Null Space Method for Single Degree of Freedom Piecewise-Smooth Dynamical Systems

Piecewise-linear forced systems are common in engineering applications but challenging to analyze due to non-smooth or discontinuous nonlinearities and complex bifurcation structures. This paper proposes an efficient null space method for computing periodic solutions and bifurcations in such systems. The method lifts the periodic-solution problem by introducing the excitation phase to lift the non-autonomous system into an autonomous form, using state transition matrices to represent segment-wise evolution, and expressing segment boundaries and periodic closure conditions as linear constraints. Rather than solving directly for initial conditions and segment times, the method transforms the problem into a small-scale null space computation combined with low-dimensional optimization over segment times. The approach relies solely on matrix exponentials and null space calculations, enabling simultaneous acquisition of stable and unstable periodic solutions. Stability is determined from Floquet multipliers of the monodromy matrix. The method can extend naturally to multi-degree-of-freedom and multi-segment cases. In an active magnetic bearing (AMB)-rotor system, multi-segment behavior corresponds to rub-impact phenomena. Validation is carried out on the AMB-rotor system via numerical simulations, static excitation tests, and rotating tests. All results are in good agreement, which verifies the accuracy and engineering applicability of the method for predicting period-doubling bifurcation intervals and analyzing periodic dynamics.