# Assembly Tolerance Design Based on Skin Model Shapes Considering Processing Feature Degradation
## 4 Case Study
### Tolerance Synthesis of Example Rotary Feed Component  
Numerical experiments were implemented with tolerance models of the shaft part and the hole part (in Figure 7a). Machined surface degradation prediction was combined as the model of the outsourced shaft part. Testing models for the hole part were generated based on MGP with given tolerance indices \(e_{1}\), \(e_{2}\), and \(e_{3}\). Using the method in Section 3.1, assembling conditions and contact points were estimated based on ICP when the assembly entity constraints and assembly force constraints were checked during iterations. It is worth noting that the location of contact points varies according to the assembling contact. There are mainly three basic patterns from the perspective of the  
Figure 6: Comparison of overall location residuals of the predictive machined surface using the multivariate Gaussian process and using the method proposed in this paper. position distribution of contact points (illustrated in Figure 7a-c), which is the probability influenced by the scale of planar form errors, cylindrical form errors, and perpendicularity. The target assembly tolerance index, the runout error \(e_{f}\) of surface F, was estimated by analyzing the transformed point cloud in assembling condition. A four-dimensional solution space was generated as an explanation of tolerance synthesis, constructed with \(e_{1}\), \(e_{2}\), and \(e_{3}\) as input and \(e_{f}\) as an output. According to the functional requirement of the assembly, the nominal value of \(e_{f}\) is 0.020 mm, which means the upper limit of \(e_{f}\) is 0.020 mm in all the adopted tolerance schemes. However, an unnecessarily narrow tolerance interval arrangement leads to extremely high costs in fabrication and maintenance, while a combination of wide tolerance intervals causes a high ratio of assembling failure. As a result, the influence of manufacturing cost and assembly probability based on an empirical formula and experimental statistics is considered in tolerance synthesis. Assembly examples with acquired functional requirements are called \"successful\" assemblies. Assembling probability is estimated by the proportion of successful assemblies of all the experimental assemblies. According to industrial practice, a yield rate of 95% is accepted, which means that the assembling probability should be more than 95%. The region of interest in the solution space is generated following this rule. The empirical relationship between the tolerance interval and manufacturing cost based on historical industrial practice is shown in Equation (19), where \(\alpha_{1}\), \(\alpha_{2}\), and \(\alpha_{3}\) are manufacturing difficulty coefficients, and K is a scaling factor:  
\[C=K(\frac{\alpha_{1}}{e_{1}^{2}}+\frac{\alpha_{2}}{e_{2}^{2}}+\frac{\alpha_{3 }}{e_{3}^{2}}) \tag{19}\]  
The influences of \(e_{1}\), \(e_{2}\), and \(e_{3}\) on the assembly probability and manufacturing cost are shown in Figure 8a. A Monte Carlo sampling method was applied in the four-dimensional solution space to calculate the output of all the sampled inputs. High-density sampling was conducted in the region of interest to ensure searching accuracy; the density of sampling points was lower, but even outside the region of interest, it avoided falling into the local optimum. The top 1% with the lowest manufacturing cost was approximately fitted into an ellipsoid limit surface, as shown in Figure 8b, where the axes of \(e_{1}\), \(e_{2}\), and \(e_{3}\) are even, and the axis of \(e_{f}\) is uneven. Figure 7: Numerical experiments (**a**) and three basic patterns of contact point positions (**b–d**). Three examples with the lower manufacturing cost and satisfactory assembling probability constitute the designed tolerance scheme, as shown in Table 3. The designed scheme with an assembling probability larger than 95% and the lowest manufacturing cost is listed in bold. The designed value of all three to-be-designed tolerance terms \(e_{1}\), \(e_{2}\), and \(e_{3}\) were enlarged by 0.003, 0.004, and 0.004, respectively.