Selection of time step for pseudodynamic testing

Although the step degree of nonlinearity has been introduced to conduct basic analysis and error propagation analysis for the pseudodynamic testing of nonlinear systems, it cannot be reliably used to select an appropriate time step before performing a pseudodynamic test. Therefore, a novel parameter of instantaneous degree of nonlinearity is introduced to monitor the stiffness change at the end of a time step, and can thus be used to evaluate numerical and error propagation properties for nonlinear systems. Based on these properties, it is possible to select an appropriate time step to conduct a pseudodynamic test in advance. This possibility is very important in pseudodynamic testing, since the use of an arbitrary time step might lead to unreliable results or even destroy the test specimen. In this paper, guidelines are proposed for choosing an appropriate time step for accurate integration of nonlinear systems. These guidelines require estimation of the maximum instantaneous degree of nonlinearity and the solution of the initial natural frequency. The Newmark explicit method is chosen for this study. All the analytical results and the guidelines proposed are thoroughly confirmed with numerical examples.     

A family of explicit algorithms for general pseudodynamic testing

A new family of explicit pseudodynamic algorithms is proposed for general pseudodynamic testing. One particular subfamily seems very promising for use in general pseudodynamic testing since the stability problem for a structure does not need to be considered. This is because this subfamily is unconditionally stable for any instantaneous stiffness softening system, linear elastic system and instantaneous stiffness hardening system that might occur in the pseudodynamic testing of a real structure. In addition, it also offers good accuracy when compared to a general second-order accurate method for both linear elastic and nonlinear systems.     

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.     

Modified state–space procedures for pseudodynamic testing

The existing on‐line numerical integration algorithms are derived from the Newmark method, which is based on an approximation of derivatives in the differential equation. The state–space procedure (SSP), based on an interpolation of the discrete excitation signals for piecewise convolution integral, has been confirmed as more reliable than the Newmark method in terms of numerical accuracy and stability. In an attempt to enhance the pseudodynamic test, this study presents an on‐line integration algorithm (referred to as the OS–SSP method) via an integration of the state–space procedure with Nakashima's operator‐splitting concept. Numerical stability and accuracy assessment of the proposed algorithm in addition to the explicit Newmark method and the OS method were investigated via an eigenvalue, frequency‐domain and time‐domain analysis. Of the on‐line integration algorithms investigated, the OS–SSP method is demonstrated as the most accurate method with an acceptable stability (although not unconditionally stable) characteristic. Therefore, the OS–SSP method is the most desirable method for pseudodynamic testing if the numerical stability criterion (Δt/T⩽0.5) is ensured for every vibration mode involved. Copyright © 2001 John Wiley & Sons, Ltd.     

Extensions of nonlinear error propagation analysis for explicit pseudodynamic testing

Two important extensions of a technique to perform a nonlinear error propagation analysis for an explicit pseudodynamic algorithm (Chang, 2003) are presented. One extends the stability study from a given time step to a complete step-by-step integration procedure. It is analytically proven that ensuring stability conditions in each time step leads to a stable computation of the entire step-by-step integration procedure. The other extension shows that the nonlinear error propagation results, which are derived for a nonlinear single degree of freedom (SDOF) system, can be applied to a nonlinear multiple degree of freedom (MDOF) system. This application is dependent upon the determination of the natural frequencies of the system in each time step, since all the numerical properties and error propagation properties in the time step are closely related to these frequencies. The results are derived from the step degree of nonlinearity. An instantaneous degree of nonlinearity is introduced to replace the step degree of nonlinearity and is shown to be easier to use in practice. The extensions can be also applied to the results derived from a SDOF system based on the instantaneous degree of nonlinearity, and hence a time step might be appropriately chosen to perform a pseudodynamic test prior to testing.     

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.     

Phase and amplitude error indices for error quantification in pseudodynamic testing

Real-time pseudodynamic (PSD) and hybrid PSD testing methods are displacement controlled experimental techniques that are used to investigate the dynamic behaviour of complex and load rate-dependent structures. Because the imposed command displacements are not predefined but generated during the test based on measured feedback, these methods are inherently prone to error propagation, which can affect the accuracy and even the stability of the entire experiment. As a result, to have these experimental methods as reliable tools, the accuracy of the test results needs to be assessed by carefully monitoring, and if possible, quantifying the errors involved. In this paper, phase and amplitude error indices (PAEI) are introduced to identify the experimental errors through uncoupled closed-form equations. Unlike the indicators that have been previously introduced in the literature for error identification purposes, PAEI do not use test setup specific parameters in their formulation, and can quantify the errors independent of the amplitude of the command displacements. As such, PAEI can be used as standard tools for assessing the quality of the experiments performed in different laboratories or under different conditions. Additionally, because they can quantify the error, when implemented online, PAEI have the potential to be incorporated in the control law and thereby improve the actuator control during the tests. The formulation and implementation of PAEI are provided in this paper. The enhanced performance of the proposed indices is demonstrated by processing several different measured and command signals using PAEI and comparing the results with those revealed by the previous indicators. Copyright © 2011 John Wiley & Sons, Ltd.     

Analytical exploration of γ-function explicit method for pseudodynamic testing of nonlinear systems

It has been well studied that the γ-function explicit method can be effective in providing favorable numerical dissipation for linear elastic systems. However, its performance for nonlinear systems is unclear due to a lack of analytical evaluation techniques. Thus, a novel technique is proposed herein to evaluate its efficiency for application to nonlinear systems by introducing two parameters to describe the stiffness change. As a result, the numerical properties and error propagation characteristics of the γ-function explicit method for the pseudodynamic testing of a nonlinear system are analytically assessed. It is found that the upper stability limit decreases as the step degree of nonlinearity increases; and it increases as the current degree of nonlinearity increases. It is also shown that this integration method provides favorable numerical dissipation not only for linear elastic systems but also for nonlinear systems. Furthermore, error propagation analysis reveals that the numerical dissipation can effectively suppress the severe error propagation of high frequency modes while the low frequency responses are almost unaffected for both linear elastic and nonlinear systems.     

Cumulative experimental errors in pseudodynamic tests

In pseudodynamic tests, experimental feedback errors are accumulated in the step-by-step integration procedure. In this paper, the growth of cumulative experimental errors is examined. Approximate cumulative error bounds are derived for linear single- and multi-degree-of-freedom systems, based on realistic models of random and systematic feedback errors. These studies show that the rate of cumulative error growth with respect to the integration time step increases rapidly with the natural frequency of the specimen and the integration time interval used. Hence, the higher modes of a multi-degree-of-freedom system are more sensitive to experimental errors than the lower ones. Furthermore, it is shown that some systematic errors are extremely undesirable. Rational criteria for assessing the reliability of pseudodynamic test results are presented.     

Improved time integration for pseudodynamic tests

Converting the second-order differential equation to a first-order equation by integrating it with respect to time once as the governing equation of motion for a structural system can be very promising in the pseudodynamic testing. This was originally found and developed by Chang. The application of this time-integration technique to the Newmark explicit method is implimented and investigated in this paper. The main advantages of using the integral form of Newmark explicit method instead of the commonly used Newmark explicit method in a pseudodynamic test are: a less-error propagation effect, a better capability in capturing the rapid changes of dynamic loading and in eliminating the adverse linearization errors. All these improvements have been verified by theoretical studies and experimental tests. Consequently, for a same time step this time-integration technique may result in less-error propagation and achieve more accurate test results than applying the original form of Newmark explicit method in a pseudodynamic test due to these significant improvements. Thus, the incorporation of this proposed time-integration technique into the direct integration method for pseudodynamic testings is strongly recommended. © 1998 John Wiley & Sons, Ltd.     

抗震拟动力试验技术研究

在全面论述拟动力试验技术的基础上，提出了大刚度多自由度钢筋混凝土结构和砌体结构的拟动力试验技术新方法，即以力控制方式为基础的力－位移混合控制方法，这种方法能实现大刚度多自由度钢筋混凝土结构和砌体结构的拟动力试验。     

An unconditionally stable hybrid pseudodynamic algorithm

There is a significant motivation to implement an unconditionally stable scheme in the pseudodynamic test method. As more complex experiments with many degrees of freedom are tested, explicit time integration methods limit the size of time step on the basis of the highest natural frequency of the system. This is true even though the response of the structure may be dominated by a few lower frequency modes. The limit on step size is undesirable because it physically increases the duration of a test, but more importantly, because the number of steps to completion increases and error propagation problems increase with the number of steps in a test. In addition, incremental displacements within each step become smaller, introducing the potential for problems associated with stress relaxation. An unconditionally stable algorithm allows the time step to be selected to give accurate response in the modes of interest without regard for higher mode characteristics.     

Implicit time integration for pseudodynamic tests

The use of unconditionally stable implicit time integration techniques for pseudodynamic tests has been recently proposed and advanced by several researchers. Inspired by such developments, a pseudodynamic test scheme based on an unconditionally stable implicit time integration algorithm and dual displacement control is presented in this paper. The accuracy of the proposed scheme is proved with error-propagation analysis. It is shown by numerical examples and verification tests that the error-correction method incorporated can eliminate the spurious higher-mode response, which can often be excited by experimental errors. The practicality of the proposed scheme lies in the fact that the implementation is as easy as that of explicit schemes and that the convergence criteria required are compatible with the accuracy limits of ordinary test apparatus.     

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.     

Bi-directional pseudodynamic test of a full-size three-storey building

The reliability of a Pseudodynamic (PsD) test depends primarily on the accuracy of the control system. Difficulties arise mainly when the method is applied to very stiff or very heavy structures or to structures with a high number of Degrees of Freedom (DoFs). This paper describes the bi-directional PsD testing of a full-size three-storey building. The tested specimen is a composite structure with plan dimensions of 12×16 m and height of 9·5 m, made of steel columns and beams combined with composite reinforced concrete slabs. The PsD test included the application of two uncorrelated accelerograms along the horizontal directions X and Y. Since the structure was not symmetric about the Y-axis, the possibility of torsion was considered by taking into account both horizontal displacements and the yaw rotation at every floor. Three displacement-controlled hydraulic actuators were thus used at each floor to impose these three DoFs while a fourth actuator with special control strategy was added to optimize the distribution of loads among the pistons. The validity of the testing methodology was verified by performing also a dynamic random burst test on the specimen which was afterwards pseudodynamically reproduced. Copyright © 1999 John Wiley & Sons, Ltd.     

Investigations of nonlinear performances of implicit pseudodynamic algorithms

Although it has been shown that the implementation of the HHT-α method can result in improved error propagation properties in pseudodynamic testing if the equation of motion is used instead of the difference equation to evaluate the next step acceleration, this paper proves that this method might lead to instability when used to solve a nonlinear system. Its unconditional stability is verified only for linear elastic systems, while for nonlinear systems, instability occurs as the step degree of convergence is less than 1. It is worth noting that the step degree of convergence can frequently be less than 1 in pseudodynamic testing, since a convergent solution is achieved only when the step degree of convergence is close to 1 regardless of whether its value is greater or less than 1. Therefore, the application of this scheme to pseudodynamic testing should be prohibited, since the possibility of instability might incorrectly destroy a specimen. Consequently, the implementation of the HHT-α method by using the difference equation to determine the next step acceleration is recommended for use in pseudodynamic testing.     

An inelastic substructure technique for the pseudodynamic test method

An inelastic substructure technique for the pseudodynamic test method is described. This technique requires testing of only a critical component of a multi-degree-of-freedom structure, while the remaining portion is modelled using standard inelastic analytical procedures. This is an economical method to investigate the seismic behaviour of a structure, provided a critical subassembly is found. This paper describes the development of a substructure algorithm which is verified with a numerically simulated test. The method was used to evaluate the seismic performance of moment-resisting steel frames. Modelling for an eight-storey, three-bay frame is discussed, and the boundary conditions between the analytical portion and the experimental component are evaluated. The results indicate that the selection of the critical subassembly was adequate and that the pseudodynamic response was significantly dependent on the behaviour of this experimental component. Furthermore, experimental results suggest that this information would have been difficult to obtain from quasi-static testing or from standard inelastic dynamic analysis. Therefore, this substructure pseudodynamic technique was an economical tool to investigate the seismic behaviour of ductile frames.     

Improved numerical dissipation for explicit methods in pseudodynamic tests

There is no second-order accurate, dissipative, explicit method in the currently available step-by-step integration algorithms. Two new families of second-order accurate, dissipative, explicit methods have been successfully developed for the direct integration of equations of motion in structural dynamics. These two families of methods are numerically equivalent and possess the desired numerical dissipation which can be continuously controlled. These two families of algorithms are very useful for pseudodynamic tests since the favourable numerical damping can be used to suppress the spurious growth of high-frequency modes due to the presence of numerical and/or experimental errors in performing a pseudodynamic test. © 1997 by John Wiley & Sons, Ltd.     

On the accuracy of an implicit algorithm for pseudodynamic tests

The error-propagation characteristics of an implicit time integration algorithm in pseudodynamic testing are examined. It is shown that the implicit algorithm is superior to explicit integration algorithms in terms of experimental error amplification. The influence of systematic experimental errors is studied and methods for controlling these errors are examined. In spite of the fact that the implicit algorithm is unconditionally stable, it is shown that the integration time interval in a pseudodynamic test is limited by the calibration range of the electronic hardware as well as the degree of participation of the higher modes. Furthermore, the tolerance for experimental errors decreases as the integration time interval increases.     

Elimination of spurious higher-mode response in pseudodynamic tests

In a pseudodynamic test, errors in restoring-force feedback are introduced into numerical computations. Some of these errors can excite the higher-frequency response of the specimen. In this paper, the use of viscous and numerical dampings to eliminate spurious higher-frequency effects is studied. Since the tangent stiffness of a non-linear specimen cannot be measured accurately, initial-stiffness-dependent viscous damping is considered. In addition, an explicit integration algorithm with desired numerical damping properties is proposed and examined. The analytical and numerical studies presented indicate that viscous-damping properties can be substantially changed by non-linear deformations. For this reason, the use of numerical damping appears to be more advantageous.