Modal analysis-based calculation of periodic nonlinear responses of harmonically forced piecewise linear elastic systems

This paper introduces a new solution method based on modal analysis to calculate periodic nonlinear response to harmonic forcing. The method can be used for piecewise linear (PL) elastic multi-degree-of-freedom (MDOF) systems with two linear states, provided that the unloaded equilibrium position of both states coincides, and the system is in both states once during a period of forcing. The method formulates a linear system of equations, which yields the initial conditions of the periodic response as a function of the time spent in each state and the forcing phase. Error functions are constructed to secure the switching between the states of the system at the given time instances and the nonlinear equations are solved for the candidates of periodic responses with the scanning of the parameter space. Finally, the vibrations with unwanted switches are filtered out to obtain the actual periodic responses. An example with support vibration shows that the proposed method is capable to find disconnected branches of the periodic responses to a harmonic excitation. As the dimension of the scanning space is independent of the discretization, the method scales almost linearly with the increase of the number of degrees-of-freedom.