1. Introduction
Torsional stiffness is an important factor influencing the transmission quality of a reducer. It signifies the reducer’s capability to withstand torsional deformation during twisting and depends on factors like the reducer’s material and structure. The hysteresis curve method is commonly used to test the torsional stiffness of a reducer, often performed alongside the evaluation of the reducer’s lost motion.
The method for testing is illustrated in
Figure 1 [
1]. During this test, the input side of the reducer remains stationary while the output side is progressively loaded to its rated torque before being released. This process is then repeated, but with the load applied in the opposite direction up to the rated torque and subsequently released. From this, multiple sets of torque and angle values are obtained, enabling the plotting of a comprehensive torque–angle curve. The testing procedure is detailed as follows, with the five steps shown in
Figure 1: (1) load the output end towards the specified torque
; (2) unload the output end to a value of 0; (3) load the output end in the opposite direction to the specified torque
; (4) unload the output end back to 0; (5) once again, load the output end towards the specified torque
. The relationship between the specified torque
on the curve and the associated rotation angle
represents the torsional stiffness.
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. Additionally, a novel testing technique—based on the transmission error method—will be introduced. This method will be capable of assessing both the actual working-state torsional stiffness and the transmission error of the reducer. Transmission error is a fundamental concept in the field of gear transmission engineering, playing a crucial role in characterizing gear transmission quality, analyzing gear dynamic characteristics (such as vibration and noise), and guiding high-performance gear design [
14,
15,
16,
17]. However, its application in the testing of reducer torsional stiffness has not been realized, primarily due to the lack of relevant theoretical underpinning, which hinders its correlation with the hysteresis loop method. This paper aims to undertake theoretical analysis and experimental research on the dynamic torsional stiffness of reducers, investigating the coherence between the transmission error method and the hysteresis loop method. The objective is to offer guidance for the design and testing of reducer torsional stiffness.
4. Experimental Procedure
4.1. Experimental Details
This research involved conducting experiments on both small and large reducers as the test subjects. The test benches specifically designed for the small and large reducers are illustrated in
Figure 7 and
Figure 8, respectively. Detailed equipment performance parameters can be found in
Table 1.
Figure 9 displays the small reducer used in the test, measuring 40 mm × 20 mm × 40 mm overall. It utilized parallel shaft gears and employed a three-stage deceleration mechanism. The gear material used was powder metallurgy. On the other hand,
Figure 10 illustrates the large reducer, which was a cycloidal pinwheel reducer made of high-strength steel, with an outer diameter of 190 mm. The detailed parameters of the reducers are given in
Table 2.
4.2. 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.
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° to 1.9261°, and for the large reducer, it decreases from 4.3552° to 4.0473°. Correspondingly, the torsional stiffness exhibits an increasing trend. For the small reducers, it increases from 0.4969 Nm/° to 0.5192 Nm/°, and for the large reducers, it increases from 5235.1212 Nm/° to 5633.3852 Nm/°.
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.
4.3. 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
Figure 13 and
Figure 14, while the detailed test results can be found in
Table 6.
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° position, the transmission error value rises from 2.0128° to 2.8436°, leading to a corresponding decrease in torsional stiffness from 0.4964 Nm/° to 0.3517 Nm/°. In comparison, large reducers exhibit smaller variations, with transmission error values increasing from 4.3974° to 4.4529°, and corresponding torsional stiffness decreasing from 5184.8820 Nm/° to 5122.587 Nm/°. 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 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. These precision reducers exhibit significantly higher transmission precision and load-bearing capacity for components such as gears and bearings, as well as stricter requirements for overall assembly accuracy compared to small-scale reducers. Consequently, their operation is characterized by greater stability, resulting in reduced torque fluctuation amplitudes and a relatively stable actual torsional stiffness.
4.4. Torsional Stiffness Test of Reducer under All Operating Conditions
In the previous test, it was observed that the torsional stiffness of the reducer varies with changes in speed. Consequently, it becomes imperative to investigate the alterations in the torsional stiffness of the reducer under real working conditions. To accomplish this, the transmission error method is utilized to examine the torsional stiffness at different speeds and torques.
Figure 15 displays the test curve of the small reducer at the 180° position, while
Figure 16 presents the average torsional stiffness test curve within one revolution range. These curves reveal that the torsional stiffness of small reducers decreases as the rotational speed increases. This behavior reflects the dynamic characteristics of the actual torsional stiffness of the reducer. The torsional stiffness curve takes on an approximately parabolic shape, illustrating the nonlinear changes in the internal system dynamics of the reducer.
On the other hand,
Figure 17 shows the test curve of the large reducer at the 180° position, and
Figure 18 illustrates the average torsional stiffness test curve within one revolution range for this large reducer. Similar to the small reducers, the test results for the large reducer also show a decrease in torsional stiffness with increasing rotational speed. However, the torsional stiffness curve of the large reducer takes on an approximately linear shape, which is influenced by factors such as its structure and material properties.