## 7 Conclusions  
In this paper, we proposed the augmented Tikhonov regularization method in combination with the Green kernel function to identify the external dynamic loads acting on a structure. We also proposed a method for the initial values selection of the regularization parameters. An iterative algorithm for the load identification process was presented. The algorithm performance was evaluated using numerical, as well as experimental, studies. First, we used three numerical examples. The examples showed the efficiency and accuracy of the proposed method for different types of loading, including: an impact load, a sinusoidal wave, or a triangle wave load. The comparisons in these examples show that the proposed method outperforms the standard Tikhonov regularization method for load identification, even when considering noise in the input data. Moreover, the proposed method shows better stability than the standard method, which is attributed to the selection of the regularization parameters. After the numerical examples, we performed an experimental study to further validate the augmented method in a practical application. The method shows high accuracy in capturing different type of loading despite the inaccuracies in the measurement, as well as in the model, parameters. Dynamic load identification, in real conditions, is often affected by constant noise levels, which inevitably appear in the identification results. In the standard Tikhonov method, the noise can result in large fluctuations in the identified load. This is mitigated in the augmented Tikhonov method by a proposed approach to identify the initial parameters based on the noise characteristics. If the noise level does not change, the identification results are stable. Moreover, the paper also proposes a parameters selection approach where the noise level is unknown. Here, the initial parameter \(\beta_{1}\) was chosen to be close to zero, while \(\beta_{2}\) was chosen based on the slope of the residual norm \(\|Hf-y\|_{2}^{2}\). Using this selection process, the values of \(\beta_{1}\) and \(\beta_{2}\) were chosen without knowing the noise level; hence, the process can be of great significance in practical engineering application. Moreover, the regularization parameters selected by the L-curve and GCV methods are not always optimal, which can lead to large errors. This is especially the case when the input data is noisy. However, the results show that the regularization parameters, selected by the proposed method, can still lead to accurate results even with relatively high level of noise.  
This paper presents a first attempt to utilize an augmented Tikhonov regularization method combined with the Green kernel function for load identification. Such a problem can be especially challenging due to the fact that they often are ill-posed. The proposed method to identify dynamic load can have a wide range of applications in several domains, such as design optimization, diagnostics, control, and monitoring of vibrating structures.  
Conceptualization J.J. and J.C.; methodology, J.J.; software, J.C.; validation, H.T.; formal analysis, J.C.; investigation, J.J.; resources, S.L.; data curation, S.L.; writing--original draft preparation, J.J.; writing--review and editing, M.S.M.; visualization, H.T.; supervision, M.S.M.; project administration, H.T.; funding acquisition, J.J. All authors have read and agreed to the published version of the manuscript.  
This research is supported by National Natural Science Foundation of China, No. 51775270.  
Our research is supported by National Natural Science Foundation of China, No. 51775270.  
The authors declare no conflict of interest.