The Virtual Point Transformation (VPT) makes it possible to experimentally identify the full DoF FRF matrix by projecting the measured displacements onto the Interface Deformation Modes (IDMs). The VP FRFs were already successfully used in Frequency-Based Substructuring (FBS); however, the VPT is susceptible to deviations in the impact location and orientation, as well as to deviations in the sensor’s sensitivity and positioning. Uncertainties associated with the sensors can be decreased by using the expanded VPT. This expanded VPT allows the projection of a directly measured rotational response onto the Interface Deformation Modes (IDMs). The consistency of the transformation is achieved by using a rotational weighting matrix, which is formulated to minimize the norm of the overall displacements due to the rotational residual at the VP for each rotational sensor. The rotational response is measured using a direct piezoelectric rotational accelerometer. In this paper the application of the expanded virtual point transformation and the possible advantages are explored on a complex and engineering-like test structure. Both transformations, standard and expanded, are performed for each VP to enable a side-by-side comparison.
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An interface is considered to be overdetermined when more DoF are included in the coupling than are in fact needed. Consider a perfectly rigid interface, where perfect coupling can be performed with only 6 DoFs (3 translation and 3 rotational). However, with the EMPC method one would need to use 3 triaxial translation accelerometers to account for the rotational DoFs. Therefore, the interface would be overdetermined.
An explicit dependency on the frequency is omitted for the simplicity of the notation, and this will also be used for the remainder of the paper.
The sensors used for measuring the interface displacement u are usually tri-axial accelerometers, but for the simplicity of the notation, displacements are considered here.
The first number in the bracket (2,4) refers to the output position and the second to the input position in the FRFs matrix i.e. (output,input).
With the three VPs on the TS, a certain level of over-determination can be expected at the interface, since the whole block is in fact rigid in the low-frequency range. Therefore, even a small error in one of the VPTs can lead to erroneous coupling results. Applying a truncated SVD on the interface flexibility Yint = BYA|BBT (see equation (2)) can be interpreted as a weakening of the interface compatibility, which is advantageous in this configuration.
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The authors acknowledge the partial financial support from the core research funding P2-0263 and applied research project L2-1837, both financed by ARRS, Slovenian research agency.
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Bregar, T., El Mahmoudi, A., Čepon, G. et al. Performance of the Expanded Virtual Point Transformation on a Complex Test Structure. Exp Tech 45, 83–93 (2021). https://doi.org/10.1007/s40799-020-00398-1
- Frequency-based substructuring
- Expanded virtual point transformation
- Rotational accelerometer
- Interface rotation