# Distributed Dynamic Load Identification of Beam Structures Using a Bayesian Method
## 1 Introduction  
Dynamic load identification is a typical class of inverse problems aiming at determining the unknown loads acting on the mechanical structures from the measured responses and dynamical models. It is of great significance to measure the force for a large number of industrial applications, e.g., the dynamic model design and optimization [1; 2; 3; 4; 5; 6; 7; 8; 9], vibration control [10], and structural health monitoring [11]. In the past decades, numerous load identification studies have been developed, involving concentrated dynamic load identification and distributed dynamic load identification. In terms of concentrated load identification, the time history of the load, that is, the one-dimensional function of time, is obtained assuming that the excitation location of the load is known. On the contrary, the distributed dynamic load involves two-dimensional variables of space and time. Due to the continuous distribution characteristic in space and the dynamicity in time, the distributed loads are more complex compared with the concentrated load, hence, limited research is available in the area of distributed dynamic load identification. For example, line distributed dynamic load identification [12] acting on the composite structures was discussed via separating the time-space domain based on aniterative algorithm, and the identified loads are well in agreement with the real distributed loads considering the spatial distribution and the magnitude with the small cost of iteration operation. In order to reconstruct the external excitation and improve the efficiency, the spatial variable of the unknown force [13], as well as the system response, were decomposed and denoted as the local basis functions, and the feasibility was also proved by introducing the regularization to decrease the effect of the noise on load identification and improve the accuracy. Moreover, the finite-difference schemes were developed by Djamaa to carry out the local distributed load identification research acting on a cylindrical thin shell structure; however, this method can lead to a higher sensitivity of the noise when the displacements are given through integrating the velocities [14]. Then, the difference procedure can be converted to the integral form based on the virtual field method, and the robustness of distributed dynamic load identification, as well as the local dynamic lateral force of the thin plate surface, were verified by numerical simulation and experiment, which indicated the better behavior on the problem of the smooth pressure distributions through measuring the plane deflections at some definite grids [15]. The problem of identifying one-dimensional distributed dynamic loads applied on the cutting instrument point by point was also discussed in space from the displacement response based on the conjugate gradient method, and the results were evaluated with the various loads, the sensors, and the noise levels [16]. Meanwhile, considering the sensitivity of the noise to the identification of distributed dynamic loads, the correction coefficient was used in the load analysis approach, which improved the accuracy without the regularization [17]. Other distributed load identification methods were investigated by Kumar [18], Coates [19], Hwang [20], and Nakamura [21] to reconstruct the distributed loads of the engineering structures with different approaches, e.g., the equivalent nodal force method and the Kalman filter method. More recently, the identification of distributed loads were studied in the context of uncertain structures. For instance, when dealing with uncertainty systems, an interval analysis utilized the Taylor expansion and the polynomial coefficients was developed to obtain the envelope interval of unknown distributed loads for the interval analysis of multiple uncertainties structures with the cost of lower computation, including the load identification model, the achievement of the interval analysis method based on Taylor expansion (IAMPTE), and the estimation of the unknown load [22].