Analytical and numerical study of a parametrically excited 2DOF oscillator with nonlinear restoring magnetic force and rotating rectangular rod

This study investigates a detailed analytical and numerical investigation of a nonlinear two-degree-of-freedom (2DOF) mechanical oscillator subjected to parametric excitation, magnetic stiffness nonlinearities, and dry friction. The considered system consists of two coupled oscillators, both of which are connected to a rotating rectangular beam that induces a time-periodic stiffness variation. The Complex Averaging (CxA) method is employed to derive approximate analytical solutions, which are thoroughly validated through time-domain simulations and bifurcation analyses. The dynamic analysis reveals a rich spectrum of nonlinear behaviors, including periodic, quasi-periodic, and chaotic responses. Detailed bifurcation diagrams, Lyapunov exponent analysis, and Poincar\'e maps demonstrate the influence of nonlinear stiffness degree, mass symmetry, and frictional effects on system stability and response amplitude. The obtained results give a significant understanding of the dynamic behavior of coupled nonlinear systems and establish a conceptual framework for the development of complex vibration abatement strategies, energy harvesting devices, and advanced mechanical systems.