# Distributed Dynamic Load Identification of Beam Structures Using a Bayesian Method
## 1 Introduction
Subsequently, this approach was employed in the aircraft via the Taylor expansion and the particle swarm optimization algorithm, which balanced the sensor cost and the performance of load identification in the system framework. However, only the polynomial distributed load can be identified, and it is critical to discuss the intricate load identification [23]. In another work, a comprehensive method based on the Latin hypercube sampling genetic algorithm (LHS-GA) and the improved L-curve method was proposed to determine the upper and lower bounds of the distributed force on uncertain structures, which improved the ill-posedness and the anti-noise compared with the traditional regularization method [24]. Unfortunately, distributed dynamic load identification is often an ill-posed problem. A common method to bypass the problem is to apply the regularization technique to stabilize the solution. The most widely adopted regularization is the Tikhonov regularization [25; 26; 27]. For example, based on the truncated singular value decomposition (TSVD) and the L-curve, the regularization method was presented to evaluate the internal excitations and the regularization efficiency, which encouraged obtaining knowledge of the defects on the drive train models [28]. Although widely used, it should be noted that there is a significant limitation on the selection of the optimal regularization parameters. For this particular reason, extensive research efforts have been focused on the Bayesian method in recent years. For example, a sparse convex optimization model of the load identification was established with different basis functions by minimizing the components of the non-zero status in the coefficients of the sparse regularization, and the effectiveness and applicability of the cantilever thin plate structure were verified through the simulation and the experiment [29]. Furthermore, B. Jin proposed the augmented Tikhonov regularization method, namely, the Bayesian regularization method, which provided a novel idea for the selection of the regularization parameters in load identification [30; 31]. In the same vein, the superiority of the Bayesian method, using the Gaussian kernel functions to obtain the time history of unknown impact load, was presented for the impact load identification of the composite structures by organizing the simulations and the test [32; 33]. Apparently, the enormous advance was observed by making the most of the Bayesian inference and the generalized iterative weighted least squares algorithm to identify the external excitation [34; 35; 36]. In general, the aforementioned application of the Bayesian method mainly focused on the concentrated load identification, however, there was no relevant research on distributed dynamic load identification. Therefore, it is essential to develop the Bayesian framework to reconstruct the distributed dynamic load. In this paper, a novel distributed dynamic load identification based on the Bayesian method is proposed. The key of the proposed method is that the orthogonal polynomial coefficients are regarded as random variables to obtain the prior probability distribution. Then, the estimate of the posterior probability distribution is carried out by using the statistical inference. This paper is arranged as follows. In Section 2, the load identification algorithm of the proposed method is introduced, including the load identification model and the application of the Bayesian method in load identification for a simply supported beam; in Section 3, the effectiveness and the accuracy under various noise levels are discussed and compared with the Tikhonov regularization. Some conclusions are generalized in Section 4.