## 4 Conclusions  
By comparing the natural frequencies of the Timoshenko-Vlasov beam with results derived from traditional vibration theories, we observed distinct differences among various vibration modes. Axial vibration displays the smallest deviation from traditional theories, while bending vibration shows a decreasing gap with higher mode numbers. The torsional mode presents the most significant difference from traditional theories. The axial vibration frequency predicted by the Timoshenko-Vlasov beam model, which incorporates bending-torsion-axial coupling, is nearly identical to the frequency obtained by considering only the axial vibrations of the rod. This similarity likely stems from the traditional beam theory's assumptions of a continuous medium and isotropy, which naturally decouple axial motion from bending and torsional motions. Even when axial deformation is coupled in the Timoshenko-Vlasov model, the axial natural frequency remains almost unchanged compared to the axial vibration of the rod. However, bending deformation is influenced by coupled vibrations, leading to deviations from the traditional theory. In high-frequency scenarios, the results align closely with the natural frequencies obtained from studying only bending vibrations. Torsional modes are most affected by coupling, exhibiting the greatest deviation from traditional rod vibration theories. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**L (m)** & _n = 1_ & _n = 2_ & _n = 3_ \\ \hline
3 & 67.0 kN & 204.8 kN & 434.6 kN \\
10 & 25.2 kN & 37.6 kN & 58.3 kN \\
50 & 21.2 kN & 21.2 kN & 21.7 kN \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Axial load at a natural frequency of zero for the different beam lengths and mode numbers of numerical example 3. \begin{table}
\begin{tabular}{l c c} \hline \hline
**External Mode** & _L = 3 m_ & _L = 10 m_ & _L = 50 m_ \\ \hline \(n=1\) & 19.57 & 3.60 & 0.66 \\  & 469.98 & 43.88 & 1.76 \\  & 851.62 & 83.42 & 3.37 \\  & 5364.62 & 1609.39 & 321.88 \\  & 11,622.62 & 10,678.39 & 10,581.47 \\  & 15,690.35 & 15,123.08 & 15,067.49 \\ \hline \(n=2\) & 68.41 & 8.80 & 1.34 \\  & 1697.72 & 173.57 & 7.04 \\  & 2801.35 & 324.75 & 13.45 \\  & 10,729.24 & 3218.77 & 643.75 \\  & 14,133.29 & 10,972.25 & 10,593.67 \\  & 17,374.48 & 15,294.79 & 15,074.45 \\ \hline \(n=3\) & 149.45 & 16.42 & 2.04 \\  & 3357.58 & 383.51 & 15.84 \\  & 5135.78 & 701.09 & 30.21 \\  & 16,093.86 & 4828.16 & 965.63 \\  & 17,345.45 & 11,435.58 & 10,613.94 \\  & 19,766.82 & 15,574.52 & 15,086.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Natural frequencies (rad/s) of Timoshenko–Vlasov beam studied as numerical example 3. We also investigated the impact of axial load variations on the first-order natural frequency of the Timoshenko-Vlasov beam. The analysis showed that as the axial load increases, the natural frequency of the beam gradually decreases, consistent with the existing literature [4; 9; 12; 14; 15; 21]. As the axial load continues to increase, the lowest natural frequency eventually drops to zero, indicating the onset of elastic buckling, again in line with the trends in the literature [15; 21]. Moreover, for Example 1 (\(L\) = 3 m, 10 m), Example 2 (\(L\) = 3 m, 10 m), and Example 3 (\(L\) = 3 m), the axial load values that cause the natural frequency to become zero are roughly proportional to \(n^{2}\) and inversely proportional to \(L^{2}\), consistent with the classic Euler formula characteristics for the stability of a compression member. This relationship holds true across various examples studied, providing a solid theoretical and practical foundation for addressing modern structural engineering challenges, particularly in designing and analyzing thin-walled beam structures subjected to complex dynamic loads. Through rigorous theoretical analysis, this study demonstrates the robustness and accuracy of the improved Timoshenko-Vlasov beam model in calculating natural frequencies under complex loading conditions.