Design of TMD for Damped Linear Systems

Abstract eng:
Historically, an auxiliary mass-spring-damper control system attached to a primary structure was known as vibration absorber, tuned mass damper (TMD), or dynamic vibration absorber (DVA). In case of undamped primary structures, the first invented TMDhad no damping element and itwas only useful in a narrowrange of frequencies very close to the natural frequency of TMD. Ormondroyd and Den Hartog found that the TMD with viscous damper was effective to an extended range of frequencies. The damped TMD proposed by Den Hartog is now known as the Voigt type TMD where a spring element and a viscous element are arranged in parallel, and is has been considered as a standard model of TMD. Since then, there have been many optimization criteria given to design of TMD for undamped primary structures. Three typical optimization criteria are (1) H_∞ optimization (or fixed-points theory), (2) H₂ optimization, and (3) Stability maximization, and up to this time, all optimization criteria have been already solved analytically. In case of damped primary structure, it is difficult to obtain analytical solutions for the optimum parameters of the TMD. And there have been only either numerical methods or approximate analytical solutions for optimal parameters of TMD so far. In this report, two closed-form expressions for optimal tuning ratio of TMD are proposed. These results are obtained based on the equivalent linearization method with conventional criterion and dual criterion suggested by Anh, where the damped primary system is replaced equivalently by an undamped system and then we use the known results for the undamped system to give the expressions of optimal tuning ratio. The solutions proposed in the present study are validated by comparing with the results given by numerical method and by other authors. The comparisons have justified the significant accuracy of the proposed expressions for both small and large structural dampings.

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