52 T. J. Skousen and R. L. Mayes Since ¯η can be many vectors, depending on the forcing motion, Lfix is chosen to guarantee satisfaction of the constraint leading to Eq. (3.9). Lfix =null + b b (3.9) Pre- and post-multiply Eq. (3.1) using the transformationLfix gives Eq. (3.10). LT fix ω 2 free −ω 2 I Lfix¯η =0 (3.10) Solving Eq. (3.10) Produces the eigenvectors, Γ, and the eigenvalues to uncouple the fixed base modal DOFs, ¯p. Then the relationship between q and ¯pbecomes Eq. (3.11). ¯q =Lfix ¯p (3.11) This equation provides the rest of the transformation T which is written from Eqs. (3.4) and (3.11) as Eq. (3.12). T = Lfix + b b (3.12) Pre-multiplying Eq. (3.1), modified for rigid body input with, Fs, by the transpose of T and substituting Eq. (3.2) into Eq. (3.1) for ¯q yields Eq. (3.13) which transforms the equations of motion for free vibration. \ω 2 fix\ Kps KT ps Kss +jω Cpp Cps CT ps Css −ω 2 I Mps MT ps Mss ¯p ¯s = 0 Fs (3.13) With this transformation, the eigenvalue and eigenvector solution have not changed from Eq. (3.1). It has exactly as many DOFs as Eq. (3.1), but now they have been transformed to the fixed base modes associated with ¯pand the free fixture rigid body modes which are on the boundary as modal DOFs ¯s. The upper left portion of the matrices is diagonal. The size of the upper left partition of the matrices is a square matrix based on the number of elastic fixed base modes used. The size of the lower right partition of the matrices is based on the number of rigid body modes that will be used as inputs to the fixtured test. Now there are mass coupling terms, Mps, between the fixed base modes and the rigid body fixture motion. Assuming the stiffness of the rigid body modes is zero, the Kss, Kps, Css, andCps terms will all be zeros. Considering the top line of Eq. (3.13) and moving the fixture DOFs, ¯s, to the right-hand side develops equations of motion from enforced fixture motion in Eq. (3.14). \ω 2 fix\ +jw Cpp −ω 2 [I] {p}= 2 Mps {s} (3.14) Rearranging this equation, the Hps matrix can be formulated as shown in Eq. (3.15). { ¯p}=Hps {s} (3.15) Hps in Eq. (3.15) is represented as Eq. (3.16) Hps = \ω 2 fix\ +jω Cpp −ω 2 [I] −1 ω 2 Mps (3.16) This equation allows the six rigid body modes, s, to drive the fixed base elastic modes, ¯p. If the ¯p elastic responses are known, the ¯s rigid body inputs to give a best fit to the responses can be calculated. If one pre-multiplies Eq. (3.2) by , we can see Eq. (3.17) that relates the accelerometer DOFs, x, to the rigid body modal DOF of the fixture, s, and the fixed base modal DOF of component A, ¯p. x =Φ¯q =ΦT ¯ p ¯s (3.17)
RkJQdWJsaXNoZXIy MTMzNzEzMQ==