Module 9 : Forced Vibration with Harmonic Excitation; Undamped Systems and resonance; Viscously Damped                    Systems; Frequency Response Characteristics and Phase Lag; Systems with Base Excitation;                    Transmissibility and Vibration Isolation; Whirling of Shafts and Critical Speed.
Lecture 1 : Harmonic vibration of single DOF system
Forced Vibration Analysis of an undamped single d.o.f system
 

Fig 9.1.1 Single degree of freedom system under harmonic excitation
 

Fig 9.1.2 Free body diagram of SDOF system

 

Consider a spring-mass system subjected to a sinusoidal force as shown in Fig. 9.1.1. A practical situation representing this is also shown in the figure. The figure shows a machine on its foundation, where the machine is operating at a rotational speed . If the rotor has some unbalance and if ‘k' denotes the stiffness of the foundation, we can represent the system as an equivalent single d.o.f system as shown in the figure. The sinusoidal excitation corresponds to the unbalance force as indicated in the figure. We will begin our discussion with the simple spring-mass system and later, return to this practical example to study it in greater detail.

The governing equation of motion can be readily obtained from the free body diagram shown in Fig. 9.1.2 as follows:

(9.1.1)

 

 The response of the system as a solution to the differential equation of motion has two parts viz., complementary solution and particular integral. For this system, the complementary solution represents the response of the system to given initial conditions and the particular integral is the response to the particular forcing function (sinusoid in this case). The complementary solution, as obtained in eqn. () is given by:


(9.1.2)

 
The particular integral can be assumed to be of the form:
(9.1.3)

 

 Thus the total response of the system is given by:
(9.1.4)
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