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3.8. Compact “Linear and nonlinear problems in dynamics” 91
Ε
ρ
M
C
Figure 3.25
F,,
l
Figure 3.26
Longitudinal vibration are described by the wave equation:
2
∂2u
2
a
∂x
∂
u
=
2
∂t
,a2=
2
E
.
ρ
Torsional vibration of a rod is described by:
2
∂2ϕ
2
a
1
∂x
∂
ϕ
=
2
∂t
,a
2
G
2
.
=
1
ρ
We can divide a rod into several pieces and then consider it as a system of discrete masses connected by massless springs (see Fig. 3.27). Fig. 3.25 illustrates
a system with a single natural frequency, but Fig. 3.26 — one with the infinite
number of natural frequencies.
What is the difference between non - system and system problems? The two
basic types of non - system problems are represented in Fig. 3.25 and Fig. 3.26.
The problem shown in Fig. 3.25 can be transformed into a system problem if
we are interesting in both vibration of a mass supported by spring, and separate
vibration of a spring element, in case the mass of this element is taken into
account. Both these processes take place simultaneously.
92 Chapter 3
M
x
m
1
C
1C2m2
Figure 3.27
mnCnm
1−n
Difference between linear and nonlinear problems
A mathematically nonlinear vibration and physically nonlinear vibration
are not distinguished clearly in the existing theory of vibration. So let’s consider the difference between linear and nonlinear vibration not concerning with
mathematical and physical aspects.
The principal distinguishes we’ll consider for the example of the simplest
model (see Fig. 3.38).
C
)(tF
Figure 3.28
Linear problems
1. Superposition principle.
c
2
2. p
=
,p— natural frequency is determined by the parameters of design.
m
3. Single natural frequency.
m¨x + cx =
n
x =
i=1
n
i=1
F
i
m(p2− w
Fisin wit
sin w
2
)
i
t.
i
3.8. Compact “Linear and nonlinear problems in dynamics” 93
Nonlinear problems
n
x =
i=1
F
i
m(p2− w
i
2
)
sin w
t.
i
1. Superposition principle cannot be used.
2. Natural frequency depends on initial conditions and time.
2
p
= f(c, t, m),
2
= f(x0, ˙x0,c,t,m)
p
3. A number of natural frequency values.
System property of solution: a periodic nonlinear problem always has a
linear solution as the first approach.
Linear problems
4. Isochronism of vibration (a frequency doesn’t depend on an amplitude of
vibration).
5. Free vibration are damped by a friction force.
Nonlinear problems
4. Isochronism is absent.
5. Stable nonlinear vibration can occur in the condition of a friction force
acting.
Phase portrait of vibration
Linear oscillator (see Fig. 3.29)
The phase portrait is a plot of velocity versus displacement for a number of
time instants.
˙x
= x
0
sin pt,
p
2
˙x
0
2
+
0
.
2
p
x = x
˙x = −x
2
x
cos pt +
p sin pt +˙x0cos pt,
0
2
˙x
+
2
p
94 Chapter 3
x
&
x
Figure 3.29
&
xx ,
oo
The phase portraits for linear damped vibration and for nonlinear damped
vibration are shown in Fig. 3.30 and Fig. 3.31, respectively.
.
x
x
.
x
x
Figure 3.30 Figure 3.31
Nonlinear vibration of a simple pendulum (see Fig. 3.32)
The differential equation of vibration:
2
¨ϕ + p
sin ϕ =0,p2=
sin ϕ = ϕ −
ϕ
g
,
l
3
.
6
3.8. Compact “Linear and nonlinear problems in dynamics” 95
Solution:
dx
dt
= y;
dy
dt
g
= −
sin x | x = ϕ, y =˙ϕ | .
l
The curve of an energy balance (V = f(x)) correspond to function:
V (x)=z = −
g
l
cos x.
This is a curve of cosine with isolated minima at the points:
x =0; ±2π; ±4π; ···,
and with maxima at the points
x =0; ±π; ±2π; ···.
We obtain stable and instable solutions. The characteristic property of the most
linear problems is that the solution of a linear problem coincides with the first
stable solution (vibration) of a nonlinear problem.
Classical types of nonlinear vibration
The four following types of nonlinear vibration are examined frequently:
1. pseudo-harmonic vibration, the system parameters depend on an amplitude
of vibration, —
m¨x + c(x)x =0;
2. quasi-harmonic vibration, the system parameters depend on time, –
m¨x + c(t)x =0;
3. parametric vibration, a particular case of quasi-harmonic vibration;
4. auto-vibration (vibration of a system having an internal energy source).
Let’s consider parametric vibration
In case of one - degree - of - freedom system the differential equation has
the form of Matje – Hill’s equation, and its solution is graphically represented
96 Chapter 3
l
mg
Figure 3.32
in Ains – Strett’s diagram (see Fig. 3.33).
z = Acos wt,
3EJ
m¨x + cx =0,c=
m¨x +
(l0+ A cos wt)
3EJ
m¨x +
(1 −
3
l
0
We introduce dimensionless time wt =2τ and write the assumed equation
in the form of Matje’s equation
,l= l0+ A cos wt,
3
l
3EJ
3A
l
x =0,
3
cos wt)x =0.
0
2
d
x
2
dt
2
d
x
+(a −2q cos 2τ)x =0.
2
dτ
2
2
d
w
=
x
·
,
dτ
2
4
We are to chose the values of parameters a and q, and to use then Ains – Strett’s
diagram (see Fig. 3.34):
a =
12EJ
mw2l
,q=
3
0
18AEJ
mw2l
.
4
0
Theory of elasticity states that two waves extend in a deformable body. Values
of its velocities are not far from
E
2
a
,a
=
1
ρ
G
2
.
=
2
ρ
3.8. Compact “Linear and nonlinear problems in dynamics” 97
3
O
O
z
O O
2
1
l
x
Figure 3.33
Longitudinal vibration of a beam are described by the equation
2
=
2
p
x + C
a
∂
∂t
u
.
2
p
sin
x,
2
a
∂2u
2
a
∂x
Using substitution u = X · T we replace the latter equation by the set of two
differential equations having known solutions:
2
p
X
+
x =0,¨T + p2T =0,
2
a
X = C
T = D
cos
1
cos pt + D2sin pt .
1
98 Chapter 3
q
2
q
à −=
2
0
à
1
qà += 1
Figure 3.34
à −=
qà −=1
2
q
4
2
à +=
2
q
5
4
12
For boundary conditions of a fixed embedment we have:
x =0⇒ X =0⇒ C
x = l ⇒ X =0⇒ X = C
p =
2
πna
l
sin
p
l =0⇒
a
.
1
=0;
p
l = πn,
a
Vibration has a sinusoidal form:
X =sin
p
x =sin
a
πnx
l
.
Nonlinear problems are physically deal with a bifurcation of solution.
The simplest example (a column loaded by a parallel force, see Fig. 3.35).
The loss of stability (if P = P
) can occur in two stable positions:
crit.
a straight line, and a bent axial line.
Dynamic method of determining critical parameters of compression: fre-
quency w =0, absence of the elastic energy which brings the rod (column) to
the initial configuration.
Integrational mechanics has the following scheme of the theory of waves
for rods (see Fig. 3.36, 3.37).
3.8. Compact “Linear and nonlinear problems in dynamics” 99
P
r
P
r
l
The types of nonlinear vibration vary
greatly. The most interesting ones nowadays are:
a great solitary wave (soliton), which translates
a nerve impulse in a living organism; auto —
oscillations in chemical vibration; nonlinear oscillations in chaotic processes resulting in selforganization phenomena (the latter are studied by
a new science – synergetics).
System and nonlinear problems are also met
with in theory of vibration. In classical linear
and nonlinear vibration, one resonance disturbing frequency awakes one frequency of a system response (in discrete systems), or one certain length of wave (in deformable systems). But
in integrational mechanics of rods several new
waves (structural waves) can be excited by one
disturbing frequency. These waves can be physically different and of different lengths. This is an
example of system and nonlinear vibration.
Figure 3.35
RIGID BODY
TWO
INDEPENDENT
MOTIONS
(TRANSLATION,
ROTATION)
DEFORMABLE
BODY
TWO WAVES OF
DEFORMATION
(EXTENTION,
SHEAR)
NONLINEAR
THEORY OF
ROD
VIBRATION
Figure 3.36
TWO
RAUGHLEY’S
WAVES
REDUCTION TO
LOSSES OF
STABILITY
LOCAL LOSS OF
STABILITY
TOTAL LOSS OF
STABILITY
IMPULSE
RUNNING
100 Chapter 3
w
ip2
0
0,6
0,4
0,2
a =6°
0
0,2
0,4
D
C
B
A
0,6 0,8 1,0 1,2
Figure 3.37
b
3.9. Compact “Stability”
Classification of stability problem by the type of nonlinearity
Stability is one of the most important phenomena in the nature. For exam-
ple, stability of planet motions, stability of loaded buildings columns, stability
of body shapes being under dynamical loading, etc.
Problems of a heart disease (sudden death) is often connected with ”col-
lapse” of heart. All the most up - to - date scientific trends (theory of catastrophe,
nonlinear theory of synergetic) involve problems of stability. We’ll study engineering problems of stability using:
• classical mechanics of a rigid body;
• mechanics of a deformable body;
• system engineering subjects (strength of materials, theory of mechanisms,
etc.);
• system mechanics of an object;
• integrational mechanics.
All stability problems in the most cases have the points of bifurcation.
Therefore all such problems are nonlinear ones. A linear problem of stability
(stability by Archimedes) is based upon a theory of lever. Archimedes said:
“Give me point of support and I shall turn the whole world over.”
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