Vibration reduction of a non-linear ship model using positive position feedback controllers

In this paper, a four-degree-of-freedom (4-DOF) of coupled pitch-roll ship model including two positive position feedback controllers (PPF) are considered appropriate to reduce the vibration of the ship model system (coupled pitch and roll modes) at primary resonance excitation (Omega(1) congruent to omega(1), Omega(2) congruent to omega(2)) in the presence of 1:1 internal resonance ( omega(3) congruent to omega(1), omega(4) congruent to omega(2)). The multiple time scales perturbation method (MSPM) is applied to get the perturbation solution and all resonance cases. The response studies for both the open and closed-loop systems are showed numerically by using the Runge-Kutta fourth-order (RK-4) method (package ode45 in Matlab R2014a). The PPF controllers succeed in reducing the vibration amplitudes for the ship model system reaches to 0.001 with a reduction rate of 99.85% for the first mode (roll) and reaches to 0.0003 with reduction rate 99.90% for the second mode (pitch). This clarifies that the PPF controller is an excellent control to reduce the vibration for the system. Then, the frequency response equations and the stability analysis are determined according to the Routh-Hurwitz criterion. The effects of different parameters are studied numerically for the main-system and controller with the same used values of the vibrating system. Validation between numerical solutions, perturbation solution and frequency response solutions are studied. Finally, a comparison is performed with earlier published work.