Optimum viscoelastic absorbers for cubic nonlinear systems

Abstract

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.

Influence of temperature on optimum viscoelastic absorbers in cubic nonlinear systems

“…As shown in Espı´ndola and Silva (1992) and also seen in Bavastri et al (2013), an equivalent model of the secondary system, obtained by its equivalent generalized parameters, can be found through the complex rigidity on its base, k a ð, TÞ. The secondary system, as a simple absorber at a given constant temperature, is dynamically depicted in Figure 2, in correspondence with the previous view of Figure 1.…”

“…Then, by considering equations ( 7)-( 9), the generalized equivalent parameters for the absorber can be obtained as (Espı´ndola and Silva, 1992;Bavastri et al, 2013) c e ð, TÞ ¼ m a a r a ð, TÞð, TÞ" 3 a r a ð, TÞ À "…”

“…system comprising a weakly nonlinear primary system and a linear absorber was presented by Ji and Zhang (2010) and Ji (2012), with the aim of studying the effectiveness of the linear absorber in suppressing resonance vibrations. With the exception of their previous work (Bavastri et al, 2013), the authors are not aware of other cases of the use of linear viscoelastic absorbers to mitigate vibrations in nonlinear systems.…”

“…It is well-known that the dynamic properties of viscoelastic materials change, in a more or less pronounced way, when the temperature changes. In Bavastri et al (2013), a methodology was presented for the optimal design of a dynamic viscoelastic absorber acting in a cubic nonlinear system (Duffing type oscillator) at a given temperature. The aim of the current paper is to review the methodology proposed there and to study the influence of temperature variations on the optimal design.…”

A methodology for an optimal design of physical parameters, positions, and viscoelastic materials of simple dynamic absorbers for passive vibration control

“…However, the energy dissipation of viscoelastic materials depends on the temperature and frequency in which the materials are operating Torvik, 1979, 1986;Pritz, 1996;Xu et al, 2010). Bavastri et al (2013) and Febbo et al (2014) presented a methodology for an optimal design of a VDN using the fractional viscoelastic model to control a cubic nonlinear system for a given temperature. Particularly in Febbo et al (2014), the influence of temperature was considered in vibration control, but the optimal viscoelastic material was not considered in the optimization process, only the physical parameters of the VDN.…”