## 5 Conclusions  
In this paper, the model of a partially covered plate under moving loads was established based on the Lagrange equation, and the optimization of the location of the patch is investigated by means of the proposed model. The main contributions of this study are listed as follows:  
(1) The governing equations of the model are derived based on the first-order shear deformation theory, and the weight and inertia of the moving force are all taken into account. In order to make the convergence faster and more accurate, the trigonometric function and power function are used in the admissible function. At the same time, differential quadrature is used to solve the time domain of the system, which has high precision. Then, the numerical results are comprehensively verified by experiments and open literature. The error between the experimental and theoretical results is within the acceptable range. This proves the accuracy and convergence of the model proposed in this paper, and carries out the subsequent optimization research based on this model.  
(2) In order to obtain the optimal damping loss factor, the problem is transformed as: the maximum damping loss factor is obtained when the local coverage size is constant. Therefore, this paper adopts a multi population genetic method, which has a high global optimization ability. The objective of the optimization function is the damping loss factor of the system, and the design variable is the coordinate of the patch (\(a,b,c,d\)). The result of the optimization shows that the location of the patch is close to the boundary of constraint, and the damping loss factor is larger. This can provide a good reference for engineering and academics.  
(3) The time-domain response of the partially covered plate under the moving load is investigated after optimization in this paper. The influence of single order optimization on the damping loss factor and various boundary conditions can be seen through the comparison of optimization results under the different boundary conditions and coverage areas. It can be concluded that in order to reduce the dynamic response of the system under a moving load, the best optimization strategy is to optimize the lower order damping loss factor.  
The method developed in this study is suitable for guiding the design of partially covered plates in engineering, and can give consideration to calculation accuracy and speed. The method is only limited to the calculation of linear materials. With nonlinear materials, it is difficult to ensure the accuracy of the results.  
Conceptualization, Y.Q., Q.S., and Z.L.; experiments and formal analysis, Y.Q. and J.S.; writing and visualization, Y.Q., Q.S., and Z.L. All authors have read and agreed to the published version of the manuscript.  
This project was supported by the National Natural Science Foundation of China (no. 51,922,066), the Natural Science Outstanding Youth Fund of Shandong Province (Grant No. ZR2019Q19), the United Fund of Ministry of Education for Equipment Pre-research (no. 6141A02022132), the Key Research and Development Plan of Shandong Province (Grant No. 2019 JMRH0307), and the Natural Science Foundation of Shandong Province (Grant No. ZR2019 MEE061).  
Not applicable.  
The data presented in this study are available upon request from the corresponding author.  
The authors declare that they have no conflict of interest.