Stability Enhancement and Bifurcation Mitigation in Nonlinear Inner Plate Oscillations Through PD Control

Axially moving wings offer remarkable aerodynamic efficiency and adaptability; however, they are highly susceptible to detrimental vibrations that may compromise flight stability and structural integrity. Previous studies have mainly focused on simplified linear models or passive control approaches, leaving the nonlinear dynamic behavior and active vibration suppression insufficiently addressed. To overcome these limitations, this study models the wing as a simplified cantilever plate and investigates its nonlinear dynamics under varying load conditions. A proportional–derivative (PD) controller is employed, and approximate analytical solutions to the governing equations are derived using the multiple-scale perturbation method (MSPM). The system’s response under primary resonance is analyzed through frequency response and bifurcation studies, while stability is assessed using the Routh–Hurwitz criterion. Analytical findings are validated with numerical simulations in MATLAB R2023b. Furthermore, the influence of key structural parameters on system dynamics and controller performance is examined. The results demonstrate that the PD controller effectively suppresses vibrations, offering a reliable solution for enhancing the stability of axially moving wing systems.