AUTHOR’S REPLY Response to ‘Discussion of paper “Real-time hybrid testing using the unconditionally stable explicit CR integration algorithm” by Cheng Chen, James M. Ricles, Thomas M. Marullo and Oya Mercan’ in Earthquake Engineering and Structural Dynamics 2009; 38:23– 44 Received 30 June 2011; Accepted 4 July 2011 KEY WORDS: real-time; hybrid simulation; integration algorithm The authors thank Professor Chang for his discussion. However, the authors do not agree with the conclusion by Chang that ‘in a general structural dynamic problem, where the total response is dom- inated by low frequency modes, the total response might be contaminated or even destroyed by the high frequency responses due to a significant overshooting effect for CRM.’ The disagreement is based on the three points given below. First, the presented numerical cases in the discussion have no inherent damping in the structure, which is not common in numerical simulation or practical real-time hybrid simulation of structures subject to earthquakes. Rayleigh proportional damping in Figure 1 is often used to simulate the inherent damping in structures Chopra [1]. When either 2% or 5% damping is selected for two of the lower modes, the resulting damping could be very large for the higher modes, as illustrated in Figure 1. This high damping ratio would significantly suppress the spurious participation of higher modes in the structural response when the structure is subjected to dynamic excitation. This phenomena has been experimentally observed in the real-time hybrid simulation of a two- story four-bay steel moment resisting frame (MRF) with dampers shown in Figure 2 Chen and Ricles [2]; Karavasilis et al. [3]. The MRF is taken as the analytical substructure and modeled using a nonlinear finite element program with a total of 122 DOFs and 71 elements. The dampers are phys- ically tested in the laboratory as the experimental substructure. The MRF has an inherent damping of 2% for the first and second modes based on Rayleigh proportional damping. A consistent mass matrix was used to create the analytical substructure resulting in the highest natural frequency of the MRF to be around 20kHz. Successful real-time hybrid simulations were conducted with an integration time step of Δt =10/1024s. The maximum value of Ω = o n (Δt) is found to be around 1226 (where o n is the natural frequency of the system and Δt is the time step size), which is larger than the values of Ω for the cases presented in the discussion by Chang. No contamination by high frequencies was observed in the real-time hybrid simulation results. The real-time hybrid simulation results showed excellent agreement with numerical simulations performed using OpenSees [4] with the Constant Average Acceleration method of direct integration. Second, in addition to the integration algorithm, the servo-hydraulic equipment is a critical component for a successful real-time hybrid simulation. State-of-the-art servo-hydraulic systems used for real-time hybrid simulations usually have a bandwidth of 40Hz, which is typically controlled by the bandwidth of the servo-valves. For signals with a frequency beyond this bandwidth the servo-hydraulic actuator would develop considerable amplitude decay and phase delay, that is, the servo-hydraulic system acts as a low-pass filter. For a commonly used integration time step of 10/1024s, the value of Ω for the frequency of 40Hz will be close to 2.45. To achieve the values of Ω equal to 10 and 100, the signal *Correspondence to: Cheng Chen, School of Engineering, San Francisco State University, San Francisco, CA, USA. E-mail: chcsfsu@sfsu.edu Copyright © 2011 John Wiley & Sons, Ltd. EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS Earthquake Engng Struct. Dyn. 2012; 41:1065–1067 Published online 16 September 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/eqe.1160