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Chapter 19 MECHANICAL VIBRATIONS
-xm
O
Equilibrium
x
P
+xm
+
Consider the free vibration of a
particle, i.e., the motion of a particle P
subjected to a restoring force
proportional to the displacement of the
particle - such as the force exerted by
a spring. If the displacement x of the
particle P is measured from its
equilibrium position O, the resultant F
of the forces acting on P (including its
weight) has a magnitude kx and is
directed toward O. Applying Newton’s
..
second law (F = ma) with a = x, the
differential equation of motion is
..
mx + kx = 0
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..
mx + kx = 0
-xm
setting wn2 = k/m
..
x + wn2x = 0
O
Equilibrium
x
P
The motion defined by this expression
is called simple harmonic motion.
The solution of this equation, which
represents the displacement of the
particle P is expressed as
+xm
+
x = xm sin (wnt + f)
where xm = amplitude of the vibration
wn = k/m = natural circular
frequency
f = phase angle
..
x + wn2x = 0
x = xm sin (wnt + f)
-xm
O
Equilibrium
x
P
+xm
+
The period of the vibration (i.e., the
time required for a full cycle) and its
frequency (i.e., the number of cycles
per second) are expressed as
2p
Period = tn = w
n
1 wn
Frequency = fn =
= 2p
tn
The velocity and acceleration of the particle are obtained by
differentiating x, and their maximum values are
vm = xmwn
am = xmwn2
O
x
a
P
v
x
f
xm
wnt QO
am= xmwn2
Q
wnt + f
vm= xmwn
The oscillatory motion of the
particle P may be represented
by the projection on the x axis of
the motion of a point Q
describing an auxiliary circle of
radius xm with the constant
angular velocity wn. The
instantaneous values of the
velocity and acceleration of P
may then be obtained by
projecting on the x axis the
vectors vm and am representing,
respectively, the velocity and
acceleration of Q.
While the motion of a simple pendulum is not truely a simple
harmonic motion, the formulas given above may be used with
wn 2 = g/l to calculate the period and frequency of the small
oscillations of a simple pendulum.
The free vibrations of a rigid body may be analyzed by
choosing an appropriate variable, such as a distance x or an
angle q , to define the position of the body, drawing a diagram
expressing the equivalence of the external and effective forces,
and writing an equation relating the selected variable and its
second derivative. If the equation obtained is of the form
..
x + wn2x = 0
..
or
q + wn2q = 0
the vibration considered is a simple harmonic motion and its
period and frequency may be obtained.
The principle of conservation of energy may be used as an
alternative method for the determination of the period and
frequency of the simple harmonic motion of a particle or rigid
body. Choosing an appropriate variable, such as q, to define
the position of the system, we express that the total energy
of the system is conserved, T1 + V1 = T2 + V2 , between the
position of maximum displacement
(q1 = qm) and the position
.
.
of maximum velocity (q 2 = qm). If the motion considered is
simple harmonic, the two members of the equation obtained
.
consist of homogeneous quadratic
expressions in qm and qm ,
.
respectively. Substituting qm = qm wn in this equation, we may
factor out qm2 and solve for the circular frequency wn.
The forced vibration of a mechanical system
occurs when the system is subjected to a
periodic force or when it is elastically
connected to a support which has an
alternating motion. The differential
dm sin wf t
equation describing
dm
each system is
presented
wf t = 0
wf t
below.
x
Equilibrium
P = Pm sin wf t
..
mx + kx = Pm sin wf t
..
mx + kx = kdm sin wf t
x
Equilibrium
..
dm
..
wf t = 0
mx + kx = Pm sin wf t
mx + kx = kdm sin wf t
x
Equilibrium
P = Pm sin wf t
dm sin wf t
The general solution
of these equations is
obtained by adding a
particular solution of
the form
xpart = xm sin wf t
x
Equilibrium
to the general solution of the corresponding
homogeneous equation. The particular solution represents the
steady-state vibration of the system, while the solution of the
homogeneous equation represents a transient free vibration
which may generally be neglected.
wf t
..
..
mx + kx = kdm sin wf t
mx + kx = Pm sin wf t
xpart = xm sin wf t
Dividing the amplitude xm of the steadystate vibration by Pm/k in the case of a
periodic force, or by dm in the case of an
oscillating support, the magnification factor
of the vibration is defined by
x
Equilibrium
P = Pm sin wf t
dm
wf t = 0
dm sin wf t
Magnification factor =
wf t
=
Equilibrium
x
xm
Pm /k
=
xm
dm
1
1 - (wf /wn )2
..
..
mx + kx = kdm sin wf t
mx + kx = Pm sin wf t
xpart = xm sin wf t
xm
Magnification factor = P /k
m
x
Equilibrium
=
P = Pm sin wf t
dm
wf t = 0
Equilibrium
x
=
xm
dm
1
1 - (wf / wn)2
The amplitude xm of the forced vibration
becomes infinite when wf = wn , i.e., when
the forced frequency is equal to the
wf t natural frequency of the system. The
impressed force or impressed support
movement is then said to be in resonance
with the system. Actually the amplitude of
the vibration remains finite, due to
damping forces.
dm sin wf t
The equation of motion describing the damped free vibrations
of a system with viscous damping is
..
.
mx + cx + kx = 0
where c is a constant called the coeficient of viscous damping.
Defining the critical damping coefficient cc as
cc = 2m
k
= 2mwn
m
where wn is the natural frequency of the system in the absence
of damping, we distinguish three different cases of damping,
namely, (1) heavy damping, when c > cc, (2) critical damping,
when c = cc, (3) light damping, when c < cc. In the first two cases,
the system when disturbed tends to regain its equilibrium
position without oscillation. In the third case, the motion is
vibratory with diminishing amplitude.
The damped forced vibrations of a mechanical system occurs
when a system with viscous damping is subjected to a periodic
force P of magnitude P = Pm sin wf t or when it is elastically
connected to a support with an alternating motion d = d m sin wf t.
In the first case the motion is defined by the differential equation
..
.
mx + cx + kx = Pm sin wf t
In the second case the motion is defined by the differential
equation
..
.
mx + cx + kx = kdm sin wf t
The steady-state vibration of the system is represented by a
.
..
particular solution of mx + cx + kx = Pm sin wf t of the form
xpart = xm sin (wf t - f)
Dividing the amplitude xm of the steady-state vibration by Pm/k in
the case of a periodic force, or by dm in the case of an oscillating
support, the expression for the magnification factor is
xm
xm
=
=
Pm/k
dm
1
[1 - (wf / wn)2]2 + [2(c/cc)(wf / wn)]2
where wn = k/m = natural circular frequency of undamped
system
cc = 2m wn = critical damping coefficient
c/cc = damping factor
xpart = xm sin (wf t - f)
xm
xm
Pm/k = dm =
1
[1 - (wf / wn)2]2 + [2(c/cc)(wf / wn)]2
In addition, the phase difference j between the impressed force
or support movement and the resulting steady-state vibration of
the damped system is defined by the relationship
2(c/cc) (wf / wn)
tan j = 1 - (w / w )2
f
n
The vibrations of mechanical systems and the oscillations of
electrical circuits are defined by the same differential equations.
Electrical analogues of mechanical systems may therefore be
used to study or predict the behavior of these systems.