Efficient solution of modal equations with arbitrary loadings

Abstract

Commonly time-varying loadings such as wind, waves and earthquakes which act on engineering structures are specified in the form of discrete time-series. The dynamic response of a linear structural system when subjected to such inputs may be computed by modal analysis techniques. These methods require the solution of a set of uncoupled second-order linear ordinary differential equations with constant coefficients. Implicit to any solution used, however, is the requirement that the original loading be matched exactly or approximated closely by a continuous time-function. Frequently a piecewise linear interpolation of the discrete input data is adopted. This paper presents superior local interpolations which have been derived and optimised by considerations in the frequency domain. These local interpolations of the input are then subsequently used in conjunction with an exact solution to the modal equations. This implies that errors are due solely to the limitations of the interpolating process, a feature absent from numerical integration methods. By the use of discrete time-systems theory and the z transform some simple recursive algorithms are derived for the solution of the modal equations. These algorithms, which operate linearly on the loading data, can be made very accurate, are unconditionally stable and are suitable for use with a hand calculator. The paper concludes with sections on methods of processing the discrete time-series loading data so as to increase or decrease the sample time step length. Increasing the step length can economise analysis when only low frequency components are of interest. Reducing the step length by inserting intermediate values is advantageous when high frequency components in the solution are of importance and in the direct solution of the matrix equations of motion. Simple but effective digital filters are specified for these operations, and in the case of sample rate increase (digital interpolation) the filters are much superior to linear interpolation.