Two-dimensional bilinear oscillator: group-preserving scheme and steady-state motion under harmonic loading

We study the dynamic behavior of a single-mass, two-degree-of-freedom bilinear oscillator, whose restoring force obeys a bilinear elastoplastic law. For the purpose of calculation we develop a group-preserving scheme utilizing the internal symmetry group of the model in the plastic phase so as to satisfy the yield condition exactly at each time step. Then the oscillator is subjected to biaxial harmonic excitations, and the responses are displayed. Based on circular motion assumption and harmonic balance method, the closed-form formulae to estimate the steady-state responses are derived, which are then compared with the results calculated by the group-preserving scheme. The accuracy of the formulae is thus confirmed. Simple formulae to select minimum driving force amplitudes or maximum yield forces which cause the bilinear oscillator steadily oscillating in the plastic phase are derived. For the benefit of engineers’ use we also derive closed-form formulae to decide the force ratios required to sustain the user-specified dissipation per cycle.