6.1 Introduction

In the usual finite element procedure, the continuous displacement field within an element is defined in terms of its unknown nodal displacement parameters by the fundamental relationship

in which the matrix N represents the shape functions. Such a relationship, however, is strictly valid only for static loading since, for the general dynamic problem, N is not unique; it is then a function of the entire time history of the nodal displacements. In the special case of free vibration involving harmonic motion, Eq. (6.1) is valid when N is a function of the instantaneous nodal displacements and also of the frequency of such motion. The resulting stiffness and mass matrices are then obtained as functions of the frequency of the harmonic motion. ^{[1] , [2]} Thus, as an example, the equation of motion for a simple bar element in Fig. 6.1 has the following form:

in which c ² = E/ ?. The solution of Eq. (6.2) may be expressed as

where u ^e = [ u _x1 u _x2] ^T and q ^e = [ q _x1 q _x2] ^T. Substituting Eq. (6.3) into Eq. (6.2) yields

Figure 6.1: Bar element with time-varying nodal displacements.

The solution to Eq. (6.4) is

Using boundary conditions

the coefficients A and B are evaluated appropriately yielding the following expression for the shape function:

The strain displacement relations may then be obtained as

resulting...