# Free Vibration Analysis of a Thermally Loaded Porous Functionally Graded Rotor-Bearing System
## 5 Validations
### Natural Frequencies of Simply Supported Porous Functionally Graded Square Cross-Sectional Beam  
The dimensionless fundamental natural frequency \(\overline{\omega}(\overline{\omega}^{4}=\omega^{2}\text{L}^{4}\rho_{\text{ss} }/\text{E}_{\text{ss}}\text{h}^{2})\) of the simply supported functionally graded beam with porosity is calculated using Python code and is validated with the published results of [PERSON] et al. [32] in this section. The validation was performed to verify the porosity formulation and the Python code. The materials used in the porous functionally graded beam are Si\({}_{3}\)N\({}_{4}\) and SUS304 and the temperature-dependent coefficients of Young's modulus, the mass density and Poisson's ratio of these materials are tabulated in Table 3. The slenderness ratio of the beam L/h is taken as 20. The natural frequencies are validated for the different material distributions of the metal-ceramic (k = 0.5, 1, 2, 5) and porosity models (\(\alpha\) = 0, 0.1, 0.2). The validations are tabulated in Table 4. The computed values are in good agreement with the values found in the literature. Hence, based on this validation, we ensured that the porosity formulation and FE code used to compute the DNFs were accurate.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(\alpha\)} & \multicolumn{3}{c}{**k = 0.5**} & \multicolumn{3}{c}{**k = 1**} \\ \cline{2-7}  & **Present** & [30] & **Error \%** & **Present** & [30] & **Error \%** \\ \hline
0 & 4.5283 & 4.5158 & 0.28 & 3.9695 & 3.9583 & 0.28 \\
0.1 & 4.5947 & 4.5821 & 0.27 & 3.9621 & 3.9509 & 0.28 \\
0.2 & 4.6806 & 4.6678 & 0.27 & 3.9516 & 3.9406 & 0.28 \\ \hline \multirow{2}{*}{\(\alpha\)} & \multicolumn{3}{c}{**k = 2**} & \multicolumn{3}{c}{**k = 5**} \\ \cline{2-7}  & **Present** & [30] & **Error \%** & **Present** & [30] & **Error \%** \\ \hline
0 & 3.5659 & 3.5553 & 0.3 & 3.2436 & 3.2332 & 0.32 \\
0.1 & 3.5188 & 3.5082 & 0.3 & 3.1736 & 3.1634 & 0.33 \\
0.2 & 3.4599 & 3.4495 & 0.3 & 3.0888 & 3.0787 & 0.33 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Dimensionless natural frequencies of a simply supported, functionally graded square cross-sectional beam with porosity.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Nodes**} & \multicolumn{3}{c}{**k = 0.5**} & \multirow{2}{*}{**Error \%**} & \multicolumn{3}{c}{**k = 1**} & \multirow{2}{*}{**Error \%**} & \multicolumn{3}{c}{**k = 5**} & \multirow{2}{*}{**Error \%**} \\ \cline{2-2} \cline{6-9}  & [26] & **Present** & & [26] & **Present** & & [26] & **Present** & \\ \hline
1 st & 3.2059 & 3.2031 & 0.087 & 3.1859 & 3.1780 & 0.248 & 3.1505 & 3.1328 & 0.562 \\
2 nd & 3.2059 & 3.2031 & 0.087 & 3.1859 & 3.1780 & 0.248 & 3.1505 & 3.1328 & 0.562 \\
3 rd & 6.3455 & 6.3492 & 0.058 & 6.3059 & 6.2995 & 0.101 & 6.2356 & 6.2098 & 0.414 \\
4 th & 6.3455 & 6.3492 & 0.058 & 6.3059 & 6.2995 & 0.101 & 6.2356 & 6.2098 & 0.414 \\
5 th & 9.3684 & 9.3941 & 0.274 & 9.3096 & 9.3206 & 0.118 & 9.2054 & 9.1872 & 0.198 \\
6 th & 9.3684 & 9.3941 & 0.274 & 9.3096 & 9.3206 & 0.118 & 9.2054 & 9.1872 & 0.198 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Dimensionless natural frequencies of simply supported non-rotating FG shaft.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Material Properties**} & \multicolumn{1}{c}{**Stainless Steel (SS)**} & \multicolumn{1}{c}{**Zirconia (ZrO2)**} \\ \hline Young’s Modulus (GPa) & 207.8 & 168 \\ Density (kg/m\({}^{3}\)) & 8166 & 5700 \\ Poisson’s ratio & 0.3 & 0.24 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Material properties of Stainless Steel and Zirconia.