Optimum design and experimental study of multiple tuned mass dampers with limited stroke

This paper develops a two‐stage optimum design procedure for multiple tuned mass dampers (MTMD) to reduce structural dynamic responses with the limitation of MTMD's stroke. A new performance index, which is a linear combination of structural response ratio and MTMD stroke ratio by a weighting factor α, is proposed; α is in the range from 0 to 1.0. The larger the α, the more important the stroke. The case of α=1.0 indicates that MTMD is locked. The analytical results show that the MTMD's stroke can be significantly suppressed with little sacrifice of structural control effectiveness when an appropriate α is selected. To verify the design algorithm, a 360 kg‐MTMD composed of five TMD units arranged in parallel was fabricated. Shaking table tests of a large‐scale three‐story building with and without the MTMD under earthquake excitations were conducted at the National Center for Research on Earthquake Engineering (NCREE) in Taiwan. The experimental results show that MTMD is not only effective in mitigating the building responses but also is successful in suppressing its stroke. Copyright © 2010 John Wiley & Sons, Ltd.     

Design formulas for tuned mass dampers based on A perturbation technique

Modal properties of tuned mass damper (TMD)-structure two-degree-of-freedom (2DOF) linear systems are studied employing a perturbation technique. Using the perturbation solutions, formulas relevant to designing the TMD for various types of loading are obtained; they are expressed as a function of mass ratio, tuning ratio, damping ratio of the TMD and damping ratio of the structure. Equivalent additional dampings of the structure due to the TMD are derived for random and harmonic forces. Matched expressions of equivalent damping, which are valid for detuned, i.e. non-optimal, conditions are also presented. The stability boundary of TMD-structure systems subject to linear self-excited forces is derived in a closed form. Using the perturbation solutions, procedures for optimizing the TMD parameters for various types of loading are explained and the optimal values are derived. The formulas obtained in this study can be used with good accuracy for mass ratios less than 0.02.     

基于Maxwell型阻尼器的多重调谐质量阻尼器性能评价

研究了基于Maxwell型阻尼器的多重调谐质量阻尼器(MTD—MTMD)在控制结构地震反应方面的最优动力特性。利用建立的设置MTD-MTMD时结构的传递函数,定义了设置MTD—MTMD时结构的动力放大系数(DMF)。将MTD-MTMD的优化准则定义为结构最大动力放大系数的最小值的最小化(Min．Min．Max．DMF)。利用定义的优化准则,评价了Maxwell型阻尼器的松弛时间系数(RTC)对MTD—MTMD最优参数和有效性的影响。利用最大的MTD—MTMD动力放大系数(DMF),评价了RTC对MTD-MTMD冲程的影响。     

Design of multiple tuned mass dampers by using a numerical optimizer

A new method to design multiple tuned mass dampers (multiple TMDs) for minimizing excessive vibration of structures has been developed using a numerical optimizer. It is a very powerful method by which a large number of design variables can be effectively handled without imposing any restriction before the analysis. Its framework is highly flexible and can be easily extended to general structures with different combinations of loading conditions and target controlled quantities. The method has been used to design multiple TMDs for SDOF structures subjected to wide‐band excitation. Some novel results have been obtained. To reduce displacement response of the structure, the optimally designed multiple TMDs have distributed natural frequencies and distinct damping ratios at low damping level. The obtained optimal configuration of TMDs was different from the earlier analytical solutions and was proved to be the most effective. A robustness design of multiple TMDs has also been presented. Robustness is defined as the ability of TMDs to function properly despite the presence of uncertainties in the parameters of the system. Numerical examples of minimizing acceleration structural response have been given where the system parameters are uncertain and are modeled as independent normal variates. It was found that, in case of uncertainties in the structural properties, increasing the TMD damping ratios along with expanding the TMD frequency range make the system more robust. Meanwhile, if TMD parameters themselves are uncertain, it is necessary to design TMDs for higher damping ratios and a narrower frequency range. Copyright © 2004 John Wiley & Sons, Ltd.     

Seismic response of controlled structures with active multiple tuned mass dampers

Active multiple tuned mass dampers （referred to as AMTMD）, which consist of several active tuned mass dampers （ATMDs） with identical stiffness and damping coefficients but varying mass and control force, have recently been proposed to suppress undesirable oscillations of structures under ground acceleration. It has been shown that the AMTMD can remarkably improve the performance of multiple tuned mass dampers （MTMDs） and is also more effective in reducing structure oscillation than single ATMDs. Notwithstanding this, good performance of AMTMD （including a single ATMD illustrated from frequency-domain analysis） may not necessarily translate into a good seismic reduction behavior in the time-domain. To investigate these phenomena, a three-story steel structure model controlled by AMTMD with three ATMDs was implemented in SIMULINK and subjected to several historical earthquakes. Likewise, the structure under consideration was assumed to have uncertainty of stiffness, such as 4-15% of its initial stiffness, in the numerical simulations. The optimum design parameters of the AMTMD were obtained in the frequency-domain by implementing the minimization of the minimum values of the maximum dynamic magnification factors （DMF） of general structures with AMTMD. For comparison purposes, response analysis of the same structure with a single ATMD was also performed. The numerical analysis and comparison show that the AMTMD generally renders better effectiveness when compared with a single ATMD for structures subjected to historical earthquakes. In particular, the AMTMD can improve the effectiveness of a single ATMD for a structure with an uncertainty of stiffness of 4-15% of its initial stiffness.     

Optimal design theories of tuned mass dampers with nonlinear viscous damping

Optimal design theory for linear tuned mass dampers(TMD) has been thoroughly investigated,but is still under development for nonlinear TMDs.In this paper,optimization procedures in the time domain are proposed for design of a TMD with nonlinear viscous damping.A dynamic analysis of a structure implemented with a nonlinear TMD is conducted fi rst.Optimum design parameters for the nonlinear TMD are searched using an optimization method to minimize the performance index.The feasibility of the proposed optimiza...     

Active multiple tuned mass dampers for structures under the ground acceleration

Active multiple tuned mass dampers (AMTMD) consisting of many active tuned mass dampers (ATMDs) with a uniform distribution of natural frequencies have been, for the first time, proposed for attenuating undesirable vibrations of a structure under the ground acceleration.The multiple tuned mass dampers (MTMD) in the AMTMD is manufactured by keeping the stiffness and damping constant and varying the mass. The control forces in the AMTMD are generated through keeping the identical displacement and velocity feedback gain and varying the acceleration feedback gain. The structure is represented by its mode‐generalized system in the specific vibration mode being controlled using the mode reduced‐order method. The optimum parameters of the AMTMD are investigated to delineate the influence of the important parameters on the effectiveness and robustness of the AMTMD by conducting a numerical searching technique. The parameters include the frequency spacing, average damping ratio, tuning frequency ratio, total number and normalized acceleration feedback gain coefficient. The criterion, which can be stated as the minimization of the minimum values of the maximum dynamic magnification factors (i.e. Min.Min.Max.DMF), is chosen for the optimum searching. Additionally, for the sake of comparison, the results of the optimum MTMD (the passive counterpart of AMTMD) and ATMD are also taken into account in the present paper. It is demonstrated that the proposed AMTMD can be expected to significantly reduce the oscillations of structures under the ground acceleration. It is also shown that the AMTMD can remarkably improve the performance of the MTMD and has higher effectiveness than ATMD. Copyright © 2002 John Wiley & Sons, Ltd.     

大跨桥梁振动控制的杠杆主动多重调谐质量阻尼器策略

提出了适用于控制大跨桥梁风致振动的杠杆式主动多重调谐质量阻尼器（LT-AMTMD）控制策略.利用建立的LT-AMTMD结构系统的动力放大系数,评价了LT-AMTMD的性能.数值结果表明,驱动器置于质量块处的LT-AMTMD比驱动器置于其它位置的LT-AMTMD更加有效.驱动器置于质量块处的LT-AMTMD可以根据实际需要,通过改变支撑位置来调节弹簧的静伸长,而且保持其性能不变（包括冲程）.数值结果还表明,驱动器置于质量块处的LT-AMTMD可以明显地提高LT-MTMD的性能,而且比单个杠杆式主动调谐质量阻尼器（LT-ATMD）更加有效.     

Seismic structural control using semi-active tuned mass dampers

This paper focuses on how to determine the instantaneous damping of the semi-active tuned mass damper (SATMD) with continuously variable damping. An off-and-towards-equilibrium (OTE) algorithm is employed to examine the control performance of the structure/SATMD system by considering the damping as an assumptive control action. The damping modification of the SATMD is carried out according to the proposed OTE algorithm, which is formulated based on analysis of the structural movement under external excitations, and the measured responses of the structure at every time instant. As examples two numerical simulations of a five-storey and a ten-storey shear structures with a SATMD on the roof are conducted. The effectiveness on vibration reduction of MDOF systems subjected to seismic excitations is discussed. Analysis results show that the behavior of the structure with a SATMD is significantly improved and the feasibility of applying the OTE algorithm to the structural control design of SATMD is also verified.     

结构多重双重调谐质量阻尼器(MDTMD)的新控制策略

本文提出了一种新的控制策略——多重双重调谐质量阻尼器(以下简称为MDTMD)。MDTMD系统参数的可能组合形成十种MDTMD模型,本文评价其中最易制作的一种MDTMD模型。利用定义的优化目标函数,评价了MDTMD的控制性能。数值结果表明MDTMD比双重调谐质量阻尼器(DTMD)具有更好的有效性和对频率调谐的鲁棒性。但MDTMD的冲程大于DTMD的冲程。     

Experimental and analytical studies on multiple tuned mass dampers for seismic protection of porcelain electrical equipment

Porcelain electrical equipment (PEE), such as current transformers, is critical to power supply systems, but its seismic performance during past earthquakes has not been satisfactory. This paper studies the seismic performance of two typical types of PEE and proposes a damping method for PEE based on multiple tuned mass dampers (MTMD). An MTMD damping device involving three mass units, named a triple tuned mass damper (TTMD), is designed and manufactured. Through shake table tests and finite element analysis, the dynamic characteristics of the PEE are studied and the effectiveness of the MTMD damping method is verified. The adverse influence of MTMD redundant mass to damping efficiency is studied and relevant equations are derived. MTMD robustness is verified through adjusting TTMD control frequencies. The damping effectiveness of TTMD, when the peak ground acceleration far exceeds the design value, is studied. Both shake table tests and finite element analysis indicate that MTMD is effective and robust in attenuating PEE seismic responses. TTMD remains effective when the PGA far exceeds the design value and when control deviations are considered.     

Optimum multiple tuned mass dampers for structures under the ground acceleration based on DDMF and ADMF

Multiple tuned mass dampers (MTMD) consisting of many tuned mass dampers (TMDs) with a uniform distribution of natural frequencies are taken into consideration for attenuating undesirable vibration of a structure under the ground acceleration. A study is conducted to search for the preferable MTMD which performs better and is easily manufactured from the five available models (i.e. MTMD‐1 – MTMD‐5), which comprise various combinations of the stiffness, mass, damping coefficient and damping ratio in the MTMD. The major objective of the present study then is to evaluate and compare the control performance of these five models. The structure is represented by its mode‐generalized system in the specific vibration mode being controlled by adopting the mode reduced‐order approach. The optimum parameters of the MTMD‐1 – MTMD‐5 are investigated to reveal the influence of the important parameters on their effectiveness and robustness using a numerical searching technique. The parameters include the frequency spacing, average damping ratio, tuning frequency ratio, mass ratio and total number. The criteria selected for the optimum searching are the minimization of the maximum value of the displacement dynamic magnification factor (DDMF) and that of the acceleration dynamic magnification factor (ADMF) of the structure with the MTMD‐1 – MTMD‐5 (i.e. Min.Max.DDMF and Min.Max.ADMF). It is demonstrated that the optimum MTMD‐1 and MTMD‐4 yield approximately the same control performance, and offer higher effectiveness and robustness than the optimum MTMD‐2, MTMD‐3, and MTMD‐5 in reducing the displacement and acceleration responses of structures. It is further demonstrated that for both the best effectiveness and robustness and the simplest manufacturing, it is preferable to select the optimum MTMD‐1. Copyright © 2002 John Wiley & Sons, Ltd.     

Performance of multiple tuned mass dampers for attenuating undesirable oscillations of structures under the ground acceleration

Multiple tuned mass dampers (MTMDs) consisting of many tuned mass dampers (TMDs) with a uniform distribution of natural frequencies are considered for attenuating undesirable vibration of a structure. The MTMD is manufactured by keeping the stiffness and damping constant and varying the mass. The structure is represented by its mode‐generalized system in the specific vibration mode being controlled using the mode reduced‐order method. The optimum parameters of the MTMD are investigated to delineate the influence of the important parameters on the effectiveness and robustness of the MTMD by conducting a numerical searching technique in two directions. The parameters include: the frequency spacing, average damping ratio, mass ratio and total number. The criterion selected for the optimization is the minimization of the maximum value of the dynamic magnification factor (DMF) of the structure with MTMD (i.e. Min.Max.DMF). In this paper, for the sake of comparison, the MTMD(II), which is made by keeping the mass constant and varying the stiffness and damping coefficient, and a single TMD are also taken into account. It is demonstrated that the optimum frequency spacing of the MTMD is the same as that of the MTMD(II) and the optimum average damping ratio of the MTMD is a little larger than that of the MTMD(II). It is also found that the optimum MTMD is more effective than the optimum MTMD(II) and the optimum single TMD with equal mass. Copyright © 2000 John Wiley & Sons, Ltd.     

A method for tuning tuned mass dampers for seismic applications

This paper studies tuned mass dampers (TMDs) resulting in high modal damping for mechanical systems incorporating such devices for the purpose of seismic response reduction. Focusing on the determination of damping and tuning, the proposed methodology identifies a point of multiplicity of complex eigenvalues and eigenvectors, resulting in different parameters for TMDs according to their location with respect to such multiplicity condition. It is shown that significant equal modal damping and average modal damping can be induced by properly tuning highly damped TMDs, obtaining parameters intrinsic to the mechanical systems, and excitation independent. Further, it is shown that the methodology yields, as particular cases, two proposals by others using TMDs for the same purpose of seismic response abatement. Copyright © 2012 John Wiley & Sons, Ltd.     

非对称结构扭转振动多重调谐质量阻尼器(MTMD)控制的最优位置

研究了非对称结构扭转振动多重调谐质量阻尼器(MTMD)控制的最优位置。本文采用的MTMD具有相同的刚度、阻尼，但质量不同。基于导出的设置MTMD时非对称结构扭转角位移传递函数，建立了扭转角位移动力放大系数解析式。MTMD最优参数的评价准则定义为：非对称结构最大扭转角位移动力放大系数的最小值的最小化。MTMD的有效性评价准则定义为：非对称结构最大扭转角位移动力放大系数的最小值的最小化与未设置MTMD时非对称结构最大扭转角位移动力放大系数的比值。基于定义的评价准则，研究了非对称结构的标准化偏心系数(NER)和扭转对侧向频率比(TTFR)对不同位置MTMD最优参数和有效性的影响。     

结构双层多重调谐质量阻尼器(DMTMD)控制策略的鲁棒性评价

基于定义的二类优化目标函数,评价双层多重调谐质量阻尼器(DMTMD)控制策略对漂移频率系数(DFR)摄动的鲁棒性。数值研究表明,使用第二类优化准则设计的DMTMD、双重调谐质量阻尼器(DTMD)和多重调谐质量阻尼器(MTMD)比使用第一类优化准则设计的DMTMD、DTMD和MTMD具有更高的对DFR摄动的鲁棒性。而且,使用第二类优化准则设计的总数为4的DMTMD、DTMD和总数为11的MTMD具有近似相同的对DFR摄动的鲁棒性。     

结构主动与被动调谐质量阻尼器的减震性能

使用Kanai-Tajimi地震动模型,建立了主动调谐质量阻尼器(ATMD)结构系统的传递函数。将ATMD最优参数的评价准则定义为：设置ATMD结构均方根位移(解析式)的最小值的最小化。将ATMD有效性的评价准则定义为：设置ATMD结构均方根位移的最小值的最小化与未设置ATMD结构的均方根位移之比。根据逃择的评价准则,评价了地震卓越频率系数(EDFR)对ATMD抗震控制性能的影响。同时也评价了EDFR对被动调谐质量阻尼器(PTMD)抗震控制性能的影响。     

On tuned mass dampers for reducing the seismic response of structures

This paper presents an energy‐based theoretical model for a two degree‐of‐freedom mechanical system. After a general formulation in Appendix A, the model is specialized to study tuned mass dampers as a means to substantially increase modal damping in order to induce a consequential decrease of the seismic response of the structures thus provided. Although approximate since it neglects coupling due to damping, it is shown that the model yields a first‐order approximation to the exact frequencies, providing values of optimum damping that closely match exact results proposed by others. In view of this, it is proposed that the model be applied through an iterative numerical procedure that identifies the pertinent optimum parameters. It is also shown that for certain particular benchmark cases the model provides closed‐form equations for the parameters defining the dynamic states related to these special conditions. Despite its approximate nature the model presented in this paper is rational, and due to its explicit consideration of energy balance and overall simplicity, it provides a convenient platform for the study of tuned mass dampers, as well as for other methods of structural passive control. Copyright © 2005 John Wiley & Sons, Ltd.     

Influence of ground motion intensity on the effectiveness of tuned mass dampers

The effectiveness of Tuned Mass Dampers (TMD) on buildings subjected to moderate and high-intensity motions is analysed. First, the response of a 22-storey four-bay reinforced concrete non-linear frame with a TMD is studied for motions with different intensities. Several values of the relevant parameters are assumed in the analyses. Then, equivalent single-degree-of-freedom systems with TMDs and without them are defined and analysed under the action of ground motions with intensities associated with different return intervals at the site where the structures are located. Vulnerability curves for the systems are obtained based on the probabilities of reaching two different performance limit states. The expected annual rate of exceedance of each limit state is calculated. The results show that the effectiveness of TMDs is higher for systems with small non-linearity produced by small and moderate earthquakes, than for systems with high non-linear behaviour, generally associated with high-intensity motions. Some recommendations about the applicability of TMD are given. Copyright © 1999 John Wiley & Sons Ltd.