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)