A large number of numerical integration methods, either explicit or implicit, have been developed for pseudo dynamic testing which is conducted with an extended time-scale. Implicit algorithms are preferred over explicit algorithms because they are unconditionally stable. However, implicit integration algorithms involve iteration within a time step and would take too much computational time for real-time hybrid testing. Therefore, explicit integration algorithms have been studied and proposed by the researchers to meet the stability and accuracy requirements for real-time hybrid testing. This section gives a brief review of the literatures focused on the development and application of integration algorithms used for real-time hybrid testing.
2.1.1 Nakashima and Masaoka
Nakashima and Masaoka conducted an online real-time hybrid test of a MDOF structural system (Nakashima and Masaoka, 1999). The central difference method was used to solve the equation of motion. Third-order polynomial extrapolation was carried out for delay compensation. Besides, extrapolation and interpolation procedures using present and past target displacements were developed to create
displacement signals successively without being interrupted by the computation of equations of motion. By their approach, it was able to perform a 10 DOFs hybrid test with reasonable accuracy.
2.1.2 Wu et al.
Wu et al. conducted stability and accuracy analysis of the central difference method for real-time substructure testing. A modified central difference method was proposed to provide explicit target velocity for rate-dependent real-time hybrid testing (Wu et al.
2005). Besides, an explicit acceleration formulation was assumed for real-time hybrid tests considering specimen mass. It was found that the stability decreases with increasing specimen mass that provides pure inertia force. The structure became unstable when the mass of the specimen was equal or greater than that of the numerical model. Moreover, the operator-splitting method was modified by using a target velocity formulation as well (Wu et al. 2006). This modified operator-splitting method was proved to be unconditionally stable when the specimens are of the softening type.
2.1.3 Bonnet et al.
Bonnet et al. developed a mathematical multi-tasking strategy to meet the requirements of control and numerical issues. A large number of both explicit and implicit integration algorithms were evaluated using this strategy (Bonnet et al., 2008).
It was observed that the computation of implicit algorithms takes too much time for MDOF real-time hybrid testing. For explicit algorithms, the Newmark-Chang method (Chang, 2002) was considered more efficient than the operator-splitting method to meet the stability requirement of real-time hybrid testing. Furthermore, the α-operator-splitting method was recommended if numerical dissipation of higher modes is considered. Finally, the Newmark explicit integration algorithm with adaptive outer-loop controllers based on the minimal controller synthesis was adopted to perform real-time hybrid testing.
2.1.4 Zhang et al.
Zhang et al. proposed an explicit predictor-corrector method for real-time hybrid testing (Zhang et al., 2005). The state-space representation (see chapter 3.2.1) was adopted to form a set of first-order differential equations. By discretizing the analytical solution in state space, an accurate model in discrete time was obtained. A series of numerical simulations were carried out to compare the explicit predictor-corrector method with explicit Newmark, central difference, operator-splitting, and operator-splitting structural state procedure methods for both linear and nonlinear systems. It was shown that the predictor-corrector method is conditionally stable depending on the artificial stiffness matrix.
2.1.5 Chen and Ricles
Chen and Ricles developed a direct integration algorithm, CR integration algorithm, for real-time hybrid tests using discrete control theory (Chen and Ricles, 2009). The algorithm is explicit for both displacement and velocity. The root locus (see chapter 3.1.4.2) method was used to investigate the stability of the integration algorithm. It was found that the CR integration algorithm is unconditionally stable for responses of linear elastic and nonlinear with softening stiffness. Moreover, Chen and Ricles also demonstrated that the HHT α-algorithm with a fixed number of sub-step iterations is unconditionally stable for linear elastic structures (Chen and Ricles 2012). However, it provides only conditional stability of nonlinear softening or hardening structures.
2.1.6 Bursi et al.
Bursi et al. (2008) proposed Rosenbrock-W integration method for real-time hybrid tests using an approximation of the Jacobian matrix. This method requires an initial estimate of stiffness and damping properties of the physical specimen. It was proved that the stability and accuracy of the approach are improved by introducing the properties of physical specimens in the integration process. Besides, the Rosenbrock-W integration method was extended for specimens with nonlinear responses (Bursi et al., 2010). The performance was demonstrated by both numerical
simulations and real-time substructuring tests. Furthermore, a sub-cycling strategy was considered in the process of the proposed algorithm to provide stable, accurate and robust numerical integration.
2.1.7 Other Researches
Hung and El-Tawil (2009) used predicted accelerations to compute explicit displacement and velocity in the predictor step and a full operator method to suppress error propagation in the corrector step. The performance of the full operator method was demonstrated via numerical simulations. Kim et al. (2011) proposed the convolution integral method for real-time hybrid simulation regardless of the size and complexity of the numerical model. Real-time hybrid testing with a magneto-rheological (MR) fluid damper as the physical specimen was conducted.
Experimental results indicated that the convolution integral method shows good agreement with the traditional approach using integration time-stepping method.
Sajeeb et al. (2007) used a high-order multi-step transversal linearization scheme to solve to equation of motion and a reproducing kernel based extrapolation technique to predict the interactive displacements over multiple time steps ahead.