## 5 Numerical Results
### Single Force Identification on the Cantilever Beam  
Next, to investigate the proposed augmented method for continuous system, we study the load identification on cantilever beam. The cantilever model is shown in Figure 22, and the parameters of the cantilever beam are: length \(l=1\,\mathrm{m}\), cross-sectional area \(A=0.0004\,\mathrm{m}^{2}\), moment of inertia \(l=2.13\times 10^{-9}\,\mathrm{m}^{4}\), Modulus of Elasticity \(E=209\,\mathrm{Gpa}\), density \(\rho=7800\,\mathrm{kg/m}^{3}\), and Poisson's ratio 0.3. The unknown load is applied at point 15, and the response is measured at point 17. Both points are displayed in Figure 22. Again, sinusoidal and impact loads are considered with the two previously considered noise levels, i.e., SNR = 30 and 40 dB. The system starts with a zero initial state. The time sampling interval is 1/1024 s, and the simulation time span is 1 s. For demonstration, the measured system response to the impulse load is shown in Figure 23. The identification results are listed in Figures 24-27, and the accuracy of the load identification results are shown in Table 4.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**SNR**} & \multirow{2}{*}{**Force**} & \multicolumn{2}{c}{**Tikhonov + L Curve**} & \multicolumn{2}{c}{**Tikhonov + GCV**} & \multicolumn{2}{c}{**Augmented Tikhonov**} \\ \cline{3-8}  & & \(r\) & \(RE\) & \(r\) & \(RE\) & \(r\) & \(RE\) \\ \hline \multirow{2}{*}{40 dB} & Force 1 & 96.91\% & 25.22\% & 98.69\% & 16.14\% & 99.38\% & 11.24\% \\  & Force 2 & 97.81\% & 21.25\% & 99.06\% & 13.78\% & 99.55\% & 9.52\% \\ \hline \multirow{2}{*}{30 dB} & Force 1 & 96.33\% & 27.16\% & 97.03\% & 24.26\% & 98.21\% & 19.02\% \\  & Force 2 & 97.02\% & 24.92\% & 97.65\% & 22.01\% & 98.72\% & 16.12\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: Identification effect of two sinusoidal loads.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**SNR**} & \multirow{2}{*}{**Force**} & \multicolumn{2}{c}{**Tikhonov + L Curve**} & \multicolumn{2}{c}{**Tikhonov + GCV**} & \multicolumn{2}{c}{**Augmented Tikhonov**} \\ \cline{3-8}  & & \(r\) & \(RE\) & \(r\) & \(RE\) & \(r\) & \(RE\) \\ \hline \multirow{2}{*}{40 dB} & Force 1 & 98.49\% & 16.75\% & 98.49\% & 16.75\% & 99.18\% & 12.29\% \\  & Force 2 & 98.72\% & 15.39\% & 98.72\% & 15.39\% & 99.33\% & 11.11\% \\ \hline \multirow{2}{*}{30 dB} & Force 1 & 98.09\% & 18.65\% & 97.11\% & 22.98\% & 98.44\% & 17.30\% \\  & Force 2 & 98.47\% & 16.80\% & 97.59\% & 21.19\% & 98.93\% & 15.46\% \\ \hline \hline \end{tabular}
\end{table}
Table 3: Identification effect of two impact loads.  
Figure 22: Cantilever model.  
Figure 23: Unit impulse response diagram.  
Figure 24: Identification result of sinusoidal load at SNR 40 dB on cantilever: (**a**) Tikhonov method; (**b**) the proposed method.  
Figure 25: Identification result of sinusoidal load at SNR 30 dB on cantilever: (**a**) Tikhonov method; (**b**) the proposed method.  
The results show that the proposed method can accurately identify the unknown loads on a continuous system, such as the considered cantilever beam. Consistent with the previous examples, the augmented Tikhonov method is more appropriate for solving the ill-posed problem of the load identification than the standard Tikhonov method. The sinusoidal load, under the two considered noise conditions, is identified with much better accuracy with the augmented method compared to the standard Tikhonov regularization method. The correlation coefficient and relative error of the results identified by tradition Tikhonov method are not as good as the proposed method, which suggests that the proposed method makes an improvement in the regularization parameters selected.