Quality and analysis indicators

1 Mode complexity indicators

The following indicators can be used to assess the complexity of the identified modes.

1.1 Mode Overcomplexity Value

The Mode Overcomplexity Value (MOV) of a mode \(i\) is defined as: \[ MOV_i = \frac{\sum_{n = 1}^N s_{ni} \vert\phi_{ni}\vert^2}{\sum_{n = 1}^N \vert\phi_{ni}\vert^2}, \] where \(\phi_{ni}\) is the mode shape value at degree of freedom \(n\) for mode \(i\), and \(s_{ni}\) is a weighted factor. The latter is given by: \[ s_{ni} = \begin{cases} 0 & \text{if } \text{Im}(-\lambda_i^2\phi_{ni}^2 c_i) > 0 \\ 1 & \text{if } \text{Im}(-\lambda_i^2\phi_{ni}^2 c_i) \leq 0 \end{cases}, \] where \(\lambda_i\) is the complex eigenvalue for mode \(i\), and \(c_i\) is the corresponding scaling factor.

A value close to 1 indicates physical modes, while a value close to 0 indicates spurious modes (or computational modes).

mov

mov(poles, ms, ci)

Compute the mode overcomplexity value

This indicator is a weighted percentage of the degrees of freedom of the response for which adding mass leads to a negative frequency shift. A value close to 1 indicates physical modes, while a low value indicates non-physical modes (numerical or noise-related)

Inputs

  • poles: Poles of the system

  • ms: Mode shapes (each column corresponds to a mode)

  • ci: Scaling factors associated to the mode shapes

Output

  • mov: Mode overcomplexity values

Reference

[1] M+P Analyzer manual. Rev. 5.1. 2017

1.2 Mode Phase Collinearity

The Mode Phase Collinearity (MPC) aims at determining whether a mode is real or complex. It is an indicator defined in \([0, 1]\). For a mode \(i\), it is given by: \[ MPC_i = \left(\frac{\lambda_1 - \lambda_2}{\lambda_1 + \lambda_2}\right)^2, \] where \(\lambda_1 = \frac{S_{xx} + S_{yy}}{2} + \sqrt{\left(\frac{S_{xx} - S_{yy}}{2}\right)^2 + S_{xy}^2}\) and \(\lambda_2 = \frac{S_{xx} + S_{yy}}{2} - \sqrt{\left(\frac{S_{xx} - S_{yy}}{2}\right)^2 + S_{xy}^2}\) are the eigenvalues of the covariance matrix \(\mathbf{S}\): \[ \mathbf{S} = \begin{bmatrix} S_{xx} & S_{xy} \\ S_{xy} & S_{yy} \end{bmatrix} \text{ with } \begin{cases} S_{xx} = \text{Re}(\tilde{\phi}_i)^T\text{Re}(\tilde{\phi}_i) \\ S_{yy} = \text{Im}(\tilde{\phi}_i)^T\text{Im}(\tilde{\phi}_i) \\ S_{xy} = \text{Re}(\tilde{\phi}_i)^T\text{Im}(\tilde{\phi}_i) \end{cases} \] Here, \(\tilde{\phi}_i\) is the corrected mode shape vector for the mode \(i\) such that: \[ \tilde{\phi}_{ni} = \phi_{ni} - \frac{\sum_{m = 1}^N \phi_{mi}}{N}. \]

A value close to 1 indicates a real mode, while a value close to 0 indicates a complex mode.

mpc

mpc(ms)

Compute the mode phase collinearity.

This indicator aims to measure the complexity of a mode. Its value ranges from 0 (no collinearity) to 1 (perfect collinearity). For real modes, mpc tends towards 1.

Input

  • ms: Mode shapes (each column corresponds to a mode)

Output

  • mpc: Mode phase collinearity values

References

[1] J.-N. Juang and R. Pappa: "An eigensystem realization algorithm for modal parameter identification and model reduction", Journal of Guidance, Control, and Dynamics, Vol. 8, No. 5, Sept.-Oct. 1985, pp. 620-627.

[2] M+P Analyzer manual. Rev. 5.1. 2017

1.3 Mode Complexity Factor

The Mode Complexity Factor (MCF) for a mode \(i\) is defined as: \[ MCF_i = 1 - \frac{(S_{xx} - S_{yy})^2 + 4S_{xy}^2}{(S_{xx} + S_{yy})^2} \]

A value close to 0 indicates a real mode, while a value close to 1 indicates a complex mode.

1.4 Mean Phase Deviation

The Mean Phase Deviation (MPD) indicator aims to measure the complexity of a mode. Its value is close to 0 for real modes, and increases as the mode becomes more complex. It is defined as: \[ MPD_i = \frac{\sum_{n = 1}^N\vert\phi_{ni}\vert\; \text{acos}\left\vert\frac{\text{Re}(\phi_{ni})V_{22} - \text{Im}(\phi_{ni})V_{12}}{\sqrt{V_{12}^2 + V_{22}^2}\;\vert\phi_{ni}\vert}\right\vert}{\sum_{n = 1}^N \vert\phi_{ni}\vert}, \] where \(V_{12}\) and \(V_{22}\) are the components of the right singular vector associated to the singular value decomposition of the 2 \(\times\) 2 matrix \(\mathbf{P}\) expressed as: \[ \mathbf{P} = \begin{bmatrix} \text{Re}(\tilde{\phi}_i) & \text{Im}(\tilde{\phi}_i) \end{bmatrix}. \]

For real modes, the indicator is close to 0, while it increases for complex modes.

mpd

mpd(ms)

Compute the mode phase deviation.

This indicator aims to measure the complexity of a mode. Its value is close to 0 for real modes, and increases as the mode becomes more complex.

Input

  • ms: Mode shapes (each column corresponds to a mode)

Output

  • mpd: Mode phase deviation values

Reference

[1] E. Reynders, J. Houbrechts and G. De Roeck. Fully automated (operational) modal analysis. Mechanical Systems and Signal Processing. 29: 228-250. 2012

[2] A. C. Dederichs and O. Oiseth. Experimental comparison of automatic operational modal analysis algorithms for application to long-span road bridges. Mechanical Systems and Signal Processing. 199: 110485. 2023

2 Correlation indicators

The following indicators can be used to assess the correlation between measured and reconstructed quantities.

2.2 Coordinate Modal Assurance Criterion

The Coordinate Modal Assurance Criterion (COMAC) is a variant of the MAC that considers the correlation between the coordinates of two mode shapes. For a coordinate \(p\) of two mode shapes \(\phi_p\) and \(\psi_p\) (row vectors), it is defined as: \[ COMAC_p = \frac{\vert\phi_{pi}\psi_{pi}^H\vert^2}{(\phi_{pi}\phi_{pi}^H) (\psi_{pi} \psi_{pi}^H)}, \] where the superscript \(H\) denotes the Hermitian transpose.

Note

The computation of the COMAC indicator requires that the two mode shapes have the same number of coordinates and similar scaling. For the the latter, one of the mode shapes is multiplied by the modal scaling factor: \[ MSF_i = \frac{\phi_i^H\psi_i}{\phi_i^H\phi_i}. \] The previous equation assumes that \(\phi_i\) is the reference mode shape to which \(\psi_i\) is scaled.

A value close to 1 indicates a strong correlation between the two mode shapes, while a value close to 0 indicates no correlation.

comac

comac(ms_exp, ms_th)

Compute the coordinate modal assurance criterion (COMAC) between experimental and theoretical mode shapes.

Inputs

  • ms_exp: Experimental mode shapes (nmes x nmodes array)

  • ms_th: Theoretical mode shapes (nmes x nmodes array)

Output

  • comac: Coordinate modal assurance criterion values (nmes array)

Reference

[1] R. J. Allemang. The modal assurance criterion twenty years of use and abuse. Sound & Vibration. 37 (8): 14-23. 2003

2.3 Enhanced Coordinate Modal Assurance Criterion

The Enhanced Coordinate Modal Assurance Criterion (ECOMAC) extends the COMAC to take into account potential calibration scaling errors and/or sensor orientation errors by compouting the mean deviations of mode amplitudes of each coordinate. It is defined as: \[ ECOMAC_p = \frac{\sum_{i=1}^N \vert \phi_{pi} - \psi_{pi} \vert}{2 N}, \] where \(N\) is the number of modes.

Note

As for the COMAC indicator, the computation of the ECOMAC indicator requires that the two mode shapes have the same number of coordinates and similar scaling.

Low values of the indicator indicates a strong correlation between the two mode shapes, while high values indicate very little correlation.

ecomac

ecomac(ms_exp, ms_th)

Compute the enhanced coordinate modal assurance criterion (eCOMAC) between experimental and theoretical mode shapes.

Inputs

  • ms_exp: Experimental mode shapes (nmes x nmodes array)

  • ms_th: Theoretical mode shapes (nmes x nmodes array)

Output

  • ecomac: Enhanced coordinate modal assurance criterion values (nmes array)

Reference [1] D. L. Hunt. Application of an Enhanced Coordinate Modal Assurance Criterion (ECOMAC). Proceedings of International Modal Analysis Conference, pp. 66-71, 1992.

[2] G. Martin, E. Balmes and T. Chancelier. Improved Modal Assurance Criterion using a quantification of identification errors per mode/sensor. Proceedings of ISMA 2014, pp. 2509-2519. 2014.

2.4 Frequency response assurance criterion

The Frequency Response Assurance Criterion (FRAC) is an indicator that quantifies the correlation between two frequency response functions (FRFs) over a given frequency range for a given measurement point \(p\) and a given excitation point \(q\). For two FRFs \(H_{pq}(\omega)\) and \(G_{pq}(\omega)\), it is defined as: \[ FRAC_{pq} = \frac{\vert \sum_{\omega = \omega_1}^{\omega_2} H_{pq}(\omega) G_{pq}^\ast(\omega) \, \vert^2}{\left( \sum_{\omega = \omega_1}^{\omega_2} \vert H_{pq}(\omega) \vert^2\right) \left( \sum_{\omega = \omega_1}^{\omega_2} \vert G_{pq}(\omega) \vert^2 \right)}, \] \(G_{pq}^\ast(\omega)\) is the complex conjugate of \(G_{pq}(\omega)\), and \(\omega_1\) and \(\omega_2\) define the frequency range of interest.

frac

frac(frf_exp, frf_th)

Compute the frequency response assurance criterion (FRAC) between experimental and theoretical frequency response functions.

Inputs

  • frf_exp: Experimental frequency response functions (nmes x nexc x nf array

  • frf_th: Theoretical frequency response functions (nmes x nexc x nf array)

Output

  • frac: Frequency response assurance criterion values (nmes x nexc array)

Reference

[1] R. J. Allemang. The modal assurance criterion twenty years of use and abuse. Sound & Vibration. 37 (8): 14-23. 2003

3 Indicator functions

The following indicator functions can be used to estimate the number of modes of a structure.

3.1 Complex Mode Indicator Function

The Complex Mode Indicator Function (CMIF) is a global indicator that helps identify the number of modes in a system. It is defined as the singular values of the frequency response function (FRF) matrix \(\mathbf{H}(\omega)\) over a frequency range of interest. The FRF matrix is constructed by stacking the FRFs for all measurement points and excitation points.

When admittance or accelerance are used, the CMIF is computed as the singular values of the imaginary part of the FRF matrix, while for mobility, the real part is used. From a general standpoint, the CMIF is given by: \[ CMIF(\omega) = \mathbf{\Sigma}(\omega), \] where \(\mathbf{\Sigma}(\omega)\) is the vector of singular values obtained from the singular value decomposition of the FRF matrix.

cmif

cmif(frf; type = :dis)

Compute the complex mode indicator function (CMIF)

Inputs

  • frf: Frequency response functions (nmes x nexc x nf array)

  • type: Type of FRF (:dis for displacement, :vel for velocity, :acc for acceleration)

Output

  • cmif: Complex mode indicator function (min(nmes, nexc) x nf array)

Reference

[1] R. J. Allemang and D. L. Brown. A Complete Review of the Complex Mode Indicator Function (CMIF) with Applications. ISMA 2006. 2006

3.2 Power spectrum indicator function

The Power Spectrum Indicator Function (PSIF) is another global indicator that helps identify the number of modes in a system. It is defined as the sum of the squared magnitudes of the FRFs over all measurement points and excitation points. The PSIF is given by: \[ PSIF(\omega) = \sum_{p=1}^{N_p} \sum_{q=1}^{N_q} \vert H_{pq}(\omega) \vert^2, \] where \(N_p\) is the number of measurement points, \(N_q\) is the number of excitation points, and \(H_{pq}(\omega)\) is the FRF between measurement point \(p\) and excitation point \(q\).

psif

psif(frf)

Compute the power spectrum indicator function (PSIF).

Input

  • frf: Frequency response functions (nmes x nexc x nf array or nmes x nf array)

Output

  • psif: Power spectrum indicator function (nf array)

Reference

[1] M+P Analyzer manual. Rev. 5.1. 2017