Different control methods of an oscillatory cantilever beam excited transversely at its free end

This paper examines the dynamic behavior and active management of a uniform cantilever beam under simultaneous principal parametric and primary resonance case Ω1≈2ω,Ω2≈ω, when it is subjected to transverse excitations at its free end. Multiple control strategies including proportional derivative (PD), proportional-integral-derivative (PID), linear quadratic regulator (LQR), sliding mode control (SMC), and a nonlinear proportional derivative cubic velocity feedback (NPDCVF) controller are implemented to suppress vibrations and enhance system stability. The mathematical model of the beam is established via Euler-Bernoulli beam theory, capturing the essential dynamics under both single and dual-frequency excitations. The motion equations (ME) are investigated using the multiple time scales method (MTSM), and the approximation solutions (AS) are numerically validated with the fourth-order Runge-Kutta (4RK) method. Numerical simulations are conducted to evaluate the performance of each control technique, focusing on displacement amplitude reduction, settling time, and control effort. The results demonstrate that nonlinear and hybrid control approaches, such as NPDCVF provide superior damping and faster stabilization compared to linear controllers, while PD, PID, SMC, and LQR achieve effective vibration mitigation with lower computational complexity. A comparative analysis of these strategies offers guidance for selecting optimal controllers in practical structural vibration suppression applications. All numerical results were obtained and cross-validated using MATLAB Software.