3 Mechanical Environment Test Specifications Derived from Equivalent Energy in Fixed Base Modes 53 In this equation, we can see that the transformed fixed base and rigid body mode shapes are given by T. In this work, we measure the responses, x, of field hardware and estimate the ¯presponses from Eq. (3.16). Equation (3.16) is approximate and subject to modal truncation errors. The entire process utilizing Eq. (3.1) through Eq. (3.17) could be accomplished with a finite element model as well as a modal test. A.1.2 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 crossspectra 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 calledSxx. 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 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 sixth 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 6 DOF shaker. To continue, a few more terms need to be defined. T from Eq. (3.12) transforms the mode shapes to fixed base (F) and rigid (R) mode shapes F/R in Eq. (3.18). ΦF/R =ΦT (3.18) Now we defineHxs using Hpsfrom 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 =ΦF/R! Hps I " (3.19) With Hxs defined, the accelerometer DOF spectral density, SxxMATV, 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+xsSxxMATVH+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 =HpsSssTestHT ps (3.21) SxxTest =HxsSssTestHT xs (3.22) Figures 3.16 and 3.17 show the physical sensor responses of the RC comparing the auto spectral density responses from MATV, SxxMATV, to the responses from the 6 DOF base excitation test, SxxTest, which match very well at most frequencies.
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