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points, it is assumed that the slope of the load curve remains constant. The entire load history is represented by piecewise linear curve, and the error of this approach can be minimized by reducing the length of the time steps. Description of this procedure is given in Clough and Penzien (1993).

Other techniques employed include Fourier analysis of the forcing function followed by solution for Fourier components in the frequency domain. For random forces, random vibration theory and spectrum analysis may be used (Warburton 1976; Dowrick 1988).

### 2.13.6 Multiple Degree Systems

In multiple degree systems, an independent differential equation of motion can be written for each degree of freedom. The nodal equations of a multiple degree system consisting of n degrees of freedom may be written as

where [m] is a symmetrical n x n matrix of mass, [c] is a symmetrical n x n matrix of damping coefficient, and {F(t)| is the force vector, which is zero in the case of free vibration.

Consider a system under free vibration without damping. The general solution of Equation 2.386 is assumed in the form

' x1 ' |

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