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.     

Simplified discrete systems for dynamic analysis of structures on footings and piles

A simplified discrete system in the form of a simple oscillator is developed to simulate the dynamic behavior of a structure founded through footings or piles on compliant ground, under harmonic excitation. Exact analytical expressions for the fundamental natural period and the corresponding damping coefficients of the above system are derived, as function of geometry and the frequency-dependent foundation impedances. In an effort to quantify the coupling between swaying and rocking oscillations in embedded foundations such as piles, the reference system is translated from the footing–soil interface to the depth where the resultant soil reaction is applied, to ensure a diagonal impedance matrix. The resulting eccentricity is a measure of the coupling effect between the two oscillation modes. The amounts of radiation damping generated from a single pile and a surface footing are evaluated. In order to compare the damping of a structure on a surface footing and a pile, the notion of static and geometric equivalence is introduced. It is shown that a pile may generate significantly higher radiation damping than an equivalent footing, thus acting as an elementary protective system against seismic action.     

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.     

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.     

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.     

System identification of linear structures based on Hilbert–Huang spectral analysis. Part 1: normal modes

Based on the Hilbert–Huang spectral analysis, a method is proposed to identify multi‐degree‐of‐freedom (MDOF) linear systems using measured free vibration time histories. For MDOF systems, the normal modes have been assumed to exist. In this method, the measured response data, which are polluted by noises, are first decomposed into modal responses using the empirical mode decomposition (EMD) approach with intermittency criteria. Then, the Hilbert transform is applied to each modal response to obtain the instantaneous amplitude and phase angle time histories. A linear least‐square fit procedure is proposed to identify the natural frequency and damping ratio from the instantaneous amplitude and phase angle for each modal response. Based on a single measurement of the free vibration time history at one appropriate location, natural frequencies and damping ratios can be identified. When the responses at all degrees of freedom are measured, the mode shapes and the physical mass, damping and stiffness matrices of the structure can be determined. The applications of the proposed method are illustrated using three linear systems with different dynamic characteristics. Numerical simulation results demonstrate that the proposed system identification method yields quite accurate results, and it offers a new and effective tool for the system identification of linear structures in which normal modes exist. Copyright © 2003 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.     

Damping‐adjusted combination rule for the response spectrum analysis of base‐isolated buildings

The classical response spectrum method continues to be the most popular approach for designing base‐isolated buildings, therefore avoiding computationally expensive nonlinear time‐history analyses. In this framework, a new method for the seismic analysis and design of building structures with base isolation system (BIS) is formulated and numerically validated, which enables one to overcome the main shortcomings of existing techniques based on the response spectrum method. The main advantages are the following: first, reduced computational effort with respect to an exact complex‐valued modal analysis, which is obtained through a transformation of coordinates in two stages, both involving real‐valued eigenproblems; second, effective representation of the damping, which is pursued by consistently defining different viscous damping ratios for the modes of vibration of the coupled BIS‐superstructure dynamic system; and third, ease of use, because a convenient reinterpretation of the combination coefficients leads to a novel damping‐adjusted combination rule, in which just a single response spectrum is required for the reference value of the viscous damping ratio. The proposed approach is specifically intended for design situations where (i) the dynamic behaviour of seismic isolators can be linearised and (ii) effects of nonproportional damping, as measured by modal coupling indexes, are negligible in the BIS‐superstructure assembly. Copyright © 2012 John Wiley & Sons, Ltd.     

Modal damping for torsionally coupled buildings on elastic foundation

A method to determine the approximate normal modes and the modal damping for torsionally coupled buildings on an elastic foundation is presented. The modal damping is determined by an iterative procedure which matches the approximate normal mode solution with the rigorous solution. The response quantity to be matched is selected in a consistent and logical manner. The normal modes and the damping ratios thus found are then used to determine the seismic response of the interaction system by the response spectrum technique.     

Seismic response of heavily damped base isolation systems

The development of an efficient energy-dissipating mechanism that works in conjunction with laminated elastomeric bearings in order to reduce the lateral deformation of the isolation system has always been a goal of base isolation research. Theoretically, this deformation will be reduced to the minimum if damping augmentation of the isolation system can reach a critical value. However, augmenting the isolation damping may cause some unwanted side effects. The purpose of this paper is to study the influence of isolation damping on the seismic response of heavily damped base-isolated buildings. The base isolation system is assumed to be linearly viscoelastic and is analysed using the complex mode method. Solutions derived by using perturbation techniques for a two-degree-of-freedom system and the computer simulation for a multiple-degree-of-freedom system reveal that augmenting the isolation damping can reduce efficiently the deformation of the isolation system, but at the price of increasing the high-frequency vibration in the superstructure. When the damping ratio of the isolation system is beyond some level, increasing the isolation damping will enlarge the extreme values of the base and superstructural accelerations. It is also found that approximate solutions derived from the use of classical damping and classical modes of vibration in the seismic analysis of heavily damped base isolation systems can be substantially in error.     

Forced vibration of a tall steel-frame building

The dynamic properties of the twenty-two-story, steel-frame San Diego Gas and Electric Company Building in San Diego, California, have been determined experimentally in a series of co-operative tests between the California Institute of Technology and the University of California at Los Angeles. The building was vibrated by two eccentric mass exciters capable of frequencies up to 10 c/sec and forces as much as 5000 Ib each. The natural frequencies, associated mode shapes and the amounts of damping were determined for the first six modes of vibration in each of the two translational directions, and also in torsion. The mode shapes and frequencies showed, in general, the regularity and uniformity that appears typical of many tall buildings, but the three fundamental modes (nominally NS, EW and torsion) of the structure showed a coupling of translational and rotational components to a degree that was unexpected in a building whose structural frame is essentially symmetric. It is believed that this may be a consequence of the exceptionally small differences among the three fundamental frequencies. The damping in the first eighteen modes of the structure varied from 1.6 to 4.4 per cent, with a slight tendency for the larger values to be associated with the higher modes. Of the simpler damping models that might be used for analysis of the building, constant modal damping appears most appropriate and stiffness or mass proportional damping would not be realistic.     

Hilbert-Huang变换在密频结构阻尼识别中的应用

Hilbert—Huang变换是一种新的数据处理方法，由经验模分解(Empirical Mode Decomposition)技术及Hilbert变换两部分组成。本文研究此方法对于密频结构阻尼识别的应用。首先对于两自由度系统模型，说明该方法用于阻尼识别的步骤。进而研究存在频率密集现象的高层建筑的阻尼识别问题。上述结果与理论值及由半功率带宽法的识别值进行了比较，对比显示Hilbert．Huang方法较传统方法具有良好的识别密频结构阻尼的性能，适用于大型结构的系统识别。     

特高压互连电气设备减震性能振动台试验

为研究铅金属减震器对特高压互连电气设备的减震效果,进行由硬管母线连接的特高压避雷器设备和电容式电压互感器设备组成的互连耦合体系的地震模拟振动台试验。通过白噪声扫频、抗震及减震试验,测定互连耦合体系抗震结构及减震结构的自振频率以及关键部位的应变、加速度响应。试验结果表明:抗震结构中避雷器设备的最大应变响应大于互感器设备,互连耦合体系中的避雷器设备属于易损设备;安装减震器后互连设备频率降低幅度较小,减震器基本不会影响电气设备的正常运行;安装减震器后互连耦合体系中的避雷器设备和互感器设备在较大峰值加速度地震波作用下,设备顶端加速度响应和最大应变响应均有较大幅度的降低,避雷器设备和互感器设备的最大应变响应的减震效率分别为75%和50%,减震效果显著,减震器的应用大幅提升了互连耦合体系的抗震能力。     

Prediction of earthquake energy input from smoothed fourier amplitude spectrum

This paper attempts to show analytically that the energy-input spectra of damped SDOF systems and undamped MDOF systems excited by an earthquake motion can be predicted by smoothing the Fourier amplitude spectrum of the base acceleration. The spectral window for smoothing in the frequency domain for a damped SDOF system is identical with the probability density function of the time-variant or instantaneous vibration frequency resulting from non-linear hysteresis. The spectral window for an undamped MDOF system is identical with the set of squared participation factors associated with vibration modes. It was found that the increase in damping factor and the increase in participation of higher modes provide wider spectral windows, resulting in more flattened or unaltered energy-input spectra due to enhanced smoothing effects.     

新型组合连梁控制结构体系

分析了连梁在水平力作用下的破坏机制，对连梁结构体系的发展及结构抗震控制进行了评述，在此基础上提出一种新型组合连梁控制结构体系，并对其进行了概念分析，提出了有待研究的若干关键问题。     

Evaluation of the half‐power bandwidth method to estimate damping in systems without real modes

The objective of this work was to assess the significance of the values of damping obtained applying the half‐power bandwidth method to the frequency response records of the steady‐state response of a system that does not possess real modes either because the damping matrix does not satisfy the orthogonality condition or because its parameters are functions of frequency. A multi‐degree of freedom system with real modes and different types of damping is considered first. A two degree of freedom system with an arbitrary damping matrix, a rigid mass on an elastic foundation subjected to vertical and coupled horizontal/rocking vibrations, and a single degree of freedom model of a building accounting for inertial soil structure interaction effects are considered next in more detail. The results show that the predictions of the method, when applicable compare very well with those provided by approximate formulae and procedures used in practice. Copyright © 2010 John Wiley & Sons, Ltd.     

平扭耦联隔震体系隔震支座阻尼分布影响分析

为了研究铅芯橡胶隔震支座的阻尼分布对平扭耦联隔震体系隔震效果的影响,本文对一个三层两跨钢框架,通过调整上部结构负重块位置及下部铅芯橡胶隔震支座的分布位置,进行不同偏心工况下平扭耦联隔震体系地震模拟振动台试验,获得了不同偏心工况、不同阻尼分布情况下结构位移和加速度的时程曲线。试验和分析表明:隔震层阻尼中心与上部结构的质心位置接近,或者增大隔震层的阻尼半径,可显著地降低上部结构的扭转反应。     

An incremental mode-superposition for non-linear dynamic analysis

This paper uses an incremental mode-superposition procedure to compute the inelastic dynamic response of multi-degree-of-freedom systems. A damping matrix proportional to the instantaneous properties is used throughout the analysis. The non-linear response of several shear type plane and space frames with elastic-plastic and bilinear column properties subjected to ground excitation was computed by both the incremental mode-superposition and the direct integration of the coupled equations of motion. When all modes are considered, the responses computed by the incremental mode-superposition are identical to those from the direct integration. Fewer modes can also be used to compute the response with reasonable accuracy by performing the modal truncation for each time increment. The study shows that incorporating instantaneous damping in non-linear dynamic analysis is relatively simple and requires less computational time than the direct integration.     

The baroclinic response to wind in a small two-basin lake

Field data and two linear layered models were used to examine the baroclinic response to wind in a small elongated two-basin lake, Amisk Lake (Alberta, Canada). For the first vertical baroclinic mode, wind-forced horizontal modes were simulated using a dynamic two-layer variable cross-section (TVC) model. The first horizontal mode, H1, was found to dominate the exchange between the two basins of Amisk Lake. The highest velocities associated with H1 occurred in the constricted channel connecting the two basins. This high velocity led to strong damping which brought H1 in near-resonance with the diurnal wind despite a difference in periods. Along with H1, the second horizontal mode, H2, was detected at a thermistor mooring. The TVC showed that H2 resulted from the coupling between along-thalweg wind distribution and the bathymetry of Amisk Lake. The damping for H2 was found to be weaker than for H1, likely because of weaker bottom drag in the connecting channel. The evolution of vertical H1 modes were simulated using a multi-layered box model. In response to wind pulses, the first vertical mode V1H1 initially dominated over higher vertical modes causing two-layer exchange. Following the faster decay of V1H1, higher vertical modes shifted the exchange to three and more layers. Our study shows the importance of the coupling between wind stress distribution and lake bathymetry in exciting horizontal modes, reveals the effect of damping on resonance with wind, and explains the evolution of exchange associated with vertical modes.     

灰岩裸露区检波器三自由度耦合系统理论的研究

中国南方海相碳酸盐岩沉积盆地的分布幅员辽阔,地下油气等资源极为丰富,但是由于大面积坚硬灰岩的出露,给检波器完好耦合带来了很大困难,从而降低了地震采集资料的品质,严重制约了该地区未来的油气勘探.目前野外检波器耦合试验缺乏较为合适的理论指导,带有较大的盲目性.为此,本文考虑到检波器与灰岩耦合时增加了耦合介质（石膏、泥饼等）的实际情况,提出了检波器-灰岩三自由度耦合系统的理论,研究了耦合介质、尾锥和阻尼等耦合因素对检波器-灰岩耦合系统传输函数的影响规律.发现检波器-灰岩耦合系统是具有三个谐振频率的谐振系统,通过增加耦合介质的弹性模量、减小耦合介质的底面积和高度、选用介质密度较小的尾锥、减小尾锥的高度、增加尾锥的底面积等,可以提高耦合谐振频率.通过适当增加耦合系统的阻尼,可以减小耦合系统的窄频带“带通滤波”的影响.最后,通过振动台实验初步验证了检波器-灰岩三自由度耦合系统理论模型.