## 4 Geometrical Analysis
### Dimensional Analysis  
Figure 7 shows the diameter deviations with respect to the nominal value for each of the machined specimens. These results have been obtained from the average error measured across 16 equidistant points along the length of the part. The figure compiles results from different combinations of cutting parameters and nominal diameters, enabling  
a comparative analysis of the influence of cutting conditions and specimen slenderness on dimensional accuracy. It is important to note that the horizontal axis (\(Z\)) in the figure represents the longitudinal position along the specimen, where \(Z\) = 0 mm corresponds to the section clamped in the chuck, and \(Z\) = 150 mm corresponds to the section supported by the tailstock. This reference allows for the spatial interpretation of how the deviations evolve along the machining axis, particularly in terms of deflection or bending effects. In general, regardless of the \(f\) and \(D\) values, higher \(v_{c}\) leads to lower deviations. This behavior is attributed to the improved stability of the process at higher \(v_{c}\), which reduces both the magnitude and variability of the cutting forces. Conversely, at low \(v_{c}\), the interaction between the tool and the material induces greater deflection of the specimen, particularly in its central region, leading to higher deviations. Regarding the \(f\) influence, it is observed that, increasing its value, the machining tends to result in slightly higher diameter errors and greater variation along the length of the part. This effect is more pronounced in the slenderest specimens, where the higher mechanical demand imposed by the increased uncut chip thickness leads to a more significant deflection under load. The combined effect of low \(v_{c}\) and high \(f\) produces the least favorable dimensional outcomes. As for the influence of \(D\), the results confirm that dimensional deviations are inversely proportional to the final diameter. The smallest specimens (Figure 7c) consistently exhibit the highest deviations, with average values close to 0.20 mm and maximum errors above 0.30 mm in some cases. These deviations are notably higher in the middle section of the  
Figure 7: Dimensional errors measured for the samples as a function of \(v_{c}\),\(f\), and \(D\): (**a**) \(D\) = 18 mm; (**b**) \(D\) = 16 mm; and (**c**) \(D\) = 14 mm. specimen, suggesting a flexural deformation mode due to reduced stiffness. In contrast, the 18 mm (Figure 7a) specimens show the lowest deviations, typically below 0.12 mm, and a much more uniform error distribution along their length. These results highlight the combined influence of cutting forces and structural rigidity on dimensional accuracy, particularly when machining slender parts produced by material extrusion-based additive manufacturing processes. In order to justify the influence of slenderness on the dimensional deviations, a normalized deflection model was developed to evaluate the bending deformation experienced by the specimen during machining. The analysis is related with the mechanical analogy of a cantilever-supported beam, which represents the boundary conditions of the part during turning, fixed at the chuck (\(Z=0\) mm) and simply supported at the tailstock (\(Z=150\) mm). The tool is considered to apply a concentrated force at a specific axial position, simulating the real interaction that occurs during the cutting process. For each machining pass, the deflection was calculated at the position where the tool is engaged with the material, considering the applied cutting force and the geometric and mechanical properties of the specimen (Figure 8). The resulting deflection values were then normalized with respect to the diameter of the part, enabling a comparative assessment across different diameters. This approach provides a quantitative framework to correlate structural flexibility with the deviations measured along the part and to support the idea that higher slenderness leads to greater deflection and consequently lower dimensional accuracy.