Computational formulation for periodic vibration of geometrically nonlinear structures .1. Theoretical background

A general computational formulation for geometrically nonlinear structures excited by harmonic forces and executing periodic motion in a steady-state is presented. The equations of both continuous and discretized models are reformulated to obtain the motion equation in a more suitable form to a further analysis. The multi-harmonic solution of motion equation is written in a form of truncated Fourier series. Next, the Galerkin, Ritz and the harmonic balance method are discussed in a context of their equivalency in derivation of the matrix amplitude equation. The matrix amplitude equation as well as the associated tangent matrix are given in an explicit form. The stability of steady-state solution is discussed by using the Floquet theory. The numerical algorithm and an example application are described in a companion paper by Lewandowski [Lewandowski, R. Computational formulation for periodic vibration of geometrically nonlinear structures-Part 2: Numerical strategy and numerical examples. International Journal of Solids and Structures. (in preparation)]. (C) 1997 Elsevier Science Ltd.