Chapter 9
Forced Vibration of Continuous Systems
9.1 Introduction
In the previous chapter, we have considered the free vibration analysis of continuous
systems. We pass on to the forced vibration analysis of continuous systems in this
chapter. Though we start with axial vibration problem, the major emphasis will
be placed on flexural vibrations of beams. It might have been noted by the readers
while going through the previous chapter that the natural frequencies associated with
flexural vibrations are of much lower magnitude than those of torsional and axial
vibrations. Hence, for practical problems they become much more important. The
entire treatment of this chapter is based on mode summation procedure [1–4].
9.2 Forced Axial Vibration of Bars
Let us consider the bar of Fig. 9.1, which is fixed at one end, and at the free end
an exciting force P(t) is applied. As discussed in Article 8.2, the forced vibration
equation is given by
E A
∂
2 u
∂ x 2 − ρ A
∂
2 u
∂t 2 = −P(t)
(9.1)
Let us express the displacement u in terms of normal coordinates.
Therefore,
u(x, t) =
U r (x) ξ r (t)
(9.2)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
M. Mukhopadhyay, Structural Dynamics,
https://doi.org/10.1007/978-3-030-69674-0_9
371