256 G. James et al. [VY] =[[VYnres] [VY12]] = [VYXnres] [VYX12] [VYFnres] [VYF12] . (24.43) Previous work on these systems directly applied Eqs. (24.39), (24.40), and (24.41) with the role of accelerations and forces reversed, but that option has not been exercised in this work reported in this paper [9]. Estimate Delta X For the full size FRF approach, the estimated forces and FRF can be used to reconstruct the measured sensors (X) and the resulting residuals ( X): [H4 (ω)] [F (ω)] =[Xs (ω)] =[X(ω)] +[ΔX(ω)] . (24.44) The full time history of the synthesized accelerations would use the process covered in Eqs. (24.1), (24.2), and (24.3). Subsequent iterations would attempt to drive the residual from Eq. (24.44) towards zero and therefore reproduce the measured accelerations. Estimate Delta F The inverse FRF approach can be used to estimate updates to the 12 DOF forcing functions using the 1st order assumed FRF process (for the ith iteration): [Fi+1 (ω)] =[H4 (ω)]−1 [X(ω)] =[G4 (ω)] [X(ω)] =[Fi (ω)] +[ΔFi (ω)] . (24.45) In a similar fashion, the full time history of the updated Below-JEL forces use the same collector/averaging process as was used for the Step #0 acceleration measurement filtering and the Step #10 acceleration synthesis. After conversion of each increment to the time domain, the winsize length time records are weighted and added to a collector vector for each time record. The collector is a running weighted sum of the estimates of the updated forces at each time step (denoted as j): FC tj : tj +Δt =FC tj : tj +Δt +weight ∗Fj; (24.46) where, FC(tj : tj + t) is the slice of the force time history collector associated with the j th increment; Fj is the force time history computed for the j th increment; j is the increment number; tj is the time step associated with the j th increment; t is the time increment; and : is a function symbol for a sweep over the number of time steps in a time increment. A numerical counter of the same length as each time history is updated when each time increment is added to the collector to track how many weighted estimates have been added to the collector at each time step: numC tj : tj +Δt =numC tj : tj +Δt +weight. (24.47) The process is repeated by shifting the original time increment window forward in time by the chosen number of samples. The ideal shift is one time sample per increment, which creates a slow process but eliminates periodic data spikes resulting in numerical frequency content due to the periodic window shifts. However, the weighting process reduces the data spikes by giving more weight to the estimates in the middle of each segment. At the end of the process, the collector value at each time step is divided by the weighted numerical counter value at that time step to average the weighted estimates and generate an updated force time history: Fi+1(t) = FC numC . (24.48)
RkJQdWJsaXNoZXIy MTMzNzEzMQ==