Closed‐form solution for seismic response of adjacent buildings with linear quadratic Gaussian controllers

Closed‐form solution for seismic response of adjacent buildings connected by hydraulic actuators with linear quadratic Gaussian (LQG) controllers is presented in this paper. The equations of motion of actively controlled adjacent buildings against earthquake are first established. The complex modal superposition method is then used to determine dynamic characteristics, including modal damping ratio, of actively controlled adjacent buildings. The closed‐form solution for seismic response of the system is finally derived in terms of the complex dynamic characteristics, the pseudo‐excitation method and the residue theorem. By using the closed‐form solution, extensive parametric studies can be carried out for the system of many degrees of freedom. The beneficial parameters of LQG controllers for achieving the maximum response reduction of both buildings using reasonable control forces can be identified. The effectiveness of LQG controllers for this particular application is evaluated in this study. The results show that for the adjacent buildings of different dynamic properties, if the parameters of LQG controllers are selected appropriately, the modal damping ratios of the system can be significantly increased and the seismic responses of both buildings can be considerably reduced. Copyright © 2001 John Wiley & Sons, Ltd.     

粘弹性阻尼器连接的相邻结构非线性随机地震反应分析

本文用随机等价线性化方法探讨了相邻结构之间用粘弹性阻尼器连接后的非线性随机地震反应，分析发现：在小震作用下，粘弹性阻尼器对相邻结构可以同时达到较好的控制效果；但是在强烈地震作用下，安装粘弹性阻尼器有可能会在减少一个结构的地震反应的同时，增大另外一个结构的地震反应。     

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     

Experimental investigation of adjacent buildings connected by fluid damper

Dynamic characteristic and harmonic response of adjacent buildings connected by fluid damper were experimentally investigated using model buildings and fluid damper. Two building models were constructed as two three-storey shear buildings of different natural frequencies. Model fluid damper connecting the two buildings was designed as linear viscous damper of which damping coefficient could be adjusted. The two buildings without fluid dampers connected were first tested to obtain their individual dynamic characteristics and responses to harmonic excitation. The tests were then carried out to determine modal damping ratios of the adjacent buildings connected by the fluid damper of different damping coefficients and at different locations. Optimal damper damping coefficient and location for achieving the maximum modal damping ratio were thus found. The measured modal damping ratios and harmonic responses of the building-fluid damper system were finally compared with those from the individual buildings. The comparison showed that the fluid damper of proper parameter could significantly increase the modal damping ratio and tremendously reduce the dynamic response of both buildings. Copyright © 1999 John Wiley & Sons, Ltd.     

A method of calculating the non‐linear seismic response of a building braced with viscoelastic dampers

Buildings are continually subject to dynamic loads, such as wind load, seismic ground motion, and even the load from internal utility machines. The recent trend of constructing more flexible high‐rise buildings underscores the importance of including viscoelastic dampers in building designs. Viscoelastic dampers are used to control the dynamic response of a building. If the seismic design is based only on the linear response spectrum, considerable error may occur when calculating the seismic response of a building; rubber viscoelastic dampers show non‐linear hysteretic damping that is quite different from viscous damping. This study generated a non‐linear response spectrum using a non‐linear oscillator model to simulate a building with viscoelastic dampers installed. The parameters used in the non‐linear damper model were obtained experimentally from dynamic loading tests. The results show that viscoelastic dampers effectively reduce the seismic displacement response of a structure, but transmit more seismic force to the structure, which essentially increases its seismic acceleration response. Copyright © 2003 John Wiley & Sons, Ltd.     

地震作用下智能相邻建筑的主动锚索控制

研究了应用主动锚索控制装置控制相邻建筑地震响应的有效性。首先，建立了主动锚索－相邻建筑系统的运动方程；然后应用广义振型分析方法，寻求了主动锚索联结的相邻建筑的力学特性，特别是振型阻尼比；最后，在频率域内应用广义振型分析与虚拟激励相结合的方法，建立了主动锚索控制装置联结相邻建筑的随机地震响应的分析方法。应用本文建立的公式，我们编制了计算机程序，进行了广泛的参数研究，以评价控制装置的有效性，并确定最优传感器类型及控制装置参数。研究表明，如果应用速度传感器，并能适当地选择主动锚索控制器的参数，则可以显著地提高系统的振型阻尼比，减小两个结构的地震响应。     

A METHOD OF ESTIMATING THE PARAMETERS OF TUNED MASS DAMPERS FOR SEISMIC APPLICATIONS

The optimum parameters of tuned mass dampers (TMD) that result in considerable reduction in the response of structures to seismic loading are presented. The criterion used to obtain the optimum parameters is to select, for a given mass ratio, the frequency (tuning) and damping ratios that would result in equal and large modal damping in the first two modes of vibration. The parameters are used to compute the response of several single and multi-degree-of-freedom structures with TMDs to different earthquake excitations. The results indicate that the use of the proposed parameters reduces the displacement and acceleration responses significantly. The method can also be used in vibration control of tall buildings using the so-called ‘mega-substructure configuration’, where substructures serve as vibration absorbers for the main structure. It is shown that by selecting the optimum TMD parameters as proposed in this paper, significant reduction in the response of tall buildings can be achieved. © 1997 John Wiley & Sons, Ltd.     

The role of damping in seismic isolation

In the current code requirements for the design of base isolation systems for buildings located at near-fault sites, the design engineer is faced with very large design displacements for the isolators. To reduce these displacements, supplementary dampers are often prescribed. These dampers reduce displacements, but at the expense of significant increases in interstorey drifts and floor accelerations in the superstructure. An elementary analysis based on a simple model of an isolated structure is used to demonstrate this dilemma. The model is linear and is based on modal analysis, but includes the modal coupling terms caused by high levels of damping in the isolation system. The equations are solved by a method that avoids complex modal analysis. Estimates of the important response quantities are obtained by the response spectrum method. It is shown that as the damping in the isolation system increases, the contribution of the modal coupling terms due to isolator damping in response to the superstructure becomes the dominant term. The isolator displacement and structural base shear may be reduced, but the floor accelerations and interstorey drift are increased. The results show that the use of supplemental dampers in seismic isolation is a misplaced effort and alternative strategies to solve the problem are suggested. Copyright © 1999 John Wiley & Sons, Ltd.     

Seismic analysis of structures with a fractional derivative model of viscoelastic dampers

Viscoelastic dampers are now among some of the preferred energy dissipation devices used for passive seismic response control. To evaluate the performance of structures installed with viscoelastic dampers, different analytical models have been used to characterize their dynamic force deformation characteristics. The fractional derivative models have received favorable attention as they can capture the frequency dependence of the material stiffness and damping properties observed in the tests very well. However, accurate analytical procedures are needed to calculate the response of structures with such damper models. This paper presents a modal analysis approach, similar to that used for the analysis of linear systems, for solving the equations of motion with fractional derivative terms for arbitrary forcing functions such as those caused by earthquake induced ground motions. The uncoupled modal equations still have fractional derivatives, but can be solved by numerical or analytical procedures. Both numerical and analytical procedures are formulated. These procedures are then used to calculate the dynamic response of a multi-degree of freedom shear beam structure excited by ground motions. Numerical results demonstrating the response reducing effect of viscoelastic dampers are also presented.     

Simplifications of CQC method and CCQC method

The response-spectrum mode superposition method is widely used for seismic response analyses of linear systems. In using this method, the complete quadratic combination （CQC） is adopted for classically damped linear systems and the complex complete quadratic combination （CCQC） formula is adopted for non-classically damped linear systems. However, in both cases, the calculation of seismic response analyses is very time consuming. In this paper, the variation of the modal correlation coefficients of displacement, velocity and displacement-velocity with frequency and damping ratios of two modes of interest are studied, Moreover, the calculation errors generated by using CQC and square-root-of-the-sum-of-thesquares （SRSS） methods （or CCQC and CSRSS methods） for different damping combinations are compared. In these analyses, some boundary lines for classically and non-classically damped systems are plotted to distinguish the allowed minimum frequency ratio at given geometric mean of the damping ratios of both modes if their relativity is neglected. Furthermore, the simplified method, which is a special mode quadratic combination method considering only relativity of adjacent modes in CQC method and named simplified CQC or partial quadratic combination （PQC） method for classically damped linear system, is proposed to improve computational efficiency, and the criterion for determination of how many correlated modes should be adopted is proposed. Similarly, the simplified CCQC or complex partial quadratic combination （CPQC） method for the non-classically damped linear system and the corresponding criterion are also deduced. Finally, a numerical example is given to illustrate the applicability, computational accuracy and efficiency of the PQC and CPQC methods.     

基于高阻尼纳米复合材料的直角型黏弹性阻尼器的数值模拟

为提高装配式钢结构梁柱节点的抗震性能,首先通过熔融共混法制备高阻尼性能纳米偏高岭土/氟橡胶(NanoGmetakaolin/Fluororubber,NMK/FKM)复合材料,并对该新型材料进行4种频率下的动态力学性能试验和静态力学试验,然后以 NMK/FKM 纳米复合材料为核心耗能材料,对所提出的直角型黏弹性阻尼器进行 ABAQUS有限元模拟分析.研究结果表明:当频率为1．5Hz 时,NMK/FKM 纳米复合材料宽阻尼温域和 TA 值皆达到峰值,该工况下材料的阻尼性能最佳;直角型黏弹性阻尼器表现出刚度随位移幅值的增大而增大的动力特性;当频率为0．5、1．0和1．5Hz 时,阻尼器滞回特性表现出非线性特征,当频率升至2．0Hz时,滞回特性则为线性.随着黏弹性材料层厚度的增大,滞回环面积、阻尼器刚度和最大阻尼力逐渐减小;随着高跨比的增大,阻尼器耗能性能提升.通过调整阻尼器的高跨比和阻尼材料层厚度,可以进一步提高直角型阻尼器的动态响应.     

Stochastic seismic response of structures with added viscoelastic dampers modeled by fractional derivative

Viscoelastic dampers, as supplementary energy dissipation devices, have been used in building structures under seismic excitation or wind loads. Different analytical models have been proposed to describe their dynamic force deformation characteristics. Among these analytical models, the fractional derivative models have attracted more attention as they can capture the frequency dependence of the material stiffness and damping properties observed from tests very well. In this paper, a Fourier-transform-based technique is presented to obtain the fractional unit impulse function and the response of structures with added viscoelastic dampers whose force-deformation relationship is described by a fractional derivative model. Then, a Duhamel integral-type expression is suggested for the response analysis of a fractional damped dynamic system subjected to deterministic or random excitation. Through numerical verification, it is shown that viscoelastic dampers are effective in reducing structural responses over a wide frequency range, and the proposed schemes can be used to accurately predict the stochastic seismic response of structures with added viscoelastic dampers described by a Kelvin model with fractional derivative.     

A two‐step direct method for estimating the seismic response of nonlinear structures equipped with nonlinear viscous dampers

The insertion of fluid viscous dampers in building structures is an innovative technology that can improve significantly the seismic response. These devices could be very useful also in the retrofit of existing buildings. The effect of this typology of damping system is usually identified with an equivalent supplemental damping ratio, which depends on the maximum displacement of the structure, so that iterative procedures are required. In this paper, a simplified direct assessment method for nonlinear structures equipped with nonlinear fluid viscous dampers is proposed. The method proposed in this study is composed by two steps. The first one yields the direct estimate of the supplemental damping ratio provided by nonlinear viscous dampers in presence of a linear elastic structural response. The second step extends the procedure to structures with nonlinear behavior. Both graphical and analytical approaches have been developed. The proposed method has then been verified through several applications and comparisons with nonlinear dynamic analyses. Moreover, an investigation has been performed with regard to the influence of the relations that define the damping reduction factor and the hysteretic damping. Copyright © 2014 John Wiley & Sons, Ltd.     

结构附加粘滞阻尼器的抗震设计

本文结合抗震设计规范反应谱,给出了一个附加非线性流体粘滞阻尼器结构的抗震设计方法。研究了非线性阻尼器的力学特性,引入了非线性流体阻尼器的等效线性阻尼比,给出了计算最大加速度时刻附加非线性流体阻尼器结构反应的荷载组合系数,提出了按阻尼力的水平力分量与楼层剪力成正比的原则分配阻尼器阻尼系数的方法。同时给出了基于抗震规范设计反应谱附加非线性阻尼器结构的设计流程,通过一个算例说明了使用该方法设计附加非线性粘滞阻尼器结构的全过程。算例分析表明,这种设计方法适合于手算,便于设计人员掌握,在初步设计阶段可以快速、有效地设计满足给定性能水平的附加非线性流体阻尼器体系。     

Simplified design procedure for frame buildings with viscoelastic or elastomeric structural dampers

A simplified design procedure (SDP) for preliminary seismic design of frame buildings with structural dampers is presented. The SDP uses elastic‐static analysis and is applicable to structural dampers made from viscoelastic (VE) or high‐damping elastomeric materials. The behaviour of typical VE materials and high‐damping elastomeric materials is often non‐linear, and the SDP idealizes these materials as linear VE materials. With this idealization, structures with VE or high‐damping elastomeric dampers can be designed and analysed using methods based on linear VE theory. As an example, a retrofit design for a typical non‐ductile reinforced concrete (RC) frame building using high‐damping elastomeric dampers is developed using the SDP. To validate the SDP, results from non‐linear dynamic time history analyses (NDTHA) are presented. Results from NDTHA demonstrate that the SDP estimates the seismic response with sufficient accuracy for design. It is shown that a non‐ductile RC frame building can be retrofit with high‐damping elastomeric dampers to remain essentially elastic under the design basis earthquake (DBE). Copyright © 2005 John Wiley & Sons, Ltd.     

On a modal damping identification model for non-classically damped linear building structures subjected to earthquakes

A simple modal damping identification model developed by the present authors for classically damped linear building frames is extended here to the non-classically damped case. The modal damping values are obtained with the aid of the frequency domain modulus of the roof-to-basement transfer function and the resonant frequencies of the structure (peaks of the transfer function) as well as the modal participation factors and mode shapes of the undamped structure. The assumption is made that the modulus of the transfer function of the non-classically damped structure matches the one of the classically damped structure in a discrete manner, i.e., at the resonant frequencies of that function modulus. This proposed approximate identification method is applied to a number of plane building frames with and without pronounced non-classical damping under different with respect to their frequency content earthquakes and its limitations and range of applicability are assessed with respect to the accuracy of both the identified damping ratios and that of the seismic structural response obtained by classical mode superposition and use of those identified modal damping ratios.     

Optimal design of viscoelastic dampers using eigenvalue assignment

In this study a procedure for determining the optimum size and location of viscoelastic dampers is proposed using the eigenvalue assignment technique. Natural frequencies and modal damping ratios, required to realize a given target response, are determined first by the convex model. Then the desired dynamic structural properties are realized by optimally distributing the damping and stiffness coefficients of viscoelastic dampers using non‐linear programming based on the gradient of eigenvalues. This optimization method provides information on the optimal location as well as the magnitude of the damper parameters. The proposed procedure is applied to the retrofit of a 10‐story shear frame, and to a three‐dimensional structure with an asymmetric plan. The analysis results confirm that the responses of model structures retrofitted by the proposed method correspond well with the given target response. 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.     

Earthquake induced pounding between adjacent buildings considering soil-structure interaction

Many closely located adjacent buildings have suffered from pounding during past earthquakes because they vibrated out of phase.Furthermore,buildings are usually constructed on soil;hence,there are interactions between the buildings and the underlying soil that should also be considered.This paper examines both the interaction between adjacent buildings due to pounding and the interaction between the buildings through the soil as they affect the buildings’ seismic responses.The developed model consists of adjacent shear buildings resting on a discrete soil model and a linear viscoelastic contact force model that connects the buildings during pounding.The seismic responses of adjacent buildings due to ground accelerations are obtained for two conditions:fixed-based(FB) and structure-soil-structure interaction(SSSI).The results indicate that pounding worsens the buildings’ condition because their seismic responses are amplified after pounding.Moreover,the underlying soil negatively impacts the buildings’ seismic responses during pounding because the ratio of their seismic response under SSSI conditions with pounding to those without pounding is greater than that of the FB condition.     

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.