## 3 Model Validation
### Dynamic Validation of the Free Oscillation of Flexible Beams  
The example analyzed in this section depicts the free fall of a flexible pendulum under its own weight. The two-dimensional beam diagram can be seen in Figure 7. The beam, hinged on the left side to allow free rotation, has its parameters detailed in Table 4 below. The beam's initial position is horizontal with zero initial velocity, and it undergoes free falling under the influence of gravity. [PERSON] et al. (2014) employed 6 and 12 elements for calculation, respectively, lending significant authority to the study. To ensure comparability, this paper employs a consistent number of 6 elements for calculation, with a computation time of 1.1 s, a time step of \(10^{-4}\) s, and a convergence error threshold set at \(10^{-7}\).  
Figure 8 depicts the vertical displacement of the right endpoint as calculated by [PERSON] and [PERSON]'s model, the [PERSON] (1955) beam model, and the ANCF beam model as proposed in this paper. From the results in Figure 8, good consistency is observed among the three models, demonstrating the correctness of the ANCF modeling method.  
In order to further validate the numerical stability of this method and simulate the large deformation characteristics of the beam, the gravitational acceleration \(g\) is increased to 50 m/s\({}^{2}\), while the other parameters remain constant, and each moment is meticulously recorded.  
\begin{table}
\begin{tabular}{c c c} \hline
**Definition** & **Numerical** & **Unit** \\ \hline Initial Length & 1.2 & m \\ Cross-sectional Area & 0.0016 & m\({}^{2}\) \\ Second Moment of Inertia & 8.533 \(\times\) 10\({}^{-6}\) & m\({}^{6}\) \\ Mass Density & 5540 & kg/m\({}^{3}\) \\ Poisson’s Ratio & 0.3 & / \\ Young’s Modulus & 0.7 \(\times\) 10\({}^{6}\) & Pa \\ Gravitational Acceleration & 9.81 & m/s\({}^{2}\) \\ \hline \end{tabular}
\end{table}
Table 4: Parameters of the freely falling flexible pendulum.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**End Position and Deformation Angle**} \\ \cline{2-4}
**Number of Elements** & \multicolumn{3}{c}{**(m and rad)**} \\ \cline{2-4}  & \multicolumn{3}{c}{**End Position (x, y)**} & **Deformation Angle** \\ \hline
4 & 19.977927 & \(-\)1.983935 & \(-\)0.098982 \\
12 & 19.923734 & \(-\)1.965861 & \(-\)0.098351 \\
16 & 19.904153 & \(-\)1.959780 & \(-\)0.098145 \\ ANSYS & 19.977936 & \(-\)1.957750 & \(-\)0.097684 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Deformation position of the endpoint of the long and slender cantilever beam, S\({}_{\text{L}}\) = 500.  
Figure 7: Flexible free oscillation beam model.  
Figure 9 depicts the swinging trajectory of the beam simulated by six ANCF elements. Under the extreme condition of increasing the gravitational constant to 50 m/s\({}^{2}\), higher acceleration values and a relatively higher mass contribute to greater inertia forces, resulting in larger deformations and higher angular velocities. The flexible beam begins swinging freely from the right end to the top position on the left side, then swings back, with the beam end curling due to inertia, which is consistent with general patterns. The computational results align with those of [PERSON] et al. [45] and others, demonstrating the accuracy and robustness of this model in the dynamic analysis of large deformation structures.  
Figure 8: Time history diagram of vertical displacement of the free end.  
Figure 9: Falling flexible pendulum at different time steps.