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.     

Real‐time hybrid experiments with Newmark integration,MCSmd outer‐loop control and multi‐tasking strategies

Real‐time hybrid testing is a promising technique for experimental structural dynamics, in which the structure under consideration is split into a physical test of key components and a numerical model of the remainder. The physical test and numerical analysis proceed in parallel, in real time, enabling testing of critical elements at large scale and at the correct loading rate. To date most real‐time hybrid tests have been restricted to simple configurations and have used approximate delay compensation schemes. This paper describes a real‐time hybrid testing approach in which non‐linearity is permitted in both the physical and numerical models, and in which multiple interfaces between physical and numerical substructures can be accommodated, even when this results in very stiff coupling between actuators. This is achieved using a Newmark explicit numerical solver, an advanced adaptive controller known as MCSmd and a multi‐tasking strategy. The approach is evaluated through a series of experiments on discrete mass–spring systems. Copyright © 2006 John Wiley & Sons, Ltd.     

Stability and accuracy analysis of outer loop dynamics in real‐time pseudodynamic testing of SDOF systems

Real‐time pseudodynamic (PSD) testing is an experimental technique for evaluating the dynamic behaviour of a complex structure. During the test, when the targeted command displacements are not achieved by the test structure, or a delay in the measured restoring forces from the test structure exists, the reliability of the testing method is impaired. The stability and accuracy of real‐time PSD testing in the presence of amplitude error and a time delay in the restoring force is presented. Systems consisting of an elastic single degree of freedom (SDOF) structure with load‐rate independent and dependent restoring forces are considered. Bode plots are used to assess the effects of amplitude error and a time delay on the steady‐state accuracy of the system. A method called the pseudodelay technique is used to derive the exact solution to the delay differential equation for the critical time delay that causes instability of the system. The solution is expressed in terms of the test structure parameters (mass, damping, stiffness). An error in the restoring force amplitude is shown to degrade the accuracy of a real‐time PSD test but not destabilize the system, while a time delay can lead to instability. Example calculations are performed for determining the critical time delay, and numerical simulations with both a constant delay and variable delay in the restoring force are shown to agree well with the stability limit for the system based on the critical time delay solution. The simulation models are also used to investigate the effects of a time delay in the PSD test of an inelastic SDOF system. The effect of energy dissipation in an inelastic structure increases the limit for the critical time delay, due to the energy removed from the system by the energy dissipation. 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.     

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 of MDOF real‐time dynamic hybrid testing systems using the discrete‐time root locus technique

The time delay resulting from the servo hydraulic systems can potentially destabilize the real‐time dynamic hybrid testing (RTDHT) systems. In this paper, the discrete‐time root locus technique is adopted to investigate the delay‐dependent stability performance of MDOF RTDHT systems. Stability analysis of an idealized two‐story shear frame with two DOFs is first performed to illustrate the proposed method. The delay‐dependent stability condition is presented for various structural properties, time delay, and integration time steps. Effects of delay compensation methods on stability are also investigated. Then, the proposed method is applied to analyze the delay‐dependent stability of a single shaking table RTDHT system with an 18‐DOF finite element numerical substructure, and corresponding RTDHTs are carried out to verify the theoretical results. Furthermore, the stability behavior of a finite element RTDHT system with two physical substructures, loaded by twin shaking tables, is theoretically and experimentally investigated. All experimental results convincingly demonstrate that the delay‐dependent stability analysis on the basis of the discrete‐time root locus technique is feasible. Copyright © 2014 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 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.     

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.     

Dual compensation strategy for real‐time hybrid testing

Real‐time hybrid testing is an experimental technique for evaluating the dynamic responses of structural systems under seismic loading. Servo‐hydraulic actuators, by nature, induce inevitable time delay between the command and the achieved displacements. This delay would lead to incorrect test results and even cause instability of the system; therefore, delay compensation is critical for stability and accuracy of hybrid simulations of structural dynamic response. In this paper, a dual delay compensation strategy is proposed by a combination of a phase lead compensator and a restoring force compensator. An outer‐loop feed‐forward phase lead compensator is derived by introducing the inverse model in the z domain. The adaptive law based on the gradient algorithm is used to estimate the system delay in the format of parametric model during the test. It is shown mathematically that the parameter in the delay estimator is guaranteed to converge. The restoring force compensator is adopted to improve the accuracy of experimental results especially when the structure is subjected to high frequency excitations. Finally, analytical simulations of an inelastic SDOF structure are conducted to investigate the feasibility of the proposed strategy. The accuracy of the dual compensation strategy is demonstrated through several shaking table tests. Copyright © 2012 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.     

Kinematic transformations for planar multi‐directional pseudodynamic testing

The pseudodynamic (PSD) test method imposes command displacements to a test structure for a given time step. The measured restoring forces and displaced position achieved in the test structure are then used to integrate the equations of motion to determine the command displacements for the next time step. Multi‐directional displacements of the test structure can introduce error in the measured restoring forces and displaced position. The subsequently determined command displacements will not be correct unless the effects of the multi‐directional displacements are considered. This paper presents two approaches for correcting kinematic errors in planar multi‐directional PSD testing, where the test structure is loaded through a rigid loading block. The first approach, referred to as the incremental kinematic transformation method, employs linear displacement transformations within each time step. The second method, referred to as the total kinematic transformation method, is based on accurate nonlinear displacement transformations. Using three displacement sensors and the trigonometric law of cosines, this second method enables the simultaneous nonlinear equations that express the motion of the loading block to be solved without using iteration. The formulation and example applications for each method are given. Results from numerical simulations and laboratory experiments show that the total transformation method maintains accuracy, while the incremental transformation method may accumulate error if the incremental rotation of the loading block is not small over the time step. A procedure for estimating the incremental error in the incremental kinematic transformation method is presented as a means to predict and possibly control the error. Copyright © 2009 John Wiley & Sons, Ltd.     

Pseudo‐dynamic testing of a concrete gravity dam

Inspired from the simplified single degree of freedom modeling approach used in the preliminary design of concrete gravity dams, a pseudo‐dynamic testing method was devised for the seismic testing of a concrete gravity dam section. The test specimen was a 1/75 scaled section of the 120‐m‐high monolith of the Melen Dam, one of the highest concrete gravity dams to be built in Turkey. The single degree of freedom idealization of the dam section was validated in the first stage of the study using numerical simulations including the dam–reservoir interaction. Afterwards, pseudo‐dynamic testing was conducted on the specimen using three ground motions corresponding to different hazard levels. Lateral displacement and base shear demands were measured. The crack propagation at the base of the dam was monitored with the measurement of the crack widths and the base sliding displacements. After the pseudo‐dynamic loading, a static pushover test was conducted to determine the reserve capacity of the test specimen. Despite major cracking at the base of the monolith, neither significant sliding nor a stability problem that might jeopardize the stability of the dam was observed. Copyright © 2015 John Wiley & Sons, Ltd.     

On the acceleration‐based adaptive inverse control of shaking tables

Accurate reproduction of time series with diverse frequency characteristics is a central issue in structural testing. This is true not only for simple experimental tests performed by reaction walls or shaking tables but also for more sophisticated ones, such as hybrid testing. Especially in the latter case, where actual feedback from an ongoing test is used in the calculation of the next excitation value, any possible mismatch may be fatal for both the validity of the test and the safety. The objective of this study is to propose a framework for the adaptive inverse control of shaking tables, which succeeds in this matching to a certain degree. By formulating a critical set of design specifications that correspond to safety, implementation, robustness and ease of use, the conducted research results in a design that is based on a modified version of the filtered‐X algorithm with very competitive features. These are the following: (i) default operation in hard real‐time and acceleration mode; (ii) very low hardware requirements; (iii) effective cancelation of the shaking table's dynamics; and (iv) robustness against specimen dynamics. For its practical evaluation, the method is applied to shaking table waveform replication tests under the installation of an approximately linear specimen of sufficiently high mass and complex geometry. The results are promising and suggest further research toward this field, especially in conjunction with hybrid testing, as the method retains certain global applicability attributes and it can be easily extended to other transfer systems, apart from shaking tables. Copyright © 2014 John Wiley & Sons, Ltd.     

Experimental investigation on dynamic and quasi‐static behavior of low‐rise reinforced concrete walls

Experimental evidence supporting the fact that results from quasi‐static (QS) test of low‐rise reinforced concrete walls may be safely assumed as a lower limit of strength and displacement, and energy dissipation capacities are still scarce. The aim of this paper is to compare the seismic performance of 12 reinforced concrete walls for low‐rise housing: six prototype walls tested under QS‐cyclic loading and six models tested under shaking table excitations. Variables studied were wall geometry, type of concrete, web steel ratio, type of web reinforcement and testing method. Comparison of results from dynamic and QS‐cyclic tests indicated that stiffness and strength properties were dependent on the loading rate, the strength mechanisms associated with the failure mode, the low‐cycle fatigue, and the cumulative parameters, such as displacement demand and energy dissipated. Copyright © 2012 John Wiley & Sons, Ltd.     

On the testing of fully integrated surface–subsurface hydrological models

Studies employing integrated surface–subsurface hydrological models (ISSHMs) have utilized a variety of test cases to demonstrate model accuracy and consistency between codes. Here, we review the current state of ISSHM testing and evaluate the most popular ISSHM test cases by comparing the hydrodynamic processes simulated in each case to the processes found in well‐characterized, real‐world catchments and by comparing their general attributes to those of successful benchmark problems from other fields of hydrogeology. The review reveals that (1) ISSHM testing and intercode comparison have not adopted specific test cases consistently; (2) despite the wide range of ISSHM metrics available for model testing, only two model performance diagnostics are typically adopted: the catchment outflow hydrograph and the catchment water balance; (3) in intercode comparisons, model performance is usually judged by evaluating only one performance diagnostic: the catchment outflow hydrograph; and (4) ISSHM test cases evaluate a small number of hydrodynamic processes that are largely uniform across the model domain, representing a limited selection of the processes of interest in well‐characterized, real‐world catchments. ISSHM testing would benefit from more intercode comparisons using a consistent set of test cases, aimed at evaluating more catchment processes (e.g. flooding) and using a wider range of simulation diagnostics (e.g. pressure head distributions). To achieve this, a suite of test case variations is required to capture the relevant catchment processes. Finally, there is a need for additional ISSHM test problems that compare model predictions with hydrological observations from intensively monitored field sites and controlled laboratory experiments. Copyright © 2012 John Wiley & Sons, Ltd.     

Real‐time force control for servo‐hydraulic actuator systems using adaptive time series compensator and compliance springs

Servo‐hydraulic actuators have been widely used for experimental studies in engineering. They can be controlled in either displacement or force control mode depending on the purpose of a test. It is necessary to control the actuators in real time when the rate‐dependency effect of a test specimen needs to be accounted for under dynamic loads. Real‐time hybrid simulation (RTHS) and effective force testing (EFT) method, which can consider the rate‐dependency effect, have been known as viable alternatives to the shake table testing method. Due to the lack of knowledge in real‐time force control, however, the structures that can be tested with RTHS and EFT are fairly limited. For instance, satisfying the force boundary condition for axially stiff members is a challenging task in RTHS, while EFT has a difficulty to be implemented for nonlinear structures. In order to resolve these issues, this paper introduces new real‐time force control methods utilizing the adaptive time series (ATS) compensator and compliance springs. Unlike existing methods, the proposed force control methods do not require the structural modeling of a test structure, making it easy to be implemented especially for nonlinear structures. The force tracking performance of the proposed methods is evaluated for a small‐scale steel mass block system with a magneto‐rheological damper subjected to various target forces. Accuracy, time delay, and resonance response of these methods are discussed along with their force control performance for an axially stiff member. Overall, a satisfactory force tracking performance was observed by using the proposed force control methods.     

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.     

Considerations for the development of real‐time dynamic testing using servo‐hydraulic actuation

This paper presents a study of the use of servo‐hydraulic systems in the implementation of real‐time large‐scale structural testing methods in force control such as effective force testing (EFT) and in displacement control such as real‐time pseudodynamic testing (RPsD). Mathematical models for both types of control systems are presented and used to investigate the influences of servo‐systems on the overall system performance. Parameters investigated include the overall system dynamics, nonlinearities of servo‐systems, actuator damping, system mass including piston mass, and system response delay. Results of both numerical simulations and experiments showed that many of the influences of the servo‐hydraulic system that significantly affect the real‐time dynamic tests can be properly compensated through control schemes identified in this paper. Copyright © 2003 John Wiley & Sons, Ltd.     

Stability and accuracy analysis of the central difference method for real‐time substructure testing

The central difference method (CDM) that is explicit for pseudo‐dynamic testing is also believed to be explicit for real‐time substructure testing (RST). However, to obtain the correct velocity dependent restoring force of the physical substructure being tested, the target velocity is required to be calculated as well as the displacement. The standard CDM provides only explicit target displacement but not explicit target velocity. This paper investigates the required modification of the standard central difference method when applied to RST and analyzes the stability and accuracy of the modified CDM for RST. Copyright © 2005 John Wiley & Sons, Ltd.