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In order to reduce the vibrations in the revolving machines and the structures in general, the dynamic absorbers are often used in different mechanical and civil applications (Crankshaft, rotor of the wings of a helicopter, beams, etc). The absorber is a mechanical oscillator called auxiliary system and it’s added to the main vibrating structure, in order to carry out a transfer of energy from the main system to the auxiliary system. This latter can either be added on the main system, or directly envisaged in design. A linear model is often insufficient to describe correctly the dynamical behaviour of a system, and then it’s natural to introduce non linear structure models. To solve these non linear problems, iterative numerical algorithms are often used. In order to approach the non linear dynamic behaviour of mechanical systems, many methods were proposed in the literature. As for the mechanical systems assimilated to a single dof system, one can mentioned different works. Whiston [1] studied the linear response of a mechanical oscillator with a single dof preloaded and subjected to a harmonic excitation. Natsiavas [2] studied the response of a vibration absorber applied to a machine with a nonlinear cubic stiffness of the Duffing type. In this chapter, numerical results are given with many parameter combinations. This leads to the improvement of the vibration absorber. The stability response, for periodic motion of harmonically excited single dof linear system with piecewise linear characteristics was studied by Natsiavas [3]. Chung et al.[4] carried out the extension of the method of incremental perturbation on a strongly nonlinear non-autonomous oscillator. Concerning the discrete systems with several dof, many works were interested in the nonlinearity. Vakakis and Paipetic [5] studied the effect of the absorber viscosity on a conservative system with several dof. Optimal values for the parameters describing the behaviour of the absorber are determined. Natsiavas [6] discussed the response of a strongly

Parametric and Robust Optimization of a Vibration Absorber with a Generalized Cubic, Quadratic and Non Integer Nonlinearities of Damping and Stiffness