# Dimensional Tolerances in Mechanical Assemblies: A Cost-Based Optimization Approach
## 3 A Proposal for Tolerance Synthesis
In the results tables, predetermined tolerances are highlighted in italics to ensure proper identification. The functional constraint values were determined using the concept of economic safety factor, which takes into account the loss of quality function introduced by [20]. For the design of multi-V belts, an upper limit value of \(\alpha_{0}=4.5^{\circ}\) was adopted for the extension angle to prevent the belts from disengaging from the pulley grooves [47]. The relationship between factors is shown in Figure 9, while Figure 6 provides the definitions of the lengths \(L1\), \(L2\), \(L3\), and \(L4\). By utilizing the design data and referring to Figure 6, the following relationships can be established for the primary chain (\(CRS\Leftrightarrow ALT\)):

\[L=L1=250\text{ mm}\text{: }\Delta_{01}=250\times\tan(4.5^{\circ})=19.7\quad \text{mm}\]

Regarding the secondary chain (\(ALT\Leftrightarrow HP\)),

\[L=L2+L3+L4=200\text{ mm}\text{: }\Delta_{02}=200\times\tan(4.5^{\circ})=15.7 \quad\text{mm}\]

Figure 5: An example of an automotive belt power drive system. Credits. The estimation of the mean financial loss is based on the functional limits. If these limits are exceeded, it would compromise the functioning of both the power steering system and the alternator, potentially resulting in accidents that could cause material and human health damage. The estimated cost of this damage is \(A_{0}=~{}500,\!000.00\). Furthermore, the cost of repairing a noise complaint during the warranty period of the vehicle includes expenses such as parts replacement, labor, administrative costs, and other associated expenses, totaling \(A=~{}800.00\). The economic safety factor of the data from the former items is

\[\phi=\sqrt{\frac{A_{0}}{A}}=\sqrt{\frac{500,\!000}{800}}=25.\]

The functional constraint limits for the main and secondary chains are determined as follows:

\[\Delta_{1}=\frac{\Delta_{01}}{\phi}=\frac{19.7}{25}=0.79\quad\text{mm}\quad \text{and}\quad\Delta_{2}=\frac{\Delta_{02}}{\phi}=\frac{15.7}{25}=0.63\quad \text{mm}.\]

The value \(\Delta_{1}\) can be increased to account for the wear caused by the product. However, in the current example, this possibility will not be considered. This is because the difference between the specified value and the maximum recommended value for misalignment [46] is 0.58 mm per 100 mm, resulting in a total of \(0.58\times 2.5=1.45\quad\text{mm}\) for this case. It should be noted that 1.45 mm is significantly greater than 0.58 mm, indicating that wear effects are

Figure 4.0 The belt power transmission system considered in the proposal example. ## 5 Results

In this section, the results of the tolerance analysis and synthesis processes are presented. These results are obtained through the application of Algorithm 1. Due to the similarity in the machining characteristics and cost within the case, the sensitivity test mentioned above was not conducted for the sake of simplification. ### Tolerance Analysis Process

To simulate a real situation in the application of the proposed method, the tolerance classes specified in the international standard [48], which provide tolerance values based on component dimensions, were adopted. These classes are presented in Figure 4.1 Additionally, the column \(Spec\) provides higher values compared to class IT14. The results of the calculation for the tolerance analysis process, including the tolerances of the primary chain (\(T1\)) and the secondary chain (\(T2\)), their respective costs (\(C1\) and \(C2\)), and the total assembly cost (\(CT\)), are summarized in the lower lines of the table for each column. Different processes present different accuracy ranges, but the same process can present a wide range of possible tolerances--several IT values--for the same size; classes for turning can vary from IT6 to IT13 [49]. For the primary chain,

Figure 4.2 Figure 4.3

For the secondary chain,

Figure 4.4 Figure 4.5

In the optimization calculation of the algorithm, the secondary chain takes priority due to its lower value functional constraint.