3 Mechanical Environment Test Specifications Derived from Equivalent Energy in Fixed Base Modes 51 Fig. 3.15 Example component on test fixture Given this test configuration, the modal parameters from a free modal test of the test article and fixture produce the following equations of motion in Eq. (3.1). ω 2 free +j2ωωfreeζfree −ω 2 I ¯q =0 (3.1) In this equation, the subscript “free” represents the set of modes obtained from the experimental modal test of the test article and the fixture. The mass-normalized mode shapes derived from the test with the analytically calculated rigid body modes will be contained in the measured mode shape matrix, Φ. The goal now is to derive a square matrix transformation, T, that will convert Eq. (3.1) to a modal CB form. We now define the generalized coordinates, ¯p, as the fixed base elastic modal coordinates and the generalized coordinates, ¯s, as the coordinates that account for the rigid body motion of the fixture as shown in Eq. (3.2). ¯q =T ¯ p ¯s (3.2) To construct T, first consider a constraint that ties the free rigid body modes of the fixture to the component. Now, use the modal approximations to set the motion of the experiment on the fixture to match the free rigid body modal motion of the fixture with the relationship in Eq. (3.3). b ¯q ≈ b¯s (3.3) In this equation, the subscript b represents the degrees-of-freedom (DOFs) of the fixture boundary where measurements aremade, Φis the experimental mode shape, and Ψ is the set of rigid body modes of the fixture. Ψ can come from a model of the fixture or experimental measurements. Then the relation between ¯q and ¯s is defined in Eq. (3.4). ¯q = + b b¯s (3.4) For this equation, the “+” sign represents the Moore-Penrose pseudo inverse. This provides the right-hand partition of the transformation, T, associated with the ¯s DOFs. To continue constructing T, next we focus on the fixed base modal DOFs, ¯p, describing the elastic motion of component A, with the fixed boundary DOFs as shown in Eq. (3.5). xb = b ¯q =0 (3.5) Previous work has shown that a practical way to accomplish Eq. (3.5) is to fix the fixture DOF as shown in Eq. (3.6) [6]. + b b ¯q =s =0 (3.6) Using Rixen’s primal assembly, the modal DOFs are replaced as shown in Eq. (3.7) [7]. ¯q =Lfix¯η (3.7) This equation is substituted back into Eq. (3.6) to obtain Eq. (3.8). + b bLfix¯η =0 (3.8)
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