Fast convergence modal analysis for continuous systems

Abstract

A continuous system has an infinite number of degrees of freedom (n.d.o.f.) in a dynamic analysis. The dynamic stiffness method is able to produce an infinite number of natural modes with use of only a finite number of co-ordinates. The associated modal analysis is the only widely applicable approximate method for computing the response without discretizing the continuous system by methods such as the finite element method, in which the infinite n.d.o.f. is not retained. However, this modal analysis converges very slowly as the number of modes is increased if the loading distribution does not follow the patterns of the first few modes. A method is suggested in this paper to accelerate the convergence. A mixed mass matrix is introduced according to the reciprocal theorem to evaluate the initial transient while retaining the infinite n.d.o.f. with a finite number of co-ordinates. Explicit formulae are given for space frames.