Research In Key Technicals In Frequency-domain Evaluation Approach For Real-time Hybrid Simulation

Frequency-domain evaluation index(FEI)interprets test error into amplitude error and phase error for real-time hybrid simulation(RTHS).The time delay can be calculated using the concept of equivalent frequency.Compared with other existing evaluation methods in time domain,FEI provides better understanding of actuator tracking during a RTHS.However,the accuracy and the scope of FEI require further study as a kind of new method,even though previous theoretical and experimental findings have demonstrated the potential of this method.This study aims at solving key tackle these unsolved questions of the FEI through theoretical analysis,numerical simulation and experimental validation,which includes the physical meaning of FEI,compensation method evaluation,local evaluation as well as uncertainty quantification.The scope of the work and main conclusions of this study are summarized as follows:(1)Study the physical meaning and improve the performance of FEIThe amplitude,phase,time delay as well as weight are calculated for a linear structure through numerical simulation and experbimental test to study the physical meaning of the FEI.Research indicates the amplitude,phase,time delay of the entire test can be calculated based on the amplitude,phase,time delay at each frequency,and the amplitude and delay for each frequency can weight to the amplitude and delay of the entire test.It is concluded that the frequency-domain evaluation method in nature is to calculate the amplitude and phase(delay)of each frequency component and to take the-weighted sum as amplitude and phrase(delay)of the test.Due to different weight of various frequencies,results of the evaluation method are often dominated by low frequencies.However,the time delay calculated is different from theoretical value at low frequencies due to spectrum leakage and spurious delay.To solve this,the data are weighted from the fourth frequency or per-minus the average of the signals before conducted FFT.(2)Evaluate the performance of delay compensation methodsIn this study,the polynomial extrapolation,the linear acceleration extrapolation,the inverse compensation and the adaptive inverse compensation are selected and compared.The Bode plot is first used to analyze the performance of the delay compensation methods,which the time delay and amplitude are calculated for constant delay compensation methods.The numerical simulations and real-time tests predefined tests with predefined displacements are conducted to calculate the amplitude and time delay using FEI.Researches show that FEI applies to not only constant delay compensation methods,but also variable delay compensation methods.(3)Study the localized evaluation method using FEIFor a typical seismic test,peak structural response often occurs within a short time window.Accurate actuator tracking during this short time window is also critical for replicating structural responses under earthquakes.Numerical simulation shows that the localized evaluation cannot balance time and frequency resolution at same time.To improve the accuracy,the window size should be as large as possible.Thus,two window techniques are proposed namely window with overlap and window without overlap.Window without overlap is computationally efficient and suitable for high frequency linear structure.Window with overlap has better time resolution and particularly suitable for nonlinear structure and low frequency structure.Moreover,window without overlap has potential for future online evaluation.(4)Study delay effect and evaluate when numerical substructure contains uncertaintiesThe DDE model,uncertainty model parameters and maximum displacement are regarded as research model,input and output respectively to build the polynomial chaos expansion(PCE)between input and output,which can be regarded as a surrogate model to represent DDE model.Sobol's index is used to evaluate the influence of the uncertainty parameter on maximum displacement.Researches show that the delay can increase the uncertainty of the test when numerical substructure contains uncertainty parameters.Sensitivity analyses show the coupling effect between variables is increasing with the time delay.Time delay can change the variable that controls the uncertainty of the model output.(5)Uncertainty quantification for RTHS when actuator dynamics contains uncertaintiesThe stochastic characteristics of the actuator dynamics can be represented by the FEI indices.When a number of repeated tests are conducted,the A and d can be calculated for each test,which can be used to build the PCE model between experimental results and actuator tracking error.Thus,the random actuator dynamics can be quantified.Research in this study shows that the A and d calculated by FEI follows Gaussian distribution when both time delay and amplitude have the same mean and variance in the time domain,where Hermite polynomials can be used as the bases for the PCE.Mean values of the A and d calculated by FEI are determined by the corresponding time indices,while the variance are determined by the coupling of the two indices.When quantifying the uncertainty of the actuator dynamics,the order and sample points should be first determined.Then,the A and d for each test can be calculated,on which the PCE model can be built.(6)Reliability analysis for RTHS when excitation contains uncertaintiesIn this paper,the correlation between the accuracy of real-time hybrid simulation results and the FEI indices is investigated for the worst-case scenarios of single-degree-of-freedom(SDOF)linear elastic structures.Correlation is established between the FEI indices and the reliability of the test by an ellipse for the worst-case scenarios.Probabilistic analysis is used to account for the effect of various ground motion inputs,different natural frequencies and damping ratios.A 5%RMSR error is used to demonstrate the proposed methodology for reliability assessment through the FEI indices derived through the calculated and measured displacements.Studies presented herein show that parameters a and b of the ellipse function are linearly correlated when the frequency of the structure is fixed,which indicates the external excitation and damping ratio can influence the reliability boundary for both amplitude error and time delay.On the contrary,the frequency only influence on b,which has little effect on amplitude reliability boundary but can significantly influence delay reliability boundary.Based on the criterion,the reliability of the real-time hybrid simulation can be easily calculated,which the test can be considered reliable if the amplitude error and time delay fulfilled the requirement of the criterion under worst-case Scenarios.Since the nonlinearity can help stable the test,this criterion can be also used for nonlinear structures.