# Dynamic Torsional Stiffness of Reducers and Its Testing Method
###### Abstract  
The torsional stiffness of the reducer is commonly assumed to be static and tested using the hysteresis curve method. However, practical engineering experience shows that the torsional stiffness is dynamic rather than static, and the hysteresis curve method cannot accurately represent the reducer's torsional stiffness during actual operation. To address these issues, this study conducts theoretical research on the dynamic characteristics of torsional stiffness, introduces a new concept called dynamic torsional stiffness, and provides its analytical formula. A dynamic torsional stiffness testing method based on the transmission error method is proposed, and its feasibility and consistency with the hysteresis curve method are theoretically demonstrated. This method allows the torsional stiffness information of the reducer to be obtained under all operating conditions. Through experimental research, the limitations of the hysteresis curve method are explained, and the dynamic characteristics of the reducer's torsional stiffness, along with the effectiveness of the transmission error method, are verified. Consequently, the torsional stiffness information of the reducer under all operating conditions is obtained. In conclusion, this method is valuable for both theoretical and engineering applications, as it offers a more accurate representation of the dynamic torsional stiffness of the reducer in real-world scenarios. ed 2023 13 927 10.3390/app1316927 10. Researchers from various fields have different understandings of the static and dynamic characteristics of torsional stiffness. For instance, as mentioned in Reference [2], it is noted that the torsional stiffness of concrete structures, such as load-bearing beams, is dynamic in nature and can be determined through modal torsional rates. Reference [3] proposes a dynamic torsional stiffness model where a magneto-sensitive circular annular rubber bushing is presented with influences of frequency, amplitude and magnetic field dependence included. In Reference [4], it is indicated that the dynamic torsional stiffness of rubber couplings is related to the transmitted torque, increasing with higher transmitted torques, and exhibiting a trend of initially decreasing and then increasing with increasing frequency. The torsional stiffness of reducers is conventionally regarded as a constant value, unaffected by other factors [5;6]. When conducting calculations and simulation analyses, it is assumed to possess static characteristics [7; 8; 9; 10]. However, based on extended engineering observations, the actual torsional stiffness of reducers, when exposed to varying operational conditions, does not always align with values derived from the hysteresis curve method. This discrepancy arises due to several reasons: Real-world operational challenges, such as speed and torque variations, cause internal system dynamics within the reducer to be unstable. This leads to fluctuations in torsional stiffness. For instance, according to Reference [11], a dual-rigid-wheel harmonic drive is subject to the influence of manufacturing errors, causing its torsional stiffness to become time-varying and exhibit dynamic characteristics. Additionally, Reference [12] highlights that the torsional stiffness of a helicopter transmission system is dynamic, subsequently affecting the system's steady-state characteristics. The hysteresis curve method is inherently static, offering a snapshot of the reducer's torsional stiffness in a quasi-static scenario. This can be different from the reducer's behavior in real operational settings. Reducer hysteresis exhibits a dependency on the loading rate [13]. Therefore, the hysteresis curve, when drawn for different loading rates, can yield varying results, creating inconsistent torsional stiffness measurements. To address these gaps, this paper investigated the dynamic characteristics of the torsional stiffness of reducers. It will introduce the concept of \"dynamic torsional stiffness\" to better represent reducer performance.