# Validation of a Manual Methodology for Measuring Roundness and Cylindicity Tolerances
## 3 Results  
The equipment is expected to be able to perform a roundness analysis on at least five sections of every part, obtaining the roundness tolerance by programming the analytical method. The objective of the research is to draw one of the analyzed sections based on the eight registered points and compare it with the ideal circle obtained by least squares regression. In this way, it is expected to visualize which parts of the section exhibit the largest deviations from LSC, letting the tolerance zone be described.  
It is essential to compare the results obtained by MM with those obtained by CME. Figure 7 shows the profile of a section of a part measured by both methods, MM and CME. As it can be observed, these methods throw a different roundness tolerance value due to the low number of profile points taken into consideration by MM (red points). The results obtained by MM are significantly lower than those recorded by the automated machine (blue profile). This causes variations in the tolerance results, although the general shape obtained by both methods is similar. The positions of the eight points selected for MM are placed out of line from the profile registered by the machine due to the reference value of the dial indicator. This causes the least squares circles (green for the manual method and black for the CME) to be offset by the same distance. However, this relative displacement does not change the tolerance results, as the calculation of the roundness tolerance refers to the relative distance between the center of the least squares circle and the points used to determine it, independently of their absolute location.  
Moreover, MM carried out on a granite surface plate leads to obtaining a tolerance error of less than 10% (30 \(\upmu\)m for this example). Conclusions can be drawn from the resulting accuracy related to the suitability of the methodology, which depends on the quality requirements of the workpieces.  
Similarly, the value of the cylindricity tolerance of every part from all the sections considered for roundness must be obtained. However, MM has limitations on this point, due to the difficulty of 3D graphing, regarding the plot of the part deviations with respect to the adjusted least squares cylinder. Nevertheless, this aspect will be supported by the 3D representation obtained by CME. For the example shown in Figure 8, the manual measurement procedure of seven cross sections of the cylinder yields a cylindricity tolerance, \(T_{cylindricity}\), of 303.66 \(\upmu\)m according to Equations (17)-(19). The same cross sections measured by CME lead to a cylindricity tolerance of 349.68 \(\upmu\)m. Thus, the error found is 46 \(\upmu\)m, which represents a relative value of approximately 13%. Again, this may be explained by the  
Figure 7: Expected representation for the roundness analysis of any cross section. Comparison between the two methods used.  
smaller number of points analyzed, which does not allow the real maximum and minimum points of the profile to be identified.  
To verify the proposed analytical method, the coordinates of the points measured by CME were exported for the seven cross sections, carrying out the analytical calculation with the same 3600 points used with the machine for each section. A value of \(Tc_{ylindricity}\) of 349.77 \(\upmu\)m validates the analytical procedure. The small difference, less than 0.1 \(\upmu\)m, is because the machine uses approximation algorithms to determine the axis of the least squares cylinder, rather than performing an exact analytical calculus.