关于土层地震反应分析中阻尼矩阵建模方式的讨论

阻尼矩阵的选择对土层时域内地震反应计算的准确性有重要意义。本文构造了质量比例阻尼、刚度比例阻尼、Rayleigh阻尼、Caughey阻尼和Clough阻尼矩阵,研究各阻尼对土层地震反应分析精度的影响。以苏通大桥5号桥塔基础处的工程场地为例,分别以人工合成基岩波,Northridge波、Parkfield波和汶川波为输入,得到不同阻尼模型下土层的地震反应,结果表明Clough阻尼在计算精度和误差稳定性上要优于其他阻尼模型,针对Clough阻尼的不足之处提出了改进方法,算例表明改进后的Clough阻尼能在保证计算精度的前提下减少计算机的存储量和计算量。     

Modeling viscous damping in nonlinear response history analysis of buildings for earthquake excitation

The Rayleigh damping model, which is pervasive in nonlinear response history analysis (RHA) of buildings, is shown to develop ‘spurious’ damping forces and lead to inaccurate response results. We prove that a viscous damping matrix constructed by superposition of modal damping matrices—irrespective of the number of modes included or values assigned to modal damping ratios—completely eliminates the ‘spurious’ damping forces. This is the damping model recommended for nonlinear RHA. Replacing the stiffness‐proportional part of Rayleigh damping by the tangent stiffness matrix is shown to improve response results. However, this model is not recommended because it lacks a physical basis and has conceptual implications that are troubling: hysteresis in damping force–velocity relationship and negative damping at large displacements. Furthermore, the model conflicts with the constant‐damping model that has been the basis for fundamental concepts and accumulated experience about the inelastic response of structures. With a distributed plasticity model, the structural response is not sensitive to the damping model; even the Rayleigh damping model leads to acceptable results. This perspective on damping provides yet another reason to employ the superior distributed plasticity models in nonlinear RHA. OpenSees software has been extended to include a damping matrix defined as the superposition of modal damping matrices. Although this model leads to a full populated damping matrix, the additional computational demands are demonstrated to be minimal. Copyright © 2015 John Wiley & Sons, Ltd.     

Errors in response calculations for non-classically damped structures

The classical normal mode method of determining response is extremely useful for practical calculations, but depends upon the damping matrix being orthogonal with respect to the modal vectors. Approximations that allow the method to be used when this condition is not satisfied have been suggested; the simplest approach is to neglect off-diagonal terms in the triple matrix product formed from the damping and modal matrices. In this paper the errors in response caused by this approximation are determined for several simple structures for a wide range of damping parameters and different types of excitation. Based on these results a criterion, relating modal damping and natural frequencies, is formulated; if this is satisfied, the errors in response caused by this diagonalization procedure are within acceptable limits.     

基于复振型分解的多自由度非线性体系动力可靠性研究

提出了基于复模态理论的多自由度非线性体系动力可靠性分析方法。该方法首先采用等效线性化的方法处理体系的非线性问题,然后采用复模态分析处理非经典的等效线性阻尼矩阵,将具有非经典阻尼的等效多自由度线性体系按复振型分解,将多自由度体系的随机反应分解为一系列一阶体系的复模态反应,从而求得体系的随机反应,最后进行体系的动力可靠度计算。通过算例验证,表明该方法概念明确、思路清晰,为一般多自由度非线性体系提供了一个普遍适用的动力可靠性分析方法。     

Dynamic characteristics of non-classically damped structures

The equations of motion of a structure in undamped modal coordinates may have non-zero off-diagonal terms in the damping matrix. Although these terms are commonly neglected, studies have shown that they may have a significant influence on the response to dynamic loads. In this paper, two independent criteria are developed to determine when these damping terms will affect the structure's modal properties and response. It is found that even small off-diagonal damping values can be significant if the structure has closely spaced natural frequencies. To quantify and understand the influence of these damping terms, closed-form analytical expressions are derived for the modal properties and harmonic and stochastic response of structures with closely spaced natural frequencies. One conclusion is that off-diagonal damping terms will decrease a modal damping ratio for each pair of closely spaced modes. This is significant, since a response analysis performed by neglecting these off-diagonal terms will underestimate the true response.     

Simplified analysis of asymmetric structures with supplemental damping

This study investigated the effects of neglecting off‐diagonal terms of the transformed damping matrix on the seismic response of non‐proportionally damped asymmetric‐plan systems with the specific aim of identifying the range of system parameters for which this simplification can be used without introducing significant errors in the response. For this purpose, a procedure is presented in which modal damping ratios computed by neglecting off‐diagonal terms of the transformed damping matrix are used in the traditional modal analysis. The effects of the simplification are evaluated first by comparing the aforementioned modal damping ratios with the apparent damping ratios obtained from the complex‐valued eigenanalysis. The variation of a parameter that was defined by Warburton and Soni as an indicator of the errors introduced by the simplification is examined next. Finally, edge deformations obtained from the simplified procedure are compared with those obtained from the direct integration of the equations of motion. It is found that the simplified procedure may be used without introducing significant errors in response for most practical values of the system parameters. Furthermore, estimates of the edge deformations, in general, tend to be on the conservative side. Copyright © 2001 John Wiley & Sons, Ltd.     

Simplified seismic analysis of one‐way asymmetric elastic systems with supplemental damping

This study investigates the effectiveness of the modal analysis using two‐degree‐of‐freedom (2DOF) modal stick to deal with the seismic analysis of one‐way asymmetric elastic systems with supplemental damping. The 2DOF modal stick possessing the non‐proportional damping property enables the modal translation and rotation to not be proportional even at elastic state. The analytical results of one‐storey and three‐storey buildings obtained by the proposed method are compared with those obtained by direct integration of the equation of motion and conventional approximate method, which neglects the off‐diagonal elements in the transformed damping matrix. It is found that the proposed simplified method, compared to conventional approximate methods, can significantly improve the accuracy of the analytical results and, at the same time, without obviously increasing computational efforts. Copyright © 2006 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.     

Appropriate viscous damping for nonlinear time‐history analysis of base‐isolated reinforced concrete buildings

There is no consensus at the present time regarding an appropriate approach to model viscous damping in nonlinear time‐history analysis of base‐isolated buildings because of uncertainties associated with quantification of energy dissipation. Therefore, in this study, the effects of modeling viscous damping on the response of base‐isolated reinforced concrete buildings subjected to earthquake ground motions are investigated. The test results of a reduced‐scale three‐story building previously tested on a shaking table are compared with three‐dimensional finite element simulation results. The study is primarily focused on nonlinear direct‐integration time‐history analysis, where many different approaches of modeling viscous damping, developed within the framework of Rayleigh damping are considered. Nonlinear direct‐integration time‐history analysis results reveal that the damping ratio as well as the approach used to model damping has significant effects on the response, and quite importantly, a damping ratio of 1% is more appropriate in simulating the response than a damping ratio of 5%. It is shown that stiffness‐proportional damping, where the coefficient multiplying the stiffness matrix is calculated from the frequency of the base‐isolated building with the post‐elastic stiffness of the isolation system, provides reasonable estimates of the peak response indicators, in addition to being able to capture the frequency content of the response very well. Furthermore, nonlinear modal time‐history analyses using constant as well as frequency‐dependent modal damping are also performed for comparison purposes. It was found that for nonlinear modal time‐history analysis, frequency‐dependent damping, where zero damping is assigned to the frequencies below the fundamental frequency of the superstructure for a fixed‐base condition and 5% damping is assigned to all other frequencies, is more appropriate, than 5% constant damping. Copyright © 2013 John Wiley & Sons, Ltd.     

地震作用下多层剪切型组合结构的合理综合阻尼比取值探讨

对于组合结构,提出了一种新的基于单元瑞利阻尼模型的应变能振型阻尼比,并证明了其在特定整体阻尼矩阵下与强迫解耦法的等价性;推导了具有明确理论依据的综合阻尼比计算公式,并基于相应的应变能振型阻尼比得出了结构的刚度综合阻尼比和瑞利综合阻尼比。分别采用复振型分解法和振型分解法对算例结构进行了地震荷载作用下的弹性时程分析,结果表明,瑞利综合阻尼比对于以剪切变形为主的多层组合结构具有良好的计算精度和适用性。     

基于复阻尼理论的混合结构Rayleigh阻尼模型

混合结构的阻尼矩阵不满足经典阻尼条件,导致传统的模态叠加法无法适用。复阻尼理论无法适用于时域计算,其自由振动响应中存在发散现象。针对混合结构的阻尼矩阵非比例性和复阻尼理论的时域发散性,基于频域等效原则构建了求解Rayleigh阻尼系数的数学优化模型,进而得到与复阻尼理论等效的Rayleigh阻尼运动方程。算例分析表明:依据位移时程响应和结构等效阻尼比可证明Rayleigh阻尼运动方程的正确性。基于本文研究成果,等效复阻尼理论的混合结构Rayleigh阻尼运动方程可直接采用模态叠加法,结合其确定的结构等效阻尼比,为混合结构的振型分解反应谱法提供理论依据。     

多级串联非比例阻尼隔震结构地震响应分析

建立了多级串联非比例阻尼隔震结构动力分析模型;引用分区瑞利阻尼模型将非比例阻尼矩阵分解为瑞利阻尼矩阵和体现非比例阻尼的余项阻尼矩阵,推导出结构的阻尼矩阵;并编制了MATLAB动力时程分析程序,对一实际隔震工程进行地震响应分析.结果表明:随着下部结构刚度的增加,结构的层剪力比和隔震层位移响应峰值均趋向于基础隔震结构的对应值;当下部结构为一层,且层间刚度大于上部结构底层层间刚度4～6倍时,可以近似按基础隔震结构进行动力分析.     

A numerical study of damping

Diagonal damping matrices were computed for three systems which have non-proportional damping matrices. These diagonal damping matrices were computed on three bases, as follows: 1. After normalizing the equations of motion by the modal matrix, the diagonal terms are retained ignoring the non-diagonal terms. 2. Diagonal damping matrix is established by the optimization algorithm which minimizes the mean square error of the frequency response. 3. Diagonal damping is determined from the normalized differential equation by matching the peaks of the coupled and uncoupled system. The frequency responses for the three cases of one of the three systems are presented together with a comparison of the energy dissipation.     

MODAL ANALYSIS OF SOFT-SOIL SITES INCLUDING RADIATION DAMPING

For the one-dimensional analysis of soft-soil layers on an elastic half-space, a general form of analytical solution is developed for converting radiation damping due to energy leaking back to the half-space into equivalent modal damping, allowing the modal analysis technique to be extended to a site where radiation damping has to be accounted for. Closed-form solutions for equivalent modal damping ratios and effective modal participation factors are developed for a single layer with a shear wave velocity distribution varying from constant to linearly increasing with depth. Compact and recursive forms of solutions for equivalent modal damping ratios are developed for a system with an arbitrary number of homogeneous layers on an elastic half-space. Comparisons with numerical solutions show that the modal solutions are accurate. The nominal frequency of a site, i.e. the inverse of four times the total shear wave travel time through the layers, is an important parameter for estimating the high mode frequencies. A parameter study shows that for the same impedance ratio of the bottom layer to the elastic half-space, a system of soil layers with an increasing soil rigidity with depth has, in general, larger peak modal amplifications at the ground surface than does a single homogeneous layer on an elastic half-space, while a system with a decreasing soil rigidity with depth has smaller modal peak amplifications. © 1997 by John Wiley & Sons, Ltd.     

Seismic behaviour of asymmetric buildings with supplemental damping

This paper investigates the response of asymmetric‐plan buildings with supplemental viscous damping to harmonic ground motion using modal analysis techniques. It is shown that most modal parameters, except dynamic amplification factors (DAFs), are affected very little by the plan‐wise distribution of supplemental damping in the practical range of system parameters. Plan‐wise distribution of supplemental damping significantly influences the DAFs, which, in turn, influence the modal deformations. These trends are directly related to the apparent modal damping ratios; the first modal damping ratio increases while the second decreases as CSD moves from right to left of the system plan, and their values increase with larger plan‐wise spread of the supplemental damping. The largest reduction in the flexible edge deformation occurs when damping in the first mode is maximized by distributing the supplemental damping such that the damping eccentricity takes on the largest value with algebraic sign opposite to the structural eccentricity. Copyright © 2000 John Wiley & Sons, Ltd.     

Damping estimation of high-rise buildings considering structural modal directions

One widespread problem in damping estimation of high-rise buildings is the neglect of structural modal directions, which may induce beating in measured dynamic responses along building geometric axes and thereby induce errors in damping estimations to some extent. Based on a proposed two degrees of freedom (2-DOF) simulation model, the effects of neglecting the modal directions on damping estimate are systematically investigated. The results show that the angular differences between the modal directions and the building geometric axes, as well as the frequency difference between the involved modes, both have significant effects on the damping estimate of high-rise buildings. This paper proposes a spectral method to determine the modal directions of high-rise buildings and further validate this method by an analysis of full-scale measurements from four skyscrapers. The damping ratios estimated based on the responses along the identified modal directions are more accurate than those based on those measured along the building geometric axes. Furthermore, an empirical prediction model for damping ratio of high-rise buildings with heights over 200 m is proposed based on the field measured damping results of several buildings with consideration of the modal directions. The objective of this study is to improve the accuracy of damping estimation of high-rise buildings and therefore provide useful information for the structural design of future skyscrapers.     

Tracking modal parameters from building seismic response data using recursive subspace identification algorithm

Tracking modal parameters and estimating the current structural state of a building from seismic response measurements, particularly during strong earthquake excitations, can provide useful information for building safety assessment and the adaptive control of a structure. Therefore, online or recursive identification techniques need to be developed and implemented for building seismic response monitoring. This paper develops and examines different methods to track modal parameters from building seismic response data. The methods include recursive data‐driven subspace identification (RSI‐DATA) using Givens rotation algorithm, and RSI‐DATA using Bona fide algorithm. The question on how well the results of RSI‐DATA reflect the real condition is investigated and verified with a bilinear SDOF simulation study. Time‐varying modal parameters of a four‐story reinforced concrete school building are identified based on a series of earthquake excitations, including several seismic events, large and small. Discussions on the different methods' ability to track the time‐varying modal parameters are presented. The variation of the identified building modal frequencies and damping ratios from a series of event‐by‐event seismic responses, particularly before and after retrofitting of the building is also discussed. Copyright © 2017 John Wiley & Sons, Ltd.     

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.     

高速铁路客站模态特征分析与阻尼模型的注记

针对"房桥合一"高速铁路客站质量、刚度分布严重不均匀、不同阻尼比构件繁多等特点,对其模态特征和用于时程法阻尼模型的确定进行了研究。通过位能加权激励原理和位能公式的阐述、天津西站II区的模态分析与振型分解时程法的应用,并以此为标准进行了5种不同瑞利阻尼比例系数时程法的响应比较。结果表明:位能加权法激励为各模态的振型响应,求得阻尼为振型阻尼,结构模态质量累计数突变发生在第9阶,选择第1,9阶振型确定的瑞利阻尼比例系数较合理。振型分解时程法的振型阻尼可基于振型响应的位能加权法确定,直接时程法的瑞利阻尼宜选择第1阶与模态质量累计数发生突变的振型来确定,可供结构设计与分析参考。     

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.