Frequency Response Assurance Criterion (FRAC)

The Frequency Response Assurance Criterion (FRAC) is a factor which identifies the level of correlation between the reference and work models based on frequency functions. This factor is a scalar value between zero and one. A value near one indicates a high degree of FRF correlation.

The FRAC for two complex FRFs, reference FRF H_{pq}(\omega) and work FRF \hat{H}_{pq}(\omega), representing the relation between output DOFp and input DOFq for each frequency point \omega, is computed as:

FRAC_{pq} = \frac{|\sum_{\omega=\omega_1}^{\omega_2}H_{pq}(\omega)\hat{H}{pq}^*(\omega)|^2}{\sum{\omega=\omega_1}^{\omega_2}H_{pq}(\omega)H_{pq}^{}(\omega):\sum_{\omega=\omega_1}^{\omega_2}\hat{H}{pq}(\omega)\hat{H}{pq}^{}(\omega)}

where the superscript * indicates the complex conjugate value.

This function is very sensitive to changes of mass and stiffness. If a stiffness factor \alpha is applied to the structure, then the frequency shifts by \sqrt{\alpha}. By defining a stretch factor \beta = \sqrt{\alpha} and computing the FRAC values for a given range of \beta values, the \beta value at the maximum value of FRF indicates the frequency shift and the global stiffness factor \alpha =\beta^2 required to improve the correlation between the reference and work FRFs. Including the stretch factor \beta into the previous equation, FRAC is computed as follows:

FRAC_{pq} = \frac{|\sum_{\omega=\omega_1}^{\omega_2}H_{pq}(\omega)\hat{H}{pq}^*(\omega/\beta)|^2}{\sum{\omega=\omega_1}^{\omega_2}H_{pq}(\omega)H_{pq}^{}(\omega):\sum_{\omega=\omega_1}^{\omega_2}\hat{H}{pq}(\omega/\beta)\hat{H}{pq}^{}(\omega/\beta)}

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Frequency Response Assurance Criterion (FRAC), Simcenter 3D 2021.1 Series

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