## 5 Numerical Results
### Single Force Identification Simulation  
We first aim to identify the load applied on a four-degrees-of-freedom system. The system is shown in Figure 4, and the system parameters are listed underneath. \[ \begin{array}{c}m_{1}=m_{2}=m_{3}=m_{4}=10\,\mathrm{kg}\\ k_{1}=k_{2}=k_{3}=k_{4}=k_{5}=5000\,\mathrm{N/m}\\ c_{1}=c_{2}=c_{3}=c_{4}=c_{5}=100\,\mathrm{N\cdot s/m}\end{array}  \tag{31}\]  
The regularization parameters are obtained following the procedure described in Section 4. Three types of force are studied: simple harmonic excitation, triangular wave excitation, and impact load. The load is only applied to the first degree of freedom. We assume the system initially has a stationary state, i.e., zero initial conditions. The sampling time interval is 0.001 s. The considered time span is 0.5 s. The load identification results of the three loads with the SNR of 40 dB, 30 dB, and 20 dB are shown in Figures 5-13. On the left side of these figures, we show the results obtained with the Tikhonov regularization method, and, on the right, we show the result of the proposed augmented Tikhonov regularization method. The identification result errors are shown in Table 1.  
Figure 4: Four-degree-of-freedom system. Figure 5: Identification result of sinusoidal load at SNR 40 dB: (**a**) Tikhonov method; (**b**) the proposed method. Figure 8: Identification result of triangle wave load at SNR 40 dB: (**a**) [PERSON] method; (**b**) the proposed method. Figure 6: Identification result of sinusoidal load at SNR 30 dB: (**a**) Tikhonov method; (**b**) the proposed method. Figure 7: Identification result of sinusoidal load at SNR 20 dB: (**a**) [PERSON] method; (**b**) the proposed method. Figure 11: Identification result of impact load at SNR 40 dB: (**a**) [PERSON] method; (**b**) the proposed method. Figure 10: Identification result of triangle wave load at SNR 20 dB: (**a**) [PERSON] method; (**b**) the proposed method. Figure 9: Identification result of triangle wave load at SNR 30 dB: (**a**) [PERSON] method; (**b**) the proposed method. Table 1 shows that the relative error and the correlation coefficient for the results of the augmented Tikhonov regularization method are significantly improved when compared to the Tikhonov method for both considered noise levels, i.e., SNR = 40 dB, 30 dB, and 20 dB. The figures show that the augmented Tikhonov regularization method leads to better identification accuracy for sinusoidal, impact and triangular wave load loads. They also show that the augmented method accurately identify the peak position of the triangular wave load, as well as the impact load. In general, the augmented Tikhonov regularization method shows much better accuracy for identifying sinusoidal loads under all considered noise conditions. For the impact load, the augmented Tikhonov regularization method tends to suppress the oscillation of the entire load, so the identification accuracy is very good when compared to the original Tikhonov regularization method. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Force**} & \multirow{2}{*}{**SNR**} & \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 sinusoidal & 40 & 87.83\% & 20.50\% & 95.30\% & 11.25\% & 99.74\% & 7.18\% \\  & 30 & 82.34\% & 23.01\% & 89.10\% & 18.21\% & 99.84\% & 7.94\% \\  & 20 & 96.93\% & 22.31\% & 90.44\% & 23.58\% & 98.30\% & 18.38\% \\ \hline Triangle & 40 & 90.99\% & 18.13\% & 97.50\% & 8.87\% & 99.82\% & 6.03\% \\  & 30 & 97.25\% & 12.69\% & 99.50\% & 9.42\% & 99.65\% & 8.43\% \\  & 20 & 98.29\% & 17.86\% & 96.55\% & 23.41\% & 99.18\% & 14.06\% \\ \hline Impact & 40 & 97.91\% & 16.58\% & 96.03\% & 16.58\% & 99.64\% & 8.47\% \\  & 30 & 96.16\% & 21.04\% & 96.13\% & 23.32\% & 98.45\% & 16.99\% \\  & 20 & 80.48\% & 28.96\% & 80.51\% & 44.53\% & 96.02\% & 27.82\% \\ \hline \hline \end{tabular}
\end{table}