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8

Structural Vibration and Dynamics

8.1 Introduction
Vibration and dynamics are fundamental subjects in engineering. Serious problems may
arise from vibration when a structure is not carefully designed for its dynamic integrity.
Vibration can cause malfunction or break down of machines that exhibit unbalance or mis-
alignment. It can also lead to massive engineering failures such as the collapse of a bridge.
Simulation plays an important role in our ability to understand a structure’s dynamic
behavior. Through modeling, the dynamic characteristics of a structure can be captured
and improved before being put into actual use.
In this chapter, we first review the basic equations and their solutions for structural
vibration and dynamic analysis. Then, we discuss the FEA formulations for solving vibra-
tion and dynamic responses. Guidelines in modeling and solving such problems are pro-
®
vided, along with a case study in ANSYS Workbench.
A structure vibrates about an equilibrium position when excited by periodic or arbi-
trary inputs. There are three main types of problems for structural vibration and dynamic
analyses:

• Modal analysis (f(t) = 0)
• Frequency response analysis (f(t) = F sin ωt)
• Transient response analysis (f(t) is arbitrary)

where f(t) is the dynamic force applied on the structure, t the time, and ω the circular fre-
quency (Figure 8.1).

8.2 Review of Basic Equations
We begin with a review of vibration of a single-DOF system, consisting of a mass, a
spring, and a damper as shown in Figure 8.2. Then we will review basic equations
for a multi-DOF system, such as the discrete system resulting from the finite element
discretization.

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