Extension of dual equivalent replacement method to optimization of damped primary structure with tuned mass damper

Abstract

Optimization of systems of damped primary structures with tuned mass dampers (TMDs) remains a challenge for analytical methods to date. Due to the existence of the primary structure damping, the well-known fixed point theory is not directly applicable. This paper deals with this problem. First, it is found that there exist two so-called semi-fixed points of which the abscissas are independent of TMD’s damping even in the case of damped primary structures. Next, the Dual equivalent replacement method (DER) is introduced with the formulation in the form of the Weighted dual criterion (WDC). Based on DER, the original system can be equivalently replaced with a system of undamped primary structure with the same TMD, to which the fixed point theory can be applied. It turns out that the damping ratio of the original system is the same as the optimal one of the equivalent system derived from the fixed point theory. Meanwhile, the tuning ratio depends on the embedding weight of WDC. A condition involving semi-fixed points is proposed to find an explicit weight, followed by the optimal tuning ratio. Comparison of the obtained result with those provided by several approximate techniques is then performed. It is shown that the proposed method gives better results than those methods.