# Distributed Dynamic Load Identification of Beam Structures Using a Bayesian Method
## 3 Numerical Simulation
### Test Example 1  
To test the accuracy of this numerical simulation in the case of low-level noise, the load acting on a simply supported beam is given by: \(F^{r}(x)=\cos\frac{2\pi x}{l}+\sin\frac{\pi x}{2l},F^{i}(x)=0\) with the excitation frequency \(\omega_{0}=20\). Meanwhile, the measuring points, \(10\), are uniformly distributed and the orders of polynomials are \(8\) to accurately identify the load. Moreover, this method is compared with the Tikhonov regularization to evaluate the superiority. To analyze the resistance to the various noise levels, Gaussian noise was added with RMS amplitudes of \(1\%\) and \(5\%\) of the response amplitudes. All the computations are run on a Windows PC with an Intel Core(TM) i5 1.6 GHz CPU and 8 GB of RAM. The identification results are shown in Figures 3 and 4 and the error analysis is shown in Table 2. Furthermore, to investigate the operation efficiency of the proposed method, the CPU time of the algorithm and the Tikhonov regularization [37] are compared in Table 3.  
Figure 4: _Cont._  
Figure 3: The identification results with 1% 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 3 and 4 show the identification results of the 3 methods in the case of 1% and 5% Gaussian noise, respectively. Figures 3a and 4a show the best approximation of the proposed method. In addition, compared with the Bayesian method, some fluctuations are observed based on the Tikhonov regularization at both ends of the simply supported beam. This suggests the improved algorithm in this paper is accurate even while the responses are contaminated. Table 2 shows the various errors, e.g., RE, PRE, and ARE. The RE and PRE of the proposed method with 1% noise are 0.64% and 1.12%, respectively, while the RE and PRE of the proposed method with 5% noise are 1.13% and 1.41%, respectively. Although these 3 methods are all in low noise conditions, the 2 errors of the Tikhonov method with GCV regularization with 5% noise are 11.4% and 31.5%, which indicates that the errors are more greatly magnified than the proposed method at the identical noise level. Table 3 indicates that the operation time of the Tikhonov regularization is nearly three times  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Noise Level**} & \multicolumn{2}{c}{**Proposed Method**} & \multicolumn{3}{c}{**Tikhonov with L–Curve Regularization**} & \multicolumn{3}{c}{**Tikhonov with GCV Regularization**} \\ \cline{2-10}  & **RE** & **PRE** & **ARE** & **RE** & **PRE** & **ARE** & **RE** & **PRE** & **ARE** \\ \hline
1\% & 0.64\% & 1.12\% & 0.21\% & 9.3\% & 25.8\% & 1.71\% & 8.86\% & 24.6\% & 1.62\% \\
5\% & 1.13\% & 1.41\% & 0.41\% & 15.08\% & 41.4\% & 2.82\% & 11.4\% & 31.5\% & 2.11\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: Identification error in Example 1.  
Figure 4: 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} \hline \hline \multirow{2}{*}{**Proposed Method**} & \multicolumn{3}{c}{**Tikhonov with L–Curve Regularization**} & \multicolumn{3}{c}{**Tikhonov with GCV Regularization**} \\ \cline{2-5}  & **RE** & **PRE** & **ARE** & **RE** & **PRE** & **ARE** & **RE** & **PRE** & **ARE** \\ \hline
1\% & 0.64\% & 1.12\% & 0.21\% & 9.3\% & 25.8\% & 1.71\% & 8.86\% & 24.6\% & 1.62\% \\
5\% & 1.13\% & 1.41\% & 0.41\% & 15.08\% & 41.4\% & 2.82\% & 11.4\% & 31.5\% & 2.11\% \\ \hline \hline \end{tabular}
\end{table}
Table 3: Calculation time in Example 1.  
that of the Bayesian method, which regards the unknown variable as a prior probability distribution; hence, it is superior in increasing the operational efficiency.