In modern industrial applications, such as robotics and machine tools, the rotary vector reducer plays a critical role due to its high precision, compact design, and ability to handle significant torque loads. As a key component, the rotary vector reducer relies on various bearings, including crank support bearings and turning arm bearings, to support and transmit forces and torque. However, the force distribution within these bearings is complex, and failures often originate from bearing issues, making it essential to understand the factors influencing their stress states. In this study, we investigate the variation laws of forces on crank support bearings and turning arm bearings in rotary vector reducers, using a comprehensive simulation approach to analyze parameters related to working conditions, reducer structure, bearing design, and installation. The goal is to provide insights for optimizing rotary vector reducer performance and extending bearing life.
The rotary vector reducer, commonly referred to as an RV reducer, integrates a two-stage transmission system: a planetary gear stage and a cycloidal pin gear stage. This combination allows for high reduction ratios in a compact space, but it also introduces complex force interactions among components. Bearings within the rotary vector reducer, such as the crank support bearings (typically tapered roller bearings) and turning arm bearings (often needle roller bearings), are subjected to varying loads during operation. Understanding these forces is crucial for design improvements and reliability enhancements. We begin by exploring the fundamental principles of the rotary vector reducer to establish a baseline for force analysis.
The transmission ratio of a rotary vector reducer is derived from the combination of the planetary and cycloidal stages. For the first stage, the planetary gear transmission ratio, denoted as \(i_1\), is given by:
$$i_1 = -\frac{Z_x}{Z_t}$$
where \(Z_x\) is the number of teeth on the planetary gear (crank shaft gear) and \(Z_t\) is the number of teeth on the sun gear (input shaft gear). The negative sign indicates opposite rotation directions. For the second stage, the cycloidal pin gear transmission ratio, \(i_2\), is expressed as:
$$i_2 = \frac{Z_c}{Z_b}$$
Here, \(Z_c\) represents the number of pins on the pin wheel (fixed to the housing), and \(Z_b\) is the number of teeth on the cycloidal gear (connected to the output). However, the overall transmission ratio \(i\) of the rotary vector reducer is not simply the product of \(i_1\) and \(i_2\); instead, it is calculated as:
$$i = 1 + \frac{Z_x Z_c}{Z_t (Z_c – Z_b)}$$
This formula highlights the interdependence of gear teeth counts in determining the reduction capability of the rotary vector reducer. From this, the output speed \(n_c\) can be derived from the input speed \(n_r\):
$$n_c = \frac{n_r}{i} = \frac{n_r}{1 + \frac{Z_x Z_c}{Z_t (Z_c – Z_b)}}$$
This relationship is fundamental for analyzing how operational parameters affect bearing forces in the rotary vector reducer.
To assess the forces on bearings, we focus on the turning arm bearings as an example, given their vulnerability in rotary vector reducer systems. The output torque \(T_c\) of the rotary vector reducer is related to the input power \(P_r\) and output speed \(n_c\) by:
$$T_c = \frac{30 P_r}{n_c \pi}$$
Since the pin wheel is fixed to the housing, its output torque \(T_z\) is approximately equal to \(T_c\). By analyzing the forces on the pin wheel and cycloidal gear, the resultant force on the turning arm bearing can be derived. For a rotary vector reducer with \(N\) crank shafts, the force \(F_z\) on the turning arm bearing is given by:
$$F_z = \frac{30 (d_1 + d_2) P_r}{\pi N D n_c d_2 \cos \alpha}$$
where \(d_1\) is the pitch circle diameter of the cycloidal gear, \(d_2\) is the distribution circle diameter of the crank shafts, \(D\) is the pitch circle diameter of the pin wheel, and \(\alpha\) is the pressure angle of the cycloidal gear. Similarly, the force \(F_q\) on the crank support bearing, assuming radial stiffness \(k_1\) and deformation \(c_1\), is:
$$F_q = k_1 c_1 = \frac{30 b_1 P_r}{\pi N D n_c b_2 \cos \alpha}$$
with \(b_1\) as the distance between two turning arm bearings and \(b_2\) as the distance between two crank support bearings. These equations form the theoretical basis for our simulation studies on the rotary vector reducer.
In this research, we employed Romax software to construct a detailed simulation model of the entire rotary vector reducer, including all critical components like gears, shafts, and bearings. This approach allows us to account for interactions between parts, which are often neglected in isolated analyses. The model incorporates specific bearing types: crank support bearings as tapered roller bearings (model 30206JR) and turning arm bearings as needle roller bearings (model HK4516). Parameters such as gear teeth counts, bearing dimensions, and installation settings were defined based on a typical rotary vector reducer design, as summarized in Table 1.
| Parameter | Value |
|---|---|
| Input speed, \(n_r\) (r/min) | 1000 |
| Input power, \(P_r\) (kW) | 3 |
| Number of cycloidal gear teeth, \(Z_b\) | 18 |
| Pitch circle diameter of cycloidal gear, \(d_1\) (mm) | 322 |
| Pressure angle, \(\alpha\) (degrees) | 20 |
| Number of crank shafts, \(N\) | 3 |
| Distribution circle diameter of crank shafts, \(d_2\) (mm) | 182 |
| Pitch circle diameter of pin wheel, \(D\) (mm) | 322 |
| Planetary gear transmission ratio, \(i_1\) | 2.5 |
Bearing-specific parameters are listed in Table 2. The model was validated by comparing theoretical calculations with simulation results for output speed and torque, ensuring accuracy for subsequent force analyses in the rotary vector reducer.
| Parameter | Value |
|---|---|
| Number of rollers in turning arm bearing | 30 |
| Length of rollers in turning arm bearing (mm) | 14 |
| Assembly interference for turning arm bearing (μm) | 80 |
| Installation preload for crank support bearing (N) | 500 |
Validation involved computing output speed \(n_c\) for various input speeds \(n_r\) using the derived formulas and comparing them with Romax simulation outputs. As shown in Table 3, the differences were minimal, confirming the model’s reliability for the rotary vector reducer. Similarly, forces on turning arm bearings were compared, with slight discrepancies due to simulation factors like preload and interference, which theoretical models often overlook.
| Input Speed, \(n_r\) (r/min) | Calculated Output Speed, \(n_c\) (r/min) | Simulated Output Speed, \(n_c\) (r/min) |
|---|---|---|
| 1000 | 20.6 | 21.0 |
| 2000 | 41.2 | 41.0 |
| 3000 | 61.8 | 62.0 |
| 4000 | 82.5 | 82.0 |
| 5000 | 103.1 | 103.0 |
With the validated model, we conducted extensive simulations to analyze how different parameters influence forces on crank support bearings and turning arm bearings in the rotary vector reducer. The study is divided into four categories: working condition parameters, rotary vector reducer structure parameters, bearing structure parameters, and bearing installation parameters. Each category is explored in detail below, with tables and formulas summarizing key findings.
First, we examine the impact of working condition parameters, specifically input power \(P_r\) and input speed \(n_r\), on bearing forces in the rotary vector reducer. By varying these parameters while holding others constant, we observed clear trends. The force on both crank support bearings and turning arm bearings increases linearly with input power, as expressed by the proportional relationships in the derived equations. For instance, doubling \(P_r\) results in approximately double the force, highlighting the direct load dependency in the rotary vector reducer. Conversely, increasing input speed reduces bearing forces in an inverse proportion manner. This is because higher speeds lead to greater output speeds \(n_c\), which decrease the torque and subsequently the forces, as shown in the formula for \(F_z\). Table 4 illustrates these trends with sample data from simulations, emphasizing the importance of optimizing input conditions for rotary vector reducer applications.
| Input Power, \(P_r\) (kW) | Input Speed, \(n_r\) (r/min) | Force on Crank Support Bearing, \(F_q\) (N) | Force on Turning Arm Bearing, \(F_z\) (N) |
|---|---|---|---|
| 1 | 1000 | 1518.5 | 1352.3 |
| 3 | 1000 | 4555.5 | 4056.9 |
| 5 | 1000 | 7592.5 | 6761.5 |
| 3 | 2000 | 2277.8 | 2028.5 |
| 3 | 3000 | 1518.5 | 1352.3 |
Mathematically, these relationships can be summarized as:
$$F_q \propto P_r \quad \text{and} \quad F_q \propto \frac{1}{n_r}$$
and similarly for \(F_z\). Thus, in practical rotary vector reducer operations, using lower power and higher speed inputs can minimize bearing forces, potentially extending service life.
Next, we investigate how structural parameters of the rotary vector reducer affect bearing forces. These include the planetary gear transmission ratio \(i_1\), the number of cycloidal gear teeth \(Z_b\), and the number of crank shafts \(N\). Simulation results indicate that increasing \(i_1\) or \(Z_b\) leads to higher forces on both crank support bearings and turning arm bearings. This is because larger values reduce the overall transmission ratio \(i\), resulting in lower output speeds \(n_c\) and higher torques, as per the equations. For example, from the formula for \(F_z\), a decrease in \(n_c\) increases the force, explaining the observed trend. On the other hand, increasing the number of crank shafts \(N\) distributes the load more evenly, effectively reducing the force on individual bearings. Table 5 provides quantitative insights from our simulations on the rotary vector reducer, showing force variations with these parameters.
| Parameter | Value | Force on Crank Support Bearing, \(F_q\) (N) | Force on Turning Arm Bearing, \(F_z\) (N) |
|---|---|---|---|
| Planetary gear transmission ratio, \(i_1\) | 2.0 | 4200.2 | 3740.1 |
| 2.5 | 4555.5 | 4056.9 | |
| 3.0 | 4910.8 | 4373.7 | |
| Number of cycloidal gear teeth, \(Z_b\) | 16 | 4300.3 | 3830.5 |
| 18 | 4555.5 | 4056.9 | |
| 20 | 4810.7 | 4283.3 | |
| Number of crank shafts, \(N\) | 2 | 6833.3 | 6085.4 |
| 3 | 4555.5 | 4056.9 | |
| 4 | 3416.6 | 3042.7 |
The relationships can be expressed as:
$$F_q \propto i_1, \quad F_q \propto Z_b, \quad \text{and} \quad F_q \propto \frac{1}{N}$$
with similar proportionality for \(F_z\). Therefore, in designing a rotary vector reducer, reducing \(i_1\) and \(Z_b\) while increasing \(N\) within spatial and cost constraints can alleviate bearing loads.
We then analyze bearing structure parameters, focusing on the number and length of rollers in the turning arm bearings. Interestingly, variations in these parameters have minimal impact on the magnitude of forces acting on both crank support bearings and turning arm bearings in the rotary vector reducer. This is because changes in roller count or length primarily alter the bearing stiffness, which in turn affects the overall system dynamics subtly without drastically changing external loads. However, these parameters significantly influence the maximum contact stress on the bearing raceways. Specifically, increasing the number of rollers or their length reduces the contact stress by distributing the load over a larger area, which is crucial for preventing fatigue failures. Table 6 summarizes simulation results, highlighting the contrast between force and stress responses in the rotary vector reducer.
| Number of Rollers | Roller Length (mm) | Force on Crank Support Bearing, \(F_q\) (N) | Force on Turning Arm Bearing, \(F_z\) (N) | Max Contact Stress on Turning Arm Bearing (MPa) |
|---|---|---|---|---|
| 25 | 14 | 4560.2 | 4060.5 | 1250.3 |
| 30 | 14 | 4555.5 | 4056.9 | 1150.8 |
| 35 | 14 | 4550.8 | 4053.3 | 1050.4 |
| 30 | 12 | 4557.0 | 4057.5 | 1200.6 |
| 30 | 14 | 4555.5 | 4056.9 | 1150.8 |
| 30 | 16 | 4554.0 | 4056.3 | 1100.2 |
It is noteworthy that while the force on the crank support bearing is generally higher than that on the turning arm bearing in the rotary vector reducer, the maximum contact stress on the turning arm bearing often exceeds that of the crank support bearing by up to 1.4 times. This disparity underscores the vulnerability of turning arm bearings, making them a critical focus for optimization in rotary vector reducer systems.
Finally, we explore bearing installation parameters, namely the preload applied to crank support bearings and the assembly interference for turning arm bearings in the rotary vector reducer. Preload is essential for tapered roller bearings to eliminate initial clearance and prevent slippage under load. Our simulations show that as preload increases, forces on both crank support bearings and turning arm bearings first decrease and then increase, with an optimal point around 1500 N where forces are minimized. For instance, at 1500 N preload, \(F_q\) reaches 5952.0 N and \(F_z\) is 5313.3 N, representing the lowest values in the studied range. This trend can be attributed to the balance between improved load distribution and excessive internal stresses at higher preloads in the rotary vector reducer.
Regarding assembly interference, which affects the radial clearance of turning arm bearings, increasing the interference reduces bearing forces slightly—by approximately 290 N in our simulations—due to changes in system deformation and dynamics. More importantly, interference impacts bearing life significantly. As shown in Table 7, there exists an optimal interference of 95 μm that maximizes the life of the turning arm bearing in the rotary vector reducer, corresponding to a slight negative clearance of -4.87 μm. This finding emphasizes the importance of precise installation for enhancing durability.
| Assembly Interference (μm) | Working Clearance (μm) | Bearing Life (hours) | Force on Turning Arm Bearing, \(F_z\) (N) |
|---|---|---|---|
| 80 | 3.01 | 850.5 | 4056.9 |
| 90 | -1.43 | 920.8 | 3950.2 |
| 95 | -4.87 | 950.2 | 3880.1 |
| 100 | -8.30 | 900.4 | 3810.0 |
| 110 | -12.74 | 800.7 | 3760.5 |
The relationship between preload \(P_{pre}\) and bearing force \(F\) can be modeled as a quadratic function, indicating an optimum, while interference \(\delta\) affects clearance \(C\) linearly: \(C = C_0 – \delta\), where \(C_0\) is the initial clearance. These insights guide installation practices for rotary vector reducers.
In conclusion, our comprehensive study on the rotary vector reducer reveals several key patterns in the variation of forces on crank support bearings and turning arm bearings. Input power and speed directly influence bearing forces linearly and inversely, respectively. Structural parameters like planetary gear ratio, cycloidal gear teeth count, and crank shaft number significantly affect load distribution, with opportunities for optimization by reducing the former two and increasing the latter. Bearing design parameters, such as roller count and length, have minimal impact on force magnitude but are crucial for reducing contact stress, particularly in turning arm bearings of the rotary vector reducer. Installation parameters, including preload and interference, play a vital role in minimizing forces and maximizing bearing life, with identified optimal values. These findings provide a foundation for improving the reliability and efficiency of rotary vector reducers in industrial applications. Future work could explore dynamic effects or material variations to further enhance performance.
Throughout this analysis, the rotary vector reducer has been central to our discussions, highlighting its complexity and the need for integrated simulation approaches. By leveraging tools like Romax, we can better understand and mitigate bearing failures, ensuring the longevity of rotary vector reducer systems. The repeated emphasis on the rotary vector reducer in this text underscores its importance in modern machinery, and we hope this research contributes to advancements in its design and application.