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.     

Modal decomposition method for stationary response of non-classically damped systems

The stationary response of multi-degree-of-freedom non-classically damped linear systems subjected to stationary input excitation is studied. A modal decomposition procedure based on the complex eigenvectors and eigenvalues of the system is used to derive general expressions for the spectral moments of response. These expressions are in terms of cross-modal spectral moments and explicitly account for the correlation between modal responses; thus, they are applicable to structures characterized with significant non-classical damping as well as structures with closely spaced frequencies. Closed form solutions are presented for the important case of response to white-noise input. Various quantities of response of general engineering interest can be obtained in terms of these spectral moments. These include mean zero-crossing rate and mean, variance and distribution of peak response over a specified duration. Examples point out several instances where non-classical damping effects become significant and illustrate the marked improvement of the results of this study over conventional analysis based on classical damping approximations.     

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.     

Dynamic characteristics of multiple substructures with closely spaced frequencies

This paper considers a main structure supporting a large number of substructures. The substructures have closely spaced natural frequencies, and the combined main structure/multiple substructures system is subjected to harmonic or wide-band forces. The goal is to characterize the effects of the substructures on the response of the main structure. A special, fundamental case is studied in detail, where the substructures are oscillators with equal stiffnesses and equally spaced natural frequencies. The exact response expressions for the combined system are in terms of a complicated rational polynomial. However, by taking the limit where the number of substructures becomes large, the response expressions reduce to simple, physically meaningful results. It is found that the multiple substrutures are equivalent to a single viscous damping which is added to the damping of the main structure. An example illustrates how the results can be applied to passive vibration control of large structures.     

A response spectrum method for random vibration analysis of mdf systems

A response spectrum method for stationary random vibration analysis of linear, multi-degree-of-freedom systems is developed. The method is based on the assumption that the input excitation is a wide-band, stationary Gaussian process and the response is stationary. However, it can also be used as a good approximation for the response to a transient stationary Gaussian input with a duration several times longer than the fundamental period of the system. Various response quantities, including the mean-squares of the response and its time derivative, the response mean frequency, and the cumulative distribution and the mean and variance of the peak response are obtained in terms of the ordinates of the mean response spectrum of the input excitation and the modal properties of the system. The formulation includes the cross-correlation between modal responses, which is shown to be significant for modes with closely spaced natural frequencies. The proposed procedure is demonstrated for an example structure that is subjected to an ensemble of earthquake-induced base excitations. Computed results based on the response spectrum method are in close agreement with simulation results obtained from time-history dynamic analysis. The significance of closely spaced modes and the error associated with a conventional method that neglects the modal correlations are also demonstrated.     

Tuned mass dampers for structures with closely spaced natural frequencies

Analytical results are developed for vibration control of structures with one or more Tuned Mass Dampers (TMDs). The input is a harmonic load with a range of possible frequencies. The control objective is to reduce the maximum amplitude of the structural response. Perturbation theory is used with three sets of small parameters: the ratio of TMD and structural modal masses, the damping of the system, and the differences between the structural and loading frequencies. It is shown analytically that for structures with widely spaced natural frequencies, the response can be approximated accurately by the response of the well-known single-mode structure/TMD system. For structures with p closely spaced natural frequencies, more general analytical results are developed to describe the coupling between the motions of the p modes of the structure and the multiple TMDs. The results show that at least p TMDs with properly placed attachments to the structure are necessary to control the response. If fewer TMDs are used, the maximum frequency response has a lower bound which is independent of the properties of the TMDs. The TMD placement is shown to be always important, regardless of the spacing of the structure's natural frequencies. The results are illustrated for both lumped-mass and continuous structures.     

The complete SRSS modal combination rule

The complete Square‐Root‐of‐Sum‐of‐Squares (c‐SRSS) modal combination rule is presented. It expresses the structural response in terms of uncoupled SDOF modal responses, yet accounting fully for modal response variances and cross‐covariances. Thus, it is an improvement over the classical SRSS rule which neglects contributions from modal cross‐covariances. In the c‐SRSS rule the spectral moments of the structural response are expressed rigorously in terms of the spectral moments of uncoupled modal responses and of some coefficients that can be computed straightforwardly as a function of modal frequencies and damping, without involving the computation of cross‐correlation coefficients between modal responses. An example shows an application of the c‐SRSS rule for structural systems with well separated and closely spaced modal frequencies, subjected to wide‐band and narrow‐band excitations. Comparisons with response calculations using the SRSS and the Complete Quadratic Combination rules are given and discussed in detail. Based on the c‐SRSS rule a response spectrum formulation is introduced to estimate the maximum structural response. An example considering a narrow‐band excitation from the great Mexico earthquake of September 19, 1985, is given and the accuracy of the response spectrum formulation is examined. Copyright © 2010 John Wiley & Sons, Ltd.     

Some methods for the analysis of equipment-structure interaction

We present some numerical methods which simplify the analysis of equipment-structure systems composed of an equipment component which is light with respect to the structure component. We consider completely general forms of equipment-structure interaction. In particular, we consider that an arbitrary number of natural frequencies of the equipment and the structure are nearly equal (or equal) to each other. We show that this coincidence of natural frequencies implies that the equipment-structure system will itself have several closely spaced natural frequencies. The essence of equipment-structure interaction is the formation of beats, with slowly varying amplitude in each degree of freedom, between these natural frequencies-at least in response to a δ-function ground motion. We derive some differential equations, the solutions of which are a slowly varying envelope function (a vector) which yields the above slowly varying amplitudes. It is obtained by numerical integration using some conventional, but modified, methods and using a step size appropriate to capture its slowly varying nature. This envelope function defines the response to a δ-function ground motion. We show how it may be combined with the properties of an arbitrary ground motion (acceleration record or response spectrum) to yield the response to such a ground motion.     

Conservatism in summation rules for closely spaced modes

It is shown that the method recommended by the Nuclear Regulatory Commission to be used to combine spectral response in the case of closely spaced modes is unnecessarily conservative for certain systems. Closely spaced modes arise in structures from symmetry and where there is a light appendage with a frequency close to one of the natural frequencies of the structure. In the former case, the closely spaced modes do not involve significant interaction between components of the system and the Nuclear Regulatory Commission Guide is reasonable. The latter case, that is when there are closely spaced modes where interaction of components occurs as in the examples of light appendages and torsionally unbalanced buildings, must be treated by consideration of the interacting components. The approach proposed here is that the modes that are not closely spaced be treated by modal analysis and the closely spaced modes, in the case of two closely spaced modes, be treated as a coupled two-degree-of-freedom system. If this is done, the beat phenomenon, the most important characteristic of the interaction, is evident, as is the associated result that the peak response of the coupled system is developed much later than the peak responses obtained in the individual modes. It is shown that the square root of the sum of the squares procedure underestimates, as expected, the response for undamped and very lightly damped systems, but for damped systems the square root of the sum of the squares method can be extremely conservative. It follows that the other methods specified by the Nuclear Regulatory Commission for closely spaced modes must be even more conservative.     

Optimum absorber parameters for simple systems

In the classical problem a damped one degree-of-freedom absorber system is attached to a main system, which has one degree of freedom and is undamped. The optimum values of absorber stiffness and damping, which will minimize the resonant response of the main mass, are well known. In this paper the effect on these optimum conditions of light damping in the main system is studied. The authors show that optimum parameters for absorbers, which are attached to beams and plates, can be obtained simply and accurately from those for an equivalent one degree-of-freedom main system. This depends upon the concept of an effective mass for the elastic body and the representation of its response by the single relevant mode. It will be shown in a later paper that for more complex elastic bodies such as cylindrical shells, for which the natural frequencies are more closely spaced, these simple concepts do not predict accurately optimum absorber parameters.     

Earthquake‐induced structural response output‐only identification by two different Operational Modal Analysis techniques

Output‐only system identification is developed here towards assessing current modal dynamic properties of buildings under seismic excitation. Earthquake‐induced structural response signals are adopted as input channels for two different Operational Modal Analysis (OMA) techniques, namely, a refined Frequency Domain Decomposition (rFDD) algorithm and an improved Data‐Driven Stochastic Subspace Identification (SSI‐DATA) procedure. Despite that short‐duration, non‐stationary, earthquake‐induced structural response signals shall not fulfil traditional OMA assumptions, these implementations are specifically formulated to operate with seismic responses and simultaneous heavy damping (in terms of identification challenge), for a consistent estimation of natural frequencies, mode shapes, and modal damping ratios. A linear ten‐storey frame structure under a set of ten selected earthquake base‐excitation instances is numerically simulated, by comparing the results from the two identification methods. According to this study, best up‐to‐date, reinterpreted OMA techniques may effectively be used to characterize the current dynamic behaviour of buildings, thus allowing for potential Structural Health Monitoring approaches in the Earthquake Engineering range.     

A note on distribution of amplitudes of peaks in structural response including uncertainties of the exciting ground motion and of the structural model

This note is an extension of earlier works that presented probability distribution functions for amplitudes of the peaks (the highest, the second highest … the m-th highest) in response of deterministic single degree-of-freedom (SDOF) and multi degree-of-freedom (MDOF) structures to ground motion, with deterministic Fourier spectrum and duration. It shows how these probability distribution functions can be evaluated if the Fourier spectrum and duration of the excitation are random variables specified via distribution functions. Two cases are considered: (l) when the structural model is deterministic, and (2) when the modal frequencies are random variables. The procedure presented here approximates the transfer function of the structural response by Dirac delta functions at the modal frequencies, and is applicable to multi-storey buildings with small modal damping, and with natural frequencies that are not too close. The resulting probability distribution functions are needed in seismic hazard calculations of peak response amplitudes of SDOF and MDOF structures that will not be exceeded with given confidence during the service time of the structure from any earthquake at all known faults within certain distance from the structure.     

Seismic response of base-isolated structures with vertical-rocking coupling

A base-isolated building is liable to have a small horizontal eccentricity between the centre of mass of the superstructure and the centre of rigidity of the supporting bearings. In seismic analysis, the structure is modelled as a rigid block with tributary masses supported on massless elastomeric rubber bearings placed at a constant elevation below the centre of mass. This simplified system has three degrees of freedom: two translations and one rotation in the vertical plane. The investigation of the dynamic behaviour of a base-isolated building is carried out for both the detuned and the perfectly tuned cases. In the detuned case, the natural frequencies of the system are assumed to be well separated. In the perfectly tuned case, the uncoupled rocking frequency is assumed to be identical to the vertical translational frequency, which may result from an unusual mass distribution and/or an extreme aspect ratio of the superstructure. Perturbation methods are implemented in finding the dynamic characteristics for both cases. However, the dynamic response of the perfectly tuned case is the major concern in this investigation. The Green's functions for the displacement response of the three-degree-of-freedom system are derived for both the undamped and damped conditions. The response spectrum modal superposition method is used in estimating the maximum acceleration response. A simple method, accounting for the effect of closely spaced modes, is proposed for combining modal maxima and results in an approximate solution corresponding to a single-degree-of-freedom system. This approximate solution may be used for the preliminary design of a base-isolated structure. Numerical results for a base-isolated building subjected to the vertical component of the El Centro earthquake of 1940 were carried out for comparison with these analytical results. The proposed modal combination method showed superiority over the conventional Square Root of the Sum of the Squares method in estimating maximum responses. The results also indicated that the approximate single-degree-of-freedom system yields accurate estimations. It is shown that the effect of rocking coupling on the vertical response of base-isolated structures subjected to transient loadings, such as earthquake motions, can generally be neglected as a result of the combined effects of the time lag between the maximum translational and rotational responses and the influence of damping in the isolation system, which for elastomeric bearings can be as high as 8 to 10 per cent of critical.     

Identification of structural and soil properties from vibration tests of the Hualien containment model

Measurements of the response of the ¼‐scale reinforced concrete Hualien (Taiwan) containment model obtained during forced vibration tests are used to identify some of the characteristics of the superstructure and the soil. In particular, attempts are made to determine the fixed‐base modal frequencies, modal damping ratios, modal masses and participation factors associated with translation and rocking of the base. The shell superstructure appears to be softer than could have been predicted on the basis of the given geometry and of test data for the properties of concrete. Estimates of the shear‐wave velocity and damping ratio in the top layer of soil are obtained by matching the observed and theoretical system frequency and peak amplitude of the response at the top of the structure. The resulting models for the superstructure and the soil lead to theoretical results for the displacement and rotations at the base and top of the structure which closely match the observed response. Copyright © 2004 John Wiley & Sons, Ltd.     

Mode-based equivalent multi-degree-of-freedom system for one-dimensional viscoelastic response analysis of layered soil deposit

Discrete models such as the lumped parameter model and the finite element model are widely used in the solution of soil amplification of earthquakes. However, neither of the models will accurately estimate the natural frequencies of soil deposit, nor simulate a damping of frequency independence. This research develops a new discrete model for one-dimensional viscoelastic response analysis of layered soil deposit based on the mode equivalence method. The new discrete model is a one-dimensional equivalent multi-degree-of-freedom (MDOF) system characterized by a series of concentrated masses, springs and dashpots with a special configuration. The dynamic response of the equivalent MDOF system is analytically derived and the physical parameters are formulated in terms of modal properties. The equivalent MDOF system is verified through a comparison of amplification functions with the available theoretical solutions. The appropriate number of degrees of freedom (DOFs) in the equivalent MDOF system is estimated. A comparative study of the equivalent MDOF system with the existing discrete models is performed. It is shown that the proposed equivalent MDOF system can exactly present the natural frequencies and the hysteretic damping of soil deposits and provide more accurate results with fewer DOFs.     

采用双向调谐质量阻尼器的大跨度桥梁风振控制仿真分析

传统的调谐质量阻尼器（TMD）设计均仅针对结构某一阶模态单独设置,当用于密频结构减振时会导致附加质量过多。为减小TMD的附加质量,结合大跨度斜拉桥结构的密频与风致耦合振动特点,提出了一种新型的双向共享质量与电涡流阻尼式TMD。具体实现方式是：水平、竖向TMD的刚度构件分别采用悬臂梁与压簧,将水平向TMD整体置于压簧上面,从而构成竖向TMD的质量;导体板固定不动,使安装在TMD质量块的永磁体阵列随质量块竖向或水平方向运动,从而分别产生竖向与水平向的电磁涡流阻尼。研究结果表明：（1）电涡流阻尼可以很好地实现双向TMD装置的共享阻尼,且电涡流阻尼的大小可以很方便地调节;（2）采用双向TMD进行斜拉桥的风致振动控制减振效果良好,可行性强。     

Ambient vibration measurements on a cable-stayed bridge

An extensive programme of full-scale ambient vibration tests has been conducted to measure the dynamic response of a 542 m (centre span of 274 m) cable-stayed bridge—the Quincy Bayview Bridge in Illinois. A microcomputer-based system was used to collect and analyse the ambient vibration data. A total of 25 modal frequencies and associated mode shapes were identified for the deck structure within the frequency range of 0–2 Hz. Also, estimations were made for damping ratios. The experimental data clearly indicated the occurrence of many closely spaced modal frequencies and spatially complicated mode shapes. Most tower modes were found to be associated with the deck modes, implying a considerable interaction between the deck and tower structure. No detectable levels of motion were evident at the foundation support of the pier. The results of the ambient vibration survey were compared to modal frequencies and mode shapes computed using a three-dimensional finite element model of the bridge. For most modes, the analytic and experimental modal frequencies and mode shapes compare quite well, especially for the vertical modes. Based on the findings of this study, a linear elastic finite element model appears to be capable of capturing much of the complex dynamic behaviour of the bridge with very good accuracy, when compared to the low-level dynamic responses induced by ambient wind and traffic excitations.     

Influence of lateral-load-resisting system on the earthquake response of structures—a system identification study

Analysis and comparison of the dynamic responses of three well instrumented (with accelerographs) high-rise buildings shaken during the 1984 Morgan Hill earthquake are presented. The buildings examined in the present work are (i) the Town Park Towers Apartment building, a 10-storey, concrete shear wall building; (ii) the Great Western Savings and Loan building, a 10-storey building with concrete frames and shear walls; and (iii) the Santa Clara County Office building, a 13-storey, moment-resistant steel frame building. The structures are located within 2 km of each other and, as may be confirmed by visual inspection of the recorded seismograms, experienced similar ground motions. One-dimensional and three-dimensional linear structural models are fitted to the observations using the modal minimization method' for structural identification, in order to determine optimal estimates of the parameters of the dominant modes of the buildings. The time-varying character of these parameters over the duration of the response is also investigated. Comparison of the recorded earthquake response of the structures reveals that the type of lateral-load-resisting system has an important effect on the dynamic behaviour of the structures because it controls the spacing of the characteristic modes on the frequency axis. The Santa Clara County Office building has closely spaced natural frequencies and exhibits strong torsional response and modal coupling. Its dynamic behaviour is contrasted with that of the Great Western Savings and Loan building which has well separated natural frequencies and exhibits small torsional response and no modal coupling. Strong modal coupling causes a beating-type phenomenon and makes earthquake response of structures different from that envisioned by codes.     

An effective fixed-base model with classical normal modes for soil–structure interaction systems

To simplify the analysis of soil–structure interaction systems, various fixed-base models have recently been proposed by the author to efficiently represent the SSI system and have been shown to have good accuracy. However, the modified mass and damping matrices of these models do not hold the properties of symmetry and orthogonality. Difficulties may consequently be induced for these models in applying conventional computer codes to carry out dynamic analysis. In the present paper, this problem is further explored to establish a fixed-base model possessing classical normal modes. Formulated in the modal space, this fixed-base model is constructed through applying an iteration algorithm to incorporate the Gram–Schmidt orthogonalization process. The convergent real orthogonal mode vectors, natural frequencies, and modal damping ratios are directly determined for this model. It is demonstrated with a numerical example that this new fixed-base model retains excellent accuracy. Accordingly, the complicated SSI systems can be directly analyzed using conventional computer codes for structural dynamics with the fixed-base model developed in this study.     

Response spectrum analysis for non-classically damped linear system with multiple-support excitations

A new response spectrum method, which is named complex multiple-support response spectrum (CMSRS) method in this article, is developed for seismic analysis of non-classically damped linear system subjected to spatially varying multiple-supported ground motion. The CMSRS method is based on fundamental principles of random vibration theory and properly accounts for the effect of correlation between the support motions as well as between the modal displacement and velocity responses of structure, and provides an reasonable and acceptable estimate of the peak response in term of peak seismic ground motions and response spectra at the support points and the coherency function. Meanwhile, three new cross-correlation coefficients or cross covariance especially for the non-classically damped linear structures with multiple-supports excitations are derived under the same assumptions of the MSRS method of classically damped system. The CMSRS method is examined and compared to the results of time history analyses in two numerical examples of non-classically damped structures in consideration of the coherences of spatially variable ground motion. The results show that for non-classically damped structure, the cross terms representing the cross covariance between the pseudo-static and dynamic component are also quite small just as same as classically damped system. In addition, it is found that the usual way of neglecting all the off-diagonal elements in transformed damping matrix in modal coordinates in order to make the concerned non-classically damped structure to become remaining proportional damping property will bring some errors in the case of subjected to spatially excited inhomogeneous ground motion.