Generalization of the Den Hartog model and rule-of-thumb formulas for optimal tuned mass dampers

In recent years, the need of improving safety standards for both existing and new buildings against earthquake and wind loads has created a growing interest in the use of the so-called tuned mass dampers, exploited to control, in active or passive way, the dynamic response of structures. To design and optimize tuned mass damper systems, the effective analytical procedure proposed by Den Hartog in his seminal work (Den Hartog, 1985) has been widely adopted over the years, without including damping of the main structure. However, in many cases of engineering interest, the damping of the primary system plays a key role in the overall mechanical response, with the result of an increase in complexity of the related optimization problem, which are in fact solved in the vast majority by means of ad hoc numerical procedures. In the present work, we recover the classical optimization strategy by Den Hartog and generalize it by including the main system’s damping, providing new analytical solutions whose results are consistent with the few ones obtained through alternative mathematical methods by Thompson (1981) and Krenk (2005) and subsequently by Fang et al. (2019). In particular, some closed-form solutions and helpful rule-of-thumb formulas for the optimal setting of the design parameters are derived, by considering two types of incoming excitations, i.e. the force acting on the main mass and the ground motion input, both of interest for engineering applications. Finally, theoretical outcomes are compared with consolidated data from the literature. • Generalization of the Den Hartog model and analytical derivation of optimal tuning frequency ratio and damping of the absorber in Tuned Mass Dampers subjected to force or ground motion inputs. • Definition of rule-of-thumb formulas for design purposes. • Comparison of results from proposed analytical approach and numerical analyses to show advantages and effectiveness of the method, also for design parameters out of classical ranges. • Explicit expressions for optimal Tuned Mass Damper configurations in cases of stochastic ground motion input.