Vibration Reduction and Stability of Non-Linear System Subjected to External and Parametric Excitation Forces under a Non-Linear Absorber

Vibrations are usually undesired phenomena as they may cause discomfort, disturbance, damage, and sometimes destruction of machines and structures. It must be reduced or controlled or eliminated. One of the most common methods of vibration control is the dynamic absorber. In this paper, the non-linear dynamics of a two-degree-of freedom vibration system including quadratic and cubic non-linearities subjected to external and parametric excitation forces is investigated. The system consists of the main system and the absorber which represents many applications in machine tools, ultrasonic cutting process, etc. The vibration of the main system can be controlled applying non-linear absorber (passive control). The stability of the system is investigated using both frequency response curves and phase-plane trajectories. Multiple time scales perturbation method is applied to differential equations describing the system to obtain the analytical solution. All possible resonance cases are extracted. The effects of different parameters of the system are studied numerically. There exist multi-valued solutions which increase or decrease by the variation of some parameters. The solution loses stability on increasing negative value of 2 α . The Numerical simulations show the system exhibits periodic motions and chaotic motions. A comparison is made with the available published work. 1052 M. Sayed, Y. S. Hamed and Y. A. Amer