## 4 Sensitivity Analysis for Eigenfrequency Problems Using HYPAD
### Free Vibration of a Cantilever Beam  
The second example corresponds to a numerical implementation of the methodology for obtaining sensitivities in eigenfrequency problems using HYPAD. In this section, the methodology is applied to the analysis of the free vibration of a uniform elastic cantilever beam, illustrated in Figure 4a. The beam has a length \(L\), stiffness \(E\), mass density \(\rho\), and a constant rectangular cross-sectional area \(A\) with width \(b\), and height \(d\). The analytical expressions for the natural frequencies and shapes of the free vibration modes correspond to [76]:  
\[\lambda_{i}= \begin{cases}\lambda_{iy}=\gamma_{i}^{4}\frac{E_{iy}}{\rho A^{4}}\\ \lambda_{iz}=\gamma_{i}^{4}\frac{E_{iz}}{\rho A^{4}}\end{cases} \textit{where }I_{y}=\frac{ bd^{3}}{12},I_{z}=\frac{db^{3}}{12},A=bd \tag{29}\]  
\[\phi_{i}(x)=\frac{1}{\sqrt{\rho AL}}\left[\frac{(\sinh(\gamma_{i})+\sin( \gamma_{i}))\left\{\cosh(\frac{\gamma_{i}^{2}}{L})-\cos(\frac{\gamma_{i}^{2}}{L })\right\}}{\sinh(\gamma_{i})+\sin(\gamma_{i})}-\frac{(\cosh(\gamma_{i})+\cos (\gamma_{i}))\left\{\sinh\left(\frac{\gamma_{i}^{2}}{L}\right)-\sin\left(\frac {\gamma_{i}^{2}}{L}\right)\right\}}{\sinh(\gamma_{i})+\sin(\gamma_{i})} \right] \tag{30}\]  
where \(\gamma_{i}\) are given by:  
\[\cosh(\gamma_{i})\cos(\gamma_{i})+1=0\ \ \ \forall(i=1,2,\ldots) \tag{31}\]  
The three first roots of Equation (31) are:  
\[\gamma_{1}=1.8751\ \gamma_{2}=4.6941\ \gamma_{3}=7.8547\]  
Since the beam has two axes of symmetry (i.e., \(y\) and \(z\)), the eigenvalues will appear alternated in the analytical solution. Therefore, the modes will be classified between those in the \(y\)-direction as \(\lambda_{iy}\), and those in the \(z\)-direction as \(\lambda_{iz}\). The system can have repeated or distinct eigenvalues depending on the characteristics of the cross-sectional area. The first case of repeated eigenvalues is evident in symmetric cross-sections when the cross-section has the same first moment of area with respect to two or more axes. For instance, for a square cross-section, the inertia in directions \(y\) and \(z\) are the same; therefore, \(\lambda_{iy}=\lambda_{iz}\). On the contrary, if a non-symmetric cross-section is used, as in the case of a rectangle cross-section, the inertia in \(y\) and \(z\) are different; therefore, \(\lambda_{iy}\
eq\lambda_{iz}\). Here, both situations are considered. The methodology was implemented in FORTRAN using the LAPACK [77] library to solve the GEP and to obtain the eigenvectors with mass normalization using the procedure described in Figure 1a. The code is made available to the reader as Supplementary Material to the article. Up to third-order sensitivities (\(m=3\)) were computed with respect to the material and geometrical parameters for the first six natural modes of the cantilevered beam. As an initial step, the parameters of the model (\(E,\rho,L,b\) and \(d\)) were transformed into their corresponding multidual representation to become \(E^{*},\rho^{*},L^{*},b^{*}\) and \(d^{*}\), depending on the derivatives calculated in each case. Subsequently, unit perturbations were added along the different multidual imaginary axes of the variables depending on the specific sensitivities  
Figure 4: (**a**) Schematic of the cantilever beam for the free vibration system; (**b**) Finite element formulation for standard Euler–Bernoulli elements. that were to be calculated. The beam was discretized using standard Euler-Bernoulli elements, with each node containing four DOF, two translational and two rotational, as shown in Figure 4b. A total of one hundred equal-length Euler-Bernoulli elements were used. The element stiffness \(\left[K^{\epsilon}\right]^{*}\) and mass \(\left[M^{\epsilon}\right]^{*}\) matrices were calculated by using the multidual variables and the formulation in Equations (32)-(33), respectively. In addition, the global mass and stiffness matrices were assembled following a traditional finite element method scheme. \[\left[K^{\epsilon}\right]^{*}=\frac{E^{*}}{L^{*}}\left[