## 4 Sensitivity Analysis for Eigenfrequency Problems Using HYPAD
### Free Vibration of a Cantilever Beam
#### 4.2.1 Sensitivity Analysis of a Cantilever Beam with Distinct Eigenvalues  
A cantilever beam was analyzed with properties \(L=5\) m, \(E=68.9\) GPa, \(\rho=2770\) kgm\({}^{-3}\), \(b=0.01\) m and \(d=0.015\) m. The characteristics of the cross-section area yielded distinct eigenvalues. Higher-order sensitivities with respect to the material (\(E\), \(\rho\)) and geometrical (\(L\)) input design parameters were calculated. Three specific cases are presented, each representing an independent pass throughout the whole methodology. Third-order sensitivities with respect to the density \(\rho\), third-order sensitivities with respect to the length of the beam \(L\), and mixed third-order sensitivities with respect to the three parameters. This requires using tri-dual numbers with \(p=7\) imaginary axes. In the first case, perturbations were added only to \(\rho\) on its three independent dual imaginary axes, becoming \(\rho^{*}=2770+e_{1}+e_{2}+e_{3}\). In the second case, perturbations were added only to \(L\) on its three independent dual imaginary axes, becoming \(L^{*}=5+e_{1}+e_{2}+e_{3}\). In the third case, the perturbations were added to the three variables, these becoming \(\rho^{*}=2770+e_{1}\), \(E^{*}=68\times 10^{9}+e_{2}\) and \(L^{*}=5+e_{3}\). The axes not displayed here contain a value of zero for all cases. Table 1 shows the results corresponding to the eigenvalues and Figure 5 the results for the eigenvectors. Excellent agreement is found between the analytical solution and the numerical estimates, resulting in a maximum RE of \(9.239\times 10^{-7}\%\) in the eigenvalues, and an average RE of \(2.750\times 10^{-6}\%\) for the eigenvectors. Tables 2-4 and Figure 6 show the results of the sensitivity analysis of the structure. A good agreement is found with a maximum percentual error of \(9.371\times 10^{-7}\%\) in the sensitivities of the eigenvalues and an average RE of \(7.066\times 10^{-6}\%\) for the eigenvectors. These results preserve the same error magnitudes as the real solution, reflecting that HYPAD does not induce truncation nor subtraction cancellation errors. In addition, the \begin{table}
\begin{tabular}{c c} \hline
**Mode i** & \(\lambda_{i}\) **(\%Error \(\times\) 10\({}^{-8}\))** \\ \hline
1 & 4.100 (92.390) \\
2 & 9.225 (2.775) \\
3 & 161.022 (1.616) \\
4 & 362.299 (0.842) \\
5 & 1262.435 (5.348) \\
6 & 2840.480 (5.243) \\ \hline \end{tabular}
\end{table} 
Table 1: Natural Frequencies of the first six modes for a Cantilever Beam with a Rectangular Cross-Sectional Area (Distinct Eigenvalues). Figure 5: Natural shape modes for a Cantilever Beam with a Rectangular Cross-Sectional Area (Distinct Eigenvalues). Solid Black lines represent the analytical solution, red represents the first and second mode, magenta the third and fourth mode, and green the fifth and sixth mode. same behavior is found for all the sensitivities regardless of the parameter of interest. Other combinations of sensitivities are presented in the Supplementary Material. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mode i** & \begin{tabular}{c} \(\lambda_{i,\rho}\times\mathbf{10^{-2}}\) \\ **(\%Error \(\times\mathbf{10^{-8}}\))** \\ \end{tabular} & \
\begin{tabular}{c} \(\lambda_{i,\rho\rho}\times\mathbf{10^{-5}}\) \\ **(\%Error \(\times\mathbf{10^{-8}}\))** \\ \end{tabular} &
\begin{tabular}{c} \(\lambda_{i,\rho\rho}\times\mathbf{10^{-8}}\) \\ **(\%Error \(\times\mathbf{10^{-8}}\))** \\ \end{tabular} \\ \hline
1 & \(-0.1480\) (92.390) & \(0.107\) (93.180) & \(-0.116\) (93.716) \\
2 & \(-0.333\) (2.775) & \(0.240\) (2.806) & \(-0.260\)(2.826) \\
3 & \(-5.813\) (1.616) & \(4.197\) (1.620) & \(-4.545\) (1.623) \\
4 & \(-13.079\) (0.842) & \(9.444\) (0.842) & \(-10.228\) (0.843) \\
5 & \(-45.575\) (5.348) & \(32.906\) (5.348) & \(-35.639\) (5.348) \\
6 & \(-102.540\) (5.243) & \(74.039\) (5.243) & \(-80.187\) (5.243) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Eigenvalue sensitivity with respect to the density for a system with distinct eigenvalues. The subscripts imply derivation.