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A computationally efficient one-parameter family of explicit (i.e., non iterative for nonlinear systems) direct integration algorithms featuring second-order accuracy, unconditional stability for linear systems, and controllable numerical dissipation is developed for solving inertial problems. Unconditional stability is attained within an explicit formulation using the concept of model-based algorithms in which the algorithmic parameters are functions of the model matrices. The proposed method improves the overshoot and nonlinear stability characteristics of the KR- method, an existing explicit-model-based algorithm. Numerical characteristics of the proposed method are compared with those of the KR- method and further demonstrated through representative numerical examples. [ABSTRACT FROM AUTHOR]

Summary: This paper presents the authors' response to the discussion by Dean J. Maxam and Kumar K. Tamma of the paper titled "Development of a family of unconditionally stable explicit direct integration algorithms with controllable numerical energy dissipation." [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

SUMMARY 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. [ABSTRACT FROM AUTHOR]

The system was subjected to initial displacement HT ht and initial velocity HT ht , where the two mode shapes are HT ht and HT ht . 2 for HT ht and 0.5. [Extracted from the article]

Real-time hybrid simulations using large-scale magnetorheological (MR) dampers were conducted to evaluate the performance of various structural control strategies to control the seismic response of a three-story steel-frame building. Magnetorheological dampers were installed in the building to limit the story drift to less than 1.5% under the design-basis earthquake (DBE). The laboratory specimens, referred to as experimental substructures, were two individual MR dampers, with the remainder of the building modeled as a nonlinear analytical substructure. The experimental technique enables an ensemble of ground motions to be applied to the building, resulting in various levels of damage, without the need to repair the experimental substructures because the damage will be within the analytical substructure. Five different damper control algorithms, including passive and semiactive control algorithms, were selected. An ensemble of five ground motions scaled to the DBE was used for the real-time hybrid simulations to obtain statistical responses of the structure for each control. The real-time hybrid simulation results show that the MR dampers can control the drift, enabling the performance objective of 1.5% maximum story drift to be achieved. Although some semiactive controllers show better performance for a specific ground motion, the response statistics from the real-time hybrid simulations show that the overall performance of the semiactive control algorithms with the selected user-defined parameters is similar to that for the passive controller for the three-story building used in this study. A comparison of real-time hybrid simulation results with numerical simulation results using OpenSees was conducted to further gain insight into the performance of the damper control algorithms observed in the real-time hybrid simulations. [ABSTRACT FROM AUTHOR]

Direct integration algorithms are typically used to solve temporally discretized equations of motion to evaluate the performance of structures under dynamic loading. The stability of these direct integration algorithms are usually investigated for linear elastic structures. However integration algorithms are often applied to structures with nonlinear behavior. This paper presents a procedure based on discrete control theory to investigate the stability of direct integration algorithms applied to multidegree-of-freedom (MDOF) nonlinear structures. The discrete root locus approach is used to investigate properties of the poles of the discrete transfer function matrix representing the nonlinear structural dynamics and to assess the stability of the integration algorithm. The procedure is illustrated using a nonlinear shear building MDOF system to investigate the stability of popular direct integration algorithms, including the Newmark family of integration algorithms, the Hilber-Hughes-Taylor α-method, and two newly developed explicit integration algorithms. Stability limits are derived for the direct integration algorithms that are found to be conditionally stable. [ABSTRACT FROM AUTHOR]

Real-time hybrid simulation combines experimental testing and numerical simulation by dividing a structural system into experimental and analytical substructures. Servohydraulic actuators are typically used in a real-time hybrid simulation to apply command displacements to the experimental substructure(s). Servohydraulic actuators may develop a time delay due to inherent actuator dynamics that results in a desynchronization between the measured restoring force(s) and the integration algorithm in a real-time hybrid simulation. Inaccuracy or even instability will occur in a hybrid simulation if actuator delay is not compensated properly. This paper presents an adaptive compensation method for actuator delay. An adaptive control law is developed using an error tracking indicator to adapt a compensation parameter used in the proposed compensation method. Laboratory tests involving large-scale real-time hybrid simulations of a single degree of freedom moment resisting frame with an elastomeric damper are conducted to experimentally demonstrate the effectiveness of the proposed adaptive compensation method. The actuator tracking capability is shown to be greatly improved and exceptional experimental results are still achieved when a good estimate of actuator delay is not available. [ABSTRACT FROM AUTHOR]

Direct integration algorithms are often used to solve the temporally discretized equations of motion for structural dynamic problems. Numerous studies have been conducted to investigate the stability of integration algorithms for linear elastic structures. Studies involving the stability analysis of integration algorithms for nonlinear structures are limited. This paper utilizes discrete control theory to investigate the stability of direct integration algorithms for nonlinear structural dynamics. The direct integration algorithms are represented by a closed-loop block diagram, where the nonlinear restoring force of the structure is related to a varying feedback gain. The root locus method is used to analyze the stability of the closed-loop system for various degrees of nonlinear structural behavior. The well-known methods of the Newmark family of integration algorithms and the Hilber–Hughes–Taylor α method, as well as a newly developed integration algorithm, referred to as the CR integration algorithm, are analyzed using the proposed method. It is shown that the stability of an integration algorithm under nonlinear structural behavior is dependent on the poles and zeros of its open-loop discrete transfer function. An unconditionally stable integration algorithm for linear elastic structures is shown not to always remain stable under nonlinear structural behavior. [ABSTRACT FROM AUTHOR]

In structural dynamics, integration algorithms are often used to obtain the solution of temporally discretized equations of motion at selected time steps. Various time integration algorithms have been developed in the time domain using different methods. In order for an integration algorithm to be reliable it must be stable and accurate. A discrete transfer function is used to study the properties of integration algorithms. A pole mapping rule from control theory in conjunction with a discrete transfer function is used to develop new integration algorithms for obtaining solutions to structural dynamics problems. A new explicit integration algorithm, called the CR (Chen and Ricles) algorithm, is subsequently developed based on the proposed method. The properties of the algorithm are investigated and compared with other well established algorithms such as the Newmark family of integration algorithms. By assigning proper stable poles to the discrete transfer function the newly developed CR explicit algorithm is unconditionally stable and has the same accuracy as the Newmark method with constant acceleration. In addition, the CR algorithm is based on expressions for displacement and velocity that are both explicit in form, making it an appealing integration algorithm for solving structural dynamics problems. [ABSTRACT FROM AUTHOR]

In implementing a substructure shake table testing (SSTT), the complete structure is divided into the experimental and the analytical substructures, which are driven by shake table and numerically simulated in a computer, respectively. A number of unconditionally stable and explicit integration algorithms referred to as model-based algorithms with highly computational efficiency have been developed to meet the requirement of real-time in conducting SSTT. This study aims to comprehensively investigate the stability of the SSTT system using two recently proposed model-based integration algorithms with controllable numerical energy dissipation, i.e., Kolay-Ricles-α (KR-α) algorithms and modified KR-α (MKR-α) algorithms. The SSTT system for the 2-degree-of-freedom (2-DOF) structures are firstly formulated. In order to take the contribution of the experimental substructure into consideration, the dynamic condensation is adopted to calculate the integration parameters for the analytical substructure. The discrete transfer function of the 2-DOF-SSTT system are derived. The influences of the mass ratio, the frequency ratio and the time step on the stability of the SSTT system are comprehensively investigated by using the discrete control theory. The results show that for larger values of the mass ratio, the frequency ratio and the time step have negative impact on the stability of the SSTT system. In addition, a subfamily of the MKR-α algorithms along with the dynamic condensation can always ensure the SSTT system stable. • The 2-DOF-SSTT systems using two model-based algorithms are formulated. • Dynamic condensation is adopted to calculate the integration parameters for the analytical substructure. • The discrete transfer function of the 2-DOF-SSTT systems are derived. • The influences of mass ratio, frequency ratio and time step on the SSTT systems are investigated. [ABSTRACT FROM AUTHOR]