## 3 Error Estimation for a Reduced Flexible Body
Therefore, the inequality in Equation (37) is clear with \(\kappa_{n}^{i}<\kappa_{N_{d}+1}^{i}\). We also found that \(\|[\mathbf{\Lambda}_{r}^{i}]^{-1}\|\) becomes smaller than the case when all residual eigenvalues located at the diagonal terms of \(\mathbf{\Lambda}_{r}^{i}\) are \(\kappa_{N_{d}+1}^{i}\). Assuming that all of the diagonal components are \(\kappa_{N_{d}+1}^{i}\) of \([\mathbf{\Lambda}_{r}^{i}]^{-1}\), it can be written as \((1/\kappa_{N_{d}+1}^{i})\mathbf{I}\). We then have the following equation:  
\[\|\mathbf{\Phi}_{r}^{i}[\mathbf{\Lambda}_{r}^{i}]^{-2}\mathbf{\[PERSON]}_{r}^{i} \|\leq\frac{1}{\kappa_{N_{d+1}}^{i}}\|\mathbf{F}_{rs}^{i}\|. \tag{38}\]  
From Equations (37) and (38), we have:  
\[\|\mathbf{\Phi}_{r}^{i}[\mathbf{\Lambda}_{r}^{i}]^{-2}\mathbf{\Phi}_{r}^{i} \|\leq\frac{1}{\kappa_{N_{d+1}}^{i}}\|\mathbf{F}_{rs}^{i}\|<\frac{1}{\kappa_{ n}^{i}}\|\mathbf{F}_{rs}^{i}\|. \tag{39}\]  
Replacing \(\mathbf{\Phi}_{r}^{i}[\mathbf{\Lambda}_{r}^{i}]^{-2}\mathbf{\Phi}_{r}^{i} \) with \((1/\kappa_{n}^{i})\mathbf{F}_{rs}^{i}\) in Equation (36) and using the inequality in Equation (39), the new error estimator \((\tilde{\xi}_{n}^{i})\) at the upper bound of the relative eigenvalue error \((\tilde{\xi}_{n}^{i})\) can be derived as:  
\[\tilde{\xi}_{n}^{i}\leq\tilde{\xi}_{n}^{i}=\Xi_{n}^{i}\left(\overline{\chi}_{ n}^{i} \begin{bmatrix}\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{X}_{b}^{i}\end{bmatrix} +\frac{[\overline{\chi}_{n}^{i}]^{2 }}{\kappa_{n}^{i}} \begin{bmatrix}\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{X}_{b}^{i}\end{bmatrix} \right)\Xi_{n}^{i}. \tag{40}\]  
\(\tilde{\xi}_{n}^{i}\) can be written as:  
\[\tilde{\xi}_{n}^{i}=(\overline{\chi}_{n}^{i}+\frac{\overline{\chi}_{n}^{i}}{ \kappa_{n}^{i}}^{2})\Xi_{n}^{i}\left(\overline{\mathbf{0}} \begin{bmatrix} \mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{X}_{b}^{i}\end{bmatrix} \[PERSON]_{n}^{i},\quad\[PERSON]_{n}^{i}= \begin{bmatrix}(\[PERSON]_{n}^{i})_{s}\\ (\[PERSON]_{n}^{i})_{b}\end{bmatrix} , \tag{41}\]  
and considering component matrices' operation, the new error estimator \(\tilde{\xi}_{n}^{i}\) is simplified as:  
\[\tilde{\xi}_{n}^{i}=(\overline{\chi}_{n}^{i}+\frac{\overline{\chi}_{n}^{i}}{ \kappa_{n}^{i}}^{2})(\[PERSON]_{n}^{i})_{b}^{T}\mathbf{\Lambda}^{i}\mathbf{\Gamma}_ {rs}^{i}\mathbf{\Lambda}^{i}(\[PERSON]_{n}^{i})_{b}. \tag{42}\]  
For the lumped mass case (\(\mathbf{M}_{c}^{i}=\mathbf{0}\)) [30], the error estimator \(\tilde{\xi}_{n}^{i}\) is defined as:  
\[\tilde{\xi}_{n}^{i}=(\overline{\chi}_{n}^{i}+\frac{\overline{\chi}_{n}^{i}}{ \kappa_{n}^{i}}^{2})(\[PERSON]_{n}^{i})_{b}^{T}\mathbf{\Sigma}^{i}\mathbf{\Gamma}_ {rs}^{i}\mathbf{\Sigma}^{i}(\Xi_{n}^{i})_{b},\quad\mathbf{\Sigma}^{i}=\mathbf{ M}_{s}^{i}\mathbf{\Gamma}^{i}. \tag{43}\]  
The lumped mass matrix \(\mathbf{M}_{s}^{i}\) is a diagonal matrix, then Equation (43) can be efficiently computed compared with Equation (42). The components \((z^{i})_{nm}\) of the matrix \(\mathbf{\Sigma}^{i}\) can be rewritten by the following tensor notation as:  
\[(z^{i})_{nm}=(m_{s}^{i})_{nm}(\psi^{i})_{nm},\quad n=1,2,\cdots N_{s}^{i},\ m=1,2, \cdots N_{b}^{i}. \tag{44}\]  
In addition, as we described, neglecting \(\overline{\mathbf{\Gamma}_{r}^{i}}^{T}\mathbf{M}_{f}^{i}\overline{\mathbf{ \Gamma}_{r}^{i}}\), the previous error estimator [23] was derived as:  
\[\eta_{n}^{i}=\overline{\chi}_{n}^{i}(\Xi_{n}^{i})_{b}^{T}\mathbf{\Sigma}^{i} \mathbf{\Gamma}_{rs}^{i}\mathbf{\Sigma}^{i}(\Xi_{n}^{i})_{b}. \tag{45}\]Therefore, the previous and proposed error estimators may become lower and upper bounds of the exact relative eigenvalue errors, respectively. This could be investigated numerically through example problems. The proposed error estimators in Equations (42) and (43) only need the eigensolutions obtained from the reduced flexible models in Equation (18), and thus, those could be efficiently computed.