Effects of causality and joint conditions on method of reverberation-ray matrix

Abstract

We reexamine the approximations for amplitude vectors of steady-state waves in framed structures as reported previously by Pao et al. (Pao, Y. H., Howard, S. M., and Keh, D. C., "Dynamic Response and Wave Propagation in Plane Trusses and Frames," AIAA Journal, Vol. 37, No. 5, 1999, pp. 594-603). The arrival and departure wave vectors have been determined exactly in the reverberation-ray matrix from the joint condition of the structure and the phase relation between them within each member. In the approximations the departure vector is obtained by truncating an infinite power series of reverberation-ray matrix to a polynomial of N degree. The arrival vector is given by another polynomial of N degree if it is calculated from the phase relation as in a previous paper, but by polynomials of (N - 1) degree if calculated from the joint conditions as in this study. The newly found set of approximation is preferred to the previous one as it satisfies the causal condition as well as the joint conditions in successive orders of approximation. The differences between these two solutions are, however, very small numerically for large orders of N.