# Research on Part Tolerances for RV-40E Reducer Based on Virtual Prototype Technology
## 2 Materials and Methods
### Three-Dimensional Modeling of the Virtual Prototype  
The RV-40E reducer is a multi-part assembly, and its output shaft rotation error is influenced by various factors, including machining errors of transmission parts, elastic-plastic deformation, and thermal deformation. Due to the complexity of these factors, it is challenging to accurately and reliably calculate the transmission accuracy using direct formulas. Therefore, a virtual prototype is typically constructed to evaluate its transmission accuracy. The basic technical parameters are listed in Table 1. SolidWorks software was employed to perform stretching, cutting, scanning, rotation, and other operations to create the models of individual parts. Consolidation was used to replace pin, key, and bolt connections. These parts were then assembled to form a three-dimensional model of the RV-40E reducer, which was saved in a file format compatible with RecurDyn software. The planetary gears and crankshaft of the involute spur gears were modeled using the Maidi tool set for rapid modeling. Additionally, the cycloid wheel was modeled parametrically based on the following cycloid Equation (1):  
\[x = (r_{p}+\Delta r_{p})\times\left(\sin(360\times t)-\frac{k}{z_{b} }\times\sin(z_{c}\times 360\times t)\right)+\frac{(r_{p}+\Delta r_{rp})\times (k\times\sin(z_{c}\times 360\times t)-\sin(360\times t))}{\sqrt{1+k^{2}-2\times k\times \cos(z_{c}\times 360\times t)}}\] \[y = (r_{p}+\Delta r_{p})\times\left(\cos(360\times t)-\frac{k}{z_{b} }\times\cos(z_{c}\times 360\times t)\right)-\frac{(r_{p}+\Delta r_{rp}) \times(k-\cos(z_{c}\times 360\times t)-\cos(360\times t))}{\sqrt{1+k^{2}-2 \times k\times\cos(z_{c}\times 360\times t)}}\] \[k = \frac{a\times z_{c}}{r_{p}+\Delta r_{p}} \tag{1}\]  
where \(t\) is the system variable in the relation (\(0\leq t\leq 1\)), \(r_{p}\) is the radius of the center circle of the needle tooth, \(z_{b}\) is the number of teeth of the needle tooth, \(r_{rp}\) is the radius of the  
\begin{table}
\begin{tabular}{c c c c c} \hline
**No.** & **Part Name (qty.)** & **Ess. Param.** & **Char. Param.** & **Tolerance Values** \\ \hline
1 & input shaft (1) & central gear \(z_{1}=16\) & pitch circle runout tolerance \(\Delta r_{1}\) & 0.012 (\(12^{0}_{-0.012}\)) \\ \hline
2 & planetary gear (2) & planetary gear \(z_{2}=16\) & pitch circle runout tolerance \(\Delta r_{2}\) & 0.015 (\(24^{0}_{-0.015}\)) \\ \hline
3 & crankshaft (2) & eccentric distance e = 1.25 & tolerance \(\Delta e\) & 0.006 (\(1.25^{+0.003}_{-0.003}\)) \\ \hline
4 & cycloid wheel (2) & cycloid wheel \(z_{c}=39\) & tolerance \(\Delta r_{p}\) & 0.002 (\(0.012^{+0.001}_{-0.001}\)) \\  & moved dist. \(r_{p}=0.012\) & tolerance \(\Delta r_{rp}\) & 0.004 (\(0.020^{+0.002}_{-0.002}\)) \\  & equidistance \(r_{rp}=0.020\) & tolerance \(\Delta a_{c}\) & 0.012 (\(72^{+0.006}_{-0.006}\)) \\  & hole distance \(a_{c}=72\) & & & \\ \hline
5 & pin-wheel housing (1) & pin wheel \(z_{b}=40\) & tolerance \(\Delta r_{b}\) & 0.006 (\(66^{+0.003}_{-0.003}\)) \\ \hline
6 & pin wheel (40) & needle radius \(r_{r}=3\) & tolerance \(\Delta r_{r}\) & 0.002 (\(3^{+0.001}_{-0.001}\)) \\ \hline
7 & input-end planet carrier (1) & hole distance \(a_{k}=72\) & tolerance \(\Delta a_{k}\) & 0.012 (\(72^{+0.006}_{-0.006}\)) \\ \hline
8 & output-end planet carrier (1) & hole distance \(a_{k}=72\) & tolerance \(\Delta a_{k}\) & 0.012 (\(72^{+0.006}_{-0.006}\)) \\ \hline \end{tabular}
\end{table}
Table 1: Characteristic tolerances of RV-40E reducer parts.  
needle tooth, \(z_{c}\) is the number of teeth of the cycloid wheel, \(a\) is the eccentric distance, \(\Delta r_{p}\) is the shifting trimming amount, and \(\Delta r_{rp}\) is the isometric trimming amount.  
Subsequently, interference checking was performed on the assembly to ensure the absence of interference and to guarantee the accuracy and reliability of the dynamics model in subsequent stages. Finally, the three-dimensional model of the RV-40E reducer was obtained, as illustrated in Figures 2 and 3.