# Dynamic Torsional Stiffness of Reducers and Its Testing Method
## 4 Experimental Procedure
### Transmission Error Method for Testing Torsional Stiffness  
In this research, the torsional stiffness of two reducers was examined using the transmission error method. The small reducer was subjected to a rated torque of 1.0 Nm, while the large reducer was loaded with a rated torque of 380.0 Nm. The test speeds were within the rated speed range of their respective output shafts, and the specific test conditions are presented in Table 5. The obtained test curves are displayed in Figures 13 and 14, while the detailed test results can be found in Table 6.  
Figure 11: Static torsional stiffness test results of a small-size reducer. Figure 12: Static torsional stiffness test results of a large reducer. Based on the test results, the following observations can be made: At low output shaft speeds, the torsional stiffness results obtained using the transmission error method closely match those obtained from the hysteresis curve method for low loading rates. As the speed increases for the small reducer at the 180\({}^{\circ}\) position, the transmission error value rises from 2.0128\({}^{\circ}\) to 2.8436\({}^{\circ}\), leading to a corresponding decrease in torsional stiffness from 0.4964 Nm/\({}^{\circ}\) to 0.3517 Nm/\({}^{\circ}\). In comparison, large reducers exhibit smaller variations, with transmission error values increasing from 4.3974\({}^{\circ}\) to 4.4529\({}^{\circ}\), and corresponding torsional stiffness decreasing from 5184.8820 Nm/\({}^{\circ}\) to 5122.587 Nm/\({}^{\circ}\). These results reveal that when using the transmission error method to test the torsional stiffness of the reducer, the test results differ at various speeds. This discrepancy is attributed to torque fluctuations, elastic deformation due to friction, and gear disengagement inside the reducer as the speed increases. These factors cause changes in the internal dynamics of the system, leading to variations in the actual torsional stiffness. Furthermore, it is evident that the torsional stiffness of small reducers is significantly influenced by rotational speed, whereas the torsional stiffness of large reducers remains relatively stable. This disparity is attributed to variations in the precision of gear processing and installation between small and large reducers, resulting in different dynamic performance under actual working conditions. 1. Small reducers are typically employed in scenarios with lower demands for transmission precision or load-bearing capacity, such as in applications like toy robots or food service robots. Consequently, the performance requirements of transmission \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Type** & **Result 1** & **Result 2** & **Result 3** \\ \hline Small-size reducer & 0.4964 Nm/\({}^{\circ}\) & 0.3880 Nm/\({}^{\circ}\) & 0.3517 Nm/\({}^{\circ}\) \\ Large precision & 5184.8820 Nm/\({}^{\circ}\) & 5179.6992 Nm/\({}^{\circ}\) & 5120.2587 Nm/\({}^{\circ}\) \\ reducer & 5120.2587 Nm/\({}^{\circ}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Test results of dynamic torsional stiffness. Figure 14: Dynamic torsional stiffness test results of a large reducer. Figure 13: Dynamic torsional stiffness test results of a small-size reducer. components such as gears, gear shafts, bearings, and other related parts, as well as the precision requirements for overall assembly, tend to be less stringent. This results in a relatively moderate transmission performance for small-scale reducers, making them susceptible to occurrences such as gear mesh impacts, gear disengagement, and similar phenomena. As a consequence, this can lead to more pronounced torque fluctuations, further contributing to significant variations in the actual torsional stiffness. 2. The large reducers mentioned in this study, on the other hand, are designed for precision applications in the fields of industrial robotics, collaborative robotics, and similar domains.