Evaluation of numerical time‐integration schemes for real‐time hybrid testing

We present a comparison of methods for the analysis of the numerical substructure in a real‐time hybrid test. A multi‐tasking strategy is described, which satisfies the various control and numerical requirements. Within this strategy a variety of explicit and implicit time‐integration algorithms have been evaluated. Fully implicit schemes can be used in fast hybrid testing via a digital sub‐step feedback technique, but it is shown that this approach requires a large amount of computation at each sub‐step, making real‐time execution difficult for all but the simplest models. In cases where the numerical substructure poses no harsh stability condition, it is shown that the Newmark explicit method offers advantages of speed and accuracy. Where the stability limit of an explicit method cannot be met, one of the several alternatives may be used, such as Chang's modified Newmark scheme or the α‐operator splitting method. Appropriate methods of actuator delay compensation are also discussed. Copyright © 2008 John Wiley & Sons, Ltd.     

Stability analysis for real‐time pseudodynamic and hybrid pseudodynamic testing with multiple sources of delay

Real‐time pseudodynamic (PSD) and hybrid PSD test methods are experimental techniques to obtain the response of structures, where restoring force feedback is used by an integration algorithm to generate command displacements. Time delays in the restoring force feedback from the physical test structure and/or the analytical substructure cause inaccuracies and can potentially destabilize the system. In this paper a method for investigating the stability of structural systems involved in real‐time PSD and hybrid PSD tests with multiple sources of delay is presented. The method involves the use of the pseudodelay technique to perform an exact mapping of fixed delay terms to determine the stability boundary. The approach described here is intended to be a practical one that enables the requirements for a real‐time testing system to be established in terms of system parameters when multiple sources of delay exist. Several real‐time testing scenarios with delay that include single degree of freedom (SDOF) and multi‐degree of freedom (MDOF) real‐time PSD/hybrid PSD tests are analyzed to illustrate the method. From the stability analysis of the real‐time hybrid testing of an SDOF test structure, delay‐independent stability with respect to either experimental or analytical substructure delay is shown to exist. The conditions that the structural properties must satisfy in order for delay‐independent stability to exist are derived. Real‐time hybrid PSD testing of an MDOF structure equipped with a passive damper is also investigated, where observations from six different cases related to the stability plane behavior are summarized. Throughout this study, root locus plots are used to provide insight and explanation of the behavior of the stability boundaries. Copyright © 2008 John Wiley & Sons, Ltd.     

Rosenbrock‐based algorithms and subcycling strategies for real‐time nonlinear substructure testing

In this paper, Rosenbrock‐based algorithms originally developed for real‐time testing of linear systems with dynamic substructuring are extended for use on nonlinear systems. With this objective in mind and for minimal overhead, both two‐ and three‐stages linearly implicit real‐time compatible algorithms were endowed with the Jacobian matrices requiring only one evaluation at the beginning of each time step. Moreover, these algorithms were improved with subcycling strategies. In detail, the paper briefly introduces Rosenbrock‐based L‐Stable Real‐Time (LSRT) algorithms together with linearly implicit and explicit structural integrators, which are now commonly used to perform real‐time tests. Then, the LSRT algorithms are analysed in terms of linearized stability with reference to an emulated spring pendulum, which was chosen as a nonlinear test problem, because it is able to exhibit a large and relatively slow nonlinear circular motion coupled to an axial motion that can be set to be stiff. The accuracy analysis on this system was performed for all the algorithms described. Following this, a coupled spring‐pendulum example typical of real‐time testing is analysed with respect to both stability and accuracy issues. Finally, the results of representative numerical simulations and real‐time substructure tests, considering nonlinearities both in the numerical and the physical substructure, are explored. These tests were used to demonstrate how the LSRT algorithms can be used for substructuring tests with strongly nonlinear components. Copyright © 2010 John Wiley & Sons, Ltd.     

Large‐scale real‐time hybrid simulation of a three‐story steel frame building with magneto‐rheological dampers

A series of large‐scale real‐time hybrid simulations (RTHSs) are conducted on a 0.6‐scale 3‐story steel frame building with magneto‐rheological (MR) dampers. The lateral force resisting system of the prototype building for the study consists of moment resisting frames and damped brace frames (DBFs). The experimental substructure for the RTHS is the DBF with the MR dampers, whereas the remaining structural components of the building including the moment resisting frame and gravity frames are modeled via a nonlinear analytical substructure. Performing RTHS with an experimental substructure that consists of the complete DBF enables the effects of member and connection component deformations on system and damper performance to be accurately accounted for. Data from these tests enable numerical simulation models to be calibrated, provide an understanding and validation of the in‐situ performance of MR dampers, and a means of experimentally validating performance‐based seismic design procedures for real structures. The details of the RTHS procedure are given, including the test setup, the integration algorithm, and actuator control. The results from a series of RTHS are presented that includes actuator control, damper behavior, and the structural response for different MR control laws. The use of the MR dampers is experimentally demonstrated to reduce the response of the structure to strong ground motions. Comparisons of the RTHS results are made with numerical simulations. Based on the results of the study, it is concluded that RTHS can be conducted on realistic structural systems with dampers to enable advancements in resilient earthquake resistant design to be achieved. Copyright © 2014 John Wiley & Sons, Ltd.     

Performance of a real‐time pseudodynamic test system considering nonlinear structural response

This paper presents the implementation details of a real‐time pseudodynamic test system that adopts an implicit time integration scheme. The basic configuration of the system is presented. Physical tests were conducted to evaluate the performance of the system and validate a theoretical system model that incorporates the dynamics and nonlinearity of a test structure and servo‐hydraulic actuators, control algorithm, actuator delay compensation methods, and the flexibility of an actuator reaction system. The robustness and accuracy of the computational scheme under displacement control errors and severe structural softening are examined with numerical simulations using the model. Different delay compensation schemes have been implemented and compared. One of the schemes also compensates for the deformation of an actuator reaction system. It has been shown that the test method is able to attain a good performance in terms of numerical stability and accuracy. However, it has been shown that test results obtained with this method can underestimate the inelastic displacement drift when severe strain softening develops in a test structure. This can be attributed to the fact that the numerical damping effect introduced by convergence errors becomes more significant as a structure softens. In a real‐time test, a significant portion of the convergence errors is caused by the time delay in actuator response. Hence, a softening structure demands higher precision in displacement control. Copyright © 2007 John Wiley & Sons, Ltd.     

Real‐time hybrid testing of a smart base isolation system

Real‐time hybrid testing is a very effective technique for evaluating the dynamic responses of rate‐dependent structural systems subjected to earthquake excitation. A smart base isolation system has been proposed by others using conventional low‐damping isolators and controllable damping devices such as magnetorheological (MR) dampers to achieve specified control target performance. In this paper, real‐time hybrid tests of a smart base isolation system are conducted. The simulation is for a base‐isolated two‐degrees‐of‐freedom building model where the superstructure and the low‐damping base isolator are numerically simulated, and the MR damper is physically tested. The target displacement obtained from the step‐by‐step integration of the numerical substructure is imposed on the MR damper, which is driven by three different control algorithms in real‐time. To compensate the actuator delay and improve the accuracy of the test, an adaptive phase‐lead compensator is implemented. The accuracy of each test is investigated by using the root mean square error and the tracking indicator. Experimental results demonstrate that the hybrid testing procedure using the proposed actuator compensation techniques is effective for investigating the control performance of the MR damper in a smart base isolation system. Copyright © 2013 John Wiley & Sons, Ltd.     

Real‐time hybrid testing using the unconditionally stable explicit CR integration algorithm

Real‐time hybrid testing combines experimental testing and numerical simulation, and provides a viable alternative for the dynamic testing of structural systems. An integration algorithm is used in real‐time hybrid testing to compute the structural response based on feedback restoring forces from experimental and analytical substructures. Explicit integration algorithms are usually preferred over implicit algorithms as they do not require iteration and are therefore computationally efficient. The time step size for explicit integration algorithms, which are typically conditionally stable, can be extremely small in order to avoid numerical stability when the number of degree‐of‐freedom of the structure becomes large. This paper presents the implementation and application of a newly developed unconditionally stable explicit integration algorithm for real‐time hybrid testing. The development of the integration algorithm is briefly reviewed. An extrapolation procedure is introduced in the implementation of the algorithm for real‐time testing to ensure the continuous movement of the servo‐hydraulic actuator. The stability of the implemented integration algorithm is investigated using control theory. Real‐time hybrid test results of single‐degree‐of‐freedom and multi‐degree‐of‐freedom structures with a passive elastomeric damper subjected to earthquake ground motion are presented. The explicit integration algorithm is shown to enable the exceptional real‐time hybrid test results to be achieved. Copyright © 2008 John Wiley & Sons, Ltd.     

Improved control algorithm for real‐time substructure testing

Real‐time substructure testing is a novel method of testing structures under dynamic loading. The complete structure is separated into two substructures, one of which is tested physically at large scale and in real time, so that time‐dependent non‐linear behaviour of the substructure is realistically represented. The second substructure represents the surrounding structure, which is modelled numerically. In the current formulation this numerical substructure is assumed to remain linear. The two substructures interact in real‐time so that the response of the complete structure, incorporating the non‐linear behaviour of the physical substructure, is accurately represented. This paper presents several improvements to the linear numerical modelling of substructures for use in explicit time‐stepping routines for real‐time substructure testing. An extrapolation of a first‐order‐hold discretization is used which increases the accuracy of the numerical model over more direct explicit methods. Additionally, an integral form of the equation of motion is used in order to reduce the effects of noise and to take into account variations of the input over a time‐step. In order to take advantage of this integral form, interpolation of the model output is performed in order to smooth the output. The improvements are demonstrated using a series of substructure tests on a simple portal frame. While the testing approach is suitable for cases in which the physical substructure behaves non‐linearly, the results presented here are for fully linear systems. This enables comparisons to be made with analytical solutions, as well as with the results of tests based on the central difference method. Copyright © 2001 John Wiley & Sons, Ltd.     

Equivalent force control method for substructure pseudo‐dynamic test of a full‐scale masonry structure

The effectiveness of equivalent force control (EFC) method has been experimentally validated through hybrid tests with simple specimens. In this paper, the EFC method is applied for the MDOF pseudo‐dynamic substructure tests in which a three‐storey frame‐supported reinforced concrete masonry shear wall with full scale is chosen as physical substructure. The effects of equivalent force controller parameters on the response performance are studied. Analytical expressions for the controller parameter ranges are derived to avoid response overshooting or oscillation and are verified by numerical simulation. The controller parameters are determined based on analytical and numerical studies and used in the actual full‐scale pseudo‐dynamic test. The test results show good tracking performance of EFC, which indicates a successful test. Copyright © 2013 John Wiley & Sons, Ltd.     

Stability of central difference method for dynamic real‐time substructure testing

This paper studies the stability of the central difference method (CDM) for real‐time substructure test considering specimen mass. Because the standard CDM is implicit in terms of acceleration, to avoid iteration, an explicit acceleration formulation is assumed for its implementation in real‐time dynamic substructure testing. The analytical work shows that the stability of the algorithm decreases with increasing specimen mass if the experimental substructure is a pure inertia specimen. The algorithm becomes unstable however small the time integration interval is, when the mass of specimen equal or greater than that of its numerical counterpart. For the case of dynamic specimen, the algorithm is unstable when there is no damping in the whole test structure; a damping will make the algorithm stable conditionally. Part of the analytical results is validated through an actual test. Copyright © 2009 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.     

等效力控制方法在拟动力试验中的应用

为了避免用复杂的迭代求解拟动力试验隐式逐步积分方法的非线性运动方程,作者提出了一种用反馈控制求解的方法—等效力控制方法。这篇文章首先介绍等效力控制方法的原理;然后通过数值模拟和试验验证此方法的有效性。数值模拟和试验的结果都表明,当选用合理的等效力控制器参数时,此方法可以取得非常好的稳定性和精度。数值模拟的结果还表明尽管等效力控制方法是以力作为反馈控制的对象,此方法能有效完成对具有负刚度试件的加载。     

Seismic performance evaluation of high voltage disconnect switches using real‐time hybrid simulation: I. System development and validation

This paper presents the development and validation of a real‐time hybrid simulation (RTHS) system for efficient dynamic testing of high voltage electrical vertical‐break disconnect switches. The RTHS system consists of the computational model of the support structure, the physical model of the insulator post, a small shaking table, a state‐of‐the‐art controller, a data acquisition system and a digital signal processor. Explicit Newmark method is adopted for the numerical integration of the governing equations of motion of the hybrid structure, which consists of an insulator post (experimental substructure) and a spring‐mass‐dashpot system representing the support structure (analytical substructure). Two of the unique features of the developed RTHS system are the application of an efficient feed‐forward error compensation scheme and the ability to use integration time steps as small as 1 ms. After the development stage, proper implementation of the algorithm and robustness of the measurements used in the calculations are verified. The developed RTHS system is further validated by comparing the RTHS test results with those from a conventional shaking table test. A companion paper presents and discusses a parametric study for a variety of geometrical and material configurations of these switches using the developed RTHS system. Copyright © 2013 John Wiley & Sons, Ltd.     

Implementation and application of the unconditionally stable explicit parametrically dissipative KR‐α method for real‐time hybrid simulation

In real‐time hybrid simulations (RTHS) that utilize explicit integration algorithms, the inherent damping in the analytical substructure is generally defined using mass and initial stiffness proportional damping. This type of damping model is known to produce inaccurate results when the structure undergoes significant inelastic deformations. To alleviate the problem, a form of a nonproportional damping model often used in numerical simulations involving implicit integration algorithms can be considered. This type of damping model, however, when used with explicit integration algorithms can require a small time step to achieve the desired accuracy in an RTHS involving a structure with a large number of degrees of freedom. Restrictions on the minimum time step exist in an RTHS that are associated with the computational demand. Integrating the equations of motion for an RTHS with too large of a time step can result in spurious high‐frequency oscillations in the member forces for elements of the structural model that undergo inelastic deformations. The problem is circumvented by introducing the parametrically controllable numerical energy dissipation available in the recently developed unconditionally stable explicit KR‐α method. This paper reviews the formulation of the KR‐α method and presents an efficient implementation for RTHS. Using the method, RTHS of a three‐story 0.6‐scale prototype steel building with nonlinear elastomeric dampers are conducted with a ground motion scaled to the design basis and maximum considered earthquake hazard levels. The results show that controllable numerical energy dissipation can significantly eliminate spurious participation of higher modes and produce exceptional RTHS results. Copyright © 2014 John Wiley & Sons, Ltd.     

磁流变阻尼器的半主动控制及实时混合模拟试验研究

磁流变阻尼器作为一种比较典型的半主动控制元件,具有构造简单、响应速度快、耐久性好、阻尼力大且连续可调等优点。即使地震中能源中断,磁流变阻尼器仍可以作为被动耗能装置继续工作发挥作用,可靠性高。设计合理有效的磁流变阻尼器半主动控制方法,对于整体结构的减震效果尤其重要。提出一种改进的磁流变阻尼器的半主动控制策略-改进的Bang-Bang控制策略,对装有磁流变阻尼器的减震控制3层框架结构进行了一系列的实时混合模拟试验,对多种半主动控制方法下的振动控制效果进行试验分析。试验结果表明:磁流变阻尼器对框架结构的减震效果显著,并验证了提出的磁流变阻尼器半主动控制策略的有效性。     

Comparison between real‐time dynamic substructuring and shake table testing techniques for nonlinear seismic applications

Results from real‐time dynamic substructuring (RTDS) tests are compared with results from shake table tests performed on a two‐storey steel building structure model. At each storey, the structural system consists of a cantilevered steel column resisting lateral loads in bending. In two tests, a slender diagonal tension‐only steel bracing member was added at the first floor to obtain an unsymmetrical system with highly variable stiffness. Only the first‐storey structural components were included in the RTDS test program and a Rosenbrock‐W linearly implicit integration scheme was adopted for the numerical solution. The tests were performed under seismic ground motions exhibiting various amplitude levels and frequency contents to develop first and second mode‐dominated responses as well as elastic and inelastic responses. A chirp signal was also used. Coherent results were obtained between the shake table and the RTDS testing techniques, indicating that RTDS testing methods can be used to successfully reproduce both the linear and nonlinear seismic responses of ductile structural steel seismic force resisting systems. The time delay introduced by actuator‐control systems was also studied and a novel adaptive compensation scheme is proposed. Copyright © 2010 John Wiley & Sons, Ltd.     

Operator‐splitting method for real‐time substructure testing

It has been shown that the operator‐splitting method (OSM) provides explicit and unconditionally stable solutions for quasi‐static pseudo‐dynamic substructure testing. However, the OSM provides only an explicit target displacement but not an explicit target velocity, so that it is essentially an implicit method for real‐time substructure testing (RST) when the velocity‐dependent restoring force is considered. This paper proposes a target velocity formulation based on the forward difference of the predicted displacements so as to render the OSM explicit for RST. The stability and accuracy of the resulting OSM‐RST algorithm are investigated. It is shown that the OSM‐RST is unconditionally stable so long as the non‐linear stiffness and damping are of the softening type (i.e. the tangent stiffness and damping never exceed the initial values). The stability of the OSM‐RST for structures with infinite tangent damping coefficient or stiffness is also proved, and the stability of the method for MDOF structures with a non‐classical damping matrix is demonstrated by an energy criterion. The effects of actuator delay and compensation are analysed based on the bilinear approximation of the actuator step response. Experiments on damped SDOF and MDOF structures verify that the stability of the OSM‐RST is preserved when the experimental substructure generates velocity‐dependent reaction forces, whereas the stability of real‐time substructure tests based on the central difference method is worsened by the damping of the specimen. Copyright © 2005 John Wiley & Sons, Ltd.     

Parameter identification for on‐line model updating in hybrid simulations using a gradient‐based method

To improve the efficiency of model fitting, parameter identification techniques have been actively investigated. Recently, the applications of parameter identification migrated from off‐line model fitting to on‐line model updating. The objective of this study is to develop a gradient‐based method for model updating to advance hybrid simulation also called hybrid test. A novel modification of the proposed method, which can reduce the number of design variables to improve the identification efficiency, is illustrated in detail. To investigate the model updating, simulated hybrid tests were conducted with a 5‐story steel frame equipped with buckling‐restrained braces (BRBs) utilized in the shaking table tests conducted in E‐Defense in Japan in 2009. The calibrated analytical model that was verified with the test results can serve as the reference model. In the simulated hybrid tests, the physical BRB substructure is numerically simulated by utilizing a truss element with the 2‐surface model identical to the part of the reference model. Such numerical verification allows simulation of measurement errors for investigation on the performance of the proposed method. Moreover, the feasibility of sharing the identified parameter values, which were obtained from the physical substructure responses, with the relevant numerical models is also verified with the artificial component responses derived from the physical experiments.     

Adaptive multi‐rate interface: development and experimental verification for real‐time hybrid simulation

Real‐time hybrid simulation (RTHS) is a powerful cyber‐physical technique that is a relatively cost‐effective method to perform global/local system evaluation of structural systems. A major factor that determines the ability of an RTHS to represent true system‐level behavior is the fidelity of the numerical substructure. While the use of higher‐order models increases fidelity of the simulation, it also increases the demand for computational resources. Because RTHS is executed at real‐time, in a conventional RTHS configuration, this increase in computational resources may limit the achievable sampling frequencies and/or introduce delays that can degrade its stability and performance. In this study, the Adaptive Multi‐rate Interface rate‐transitioning and compensation technique is developed to enable the use of more complex numerical models. Such a multi‐rate RTHS is strictly executed at real‐time, although it employs different time steps in the numerical and the physical substructures while including rate‐transitioning to link the components appropriately. Typically, a higher‐order numerical substructure model is solved at larger time intervals, and is coupled with a physical substructure that is driven at smaller time intervals for actuator control purposes. Through a series of simulations, the performance of the AMRI and several existing approaches for multi‐rate RTHS is compared. It is noted that compared with existing methods, AMRI leads to a smaller error, especially at higher ratios of sampling frequency between the numerical and physical substructures and for input signals with high‐frequency content. Further, it does not induce signal chattering at the coupling frequency. The effectiveness of AMRI is also verified experimentally. Copyright © 2016 John Wiley & Sons, Ltd.     

基于钢筋混凝土足尺柱拟静力试验的离线模型更新混合试验方法

离线模型更新混合试验对构件拟静力数据进行恢复力模型参数识别,并更新数值子结构中相应构件的模型参数来提高混合试验精度,但该方法尚缺少真实试验的验证。本文基于课题组开展的足尺RC柱拟静力试验,取恢复力模型为集中塑性铰Ibarra-Medina-Krawinkler（IMK）模型,进行框架结构离线模型更新混合试验研究。结果表明,当物理子结构取为RC足尺柱时,离线模型更新混合试验能获得接近于真实试验情况下结构的地震响应,从而对该方法的有效性进行了试验验证。利用IMK经验公式,将真实试验模型参数识别值按轴压比进行对照修正,应用于不同层数的框架结构地震响应模拟,实现了试验数据的重复利用。