Nonlinear vibration of plane structures by finite element and incremental harmonic balance method

A nonlinear steady state vibration analysis of a wide class of plane structures is analyzed. Both the finite element method and incremental harmonic balance method are used. The usual beam element is adopted in which the nonlinear effect arising from longitudinal stretching has been taken into account. Based on the geometric nonlinear finite element analysis, the nonlinear dynamic equations including quadratic and cubic nonlinearities are derived. These equations are solved by the incremental harmonic balance (IHB) method. To show the effectiveness and versatility of this method, some typical examples for a wide variety of vibration problems including fundamental resonance, super- and sub-harmonic resonance, and combination resonance of plane structures such as beams, shallow arches and frames are computed. Most of these examples have not been studied by other researchers before. Comparison with previous results are also made.