This article contemplates the demeanor of the giant magnetostrictive actuator (GMA) when a positive position feedback (PPF) damper is used to enable tight control over its vibration. The methodology followed here mathematically searches for the approximate solution for the motion equations of the GMA with the PPF damper, which has been accomplished by using one of the most famous perturbation methods. The multiple scale perturbation technique (MSPT) of the second-order approximation is our strategy to obtain the analytical results. The stability of the system has also been investigated and observed by implementing frequency response equations to close the concurrent primary and internal resonance cases. By utilizing Matlab and Maple programs, all numerical discussions have been accomplished and explained. The resulting influence on the amplitude due to changes in the parameters' values has been studied by the frequency response curves. Finally, a comparison between both the analytical and numerical solutions using time history and response curves is made. In addition to the comparison between our PPF damper's effect on the GMA, previous works are presented. To get our target in this article, we have mentioned some important applications utilized in the GMA system just to imagine the importance of controlling the GMA vibration.

This paper presents a novel nonlinear proportional-derivative cubic velocity feedback (NPDVF) controller for controlling vibrations in systems with both mechanical and electrical components subjected to mixed forces. The proposed controller aims to address the challenges posed by nonlinear bifurcations, unstable motion, and vibrations. The effectiveness of the controller demonstrated through numerical simulations, where it shown to significantly reduce harmful vibrations and stabilize the system under varying operating conditions. To analyze the system, a perturbation technique employed to derive approximate solutions to the system's equations up to the second order at simultaneous resonance case ( ${\Omega }_2\cong {\omega }_1,{\Omega }_4\cong {\omega }_2$). A comparative analysis with other control strategies, such as proportional-derivative (PD) control, sliding mode control (SMC), and model predictive control (MPC), the superior robustness, computational efficiency, and control signal amplitude of the NPDVF controller. Results indicate that the proposed approach not only outperforms traditional methods in terms of energy efficiency and computational cost but also maintains robust performance even in the presence of nonlinearities and parameter uncertainties. The findings support the potential application of the NPDVF controller in real-time vibration control systems.

Vibration-based warning and sensing devices rely on compact nonlinear structures capable of operating reliably under combined parametric and harmonic disturbances. To address these engineering demands, we developed a two-degree-of-freedom (2-DOF) lumped-parameter model that captures the coupled dynamics of a mass-based multi-warning unit integrated into a host structure. An Ⅱ-shaped coupled beam was proposed as a practical realization of the concept, where an auxiliary "warning mass" interacts dynamically with a primary supporting beam through nonlinear oscillations. The governing nonlinear differential equations were analytically solved using the multiple-time-scale technique (MTST), with detailed emphasis on primary and internal resonance conditions ( ${\omega }_d\cong {\omega }_1,{\omega }_e\cong {\omega }_2$) that critically influence device functionality and stability. To mitigate large-amplitude oscillations and enhance robustness under strong nonlinearities, a new control law, Nonlinear Proportional-Derivative Control with Negative Cubic Velocity Feedback (NPDCVF), was introduced. This controller combines a nonlinear PD term with a cubic dissipative component to improve damping and suppress resonance amplification. Control performance was assessed using the worst-case resonance scenario of the primary mass. Comparative simulations against Integral Resonance Control (IRC), Positive Position Feedback (PPF), Proportional-Integral-Derivative (PID), and Nonlinear Integral PPF (NIPPF) showed that the NPDCVF controller achieves up to 35–60% reduction in vibration amplitude, ensures faster settling, and maintains stable operation even near low natural frequencies where conventional controllers deteriorate. Frequency-response-based stability analysis revealed the boundaries of safe operation and highlighted instability regions relevant for design considerations. Numerical simulations using MATLAB/Simulink confirmed that the NPDCVF controller significantly enhances vibration suppression and flexibility against nonlinear disturbances. These results demonstrated the potential of the proposed system and control strategy for engineering applications such as structural warning devices, precision actuators, robotic manipulators, and advanced motorcycle suspension systems.

In this study, I investigated the energetic behavior of a twelve-pole active magnetic bearing (AMB) framework with a nonlinear proportional derivative cubic velocity feedback (NPDCVF) controller in the presence of mixed excitations and primary resonance ( $\Omega \cong {\omega }_1,\Omega \cong {\omega }_2$). The controller combines classic proportional-derivative (PD) control with nonlinear cubic velocity feedback to improve stability, reduce rotor oscillations, and increase robustness. A detailed mathematical framework of the 12-pole AMB system was developed, accounting for magnetic force nonlinearity, dynamic interactions between poles, and the effects of rotor eccentricity. The motion equations (ME) were investigated using the multiple time scales approach (MTSA), and the approximation solutions (AS) were numerically validated with the fourth-order Runge-Kutta (4RK) method. Simulations were performed to compare five controllers: PD, integral resonant controller (IRC), positive position feedback (PPF), nonlinear integral positive position feedback (NIPPF), and the proposed NPDCVF scheme. MATLAB 18.2 numerical simulations (4RK) were employed to analyze time-history responses, the effects of system parameters, and the performance of controllers. The time-domain results showed that the NPDCVF controller delivers the quickest vibration reduction, the least overshoot, and increased robustness to disturbances and parameter changes. Time-domain, frequency-response, and phase-plane analyses confirmed wider stability margins and increased damping effectiveness. Additional nonlinear dynamical assessments, such as bifurcation charts, frequency response curves, and stable and unstable zones, showed that nonlinear oscillations have been successfully reduced and the AMB system has stabilized reliably.