This page discusses some more practical issues related to the virtual point experiment. Proper positioning of the measurement and excitation points is essential to obtain a high quality model. Care should be taken to ensure that all 6 DoFs per virtual point can be described independently. This has implications for both sensor placement and impact positions.
The use of triaxial accelerometers has become standard practice in experimental testing. A single accelerometer hosts 3 sensing devices in orthogonal directions, that can be assumed to be collocated. Although one pair of those sensors measures a total of 6 channels, it is normally not sufficient to describe all 6 DoFs of the virtual point. One linear dependence will appear in the virtual points DoFs: the triaxial sensors are unable to describe the rotation over the axis spanned between both sensors, as illustrated by the following figure:
A) Two sensors spanning a line: one dependency exists between the rotational axis.
B) Three sensors spanning a surface: all rotations are fully determined.
Introduction of a third triaxial sensor enables the three sensors with a total of 9 DoFs to describe all 6 DoFs of the virtual point properly.The additional benefit of the third triaxial sensor is the overdetermination of the interface problem. Applying the virtual point transformation reduces the effects of uncorrelated measurement noise, as well as bias errors due to misalignment of the sensors. The use of at least three triaxial sensors per virtual point can therefore be held as rule of thumb, if the aim is to describe the virtual point by 6 independent coordinates.
Finally, considering the rigid interface assumption, it seems logical to place the sensors as close as possible to the virtual point. The smaller the distances are, the lesser the effects of flexible interface motion compared to the rigid interface motion. On the other hand, for smaller distances, the virtual point transformation gets more sensitive to absolute errors on the position. In general, one should approach this aspect with some engineering judgement or foreknowledge about the system and measurement equipment.
Unlike the 6 accelerations measured by 2 triaxial sensors, 6 well-positioned hammer impacts are potentially sufficient to fully determine the 6 generalised loads of the virtual point. Still, for the same reason, it is preferable to use more excitation points and overdetermine the force transformation. Similar to the sensor placement, three impacts in each direction (x, y, z) can be used as minimum, creating 9 columns in the FRF matrix for each coupling point. However, as one is not restricted by the available measurement equipment (as is the case for the simultaneous sensor positions), it is advised to excite at much more points.
Care should be taken to include excitation directions that do not point straight to the virtual point, in order to generate “moment” along the rotational axes. Besides that, it is not suggested to impact the faces of sensors, as FRFs obtained at the sensors’ faces exhibit poor coherence, especially for the cross-directional FRFs of one sensor. Also, the sensor is easily driven in overload. As the virtual point transformation does not require physical driving-point measurement (driving-point FRFs are correctly rendered by the two transformations), this type of excitation is discouraged.
With respect to the distance of impact points from the virtual point, a similar reasoning applies as for the sensors. However, remind that the uncertainty of hammer impacts is highly subject to the skill of the experimentalist. Position errors are always made, typically in the order of a few millimetres. To minimise their effect for the transformation, it is advised to excite at some more distance from the virtual point. This is possible as long as the local rigidness assumption is still justified, which is normally a matter of the targeted frequency bandwidth.
DIRAC offers quality indicators that assess if the sensor and impact placements are sufficient for the VP calculations. For deeper dive please be referred to the DIRAC manual.
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