Hybrid Simulation CURRENT ASSESSMENT MEASURES (HSCAM)

Contents

Hybrid Simulation Current Assessment Measures, Prepared by:CARLOS ANDRES RIASCOS GONZALEZ, Ge(Gaby) Ou, and Shirley Dyke, code published on: 10/15/2014 %

1. Test

- Dynamics Properties of the Structure

- Time Domain Response

2. Peak Error

The Peak Error (e) captures the maximum error between the target displacement and the measured displacement. eDM is the error between the peak target displacement and the peak measured displacement and eMD is the maximum difference error between target and measured displacements.

3. Tracking Indicator (TI)

The Tracking Indicator (TI) is used to quantify the difference between the target and measured displacement of the actuator at each time step. The outcome provides a measure of the error at each time step in the hybrid simulation.

$$ TI(i+1)=\frac {A(i+1)-TA(i+1)}{2}$$

where:

A(i+1)=A(i)+ \frac {[x_c(i+1)+x_c(i)] [x_m(i+1)+x_m(i)]}{2}

4. Normalized Root Mean Square Error (NRMSerror)

The Normalized Root Mean Square (NRMS) can be used to obtain a single value representing this difference between the measured and target displacements of the actuator. These two measures are time domain measures of the actuator tracking.

$$ NRMS_{error} = \sqrt { \frac {\sum_{i=1}^N [x_m(i)-x_c(i)]^2} {\sum_{i=1}^N x_c(i)^2} } $$

5. Frequency Evaluation Index (FEI)

Frequency Evaluation Index (FEI) is a frequency domain measure of the actuator tracking

$$ FEI= \sum_{i=i}^N \frac{fft(x_m)}{fft(x_c)} \frac{|fft(x_m)|^l}{ \sum_{i=i}^N|fft(x_c)|^l} $$

$$ f^{eq} = \frac{\sum_{i=i}^N {|fft(x_c)|^l f_j} } {\sum_{i=i}^N {|fft(x_c)|^l} }$$

$$ A_0=|FEI|$$

$$ \phi=arctan (\frac {Im(FEI)}{Re(FEI)})$$

$$ \delta=-\frac {\phi}{2\pi f^{eq}}$$

6. Cross Correlation

The Cross Correlation (CC) between the target and measured displacement can also provide frequency domain insight into the performance of actuator tracking. Cross-correlation quantifies the degree of similarity between two time series. In hybrid simulation, cross-correlation between the target and measured displacements provides a reasonable estimate of time delay in the actuator.

$$[x_c*x_m](\tau) = \int_{-\infty}^{\infty} x_c(\tau)x_m(t-\tau) d\tau $$

$$ \tau_{est}=argmax{[x_c*x_m](\tau)} $$

7. Energy Error

Mosqueda et al. (2007) proposed the use of energy to quantify the difference between actual experimental behavior (measured forces versus measured displacements) and that observed in explicit numerical simulations (measured forces versus target displacements).

$$ E_E^{err}=E_E^c-E_E $$

8. Hybrid Simulation Energy Monitor (HSEM)

Mosqueda et al. (2007b) showed that one can limit the displacement and force errors of a hybrid simulation by limiting the amount of the HSEM. Since the majority of errors in a hybrid simulation are likely from experimental sources, the HSEM can be a suitable choice for monitoring the simulation quality.

$$ HSEM = \frac{E_E^{err}}{E_I-E_E^{max}} $$

where:

$$ E_I=\int F_m^T dx $$

$$ E_E^{err}=\frac {1}{2}x_0^TK^0x_0 $$

9. Nomenclature


Published with MATLAB® R2012b