## 2 Background  
### First Order Sensitivities of Eigenvalues and Eigenvectors  
In this section, the methodologies derived by [PERSON] and [PERSON] [17] to calculate eigenvalues' sensitivities, and from [PERSON] and [PERSON] [33] for eigenvectors' sensitivities are reviewed. Consider the equation of motion in the frequency domain for a linear undamped discrete system subjected to harmonic loading:  
\[[F_{i}]\phi_{i}=[K-\lambda_{i}M]\phi_{i}=0 \tag{1}\]  
Equation (1) corresponds to the Generalized Eigenvalue Problem (GEP), where \([F_{i}]=[K-\lambda_{i}M]\), \(\lambda_{i}\) and \(\phi_{i}\) are the \(i^{th}\) eigenvalue and right eigenvectors, respectively, \(i=1,2,\ldots,n\) corresponds to the identifier of the \(i^{th}\) mode of vibration, and \(n\) identifies the number of DOF in the system. In structural problems, \([M]\) and \([K]\) correspond to \(n\times n\) self-adjoint matrices representing the mass and stiffness, respectively. These matrices are symmetric, and their components are functions of the \(j\) number of input parameters of the model, \(\mathbf{\alpha}=\{\alpha_{1},\ldots,\alpha_{j}\}\), such as elastic modulus, geometry, and density. Depending on the geometrical characteristics of the system, Equation (1) can produce repeated or distinct eigenvalues. In the first case, there is a unique eigenvector \(\phi_{i}\) for each eigenvalue \(\lambda_{i}\), and in the second case, each eigenvalue can have two or more eigenvectors associated (e.g., systems with geometrical symmetries) [35]. Equation (1) has a non-trivial solution for the orthogonal eigenvectors \(\phi_{i}\); therefore, these can be scaled by any magnitude and still be a basis of the system. For this reason, normalization is required to guarantee uniqueness in the solution [66]. In this work, mass normalization is used for all eigenvectors, satisfying \(\phi_{i}^{T}[M]\phi_{i}=1\). This specific type of normalization is required to make the expressions presented below valid. Taking the GEP defined in Equation (1), and premultiplying by \(\phi_{i}\), gives:  
\[\phi_{i}^{T}[F_{i}]\phi_{i}=0 \tag{2}\]  
Equation (2) is differentiated to calculate the sensitivity of the eigenvalues with respect to any input parameter of the model \(\alpha_{j}\). By making use of the conditions of the orthogonality of the eigenvectors and the mass normalization, the first-order derivative of the \(i^{th}\) eigenvalue is calculated as:  
\[\frac{\partial\lambda_{i}}{\partial\alpha}=\phi_{i}^{T}\Big{[}\frac{\partial K }{\partial\alpha}-\lambda_{i}\frac{\partial M}{\partial\alpha}\Big{]}\phi_{i} \tag{3}\]  
where Equation (3) corresponds to the exact solution for the first-order sensitivities of eigenvalues in structural systems as previously reported by [PERSON] and [PERSON] [17]. Note that this equation requires knowledge of the sensitivities of the stiffness and the mass matrices, \([\partial K/\partial\alpha]\) and \([\partial M/\partial\alpha]\). In the case of eigenvectors' sensitivities, [PERSON] and [PERSON] derived exact expressions for the first- and second-order derivatives for distinct and repeated eigenvalues [33]. The first-order derivative of the \(i^{th}\) eigenvector with respect to a design parameter \(\alpha_{j}\) is:  
\[\frac{\partial\phi_{i}}{\partial\alpha}=\Big{[}\Theta_{i}]^{-1}[\Pi_{i}]\phi_ {i} \tag{4}\]  
where  
\[[\Theta_{i}]=[K]-\lambda_{i}[M]+\lambda_{i}\phi_{i}\phi_{i}^{T}[M]\]  
\[[\Pi_{i}]=\frac{\partial\lambda_{i}}{\partial\alpha}[M]+\lambda_{i}\bigg{[} \frac{\partial M}{\partial\alpha}\bigg{]}-\bigg{[}\frac{\partial K}{\partial \alpha}\bigg{]}-\beta_{i}\lambda_{i}\]  
\[\beta_{i}=\frac{1}{2}\phi_{i}^{T}\bigg{[}\frac{\partial M}{\partial\alpha} \bigg{]}\phi_{i}\]  
Higher-order sensitivities are obtained by differentiating Equation (4) up to arbitrary-order; however, the expressions become lengthy and cumbersome to implement.