## 4 Design and Strength Analysis of an HTP Tooth Profile Reducer
### Strength Analysis of the Reducer  
The strength analysis of the reducer was performed using the engineering analysis tool Ansys 2023 R2. In the analysis model, a fixed support condition was applied to the outer surface of the ring gear, and a torque of 47 Nm was applied to the eccentric shaft to evaluate the equivalent stress, bending stress, and total deformation in the area where tooth meshing occurs. The mesh was automatically generated based on the default settings of Ansys, but local refinement was manually applied to the gear tooth contact regions and the areas where stress concentration is expected. Element types used in parallel were SOLID186, a 20-node hexahedral element, and SOLID187, a 15-node tetrahedral element. To ensure the accuracy of the numerical analysis, the built-in convergence criteria of Ansys were used. For convenience, the material properties used in the analysis were those of general steel: density, 7.58 \(\times\) 10\({}^{-6}\) kg/mm\({}^{3}\); Young's modulus, 2 \(\times\) 10\({}^{5}\) MPa; and tensile ultimate strength, 460 MPa.  
The distribution of bending stress and equivalent stress generated in the reducer is shown in Figure 14 for when a torque of 47 Nm is applied clockwise to the input shaft. At this time, the maximum value of the bending stress is 88.335 MPa, which occurs at the upper part, where tooth engagement takes place. The maximum value of the equivalent stress is 122.89 MPa and occurs in the same region as the maximum bending stress.  
As a result of the analysis of the stress distribution on the ring gear tooth surface, where the bending stress is highest, the stress distribution in the x and y directions is shown in Figure 15. The stress value in the X-direction of the tooth profile cross-section is 63.7% (56.232 MPa) of the maximum bending stress, while the stress in the Y-direction is 3.964 MPa, which is relatively small.  
Figure 14: Stress distribution in Reducer R21_23.  
\begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Design Value** \\ \hline Module (mm) & 2.0 \\ \hline \multirow{2}{*}{Number of teeth} & Ring gear & 21 \\ \cline{2-3}  & External gear & 18 \\ \hline \multirow{2}{*}{Tooth profiles} & Hybrid tooth profile \\ \hline Reducer ratio & 1/6 \\ \hline Eccentricity (mm) & 3.0 \\ \hline Pressure angle (\({}^{\circ}\)) & 20 \\ \hline \end{tabular}
\end{table}
Table 3: Reducer design specifications.  
In addition, the total deformation of the R21_z3 reducer is 0.0037 mm, and the maximum deformation occurs in the direction opposite to the gear engagement region. The total deformation of the ring gear and external tooth gear is shown in Figure 16.  
When a load torque of 47 Nm is applied to the R21_z3 reducer, the maximum contact stress acting on the tooth surface is 292.61 MPa, and this occurs on the ring gear, where tooth engagement takes place.  
Figure 17 shows the contact condition and frictional stress distribution between the ring gear and external gear during meshing. As shown in the figure, as the clockwise load torque increases, the inter-gear tooth bite also increases until full engagement is achieved, after which the tooth bite gradually decreases.  
Figure 16: Total deformation distribution of Reducer R21_z3.  
Figure 17: Contact status of Reducer R21_z3.  
Figure 15: Bending stress distribution on the tooth surface of the Reducer R21_z3.  
Figure 18 shows a graph representing the main stresses in the R21_z3 reducer under varying load torque conditions. As shown in the figure, both bending stress and equivalent stress increase with the applied torque, and the equivalent stress shows a relatively larger increase with respect to the load.