Exploring static bifurcations in a controlled dynamical system with cubic and quadratic nonlinearities: 2D and 3D visualization

Abstract

This work delves into the investigation of static bifurcation control and vibration reduction of a two-degree-of-freedom dynamical system. The system under study simulates the lateral oscillations of rotating machinery and encompasses both cubic and quadratic nonlinearities. The nonlinear system is augmented with a magnetic bearing actuator, incorporating a novel control strategy that combines two first-order filters. The system model is derived based on classical mechanics and electromagnetic theories. Then, an analytical solution is extracted for the obtained dynamical model. The solutions obtained have been utilized to visualize the static bifurcations of the system in both two and three-dimensional spaces, using various system parameters as bifurcation variables. The mono-stable and multiple-stable solution regions have been distinguished in two-parameter space. Subsequently, an investigation has been conducted to evaluate the effectiveness of the introduced control technique in eliminating the catastrophic bifurcation of the rotor and suppressing undesirable vibrations. Furthermore, as a precautionary measure, the impact of the controller’s sudden malfunction on the stability of the system was explored. The main findings revealed that the implemented control approach effectively eliminates dangerous bifurcation characteristics and induces the nonlinear rotor to exhibit a response like a linear system with minimal vibration amplitudes. Furthermore, it was observed that the abrupt failure of the controller does not affect the stability of the system; however, the nonlinearities regain dominance in the system’s response