## 3 Error Estimation for a Reduced Flexible Body
Substituting Equation (26a) into Equation (25), we have:  
\[\boldsymbol{\varphi}_{n}^{i}=\overline{\mathbf{T}}_{1}^{i}\Xi_{n}^{i}+\delta \boldsymbol{\varphi}_{n}^{i}. \tag{27}\]  
Substituting Equation (27) into Equation (24), the original eigenvector \(\boldsymbol{\varphi}_{n}^{i}\) can be represented as:  
\[\frac{1}{\chi_{n}^{i}}[\overline{\mathbf{T}}_{1}^{i}\[PERSON]_{n}^{i}+\delta \boldsymbol{\varphi}_{n}^{i}]^{T}\mathbf{K}_{F}^{i}[\overline{\mathbf{T}}_{1}^ {i}\[PERSON]_{n}^{i}+\delta\boldsymbol{\varphi}_{n}^{i}]=[\overline{\mathbf{T}}_{1} ^{i}\[PERSON]_{n}^{i}+\delta\boldsymbol{\varphi}_{n}^{i}]^{T}\mathbf{M}_{F}^{i}[ \overline{\mathbf{T}}_{1}^{i}\[PERSON]_{n}^{i}+\delta\boldsymbol{\varphi}_{n}^{i}]. \tag{28}\]  
If the approximated eigenvector is close to the exact eigenvector in Equation (25), the error term (\(\delta\boldsymbol{\varphi}_{n}^{i}\)) becomes small, and then, all the terms with respect to \(\delta\boldsymbol{\varphi}_{n}^{i}\) could be neglected to approximate Equation (28). In addition, using Equation (26a), Equation (28) is written as:  
\[\frac{1}{\chi_{n}^{i}}\boldsymbol{\Xi}_{n}^{i}{}^{T}[\overline{\mathbf{T}}_{0 }^{i}+\omega^{2}\overline{\mathbf{T}}_{r}^{i}]^{T}\mathbf{K}_{F}^{i}[ \overline{\mathbf{T}}_{0}^{i}+\omega^{2}\overline{\mathbf{T}}_{r}^{i}]\[PERSON]_{n} ^{i}\approx\boldsymbol{\Xi}_{n}^{i}{}^{T}[\overline{\mathbf{T}}_{0}^{i}+ \omega^{2}\overline{\mathbf{T}}_{r}^{i}]^{T}\mathbf{M}_{F}^{i}[\overline{ \mathbf{T}}_{0}^{i}+\omega^{2}\overline{\mathbf{T}}_{r}^{i}]\[PERSON]_{n}^{i}. \tag{29}\]  
Using the mass orthornornality and stiffness orthogonality conditions of the reduced flexible body (\(\boldsymbol{\Xi}_{n}^{i}{}^{T}\overline{\mathbf{K}}_{F}^{i}\Xi_{n}^{i}= \overline{\chi}_{n}^{i}\) and \(\boldsymbol{\Xi}_{n}^{i}{}^{T}\overline{\mathbf{M}}_{F}^{i}\Xi_{n}^{i}=1\)), Equation (29) is expressed as:  
\[ \begin{split}\frac{\overline{\chi}_{n}^{i}}{\chi_{n}^{i}}-1\approx \boldsymbol{\Xi}_{n}^{i}{}^{T}\left[\omega^{2}\overline{\mathbf{T}}_{0}^{i}{}^{T }\mathbf{M}_{F}^{i}\overline{\mathbf{T}}_{r}^{i}+\omega^{2}\overline{\mathbf{T} }_{r}^{i}{}^{T}\mathbf{M}_{F}^{i}\overline{\mathbf{T}}_{0}^{i}+\omega^{4} \overline{\mathbf{T}}_{r}^{i}{}^{T}\mathbf{M}_{F}^{i}\overline{\mathbf{T}}_{r}^{i }\right]\Xi_{n}^{i}\\ -\frac{1}{\chi_{n}^{i}}\boldsymbol{\Xi}_{n}^{i}{}^{T}\left[\omega^ {2}\overline{\mathbf{T}}_{0}^{i}{}^{T}\mathbf{K}_{F}^{i}\overline{\mathbf{T}}_ {r}^{i}+\omega^{2}\overline{\mathbf{T}}_{r}^{i}{}^{T}\mathbf{K}_{F}^{i} \overline{\mathbf{T}}_{0}^{i}+\omega^{4}\overline{\mathbf{T}}_{r}^{i}{}^{T} \mathbf{K}_{F}^{i}\overline{\mathbf{T}}_{r}^{i}\right]\[PERSON]_{n}^{i}.\end{split}  \tag{30}\]where \(\omega^{2}\) is \(\chi_{n}^{i}\), we have:  
\[\frac{\overline{\chi}_{n}^{i}}{\chi_{n}^{i}}-1\approx 2\[PERSON]_{n}^{i}{}^{T}\overline{ \mathbf{T}}_{0}^{i}{}^{T}\left[\chi_{n}^{i}\mathbf{M}_{F}^{i}-\mathbf{K}_{F}^{ i}\right]\overline{\mathbf{T}}_{r}^{i}\Xi_{n}^{i}+\Xi_{n}^{i}{}^{T}\overline{ \mathbf{T}}_{r}^{i}{}^{T}\left[[\chi_{n}^{i}]^{2}\mathbf{M}_{F}^{i}-\chi_{n}^ {i}\mathbf{K}_{F}^{i}\right]\overline{\mathbf{T}}_{r}^{i}\Xi_{n}^{i}. \tag{31}\]  
The left side of Equation (31) is the relative eigenvalue error \(\zeta_{n}^{i}\) defined in Equation (23). Therefore, the right side of Equation (31) is a direct approximation of \(\zeta_{n}^{i}\). In the same manner as Equation (25), \(\chi_{n}^{i}\) is decomposed as:  
\[\chi_{n}^{i}=\overline{\chi}_{n}^{i}+\delta\chi_{n}^{i}. \tag{32}\]  
In this derivation, we assumed that the approximated eigenvector is close to the exact eigenvector in Equation (25). This also implies that \(\delta\chi_{n}^{i}\) is very small, and then, \(\chi_{n}^{i}\) can be approximated as \(\overline{\chi}_{n}^{i}\).