Vibration Mitigation in a Pitch–Roll Ship Motion Under Multi-Parametric Excitations Using Proportional–Derivative Controllers

Vessel vibrations have serious safety risks and must be effectively mitigated. This study investigates the reduction in ship pitch–roll vibrations modeled as a two degrees of freedom of nonlinear spring–pendulum system subjected to multi-parametric excitation, using proportional–derivative controller. The main objective is to develop a rapid and efficient analytical approach to nonlinear vibration analysis. A non-perturbative approach is employed to transform weakly nonlinear oscillators of ordinary differential equations into equivalent linear ones without using Taylor expansions. He’s frequency formula plays a central role in this transformation. The resulting parametric solutions are validated using Mathematica Software (v13) and show a strong agreement with the original nonlinear model. The effects of various parameters on stability are examined. Theoretical analysis is conducted using the multiple time scales method to identify worst resonance conditions and derive frequency response equations. Stability near simultaneous sub-harmonic resonance is assessed using Routh–Hurwitz criterion. Numerical simulations based on the fourth-order Runge–Kutta method confirm the effectiveness of proportional–derivative control. Excellent agreement between analytical and numerical results demonstrates the accuracy, efficiency, and practical applicability of the proposed method.