## Appendix A Expansion Terms in the Expression from Equation (26a)  
The terms for evaluating Equation (26a) are displayed below. \[ \begin{array}{c}\left[\Theta_{\text{i}}\right]^{*}=\Big{[}\mathcal{C}^{Re}+ \frac{\partial K}{\partial c}\varepsilon_{1}\Big{]}-\Big{(}\lambda_{1}^{Re}+ \frac{\partial\lambda_{i}}{\partial c}\varepsilon_{1}\Big{)}\Big{[}m^{Re}+ \frac{\partial M}{\partial c}\varepsilon_{1}\Big{]}\\ \hskip 14.226378 pt+\Big{(}\lambda_{1}^{Re}+\frac{\partial\lambda_{i}}{ \partial c}\varepsilon_{1}\Big{)}\Big{(}\phi_{1}^{Re}+\frac{\partial\phi_{1}} {\partial c}\varepsilon_{1}\Big{)}\Big{(}\phi_{1}^{Re}+\frac{\partial\phi_{1}} {\partial c}\varepsilon_{1}\Big{)}^{T}\Big{[}m^{Re}+\frac{\partial M}{ \partial c}\varepsilon_{1}\Big{]}\\ \left[\Pi_{\text{i}}\right]^{*}=\Big{(}\frac{\partial\lambda_{1}}{\partial \vartheta}^{*}\Big{)}\Big{[}m^{Re}+\frac{\partial M}{\partial c}\varepsilon_{ 1}\Big{]}+\Big{(}\lambda_{1}^{Re}+\frac{\partial\lambda_{i}}{\partial c} \varepsilon_{1}\Big{)}\Big{[}\frac{\partial M}{\partial\rho}+\frac{\partial M }{\partial\rho\partial c}\varepsilon_{1}\Big{]}\\ \hskip 14.226378 pt-\beta^{*}\Big{(}\lambda_{1}^{Re}+\frac{\partial\lambda_{1}}{ \partial c}\varepsilon_{1}\Big{)}\\ \hskip 14.226378 pt\beta^{*}=\frac{1}{2}\Big{(}\phi_{1}^{Re}+\frac{\partial\phi_ {1}}{\partial c}\varepsilon_{1}\Big{)}^{T}\Big{[}\frac{\partial M}{\partial \rho}+\frac{\partial M}{\partial\rho c}\varepsilon_{1}\Big{]}\Big{(}\phi_{1}^ {Re}+\frac{\partial\phi_{1}}{\partial c}\varepsilon_{1}\Big{)}\end{array} \]  
Appendix B Analytical Solution for the Derivatives of the Eigenvalues and Eigenvectors for a Single DOF System  
The eigenvalues and eigenvectors of the single DOF spring mass system are:  
\[\lambda_{1}=\frac{c}{\rho}\]  
\[\phi_{1}=\frac{1}{\sqrt{\rho}}\]  
The first-order derivatives with respect to the two model input parameters (i.e., \(c,\rho\)) are:  
\[\frac{\partial\lambda_{1}}{\partial\rho}=-\frac{k}{\rho^{2}}\]  
\[\frac{\partial\phi_{1}}{\partial\rho}=-\frac{1  
* (3) [PERSON]; [PERSON] Second-Order Sensitivity Analysis of Multimodal Eigenvalues and Related Optimization Techniques. _J. Struct. Mech._**1986**, _14_, 421-436. [CrossRef]
* (4) [PERSON]; [PERSON]; [PERSON] A state-of-the-art review on theory and engineering applications of eigenvalue and eigenvector derivatives. _Mech. Syst. Signal Process._**2020**, _138_, 106536. [CrossRef]
* (5) [PERSON]; [PERSON]; [PERSON] The sensitivity method in finite element model updating: A tutorial. _Mech. Syst. Signal Process._**2011**, _25_, 2275-2296. [CrossRef]
* (6) [PERSON]; [PERSON]; [PERSON]; [PERSON] Highly efficient general method for sensitivity analysis of eigenvectors with repeated eigenvalues without passing through adjacent eigenvectors. _Int. J. Numer. Methods Eng._**2020**, _121_, 4473-4492. [CrossRef]
* (7) [PERSON]; [PERSON]; [PERSON] Projection methods for stochastic dynamic systems: A frequency domain approach. _Comput. Methods Appl. Mech. Eng._**2018**, _338_, 412-439. [CrossRef]
* (8) [PERSON]. The QR Transformation A Unitary Analogue to the LR Transformation--Part 1. _Comput. J._**1961**, \(4\), 265-271. [CrossRef]
* (9) [PERSON]. The QR Transformation--Part 2. _Comput. J._**1962**, \(4\), 332-345. [CrossRef]
* (10) [PERSON] Implicit Application of Polynomial Filters in a k-Step Arnoldi Method. _SIAM J. Matrix Anal. Appl._**1992**, _13_, 357-385. [CrossRef]
* (11) [PERSON] The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices. _J. Comput. Phys._**1975**, _17_, 87-94. [CrossRef]
* (12) [PERSON] Iterative Computation of Derivatives of Eigenvalues and Eigenvectors. _IMA J. Appl. Math._**1979**, _24_, 209-218. [CrossRef]
* (13) [PERSON]; [PERSON] Computing Derivatives of Eigenvalues and Eigenvectors by Simultaneous Iteration. _IMA J. Numer. Anal._**1989**, \(9\), 111-122. [CrossRef]
* (14) [PERSON]. Some acceleration methods for iterative computer of derivatives of eigenvalues and eigenvectors. _Int. J. Numer. Methods Eng._**1989**, _28_, 1505-1519.