Dynamic analysis using a reduced basis of exact modes and Ritz vectors

The problem of obtaining an approximate solution for the response of a structure under dynamic forces is addressed. The structure is modeled using mass and stiffness matrices and has n degrees of freedom. An ///-dimensional approximation is sought in a subspace defined by an arbitrary M-orthogonal array £. A residual vector, which is smaller in magnitude the more accurate the approximation, is derived. The residual contains two terms: one is the error involved representing the force vector in the £ subspace; the other is smaller the better the £ array leads to m approximate solutions of the exact eigenproblem. Numerical experiments are performed using £ arrays that are combinations of exact eigenvectors and of Ritz vectors. In one case, when 11 different 10-dimensional bases are used to approximate the solution of a 60 degree of freedom problem, it is found that a root-mean-square measure of relative error is smallest for the basis formed using the 6 lowest exact eigenvectors plus 4 Ritz vectors generated from the spatial distribution of the loading.