Non-linear, steady-state vibration of structures by harmonic balance/finite element method

A steady-state response of a wide class of geometrically non-linear structures subjected to harmonic excitation is analyzed. Both the finite element method and harmonic balance method are used to obtain an approximate solution. Adopting a total Lagrangian formulation for structures undergoing large displacements, small strains and rotation a set of non-linear differential equations in time and space variables is derived in a form most suitable for further study. The nodal displacements are expressed by the Fourier series in time and with help of the harmonic balance method a system of non-linear algebraic equations is obtained with the Fourier nodal coefficients as unknowns. A systematic procedure to formulate the resulting system of non-linear equations and their incremental counterparts is described. The method is suitable for parametric studies, various kinds of resonances can be analyzed. The Newton-type algorithm with the arc-length procedure is adopted to determine the response curves and a computational procedure is described in detail. A simple and numerically very efficient approximate stability criterion for a steady-state solution is also suggested. A numerical example is given and the results obtained by Newmark and the present methods are compared.