# Dynamic Torsional Stiffness of Reducers and Its Testing Method
## 3 Measurement of Torsional Stiffness Using Transmission Error Method
### Unification with Hysteresis Curve Method  
Assuming the reducer is subjected to torsional stiffness testing using a hysteresis curve at the output \(\theta_{out1}\) position, with a loading rate of \(\tilde{\tau}_{1}\) and torque values at the endpoints A as \(\tau_{1}\), then \(\theta_{outA}\) is shown in Figure 4 and represented by Equation (15).  
\[\theta_{outA}(\tau_{1},\tilde{\tau}_{1},0)=\theta_{s+}(\tau_{1},0)+\theta_{f+ }(\tau_{1},\tilde{\tau}_{1},0)+\theta_{g+}(\tau_{1})+\frac{\theta_{in}}{R} \tag{15}\]  
Equation (14) can be transformed into Equation (16).  
\[\theta_{outA}(\tau_{1},\dot{\tau}_{1},0)-\frac{\theta_{in}}{R}=\theta_{s+}( \tau_{1},0)+\theta_{f+}(\tau_{1},\dot{\tau}_{1},0)+\theta_{g+}(\tau_{1}) \tag{16}\]  
When the test is conducted at the \(\theta_{out1}\) position of the reducer, the angle value of point A at torque \(\tau_{1}\) is given in Equation (17).  
\[\theta_{outA}(\tau_{1},\dot{\tau}_{1},0)=\theta_{out1}(\tau_{1},\dot{\tau}_{1},0) \tag{17}\]  
Figure 4: Schematic diagram of hysteresis curve.  
Figure 3: Measuring principle of transmission error.  
By substituting Equation (17) into Equation (16), we obtain Equation (18).  
\[\theta_{out1}(\tau_{1},\dot{\tau}_{1},0)-\frac{\theta_{in}}{R}=\theta_{s+}(\tau_ {1},0)+\theta_{f+}(\tau_{1},\dot{\tau}_{1},0)+\theta_{\mathrm{g}+}(\tau_{1}) \tag{18}\]  
In an ideal state, during the test at a loading rate \(\dot{\tau}_{1}=0\), Equation (18) simplifies to Equation (19).  
\[\theta_{out1}(\tau_{1},0,0)-\frac{\theta_{in}}{R}=\theta_{s+}(\tau_{1},0)+ \theta_{f+}(\tau_{1},0,0)+\theta_{\mathrm{g}+}(\tau_{1}) \tag{19}\]  
By introducing Equation (19) into Equation (12), we can express the static torsional stiffness \(K_{s}\) of the reducer as Equation (20).  
\[K_{s}=\frac{\tau_{1}}{\theta_{out1}(\tau_{1},0,0)-\frac{\theta_{in}}{R}}=\frac {\tau_{1}}{\theta_{s+}(\tau_{1},0)+\theta_{f+}(\tau_{1},0,0)+\theta_{\mathrm{ g}+}(\tau_{1})} \tag{20}\]  
When using transmission error for testing and assuming a speed of 0, Equation (13) can be rewritten as Equation (21).  
\[TE_{z}(\theta_{out1}(\tau_{1},\;0,\;0))=\theta_{out1}(\tau_{1},\;0,\;0)-\frac{ \theta_{in}}{R} \tag{21}\]  
At this point, the torsional stiffness represents the dynamic torsional stiffness \(K_{d}\) of the reducer, which can be expressed as Equation (22).  
\[K_{d}=\frac{\tau_{1}}{\theta_{out1}(\tau_{1},\;0,\;0)-\frac{\theta_{in}}{R}}= \frac{\tau_{1}}{TE_{z}(\theta_{out1}(\tau_{1},\;0,\;0))} \tag{22}\]  
The relationship between them can be observed in Equations (23) and (24).  
\[TE_{z}(\theta_{out1}(\tau_{1},\;0,\;0))=\theta_{s+}(\tau_{1},0)+\theta_{f+}( \tau_{1},\;0,\;0)+\theta_{\mathrm{g}+}(\tau_{1}) \tag{23}\]  
\[\frac{\tau_{1}}{TE_{z}(\theta_{out1}(\tau_{1},0,0))}=\frac{\tau_{1}}{\theta_{ s+}(\tau_{1},0)+\theta_{f+}(\tau_{1},0,0)+\theta_{\mathrm{g}+}(\tau_{1})} \tag{24}\]  
At this point, the relationship between \(K_{d}\) and \(K_{s}\) is shown in Equation (25).  
\[K_{d}=K_{s} \tag{25}\]  
Based on the theoretical analysis above, it becomes apparent that when using the transmission error method for torsional stiffness testing, if the testing speed is 0, the dynamic torsional stiffness \(K_{d}\) obtained equals the static torsional stiffness \(K_{s}\) at a loading rate of 0. This indicates the alignment of the transmission error method and the hysteresis curve method. However, having both a speed of 0 and a loading rate of 0 are ideal situations, but they are not achievable in reality. Nevertheless, the transmission error method employs a loading rate of 0, minimizing the impact of loading rate dependence on torsional stiffness testing. Simultaneously, the method also takes into account the influence of rotational speed, ensuring that the obtained dynamic torsional stiffness more accurately reflects the torsional stiffness in the actual working state of the reducer.