Civil Engineering Reference
In-Depth Information
10
CHAPTER
Structural Dynamics
10.1 INTRODUCTION
In addition to static analyses, the finite element method is a powerful tool for
analyzing the dynamic response of structures. As illustrated in Chapter 7, the
finite element method in combination with the finite difference method can be
used to examine the transient response of heat transfer situations. A similar
approach can be used to analyze the transient dynamic response of mechanical
structures. However, in the analysis of structures, an additional tool is available.
The tool, known as
modal analysis,
has its basis in the fact that every mechani-
cal structure exhibits natural modes of vibration (dynamic response) and these
modes can be readily computed given the elastic and inertia characteristics of the
structure.
In this chapter, we introduce the concept of natural modes of vibration via the
simple harmonic oscillator system. Using the finite element concepts developed
in earlier chapters, the simple harmonic oscillator is represented as a finite element
system and the basic ideas of natural frequency and natural mode are introduced.
The single degree of freedom simple harmonic oscillator is then extended to mul-
tiple degrees of freedom, to illustrate the existence of multiple natural frequencies
and vibration modes. From this basis, we proceed to more general dynamic analy-
ses using the finite element method.
10.2 THE SIMPLE HARMONIC OSCILLATOR
The so-called simple harmonic oscillator is a combination of a linear elastic
spring having free length
L
and a concentrated mass as shown in Figure 10.1a.
The mass of the spring is considered negligible. The system is assumed to be
subjected to gravity in the vertical direction, and the upper end of the spring is
attached to a rigid support. With the system in equilibrium as in Figure 10.1b, the
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