energy.
If we concern with constraints, then the two types of mechanics are also
distinguished: mechanics of a free body, mechanics of a constrained body.
F = m¨r – mechanics of a free body,
F +R = m¨r – mechanics of a constrained body.
In both cases the introduction of D’Alembert’s principle allows to reduce
the equation of motion to the statics equation:
Φ=−m¨r,
F +R =0
— mechanics of a free body,
F +R +Φ=0 (3.8)
— mechanics of a deformable body.
Equations (3.7) and (3.8) — Newton’s mechanics. It has a physical sense.
Let’s use the information operator
[ Mathematic — Physics — Dialectics ],
that is formalize mechanics and a physical component. It was done by
Lagrange.
(3.7)
Analytical mechanics
Analytical mechanics – special coordinates, notion of ideal constraints, virtual displacements δr are introduced.
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