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Chapter 19 MECHANICAL VIBRATIONS
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
O
+xm
Equilibrium^ P
mx + kx = 0
x
O
+xm
Equilibrium^ P
mx + kx = 0
setting ωn^2 = k/m
x + ωn^2 x = 0
The motion defined by this expression is called simple harmonic motion.
x = xm sin (ωnt + φ )
The solution of this equation, which represents the displacement of the particle P is expressed as
where xm = amplitude of the vibration ωn = k/m = natural circular frequency φ = phase angle
x
O
+xm
Equilibrium P
x + ωn^2 x = 0
x = xm sin (ωnt + φ)
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
Period = τn =
ωn
Frequency = f n = =
ωn
τn
The velocity and acceleration of the particle are obtained by differentiating x, and their maximum values are
vm = xmωn am = xmωn^2
x
O
P
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 ωn. The instantaneous values of the velocity and acceleration of P may then be obtained by projecting on the x axis the vectors v m and a m representing, respectively, the velocity and acceleration of Q.
xm
a m= xmωn^2
φ
ωnt
v m= xmωn
ωnt + φ
x QO Q
v
a
x
x Equilibrium
P = Pm sin ωf t
ωf t = 0
δm
δm sin ωf t
ωf t
x Equilibrium
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 equation describing each system is presented below.
mx + kx = Pm sin ωf t
mx + kx = kδm sin ωf t
x Equilibrium
P = Pm sin ωf t
ωf t = 0
δm
δm sin ωf t
ωf t
x Equilibrium
mx + kx = Pm sin ωf t
mx + kx = kδm sin ωf t
The general solution of these equations is obtained by adding a particular solution of the form
xpart = xm sin ωf t
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.
x Equilibrium
P = Pm sin ωf t
ωf t = 0
δm
δm sin ωf t
ωf t
Equilibrium^ x
mx + kx = Pm sin ωf t
mx + kx = kδm sin ωf t
xpart = xm sin ωf t
Dividing the amplitude xm of the steady- state vibration by Pm/k in the case of a periodic force, or by δm in the case of an oscillating support, the magnification factor of the vibration is defined by
Magnification factor = =
xm
Pm /k
xm
δm
1 - (ωf /ωn )^2
x Equilibrium
P = Pm sin ωf t
ωf t = 0
δm
δm sin ωf t
ωf t
Equilibrium^ x
mx + kx = Pm sin ωf t
mx + kx = kδm sin ωf t
xpart = xm sin ωf t
Magnification factor = =
xm Pm /k
xm δm 1 1 - (ωf / ωn)^2
The amplitude xm of the forced vibration becomes infinite when ωf = ωn , i.e., when the forced frequency is equal to the 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.
The steady-state vibration of the system is represented by a particular solution of mx + cx + kx = Pm sin ωf t of the form
xpart = xm sin (ωf t - φ )
Dividing the amplitude xm of the steady-state vibration by Pm/k in the case of a periodic force, or by δm in the case of an oscillating support, the expression for the magnification factor is
xm
Pm/k
xm
δ m
[1 - (ωf / ωn)^2 ]^2 + [2(c/cc)(ωf / ωn)]^2
where ωn = k/m = natural circular frequency of undamped system cc = 2m ωn = critical damping coefficient c/cc = damping factor
In addition, the phase difference ϕ between the impressed force or support movement and the resulting steady-state vibration of the damped system is defined by the relationship
tan ϕ =
2(c/cc) (ωf / ωn)
1 - (ωf / ωn)^2
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.
xm
Pm/k
xm
δ m
[1 - (ωf / ωn)^2 ]^2 + [2(c/cc)(ωf / ωn)]^2
xpart = xm sin (ωf t - φ )
