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 Poincaré 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.

💡 Research Summary

This paper presents a comprehensive analytical and numerical investigation of a parametrically excited two‑degree‑of‑freedom (2DOF) mechanical oscillator that incorporates nonlinear magnetic restoring forces and dry (Coulomb) friction. The physical configuration consists of two masses sliding on low‑friction guides, each equipped with a pair of repulsive permanent magnets that generate a highly nonlinear stiffness characteristic. The masses are coupled by a rectangular cross‑section beam whose orientation is controlled by a stepper motor; the beam rotates at a constant angular speed Ω, causing the coupling stiffness to vary periodically in time as K(t)=k_ξ+k_η/2+(k_ξ−k_η)/2 cos 2Ωt.

The governing equations of motion are first expressed in dimensional form, including viscous damping (c₁, c₂), Coulomb friction (T₁, T₂) modeled by a set‑valued sign function, and magnetic forces approximated by a fifth‑order polynomial F_si(x_i)=k_i1 x_i+k_i3 x_i³+k_i5 x_i⁵. To facilitate analysis, the system is nondimensionalized using the natural frequency of the first mass (with the second mass fixed) ω_n=√