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1.4. Classification according to the type of nonlinearity 21
Table 1.1 contains ten different problems. The classical course of mechanics and the course of mechanics of deformable body cover problems 1, 2, 3; the
system analysis — problem 4, 5, 6. Integrational mechanics is intended to solve
problems 7, 8, 9, 10. This course of integrational mechanics has classical mechanics as an object. We shall also consider on case when mathematics, physics,
and philosophy are in unit for one object. The possibility of this fact was noted
by A. Poincare, famous mathematician, physicist, and philosopher.
Tab le 1.1
1
4
2
5
3
6
7
9
8
10
Chapter 2
SYSTEM APPROACH IN STATICS
AND KINEMATICS
2.1. Information operator of null action and axioms of statics
We use the axioms of statics from textbook [1, 2]. The information operator
of null action and its quantitative analog — the general operator of mechanics —
explain all axioms of statics.
The first axiom: a system of two equal opposite forces having the same line
of action and applied to the same rigid body, is equivalent to zero. This axiom
is the mathematical component (a null-vector) of the information null-operator.
The null-vector is widely used in [5].
The second axiom: a mechanical state of a body will not be disturbed if
a force system equivalent to zero is added (or subtracted) to the force system
acting on the body. This axiom is the information null-vector, having the same
dimension can be added or subtracted.
The third axiom: action equals reaction. In case of a system of two bodies
we obtain again the information operator of null action.
The fourth axiom: a system of two forces applied at the same point of a
rigid body can be replaced by one resultant force.
This axiom can be formulated inversely: a force can be resolved by infinity
ways onto two forces applied at any point on the line of action of the original
force. This axiom is a pure mathematical component of the information operator
of null action.
The fifth axiom: effect done on a rigid body by constraints is the same as
one done by some additional forces applied to the body instead of constraints.
This is a system component of the information operator of null action. The
axiom implies an unchanged action, and the replacement of constraints by additional forces permits to change the information only but not the action.
The sixth axiom (axiom of “solidification”): an equilibrium of mechanical
system will not change by applying new constraints; in particular, an equilibrium
of a system will not change if all parts of the system are rigidly connect each
other. New constraints give new information but action (the equilibrium of a
force system) remains constant.
2.2. System of concurrent forces 23
The technique of extending of body’s overall sizes is a system technique
too. The information about sizes is changed but the action is unchanged since
the whole space is frozen.
One can obtain all statics axiom from the information operator of null action
using the operator of inversion. Therefore the system mechanics does not give
up static’s axioms since they cleanly explain some parts and notions of statics.
But we make our understanding statics deeper if we consider the axioms in unity.
2.2. System of concurrent forces
A system of forces, whose lines of action intersect at the same point, is
applied to a rigid body (see Fig. 2.1). The physical component for a rigid body
does not change an information translated; the original information (by definition) gives that the lines of action of forces after sliding them along lines of
action to the common point is the pure mathematical component of the general
operator and also of the information operator of null action.The philosophy component (we’ll also use the term “system component”) realizes the principle of
compression of an information, i.e. sign “”.
R
O
A
n
F
1
F
1
A
2
F
2
A
F
n
3
F
3
Figure 2.1. Resultant of a system of concurrent forces
The condition of equilibrium of a system of concurrent forces is zero resultant force (R):
R =0;
n
Fix=0,
i=1
R =
n
i=1
n
i=1
F
Fiy=0,
n
i +
ix
F
i=1
n
j +
iy
n
i=1
k;
F
iz
Fiz=0.
i=1
(2.1)
24 Chapter 2
The basic mechanics operator for statics is illustrated in Fig. 2.2. Condition
has the effect of null action for all three components: mathematical null does
not change an action; then, there is no action for a rigid body rests in a frozen
space, but an information can change — this is a physical null; system null is in
the formulation of the equilibrium condition, it gives the freeze (solidification)
of the whole space. One can see that the system mechanics gives a rather
complicated explanation of simple expression (2.1), but the information operator
of null action is a unit operator for the whole statics.
Information operator
Information operator of null action
Basic operator of mechanics
= ; 0 ; · ; Σ ;
Figure 2.2. How the basic operator of mechanics for statics is formed
2.3. Moment of a force about a point and an axis
An algebraic moment of a force about a point (see Fig. 2.3) is (2.2):
(F )=±Fh. (2.2)
M
0
It does not depend on moving a force along its line of action (this is the
physical property of an object), it compresses an information (two informational
values ofF and h are replaced by a sign M
in degree(a mathematical component):
M
(F )=±2SΔOAB. (2.3)
0
(F ). An algebraic moment equals
0
2.3. Moment of a force about a point and an axis 25
B
F
h
O
Figure 2.3. Algebraic moment of a force about a point
A
where SΔOAB is the area of triangle OAB. A vector moment of a force about
a point (see Fig 2.4) is an operator of compression of information.
B
F
M
0
A
r
h
O
Figure 2.4. Vector moment of a force about a point
Sliding vectorF involves three informational invariants: a line of action
(rigid body does not distort this line), a direction, a module of force.
The vector moment of a force about a point carries the invariants in a
changed form: the area of triangle AOB remains constant, it corresponds to
moving force F along its line of action; line of action ofF forms with point O a
plane; direction of forceF gives a rotation ofF about O in the plane, the
position of this plane is determined by point O, being point of application of
new vectorM
absorbing the whole information.
0
If the information concern s with some axis perpendicular to the plane of
26 Chapter 2
the force and moving through point O, then the information operator of null
action states that an algebraic moment of a force about the axis equals one about
any point on this axis.
2.4. Reduction of two parallel forces. Couple
A couple of forces arises from the null-operator of forceF (see Fig. 2.5),
when point O of application.
−F
O
Figure 2.5. Forming of a couple
F
−F
O
h
O
F
A couple of forces arises from the null-operator of forceF (see Fig. 2.5)),
when point O of application of two equal and opposite forces splits into two
points. Then these two nulls O and O
became be distanced h apart. In statics
each point of a rigid body and its frozen closely space is a source of the information operator of null action and of its particular case — the operator of null
information. We “develop” the operator of null information of a force at point D.
The aim is to turn temporarily the parallel forces to intersect them. Then the operator of null information of a force should be cancel, and parallel forces should
be added (Fig. 2.6). Like any notion of an equivalence the equivalence of two
couples carries the sign “=”, i. e. it is the information operator of null action.
Different couples acting in one plane are equivalent to each other if they have
a constant action, this condition is quantitatively determined by the equality of
their algebraic moments. Mathematically a couple represents a free vector which
involves the following invariants: plane of action,direction of rotation, constant
algebraic moment of couple. A couple can be moved to the parallel plane, it can
be turned and moved in its own plane,besides, one can change the magnitudes
of forces and the arm of couple in such way that its algebraic moment and plane
of action will remain the same. These operations do not change the couple action on a rigid body (we consider a conventional rotation with unfixed axis of
rotation).
2.5. The basic theorem of statics 27
S
S
1
2
F
1
R
S
A
1
F
1
S
S
D
1
2
R
S
B
2
F
2
Figure 2.6. Adding of two parallel forces
The theorem about sum of couple moments states that the sum of vector
moments of forces, forming a couple, about any point does not depends on the
choice of a point and is equal to the vector moment of the couple. If shows that
the plane of a couple is equivalent to the line of action of a force. The adding of
two couples placed in the intersecting planes corresponds to the adding of two
intersecting forces. The condition of couples equilibrium
n
n
i=1
M =
Mix=0,
i=1
n
Miy=0,
i=1
M
;M =0, (2.4)
i
n
Miz=0. (2.5)
i=1
is the information operator of null action, since the system being under the action
of external couples is in equilibrium only if these couples equal zero.
2.5. The basic theorem of statics
Poinsot’s theorem (the basic theorem of statics) states that any arbitrary
force system acting on a rigid body can in general be reduced to the system
of one force and one couple. This theorem is the information operator of null
28 Chapter 2
action. Its physical component (the rigid – body property of translation of any
external information along the line of action or plane of action) is realized at the
point of reduction by means of the mathematical null-vector of a force.
The principal vector of a force system
n
F =
F
, (2.6)
i
i=1
changes the information but not the action, since the action is the external information, whereas we use operator R for the internal operations.
The condition of equilibrium of a force system: the system of forces applied
to a rigid body is in equilibrium, if the principal vector and principal moment of
the system are both zero for any choice of the reduction center:
R =0;L
=0. (2.7)
0
The external information is changed, but the action remains constant,i.e.
the equilibrium holds. Indeed, the information operator of null action gives all
formulas of a rigid–body statics.
2.6. Coplanar force system. Varygnon’s theorem
The reduction of a coplanar force system to the more simple one is the
technique of system approach: an information is compressed whereas the action
is not changed. The consideration of the case of a coplanar force system is important for the understanding the application of system mechanics for statics. In
this case it can be easily seen that a couple represents the equilibrium condition
for every point of a frozen space. If a coplanar force system is not in equilibrium
after reduction, we can compress the information again: in condition that prin-
cipal vectorR =0and principal momentL
and replaced by a single resultant forceR. This resultant force is fixed at the
point being distanced
d =
apart the original point of reduction.
Thus, the couple acted in the plane is hidden into this chosen point.But this
means that there is no possibility any rotation of a body now, or possibility of rotation of a plane with attached frozen space. A couple is an internal information
which can not change an external one, i.e. to act on object.
=0, the system can be simplified
0
L
o
R
(2.8)
2.7. Statically determinate and statically indeterminate problems 29
Varygnon’s theorem realizes operators “S”and“Θ” for a coplanar force
system: the algebraic moment of the resultant of a coplanar force system about
any point lying in the plane of the system, is equal to the sum of algebraic
moments of all forces about the same point.
2.7. Statically determinate and statically indeterminate
problems
In this case the mathematical component of the information operator is very
distinct, since the statically determinate problems are characterized by the equality of the number of equations and the number of forces to be determined. There
is no notion of statically indeterminate problems in system mechanics, since
the information operator of null action keeps unlimited amount of an additional
equations.
2.8. Center of gravity of bodies
The center of gravity of bodies is determined by the operator of null action.
It can be located as follows:
n
ri· P
r
i
=
c
i
, (2.9)
P
where r
— position vector of a gravity center, ri— position vector of i part of
c
a body.
The sign “” joints the multiplication P
,andalsoPi,wherePi=
iri
= P is a total weight of a body. The information can change, since subsystem,
separate parts of a body can be different, but the basic action — equilibrium
condition — remains constant.
2.9. Invariants of force system
This is a search for an information which never change. The force system
invariants can not be searched for in an internal information, since the force
system represents an external information. If something remains constant externally, then it keeps an internal information without any change.
The first force system invariant: the principal vector of a force system does
not depend on the choice of a reduction center. Hence, for centers O and O
we
1
30 Chapter 2
have
R
n
1
F
=
i
R
=R1. (2.10)
i
0
The second invariant: the scalar product of principal vector and principal
moment of a force system is the same for any center of reduction:
L
·R1=L0·R0. (2.11)
1
2.10. Statics paradoxes
System mechanics demonstrates the paradoxes of statics very clearly.In
property, mechanics consider a force moment about a point to be an attached
vector, but physicists doesn’t. In this case a mathematical component (free vector). Then, the motion of a couple is debating. Like a free vector a couple
moves in a plane to any point and turns through any angle. But a body can not
physically turn about any axis. There are two notions and physical phenomena
in a couple: couple as external information and hence as indivisible information
for a rigid body; couple(in combination with a force) as an element of internal
information without physical meaning, since couple gives the assistance for the
translation of a force to any point.
2.11. Peculiarities of kinematics as an ideal theory
Kinematics possesses of some attributes of an ideal theory. The simplicity
of kinematics deals with the absence of notion of a force. The position of kinematics is between statics and dynamics in the existing classification. But it isn’t
correct in a system approach. If the basic problem in statics is an equilibrium
state or rest, then kinematics opposes statics.There are no forces but motion in
kinematics; there are forces but no motion in statics. Kinematics gives enough
amount of an information, since if the displacements, velocities and accelerations are known in points of given masses, then one can find the dynamical
forces.
Kinematics have a certain system classification [1]: motion of a particle
with respect to a fixed system of reference, motion of a particle with respect to
a moving frame of reference, simple motions of rigid body,compound motion of
a body. There is a closely analogy between kinematics and statics, for example
in rigid body rotation about few axes.But kinematics does not possess of all
required properties of an ideal theory. Kinematics arose historically as a prelude
of more complex mechanics which operates with both forces and motion.
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