## 3 Linear Analysis
This figure also illustrates the platform deformation in mode 11, revealing similar deformation patterns across models. Table 5 presents the periods and mass participation factors for the first 30 modes across all three master node configurations. Key observations include the following:  
* The highest natural period across all models is 2.70 s along the X-axis, with mass participation factors exceeding 83%. * Along the Y-axis, the highest period is approximately 2.5 s, with mass participation factors around 79%. * In the Z-axis, the maximum mass participation factor occurs in mode 11 for Models 1 and 2, whereas for Model 3, it occurs in mode 12. These variations influence the final results, as detailed in Section 5.3. Figure 4: Distribution of stiffness and mass. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Frequencies and Generalized Mass**} & \multicolumn{6}{c}{**Mass Participation Factor Report**} \\ \cline{3-10} \multicolumn{1}{c}{**Mode**} & \multicolumn{3}{c}{**Based on Expanded Degrees of Freedom**} \\ \cline{3-10}  & \multicolumn{3}{c}{**FREQ. (CPS)**} & \multicolumn{3}{c}{**Period (SECS)**} & \multicolumn{3}{c}{**X**} & \multicolumn{3}{c}{**Y**} & \multicolumn{3}{c}{**Z**} & \multicolumn{3}{c}{**X**} & \multicolumn{3}{c}{**Y**} & \multicolumn{3}{c}{**Z**} \\ \cline{3-10}  & & & & & & & & & & \\ \hline \multirow{11}{*}{
\begin{tabular}{} \end{tabular} } & 1 & 0.3698 & 2.7045 & 0.8379 * & 0.0003 & 0.0005 & 0.8379 & 0.0003 & 0.0005 \\  & 2 & 0.3947 & 2.5336 & 0.0004 & 0.7951 * & 0.0000 & 0.8382 & 0.7954 & 0.0005 \\  & 3 & 0.4790 & 2.0877 & 0.0016 & 0.0007 & 0.0000 & 0.8398 & 0.7961 & 0.0005 \\  & 4 & 1.0989 & 0.9436 & 0.0001 & 0.1803 & 0.0002 & 0.8399 & 0.9764 & 0.0008 \\  & 5 & 1.1257 & 0.8883 & 0.1405 & 0.0003 & 0.0014 & 0.9805 & 0.9768 & 0.0021 \\  & 6 & 1.2975 & 0.7707 & 0.0013 & 0.0032 & 0.0000 & 0.9818 & 0.9800 & 0.0021 \\  & 7 & 1.4527 & 0.6862 & 0.0001 & 0.0002 & 0.0003 & 0.9819 & 0.9802 & 0.0025 \\  & 8 & 1.6618 & 0.0017 & 0.0000 & 0.0000 & 0.9000 & 0.9819 & 0.9802 & 0.0025 \\  & 9 & 2.3835 & 0.4195 & 0.0011 & 0.0014 & 0.1039 & 0.9830 & 0.9815 & 0.1064 \\  & 10 & 2.4075 & 0.4154 & 0.0000 & 0.0124 & 0.0084 & 0.9830 & 0.9940 & 0.1148 \\  & 11 & 2.4145 & 0.0142 & 0.0072 & 0.0004 & 0.4464 * & 0.9903 & 0.9944 & 0.5613 \\  & 12 & 2.5465 & 0.3927 & 0.0040 & 0.0001 & 0.3564 & 0.9943 & 0.9944 & 0.9177 \\  & 13 & 3.2880 & 0.3041 & 0.0001 & 0.0000 & 0.0016 & 0.9944 & 0.9944 & 0.9192 \\  & 14 & 3.4741 & 0.2878 & 0.0009 & 0.0000 & 0.0055 & 0.9952 & 0.9945 & 0.9247 \\  & 15 & 3.7416 & 0.2673 & 0.0000 & 0.0004 & 0.0004 & 0.9952 & 0.9949 & 0.9251 \\  & 30 & 4.1567 & 0.2406 & 0.0000 & 0.0000 & 0.0000 & 1.0000 & 0.9982 & 0.9254 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Period and frequencies, platform, and Models 1, 2, and 3.  
vertical direction. According to the Pemex technical specification \"P.2.0130.01-2015\" [36], the resistance-level analysis was conducted using acceleration spectrum loads. Damping factors significantly influenced the analysis results. A 5% structural critical damping was considered, as prescribed in API RP-2A-WSD (Section 5.3.6.3.2) [35]. Table 7 compares load results for 5% damping (as per Pemex specifications) and 2% damping, revealing that vertical loads at 2% damping (1314.98 tons) are lower than those at 5% damping (1509.55 tons). Consequently, structural analysis was performed using the more conservative 5% damping results.