Vibrations and dynamic chaos should be controlled in structures and machines. The wing of the airplane should be free from vibrations or it should be kept minimum. To do so, two main strategies are used. They are passive and active control methods. In this paper we present a mathematical study of passive and active control in some non-linear differential equations describing the vibration of the wing. Firstly, non-linear differential equation representing the wing system subjected to multi-excitation force is considered and solved using the method of multiple scale perturbation. Secondly, a tuned mass absorber (TMA) is applied to the system at simultaneous primary resonance. Thirdly, the same system is considered with 1:2 internal resonance active control absorber. The approximate solution is derived up to the fourth order approximation, the stability of the system is investigated applying both frequency response equations and phase plane methods. Previous work regarding the wing vibration dealt only with a linear system describing its vibration. Some recommendations are given by the end of the work.
"Comparison Between Passive and Active Control of a Non-Linear Dynamical System." Japan J. Indust. Appl. Math. 23 (2) 139 - 161, June 2006.