## 4 Iterative Reduction Process  
The reduced flexible body (RFlex) can decrease the computational cost compared to the full-ordered flexible body (FFlex), but it has a disadvantage in that the accuracy is relatively low because the residual modes are truncated. Therefore, it is necessary to construct a reliable RFlex when performing effective FMBD analysis. Using the frequency cut-off rule and the a posteriori error estimator derived in Equation (43), the iterative coordinate reduction process with a decision about the number of dominant modes (\(N_{d}^{i}\)) is proposed to make a reliable RFlex. The goal of the process is to find \(N_{d}^{i}\) satisfying the specified relative eigenvalue error tolerance \(e^{i}\) within the desired mode range \(N_{m}^{i}\). The initial input variables are the error tolerance \(e^{i}\), the initial number of dominant modes \(N_{init}^{i}\), and the mode range \(N_{m}^{i}\). The size of the initial RFlex is then \((N_{init}^{i}+N_{b}^{i})\times(N_{init}^{i}+N_{b}^{i})\). After constructing the reduced flexible model, its eigenvalue problem of the RFlex in Equation (18) should be solved to construct the diagonal matrices of the reduced flexible model, and then, those could also be used to estimate its relative eigenvalue error using Equation (43) for the \(n\)-th mode, where \(n=1,2,\cdots,N_{m}^{i}\). Using the estimated error, it is determined whether the relative eigenvalue error of a certain mode is greater than the error tolerance or not. If there is a mode with an error larger than the error tolerance, the dominant modes are added in the transformation matrix to create a new RFlex. If the \(n\)-th mode in the previous step did not satisfy the error tolerance, the iteration process starts from the \(n\)-th mode when determining the eigenvalue error of the newly created RFlex. Because adding dominant modes means improving the accuracy of RFlex, the eigenvalues before the \(n\)-th mode naturally satisfy the error tolerance. The iteration process stops when the estimated errors from the first to \(N_{m}^{i}\)-th modes are less then the desired error tolerance \(e^{i}\). When the dominant modes are added \(N_{iter}^{i}\) times, the size of RFlex becomes \((N_{init}^{i}+N_{iter}^{i}+N_{b}^{i})\times(N_{init}^{i}+N_{iter}^{i}+N_{b}^{ i})\). Through this process, the size of RFlex that achieves the desired accuracy can be selected. The schematic diagram of the process is shown in Figure 1.  
After the iterative reduction process, the reliable \(N_{m}^{i}\) modes will be used to solve the EOMs in Equation (21) in the FMBD analysis. The coordinate \(\mathbf{\bar{p}}_{F}^{i}\) is then transformed with the approximated mode shape matrix \(\mathbf{\bar{\Phi}}^{i}\) solving Equation (18) for the RFlex with \(N_{m}^{i}\) dominant modes as:  
\[\mathbf{\bar{p}}_{F}^{i}=\mathbf{\bar{\Phi}}^{i}\mathbf{a}_{F}^{i},\ \ \mathbf{\bar{\Phi}}^{i}=\left[\mathbf{\Xi}_{1}^{i}, \mathbf{\Xi}_{2}^{i},\cdots,\mathbf{\Xi}_{N_{m}^{i}}^{i}\right], \tag{46}\]  
where \(\mathbf{a}_{F}^{i}\) is a modal coordinate. Then, the coordinates in Equation (21) can be defined as:  
\[ \begin{bmatrix}\mathbf{q}_{R}^{i}\\ \mathbf{\bar{p}}_{F}^{i}\end{bmatrix} =\mathbf{\bar{B}}^{i} \begin{bmatrix} \mathbf{q}_{R}^{i}\\ \mathbf{a}_{F}^{i}\end{bmatrix} ,\ \ \ \mathbf{\bar{B}}^{i}= \begin{bmatrix} \mathbf{I}&\mathbf{0}\\ \mathbf{0}&\mathbf{\Phi}^{i}\end{bmatrix} . \tag{47}\]  
Figure 1: Iterative process using the error estimator. In the process of generating the reduced model, the dominant mode addition operation is repeated to satisfy that all of the relative eigenvalue errors within the specified mode range \(N_{m}^{i}\) are less than \(e^{i}\). FMBD, flexible multibody dynamics. Using \(\tilde{\mathbf{B}}\), the EOMs in Equation (21) are additionally reduced as:  
\[ \begin{bmatrix}\mathbf{M}_{R}^{i}&\tilde{\mathbf{M}}_{RF}^{i}\\ \tilde{\mathbf{M}}_{FR}^{i}&\mathbf{I}\end{bmatrix}\begin{bmatrix}\tilde{ \mathbf{q}}_{R}^{i}\\ \tilde{\mathbf{a}}_{F}^{i}\end{bmatrix} +