Dynamic Analysis of Roll-Pitch Coupled Motion of Ship With or Without Gain and Delayed Feedback Control

Abstract

This paper presents the complex behavior of two-degree-of-freedom (2DOF) nonlinear coupled pitch-roll motions such as periodic, aperiodic, and chaotic solutions with or without active vibration control based on time-delayed feedback control under sinusoidal excitation. In the present study, the classical numerical integration technique has been applied to study the response behavior of coupled pitch-roll motion. A comprehensive numerical scheme based on numerical integration is applied to analyze all possible resonances of nonlinear couple pith-roll motion of the ship under time-delayed feedback. The numerical integration technique is applied to solve the nonlinear differential equation of two modes of a system model under sinusoidal harmonic excitation near the primary resonance, subharmonic resonance, combination resonance, internal resonance, and simultaneous resonance. The behavior of the system is studied using a frequency response curve and the different types of resonance cases (1) Ω ≅ ω1, (2) Ω ≅ 2ω1, ω2 ≅ 2ω1, (3) Ω ≅ ω1, ω1 ≅ ω2, (4) Ω ≅ ω1, ω2 ≅ 2ω1 are checked. The numerical simulation is carried out by numerical integration (NI) method and the results are presented graphically and are discussed. After the application of feedback control law, the frequency response plots have been obtained and control of response has been studied and compared. The different feedback control laws have been applied with either only displacement feedback with gain and delay or both displacement and velocity feedback with gain and delay. The gain and delay values are generally to be obtained from linear stability analysis. In the present study, the two-degree-of-freedom model is considered as weakly nonlinear coupled. The arbitrary gain and delay value have been used on the trial basis of different combinations of displacement and velocity gain-delay value to obtain the control of roll and pitch responses. The response stability of the 2DOF system is solved using MATLAB and Simulink toolbox. The frequency responses were analyzed using ode45 of MATLAB and Simulink of ode 45 and the results were compared. Characteristics of the solutions were also identified with the help of the phase plot diagram and Poincare map.