Seismic response analysis of structure–equipment systems with non-classical damping effects

A method is presented to obtain the exact complex-valued eigenproperties of a classically damped structure and equipment system. The non-classically damped character of the combined system as well as the effect of dynamic interaction between primary structure and equipment are properly included in the calculation of these eigenproperties. It is necessary only to know the classical modal properties of the structure and, of course, the equipment characteristics. The eigenvalues are obtained as the solution of a non-linear equation which can be easily solved by the Newton–Raphson algorithm. Once the eigenvalues are known, the corresponding eigenvectors are obtained from simple closed-form expressions. The method can be used equally effectively with light as well as heavy equipment. Numerical results demonstrating the effectiveness of the method are presented. A procedure which utilizes the complex-valued eigenproperties is developed for calculating the floor response spectra directly from the ground spectra. Numerical results of floor response spectra obtained from this procedure are presented. The floor spectra calculated by this approach include the structure–equipment interaction effect.     

Rosenblueth's modal combination rule for systems with non-classical damping

A simple rule is derived to combine, within the framework of a complex mode superposition, the maximum modal responses of systems such as soil-structure and structure-equipment systems, for which closely spaced natural frequencies are likely, and for which, because of the large difference in the damping values of their various components, the assumption of an orthogonal damping matrix may lead to significant errors. The rule constitutes the generalization of Rosenblueth's rule for systems with closely spaced natural frequencies and classical modes, and is expressed in terms of their complex mode shapes and natural frequencies. Its derivation is based on the theory of a complex modal analysis for systems with non-classical modes of vibration and on Rosenblueth's original derivation. As in this original derivation, earthquake ground motions are modelled as a stationary white noise process, but the formulae obtained under this assumption are modified later on to account for the transient nature of actual earthquakes. A numerical example is presented to illustrate the application of the rule, and a comparative study with numerical integration solutions is performed to assess its accuracy. In this comparative study, it predicts the numerical integration solutions with an average error of 0.3 per cent.     

Modal methods in the dynamics of systems with non-classical damping

When damping in a system is both significantly high and its distribution is non-classical the solution of dynamical problems by conventional modal analysis is complicated by the presence of coupling between the normal co-ordinates. Further, the convergence of a solution may be erratic with successive modal additions, leading to the need to include a larger number of modes than would otherwise be expected. In this paper methods of modal analysis in structural dynamics are discussed and their derivations briefly given. These include the conventional mode displacement method and the force summation method, employing normal modes, and the analogous procedures with damped modes. In the latter, dynamic response equations are not coupled. Dynamic loading solutions by the four approaches, each taking account of the non-classical damping distribution, are demonstrated with a simple model representing a structure on a compliant foundation. The results strongly suggest that the use of damped modes with force summation could be the most effective procedure when damping is non-classical.     

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.     

隔震结构中非经典阻尼影响及最佳阻尼比分析

本文采用双自由度非比例阻尼振动模型描述基础隔震体系,用拉普拉斯变换方法获得其地震时域响应的近似解析解,借助于应谱理论分析了非比例阻尼对隔震体系中的上部结构层间最大剪力及隔震层最大位移等响应值的影响,探寻了隔震系统最佳阻尼比的取值范围。     

结构-地基体系的非比例阻尼影响及随机地震响应分析

针对结构-地基体系随机地震响应分析中存在的非比例阻尼耦合问题，采用不同非比例阻尼判断准则研究了结构-地基体系在不同刚度比，不同阻尼比下结构与地基间的阻尼耦合及体系不同振型间的阻尼耦合问题，推导了用于结构-地基非比例阻尼系统随机分析的小参数随机摄动分析公式，并用于地下大型有衬砌洞室的随机地震响应分析中。     

Damped vibration mode superposition method for dynamic response analysis

In the dynamic response analysis of extremely complex structural systems in which the damping characteristics of each element are independent, the damping matrix is not always diagonalized by the use of undamped free vibration mode shapes. In the present paper, a mode-superposition method by the use of damped free vibration mode shapes is developed for such structural systems. It is also shown that the Fast Fourier Transform (FFT) procedures, that are available for the dynamic response analysis of linear structural systems, are used effectively in this mode-superposition method with good accuracy.     

Ritz vectors and generation criteria for mode superposition analysis

A generation procedure of Ritz vectors to control the inclusion of static effect and the number of vectors in mode superposition dynamic analysis is presented. The original algorithm of the Ritz vectors¹⁵ is modified to improve stability in the generation procedure and to include the use of static residual. To reject unimportant Ritz vectors, cut-off criteria, which are based on the participation of mass distribution and spatial load distribution, are proposed. Numerical examples are presented to illustrate the effectiveness of the derived Ritz vectors over the eigenvectors and the performance of the cutoff criteria in the mode superposition dynamic analysis.     

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.     

On the accuracy of mode superposition analysis in structural dynamics

It is pointed out that the number of modes which should be included in a mode superposition dynamic response analysis depends on both the frequency content and the distribution of the loading. If the loading frequency is low the effect of the higher modes can be approximated by a static analysis. A technique is described for calculating this static contribution from the higher modes; the total response is then represented by the sum of the lower mode dynamic response and the higher mode static effects. The effectiveness of the procedure is demonstrated by a numerical example.     

Nonlinear performance of classical damping

The performance of a classical damping matrix, constructed either from the use of initial structural properties or current structural properties, in the step-by-step solution of a nonlinear multiple degree of freedom (MDOF) system is analytically evaluated. The analytical results are confirmed by numerical examples. Consequently, some conclusions are drawn from these analytical results that might be considered as rough guidelines for practical applications. It is found that a classical damping matrix constructed from initial structural properties is adequate for practical applications, since it has approximately the same damping effect as obtained by current structural properties and is more efficient in terms of computing.     

悬挂阻尼控制结构体系巨型框架地震响应分析

提出一种巨型悬挂阻尼控制结构体系，采用结构动力学有限元方法对结构体系进行了地震随机振动分析、时程分析和地震反应谱分析，分析结果表明，这种结构体系能有效地减小结构的地震响应。最后讨论了影响控制效果的参数。     

Non-linear seismic response analysis using vector superposition methods

Computational algorithms based on the pseudo-force method (PFM) and the tangent spectrum method (TSM) for the seismic analysis of elasto-plastic MDOF structures by mode superposition are presented. The emphasis is put on the effect of the truncation of higher modes on the convergence of the ductility demand and energy dissipated during the earthquake. Eigenvectors and load-dependent vectors have been used in comparative analyses. Applications on a flexible 25-storey building and a stiff, 5-storey shear building indicate that, for a flexible structure, the ductility demand computed from the PFM is more sensitive to basis truncation than that computed from the TSM. However, for the stiff structure, the opposite behaviour is observed. The results indicate that the use of load-dependent vectors in inelastic analyses maintains the computational advantages found for elastic analyses in previous investigations. Although the PFM is more stable and computationally more effective than the TSM, it does not provide any information on the evolution of tangent modal properties in time that reflects dynamic response modification as the structure becomes inelastic.     

不同阻尼体系地震能量输入及阻尼能量耗散计算分析

本文给出了复阻尼体系地震能量输入及阻尼能量耗散计算公式。利用天津、E1Centro、迁安记录及Mexico地震Cale记录计算了复阻尼及粘性阻尼体系的地震能量输入及阻尼能量耗散,并绘制了相应的时程曲线。计算所用阻尼比取0．1及0．05,周期分别取0．3s,0．5s,1s,1．5s及5s。通过计算我们观察到,除迁安记录外,在其它几个地震作用下,对固有周期小于1s的短周期结构,复阻尼体系能量输入及阻尼能量耗散时程曲线值高于相应的粘性阻尼体系(对迁安记录固有周期需小于0．5s)。对中周期结构,两组曲线相近,对于长周期结构,复阻尼体系的能量时程曲线值低于相应的粘性阻尼体系值。对于每个地震记录,有一个临界值,体系的固有周期小于此值时,不同阻尼模型的能量输入、阻尼能量耗散时程曲线值无大差异,当周期大于此值时,不同阻尼模型的能量输入及阻尼能量耗散时程曲线差异较大。复阻尼体系对周期的敏感程度远大于粘性阻尼体系。当周期不变,阻尼比增大时,在峰值点之后,复阻尼体系能量时程曲线基本无大的变化,粘性阻尼体系能量时程曲线有抬高趋势。     

复阻尼多自由度系统动力分析的模态叠加法

工程实际中,复阻尼多自由度系统的瞬态响应过去一直是通过频域方法求解的。频域方法的一个突出问题是求传递函数矩阵的计算工作量过大。本文给出一种有效而实用的时域解法———实模态叠加法,此外还介绍了复模态叠加法。实模态叠加法是基于一个事实,即n维复向量空间中的复向量可以在n维实向量空间中的一组线性无关的实向量构成的基下表出;另外,就此方法还讨论了复阻尼多自由度系统初始运动条件的给出和转换问题。复模态叠加法则是通过变量替换的方法,变其中的复特征值问题为形式上的一个实特征值问题来解决的。     

Ritz method for dynamic analysis of large discrete linear systems with non-proportional damping

Real and complex Ritz vector bases for dynamic analysis of large linear systems with non-proportional damping are presented and compared. Both vector bases are generated utilizing load dependent vector algorithms that employ recurrence equations analogous to the Lanczos algorithm. The choice of static response to fixed spatial loading distribution, as a starting vector in recurrence equations, is motivated by the static correction concept. Different phases of dynamic response analysis are compared with respect to computational efficiency and accuracy. It is concluded that the real vector basis approach is approximately eight times more efficient than the complex vector basis approach. The complex vector basis has some advantages with respect to accuracy, if the excitation is of piecewise linear form, since the exact solution can be utilized. In addition, it is demonstrated that both Ritz vector bases, real and complex, possess superior accuracy over the adequate eigenvector bases.     

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 Power Spectral Density technique applied to micropulsation analysis

Summary An empirical investigation has been conducted on the practical methods of obtaining the spectra of micropulsation activity, and the interpretational limits which can be applied to the detailed use of spectra computed by the Power Spectral Density (PSD) method. Results indicate that sufficiently high resolution may be obtained reliably from the PSD method for it to be regarded as a useful improvement over handscaling methods. The limitation is shown to remain in the selection of events rather than in computational or instrumental properties.     

海洋平台结构SMA阻尼隔振振动台试验与分析

本文针对渤海JZ20-2MUQ导管架式海洋平台进行比例为1∶10的模型结构的形状记忆合金阻尼隔振振动台试验研究。首先,根据原结构的基本参数,确定试验模型的几何参数,推导出相似比关系,并对形状记忆合金阻尼器和黏滞阻尼器进行性能试验,确定其基本力学性能;其次,对模型结构进行振动台试验研究,分别考察4种地震作用下无控结构、纯隔振结构、SMA隔振结构和SMA阻尼隔振结构的反应;最后,对模型结构进行数值计算分析。结果表明,各种隔振结构方案对平台结构导管架端帽位移和平台甲板加速度均有很好的控制效果。数值分析结果与试验结果吻合较好。     

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.