Proceedings of the Canadian Society for Mechanical Engineering International Congress 2020 CSME Congress 2020 June 21-24, 2020, Charlottetown, PE, Canada AN OPTIMIZATION METHOD TO FIND THE INITIAL CATENARY CONFIGURATION BY USING A GRADIENT-BASED ALGORITHM Kee Seung Oh 1 , Rosalie Morin 1 , Ben Shiff 1 , Stephen William Knox Roper 1 , Il Yong Kim 1* 1 Department of Mechanical and Materials Engineering, Queen's University, Kingston, Canada *kimiy@queensu.ca Abstract— In this study, an optimization method is proposed to obtain an initial configuration of the catenary. To this end, a gradient-based algorithm is employed, and the sensitivity analysis is performed by introducing an alternative finite difference method (FDM). Unlike the original FDM, a proposed method can dramatically reduce the computation cost due to its simplified format. The form-finding problem is formulated as the unconstrained optimization problem with an objective function defined by half mean squared error. In the optimization process, static analysis for the catenary constructed by the 2-node beam elements is performed at each iteration calculation using commercial software. A well- defined unconstrained optimization problem is solved successfully, and the validity of the suggested optimization method is supported by the numerical results obtained for specific design conditions. Keywords- pantograph-catenary system; static analysis; optimization; finite different method I. INTRODUCTION The overhead line system for a high-speed modern train should be operated under the stable current-collection quality which can be judged by the contact loss between the catenary and pantograph. Numerical simulations are required to verify performance primarily because real-world overhead line tests are inefficient, expensive and sometimes dangerous. The simulation for the high-speed train should capture exactly the wave propagation phenomena in the catenary system because it dominates the physical behaviors between the pantograph and catenary [1]. In order to simulate the pantograph-catenary system accurately, two numerical methods have been intensively investigated so far: the absolute nodal coordinate formulation (ANCF) based finite element method (FEM) [2-9] and the standard FEM [10-16]. In a nutshell, the standard FEM considers six degrees of freedom (DOF) for the 2-node beam element, but the ANCF based FEM solves for 12 DOF for the same beam element because it additionally considers the absolute nodal coordinates to calculate the large deformation phenomenon effectively. There is another important issue in the simulation to obtain the stable current-collection quality apart from the formulation selection of the numerical methods: initial catenary form- finding problem considering pre-sag and distortion. Since the catenary configuration is exerted by the external forces such as gravitational and tensile forces, the pre-sag and distortion of the catenary are inevitable if one does not perform optimization for finding proper initial catenary configuration. Accordingly, numerous methods have been reported to overcome this issue so far. There were useful form-finding methods that considered the constraints introduced during the assembly of the catenary, pantograph and the other parts [4,7], or found the static form of the catenary configuration by using geometry variation method [11,12]. Also, Zhou et al. [13] performed the optimization process by introducing the negative sag method, Massat et al. identified the dropper length by minimizing the dropper tension error in [14], and Yang et al. [8] facilitated the catenary form- finding by controlling tension with the piecewise equations. Several types of research with respect to minimizing distance have been reported. Ambrósio et al. [15] performed minimizing the distance between the static deformed geometry of the contact wire and its specified position and Collina et al. minimized a residual function constructed by the weighted differences between the target and the design values of the tensile load in the wires and the lateral position of the steady arms in [16]. Besides, Gregori et al. [9] minimized the interaction force between the catenary wire and pantograph head by using a genetic algorithm. However, there was a lack of optimization formulation descriptions in the researches mentioned previously to the catenary form-finding problems. Even though a tailored analytical solution was proposed in [17], it was only valid for the specific geometry and parameters. Therefore, an advanced optimization scheme should be developed to obtain the proper initial configuration for the various catenary types. In this study, a new optimization method is proposed to solve the form-finding problem by using a gradient-based algorithm with an alternative finite difference method (FDM) for the sensitivity analysis. This paper is organized as follows: Section II describes the static analysis method with the governing equation and boundary conditions. Section III describes the optimization formulation for the catenary initial form-finding problem. In Section IV, the optimal results are shown with the optimization history and tabulated data. NSERC, SNC-Lavalin