A modal superposition pseudo-force method for dynamic analysis of structural systems with non-proportional damping

A modal superposition pseudo-force method for the dynamic analysis of structural systems with non-proportional damping is presented. The method combines the advantages of the classical modal superposition method and the pseudo-force method. When the system damping is non-proportional, the dynamic equilibrium equations in generalized coordinates are coupled through the damping terms. In the present method, these coupled equations are solved by an iterative process in which the coupling terms are treated as pseudo-forces. A proof of the convergence of the iterative process is given. Numerical examples show the good convergence characteristics of the process and the good accuracy of the obtained results.     

地震动永久位移对结构地震反应影响

本文推导了基于位移激励计算单自由度体系拟速度谱公式,通过构造的脉冲位移时程对公式精度进行了验证;之后利用小波变换去除强震记录噪声而保留地震动永久位移,再基于去趋势项方法和滤波方法去除永久位移后,计算拟速度谱。算例结果表明:短周期段内,不保留永久位移的位移激励拟速度谱值与保留永久位移的位移激励拟速度谱值相差很小;中长周期段内,不保留永久位移的位移激励拟速度谱值总体上小于保留永久位移的位移激励拟速度谱值,且不保留永久位移时,滤波方法引起的拟速度谱降幅大于去趋势项方法所引起的拟速度谱降幅。因此,基于位移激励计算中长周期结构的地震反应时,应保留地震动永久位移,或基于去趋势项方法去除永久位移。      

基于输出反馈的建筑结构闭开环次优控制

对传统的结构抗震闭开环控制算法进行改进。基于地面运动自回归模型,采用Kalman滤波利用可以量测到的地面加速度激励对未来时段即将发生的地面加速度激励进行预估,并在微分方程的求解中引入精确高效的精细积分算法。考虑到实际控制中量测全部状态变量的困难,改进算法仅需量测部分状态变量。数值仿真表明,基于输出反馈的闭开环次优控制策略能大大降低结构的地震响应。     

A pseudo‐force iterative method with separate scale factors for dynamic analysis of structures with non‐proportional damping

Dynamic equilibrium equations of structural systems with non‐proportional damping are coupled through the damping terms. Such coupling invalidates application of the classical modal superposition method. In this paper, a mode‐superposition pseudo‐force method is proposed. The coupled equilibrium equations are solved by an iterative process in which the coupling terms are treated as pseudo‐forces. A scale factor for each mode of the system is obtained by optimizing the iteration convergence. Through these uniquely solved scale factors, the modified modal equations not only converge much faster but also yield results with higher accuracy. A proof of the convergence of the iterative process is also presented. Copyright © 2002 John Wiley & Sons, Ltd.     

Precise integration method without inverse matrix calculation for structural dynamic equations

The precise integration method proposed for linear time-invariant homogeneous dynamic systems can provide accurate numerical results that approach an exact solution at integration points. However, difficulties arise when the algorithm is used for non-homogeneous dynamic systems due to the inverse matrix calculation required. In this paper, the structural dynamic equalibrium equations are converted into a special form, the inverse matrix calculation is replaced by the Crout decomposition method to solve the dynamic equilibrium equations, and the precise integration method without the inverse matrix calculation is obtained. The new algorithm enhances the present precise integration method by improving both the computational accuracy and efficiency. Two numerical examples are given to demonstrate the validity and efficiency of the proposed algorithm.     

Spectral analysis of systems with non-classical damping using classical mode superposition technique

A spectral method for random vibration analysis of a structural system with non-proportional damping is presented using classical (undamped) mode superposition technique. The method obtains the frequency response function of the system by solving the dynamic equilibrium equations in generalized co-ordinates through an iterative process. The iterative solution is written in closed form and the proof for convergence of the iterative process is given. Numerical examples show the convergence characteristics of the process and an excellent accuracy of the obtained results. The method turns out to be computationally more efficient than the conventional methods of spectral analysis using damped mode shapes and frequencies.     

Efficient mode superposition algorithm for seismic analysis of non-linear structures

A step-by-step integration method is proposed to compute within the framework of the conventional mode superposition technique the response of bilinear hysteretic structures subjected to earthquake ground motions. The method is computationally efficient because only a few modes are needed to obtain an accurate estimate of such a response, and because it does not require the use of excessively small time steps to avoid problems of accuracy or stability. It is developed on the basis that the non-linear terms in the equations of motion for non-linear systems may be considered as additional external forces, and the fact that by doing so such equations of motion can be interpreted as the equations of motion of an equivalent linear system, excited by a modified ground motion. These linear equations are then subjected to a conventional modal decomposition and transformed, as with linear systems, into a set of independent differential equations, each representing the system's response in one of its modes of vibration. To increase the efficiency of the method and account properly for the participation of higher modes, these independent equations are solved using the Nigam-Jennings technique in conjunction with the so-called mode acceleration method. The accuracy and efficiency of the method is verified by means of a comparative study with solutions obtained with a conventional direct integration method. In this comparative study, including only a few modes, the proposed method accurately predicts the seismic response of three two-dimensional frame structures, but requiring only, on an average, about 47 per cent less computer time than when the direct integration method is used.     

Modal time history analysis of non-classically damped structures for seismic motions

The step-by-step modal time history integration methods are developed for dynamic analysis of non-classically damped linear structures subjected to earthquake-induced ground motions. Both the mode displacement and mode acceleration-based algorithms are presented for the calculation of member and acceleration responses. The complex-valued eigenvectors are used to effect the modal decoupling of the equations of motion. However, the recursive step-by-step algorithms are still in terms of real quantities. The numerical results for the acceleration response and floor response spectra, obtained with these approaches, are presented. The mode acceleration approach is observed to be decidedly better than the mode displacement approach in as much as it alleviates the so-called missing mass effect, caused by the truncation of modes, very effectively. The utilization of the mode acceleration-based algorithms is, thus, recommended in all dynamic analyses for earthquake-induced ground motions.     

基于数学去噪后模拟地震信号测试混凝土结构抗震性能的研究

针对当前混凝土结构建筑抗震性能测试方法存在的测试结果准确率较低的问题,提出一种新的测试方法,利用小波函数的伸缩平移操作得到一个函数族,对模拟地震信号进行小波变换。当函数族组成一个标准的正交基时,从连续小波变换中对地震信号进行重构,将去噪后的地震信号作为检测混凝土结构建筑抗震性能的信号样本。利用单自由度体系的动力平衡方程对地震的位移反应、速度反应加速度反应以及恢复力进行求解,通过强度折减系数计算公式和结构最大弹性反应计算公式,计算混凝土结构建筑的强度折减系数和结构最大弹性反应,最后在震动台上对混凝土结构建筑抗震性能进行测试。测试实验结果表明,所提方法可以实现对混凝土结构建筑抗震性能的测试,且测试结果准确程度较高。     

Stability and accuracy analysis of direct integration methods

A systematic procedure is presented for the stability and accuracy analysis of direct integration methods in structural dynamics. Amplitude decay and period elongation are used as the basic parameters in order to compare various integration methods. The specific methods studied are the Newmark generalized acceleration scheme, the Houbolt method and the Wilson θ-method. The advantages of each of these methods are discussed. In addition, it is shown how the direct integration of the equations of motion is related to the mode superposition analysis.     

强震作用下超高桥墩动力稳定性理论研究

根据平截面假定,考虑超高桥墩大位移变形产生的几何非线性影响,建立超高桥墩的振动控制微分方程,利用变步长的龙格-库塔法进行求解,结合B-R运动判定准则,对超高桥墩在地震作用下的动力失稳机理进行研究。理论分析表明,超高桥墩的动力失稳与桥墩的几何尺寸、质量分布、边界条件有密切关系;桥墩动力失稳时刻随地震波加速度峰值的增大而减小;失稳时刻与失稳加速度荷载有对应关系。算例结果表明:本文方法正确,利用本文理论能够准确计算超高桥墩的失稳时刻及失稳加速度,对超高桥墩动力失稳的理论分析及工程实践有重要指导意义。     

非平稳地震激励下隔震曲线梁桥振动控制研究

曲线梁桥由于其平面不规则性导致结构在地震激励下产生弯扭耦合效应,使得隔震曲线梁桥的地震响应更加复杂。目前常用的控制方法是将隔震技术与附加减震装置相结合对曲线梁桥进行控制。本文将地震动考虑为一均匀调制非平稳随机过程,针对隔震曲线梁桥长周期、低频率的特点,选取Clough-Pension平稳地震动功率谱模型作为随机地震动输入模型,对无控(NON-C)、经典线性最优控制(COC)以及序列最优控制算法(SOC)三种状态下的曲线桥梁进行随机响应分析。通过建立曲线梁桥在随机地震动作用下的运动方程,求出减震控制结构的位移谱密度、加速度谱密度响应及时变方差。分析结果表明:序列最优控制算法(SOC)在使隔震层位移得到减小的同时,可以更有效地控制上部结构的地震响应,具有更好的控制效果。     

Sliding fragility of block‐type non‐structural components. Part 1: Unrestrained components

Motivated by the development of performance‐based design guidelines with emphasis on both structural and non‐structural systems, this paper focuses on seismic vulnerability assessment of block‐type unrestrained non‐structural components under sliding response on the basis of seismic inputs specified by current seismic codes. Two sliding‐related failure modes are considered: excessive relative displacement and excessive absolute acceleration. It is shown that an upper bound for the absolute acceleration response can be assessed deterministically, for which a simple yet completely general equation is proposed. In contrast, fragility curves are proposed as an appropriate tool to evaluate the excessive relative displacement failure mode. Sample fragility curves developed through Monte‐Carlo simulations show that fragility estimates obtained without taking into account vertical base accelerations can be significantly unconservative, especially for relatively large values of the coefficient of friction. It is also found that reasonable estimates of relative displacement response at stories other than the ground in multistorey buildings cannot in general be obtained by simply scaling the ground acceleration to the peak acceleration at the corresponding storey. Failure modes considered in this study are found to be essentially independent of each other, a property that greatly simplifies assessment of conditional limit states. Copyright © 2002 John Wiley & Sons, Ltd.     

An active mass damper designed using ARX models of a building structure

This study proposes a new design method for an active mass damper (AMD) that is based on auto‐regressive exogenous models of a building structure. The proposed method uses the results of system identification in the field of active structural control. The uncontrolled structure is identified as auto‐regressive exogenous models via measurements under earthquake excitation and forced vibration. These models are linked with an equation of motion for the AMD to introduce a state equation and output equation for the AMD–structure interaction system in the discrete‐time space; the equations apply modern control theories to the AMD design. In the numerical applications of a 10‐degree‐of‐freedom building structure, linear quadratic regulator control is used to understand the fundamental characteristics of the proposed design procedure. The feedback control law requires the AMD's acceleration, velocity and stroke; the structure's acceleration; and the ground acceleration as vibration measurements. The numerical examples confirm the high applicability and control effectiveness of the proposed method. One remarkable advantage of the proposed method is that an equation of motion for the structure becomes unnecessary for designing controllers. Copyright © 2016 John Wiley & Sons, Ltd.     

Solution of viscously damped linear systems using a set of load-dependent vectors

This paper considers a solution method for viscously damped linear structural systems which are subjected to transient loading. The equations of motion of such systems are written in a first-order form. A solution subspace is generated using the damped dynamic matrix and the static deflection from the first-order form of the equations of motion. Two convenient bases, Lanczos vectors and Ritz vectors, are constructed from this subspace. An approximate solution is then obtained by superposition of the Lanczos vectors or the Ritz vectors. In contrast to the traditional mode superposition method using complex eigenvectors, the Lanczos vectors or the Ritz vectors are less expensive to generate than the complex eigenvectors, yet yield comparable accuracy. In addition, there is no need for a static correction since the static deflection is already contained in our solution subspace. Numerical examples are presented to show the potential of using the Ritz vectors to compute responses of damped dynamic systems.     

Dynamic response of an elastic-viscoplastic system in modal co-ordinates

A dynamic analysis of elastic–viscoplastic systems, incorporating the modal co-ordinate transformation technique, is presented. The formulation results in uncoupled incremental equations of motion with respect to the modal co-ordinates. The elastic–viscoplastic model adopted allows the analysis not to involve yielding regions and loading/unloading processes. An implicit Runge–Kutta scheme together with the Newton–Raphson method are used to solve the non-linear constitutive equations. Stability and accuracy of the numerical solution are improved by utilizing a local time step sub-incrementing procedure. Applications of the analyses to multi-storey shear buildings show that good results can be obtained for the maximum displacement response by including only a few lower modes in the computation, but the prediction of the ductility factor response tends to underestimate the peak values when too few modes are used. In addition, stable and valid results can be obtained even with a sizable time step increment.     

Comparison of capability of time integration methods in capturing dynamic loading

Numerical properties of the time integration method proposed by the first author of this paper in 2007 are the same as those of the constant average acceleration method (AAM) for linear elastic systems, except that the capability to capture dynamic loading was not explored. It was found that there were different quadrature equations to predict the next step displacement increment. A modified quadrature equation of this method was derived so that the equation to determine the next step displacement was numerically equivalent to the equation used in the constant AAM. It was verified that the original form of this method, in general, had a better capability to capture dynamic loadings than the constant AAM. This excellent property, in addition to computational efficiency, will help to make this method competitive with general secondorder accurate integration methods.     

An error analysis of truncated starting conditions in step-by-step time integration: Consequences for structural dynamics

Most of the step-by-step time integration algorithms for structural dynamics require an initial acceleration vector to be specified, in addition to displacement and velocity vectors. A consistent initial acceleration vector may be calculated by solving the equations of motion at the initial time, while a truncated initial acceleration vector is obtained by setting the acceleration values to zero. Although the truncated starting procedure decreases computational effort, it is shown to affect accuracy adversely. For the structural dynamics algorithms considered herein, the rate at which the numerical solution converges to the exact solution is O(Δt) when the truncated starting procedure is used, compared to O(Δt²) when consistent initial acceleration values are used.     

Dynamic response of a three-beam system with intermediate elastic connections under a moving load/mass-spring

The objective of this research is to study the dynamic response characteristics of a three-beam system with intermediate elastic connections under a moving load/mass-spring. In this study, the finite Sine-Fourier transform was performed for the dynamic partial differential equations of a simply supported three-beam system (SSTBS) under a moving load and a moving mass-spring, respectively. The dynamic partial differential equations were transformed into dynamic ordinary differential equations relative to the time coordinates, and the equations were solved and the displacement Fourier amplitude spectral expressions were obtained. Finally, based on finite Sine-Fourier inverse transform, the expressions for dynamic response of SSTBS under the moving load and moving mass-spring were obtained. The proposed method, along with ANSYS, was used to calculate the dynamic response of the SSTBS under a moving load/mass-spring at different speeds. The results obtained herein were consistent with the ANSYS numerical calculation results, verifying the accuracy of the proposed method. The influence of the load/mass-spring’s moving speed on the dynamic deflections of SSTBS were analyzed. SSTBS has several critical speeds under a moving load/mass-spring. The vertical acceleration incurred by a change in the vertical speed of SSTBS due to the movement of mass-spring and the centrifugal acceleration produced by the movement of massspring on the vertical curve generated by SSTBS vibration could not be neglected.     

A method for stochastic seismic response analysis of non-classically damped structures

A method to calculate the stationary random response of a non-classically damped structure is proposed that features clearly-defined physical meaning and simple expression. The method is developed in the frequency domain, The expression of the proposed method consists of three terms, i.e., modal velocity response, modal displacement response, and coupled (between modal velocity and modal displacement response), Numerical results from the parametric study and three example structures reveal that the modal velocity response term and the coupled term are important to structural response estimates only for a dynamic system with a tuned mass damper. In typical cases, the modal displacement term can provide response estimates with satisfactory accuracy by itself, so that the modal velocity term and coupled term may be ignored without loss of accuracy, This is used to simplify the response computation of non-classically damped structures. For the white noise excitation, three modal correlation coefficients in closed form are derived. To consider the modal velocity response term and the coupled term, a simplified approximation based on white noise excitation is developed for the case when the modal velocity response is important to the structural responses. Numerical results show that the approximate expression based on white noise excitation can provide structural responses with satisfactory accuracy~