## 4 Sensitivity Analysis for Eigenfrequency Problems Using HYPAD
### Free Vibration of a Cantilever Beam
#### 4.2.2 Cantilever Beam with Repeated Eigenvalues  
A model with symmetries about two rotational axes was studied to demonstrate our methodology when the system exhibits repeated eigenvalues. Here, a square cross-sectional area cantilever beam with \(L=5\) m, \(E=68.9\) GPa, \(\rho=2770\) kgm\({}^{-3}\), \(b=0.01\) m, and \(b=d=0.01\) m was considered. High-order sensitivities with respect to geometrical global and directional design variables were calculated. The first case corresponded to variables whose derivatives are also repeated because the symmetry of the system was not altered. On the other hand, directional design variables correspond to variables that affect the symmetry of the system; therefore, regardless of having repeated eigenvalues, their sensitivities are not repeated. Table 5 shows the results for the eigenvalues of the system. The eigenvectors are shown in Figure 7. As in the previous example, good agreement is found between the analytical solution and the numerical estimates, resulting in a maximum RE of \(7.562\times 10^{-7}\%\) for the eigenvalues and an average of \(0.0499\%\) for the eigenvectors.  
\begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Mode i**} & \(\lambda_{i,E}\times\mathbf{10^{-9}}\) & \(\lambda_{i,E_{\rho}}\times\mathbf{10^{-13}}\) & \(\lambda_{i,E_{\rho},L}\times\mathbf{10^{-13}}\) \\  & **(\%Error \(\times\) 10\({}^{-8}\))** & **(\%Error \(\times\) 10\({}^{-8}\))** & **(\%Error \(\times\) 10\({}^{-8}\))** \\ \hline
1 & 0.0595 (0.0911) & \(-0.215\) (0.107) & 0.172 (0.831) \\
2 & 0.134 (0.137) & \(-0.483\) (0.178) & 0.3866 (0.585) \\
3 & 2.337 (0.679) & \(-8.436\) (0.679) & 6.750 (0.689) \\
4 & 5.258 (0.672) & \(-18.983\) (0.671) & 15.187 (0.675) \\
5 & 18.323 (5.286) & \(-66.147\) (5.286) & 52.918 (5.286) \\
6 & 2578 (5.285) & \(-148.830\) (5.285) & 119.060 (5.285) \\ \hline \end{tabular}
\end{table}
Table 4: Eigenvalue sensitivity with respect to Young’s Modulus and mixed partial derivatives for a system with distinct eigenvalues. The subscripts imply derivation.  
\begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Mode i**} & \(\lambda_{i,L}\times\mathbf{10^{2}}\) & \(\lambda_{i,LL}\times\mathbf{10^{2}}\) & \(\lambda_{i,LLL}\times\mathbf{10^{2}}\) \\  & **(\%Error \(\times\) 10\({}^{-8}\))** & **(\%Error \(\times\) 10\({}^{-8}\))** & **(\%Error \(\times\) 10\({}^{-8}\))** \\ \hline
1 & \(-0.0328\) (23.643) & 0.0328 (11.605) & \(-0.0394\) (6.725) \\
2 & \(-0.0738\) (2.916) & 0.0738 (0.827) & \(-0.0885\) (0.216) \\
3 & \(-1.288\) (0.883) & 1.288 (0.791) & \(-1.5458\) (0.739) \\
4 & \(-2.898\) (0.683) & 2.898 (0.691) & \(-3.478\) (0.696) \\
5 & \(-10.099\) (5.308) & 10.099 (5.295) & \(-12.119\) (5.290) \\
6 & \(-22.724\) (5.276) & 22.724 (5.282) & \(-27.269\) (5.284) \\ \hline \end{tabular}
\end{table}
Table 3: Eigenvalue sensitivity to the beam length for a system with distinct eigenvalues. The subscripts imply derivation.  
\begin{table}
\begin{tabular}{c c} \hline
**Mode i** & \(\lambda_{i}\) **(\%Error \(\times\) 10\({}^{-8}\))** \\ \hline
1, 2 & 4.100 (75.618) \\
3, 4 & 161.020 (1.152) \\
5, 6 & 1262.400 (5.330) \\ \hline \end{tabular}
\end{table}
Table 5: Natural Frequencies of the first six modes for a Cantilever Beam with a Rectangular Cross-Sectional Area (Distinct Eigenvalues).