NDF Controller-Based Stability Analysis and Vibration Mitigation of a Nonlinear Electromechanical Oscillator Under Primary Resonance

Abstract

This work examines how well a Negative Derivative Feedback (NDF) controller suppresses vibration in a nonlinear electromechanical oscillator that is subjected to mixed excitations. Coupled nonlinear ordinary differential equations are used to model the system and show how mechanical and electrical components interact. The method of multiple scales (MMS) is used to develop analytical approximate solutions up to the second order, specifically for the primary resonance scenario. This study’s main contribution is a thorough bifurcation analysis and proof of the NDF controller’s high efficacy, which effectively lowers the first and second mode resonance amplitudes by roughly 99.8% and 98%., respectively, with impressive reported effectiveness values of roughly 590 and 51.5. Additionally, the quantitative error analysis between the numerical simulation and the analytical approximation solution demonstrates a high degree of agreement, with a maximum error of less than 10−5% for the second mode and just 0.01% for the first mode. Furthermore, we present the impact of parameters on FRCs. Frequency response curves (FRCs) are used in a thorough comparison analysis to assess the behavior of the system both before and after the controller is activated. A strong degree of connection between the analytical conclusions and numerical simulations carried out using the “fourth-order Runge–Kutta method” rigorously validates the accuracy of the perturbation analysis. Additionally, a performance benchmark between different control techniques, such as the NDF controller, Positive Position Feedback (PPF), and Linear Negative Position Feedback (LNPF), is shown in the paper. When compared to alternative approaches, the NDF controller shows the greatest reduction in oscillation amplitudes and higher robustness, as shown by transient response analysis (time history) at various time intervals. The outcomes validate the NDF approach’s dependability and efficiency in stabilizing intricate nonlinear electromechanical systems. The chaotic response and system periodicity were demonstrated through bifurcation diagrams and Poincaré maps.