6 Evaluating Blocked Force quality

On-board validation

When evaluating the quality of computed Blocked Forces, an initial check that is often called on-board validation can be done. The Blocked Forces, computed on an assembly AR (source component A on a test rig R) using the indicator responses $$ \mathbf{u}_4^{\mathrm{AR}} $$, can be used to predict the vibration at the target $$ \mathbf{u}_3^{\mathrm{AR}} $$ responses on the same assembly.
These target responses $$ \mathbf{u}_3^{\mathrm{AR}} $$ were recorded parallel to $$ \mathbf{u}_4^{\mathrm{AR}} $$ during the same operational measurements, but did not contribute to the calculation of the Blocked Forces.

$$ \mathbf{f}_{2}^{\mathrm{bl}} = (\mathbf{Y}_{42}^\mathrm{AR})^\mathrm{{+}}\mathbf{u}_{4}^\mathrm{AR} $$

Subsequentlty, the target responses can also be predicted $$ (\mathbf{u}_3^{\mathrm{AR,pred}}) $$ by multiplying the Blocked Forces $$ \mathbf{f}_2^{\mathrm{bl}} $$ to the previously determined $$ \mathbf{Y}_{32}^{\mathrm{AR}} $$ FRF.

$$ \mathbf{u}_{3}^{\mathrm{AR,pred}} = \mathbf{Y}_{32}^{\mathrm{AR}} \mathbf{f}_{2}^{\mathrm{bl}} = \mathbf{Y}_{32}^{\mathrm{AR}} (\mathbf{Y}_{42}^{\mathrm{AR}})^{+}\mathbf{u}_{4}^{\mathrm{AR}} $$

Here, we can distinguish between “Characterisation FRF” $$ \mathbf{Y}_{42}^{\mathrm{AR}} $$ and the “Synthesis FRF” $$ \mathbf{Y}_{32}^{\mathrm{AR}} $$.
If the interface description is complete, this should yield vibration levels equivalent to those recorded during the measurement.

In case the description of the Blocked Forces is inappropriate, e.g. since a critical transfer path on the interface was neglected, this would manifest in bad predictability of the measured $$ \mathbf{u}_{3}^{\mathrm{AR}} $$.

Note, to perform an on-board validation, it is not always mandatory to use actual operational data. In many cases, artificial excitations on a few well-defined locations on the idle source component, e.g. by a modal hammer, can already serve as a quick and easy way to gain similar insights.

Transfer validation

When confronted with the task to derive Blocked Forces for any given active component using component-based TPA, one would initially perform an on-board validation to check if the computed Blocked Forces are valid.

However, an on-board validation can potentially give a too optimistic impression on the quality of the identified Blocked Forces. Therefore, transfer validation on a second passive subsystem, e.g., the target assembly AB or a test rig with varying dynamic properties, is recommended.

For the calculation, we multiply the previously derived Blocked Forces by $$ \mathbf{Y}_{32}^{\mathrm{AB}}  $$ an FRF of the second assembly, i.e., use the same operational data and Characterization FRF as for the on-board validation, but change the Synthesis FRF to match the second assembly:

$$ \mathbf{u}_{3}^{\mathrm{AB,pred}} = \mathbf{Y}_{32}^{\mathrm{AB}} \mathbf{f}_{2}^{\mathrm{bl}} = \mathbf{Y}_{32}^{\mathrm{AB}} (\mathbf{Y}_{42}^{\mathrm{AR}})^{+}\mathbf{u}_{4}^{\mathrm{AR}} $$

Even if the on-board validation had shown an appropriate description of the Blocked Forces $$ (\mathbf{u}_{3}^{\mathrm{AR,pred}} \cong \mathbf{u}_{3}^{\mathrm{AR}})  $$ the outcome of the transfer validation can look completely different. E.g., this can be the case if the FRFs at some frequency ranges show low singular values. Those can be interpreted as dynamically “stiff” directions, where random measurement noise is more dominant.

However, these errors might be masked within the on-board validation, since the resulting Blocked Forces are again multiplied by the same “stiff” FRFs limiting the noise contamination.

Yet, an alternate set of FRFs is applied for the transfer validation, which should show a different dynamic behavior. In this case, the noise is not masked anymore but can even be further amplified by resonances that can be present in the once “stiff” frequency range.

Thus, the Blocked Forces can be considered of good quality only when the transfer validation also shows a good prediction $$ (\mathbf{u}_{3}^{\mathrm{AB,pred}} \cong \mathbf{u}_{3}^{\mathrm{AB}}) $$.

Interface Completeness Criterion (ICC)

Another valuable check for testing the “completeness” of the interface description can be given by the Interface Completeness Criterion (ICC). It is a way to describe the degree to which the coupling interface dynamics are accounted for by the coupling interface admittance matrix $$ \mathbf{Y}_{\mathrm{31}} $$ (see the paper at this link).

This FRF can be derived from artificial excitations in two ways. First, one may derive$$ \mathbf{Y}_{\mathrm{31}} $$ from the measured forces $$ \mathbf{f}_{\mathrm{1}} $$ and responses $$ \mathbf{u}_{\mathrm{3}} $$ directly. Alternatively, one may perform a round trip over the interface to derive a secondary set of FRF $$\mathbf{\widetilde{Y}}_{\mathrm{31}}$$. The round trip method is found as 

$$ \mathbf{\widetilde{Y}}_{\mathrm{31}} = \mathbf{{Y}}_{\mathrm{32}}\left (\mathbf{{Y}}_{\mathrm{22}} \right )^{-1} \mathbf{{Y}}_{\mathrm{21}} $$


$$ \mathbf{\widetilde{Y}}_{\mathrm{31}} = \mathbf{{Y}}_{\mathrm{32}}\left (\mathbf{{Y}}_{\mathrm{42}} \right )^{+} \mathbf{{Y}}_{\mathrm{41}} $$

Comparing these two sets of FRF ($$\mathbf{{Y}}_{\mathrm{31}} $$ and $$\mathbf{\widetilde{Y}}_{\mathrm{31}}$$) allows us to understand the completeness. Two approaches for a criterion will be described in the following sections. 

MAC-like criterion

By applying an expression similar to the Modal Assurance Criterion (MAC), the ICC yields a frequency-dependent scalar value bound between 0 and 1. It is defined as: 

$$\mathrm{ICC}_{\mathrm{MAC}} = \frac{\left | \left ( \mathbf{Y}_{31} \right )_{:,j}^{\mathrm{H}} \left ( \mathbf{\widetilde{Y}}_{31} \right )_{:,j} \right |^2}{\left ( \mathbf{Y}_{31} \right )_{:,j}^{\mathrm{H}} \left ( \mathbf{Y}_{31} \right )_{:,j} \left ( \mathbf{\widetilde{Y}}_{31} \right )_{:,j}^{\mathrm{H}} \left ( \mathbf{\widetilde{Y}}_{31} \right )_{:,j} }$$

 The notation $$ \left ( \right )^\mathrm{H}$$ indicates the conjugate or Hermitian transpose of the respective matrix. 

If all coupling DoFs are accounted for, that is, $$\left ( \mathbf{\widetilde{Y}}_{31} \right )_{:,j} = \left ( \mathbf{Y}_{31} \right )_{:,j}$$, the ICC is equal to one. Similarly, if none of the coupling DoFs are accounted for, that is, $$\left ( \mathbf{\widetilde{Y}}_{31} \right )_{:,j} = 0$$, the ICC is equal to zero. Note, the ICC is calculated for a single measured impact on the source subsystem. As such, a different ICC is obtained for each artificial excitation (per column $$j$$ of the FRFs), as denoted by the subscript $$\left ( :,j \right )$$. Although the criterion, in theory, yields 1 for a complete interface description, this is not likely to be achieved by measurements, since $$\mathbf{Y}_{31}$$ and $$\mathbf{\widetilde{Y}}_{31}$$ are prone to measurement uncertainty. 

Coherence-like criterion

The MAC-like criterion described above can potentially be troublesome since it is only sensitive to the shape of the vectors and the phase of their entries. However, it is not sensitive to differences in the magnitude of the response vectors.  

Therefore, we propose another measure for computing the ICC, which is also sensitive to magnitude differences in the predicted and measured response. Basically, this one is evaluating the similarity between $$\mathbf{Y}_{31}$$ and $$\mathbf{\widetilde{Y}}_{31}$$ for every matrix entry or response/source combination, as denoted by the subscripts $$(i,j)$$. This comes down to comparing the similarity of two complex numbers at each frequency, which can be done with a coherence-like definition: 

$$\mathrm{ICC_{COH}} = \frac{\left ( \mathbf{Y}_{31} \right )_{i,j} \left ( \mathbf{\widetilde{Y}}_{31} \right )_{i,j} + \left ( \mathbf{Y}_{31} \right )_{i,j}^* \left ( \mathbf{\widetilde{Y}}_{31} \right )_{i,j}^* }{2\left ( \left ( \mathbf{Y}_{31} \right )_{i,j} \left ( \mathbf{Y}_{31} \right )_{i,j}^* + \left ( \mathbf{\widetilde{Y}}_{31} \right )_{i,j}\left ( \mathbf{\widetilde{Y}}_{31} \right )_{i,j}^* \right ) }$$

 The notation $$()^{*}$$ indicates the complex conjugate of the respective matrix entry. 

When averaging the results of a full column of the matrices, this definition produces similar results to the MAC-like criterion but is sensitive to phase and amplitude differences. 

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