Green's function of support-excited structures with tuned-mass dampers derived by A perturbation method

A tuned-mass damper is a small damped spring-mass system which vibrates in resonance with the main structure to which it is attached so as to be able to dissipate vibration energy and reduce the structural response. In this paper, explicit forms of Green's function for the transient response of main structures equipped with the tuned-mass damper and subjected to support excitation are derived by perturbation techniques and provide an insight into the characteristics of the damper. It is found that there exists a critical damping level for the tuned-mass damper. If the damper damping is higher than this critical damping level, increasing the damper damping will enhance the structural response. When the damper damping is below this critical value, something called ‘beat phenomenon’ occurs where the structure will have a smaller response in the first beat cycle, but have a higher rebound in the following beat cycles.     

Particle swarm optimization of TMD by non‐stationary base excitation during earthquake

There are many traditional methods to find the optimum parameters of a tuned mass damper (TMD) subject to stationary base excitations. It is very difficult to obtain the optimum parameters of a TMD subject to non‐stationary base excitations using these traditional optimization techniques. In this paper, by applying particle swarm optimization (PSO) algorithm as a novel evolutionary algorithm, the optimum parameters including the optimum mass ratio, damper damping and tuning frequency of the TMD system attached to a viscously damped single‐degree‐of‐freedom main system subject to non‐stationary excitation can be obtained when taking either the displacement or the acceleration mean square response, as well as their combination, as the cost function. For simplicity of presentation, the non‐stationary excitation is modeled by an evolutionary stationary process in the paper. By means of three numerical examples for different types of non‐stationary ground acceleration models, the results indicate that PSO can be used to find the optimum mass ratio, damper damping and tuning frequency of the non‐stationary TMD system, and it is quite easy to be programmed for practical engineering applications. Copyright © 2008 John Wiley & Sons, Ltd.     

Optimum viscous damper for connecting adjacent SDOF structures for harmonic and stationary white‐noise random excitations

The dynamic behaviour of two adjacent single‐degree‐of‐freedom (SDOF) structures connected with a viscous damper is studied under base acceleration. The base acceleration is modelled as harmonic excitation as well as stationary white‐noise random process. The governing equations of motion of the connected system are derived and solved for relative displacement and absolute acceleration responses of connected structures. The response of structures is found to be reduced by connecting with a viscous damper having appropriate damping. For undamped SDOF structures, the closed‐form expressions for optimum damping of viscous damper for minimum steady state as well as minimum mean square relative displacement and absolute acceleration of either of the connected SDOF structures are derived. The optimum damper damping is found to be functions of mass and frequency ratio of two connected structures. Further, numerical results had indicated that the damping of the connected structures does not have noticeable effects on the optimum damper damping and the corresponding optimized response. This implies that the derived closed‐form expressions for optimum damper damping of undamped structures can also be used in practical applications for damped structures. Copyright © 2006 John Wiley & Sons, Ltd.     

Optimum Multiple Tuned Mass Dampers for base-excited undamped system

Optimum parameters of Multiple Tuned Mass Dampers (MTMD) for an undamped system to harmonic base excitation are investigated using a numerical searching technique. The criteria selected for the optimality is the minimization of steady-state displacement response of the main system. The explicit formulae for the optimum parameters of MTMD (i.e. damping ratio, bandwidth and tuning frequency) are then derived using curve-fitting scheme that can readily be used for engineering applications. The error in the proposed explicit expressions is investigated and found to be quite negligible. The optimum parameters of the MTMD system are obtained for different mass ratios and number of dampers. Copyright © 1999 John Wiley & Sons, Ltd.     

钢梁阻尼器的最佳延性率及其控制

基于双线性和三线性的力－位移滞回线假设，导出了钢梁阻尼器隔震系统等效阻尼比与阻尼器延性率的关系；为了获得大阻尼比隔震系统，以尽量减小隔震系统的地震反应，求得了使系统阻尼比为最大的阻尼器相应延性率（即最佳延性率）所需满足的条件；给出了在给定地震地面运动输入条件下，使系统阻尼器变形位移达到最佳延性率的阻尼器初始刚度和屈服位移的确定方法。     

Optimum placement and characteristics of velocity-dependent dampers under seismic excitation

In this study,through novel drift-based equations of motion in the frequency domain,optimum placement and characteristics of linear velocity-dependent dampers are investigated.In this study,the sum of the square of the absolute values of transfer matrix elements for interstory drifts is considered as the optimization index.Optimum placement and characteristics of dampers are simultaneously obtained by minimizing the optimization index through an incremental procedure.In each step of the procedure,a predefined value is considered as the damper characteristic.The optimum story for this increment is selected such that it leads to a minimum value for the optimization index.The procedure is repeated for the next increments until the optimization index meets its target value,which is obtained according to the desired damping ratio for the overall structure.In other words,the desired overall damping ratio is the input to the proposed procedure,and the optimal placement and characteristics of the dampers are its output.It is observed that the optimal placement of a velocitydependent damper depends on the damping coefficient of the added damper,frequency of the excitation,and distribution of the mass,stiffness,and inherent damping of the main structure.     

结构最优主动调谐质量阻尼器风致振动控制的研究

基于我国现行的风荷载规范,建立了在风荷载作用下结构-主动调谐质量阻尼器(ATMD)系统的动力方程。定义ATMD最优参数准则为:结构-ATMD系统的位移或加速度响应方差的最小化。ATMD有效性的评价准则为:设置ATMD结构的最小化位移或加速度响应方差与未设置ATMD结构的位移或加速度响应方差之比(分别称为位移和加速度减振系数)。根据上述准则,在频域内数值研究了结构自振频率、标准化加速度反馈增益系数、质量比对ATMD系统的最优参数(包括最优频率比和阻尼比)、有效性和冲程的影响。此外,为了比较的目的,论文同时考虑了结构TMD风致振动控制的情况。     

Dynamic behaviour of structures with embedded foundations

A study of the dynamics of building-soil interaction is presented that includes embedding of the foundation and material damping. By considering buildings on rigid footings embedded into linear elastic soil with hysteretic damping, it is shown that the earthquake response of the building-foundation model may be found from the response to modified excitation of equivalent one-degree-of-freedom linear, viscously damped oscillators resting on rigid ground. For a single-storey building approximate formulas are developed for the modified natural frequency and damping ratio. Results show that the natural frequency and damping in the system increase with embedding. Effective damping also increases with internal friction in the soil. Ignoring these two factors may underestimate considerably the effective natural frequency and damping in the system. In spite of additional sources of energy dissipation provided by the soil, damping in the equivalent oscillator may be greater or smaller than that corresponding to the superstructure alone, depending upon the system parameters. For lightly damped superstructures, the peak amplitude of the steady-state overturning moment at the base of a building supported on flexible soil is significantly smaller than that corresponding to rigid ground. This result has practical implications for earthquake design.     

Response spectrum analysis for non-classically damped linear system with multiple-support excitations

A new response spectrum method, which is named complex multiple-support response spectrum (CMSRS) method in this article, is developed for seismic analysis of non-classically damped linear system subjected to spatially varying multiple-supported ground motion. The CMSRS method is based on fundamental principles of random vibration theory and properly accounts for the effect of correlation between the support motions as well as between the modal displacement and velocity responses of structure, and provides an reasonable and acceptable estimate of the peak response in term of peak seismic ground motions and response spectra at the support points and the coherency function. Meanwhile, three new cross-correlation coefficients or cross covariance especially for the non-classically damped linear structures with multiple-supports excitations are derived under the same assumptions of the MSRS method of classically damped system. The CMSRS method is examined and compared to the results of time history analyses in two numerical examples of non-classically damped structures in consideration of the coherences of spatially variable ground motion. The results show that for non-classically damped structure, the cross terms representing the cross covariance between the pseudo-static and dynamic component are also quite small just as same as classically damped system. In addition, it is found that the usual way of neglecting all the off-diagonal elements in transformed damping matrix in modal coordinates in order to make the concerned non-classically damped structure to become remaining proportional damping property will bring some errors in the case of subjected to spatially excited inhomogeneous ground motion.     

A hybrid analysis method for linear dynamic soil-structure interaction in time and frequency domain

IntroductionThe analysis of dynamic soil-structure interaction for important engineering project is still based on linear model (including equivalent linear model) with complex damping, and traditional frequency domain method (Lysmer, et al, 1975, 1981; DING, et al, 1999). Namely, first calculating frequency domain solution by Fourier transform, and then calculating time domain solution by Fourier inverse transform. The motion equation of a system in frequency domain is usually written as (…     

液压阻尼系统（HDS）控制立式钢制储罐地震响应分析

本文采用液压阻尼系统（HDS）控制立式圆柱钢制储罐的地震反应，建立了安装HDS储罐的力学分析模型，并进行了数值计算，从频域对安装HDS后储罐的地震响应进行了分析，研究了HDS的减震效果和HDS和HDS参数对储罐地震反应的影响。     

Application of semi‐active tuned mass dampers to base‐excited systems

A continuously variable semi‐active damper is used in a tuned mass damper (TMD) to reduce the level of vibration of a single‐degree‐of‐freedom system subjected to harmonic base excitations. The ground hook dampers as have been used in the auto‐industry are being studied here. Using these dampers a new class of tuned mass dampers, named as ground hook tuned mass dampers (GHTMD) is being introduced. In order to generalize the design properties of the GHTMDs, they are defined in terms of non‐dimensional parameters. The optimum design parameters of GHTMDs for lightly damped systems are obtained based on the minimization of the steady‐state displacement response of the main mass. These parameters are computed for different mass ratios and main system damping ratios. Frequency responses of the resulting systems are compared to that of equivalent TMDs using passive dampers. In addition, other characteristics of this system as compared to the passive TMDs are discussed. A design guide to obtain the optimum parameters of GHTMD using the developed diagrams in this paper based on non‐dimensional values is presented. Copyright © 2001 John Wiley & Sons, Ltd.     

利用单频双程波动方程计算初至走时及其振幅

通过在频率域双程波动方程模拟算法中加入一个复数频率（实部表示频率,虚部表示衰减因子）压制地震波初至走时之后的能量,从而把初至走时及其振幅的计算问题转换为单一频率波场中最大能量走时和振幅的拾取问题,然后利用单一频率域波场的相位项和振幅项分别计算初至走时及其振幅.本文还提出利用参数分析方法求取最优的复数频率,并给出数值计算例子,将本方法的计算结果与有限差分程函方程初至走时和最大能量走时振幅进行比较,结果表明,该方法具有适应于任意复杂介质和多炮多接收点走时和振幅的计算.     

Experimental study of vibration mitigation of mast arm signal structures with particle-thrust damping based tuned mass damper

Large amplitude vibration of mast arm structures due to wind loads are the primary contributing factor to the reduced fatigue life of signal support structures. To alleviate this problem of wind-induced in-plane vibration of mast arm signal structures, a particle-thrust damping based turned mass damper(PTD-TMD) device is adopted and its damping effect is characterized experimentally. The particle-thrust damping is a passive damping device that does not require electric power and is temperature independent. Based on the calibration test, an equivalent dynamic model of the PTD-TMD device is developed and used for numerical simulation study. The damping effects of this PTD-TMD device on signal support structures was investigated through both numerical analysis and laboratory testing of a 50-ft(15.24 m) mast arm structure including both free vibration and forced vibration tests. The experimental test and numerical study results show that vibration response behavior of mast arm signal support structures can be significantly reduced by installing the PTD-TMD that can increase the critical damping ratio of the mast arm signal structures to 4%. The stress range at the welded connection between the mast arm and traffic pole is also reduced.     

Tuned inerter dampers with linear hysteretic damping

This paper explores the influence of linear hysteretic damping on the performance of passive tuned-inerter devices. An inerter is a device that produces a force proportional to the relative acceleration across its two terminals; devices incorporating inerters have received widespread attention in the earthquake engineering community, because they offer the ability to improve the seismic response of structures. However, the majority of this research has assumed that the damping components within the tuned-inerter device exhibit viscous, rather than hysteretic, damping. This restriction imposes an essential question on how the hysteretic damping model will change the performance of the device compared with the viscous damping model. It is shown that the response of viscous and hysteretic inerter systems have significant differences in displacement amplitude due to the frequency dependency of the damping. Therefore, a new formulation for obtaining the optimum loss factor of the hysteretic damping in the inerter system is proposed. Next, the challenges associated with accurately predicting the time-response of a hysteretically damped system are discussed. A numerical time-integration method is extended to address these challenges, using a new formulation that has the benefit of being broadly applicable to multidegree-of-freedom hysteretic linear systems and nonstationary random signals. The results show that the earthquake responses from the hysteretic damping model can differ significantly from the ones obtained via the viscous model.     

Optimum multiple tuned mass dampers for structures under the ground acceleration based on the uniform distribution of system parameters

The five MTMD models, with natural frequencies being uniformly distributed around their mean frequency, have been recently presented by the first author. They are shown to have the near‐zero optimum average damping ratio (more precisely, for a given mass ratio there is an upper limit on the total number, beyond which the near‐zero optimum average damping ratio occurs). In this paper, the eight new MTMD models (i.e. the UM‐MTMD1～UM‐MTMD3, US‐MTMD1～US‐MTMD3, UD‐MTMD1 and UD‐MTMD2), with the system parameters (mass, stiffness and damping coefficient) being, respectively, uniformly distributed around their average values, have been, for the first time here, proposed to seek for the MTMD models without the near‐zero optimum average damping ratio. The structure is represented by the mode‐generalized system corresponding to the specific vibration mode that needs to be controlled. Through minimization of the minimum values of the maximum dynamic magnification factors (DMF) of the structure with the eight MTMD models (i.e. through the implementation of Min.Min.Max.DMF), the optimum parameters and values of Min.Min.Max.DMF for these eight MTMD models are investigated to evaluate and compare their control performance. The optimum parameters include the optimum mass spacing, stiffness spacing, damping coefficient spacing, frequency spacing, average damping ratio and tuning frequency ratio. The six MTMD models without the near‐zero optimum average damping ratio (i.e. the UM‐MTMD1～UM‐MTMD3, US‐MTMD1, US‐MTMD2 and UD‐MTMD2) are found through extensive numerical analyses. Likewise, the optimum UM‐MTMD3 offers the higher effectiveness and robustness and requires the smaller damping with respect to the rest of the MTMD models in reducing the responses of structures subjected to earthquakes. Additionally, it is interesting to note, by comparing the optimum UM‐MTMD3 with the optimum MTMD‐1 recently investigated by the first author, that the effectiveness and robustness for the optimum UM‐MTMD3 is almost identical to that for the optimum MTMD‐1 (without inclusion of the optimum MTMD‐1 with the near‐zero optimum average damping ratio). Recognizing these performance benefits, it is preferable to employ the optimum UM‐MTMD3 or the optimum MTMD‐1 without the near‐zero optimum average damping ratio, when installing the MTMD for the suppression of undesirable oscillations of structures under earthquakes. Copyright © 2003 John Wiley & Sons, Ltd.     

Asymmetric one‐storey elastic systems with non‐linear viscous and viscoelastic dampers: Simplified analysis and supplemental damping system design

Investigated is the accuracy in estimating the response of asymmetric one‐storey systems with non‐linear viscoelastic (VE) dampers by analysing the corresponding linear viscous system wherein all non‐linear VE dampers are replaced by their energy‐equivalent linear viscous dampers. The response of the corresponding linear viscous system is determined by response history analysis (RHA) and by response spectrum analysis (RSA) extended for non‐classically damped systems. The flexible and stiff edge deformations and plan rotation of the corresponding linear viscous system determined by the extended RSA procedure is shown to be sufficiently accurate for design applications with errors generally between 10 and 20%. Although similar accuracy is also shown for the ‘pseudo‐velocity’ of non‐linear VE dampers, the peak force of the non‐linear VE damper cannot be estimated directly from the peak damper force of the corresponding linear viscous system. A simple correction for damper force is proposed and shown to be accurate (with errors not exceeding 15%). For practical applications, an iterative linear analysis procedure is developed for determining the amplitude‐ and frequency‐dependent supplemental damping properties of the corresponding linear viscous system and for estimating the response of asymmetric one‐storey systems with non‐linear VE dampers from the earthquake design (or response) spectrum. Finally, a procedure is developed for designing non‐linear supplemental damping systems that satisfy given design criteria for a given design spectrum. Copyright © 2003 John Wiley & Sons, Ltd.     

黏滞阻尼器连接相邻结构减震优化分析

阻尼器参数的确定是利用阻尼器连接相邻结构进行减震设计的关键.根据随机地震反应理论,以相邻结构的频率比和质量比为参数,推导了结构位移反应均方差与连接阻尼比的关系式,得到了相邻结构的地震反应与频率比、质量比以及连接阻尼比的影响规律,从而得到了连接阻尼器的优化设计参数.根据自振频率相等的原则,探讨了将多自由度体系简化为单自由度体系的分析方法.最后在El Centro波、Taft 波及人工波激励下,对比分析了某相邻10层建筑结构有连接和无连接时的地震反应,表明黏滞阻尼器连接相邻结构具有较好的减震效果.本分析方法可供相邻结构减震设计参考.     

Higher-mode effect on the seismic responses of buildings with viscoelastic dampers

In conventional modal analysis procedures, usually only a few dominant modes are required to describe the dynamic behavior of multi-degrees-of-freedom buildings. The number of modes needed in the dynamic analysis depends on the higher-mode contribution to the structural response, which is called the higher-mode effect. The modal analysis approach, however, may not be directly applied to the dynamic analysis of viscoelastically damped buildings. This is because the dynamic properties of the viscoelastic dampers depend on their vibration frequency. Therefore, the structural stiffness and damping contributed from those dampers would be different for each mode. In this study, the higher-mode effect is referred to as the response difference induced by the frequency-dependent property of viscoelastic dampers at higher modes. Modal analysis procedures for buildings with viscoelastic dampers distributed proportionally and non-proportionally to the stiffness of the buildings are developed to consider the higher-mode effect. Numerical studies on shear-type viscoelastically damped building models are conducted to examine the accuracy of the proposed procedures and to investigate the significance of the higher-mode effect on their seismic response. Two damper models are used to estimate the peak damper forces in the proposed procedures. Study results reveal that the higher-mode effect is significant for long-period viscoelastically damped buildings. The higher-mode effect on base shear is less significant than on story acceleration response. Maximum difference of the seismic response usually occurs at the top story. Also, the higher-mode effect may not be reduced by decreasing the damping ratio provided by the viscoelastic dampers. For practical application, it is realized that the linear viscous damping model without considering the higher-mode effect may predict larger damper forces and hence, is on the conservative side. Supported by: Science Council, Chinese Taipei, grant no. 88-2625-2-002-006     

拉索式旋转电涡流阻尼器的理论模型及频域响应分析

惯容器是一种新型的振动控制装置,该装置可以将螺杆的轴向运动转换为黏滞材料和旋转质量的高速旋转运动,使阻尼器的阻尼效应和质量效应得到放大。基于惯容器原理以及电涡流原理,提出一种新型的拉索式旋转电涡流阻尼器,为结构被动控制提供了新的设计思路。为研究这种新型惯容系统的减振机理,对安装拉索式旋转电涡流阻尼器单层框架的力学模型进行了探讨。推导出了配置惯容系统单质点体系的动力表达式,并基于此表达式探讨在频域内该惯容系统关键参数对单质点体系位移、速度和加速度响应的影响。结果表明:这种拉索式旋转电涡流阻尼器可以起到放大质量的效果,惯容系统可以有效地减少单质点体系的位移、速度和加速度响应幅值。