Proceedings of the 3 rd Annual World Conference of the Society for Industrial and Systems Engineering, San Antonio, Texas, USA October 20-22, 2014 ISBN: 97819384960-2-8 340 Analysis and Modeling of Roundness Error in the Design Process of a Measuring Machine J. M. Díaz-Mendoza, J. Molina, L. Rico, and L. Vidal Universidad Autónoma of Cd. Juárez Henry Dunant 4016, Zona Pronaf Cd. Juárez Chih. México, C.P. 32310 Ph (52656-68848439) Ext. 4644 Corresponding author's Email: juan.diaz@uacj.mx Abstract: Roundness is a geometric form tolerance required for circular or cylindrical parts used in a great amount of mechanical assemblies. This form tolerance is controlled in order to assure the correct performance and lifetime of an assembly. The design and construction of a roundness machine requires the ability to measure, analyze and validate roundness on parts, to assure the measuring capability. The analysis applied in this paper uses two international methods to determine roundness error: minimum zone circle (MZC) and least square circle (LSC); ¨The calibration of a Roundness Standard of the National Bureau of Standards from USA was used to elaborate the study. The referred methods use polar coordinates data as primary source. The analysis performed using these methods help to determine the preliminary machine error in measuring roundness. Keywords: Roundness, Geometric Characteristic, Least Square Circle, Minimum Zone Circle, Polar Coordinate, Error Evaluation, Measuring Standard, Coordinate Data 1. Introduction The measuring process is a key factor to determine if product dimensional and geometric features are inside specified tolerances. The components design process is critical to product performance, whether is on a prototype phase or trial run prior to production. Since product performance is a key factor, dimensions and tolerances are relevant to meet the requirements in the final design. Roundness geometric tolerance is critical to circular machined parts (Lei, 2011), which has been a feature discussed in several researches (Dhanish, 2002). (Dhanish, 2002) also points out that Least squares has been broadly used for the calculation and analysis of roundness. However, ISO 1101 establishes that MZC is the required methodology to estimate roundness error. 1.1 Minimum Zone Circle (MZC) According to the standard ISO 1101, roundness is defined as: “two concentric circles limit the tolerance zone in the considered cross-section, with a difference in radii of t”. (DIN, 2008). However, the method to calculate the roundness error is not given by ISO (Dhanish, 2002). The description of the minimum zone circle (MZC) is as: minimum zone deviation form from ISO 1101 standard (Lai & Chen, 1996) and (Moroni & Petrò, 2007). Moroni also indicates that the method to estimate MZC is a non-linear problem, which calculation evolves with some difficulty. In Figure 1-a is shown the geometric interpretation of MZC (DIN, 2008). There has been work done to estimate MZC applying optimization and graphical methods to obtain the tolerance value. However, among these analyses, as mentioned by (Moroni & Petrò, 2007), several of the methods transform the non-linear problem into a linear system. 1.2 Least Squares Circle (LSC) The least square method has been used to determine the best fit for a line, circle or higher degree curve. It has been applied in several areas such as: statistics, finance, economics engineering and others. However, the least square method has been used longer than the MZC to determine the least square circle (LSC), since its mathematical roots is the Gauss minimum square method. LSC has been used in coordinate measuring machines (CMM) and probe machines. Nevertheless, LSC has a limitation to estimate the mean best fit for a linear function or nonlinear function from a set of points. This does