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.     

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.     

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.     

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.     

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.     

Energy-consistent integration method and its application to hybrid testing

The time-integration algorithm is an indispensable element to determine response of the boundary of the numerical as well as physical parts in a hybrid test. Instability of the time-integration algorithm may directly lead to failure of the test, so stability of an integration algorithm is particularly important for hybrid testing. The explicit algorithms are very popular in hybrid testing, because iteration is not needed. Many unconditionally stable explicit-algorithms have been proposed for hybrid testing. However, the stability analysis approaches used in all these methods are valid only for linear systems. In this paper, a uniform formulation for energy-consistent time integrations, which are unconditionally stable, is proposed for nonlinear systems. The solvability and accuracy are analyzed for typical energy-consistent algorithms. Some numerical examples and the results of a hybrid test are provided to validate the effectiveness of energy-consistent algorithms.     

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.     

Non-planar pseudodynamic testing

The pseudodynamic test method provides a means of inexpensive seismic performance testing for laboratories that do not have a shaking table. However, most pseudodynamic tests to date have used planar portions of structures subjected to a single lateral component of base excitation, mimicking the type of testing that would occur on a shaking table. There has been little work on the extension of the pseudodynamic test method to three-dimensional testing of structures under multiple components of base excitation. In this paper a three-dimensional specimen is tested under a multicomponent fixed base excitation and the response is compared to shaking table tests. The paper presents an overview of the pseudodynamic test method, including non-planar extensions, and highlights many physical problems that occurred during the testing process. Many of these problems apply to any pseudodynamic test, not just non-planar tests, but the results show that very accurate non-planar tests can be achieved with careful error control.     

Analysis of implicit HHT‐α integration algorithm for real‐time hybrid simulation

Real‐time hybrid simulation is a viable experiment technique to evaluate the performance of structures equipped with rate‐dependent seismic devices when subject to dynamic loading. The integration algorithm used to solve the equations of motion has to be stable and accurate to achieve a successful real‐time hybrid simulation. The implicit HHT α‐algorithm is a popular integration algorithm for conducting structural dynamic time history analysis because of its desirable properties of unconditional stability for linear elastic structures and controllable numerical damping for high frequencies. The implicit form of the algorithm, however, requires iterations for nonlinear structures, which is undesirable for real‐time hybrid simulation. Consequently, the HHT α‐algorithm has been implemented for real‐time hybrid simulation using a fixed number of substep iterations. The resulting HHT α‐algorithm with a fixed number of substep iterations is believed to be unconditionally stable for linear elastic structures, but research on its stability and accuracy for nonlinear structures is quite limited. In this paper, a discrete transfer function approach is utilized to analyze the HHT α‐algorithm with a fixed number of substep iterations. The algorithm is shown to be unconditionally stable for linear elastic structures, but only conditionally stable for nonlinear softening or hardening structures. The equivalent damping of the algorithm is shown to be almost the same as that of the original HHT α‐algorithm, while the period elongation varies depending on the structural nonlinearity and the size of the integration time‐step. A modified form of the algorithm is proposed to improve its stability for use in nonlinear structures. The stability of the modified algorithm is demonstrated to be enhanced and have an accuracy that is comparable to that of the existing HHT α‐algorithm with a fixed number of substep iterations. Both numerical and real‐time hybrid simulations are conducted to verify the modified algorithm. The experimental results demonstrate the effectiveness of the modified algorithm for real‐time testing. Copyright © 2011 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.     

Development of a family of unconditionally stable explicit direct integration algorithms with controllable numerical energy dissipation

The implicit dissipative generalized‐ α method is analyzed using discrete control theory. Based on this analysis, a one‐parameter family of explicit direct integration algorithms with controllable numerical energy dissipation, referred to as the explicit KR‐α method, is developed for linear and nonlinear structural dynamic numerical analysis applications. Stability, numerical dispersion, and energy dissipation characteristics of the proposed algorithms are studied. It is shown that the algorithms are unconditionally stable for linear elastic and stiffness softening‐type nonlinear systems, where the latter indicates a reduction in post yield stiffness in the force–deformation response. The amount of numerical damping is controlled by a single parameter, which provides a measure of the numerical energy dissipation at higher frequencies. Thus, for a specific value of this parameter, the resulting algorithm is shown to produce no numerical energy dissipation. Furthermore, it is shown that the influence of the numerical damping on the lower mode response is negligible. It is further shown that the numerical dispersion and energy dissipation characteristics of the proposed explicit algorithms are the same as that of the implicit generalized‐ α method. A numerical example is presented to demonstrate the potential of the proposed algorithms in reducing participation of undesired higher modes by using numerical energy dissipation to damp out these modes. Copyright © 2014 John Wiley & Sons, Ltd.     

EVALUATION OF SOME IMPLICIT TIME-STEPPING ALGORITHMS FOR PSEUDODYNAMIC TESTS

Two types of implicit time-stepping algorithms have been proposed recently for pseudodynamic tests. The first type consists of an algorithm which relies on Newton iterations to satisfy the equations of motion. The second type consists of an algorithm which is based on the Operator-Splitting technique and does not require any numerical iteration. While one or the other has been preferred by some researchers, these time-stepping algorithms have not been analysed and compared under a uniform setting. In this paper, a concise summary of these schemes is presented, and they are evaluated in a consistent manner in terms of numerical dissipation, frequency distortion and experimental errors. The analytical results are validated by numerical simulations as well as experimental results. It is shown that the algorithm based on Newton iterations can control experimental error effects effectively by means of an error-correction procedure. The algorithm based on the Operator-Splitting technique demonstrates similar performance provided the I-Modification is adopted.     

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.     

Retracted: Discussion of paper ‘Development of a family of unconditionally stable explicit direct integration algorithms with controllable numerical energy dissipation’ by Chinmoy Kolay and James M. Ricles,Earthquake Engineering and Structural Dynamics 2014; 43:1361–1380

It seems that the explicit KR‐α method (KRM) is promising for the step‐by‐step integration because it simultaneously integrates unconditional stability, explicit formulation, and numerical dissipation together. It was shown that KRM can inherit the numerical dispersion and energy dissipation properties of the generalized‐α method (GM) for a linear elastic system, and it reduces to CR method (CRM) if ρ_∞ = 1is adopted, where ρ_∞ is the spectral radius of the amplification matrix of KRM as the product of the natural frequency and the step size tends to infinity. However, two unusual properties were found for KRM and CRM, and they might limit their application to solve either linear elastic or nonlinear systems. One is the lack of capability to capture the structural nonlinearity, and the other is that it is unable to realistically reflect the dynamic loading. Copyright © 2014 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.     

Improved formulations of the CR and KR methods for structural dynamics

Although the Chen-Ricles (CR) method and the Kolay-Ricles (KR) method have been applied to conduct pseudodynamic tests, they have both been found to have some adverse numerical properties, such as conditional stability for stiffness hardening systems and an unusual overshoot in the steady-state response of a high-frequency mode. An improved formulation for each method can be achieved by using a stability amplification factor to boost the unconditional stability range for stiffness hardening systems and a loading correction term to eliminate the unusual overshoot in the steady-state response of a high-frequency mode. The details for developing improved formulations for each method are shown in this work.     

Implicit time integration for pseudodynamic tests: Convergence and energy dissipation

The convergence and energy-dissipation characteristics of an unconditionally stable implicit time integration scheme that has been adopted for pseudodynamic testing are examined in this paper. A convergence criterion is derived for general multiple-degree-of-freedom softening systems. Furthermore, it is shown that undesired loading and unloading cycles can be avoided in numerical iterations by scaling down the incremental corrections. Finally, it is proved that the total energy dissipation introduced by the residual convergence errors and proposed numerical correction is always positive for any softening structures.     

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.     

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.