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.     

Floor response spectra with structure–equipment interaction effects by a mode synthesis approach

A mode synthesis-based direct approach is presented to calculate seismic response of equipment supported on structures. The approach incorporates the effect of the dynamic interaction between the equipment and the supporting structure. The modal properties of the combined structure–equipment system are obtained by synthesizing the modal properties of the individual structures. The seismic input defined in terms of smoothed ground response spectra can be directly utilized in this approach. Both heavy and light equipment can be considered by the approach equally effectively. Numerical examples demonstrating the effectiveness of the proposed approach are presented.     

Approximate soil–structure interaction analysis by a perturbation approach: The case of soft soils

An approximate solution of the classical eigenvalue problem governing the vibrations of a relatively stiff structure on a soft elastic soil is derived through the application of a perturbation analysis. The full solution is obtained as the sum of the solution for an unconstrained elastic structure and small perturbing terms related to the ratio of the stiffness of the soil to that of the superstructure. The procedure leads to approximate analytical expressions for the system frequencies, modal damping ratios and participation factors for all system modes that generalize those presented earlier for the case of stiff soils. The resulting approximate expressions for the system modal properties are validated by comparison with the corresponding quantities obtained by numerical solution of the eigenvalue problem for a nine-story building. The accuracy of the proposed approach and of the classical normal mode approach is assessed through comparison with the exact frequency response of the test structure.     

Approximate soil-structure interaction analysis by a perturbation approach: The case of stiff soils

An approximate solution of the classical eigenvalue problem governing the vibrations of a structure on an elastic soil is derived through the application of a perturbation analysis. For stiff soils, the full solution is obtained as the sum of the solution for a rigid-soil and small perturbing terms related to the inverse of the soil shear modulus. The procedure leads to approximate analytical expressions for the system frequencies, modal damping ratios and participation factors for all system modes that generalize those presented by other authors for the fundamental mode. The resulting approximate expressions for the system modal properties are validated by comparison with the corresponding quantities obtained by numerical solution of the eigenvalue problem for a nine-story building. The accuracy of the proposed approach and of the classical normal mode approach is assessed through comparison with the exact frequency response of the test structure.     

System identification of linear structures based on Hilbert–Huang spectral analysis. Part 2: Complex modes

A method, based on the Hilbert–Huang spectral analysis, has been proposed by the authors to identify linear structures in which normal modes exist (i.e., real eigenvalues and eigenvectors). Frequently, all the eigenvalues and eigenvectors of linear structures are complex. In this paper, the method is extended further to identify general linear structures with complex modes using the free vibration response data polluted by noise. Measured response signals are first decomposed into modal responses using the method of Empirical Mode Decomposition with intermittency criteria. Each modal response contains the contribution of a complex conjugate pair of modes with a unique frequency and a damping ratio. Then, each modal response is decomposed in the frequency–time domain to yield instantaneous phase angle and amplitude using the Hilbert transform. Based on a single measurement of the impulse response time history at one appropriate location, the complex eigenvalues of the linear structure can be identified using a simple analysis procedure. When the response time histories are measured at all locations, the proposed methodology is capable of identifying the complex mode shapes as well as the mass, damping and stiffness matrices of the structure. The effectiveness and accuracy of the method presented are illustrated through numerical simulations. It is demonstrated that dynamic characteristics of linear structures with complex modes can be identified effectively using the proposed method. Copyright © 2003 John Wiley & Sons, Ltd.     

TMD结构基于双过滤白噪声随机地震激励的系列响应的简明解

针对既有方法在分析TMD结构基于双过滤白噪声激励下结构响应的解表达式复杂而导致计算效率低的问题,提出了一种简明封闭解法。首先,利用双过滤白噪声谱的滤波方程与TMD结构的地震动方程联立,可将TMD结构基于复杂的双过滤白噪声激励准确的表示为易于求解的运动方程;其次,基于复模态法获得TMD耗能结构位移、层间位移的系列响应的复特征值及复模态参与系数;然后基于随机振动理论获得了TMD结构随机地震动系列响应（相对于地面绝对位移和结构层间位移）的功率谱统一形式的二次正交解,进而获得了TMD结构系列随机响应的0-2阶谱矩和方差的简明封闭解。最后研究了基于首超破坏准、Markov过程假设及串联失效模式的TMD结构的体系动力可靠度。通过一算例分析,表明了本文方法的正确性和高效性。因此,本文方法可用于各类线性结构基于复杂的随机地震动响应的分析及其动力可靠度计算。     

Decoupled dynamic analysis of combined systems by iterative determination of interface accelerations

A dynamic analysis technique is presented that can be used to determine the response of a discrete model of a large linear structural system composed of multiple substructures. The technique circumvents the costly computation of the modal characteristics of the combined system. This is accomplished by relying on a predictor-corrector scheme to converge iteratively to the interface accelerations of the combined system, while the equations of motions of the individual structures are integrated separately. In this regard, the temporal slopes of the interface accelerations (jerks) are computed at each time point of integration to predict the interface accelerations at the next time point. The proposed technique is exemplified by conducting a space shuttle landing loads analysis; the obtained numerical data demonstrate its reliability and efficiency.     

A Rosenbrock‐W method for real‐time dynamic substructuring and pseudo‐dynamic testing

A variant of the Rosenbrock‐W integration method is proposed for real‐time dynamic substructuring and pseudo‐dynamic testing. In this variant, an approximation of the Jacobian matrix that accounts for the properties of both the physical and numerical substructures is used throughout the analysis process. Only an initial estimate of the stiffness and damping properties of the physical components is required. It is demonstrated that the method is unconditionally stable provided that specific conditions are fulfilled and that the order accuracy can be maintained in the nonlinear regime without involving any matrix inversion while testing. The method also features controllable numerical energy dissipation characteristics and explicit expression of the target displacement and velocity vectors. The stability and accuracy of the proposed integration scheme are examined in the paper. The method has also been verified through hybrid testing performed of SDOF and MDOF structures with linear and highly nonlinear physical substructures. The results are compared with those obtained from the operator splitting method. An approach based on the modal decomposition principle is presented to predict the potential effect of experimental errors on the overall response during testing. Copyright © 2009 John Wiley & Sons, Ltd.     

Modal identification of self-adjoint distributed-parameter systems

A new method is introduced for the identification of modal quantities of self-adjoint distributed-parameter systems. The method uses the temporal and spatial orthogonality properties of distributed-parameter systems to form a pseudo-Rayleigh quotient. The stationary values of the pseudo-Rayleigh quotient can be determined by the solution of an eigenvalue problem, where the eigenvalues and eigenfunctions provide the frequencies and natural modes of the distributed-parameter system. By way of formulation, the method is insensitive to zero mean measurement noise and is applicable to continuous as well as discrete systems. Numerical examples are presented in which the modal quantities of a simply-supported beam and a discrete model of a membrane are identified. The results are compared with another modal identification technique, namely, the Ibrahim time domain method.     

Implementation of online model updating in hybrid simulation

Hybrid simulation combines numerical and experimental methods for cost‐effective, large‐scale testing of structures under simulated earthquake loading. Structural system level response can be obtained by expressing the equation of motion for the combined experimental and numerical substructures, and solved using time‐stepping integration similar to pure numerical simulations. It is often assumed that a reliable model exists for the numerical substructures while the experimental substructures correspond to parts of the structure that are difficult to model. A wealth of data becomes available during the simulation from the measured experiment response that can be used to improve upon the numerical models, particularly if a component with similar structural configuration and material properties is being tested and subjected to a comparable load pattern. To take advantage of experimental measurements, a new hybrid test framework is proposed with an updating scheme to update the initial modeling parameters of the numerical model based on the instantaneously‐measured response of the experimental substructures as the test progresses. Numerical simulations are first conducted to evaluate key algorithms for the selection and calibration of modeling parameters that can be updated. The framework is then expanded to conduct actual hybrid simulations of a structural frame model including a physical substructure in the laboratory and a numerical substructure that is updated during the tests. The effectiveness of the proposed framework is demonstrated for a simple frame structure but is extendable to more complex structural behavior and models. Copyright © 2013 John Wiley & Sons, Ltd.     

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‐based decoupling criterion for multiply‐supported secondary systems

It is often infeasible to carry out coupled analyses of multiply‐supported secondary systems for earthquake excitations. ‘Approximate’ decoupled analyses are then resorted to, unless the response errors due to those are significantly high. This study proposes a decoupling criterion to identify such cases where these errors are likely to be larger than an acceptable level. The proposed criterion is based on the errors in the primary system response due to decoupling and has been obtained by assuming (i) the input excitation to be an ideal white noise process, (ii) cross‐modal correlation to be negligible, and (iii) the combined system to be classically damped. It uses the modal properties of the undamped combined system, and therefore, a perturbation approach has been formulated to determine the combined system properties in case of light to moderately heavy secondary systems. A numerical study has been carried out to illustrate the accuracy achieved with the proposed perturbation formulation. The proposed decoupling criterion has been validated with the help of two example primary‐secondary systems and four example excitation processes. Copyright © 2002 John Wiley & Sons, Ltd.     

Novel coupling Rosenbrock‐based algorithms for real‐time dynamic substructure testing

Real‐time testing with dynamic substructuring is a novel experimental technique capable of assessing the behaviour of structures subjected to dynamic loadings including earthquakes. The technique involves recreating the dynamics of the entire structure by combining an experimental test piece consisting of part of the structure with a numerical model simulating the remainder of the structure. These substructures interact in real time to emulate the behaviour of the entire structure. Time integration is the most versatile method for analysing the general case of linear and non‐linear semi‐discretized equations of motion. In this paper we propose for substructure testing, L‐stable real‐time (LSRT) compatible integrators with two and three stages derived from the Rosenbrock methods. These algorithms are unconditionally stable for uncoupled problems and entail a moderate computational cost for real‐time performance. They can also effectively deal with stiff problems, i.e. complex emulated structures for which solutions can change on a time scale that is very short compared with the interval of time integration, but where the solution of interest changes on a much longer time scale. Stability conditions of the coupled substructures are analysed by means of the zero‐stability approach, and the accuracy of the novel algorithms in the coupled case is assessed in both the unforced and forced conditions. LSRT algorithms are shown to be more competitive than popular Runge–Kutta methods in terms of stability, accuracy and ease of implementation. Numerical simulations and real‐time substructure tests are used to demonstrate the favourable properties of the proposed algorithms. Copyright © 2007 John Wiley & Sons, Ltd.     

A new simplified approach for assessing nonlinear seismic response of plan-asymmetric structures considering soil-structure interaction

An innovative approximate method is presented to consider the plan asymmetry, nonlinear structural behaviour and soil-structure interaction (SSI) effects simultaneously. The proposed method so-called Flexible base 2DMPA (F2MPA) is an extension of 2 degrees of freedom modal pushover analysis (2DMPA) approach to consider foundation flexibility in seismic response analysis of plan asymmetric structures which itself were developed based on Uncoupled Modal Response History Analysis method for inelastic fixed-base asymmetric structures. In F2MPA for each mode shape using 2DMPA procedure, the elastic and inelastic properties of 2DOF modal systems corresponding to the fixed-base structure are initially derived. Then in each time step, displacements and inelastic restoring forces of the superstructure are computed from modal equations of the flexibly-supported structure. In each time step, the nonlinear secant stiffness matrix corresponding to the n-th MDOF modal equations of soil-structure system is updated using the corresponding modal 2DOF system of fixed-base structure. To update the transformed modal stiffness matrix of the SSI system, this matrix is partitioned and it is assumed that the non-linear variation of the superstructure can be estimated from the variation of modal stiffness matrix of the fixed-base structure. Accuracy of the proposed method was verified on an 8-story asymmetric-plan building under different seismic excitations. The results obtained from F2MPA method were compared with those obtained by nonlinear response history analysis of the asymmetric soil-structure system as a reference response. It was shown that the proposed approach could predict the results of the nonlinear time history analysis with a good accuracy. The main advantage of F2MPA is that this method is much less time-consuming and useful for the practical aims such as massive analysis of a nonlinear structure under different records with multiple intensity levels.     

Optimal design of viscoelastic dampers using eigenvalue assignment

In this study a procedure for determining the optimum size and location of viscoelastic dampers is proposed using the eigenvalue assignment technique. Natural frequencies and modal damping ratios, required to realize a given target response, are determined first by the convex model. Then the desired dynamic structural properties are realized by optimally distributing the damping and stiffness coefficients of viscoelastic dampers using non‐linear programming based on the gradient of eigenvalues. This optimization method provides information on the optimal location as well as the magnitude of the damper parameters. The proposed procedure is applied to the retrofit of a 10‐story shear frame, and to a three‐dimensional structure with an asymmetric plan. The analysis results confirm that the responses of model structures retrofitted by the proposed method correspond well with the given target response. Copyright © 2003 John Wiley & Sons, Ltd.     

Damage detection of base‐isolated buildings using multi‐input multi‐output subspace identification

A damage detection algorithm of structural health monitoring systems for base‐isolated buildings is proposed. The algorithm consists of the multiple‐input multiple‐output subspace identification method and the complex modal analysis. The algorithm is applicable to linear and non‐linear systems. The story stiffness and damping as damage indices of a shear structure are identified by the algorithm. The algorithm is further tuned for base‐isolated buildings considering their unique dynamic characteristics by simplifying the systems to single‐degree‐of‐freedom systems. The isolation layer and the superstructure of a base‐isolated building are treated as separate substructures as they are distinctly different in their dynamic properties. The effectiveness of the algorithm is evaluated through the numerical analysis and experiment. Finally, the algorithm is applied to the existing 7‐story base‐isolated building that is equipped with an Internet‐based monitoring system. Copyright © 2004 John Wiley & Sons, Ltd.     

Response spectrum techniques for asymmetric buildings

A scheme is proposed to calculate the effect of torsion on each lateral load resisting element of asymmetrical buildings in the context of the response spectrum technique. The scheme consists of: (i) Obtain the modal shear and torque on the building by the response spectrum technique, (ii) Compute the total modal shear forces on each frame by resolving the modal shear and torque on the building according to principles of structural mechanics. The shears on each frame due to the lateral load effect and torsional effect are combined algebraically, (iii) Obtain the total shear force on each frame by combining the total modal shears on that frame in a root sum square manner. Since the proper phase relationship between the lateral load effect and torsional effect is accounted for on a modal basis, it is believed that the proposed scheme provides a more realistic load estimate on the frames than the conventional approach. An example of a simplified mono-symmetrical frame structure is chosen to illustrate the accuracy of the proposed scheme, using dynamic time-history analysis as a standard for comparison.     

Output-only identification of the modal and physical properties of structures using free vibration response

The viability of a complete structural characterization of civil structures is explored and discussed. In particular, the identification of modal (i.e. natural frequencies, damping ratios and modal shapes) and physical properties (i.e. mass and stiffness) using only the structure’s free decay response is studied. To accomplish this, modal analysis from free vibration response only (MAFVRO) and mass modification (MM) methodologies are engaged along with Wavelet based techniques for optimal signal processing and modal reconstruction. The methodologies are evaluated using simulated and experimental data. The simulated data are extracted from a simple elastic model of a 5 story shear building and from a more realistic nonlinear model of a RC frame structure. The experimental data are gathered from shake table test of a 2-story scaled shear building. Guidelines for the reconstruction procedure from the data are proposed as the quality of the identified properties is shown to be governed by adequate selection of the frequency bands and optimal modal shape reconstruction. Moreover, in cases where the structure has undergone damage, the proposed identification scheme can also be applied for preliminary assessment of structural health.     

Eigenproperties of non-classically damped primary structure and equipment systems by a perturbation approach

Closed-form expressions are obtained to calculate the approximate complex eigenvalues and eigenvectors of a system composed of a non-classically damped primary structure and a single degree of freedom oscillator. The expressions are obtained through a systematic second order perturbation analysis of a transformed eigenvalue problem of the combined system. The possibility of tuning between the structure and equipment is considered. The dynamic properties of the combined system are derived in terms of the complex eigenvalues and eigenvectors of the supporting structure and the frequency, mass and damping ratio of the equipment. Examples demonstrating the accuracy of the expressions for the eigenvalues and eigenvectors are presented. These eigenproperties are used for generation of floor response spectra for non-classically damped structures to incorporate the dynamic interaction effects between the structure and equipment.     

Bayesian spectral density approach for modal updating using ambient data

The problem of identification of the modal parameters of a structural model using measured ambient response time histories is addressed. A Bayesian spectral density approach (BSDA) for modal updating is presented which uses the statistical properties of a spectral density estimator to obtain not only the optimal values of the updated modal parameters but also their associated uncertainties by calculating the posterior joint probability distribution of these parameters. Calculation of the uncertainties of the identified modal parameters is very important if one plans to proceed with the updating of a theoretical finite element model based on modal estimates. It is found that the updated PDF of the modal parameters can be well approximated by a Gaussian distribution centred at the optimal parameters at which the posterior PDF is maximized. Examples using simulated data are presented to illustrate the proposed method. Copyright © 2001 John Wiley & Sons, Ltd.