Evaluating quality of your model

Being able to evaluate the quality of your model in real-time during the measurement is crucial for efficiently measuring accurate and reliable data. This page summarizes the main quality indicators that can be observed in DIRAC.

Operational Deflection Shape (ODS)

ODS are very helpful to make sense of the data. At lower frequencies, the ODS should mainly show the source moving like a rigid body in the soft mountings.

  • Therefore, this is a first visual check to spot cabling / orientation errors on sensors.

Coherence

The coherence indicates how well the excitation signal correlates with the response signal. The coherence matrix overview in DIRAC allows to check the repeatability of all impact measurements at once:

  • White/brighter rows indicate problems at specific sensor channels, e.g., broken cables or high noise level.
  • White/brighter columns indicate problems at specific impacts, e.g., bad repeatability or not enough excitation level.

Consistency

The consistency metric evaluates the validity of the virtual point transformation of the measurements. It takes geometric information and physical modelling assumptions of the virtual points into account. Thus, the consistency evaluates the integrity of the experimental model of the system (and not just the individual signal – see coherence). The consistency is a MAC like comparison of measured signal vs. signal after filtering out non-rigid motion.

Response consistency

  • The response consistency estimates how rigid the body is by comparing the similarity of the ‘filtered’ motion $$ \mathbf{\tilde{u}} $$ with the ‘unfiltered’ motion $$ \mathbf{u} $$. If the sensors are on a structure that moves like a rigid body, the response consistency equals 1.

$$ \mathbf{u = R}_{\mathrm{u}} \mathbf{q + \boldsymbol{\mu} = \tilde{u} + \boldsymbol{\mu}} $$

$$ \mathbf{\tilde{u} = R}_{\mathrm{u}} \mathbf{T}_{\mathrm{u}} \mathbf{u} $$

consistency $$ (\mathbf{u, \tilde{u}}) $$

  • A low response consistency in the low frequency range can indicate wrong sensor positioning (see ODS). If this is the case in the high frequency range only, this indicates flexibility between the respective sensors. The matrix overview in DIRAC allows to check response consistency for all sensors at once:

Force consistency

  • The force consistency estimates compares the response motion $$ \mathbf{u} $$ due to the ‘unfiltered’ excitation forces $$ \mathbf{f} $$ with the response motion $$ \mathbf{\tilde{u}} $$ due to the ‘filtered’ forces $$ \mathbf{\tilde{f}} $$ (which result from the reduction to the VP and transforming back to the original position). If these excitations cause the same responses, the force consistency equals 1.

$$ \mathbf{u = Y}_{\mathrm{uf}} \mathbf{f} $$

$$ \mathbf{\tilde{u} = Y}_{\mathrm{uf}} \mathbf{\tilde{f} = Y}_{\mathrm{uf}} \mathbf{T}_{\mathrm{f}}^ {\mathrm{T}}  \mathbf{R}_{\mathrm{f}}^{\mathrm{T}}  \mathbf{f} $$

consistency $$ (\mathbf{u, \tilde{u}}) $$

  • A low force consistency in the low frequency range can indicate a wrong excitation position / angle. If this is the case in the high frequency range only, this indicates flexibility between the respective impacts. The force consistency matrix overview in DIRAC allows to check all impact measurements at once.

Magnitude

Sensors can show signs of electrical overload below their actual maximal voltage. These overloads are not always detected by the measurement system.

  • Focusing on the maximum magnitude in the very low frequency range (0-20Hz) in the matrix overview in DIRAC allows to quickly spot the characteristic “ski-slope” signal of these overloads:

Passivity

For a passive system, the phase of a driving point accelerance FRF must be positive. This can be derived from the modal superposition of the accelerance FRF:

$$ \mathbf{Y}_{\mathrm{qm,}\mathit{ij}}({\omega }) = \frac{\mathbf{\ddot{u}}_{\mathit{i}}({\omega })}{\mathbf{f}_{\mathit{j}}({\omega })} = \sum_{s=1}^{n_{modes}} \frac{-{{\omega }^2 \mathbf{x}_{s,\mathit{i}}} {\mathbf{x}_{s,\mathit{j}}}^{\mathrm{T}}}{-{\omega}^2 {\mu}_s + \mathit{i}{\omega \beta}_s + {\gamma }} $$

  • For driving points $$ \mathit{(i = j)} $$, the numerator is always a negative real number $$ {(-{{\omega }^2 \mathbf{x}_s^2 (\mathit{i}))}} $$. The denominator is always a complex number with positive imaginary part (modal damping $$ {\beta_s} $$ should always be positive for a passive system).
  • The diagonal of the VP transformed accelerance FRF $$ \mathbf{Y}_{\mathrm{qm}} $$ represents a driving point FRF and should therefore fulfill this criterion for a passive system. Fulfilling passivity indicates a good quality of the measured model.

Reciprocity

As the virtual point FRF has collocated DoFs for forces and responses, strict reciprocity is required for the off-diagonal FRFs.

$$ \mathbf{Y}_{\mathrm{qm},\mathit{ij}} = \mathbf{Y}_{\mathrm{qm},\mathit{ji}} $$

This characteristic can also be observed from the modal superposition equation of the accelerance FRF. If the VP-transformed FRF shows symmetry, this confirms a good quality of the model.

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