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

TL;DR

This study addresses a nonlinear two-degree-of-freedom oscillator subjected to parametric excitation, magnetic stiffness nonlinearities, and dry friction, driven by a rotating beam that modulates inter-component stiffness. It develops approximate analytical solutions using the Complex Averaging (CxA) method and validates them against time-domain simulations and bifurcation analyses, revealing periodic, quasi-periodic, and chaotic dynamics. The analysis shows that dynamical behavior depends on nonlinear stiffness order, mass distribution, and friction, with distinct bifurcation structures arising in unequal versus equal-mass configurations. The findings advance understanding of coupled nonlinear oscillators and offer a framework for vibration control and energy harvesting, while outlining future work on semi-analytical methods and experimental validation.

Abstract

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.

Figures (20)

• Figure 1: Experimental setup of 2DOF mechanical parametric oscillator.
• Figure 2: Characteristics of the magnetic force: original model (\ref{['3.2b']}) and its approximations by $\text{3}^{\text{rd}}$ degree (\ref{['mag1a']}) and $\text{5}^{\text{th}}$ (\ref{['mag1b']}) degree polynomial models.
• Figure 3: Comparing the analytical results from CxA for the first mass (a) and the second mass (b) in the case of $m_1\neq m_2$.
• Figure 4: Comparing the analytical results with a Runge-Kutta sweep simulation for the first mass (a) and the second mass (b) where $\sigma_1=0$ and $m_1\neq m_2$.
• Figure 5: Comparing the analytical results with a Runge-Kutta sweep simulation for the first mass (a) and the second mass (b) where $\sigma_1=0.009106$ and $m_1\neq m_2$.