Global-local approach to the design of dynamic vibration absorber for damped structures

Abstract

The dynamic vibration absorber (DVA) has attracted attention since its invention. This paper deals with the optimization problems of the standard DVA and two other models of DVA called three-element DVA and non-traditional DVA for damped primary structures. Unlike the standard configuration, the three-element DVA contains two spring elements in which one is connected to a dashpot in a series and the other is placed in parallel. Meanwhile the non-traditional DVA has a linear viscous damper connecting the absorber mass directly to the ground. There have been some studies on the design of three-element and non-traditional dynamic vibration absorbers in the case of undamped primary structures. These studies have shown that both three-element and non-traditional DVAs perform better than the standard DVA. When the primary structure is damped, there are very few studies on the three-element and non-traditional DVAs in the literature. This article proposes a global-local approach to give approximate analytical solutions of the $H_{\infty }$ optimization for all standard, three-element and non-traditional DVAs attached to damped primary structures. The main idea of the study is based on the global-local criterion of the equivalent linearization method in order to replace approximately the original damped structure by an equivalent undamped one. Afterwards, the already derived expressions of the optimal parameters for the undamped primary system case are used with the equivalent undamped structure that was just obtained. The numerical simulations are carried out to verify the effectiveness of the obtained results. Additionally, design aids, which show the variation of the optimal design quantities for various DVA mass ratios and inherent structural damping ratios are also provided.