## 3 Numerical Examples
### Numerical Example 1  
The calculation example directly applies the simply supported beam model provided by [PERSON] in Reference [13]. The first cross-section is shown in Figure 2, characterized by its rectangular shape with an additional notch, designed to investigate the influence of geometric irregularities on the dynamic behavior of beams under axial loads and bidirectional bending. This geometry serves to elucidate how asymmetrical modifications affect structural responses and validates the accuracy of our model in simulating real-world structural elements. The geometric and material properties and parameters are listed in Appendix A (Table A1). The numerical results are shown in Table 1.  
It can be observed that for different values of \(n\), Equation (31) yields the six natural vibration frequencies of a simply supported beam. These frequencies correspond to the natural vibration modes associated with \(V_{x}\), \(V_{y}\), \(V_{z}\), \(\Theta_{y}\), \(\Theta_{z}\), and \(\Psi\). The calculations are based on the parameters provided in the coordinate system defined in [13]. To validate the theory proposed in this paper, it was compared with the classical theory. By calculating the natural frequencies and comparing them with the results from traditional methods, we aimed to further demonstrate the effectiveness and accuracy of the proposed Timoshenko-Vlasov beam approach. The traditional calculation methods used to analyze vibrations in rods and beams are introduced below. For the axial vibration of the rod, the displacement function \(V_{x}(x)=C_{V_{x}}\cos(\lambda_{n}x)\), as given in Equation (28), can be verified to satisfy the free boundary conditions, where the derivative at both \(x=0\) and \(x=L\) is zero. The corresponding frequency equation is given by  
\[\omega_{n}=\frac{n\pi}{L}\sqrt{\frac{E}{\rho}}, \tag{32}\]  
where \(E\) is [PERSON]'s modulus and \(\rho\) is the material density. In the case of the torsional vibration of the rod, the displacement function \(\Psi(x)=C_{\Psi}\sin(\lambda_{n}x)\) satisfies the boundary conditions at \(x=0\) and \(x=L\). The frequency equation for torsional vibration is then given by  
\[\omega_{n}=\frac{n\pi}{L}\sqrt{\frac{G}{\rho}}, \tag{33}\]  
where \(G\) is the shear modulus of the material. For the vibration of a Timoshenko beam with simply supported ends, considering the effect of the rotational inertia of the cross-section, the frequency equation is expressed as \begin{table}
\begin{tabular}{l c c c} \hline
**External Mode** & \(L\) = 3 m & \(L\) = 10 m & \(L\) = 50 m \\ \hline \(n=1\) & 126.92 & 12.11 & 0.96 \\  & 623.79 & 65.09 & 2.65 \\  & 928.27 & 110.83 & 4.62 \\  & 4076.94 & 1609.39 & 321.88 \\  & 4648.63 & 3487.30 & 3359.94 \\  & 5364.62 & 3532.80 & 3458.59 \\ \hline \(n=2\) & 504.83 & 46.19 & 2.47 \\  & 1897.84 & 247.51 & 10.57 \\  & 2439.76 & 399.06 & 18.38 \\  & 5361.12 & 3218.77 & 643.75 \\  & 7075.23 & 3697.01 & 3377.22 \\  & 10,729.24 & 3894.63 & 3467.12 \\ \hline \(n=3\) & 1132.85 & 102.96 & 4.80 \\  & 3351.31 & 518.71 & 23.71 \\  & 3960.61 & 786.96 & 40.98 \\  & 6830.85 & 3971.08 & 965.63 \\  & 9810.44 & 4441.75 & 3405.40 \\  & 16,093.86 & 4828.16 & 3481.55 \\ \hline \end{tabular}
\end{table} 
Table 1: Natural frequencies (rad/s) of Timoshenko–Vlasov beam studied as numerical example 1. Figure 2: Cross-section used in numerical example 1. \[\omega_{n}=\left(\frac{n\pi}{L}\right)^{2}\sqrt{\frac{EI^{\prime}}{\rho A}}\left[1+ \frac{I^{\prime}}{A}\left(\frac{n\pi}{L}\right)^{2}\right]^{-1/2}, \tag{34}\]  
where \(I^{\prime}\) is the second moment of area, \(A\) is the cross-sectional area, and \(E\) is [PERSON]'s modulus.