120 J. Harvie and D. de Klerk the data at these low frequencies. If the data are not consistent at the low frequencies, the virtual point transformation can be used to rigidify the FRFs. Using Eqs. 10.4 and 10.5, a set of FRFs can be transformed to a single 6-DoF virtual point near the center of an object, and then projected back to the original set of degrees of freedom. However, this new set of FRFs will only be accurate in the frequency range where the test object behaves rigidly. Therefore, prior knowledge of the first flexible mode of the test object should be used to select a cutoff frequency where the rigidified FRFs can be cross-faded into the original set of FRFs. Additionally, the FRF matrix will now be very poorly conditioned at the low frequencies where the data has been rigidified, containing only six substantial singular values to represent whatever FRF size has been measured. To use the new FRF matrix for in-situ TPA, or any activities involving a matrix inverse, only the first six singular values should be included in a truncated inverse. The new hybrid set of FRFs should produce more accurate TPA results at the lower frequencies while leaving the higher frequencies unaffected. Reciprocal FRFs for Mid-Frequency TPA Predictions As noted in Eq. 10.3, component-based TPA can be used to predict responses of interest (accelerations or sound pressure levels) using equivalent forces and the FRFs of the assembled system. The FRFs YAB 32 contain the responses at the locations of interest relative to the interface degrees of freedom. This FRF matrix is usually obtained by making several impacts around each interface and measuring the responses of interest. However, these FRFs can be obtained in an alternative way due to the property of reciprocity. Due to reciprocity, we can instead measure YAB 32 , with inputs at the locations where responses are desired, and outputs at the interface degrees of freedom. A review of reciprocity in this context is provided in [7, 8]. Practically, this means installing sensors around the interfaces, and then exciting the structure at the locations of interest using either an impact hammer for structural interests or a volume source for acoustic interests. The actual testing time will be greatly reduced for a test of this form. These FRFs can then be used in TPA predictions across whatever frequency range has good coherence, which is typically best in the mid-frequency range. Rotational FRFs for Mid- to High-Frequency TPA In general, it is easier to measure translational forces than rotational moments for both classical and component-based TPA. While 6-DoF force gauges exist that can be used to directly measure forces and moments, those force gauges can often be substantial in size, and therefore may not be able to measure the desired quantities exactly at the interfaces. With FRF-based TPA techniques, impact hammers are naturally only able to measure translational force inputs. Thus, TPA is often performed using only translational forces. TPA using only translational forces is likely accurate at low frequencies in most cases where the behavior of the components and systems is fairly rigid. However, with the current trends in the automotive industry shifting toward electric vehicles and components, higher frequencies are becoming more and more relevant and troublesome. At the higher frequencies where the dynamics are more complex, translational forces alone are likely inadequate for source characterization. It will be shown that including rotational degrees of freedom in the virtual point transformation can improve the results at those higher frequencies. The advantages of including rotations in source characterization have previously been discussed in [9] where the recommendation to include rotations in two upcoming ISO standards [10] [11] was introduced. Additionally, the use of an on-board validation sensor to validate the accuracy of the calculated blocked forces is also presented in [9] and will be used in this work. 10.2 Analysis 10.2.1 Rigidness Correction for Low Frequency TPA An active component was recently tested on a rigid test bench to determine blocked forces using various methods. The structure was equipped with 32 tri-axial accelerometers (96 channels) during both FRF testing and operational measurements, in order to calculate in-situ blocked forces with Eq. 10.2. The consistency of these sensors during FRF testing is shown in the left side of Fig. 10.3. As seen in the figure, the consistency is close to 100% from around 20 to 40 Hz. This means that all sensors are moving rigidly together in this frequency range. There are modes of the structure above 40 Hz (related to both the component and the test bench), so the consistency of all sensors is not expected to be high in that frequency range. However, at frequencies below 20 Hz, the structure should be moving rigidly and the sensor consistency should be higher than it is. It is believed that the poor consistency in this frequency range is primarily due to poor signal-to-noise ratio (SNR) on the very stiff test bench.
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