# Distributed Dynamic Load Identification of Beam Structures Using a Bayesian Method
## 4 Conclusions  
In this paper, we proposed distributed dynamic load identification using a Bayesian method based on the Bayesian inference to identify the distributed dynamic load. This method is discussed with the various noise levels and the irregular loads in the frequency domain. Firstly, the identification model of the simply supported beam is given by using the generalized Fourier series expansion, while the dynamic calibration matrix is also determined to obtain the orthogonal polynomial coefficient vectors. Secondly, based on the Bayesian inference, the Markov random field is employed as the prior model and the Gaussian noise is used to obtain the likelihood function. Therefore, the hierarchical Bayesian model of the load identification is derived to better determine the parameters. Finally, we select the Gamma distribution to derive the joint posteriori probability density function of the orthogonal polynomial coefficients through combining the likelihood function and the prior probability distribution, and the coefficients, as well as the unknown parameters, are solved via the numerical iteration.  
The algorithm performance is tested through a series of numerical simulations with various spatial distributions. Meanwhile, the comparison between the Tikhonov regularization and the proposed method is performed to show the advantage of the proposed method. In example 1, the results show the RE error of the proposed method is 0.64% when the 1% noise is added; on the contrary, the RE are 9.3% and 8.86% using the Tikhonov regularization, which is a higher error level compared with the proposed method at the identical noise level. This suggests that the prior information fully used in the distributed dynamic load identification achieves the better accuracy. Test example 2 indicates the suppression effect on the PRE error of the proposed method is obvious in the case of more complicated distributed load and high noise levels. In example 3, the proposed approach resilience to noise is investigated with different levels of noise, considering both the real part and imaginary parts. In these circumstances, identified distributed loads remain robust and accurate, which reflect a strong anti-noise performance. In addition, the operation accuracy based on the Bayesian method is superior contrasted with the Tikhonov regularization in the three different cases of simulation examples. This work presents a first attempt to obtain the distributed loads using the Bayesian inference and has broad application prospects.  
Conceptualization, S.L. and J.J.; methodology, J.J.; software, S.L.; validation, S.L.; formal analysis, F.Z.; investigation, F.Z.; resources, F.Z.; data curation, S.L.; writing--original draft preparation, S.L. and J.J.; writing--review and editing, M.S.M.; visualization, S.L.; supervision, M.S.M.; project administration, S.L.; funding acquisition, J.J. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Foundation of National Key Laboratory of Science and Technology on Rotocraft Aeromechanics (No. 61422202105), Qing Lan Project.