# Dynamic Torsional Stiffness of Reducers and Its Testing Method
## 4 Experimental Procedure
### Hysteresis Curve Method for Testing Torsional Stiffness  
In this study, the hysteresis curve method was employed to test the torsional stiffness of two reducers. The small reducers were subjected to a rated torque of 1.0 Nm at varying loading rates, while the large reducers were subjected to a rated torque of 380 Nm at different loading rates. The specific test conditions are provided in Table 3.  
In Table 3, Condition 1 represents the lowest loading rate typically encountered in engineering practice. To confirm the analysis presented earlier, tests were performed under various conditions. Figure 11 displays the test curve for small reducers, and Figure 12 shows the test curve for large reducers. The corresponding test results can be found in Table 4.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Type** & **Result 1** & **Result 2** & **Result 3** \\ \hline Small-size reducer & 0.4969 Nm/\({}^{\circ}\) & 0.5124 Nm/\({}^{\circ}\) & 0.5192 Nm/\({}^{\circ}\) \\ Large precision & 5235.1212 Nm/\({}^{\circ}\) & 5474.5840 Nm/\({}^{\circ}\) & 5633.3852 Nm/\({}^{\circ}\) \\ reducer & & & \\ \hline \hline \end{tabular}
\end{table}
Table 4: Test results of static torsional stiffness.  
Figure 10: Physical picture of the large reducer.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Type** & **Rate Torque** & **Max Speed** & **Lost Motion** \\ \hline Small-size reducer & 1.0 Nm & 70 rpm & 3.6\({}^{\circ}\) \\ Large precision & 380 Nm & 70 rpm & 2\({}^{\prime}\) \\ reducer & & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters of the two types of reducers used in the experiments.  
Figure 9: Physical picture of the small-sized reducer.  
Based on the test results, it is evident that as the loading rate increases, the following changes occur: The angle test results of the reducer are affected, leading to a decrease in the angle value. Specifically, for the small reducer, the angle decreases from 2.0126\({}^{\circ}\) to 1.9261\({}^{\circ}\), and for the large reducer, it decreases from 4.3552\({}^{\circ}\) to 4.0473\({}^{\circ}\). Correspondingly, the torsional stiffness exhibits an increasing trend. For the small reducers, it increases from 0.4969 Nm/\({}^{\circ}\) to 0.5192 Nm/\({}^{\circ}\), and for the large reducers, it increases from 5235.1212 Nm/\({}^{\circ}\) to 5633.3852 Nm/\({}^{\circ}\).  
From these findings, it is evident that when using the hysteresis curve method to test the torsional stiffness of the reducer, the results are affected by the loading rate dependence, resulting in variations in the test outcomes at different loading rates. The reason for this situation lies in the inconsistency between the rate of change in internal dynamics from one equilibrium point to another when the reducer is externally loaded and the rate of external input action, i.e., the loading rate. This discrepancy leads to changes in the output shaft angle as follows: At slow loading rates, the angle change can keep up with the loading rate. At fast loading rates, the loading reaches the target torque, but the turning angle is not yet in place. As a result, the results of torsional stiffness testing become inaccurate.