# Distributed Dynamic Load Identification of Beam Structures Using a Bayesian Method
## 3 Numerical Simulation
### Test Example 2  
In this example, a more complicated distributed load is considered compared with the test example 1. The load is now defined as \(F^{\prime}(x)=(x-\frac{l}{2}+1)(x-\frac{l}{2}+3)+\cos(2\pi x)\), \(F^{i}(x)=0\) with the excitation frequency of \(\omega_{0}=50\pi\). This helps to investigate the reconstruction effect in case of high noise levels and irregular loads. Other parameters of the problem remain unchanged as in Example 1. Moreover, this method is compared with the Tikhonov regularization to evaluate the superiority. We applied 10% and 15% Gaussian noise to the measurement vectors to investigate the anti-noise performance of the proposed approach and the identification results are demonstrated in Figures 5 and 6. Moreover, the identification errors are shown in Table 4 and the calculation time is compared in Table 5.  
Figure 5: 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.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Proposed Method**} & \multicolumn{3}{c}{**Tikhonov with L–Curve**} & \multicolumn{3}{c}{**Tikhonov with GCV**} \\ \cline{2-10}
**Noise Level** & \multicolumn{3}{c}{**RE**} & **PRE** & **ARE** & **RE** & **PRE** & **ARE** & **RE** & **PRE** & **ARE** \\ \hline
10\% & 3.05\% & 2.96\% & 1.54\% & 3.97\% & 5.66\% & 1.93\% & 4.09\% & 6.38\% & 1.95\% \\
15\% & 8.11\% & 7.61\% & 4.08\% & 8.53\% & 10.3\% & 4.22\% & 8.64\% & 11.1\% & 4.24\% \\ \hline \hline \end{tabular}
\end{table}
Table 4: Identification error in Example 2.  
Figure 6: 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.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Proposed Method**} & \multicolumn{3}{c}{**Tikhonov with L–Curve**} & \multicolumn{3}{c}{**Tikhonov with GCV**} \\ \cline{2-10}  & **RE** & **PRE** & **ARE** & **RE** & **PRE** & **ARE** & **RE** & **PRE** & **ARE** \\ \hline
10\% & 3.05\% & 2.96\% & 1.54\% & 3.97\% & 5.66\% & 1.93\% & 4.09\% & 6.38\% & 1.95\% \\
15\% & 8.11\% & 7.61\% & 4.08\% & 8.53\% & 10.3\% & 4.22\% & 8.64\% & 11.1\% & 4.24\% \\ \hline \hline \end{tabular}
\end{table}
Table 5: Calculation time in Example 2.  
Figures 5 and 6 show the results with the high-level noise. Although the errors still exist in identified load and the exact load, the results suggest that the identified load obtained by the proposed method is well in agreement with the exact load of the simply supported beam. Table 4 also illustrates the corresponding errors in the two considered cases. Through the error analysis of the results, we find that the RE errors of the 2 noise conditions are 3.05% and 8.11%, respectively, the PRE errors are 2.96% and 7.61%, respectively, and the ARE errors are 1.54% and 4.08%, respectively. In addition, the RE, PRE, and ARE errors are lower compared with the Tikhonov regularization method. Therefore, despite the error increasing significantly with the improvement of noise level when identifying the distributed load, the PRE errors remain reasonably small, which demonstrates the obvious suppression effect on the PRE error of the proposed method. Furthermore, the larger error is focused on both ends of the distributed load. The accuracy of the proposed distributed load identification algorithm in this paper is validated considering the perspective. Table 5 indicates that this algorithm is superior to the Tikhonov regularization method in efficiency.