## 4 Sensitivity Analysis for Eigenfrequency Problems Using HYPAD
### Free Vibration of a Cantilever Beam
\begin{array}{cccccc} \frac{12I_{z}^{*}}{L^{*2}}&0&0&\frac{6I_{z}^{*}}{L^{*}}&-\frac{12I_{z}^{*}}{L ^{*2}}&0&0&\frac{6I_{z}^{*}}{L^{*}}\\ 0&\frac{12I_{z}^{*}}{L^{*2}}&-\frac{6I_{z}^{*}}{L^{*}}&0&0&-\frac{12I_{z}^{*}}{ L^{*2}}&-\frac{6I_{z}^{*}}{L^{*}}&0\\ 0&-\frac{6I_{y}^{*}}{L^{*}}&4I_{y}^{*}&0&0&\frac{6I_{y}^{*}}{L^{*}}&2I_{y}^{*} &0\\ \frac{6I_{z}^{*}}{L^{*}}&0&0&4I_{z}^{*}&-\frac{6I_{z}^{*}}{L^{*}}&0&0&2I_{z}^{* }\\ -\frac{12I_{z}^{*}}{L^{*2}}&0&0&-\frac{6I_{z}^{*}}{L^{*}}&\frac{12I_{z}^{*}}{ L^{*2}}&0&0&-\frac{6I_{z}^{*}}{L^{*}}\\ 0&-\frac{12I_{z}^{*}}{L^{*2}}&6I_{y}^{*}&0&0&\frac{12I_{z}^{*}}{L^{*}}&6I_{y}^{* }&0\\ 0&-\frac{6I_{y}^{*}}{L^{*}}&2I_{y}^{*}&0&0&\frac{6I_{y}^{*}}{L^{*}}&4I_{y}^{* }&0\\ \frac{6I_{z}^{*}}{L^{*}}&0&0&2I_{z}^{*}&-\frac{6I_{z}^{*}}{L^{*}}&0&0&4I_{z}^{* }\\ \end{array} \right] \tag{32}\]  
where \(I_{z}^{*}=d^{*}b^{*}/12\), and \(I_{y}^{*}=d^{*3}b^{*}/12\). \[\left[M^{\epsilon}\right]^{*}=\frac{\epsilon^{\prime}A^{*}L^{*}}{420}\left[ \begin{array}{cccccc}156&0&0&22L^{*}&54&0&0&-13L^{*}\\ 0&156&-22L^{*}&0&0&54&13L^{*}&0\\ 0&-22L^{*}&4L^{*2}&0&0&-13L^{*}&-3L^{*2}&0\\ 22L^{*}&0&0&4L^{*2}&13L^{*}&0&0&-3L^{*2}\\ 54&0&0&13L^{*}&156&0&0&-22L^{*}\\ 0&54&-13L^{*}&0&0&156&22L^{*}&0\\ 0&13L^{*}&-3L^{*2}&0&0&22L^{*}&4L^{*2}&0\\ -13L^{*}&0&0&-3L^{*2}&-22L^{*}&0&0&4L^{*2}\\ \end{array} \right] \tag{33}\]  
where \(A^{*}=d^{*}b^{*}\). Subsequently, the boundary conditions were defined for the multidual matrices \(\left[K\right]^{*}\) and \(\left[M\right]^{*}\), eliminating the real and dual imaginary DOF under homogeneous Dirichlet boundary conditions. Then, the information from each imaginary axis of the multidual arrays was extracted to form \(p=7\) matrices corresponding to each of the dual imaginary axes of the tridual number. The matrices \(\left[K\right]\) and \(\left[M\right]\) constructed from the information of the real axis were used to solve the GEP and obtain the eigenvalues and eigenvectors. In addition, the information from the dual imaginary axes was used to calculate the partial derivatives of the system matrices by following Equation (7). Then, as described in Figure 2b, the eigenvalues \(\lambda_{i}\), the eigenvectors \(\phi_{i}\) for \(i=1,\dots,6\), \(m=3\), the system matrices \(\left[K\right]\) and \(\left[M\right]\), and their partial derivatives, were used to calculate the sensitivities of the eigenvalues and eigenvectors. Three iterations were processed through the algorithm in Stage 3. The relative percentage error (RE) described in Equation (34) was used to measure the error for each of the eigenvalues and their sensitivities. For the eigenvectors, an average of the RE was used as an error measurement. For both measures of error, the analytical solutions in Equations (29)-(30) were used as a reference. \[RE=\frac{\left|y_{j}^{Analytical}-y_{j}^{2 TSE}\right|}{\left|y_{j}^{Analytical} \right|} \tag{34}\]  
A semi-analytical demonstration was conducted to illustrate the present verification problem for the reader. In this case, the cantilever beam was discretized using a single Euler-Bernoulli element. Although such a small number of elements does not achieve convergence and the stability of the results, it shows a step-by-step demonstration of the methodology. An example can be found in the Supplementary Material.