## 3 Error Estimation for a Reduced Flexible Body  
The following relative eigenvalue error has been widely used to evaluate the accuracy of a reduced flexible body:  
\[\zeta_{n}^{i}=\frac{\overline{\chi}_{n}^{i}-\chi_{n}^{i}}{\overline{\chi}_{n}^ {i}}=\frac{\overline{\chi}_{n}^{i}}{\overline{\chi}_{n}^{i}}-1,\ \ n=1,2,\cdots,\overline{N}^{i}. \tag{23}\]  
The value of \(\zeta_{n}^{i}\) implies a difference between \(\overline{\chi}_{n}^{i}\) in Equation (18) and \(\chi_{n}^{i}\) in Equation (10), and thus, it can provide information about how many modes are needed for accurate reduction of flexible bodies. However, the computation of \(\zeta_{n}^{i}\) requires a huge burden because the reference eigenvalues, \(\chi_{n}^{i}\), are obtained by solving the eigenvalue problems of the original flexible bodies. To overcome this issue,Kim et al. has proposed a posteriori error estimation techniques that offer a way to directly estimate \(\xi_{n}^{i}\) without the reference eigenvalues (\(\chi_{n}^{i}\)) [22; 23]. The previous error estimators provide accurate eigenvalue error prediction, but the estimated errors are generally less than the exact errors [23]. This feature may cause excessive and inaccurate model reduction. In this work, a more reasonable error estimator toward the upper bound of the exact error is proposed from a conservative point of view. Premultiplying \(\boldsymbol{\varphi}_{n}^{i}{}^{T}\) by the eigenvalue problem of the original flexible body in Equation (10), the following scalar equation could be obtained:  
\[\frac{1}{\chi_{n}^{i}}\boldsymbol{\varphi}_{n}^{i}{}^{T}\mathbf{K}_{F}^{i} \boldsymbol{\varphi}_{n}^{i}=\boldsymbol{\varphi}_{n}^{i}{}^{T}\mathbf{M}_{F} ^{i}\boldsymbol{\varphi}_{n}^{i},\ \ n=1,2,\cdots,N_{F}^{i}. \tag{24}\]  
The original eigenvector \(\boldsymbol{\varphi}_{n}^{i}\) is then decomposed to the approximated eigenvector and the error term as:  
\[\boldsymbol{\varphi}_{n}^{i}=\overline{\boldsymbol{\varphi}}_{n}^{i}+\delta \boldsymbol{\varphi}_{n}^{i}. \tag{25}\]  
The approximated eigenvector \(\overline{\boldsymbol{\varphi}}_{n}^{i}\) of the flexible body is defined in Equation (19). Considering the residual modal compensation, \(\overline{\boldsymbol{\varphi}}_{n}^{i}\) could be more accurately approximated [4; 10] as:  
\[\overline{\boldsymbol{\varphi}}_{n}^{i}=\overline{\mathbf{T}}_{1}^{i}\Xi_{n} ^{i},\ \ \ \overline{\mathbf{T}}_{1}^{i}=\overline{\mathbf{T}}_{0}^{i}+\omega^{2} \overline{\mathbf{T}}_{r}^{i},\ \ \ \overline{\mathbf{T}}_{r}^{i}= \begin{bmatrix}\mathbf{0}& \mathbf{F}_{rs}^{i}\mathbf{A}^{i}\\ \mathbf{0}&\mathbf{0}\end{bmatrix} , \tag{26a}\] \[\mathbf{A}^{i}=\mathbf{M}_{s}^{i}\mathbf{\Psi}^{i}+\mathbf{M}_{c}^{i},\ \ \ \mathbf{F}_{rs}^{i}=\boldsymbol{\Phi}_{r}^{i}[\boldsymbol{\Lambda}_{r}^{i}]^{-1} \boldsymbol{\Phi}_{r}^{i}{}^{T}, \tag{26b}\]  
where \(\omega\) is an angular frequency. \(\mathbf{F}_{rs}^{i}\) is a flexibility matrix including the residual modes that are defined in Equation (15). \(\mathbf{\Lambda}_{r}^{i}\) is a diagonal matrix of the residual eigenvalues with respect to the residual eigenvectors \(\boldsymbol{\Phi}_{r}^{i}\). Equation (26a) shows that the approximated eigenvector in Equation (19) can be refined with the additional transformation matrix \(\overline{\mathbf{T}}_{r}^{i}\), which provides better representation of the flexible mode shapes [4; 10]. The derivation details are presented in Appendix A.