In the field of precision machinery, the rotary vector reducer plays a critical role due to its compact structure, high transmission ratio, efficiency, and longevity. It is widely used in robotics, aerospace, and measurement instruments. The performance of a rotary vector reducer heavily depends on the reliability and load conditions of its bearings, particularly the cycloid wheel bearings. This study focuses on the force analysis and life verification of cycloid wheel bearings within a rotary vector reducer, aiming to provide insights for optimal bearing design. I will present a comprehensive analysis starting from the transmission theory of the rotary vector reducer, covering force calculations, meshing characteristics, and bearing life evaluation using advanced software tools.
The rotary vector reducer is a two-stage differential gear system consisting of a first-stage involute planetary gear transmission and a second-stage cycloid-pin wheel transmission. The high-speed stage includes a gear shaft and planetary gears, while the low-speed stage comprises crankshafts, cycloid wheels, pin wheels, a pin gear housing, and a planet carrier. The working principle involves input rotation driving the gear shaft, which engages with planetary gears to achieve first-stage reduction. The crankshafts, fixed to the planetary gears, drive the cycloid wheels via eccentric motion, engaging with the pin wheels. When the pin gear housing is fixed, the cycloid wheels undergo both revolution and rotation, transmitting motion to the output shaft through support bearings for second-stage reduction. This complex mechanism underscores the importance of understanding force distributions in the rotary vector reducer.
To analyze the forces in the rotary vector reducer, specific parameters are essential. I consider a model similar to the RV-100C type, with key parameters summarized in the table below. These parameters form the basis for all subsequent calculations in this study of the rotary vector reducer.
| Parameter | Value |
|---|---|
| Rated Input Power (kW) | 2.05 |
| Rated Output Torque \( T_o \) (N·m) | 980 |
| Rated Output Speed (r/min) | 15 |
| Single Reducer Ratio | 36.75 |
| Total Reduction Ratio | 242.1176 |
| Crank Eccentricity \( e \) (mm) | 1.5 |
| Input Gear Teeth \( Z_1 \) | 17 |
| Center Gear Large Teeth \( Z_2 \) | 112 |
| Center Gear Small Teeth \( Z_3 \) | 48 |
| Planetary Gear Teeth \( Z_4 \) | 33 |
| Cycloid Wheel Teeth \( Z_5 \) | 51 |
| Pin Teeth Number \( Z_6 \) | 52 |
| Input Gear Pitch Radius \( r_1 \) (mm) | 14.875 |
| Center Gear Large Pitch Radius \( r_2 \) (mm) | 98 |
| Center Gear Small Pitch Radius (mm) | 42 |
| Planetary Gear Pitch Radius \( r_4 \) (mm) | 28.875 |
| Cycloid Wheel Pitch Radius \( r’_c \) (mm) | 76.5 |
| Planetary Gear Center Circle Radius \( a_0 \) (mm) | 70.875 |
| Pin Tooth Center Circle Radius \( r_p \) (mm) | 102 |
| Pin Tooth Radius \( r_{rp} \) (mm) | 3 |
| Cycloid Wheel Width (mm) | 11.5 |
| Motor Speed (r/min) | 3600 |
| Output Speed (r/min) | 14.8688 |
The force analysis begins with the transmission torque of the rotary vector reducer. Given the rated output torque \( T_o = 980 \, \text{N·m} \), the instantaneous maximum allowable output torque is often taken as 1.6 times \( T_o \), which is 1568 N·m. Considering efficiency, the maximum allowable input torque \( T_{i \text{max}} \) is calculated as:
$$ T_{i \text{max}} = \frac{T_o \times \text{Total Reduction Ratio}}{\eta} $$
where \( \eta \) is the transmission efficiency. For this rotary vector reducer, \( T_{i \text{max}} \) is determined to be 20238.1 N·mm. The planetary gear system distributes this torque. The torque on the center gear shaft \( T_3 \) is:
$$ T_3 = T_{i \text{max}} \times \frac{r_2}{r_1} $$
and each crankshaft experiences an equal torque \( T_4 \):
$$ T_4 = \frac{T_3}{3} $$
This forms the basis for analyzing forces on the crankshafts in the rotary vector reducer.
Next, I examine the forces on a single crankshaft within the rotary vector reducer. The crankshaft supports bearings and cage assemblies, with force balance equations established in the Y and Z directions. Let \( F_{3R}, F_{3T}, F_{4R}, F_{4T} \) represent radial and tangential forces on two tapered roller bearings, and \( F_{5R}, F_{5T}, F_{6R}, F_{6T} \) for two cage assemblies. The forces from the center gear small gear on the planetary gear are denoted as \( F_{34R} \) and \( F_{34T} \). The distances between components are \( L_1, L_2, L_3, L_4 \). The equilibrium equations are:
$$ F_{5R} + F_{4R} – F_{3R} – F_{6R} – F_{34R} = 0 \quad \text{(Y-direction)} $$
$$ F_{5T} + F_{4T} – F_{3T} – F_{6T} – F_{34T} = 0 \quad \text{(Z-direction)} $$
$$ F_{3R} L_1 – F_{6R} L_2 + F_{4R} (L_2 + L_3) – F_{34R} (L_2 + L_3 + L_4) = 0 \quad \text{(Moment in XY plane)} $$
$$ F_{3T} L_1 – F_{6T} L_2 + F_{4T} (L_2 + L_3) – F_{34T} (L_2 + L_3 + L_4) = 0 \quad \text{(Moment in XZ plane)} $$
$$ (F_{5T} + F_{6T}) e = F_{34T} r_4 = T_4 \quad \text{(Moment in YZ plane)} $$
These equations are crucial for determining bearing loads in the rotary vector reducer. The output shaft forces are derived from the overall torque balance. For the rotary vector reducer output mechanism, the moment equilibrium around the X-axis gives:
$$ 3 F_{5T} a_0 + 3 F_{6T} a_0 = T_o $$
This relates the tangential forces on the cage assemblies to the output torque of the rotary vector reducer.
Now, focusing on the cycloid wheel forces in the rotary vector reducer, the cycloid wheel is subjected to forces from the planetary gears and pin teeth. Let \( O_p \) be the pin gear center, \( O_c \) the cycloid wheel center, \( F_x \) the force from the planetary gear in the X-direction, \( F_y \) the force in the Y-direction, and \( F_i \) the force from the i-th pin tooth during meshing. The angle \( \alpha_i \) is between \( F_i \) and the X-axis, \( \phi_i \) is the angle between the line connecting the i-th pin tooth and \( O_p \) with the Y-axis, and \( l_i \) is the distance from the common normal at the meshing point to \( O_c \). The torque on the cycloid wheel \( T_c \) is:
$$ T_c = F_x r’_c = 3 F_{5T} a_0 $$
Assuming equal forces from the three cranks due to symmetry in the rotary vector reducer, we have \( F_{5R} = F_{6R} \) and \( F_{5T} = F_{6T} \). The Y-direction force balance yields:
$$ F_y = F_{5R} + 2 F_{5R} \sin 30^\circ = 2 F_{5R} $$
Using cycloid wheel meshing theory, the forces are expressed as:
$$ F_x = \frac{T_c Z_6}{K_1 r_p Z_5} $$
$$ F_y = \sum F_i \left( \frac{r’_c^2 – l_i^2}{r’_c} \right) $$
$$ l_i = r’_c \frac{\sin \phi_i}{\sqrt{1 + K_1^2 – 2 K_1 \cos \phi_i}} $$
$$ F_i = \frac{(\delta_i – \Delta \phi_i) F_{\text{max}}}{\delta_{\text{max}}} $$
$$ K_1 = \frac{e Z_6}{r_p} $$
where \( K_1 \) is the short amplitude factor, \( F_{\text{max}} \) is the maximum force when \( \phi_i \) approaches \( \arccos K_1 \), \( \delta_i \) is the total deformation at the i-th pin tooth, \( \delta_{\text{max}} \) is the maximum deformation under \( F_{\text{max}} \), and \( \Delta \phi_i \) is the initial clearance at the i-th pin tooth. For the rotary vector reducer, considering modified tooth profiles, the equal-distance modification \( \Delta r_{rp} \) is 0.15 mm and the profile shift modification \( \Delta r_p \) is 0.05 mm. The initial clearance is calculated as:
$$ \Delta \phi_i = \Delta r_{rp} \left(1 – \frac{\sin \phi_i}{\sqrt{1 + K_1^2 – 2 K_1 \cos \phi_i}} \right) – \Delta r_p \left(1 – \frac{K_1 \cos \phi_i – \sqrt{1 – K_1^2} \sin \phi_i}{\sqrt{1 + K_1^2 – 2 K_1 \cos \phi_i}} \right) $$
The curvature radius \( \rho_\phi \) at \( \phi_0 = \arccos K_1 \) is:
$$ \rho_\phi = \frac{r_p (1 + K_1^2 – 2 K_1 \cos \phi_0)^{3/2}}{K_1 (Z_6 + 1) \cos \phi_0 – (1 + Z_6 K_1^2)} + r_{rp} $$
The contact deformation factor \( c’ \) and maximum deformation \( \delta’_{\text{max}} \) for non-clearance meshing are:
$$ c’ = 4.99 \times 10^{-3} \sqrt{\frac{2(1 – \mu^2)}{E} \frac{F’_{\text{max}}}{b} \sqrt{\frac{2 \rho_\phi r_{rp}}{\rho_\phi + r_{rp}}}} $$
$$ \delta’_{\text{max}} = \frac{2(1 – \mu^2)}{E} \frac{F’_{\text{max}}}{\pi b} \left(2 + \ln \frac{16 r_{rp} \rho_\phi}{c’^2} \right) $$
where \( \mu \) is Poisson’s ratio, \( E \) is the elastic modulus, and \( b \) is the cycloid wheel width. For the rotary vector reducer, iterative calculations determine the number of meshing teeth and force distribution. Based on parameters, the meshing teeth range from tooth 4 to tooth 9, resulting in 6 teeth meshing simultaneously in the rotary vector reducer. The results for each meshing tooth are summarized in the table below.
| Tooth Number | \( \phi_i \) (°) | \( \Delta \phi_i \) (mm) | \( \delta_i \) (mm) | \( F_i \) (N) | \( l_i \) (mm) |
|---|---|---|---|---|---|
| 4 | 27.6924 | 0.002380974 | 0.013315038 | 1874.091975 | 74.04125803 |
| 5 | 34.6155 | 0.000379509 | 0.013685360 | 2280.614786 | 76.10051485 |
| 6 | 41.5386 | 0.000021593 | 0.013753101 | 2353.572260 | 76.47720778 |
| 7 | 48.4617 | 0.000662199 | 0.013632242 | 2223.057541 | 75.80514172 |
| 8 | 55.3848 | 0.002021873 | 0.013380562 | 1946.872409 | 74.40561776 |
| 9 | 62.3079 | 0.003971248 | 0.013029758 | 1552.623176 | 72.45489548 |
The maximum force \( F_{\text{max}} \) is found to be 2230.521022 N, and \( \delta_{\text{max}} \) is 0.013043874 mm for this rotary vector reducer. Combining all equations, the external loads on the cycloid wheel bearings under instantaneous maximum torque and rated torque conditions are calculated. These loads are essential for bearing design in the rotary vector reducer. The results are presented in the following table.
| Component | Radial Load at Instantaneous Max Torque (N) | Radial Load at Rated Torque (N) |
|---|---|---|
| Tapered Roller Bearing 1 | 16436.47 | 1782.14 |
| Tapered Roller Bearing 2 | 8049.34 | 2221.07 |
| Two Cage Assemblies | 14848.89 | 4105.93 |
With these external loads, I proceed to a design case for the cycloid wheel bearings in the rotary vector reducer. Considering the operational environment of the rotary vector reducer, a tapered roller bearing, such as type 30202, is designed. The axial load is approximately 200 N. Using Romax Designer software, I model and analyze the bearing under the derived loads. The internal load distribution, contact stress, and life are evaluated. For the rotary vector reducer, under rated torque conditions, the bearing raceway loads and contact stresses are analyzed. The maximum load occurs at the 90° direction vertically downward, with a value of about 534.8 N, and the maximum contact stress is approximately 1372 MPa. The life verification results for both tapered roller bearings are summarized below.
| Bearing | Condition | Rated Life (h) | Max Contact Stress (MPa) | Max Contact Load (N) |
|---|---|---|---|---|
| Tapered Roller Bearing 1 | Instantaneous Max Torque | 148.9 | 2239.5 | 1816.7 |
| Rated Torque | 9315.9 | 1372.1 | 534.8 | |
| Tapered Roller Bearing 2 | Instantaneous Max Torque | 78.2 | 2716.1 | 2728.6 |
| Rated Torque | 7106.5 | 1605.7 | 797.9 |
This analysis demonstrates that the load conditions and reliability of cycloid wheel bearings significantly impact the transmission performance of the rotary vector reducer. By thoroughly calculating and analyzing the forces in the rotary vector reducer transmission system, we can obtain accurate internal loads, contact stresses, and life estimates for the bearings. This approach effectively guides the design of cycloid wheel bearings, ensuring the durability and efficiency of the rotary vector reducer. The use of advanced software like Romax Designer enhances the precision of life verification, contributing to the optimization of rotary vector reducer components. In summary, understanding the complex force interactions in the rotary vector reducer is key to improving its performance and longevity in high-precision applications.
Further considerations for the rotary vector reducer include manufacturing tolerances, lubrication effects, and dynamic loads during operation. The assumptions made in this study, such as equal force distribution among cranks and torque sharing between cycloid wheels, provide a foundational model. However, real-world variations may necessitate adjustments. For instance, the maximum instantaneous torque is often set at 1.6 times the rated torque, and modifications like equal-distance and profile shift are applied to reduce wear and noise in the rotary vector reducer. These factors highlight the need for continuous refinement in the analysis of rotary vector reducer bearings.
In conclusion, the force analysis and life verification of cycloid wheel bearings in a rotary vector reducer involve a systematic approach from transmission torque calculations to meshing dynamics and bearing design. The rotary vector reducer’s unique two-stage mechanism requires detailed attention to force balances and deformations. By integrating theoretical formulas with software simulations, designers can achieve reliable bearing performance, ultimately enhancing the rotary vector reducer’s role in advanced mechanical systems. This study underscores the importance of comprehensive analysis in the development of robust rotary vector reducer technologies.