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3.9. Compact “Stability” 101
α
Stability of Archimedes
A gun is conveniently shown in Fig. 3.38. A shot create a turning moment
M = P · h.
Moment M should be balanced otherwise the gun will turn over after a shot.
Really the angle α of a gun inclination should be taken into account.
→
P
h
Figure 3.38
Stability of motion (stability by Lyapunov) is widely discussed in the course
of auto - control.
Stability of a deformable body (stability by Euler). It is the stability of
columns, roads, shells, rings, statically and dynamically loaded.
Stability of an object. In system mechanics of an object all the basic phys-
ical problems are considered in unity (see Fig. 3.39). An informational pyramid
has the following sides: vibration, stability, statics (strength), and impact (as a
base).
Integrational mechanics of an object considers also such types of stability
which have no bifurcation points, and combinations of bifurcation and non bifurcation problems.
Stability in integrational mechanics deals with system and nonlinear prob-
lems, i.e. with problems when their bifurcation take place. In that type of prob-
102 Chapter 3
stress
stability
vibration
impact
Figure 3.39
lems stability phenomena are explained by the subproblems of different physics
(see Fig. 3.40).
The two latter types of stability are considered in [7].
System approach in stability by Lyapunov
Let’s consider stability of motion by Lyapunov. At first we regard the
energetical method to derive a differential equation of free vibration of linear
oscillator(see Fig. 3.41).
2
T + V =const,T=
d
(T + V )=0,V=
dt
mx
cx
,
2
2
;
2
˙x(m¨x + cx)=0, ˙x =0;
¨x + p
2
x =0,p2=
c
,t=0 x = x
m
, ˙x =˙x0.
0
We have two solutions:
˙x
x = x
cos pt+
0
0
sin pt;
p
x =0.
Let’s bring some perturbation into the problem having the first solution. If
this problem obtains the second solution x =0, then we can state that the initial
3.9. Compact “Stability” 103
System – linear problems
Physical – linear
problems
Mathematical –
linear problems
Stability by
Archimedes
Stability in system mechanics of an object (combination of bifurcational and non bifurcational problems)
Physical – nonlinear
problems
Mathematical –
nonlinear problems
Stability of bifurcational
problems
System – nonlinear problems
Stability in integrational mechanics (System
nonlinear problems)
Stability of motion
(discrete systems)
Stability in bifurcational problems
Deformable
bodies
Figure 3.40
free motion was stable. This sense of Lyapunov’s method is given in terms of
integrating mechanics. The motion of stability by Lyapunov we’ll consider using
examples.
The equations of a perturbed motion are [7]:
dx
dt
= y − x
dy
3
,
= −x − y
dt
3
.
104 Chapter 3
M
x
C
Figure 3.41
Is the position x = y =0stable?
We construct Lyapunov’s function (V )
2
V = x
+ y2,
which must have a positive sign only.
∂V
=2x(y −x
∂t
∂V∂xdx
∂V
=
∂t
∂V
=2x,
∂x
3
)+2y(−x − y3)=−2x4− 2y4,
∂V
∂t
dt
< 0.
+
∂V
∂y
∂V
∂ydydt
=2y,
Hence, the position x = y =0is stable. We have a fork:
,
+V ;0;−
∂V
∂t
.
Information operator of null action and stability problems
We have two operators of null action (a rest and a rectilinear motion) for
Newton’s first law. Both these internal operators of null action are parts of the
united operator of null action.
If an action in a system is periodic (this is already some information about
operator of null action) and there is the first null operator (x = y =0,the
system is stable), then the periodic motion of the system is stable. The very
simplified analogy can be discovered in the case of general plane motion of a
body. This type of motion can be reduced a rotation (the second action) about
3.9. Compact “Stability” 105
a fixed point (the first action). If an instantaneous center of motion does not
change its position (x = y =0), then we have a stable motion (a stable action).
Stability of deformable bodies. Critical force of rod compression
We use the technique “to find the solution of the problem without solving
it”. There is no buckling (critical) force P in the assumed equation of transverse
vibration. This equation is:
EJ
∂
∂x
4
y
+ ρF
4
2
y
∂
=0. (3.69)
2
∂x
Here EI — transversal stiffness of a rod at plane where force P acts,
y — transverse displacement of a rod cross section, x — transverse displace-
ment in the direction of force P action.
We use the usual substitution
y = X ·T,
where X depends only on position in the rod, i. e. on x — coordinate, and T
depends on time only. We obtain two independent equations:
IV
X
− k4X = o,
2
¨
T + p
T =0,
2
p
4
k
ρF
=
EI
.
For the special choice of boundary conditions (hinge embedment in both
ends of a rod) the frequency of transverse vibration equals
4n4
π
2
=
p
EJ
·
ρF
.
4
l
We apply a dynamic method: buckling force of a rod compression (critical
force) P
is determined from the condition that the transverse frequency equals
cr.
zero.
We derive a formula for the frequency square with P taken into account:
4n4
π
2
=
p
EJ
·
4
l
(1 − A) , (3.70)
ρF
106 Chapter 3
where force P is presented in the dimensionless value of A. Physically P
must be placed at the numerator of A; whereas the information of EI and some
information providing the dimensionless of A must be in the denominator. Let’s
rewrite (1.68) in another form:
2
ρF p
= k2k2EJ(1 −A) .
It is enough to write k
2
EI outside the round brackets to turn the expression
of A into 0 a dimensionless one. It means mathematically that the appearance
∂2y
of P in the assumed equation corresponds to the second derivative (
∂x
);and
2
the difference between the orders of the derivatives (the fourth order in equation
(3.69) and the supposed second order) gives the required information about the
value of k
2
.
Thus,
4n4
π
2
p
=
EJ
·
ρF
· (1 −
4
l
2
Pl
π2n2EJ
) ,
whence
2n2
π
P
=
cr.
EJ
.
2
l
Nonclassical types of stability loss. Stability of bifurcation problems
A distortion of an original form is referred to such a type. For the 3 – rods
this is a distortion of a form in conditions of compression of a fixed rod [2].
Technological types of stability loss
These types can be discovered in integrating mechanics of an object. One
can chose the technological parameters of the forming of an inter — coil pressure
(in spring with one coil) such that the spring will be flat. It makes possible to
exclude an operation of grinding [2].
Operational types of stability loss
They are discovered in a complex analysis of an object [2]. Such types
represent combinations of different types of stability loss. The instable spring of
Kalashnikov’s gun is upon the mandrel. It has a distorted form (due to the loss
of stability by Euler), coils near nodes of deformation waves are concentrated.
This results in the spring spoilage.
3.10. Analytical mechanics as an “ideal” theory 107
The special types of a stability loss connected with the mechanisms char-
acteristics are discussed in this paragraph. There exist 14 types of stability loss
in spring mechanisms [2].
We considered the elements of mechanics of future in paragraphs 1.7, 1.8,
1.9. It studies all physical aspects (vibration, stress, stability, impact) in unity
and solves the problem of design, technology, exploitation of a mechanism.
3.10. Analytical mechanics as an “ideal” theory
Dynamics of constrained system as an “ideal” theory
The summary of analytical mechanics as an “ideal” theory corresponds to
course [2].
Earlier the notion of “an ideal” theory was introduced. Its basic charac-
teristics are: simplicity, enough amount of information, the overcoming of an
essential contradiction, support solutions (well - known before), turning harm
into benefit, system organization of the theory developing.
Dynamics of constrained systems is one of the examples of the ideal the-
ory in mechanics. Let’s consider the basic characteristics of an ideal theory in
application to dynamics of constrained systems.
The simplicity of the theory is ensured by the extension of ideas and meth-
ods of dynamics of free systems to the dynamics of constrained systems (i. e.
beyond the scope of applicability of the first theory).
The informativity of dynamics of constrained systems is provided by an
essential compression of information (generalized coordinates,generalized velocities, generalized forces), as well as by using its ideas for solving statics
problems (Zhukovsky’s lever).
The presence of constraints prevents the complete transfer of ideas and
methods of free-system dynamics to the dynamics of the constrained system.
The introduction a principle of removing constraints and the use of energetic
level of the theory (with some restrictions imposed upon constraints) allows to
make this transfer.
The ideas and methods of free — system dynamics are the basic solutions
in dynamics of constrained systems.
Unknown reactions are harmful for an analysis of dynamics of constraint
systems; but this harm is transformed into benefit by the going to an energetic
level and by the introduction of notions of ideal constraints and virtual displacements. The reached benefit lies in the fact that reactions of ideal constraints are
not used in the differential equations of motion and the procedure of forming
108 Chapter 3
the equations is considerably formalized. A system organization of developing
dynamics of constrained system includes:
• An introduction of generalized coordinates (a set of independent parame-
ters specifying the position of a system in space). If in Newton’s vector
dynamics linear, angular and arc coordinates are used, then in the dynamics of constrained system they are generalized. It allows to formalize the
procedure of forming differential equations.
• The use of principle of removing constraints which allows to replace con-
straints by their reactions.
• The classification of constraints by “yes-no” principle: stationary – nonsta-
tionary (dependent on time),holonomic – nonholonomic, fixed – unfixed,
ideal – nonideal.
• The introduction of imaginary infinitesimal displacements of points, which
are permitted by constraints imposed on a system; the actual displacements
caused by applied forces can be among these displacements; the imaginary infinitesimal displacements consistent with instantaneously fixed constraints, and displacements caused by change of constraints.
• The “compression” of information about the infinitesimal imaginary dis-
placements caused by changing the constraints; this permits to introduce the
notion of possible displacements of a system (some sort of virtual displacements) as the imaginary infinitesimal displacements of points, consistent
with instantaneously fixed constraints.
• The introduction of the constraints equations in Cartesian coordinates
f(t, x, y, z, ˙x, ˙y, ˙z)=0,
and in generalized coordinates
Φ(t, q
, ˙qi)=0.
i
• The introduction of the notion of a function variation:
δf =
∂f
∂x
δx +
∂f
∂y
δy +
∂f
∂z
δz ,
3.10. Analytical mechanics as an “ideal” theory 109
which distinguishes from
df =
∂f
∂t
dt +
∂f
∂x
dx +
∂f
∂y
dy +
∂f
∂z
dz .
• Since in the first case time is fixed, the virtual displacements are considered
for a fixed time; the fixing of a process represents a typical technique of
system approach.
• The introduction of the notion of ideal constraints; the sum of elementary
works of ideal constraints done on any virtual displacements of points of a
system, equals zero,
n
R
· δri=0. (3.71)
i
i=1
• The introduction of the notion generalized force Q
constraint Q
•
; if the variation of the position vector of i −th point of a
j
generalized reaction of
j
system is
k
∂r
j=1
k
j=1
∂q
(
n
i=1
i
j
δqj,
∂r
i
F
) δqi, (3.72)
i
∂q
j
and for the resultantF
n
i=1
=
δr
i
of active forces acting on i-th point we have
i
F
· δri=
i
then the generalized force is
Q
and the generalized reaction is
Q
n
∂r
i
F
=
j
i=1
n
∗
j
R
=
i=1
, (3.73)
i
∂q
j
∂r
i
, (3.74)
i
∂q
j
110 Chapter 3
If a system is constrained by ideal, holonomic (integrated) constraints,
then all generalized reactions corresponding to independent virtual displacements of a system, equal zero,ˇ-2mm
∗
=0 (j =1, ..., k) . (3.75)
Q
j
• The use of D’Alembert’s principle, which allows to obtain dynamics equations in the form of statics equations. The principle that states the sum of
active forces and inertia forces applied to the points of a constrained system
are balanced by reactions of constraints
F
+Φi+Ri=0. (3.76)
i
• The formulation of virtual displacement principle, which gives common
statics equation and also with D’Alembert’s principle gives new possibilities for the dynamics of constrained systems; this principle states that a
mechanical system at rest in an initial configuration where certain active
forces are applied, and constrained only by “fixed-guide” constraints, is in
equilibrium if, and only if, the total virtual work of all the active forces is
zero for any arbitrary kinematically admissible virtual displacement from
the initial configuration,
n
F
· δri=0. (3.77)
i
i=1
One of the applications of principle of virtual displacements for one – degree – of – freedom mechanisms is the rule of “Zhukovsky’s lever”. This
rule asserts that if a mechanism is at rest acted upon by a force system,
then its velocity diagram turned through 90
0
about a reference point and
considered as a right body loaded by the same forces at the same points as
the original mechanism, is at rest too. But how can the velocity diagram be
constructed for a body at rest? This can be done because we replace virtual
displacement δr
by actual displacement dri=Vidt in the equality
i
n
i=1
F
i
· δri=
n
Fi· δri· cos(Fi,δri) ,
i=1
since in case of “fixed – guide” ideal constraints an actual displacement is
one of virtual displacements. Then, eliminating dt we obtain the expression
n
Fi· Vi· cos(Fi,
i=1
V
)=0,
i
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