A Modal Superposition Method for the Analysis of Nonlinear Systems

In the determination of response of nonlinear structures, computational burden is always a major problem even if frequency domain methods are used. One of the methods used to decrease the computational effort is the modal superposition method for nonlinear systems where the modes of the linear system are used in the calculation. However, depending on the type of the nonlinearity, in order to obtain an accurate response, the number of modes retained in the response calculations needs to be increased, which increases the number of nonlinear equations to be solved. In this study, a method is proposed to decrease the number of modes used for systems having nonlinearities where the equivalent stiffness varies between two limiting values. For such systems, one can define different linear systems for each value of the limiting equivalent stiffness. In this study, it is proposed to use a combination of these linear mode shapes in the modal superposition method. It is shown that proper combination of mode shapes of different linear systems provides satisfactory results by keeping the number of modes used at a minimum. The method is demonstrated on case studies where describing function method is used in the analysis of the nonlinear system.