3 Evaluating the quality of your model

The most important in a nutshell:

• With DIRAC, you can measure high-quality FRFs on the first attempt based on several quality indicators.
  • You can save time by intuitively identifying the most common errors on the first impacts. 
  • The Analyze module provides an intuitive matrix overview where you can review your entire measurement at once.
• On this page, you will find a summary of the most important quality indicators for Virtual Point measurements. 

This is how to proceed:

• Signal checks:
  • After the Design of Experiment, you can perform an initial excitation and visualize the Operating Deflection Shapes (ODS). This is an initial visual check that can be used to identify cabling/alignment errors of the sensors. 
  • Coherence of measured signals checks how repeatable measurements of a particular impact were performed. 
  • Magnitude can reveal internal sensor overloads that occurred during measurement.
• VP Transformation checks:
  • Force consistency can indicate problems associated with impacts. 
  • Response consistency may reveal issues related to the sensors.
• VP quality checks:
  • Passivity indicates the physical plausibility of the measurement. 
  • Reciprocity indicates the physical plausibility of the measurement.

Evaluating 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 Shapes (ODS)

The ODS of the accelerometers (and other sensors) illustrates the motion at a specific frequency. This can help to understand the mechanism in modes of interest. At lower frequencies, the ODS should mainly show the component moving like a rigid body. 

• 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 levels.
• White/brighter columns indicate problems at specific impacts, e.g., bad repeatability or insufficient excitation level.

Consistency

The consistency metric evaluates the validity of the Virtual Point Transformation of the measurements. It takes geometric information and physical modeling 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 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 checking 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.