Vibration Reduction and Stability Investigation of Van Der Pol–Mathieu–Duffing Oscillator via the Nonlinear Saturation Controller

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 (Λ1≅ϖ1 and ϖ1≅2ϖ2). 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.