DESIGN OPTIMIZATION FOR WEIGHT REDUCTION OF LOCOMOTIVE WHEEL USING RESPONSE SURFACE METHODOLOGY

Manufacturing cost of locomotive wheel largely depends on mass of locomotive wheel and to reduce mass of wheel, design optimization is necessary. In this research the design of locomotive wheel is optimized considering hub radius and hub width as input parameters to DOE (Design of Experiments). Initially finite element analysis is performed under static structural loading conditions to determine equivalent stress and safety factor which is followed by design optimization using Response Surface Methodology. The software used for design and analysis is ANSYS.


Introduction
A train wheel or rail wheel is a type of wheel specially designed for use on rail tracks. A rolling component is typically pressed onto an axle and mounted directly on a rail car or locomotive or indirectly on a bogie, called a truck. Wheels are cast or forged (wrought) and are heat-treated to have a specific hardness. New wheels are trued, using a lathe, to a specific profile before being pressed onto an axle. All wheel profiles need to be periodically monitored to insure proper wheelrail interface. Improperly trued wheels increase rolling resistance, reduce energy efficiency and may create unsafe operation. A railroad wheel typically consists of two main parts: the wheel itself, and the tire around the outside. A rail tire is usually made from steel, and is typically heated and pressed onto the wheel, where it remains firmly as it shrinks and cools. Mono block wheels do not have encircling tires, while resilient rail wheels have a resilient material, such as rubber, between the wheel and tire. Nomenclature of different regions of locomotive wheel can be seen in figure 1 above. Tread and flange are the regions which comes in immediate contact with rail track.

Problem Description
Structural and fatigue life analysis of railway wheel is done using Finite Element method. The method involves three stages of analysis i.e. Preprocessing, solution and post-processing.
• Preprocessing stage involves CAD modeling, meshing into elements and nodes (discretization), assigning loads and boundary conditions. • Solution stage involves matrix formulations, matrix inversions and multiplication, assemblage of element stiffness matrix, global stiffness matrix. • Postprocessing stage involves viewing results, contour plots, vector plots and optimization of input parameters.
The base design reference is taken from KLW data sheet which provides range of dimensions of hub, tread, flange, rim and web. The dimension ranges of these parameters are provided in figure 2 below.

Finite Element Analysis
The CAD model of locomotive wheel and track is modeled using data reference ranges provided in figure 2. The CAD model developed is 1/4th of actual size to save computational time in meshing and solution.

Optimization Using Response Surface Methodology
Response surface methodology (RSM) is a collection of mathematical and statistical techniques for empirical model building [5]. By careful design of experiments, the objective is to optimize a response (output variable) which is influenced by several independent variables (input variables). An experiment is a series of tests, called runs, in which changes are made in the input variables in order to identify the reasons for changes in the output response. When behavior (response, y) that should be taken into consideration for design is determined as a function of multiple design variables (xi), the behavior in response surface method is expressed by the approximation as a polynomial y = f(x) on the basis of observation data. A quadratic response function with two variables with a regression model is expressed by y=β0 + β1x1 + β2x2 + β3x12 + β4x22+ β5x1x2 Where β0, β1, β2, β3, β4 and β5 are the regression coefficients.
The optimization is performed on 2 design parameters i.e. hub width (x1) and hub radius (x2) using response surface methodology. The response surface method (RSM) is a statistical and mathematical method to model approximately and analyze the response surface with the design variables, when the interesting responses are influenced by various design variables. RSM was to use regression methods based on least square methods. In the study, RSM was used to determine the optimum design for the minimization of the weight within the specific life. The significant process variables were identified by using the central composition design (CCD), which is a kind of design of experiments (DOE). Central composite design is the default DOE type. It provides a screening set to determine the overall trends of the metamodel to better guide the choice of options in Optimal Space-Filling Design. The CCD DOE type supports a maximum of 20 input parameters. In Central Composite Design (CCD), a Rotatable (spherical) design is preferred since the prediction variance is the same for any two locations that are the same distance from the design center. However, there are other criteria to consider for an optimal design setup. Among these criteria, there are two that are commonly considered in setting up an optimal design using the design matrix. The degree of non-orthogonality of regression terms can inflate the variance of model coefficients. The position of sample points in the design can be influential based on their position with respect to others of the input variables in a subset of the entire set of observations. After DOE, a response surface is generated for all the input and output values using the least squares methodology. The data points are fitted with a standard 2nd order model. The points generated on the response surface are then used to perform the optimization. The goodness of fit plots for all the subsystems are shown below. Figure 10: Goodness of fit curve "Goodness of Fit" of a linear regression model describes how well a model fits a given set of data, or how well it will predict a future set of observations. An X-Y Scatter plot illustrating the difference between the data points and the linear fit. The above graph shows safety factor at different design points (x1: hub width and x2: hub radius). The safety factor is found to be maximum at design point number 8 for which hub width is 112.5mm and hub radius is 88mm. The safety factor is minimum for point number 6 for which hub width is 112.5mm and hub radius is 72mm. The equivalent stress is found to be maximum at design point number 6 for which hub width is 112.5mm and hub radius is 72mm and minimum at design point number 8 for which hub width is 112.5mm and hub radius is 88mm.  The geometric mass of wheel is found to be maximum (678.04Kg) at design point 7 for which hub width is 135mm and hub radius is 72mm. The geometric mass is minimum(673.27Kg) at design point 8 for which hub width is 112.5mm and hub radius is 88mm. Contour plots developed through RSM analyze the effect of input variable with respect to one output variable keeping all other variables fixed. Effect of tread depth and tread width on locomotive wheel are analyzed with contour plots.   Local sensitivity graph is plotted for all the three output variables (i.e. safety factor, equivalent stress and geometric mass). For safety factor, hub width has slightly higher contribution as compared to hub radius. For equivalent stress, hub width has higher contribution as compared to hub radius. For geometric mass, hub radius has much higher contribution as compared to hub width. Maximum and minimum values of output variables (safety factor, equivalent stress, geometric mass) are generated and shown in table 2 above. The minimum geometric mass calculated from RSM 673.22 Kg and maximum geometric mass is 678.08Kg.

Conclusion
Finite Element Analysis of locomotive wheel is performed using ANSYS 18.1 software package. The design of locomotive wheel is optimized using response surface methodology and input parameters for optimization are tread depth and tread width. The output parameters are equivalent stress, safety factor and geometric mass. The minimized geometric mass is 673.22Kg.