An externally excited Duffing oscillator under feedback control is discussed and analyzed under the worst resonance case. Multiple time scales method is applied for this system to find analytic solution with the existence and nonexistence of the time delay on control loop. An appropriate stability analysis is also performed and appropriate choices for the feedback gains and the time delay are found in order to reduce the amplitude peak. Different response curves are involved to show and compare controller effects. In addition, analytic solutions are compared with numerical approximation solutions using Rung-Kutta method of fourth order.

A rotating beam at varying speed mathematical model is studied. Multiple time scales method is applied to the nonlinear system of differential equations and investigated the system behavior approximate solution in the instance of resonance case. We studied the system in case of applying the delayed control on the displacement and the velocity with Proportional–derivative (PD) controller. The consistency of the steady state solution in the near-resonance case is reviewed and analyzed using the Routh-Huriwitz approach. The factors on the steady state solution of the various parameters are recognized and discussed. Simulation effects are obtained using MATLAB software package. Different response curves are involved to show and compare controller effects at various system parameters.

Aim: This work analyzes a two-degree-of-freedom (DOF) auto-parametric system for concurrent vibration control and enhanced energy harvesting at resonance. The main objective is to achieve simultaneous vibration mitigation and energy extraction by integrating mechanical and electrical subsystems through electromechanical coupling. Methods: The investigated system consists of a damped oscillator with an attached pendulum and two distinct energy harvesting (EH) devices: a piezoelectric harvester and an electromagnetic harvester. The primary structure, composed of a vertical damped oscillator coupled with a suspended pendulum, is equipped with a negative-velocity-feedback (NVF) controller to reduce unwanted vibrations that impair system performance, especially under resonance conditions. The governing equations of motion are derived using Lagrange’s equations (LE) and solved approximately by the perturbation method, namely the multiple scales method (MSM). Validation of the obtained approximate solutions (AS)is performed through comparison with the computed numerical solutions (NS) via the Runge–Kutta fourth order (RK-4) method. Results: After classifying the resonance cases, the analysis focuses on the worst-case scenario, namely the primary external resonance. Under this condition, the influence of various system parameters is thoroughly examined. The system’s stable and unstable responses are analyzed using frequency responses (FR), time histories, Poincaré maps (PM), and bifurcation diagrams. The results show that the NVF controller effectively reduces the amplitude of unwanted vibrations while maintaining stable system behavior. Both the piezoelectric and electromagnetic harvesters successfully extract electrical energy from the system’s oscillations. The comparison between analytical and numerical results demonstrates excellent agreement, confirming the reliability of the derived solutions. Applications: This dynamical model offers a dual advantage by harvesting energy from vibrating motion through both electromagnetic and piezoelectric transducers, which convert mechanical vibration into electrical power via magnetic induction and material strain, respectively, while actively reducing unwanted oscillations using an NVF controller. It is particularly useful in systems operating near resonance, such as in vehicles, buildings, and industrial machinery. The model enhances both energy efficiency and structural reliability, making it highly suitable for smart infrastructure and self-powered monitoring applications.

One of the most essential ship reactions to waves is roll motion. Due to the intricacy of ship wave interactions and their sensitivity, predicting such a reaction is extremely challenging. Because vibration motion is an undesirable occurrence, it must be removed, decreased, or controlled. A coupled Pitch- roll ship model with negative cubic velocity feedback control subjected to parametric excitations is premeditated and solved in this paper. The method of multiple time scales is applied to scrutinize the response of the two modes of the system neighbouring the simultaneous sub-harmonic, and internal resonance situation. Besides, the steady-state solution is determined through the Rung-Kutta Method (RKM) of fourth order. Stability of the steady state solution near this resonance case is discussed and studied applying Lyapunov’s first indirect method and Routh- Hurwitz criterion. The influences of the different parameters on the steady state solution are reconnoitred and discussed. The controller effects on the stability are clarified. Simulation results are accomplished with the help of MATLAB and Maple software programs.

The goal of this article is to reduce the vibration of a hybrid oscillator with a cubic–quintic nonlinear term under internal and external forces in the worst resonance case. To eliminate the harmful vibration in the system, the following strategies are suggested: nonlinear derivative feedback control (NDF), linear negative velocity feedback control (LNVC), nonlinear integral positive position feedback (NIPPF), integral resonant control (IRC), negative velocity with time delay (TD), and positive position feedback (PPF). It is discovered that the PPF control suppresses vibration more effectively than typical controllers, which reduces the vibration to 0.0001 with an effectiveness of 99.92%. Moreover, the main advantages of the PPF controller are its low cost and the fast response. The multiple time scale perturbation technique (MSPT) is used to apply the theoretical methodology and obtain a perturbed response. In order to close the concurrent primary and internal resonance case, frequency response (FR) equations are also used to study and analyze the system’s stability. The MATLAB software is used to complete and clarify all numerical topics. The FR curves are examined to determine the amplitude’s subsequent impact from variations in the parameters’ values. Lastly, a comparison of the numerical and analytical solutions is performed utilizing time history. Along with comparing the impact of our PPF damper on the hybrid oscillator, earlier research is also provided.

This study investigates the effect of a nonlinear saturation controller (NSC) on the van der Pol–Mathieu–Duffing oscillator (VMDO). The oscillator is a single degree of freedom (DOF) system. It is driven by an external force. It is described by a nonlinear differential equation (DE). The multiple-scale perturbation method (MSPT) is applied. It gives second-order analytical solutions. The first indirect Lyapunov method is used. It provides the frequency–response equation. It also shows the stability conditions. Internal resonance is included. The analysis considers steady-state responses. It studies simultaneous primary resonance with a 1:2 internal resonance (${\Lambda }_1\cong {\varpi }_1$ and ${\varpi }_1\cong 2{\varpi }_2$). Time–response simulations are presented. They show controlled and uncontrolled systems. Numerical solutions (NSs) are obtained with the fourth-order Runge–Kutta method (RK-4). They are compared with the approximate analytical solution (AS). The agreement is strong. It confirms the perturbation method. It shows that the method captures the main system dynamics.