6.1 Introduction

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This chapter presents numerical techniques for three classes of vibration problems; the single spring

mass, the multiple spring mass, and the continuous model. The simplest oscillatory problem results when

the problem can be modeled by the single spring – mass system. This system is modeled by a second-order

constant coefficient differential equation, which can be solved analytically when either there is no forcing,

or the forcing is harmonic. When the forcing is not harmonic, the finite difference method can be used to

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solve this second-order equation or to solve the system of first-order equations, which is equivalent to the

second-order equation. A larger computational problem arises when there are multiple springs and

masses subjected to nonharmonic forcing. Again, the finite difference method can be used to compute

the full solution. It is not always necessary to compute the full solution. Often, all that is needed is

the fundamental frequency. This can be computed from the eigenvalues of a matrix problem. Both the

numerical solution to the full problem and the matrix problem will be discussed. Finally, one may want

to look at the vibration of a continuous element like a beam. Here, both the finite difference method and

the finite element method will be discussed.

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