3 Mechanical Environment Test Specifications Derived from Equivalent Energy in Fixed Base Modes 33 3.6 Extracting the Nominal Fixed Base Modal Cross Spectra from System Level Test Now that the system level acoustic test data and the free modal test data are available, we obtain the modal based cross spectra of the fixed base elastic modes of the RC due to the acoustic loads in the MATV test. The cross-spectra provide auto spectral density for each degree-of-freedom (DOF) and the relationships to the other degrees-of-freedom. The starting point is the cross-spectra of the response from the acoustic MATV test relating the sensor DOFs which will be called Sxx. The free modal test data is then used to filter Sxx from the accelerometer DOFs into modal DOFs space. In this, we are interested in the determining the fixed base modes of the RC on the plate to replicate the MATV test responses. The first five elastic modes from the modal tests involved elastic motion of the RC with the base essentially remaining fixed. The 6th mode is a twisting mode of the plate fixture and will not be included in the modal filtering of Sxx because we are focusing on the rigid body motion of the fixture on the 6DOF shaker. To continue, a few more terms need to be defined. T from Eq. (3.12) transforms the mode shapes to fixed base and rigid mode shapes F/R in Eq. (3.18). F/R = T (3.18) Now we define Hxs using Hps from Eq. (3.16) that relates the DOFs of the rigid body motion of the fixture plate, ¯s, to the MATV accelerometer DOFs, ¯x as seen in Eq. (3.19). Hxs = F/R Hps I (3.19) WithHxs defined, the accelerometer DOF spectral density, Sxx,MATV, can be transformed to the rigid body motion of the test fixture plate, SssTest as shown in Eq. (3.20). The superscript, H, denotes the Hermitian transpose. SssTest =H+xsSxxMATV H+H xs (3.20) Now the test rigid body spectral density, SssTest, can be used to find the fixed base elastic modes spectral density response, SppTest, with the transform in Eq. (3.21) and can also be used withHps from the modal test to determine the RC accelerometer spectral density responses, SxxTest, due to the test input through Eq. (3.22). SppTest =HpsSssTest HT ps (3.21) SxxTest =HxsSssTest HT xs (3.22) Figures 3.6 and 3.7 show the physical sensor responses of the RC comparing the auto spectral density responses from MATV, SxxMATV, to the responses from the 6DOF base excitation test, SxxTest, which match very well at most frequencies. 3.7 Calculated 6 DOF Base Input Specs to Ensure Conservatism on Fixed Base Modal DOF Based on Variability One of the goals of this work was to generate test specifications that would guarantee quantifiable conservatism on response of the RC comparing the responses between the MATV and the 6DOF test. One thing that we wanted to look at was unit to unit variability. As noted in the section discussing the modal testing of the RC on the fixture plate, modal testing was performed for the RC on the rigid plate fixture with three different torque values to mimic unit to unit variability. We expected that the modal frequencies and the modal damping values would differ by increasing or decreasing the torque values. Looking at the modal frequencies, it was immediately noted that the modal frequencies essentially stayed the same. Looking at the damping, the values didn’t change much either, but it was more significant than the modal frequencies. The process shown in
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