استفاده از میراگرهای جرمی تنظیم شده برای کنترل پاسخ سازه های پیچشی Tuned mass dampers for response control of torsional buildings

For seismic response control of building structures, several passive and active control schemes are being considered. Among the active and passive schemes, the latter have gained a wider acceptance and are already being used in practice. The passive systems can be divided into two basic categories: (1) base isolation systems and (2) energy dissipation systems. In energy dissipation systems, several di<erent damping systems have been considered (1) friction dampers, (2) visco-elastic dampers, (3) viscous dampers, (4) yielding metallic dampers, and (5) tuned mass dampers. (Although, the tuned mass damper is put in the category of the energy dissipation system here, the initial concept of this system was not based on the dissipation of energy but rather on the transfer of energy from the system to be protected to the tuned mass absorber.) All these systems have their special features and attributes that makethem suitablefor particular responsecontrol applications. Thefocus of this paper is on theuseof thetuned mass dampers. Since its invention in 1909 by Frahm, the concept of a tuned mass damper has attracted special and continued attention of several researchers and practitioners for its application to control vibrations caused by di<erent types of forces. Since it will not be possible to give a complete account of the vast literature available on these devices, here only a few relevant studies are cited. Den Hartog has lucidly described the working principle of the device in his monograph [1], providing simpleformulas to obtain theoptimum tuning and damping parameters of a tuned mass damper to control the displacement of an undamped single degreeof-freedom system subjected to a harmonic force. Since its early initial application to control the displacement response caused by a harmonic force, now several other excitation and response control conditions have been analyzed. Warburton [2] among others has extended the solution repertoire by covering several other cases of excitation and responses to be controlled. The use and limitation of these formulas with multi-degree of freedom systems is also discussed. Among other researchers, Tsai and Lin [3] have extended the classical solution to a damped primary system. Using curve =tting, they have provided formulas to obtain the optimal parameters. Although tuned mass dampers have been quite successfully used in several structures to control the wind induced vibrations, their use for seismic response control has not been that convincing. Some investigators [4–۶] have shown that tuned mass dampers can be used to control the seismic response, yet there are others [7–۹] that show otherwise. Without scrutinizing the details of these studies, it is not straightforward to identify the precise reason (or reasons) for these conLicting conclusions. However, Villaverde and his associate [10–۱۲] o<er, perhaps, the most convincing argument for these di<erent =ndings. They observe that the primary reason for ine<ectiveness of the dampers is the use of the classical solutions that are not necessarily optimal for the particular situation under study. They suggest that the damper parameters must be tuned such that the damping ratios of the dominant modes areincreased. For this, thedamper must bein resonancewith its supporting structureand its damping ratio must be equal to the structural damping ratio plus a term that depends on the generalized mass ratio and the modal displacement at the point where the damper is attached [11]. Several numerical results were presented to show the e<ectiveness of this tuned mass damper design procedure. Sadek et al. [13], however, further examined the approach suggested by Villaverde and Koyama [11] with an example of a tuned mass damper attached to a single degree of freedom system. They observe that that, except for very small mass ratios, Villaverde and Koyama’s approach usually leads to unequal damping ratios in the two modes of the combined system, which is not as eNcient as having two equal damping ratios in these modes. Based on an exhaustive numerical search of the eigenvalues of the state matrix of the combined structure and damper system for di<erent values of the system parameters, they were able to identify the optimum values of the tuning and damping ratio parameters that would produce two modes with nearly equal damping ratios. By curve=tting they proposesimpleformulas to calculatetheseoptimum parameters in terms of the mass ratio and the damping ratio of the primary mass. They present several sets of numerical results to demonstrate the e<ectiveness of their design procedure. The e<ective use of the tuned mass damper formulas developed for the single degree of freedom system with the multi degree of freedom systems has also been well demonstrated by several investigators, e.g. in References [10–۱۳]. In such applications, the multidegree of freedom system is represented by an equivalent single degree of freedom system. Such equivalent representations can be successfully used when the response of the multidegree of freedom system is dominated by a single mode, usually the fundamental mode. However, to further improve the robustness and reduce the sensitivity of a design caused by miss-tuning or variability in the system parameter values, and also to control structures with closely spaced frequencies, several researchers [14–۱۸] have proposed the use of a cluster of tuned mass dampers (usually called multiple tuned mass dampers). The frequencies of the cluster are distributed within a frequency bandwidth, usually centered around the frequency of the dominant mode. The objective of using a cluster of dampers is usually not to control several modes of thestructurebut to improvethecontrol characteristics of thesystem. However, Rana and Soong [19] have also examined the use of di<erent tuned mass dampers for controlling di<erent modes of a system. The building systems with accidental or intended eccentricities between their mass and sti<ness centers respond with coupled lateral and torsional motions under seismic excitation. It is of interest to study the response control of such structures using several tuned mass dampers, and this paper is precisely concerned with this topic. The writers are familiar with thestudy by Jangid and Dutta [20] and Lin et al. [21] on thecontrol of torsional systems with tuned mass dampers. Jangid and Dutta [20] study the response control of a two degrees of freedom torsional system by a cluster of multiple tuned mass dampers. The input to the main system was white noise excitation. The optimum frequency bandwidth value—the value corresponding to themaximum reduction in theroot mean squarevalueof themain system response—was obtained by a parametric variation study. Lin et al. [21] study a multi-story torsional building system with one and two tuned mass dampers. They propose a method to identify the dominant modes and critical orientation of the damper track. The optimal parameters are obtained by minimization of the root mean square response of displacement of thedominant modefor a random input. In this study, four tuned mass dampers, placed along two orthogonal directions in pairs, are considered to control the coupled lateral and torsional response of a multistory building structures subjected to bi-directional earthquake induced ground motions. The objective here is not to target a particular modeof thesystem for control, but to maximizea performance function that quanti=es reduction in a particular response or an overall system response. The performance function may be de=ned in terms of the Loor accelerations, story drifts and shears, and other response quantities of interest. The seismic motion at the base is represented in a more general form either by a stationary random model or by a set of design response spectra. A genetic algorithm is used to search for the optimal design parameters of the four dampers. Several sets of numerical results are presented to demonstrate the e<ectiveness of such optimally designed tuned mass dampers.