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” $$ \mathbf{Y}_{32}^{\mathrm{AR}} $$ and the “Synthesis FRF” $$ \mathbf{Y}_{42}^{\mathrm{AR}} $$.
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 $$ (\mathbf{u}_{3}^{\mathrm{AB,pred}} \cong \mathbf{u}_{3}^{\mathrm{AB}}) $$.

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