ISSN 2347-1921 Volume 12 Number 11 Journal of Advances in Mathematics 6758 | Page November 2016 www.cirworld.com Positive Position Feedback Controllers for Reduction the Vibration of a Nonlinear Spring Pendulum Y.A. Amer, A.T. EL-Sayed, F.T. El-Bahrawy Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt yaser31270@yahoo.com Department of Basic Sciences, Modern Academy for Engineering and Technology, Maadi, Egypt ashraftaha211@yahoo.com Department of Basic Sciences, Modern Academy for Engineering and Technology, Maadi, Egypt f_t_m_33@yahoo.com ABSTRACT In this paper, the two positive position feedback controllers (PPF) are proposed to reduce the longitudinal and angular vibrations of the nonlinear spring pendulum system which simulated the ship roll motion. This described by a four-degree- of-freedom system (4-DOF) which subjected to the external excitation force at simultaneous primary and internal resonance case. The method of multiple scale perturbation technique (MSPT) is applied to study the approximate solution of the given system. The stability of the system is investigated near the resonance case applying the frequency-response equations. Numerically, the effects of different controller’s parameter’s on the basic system behavior are studied. Keywords Vibration control; Positive position feedback; Stability; simultaneous resonance and spring pendulum. Nomenclature , ( 1, 2) j j c j   The damping coefficients of the spring pendulum modes and the PPF controllers, respectively. 1 2 3 , ,   and 4  The natural frequencies of the spring pendulum modes and PPF controllers, respectively. 1  and 2  The non-linear parameters j f The external forcing amplitudes of the main system j  The excitation frequencies of the main system  A small perturbation parameter. x Longitudinal response of the spring–pendulum  Angular response of the pendulum u Displacement of the first PPF controller. v Displacement of the second PPF controller. 1 2 ,   Control signal gains. 1 2 ,  Feedback signal gains. 1. INTRODUCTION Roll motion is one of the most significant phenomenon of a ship in waves. Therefor a lot of projects mentioned that. The roll motion of a ship can be fixed by analyzing various forms of moments acting on the ship, functional and actual mass moments of inertia roll damping moment, restoring moment, wave excitation and other moments caused by other modes of ship motion. Otherwise, in roll motion equation, there are a substantial term must be in the equation which is the damping term, that because among the other terms it controls on the magnitude of the amplitude. Though, it is the most difficult parameter to estimate because of its complex nature. The reason for its complexity is because of the non-linear nature of roll, and also because of the difficulty indetermining the roll damping characteristics.