# Distributed Dynamic Load Identification of Beam Structures Using a Bayesian Method
## 3 Numerical Simulation
### Test Example 3  
In this subsection, we aim to investigate the identification accuracy, while the real and imaginary parts are both considered. The distributed load is now rewritten as \(F_{r}=50\cos 6x+2x^{4}+40,F_{i}=\cos(2\pi x)+\sin(\pi x+\frac{\pi}{4})\) with an excitation frequency of \(\omega_{0}=80\pi\). Other parameters of the problem remain unchanged. Moreover, this method is compared with the Tikhonov regularization to evaluate the superiority. Again, the 3 different cases of noise level are all considered, e.g., 5% noise, 10% noise, and 15% noise to investigate the anti-noise performance of the proposed approach. Figures 7-9 show the identified results using the proposed algorithm and the Tikhonov method. The RE, PRE, and ARE errors are shown in Table 6. Table 7 lists the calculation time required for the proposed approach and Tikhonov to compare their operation efficiency. Figure 7: \(Cont.\)  
Figure 8: The identification results with 10% noise. (**a**) Proposed method; (**b**) identification error; (**c**) Tikhonov with L\(-\)curve regularization; (**d**) identification error; (**e**) Tikhonov with GCV regularization; (**f**) identification error. Figure 7: The identification results with 5% noise. (**a**) Proposed method; (**b**) identification error; (**c**) Tikhonov with L\(-\)curve regularization; (**d**) identification error; (**e**) Tikhonov with GCV regularization; (**f**) identification error. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Noise Level**} & \multicolumn{3}{c}{**Proposed Method**} & \multicolumn{3}{c}{**Tikhonov with L\(-\)Curve**} & \multicolumn{3}{c}{**Tikhonov with GCV**} \\  & \multicolumn{3}{c}{**Regularization**} & \multicolumn{3}{c}{**Regularization**} \\ \cline{2-10}  & **RE** & **PRE** & **ARE** & **RE** & **PRE** & **ARE** & **RE** & **PRE** & **ARE** \\ \hline
5\% & 3.67\% & 5.67\% & 1.63\% & 4.11\% & 7.18\% & 1.77\% & 4.15\% & 7.3\% & 1.78\% \\
10\% & 7.7\% & 9.79\% & 3.48\% & 8.13\% & 11.37\% & 3.62\% & 8.16\% & 11.5\% & 3.63\% \\
15\% & 9.74\% & 11.86\% & 4.4\% & 10.2\% & 13.5\% & 4.54\% & 10.2\% & 13.6\% & 4.55\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Identification error in Example 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Proposed Method** & \multicolumn{3}{c}{**Tikhonov with L\(-\)Curve**} & \multicolumn{3}{c}{**Tikhonov with GCV**} \\  & \multicolumn{3}{c}{**Regularization**} & \multicolumn{3}{c}{**Regularization**} \\ \hline Calculation time (s) & 1.4 & 4.79 & 4.80 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Calculation time in Example 3. Figure 9: The identification results with 15% noise. (**a**) Proposed method; (**b**) identification error; (**c**) Tikhonov with L\(-\)curve regularization; (**d**) identification error; (**e**) Tikhonov with GCV regularization; (**f**) identification error. Figures 7-9 show the identification results with the real and imaginary parts. It can be shown that the feasibility of this algorithm for the simulation model is tested. As we can see, considering the proposed method, the RE errors of the three noise conditions are 3.67%, 7.7%, and 9.74%, respectively, the PRE errors are 5.67%, 9.79%, and 11.86%, respectively, and the ARE errors are 1.63%, 3.48%, and 4.4%, respectively. The identifications under 15% noise conditions have some slight fluctuations in spatial distribution. However, the error is still within the acceptable range, which proves that the proposed approach can realize satisfactory identification effects under high levels of noise. On the contrary, these errors of the Tikhonov regularization with 5%, 10%, and 15% noise are higher than the algorithm proposed in this paper. Thus, the proposed distributed dynamic load identification combining Bayesian inference and numerical iteration can perform perfectly for the beam structure. Table 7 shows that the Bayesian method spends much less time and is more efficient than the Tikhonov regularization method for distributed dynamic load identification.