Time Delay-Induced Dynamics in Real-Time Hybrid Simulation: Spectral Decomposition and Energy-Based Evaluation

Real-time hybrid simulation (RTHS) inherently functions as a feedback system with intrinsic time delays, which can be accurately modeled using a delay differential equation (DDE). The presence of time delays introduces infinite-dimensional dynamics, complicating the analysis of associated errors. While time delay represents a key experimental imperfection, its quantitative influence on structural vibration remains insufficiently understood. To address this gap, we propose a spectral decomposition framework for linear RTHS systems. This method decomposes the delay system into a finite set of single-degree-of-freedom (SDOF) systems, enabling systematic analysis of delay-induced effects, including frequency shifts, spurious mode generation, and energy redistribution. We establish explicit relationships linking time delay to substructural partitioning and excitation characteristics. Based on these insights, we propose three error mitigation strategies: (1) minimizing actuator delay, (2) reducing the experimental substructure ratio, and (3) optimizing spectral alignment between external excitation and system response. Additionally, we introduce two energy-based evaluation metrics—with corresponding tolerances—to quantify the influence of time delay on total energy input and the modal concentration of input energy. The effectiveness of the proposed approach is validated through numerical simulations and physical experiments, offering novel insights into RTHS error mechanisms from modal and energetic perspectives.

Summary

• RTHS can be described as a DDE. This study introduced the spectral decomposition method for projecting the dynamic behavior of DDE to individual modes.
• From the mode and energy perspective, this method can evaluate and quantify how the energy input caused by time-delay is distributed between the inherent and the spurious modes of test system.
• Three ways are effective for error control: (1) reduce the actuator delay, (2) reduce the ratio of experimental substructure, and (3) coordinate spectrums of external excitation and system response.