24 Forcing Function Estimation for Space System Rollout 255 FCG FT −FM = T ∗U 0 0 U N −Mbj ∗U1U ∗ ⎡ ⎣ α β γ ⎤ ⎦=B2 =A2C. (24.38) The constraint equations are used to assure that the rigid body part of the FF maintains the expected force at the CG as well as to assure that the flex body and rigid body part of the Below-JEL accelerations match up with the below JEL accelerations. Eqs. (24.32), (24.33), and (24.34) are again used to solve these equation and the updated total force is reconstructed using Eq. (24.30). Equation (24.38) can be used with an initial guess of FT =FMor can be used to modify an updatedFT toassure that the constraints are met during iterative processing. Step #10 – Determine FRF Between Total Below-JEL Forces and Original Data Estimate FRF Step #10 takes the current force estimates and the measured responses to estimate a Frequency Response Function (FRF). The Total Least Squares using an SVD approach as discussed in Reference [18] is the primary approach for FRF estimation (denoted as H4) used in this work. However, other common approaches as implemented in Reference [15] can be used (denoted as H1, H2, and H3). Under the H4 approach, the frequency domain version of the input and output for each winsize/winskip0 segment are collected (each row is a different segment and each column is a different force or measured acceleration) into the same matrix representation (Y): [Y (ω)]H = [FT (ω)] H [X(ω)]H =[UY] [SY] [VY] H; (24.39) Where UY is the matrix of left singular vectors of Y; SY is the diagonal matrix of non-zero singular values of Y; and VY is the matrix of right singular vectors of Y. Therefore, the total least squares FRF is given by the following: [H4] =[VYX12] [VYF12]−1 . (24.40) where: [VY] =[[VY12] [VYn12]] = [VYF12] [VYFn12] [VYX12] [VYXn12] . (24.41) The subscripts defining the partitions of VY in Eq. (24.41) can be interpreted as follows: (1) F12 are first nref columns and nref rows; (2) X12 are the first nref columns and last nres rows; (3) Fn12 are the last nres columns and the first nref rows; and (4) Xn12 are the last nres columns and the last nres rows. The “nref ” descriptor refers to the number of assumed references or inputs (12 in the case of the data provided in this paper). The “nres” descriptor refers to the number of measured responses or outputs (63 in the case of the data provided in this paper). Estimate Inverse FRF Step #10 also estimates an inverse transfer function called G4 by performing a pseudoinverse of H4: [G4] =[VYFnres] [VYXnres]−1 . (24.42) Where:
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