Optimum viscoelastic absorbers for cubic nonlinear systems

Dynamic vibration absorbers are efficient devices used in vibration and noise control of several mechanical systems. In recent years, some studies about these control devices comprising systems with nonlinear characteristics have emerged. In those cases, either the primary system or the dynamic absorber, or even both, can be nonlinear in terms of their stiffness. On the other hand, the absorber damping is generally modeled as viscous. The viscous damping model is widely used in numerical simulations but is very difficult to achieve in real situations. An alternative is the use of viscoelastic damping models, which brings flexibility for vibration control actions. In this work, a methodology to optimally design a viscoelastic dynamic vibration absorber when attached to a nonlinear single-degree-of-freedom system will be presented. The mathematical formulation of the problem is based on the generalized equivalent parameters concept along with the harmonic balance method. The cubic nonlinearity is considered in the primary system and the viscoelastic material is represented by the four-parameter fractional derivative model. Numerical simulations to find the optimal parameters of the absorber are performed for three different types of viscoelastic materials using nonlinear optimization techniques. For some conditions, the results show that the viscoelastic absorber "linearizes" the compound system when this device is properly designed and attached to it. This is mainly due to the reaction forces introduced by the absorber and the large dissipation of vibratory energy introduced by the viscoelastic material. A study of the stability of the compound system reveals that, for most of the time, the periodic solution remains stable for the whole frequency range of concern.