**Forced Harmonic
Vibration**

Harmonic excitation is often
encountered in engineering systems. It is commonly produced by the unbalance in rotating
machinery. Although pure harmonic excitation is less likely to occur than periodic or
other types of excitation, understanding the behavior of a system undergoing harmonic
excitation is essential in order to comprehend how the system will respond to more general
types of excitation. Harmonic excitation may be in the form of a force or displacement of
some point in the system.

We will first consider a single DOF
system with viscous damping, excited by a harmonic force , as shown in Fig. 7. Its differential equation of motion is found
from the free-body diagram.

(29)

Figure 7 Viscously Damped System
with Harmonic Excitation

The solution to this equation consists
of two parts, the **complementary function**, which is the solution of the
homogeneous equation, and the **particular integral**. The complementary
function. in this case, is a damped free vibration.

The particular solution to the
preceding equation is a steady-state oscillation of the same frequency w as that of the
excitation. We can assume the particular solution to be of the form :

(30)

where X is the amplitude of
oscillation and f is the phase of the displacement with respect to the exciting force.

The amplitude and phase in the
previous equation are found by substituting Eqn. (30) into the differential equation (29).
Remembering that in harmonic motion the phases of the velocity and acceleration are ahead
of the displacement by 90° and 180°, respectively, the terms of the differential
equation can also be displayed graphically, as in Fig. 8.

Figure 8 Vector Relationship for
Forced Vibration with Damping

It is easily seen from this diagram
that

(31)

and

(32)

We now express Eqs (31) and (32) in
non-dimensional term that enables a concise graphical presentation of these results.
Dividing the numerator and denominator of Eqs. (31) and (32) by *k*, we obtain :

(33)

and

(34)

These equations can be further
expressed in terms of the following quantities:

The non-dimensional expressions for
the amplitude and phase then become

(35)

and

(36)

These equations indicate that the non
dimensional amplitude , and the phase
f are functions only of the frequency ratio , and the damping factor z and can be plotted as shown in Fig 9.

Figure 9 Plot of Eqs. (35) and
(36)