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The incremental method has been widely used in various types of nonlinear analysis, however, so far it has received little attention in the analysis of periodic nonlinear vibrations. In this paper, an amplitude incremental variational principle for nonlinear vibrations of elastic systems is derived. Based on this principle various approximate procedures can be adapted to the incremental formulation. The linear solution for the system is used as the starting point of the solution procedure and the amplitude is then increased incrementally. Within each incremental step, only a set of linear equations has to be solved to obtain the data for the next stage. To show the effectiveness of the present method, some typical examples of nonlinear free vibrations of plates and shallow shells are computed. Comparison with analytical results calculated by using elliptic integral confirms that excellent accuracy can be achieved. The technique is applicable to highly nonlinear problems as well as problems with only weak nonlinearity.

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An incremental harmonic balance method with multiple time scales is presented in this paper. As a general and systematic computer method, it is capable of treating aperiodic “steady-state” vibrations such as combination resonance, etc. Moreover, this method is not subjected to the limitation of weak nonlinearity. To show the essential features of the new approach, the almost periodic free vibration of a clamped-hinged beam is computed as an example.

A variable parameter incrementation method is proposed and then applied to the determination of parametric instability boundary of columns. Attention is particularly paid to the geometrically nonlinear problems including the instability of nonlinear vibrations. Although only beam and column problems are treated at present, the approach is believed to be general in methodology. This method is not subjected to the limitations of small exciting parameters and weak nonlinearity.