## 2 Formulation of the Proposed BMUA
### The Formulation of the Proposed BMUA
#### 2.4.1 Formulations of the New Prior PDF with \(\Delta k\)  
Let us assume \(N_{m}\) modes are measured. The prior PDF in the proposed BMUA is chosen as when updating parameters \(\mathbf{\beta}\):  
\[p_{m}(\mathbf{\lambda},\mathbf{\phi},\mathbf{\beta}|C)=p_{m}(\mathbf{\lambda},\mathbf{\phi}\,|\bm {\beta},C)\cdot p_{m}(\mathbf{\beta}|C) \tag{18}\]where \(\mathbf{\lambda}=\left[ \begin{array}{cccc}\lambda^{(1)},&\lambda^{(2)},&\cdots,& \lambda^{(N_{m})}\end{array} \right]^{T}\) and \(\mathbf{\phi}=\left[ \begin{array}{cccc}\mathbf{\phi}^{(1)},&\mathbf{\phi}^{(2)},&\cdots,& \mathbf{\phi}^{(N_{m})}\end{array} \right]^{T}\) are eigenvaluesand the eigenvector to be updated, respectively. \(p_{m}(\mathbf{\lambda},\mathbf{\phi}|\mathbf{\beta},C)\) is formulated using a Gaussian PDF with the new eigenequation error in Equation (11):  
\[p_{m}(\mathbf{\lambda},\mathbf{\phi}|\mathbf{\beta},C)=c_{0}exp\left[-\frac{\left|\left| \mathbf{\phi^{\prime}}^{T}\mathbf{M}\mathbf{\phi}-\left(\mathbf{\lambda}^{\prime}-\mathbf{\lambda} \right)^{-1}\mathbf{\phi^{\prime}}^{T}\Delta\mathbf{k}\mathbf{\phi}\right|\right|^{2}}{2 \sigma_{eq}^{2}}\right] \tag{19}\]  
where \(c_{0}\) denotes a constant; the sign \(||\cdot||\) denotes a mathematical Euclidean norm. \(\sigma_{eq}^{2}\) denotes a defined variance of the eigenequation error. Equation (19) is simplified as  
\[p_{m}(\mathbf{\lambda},\mathbf{\phi}|\mathbf{\beta},C)=c_{0}exp\left[-\frac{1}{2}I_{g,m}( \mathbf{\lambda},\mathbf{\phi};\mathbf{\beta})\right] \tag{20}\]  
where  
\[I_{g,m}(\mathbf{\lambda},\mathbf{\phi};\mathbf{\beta})=\mathbf{T_{m}}^{T}\mathbf{\Sigma}_{eq}^{-1} \mathbf{T}_{m} \tag{21}\]  
where \(\mathbf{T}_{m}=\mathbf{\phi^{\prime}}^{(m)}{}^{T}\mathbf{M}\mathbf{\phi}^{(m)}-\left(\lambda^ {\prime}{}^{(m)}-\lambda^{(m)}\right)^{-1}\mathbf{\phi^{\prime}}^{(m)}{}^{T}\Delta \mathbf{k}\mathbf{\phi}^{(m)}\). \(\mathbf{\Sigma}_{eq}=\sigma_{eq}^{2}\mathbf{I}\) denotes the covariance matrix in the prior PDF; \(\mathbf{I}\) denotes an identity matrix. \(\mathbf{\Sigma}_{eq}\) arises from the modeling error between theoretical and target FEM models. Another term of \(p_{m}(\mathbf{\beta}|C)\) in Equation (18) is approximated by a Gaussian distribution, which has a mean value of \(\mathbf{\beta}^{\eta}\) (nominal value) and a covariance matrix, \(\mathbf{\Sigma}_{\beta}=\sigma_{\beta}^{2}\mathbf{I}\), where \(\sigma_{\beta}\) is selected as a large variance to let \(p_{m}(\mathbf{\beta}|C)\) be a non-informative prior. Hence, \(p_{m}(\mathbf{\beta}|C)\) may be expressed as  
\[p_{m}(\mathbf{\beta}|C)=\exp\left[-\frac{\left|\left|\mathbf{\beta}-\mathbf{\beta}^{\eta }\right|\right|^{2}}{2\sigma_{\beta}^{2}}\right] \tag{22}\]  
Combining Equations (20) and (22), the prior PDF in Equation (18) is rewritten as  
\[p_{m}(\mathbf{\lambda},\mathbf{\phi},\mathbf{\beta}|C)=c_{0}\exp\left[-\frac{1}{2}I_{g,m} (\mathbf{\lambda},\mathbf{\phi};\mathbf{\beta})\right]\cdot\exp\left[-\frac{\left|\mathbf{ \beta}-\mathbf{\beta}^{\eta}\right|\right|^{2}}{2\sigma_{\beta}^{2}}\right] \tag{23}\]  
In terms of updating stiffness \(\mathbf{\theta}\) the same derivation procedures are employed; using the new eigenequation in Equation (17), the prior PDF has an expression as Equation (24):  
\[p_{k}(\mathbf{\lambda},\mathbf{\phi},\mathbf{\theta}|C)=c_{0}exp\left[-\frac{1}{2}I_{g,k} (\mathbf{\lambda},\mathbf{\phi};\mathbf{\theta})\right]\cdot exp\left[-\frac{\left(\mathbf{ \theta}-\mathbf{\theta}^{\eta}\right)^{2}}{2\sigma_{\theta}^{2}}\right] \tag{24}\]  
where \(\mathbf{\lambda}\) are the updated eigenvalues; \(\mathbf{\phi}\) is the updated eigenvector.