The rotary vector reducer, a high-precision speed reduction unit, is renowned for its exceptional combination of large reduction ratio, high torsional stiffness, compact size, and superior load capacity. Its predominant application lies in the joint actuators of industrial robots, where performance reliability is paramount. A critical aspect of ensuring this reliability and minimizing operational noise is the comprehensive analysis of its operating frequencies. The complex, closed differential planetary structure of the rotary vector reducer, comprising multiple gears and rolling bearing sets, gives rise to a rich spectrum of rotational and meshing frequencies. Accurately calculating these frequencies is essential for dynamic analysis, condition monitoring, and avoiding resonant conditions that could compromise the reducer’s integrity and acoustic performance.
The kinematic structure of a rotary vector reducer integrates a first-stage planetary gear train with a second-stage cycloidal-pin gear planetary mechanism. Two primary operational configurations exist, defined by which component is held stationary: the support disc (also called the output flange) or the pin wheel (ring).
In the first configuration, with the support disc fixed, the input sun gear drives the planetary gears. Since the crankshafts are rigidly connected to these planet gears, they undergo pure rotation (spin) about their own axes. This rotation, via crankpin bearings, imparts a pure translational orbit (revolution without spin) to the cycloidal discs around the central axis. The interaction between the orbiting cycloidal discs and the stationary pin wheel forces the pin wheel itself to rotate, providing the output motion. Conversely, in the second common configuration with the pin wheel fixed, the sun gear input causes the planetary gears to both spin and revolve (planetate). The connected crankshafts follow this compound motion. The crankpins then drive the cycloidal discs, which simultaneously orbit the central axis and spin due to their meshing with the fixed pins. This spin of the cycloidal discs is transmitted back through the crankshafts to the support disc, which becomes the rotating output element.
Theoretical Derivation of Operating Frequencies
The foundation for calculating all relevant frequencies in a rotary vector reducer lies in applying the fundamental principles of planetary gear train analysis and rolling bearing kinematics.
Frequencies of Transmission Components
Using the method of inversion (fixing the carrier) for planetary systems and standard gear ratios, the rotational frequencies for all major shafts and gears can be derived. Let us define the following rotational speeds: $n_1$ for the sun gear (input), $n_2$ for the crankshaft spin, $n_3$ for the crankshaft revolution (planetation), $n_4$ for the cycloidal disc spin, $n_5$ for the cycloidal disc orbit, $n_6$ for the pin wheel, and $n_7$ for the support disc. The corresponding frequencies are $f_i = n_i / 60$. The tooth numbers are $z_1$ (sun), $z_2$ (planet), $z_3$ (cycloidal disc), and $z_4$ (pin wheel).
For the Support Disc Fixed configuration, we have $n_3 = n_4 = n_7 = 0$. The crankshaft spin equals the cycloidal disc orbit ($n_2 = n_5$). The derived frequencies are:
$$
f_2 = f_5 = -\frac{z_1}{z_2} f_1
$$
$$
f_6 = -\frac{(z_4 – z_3) z_1}{z_2 z_4} f_1
$$
For the Pin Wheel Fixed configuration, $n_6 = 0$. The crankshaft revolution, cycloidal disc spin, and support disc output speeds are equal ($n_3 = n_4 = n_7$), and the crankshaft spin equals the cycloidal disc orbit ($n_2 = n_5$). The derived frequencies are:
$$
f_2 = f_5 = \frac{z_1 z_4}{(z_3 – z_4)(z_1 + z_2)} f_1
$$
$$
f_3 = f_4 = f_7 = \frac{z_1}{z_1 + z_2} f_1
$$
The meshing frequencies are equally critical. The first-stage (planetary) meshing frequency $f_{1c}$ is the rate at which gear teeth engage, and the second-stage (cycloidal) meshing frequency $f_{2c}$ is the engagement rate between the cycloidal disc lobes and the pin teeth. The complete set of formulas for the transmission component frequencies in a rotary vector reducer is summarized in the table below.
| Frequency Component | Support Disc Fixed | Pin Wheel Fixed |
|---|---|---|
| Sun Gear (Input) | $f_1$ | $f_1$ |
| Crankshaft Spin | $f_2 = -\frac{z_1}{z_2} f_1$ | $f_2 = \frac{z_1 z_4}{(z_3 – z_4)(z_1 + z_2)} f_1$ |
| Crankshaft Revolution | $f_3 = 0$ | $f_3 = \frac{z_1}{z_1 + z_2} f_1$ |
| Cycloidal Disc Spin | $f_4 = 0$ | $f_4 = \frac{z_1}{z_1 + z_2} f_1$ |
| Cycloidal Disc Orbit | $f_5 = -\frac{z_1}{z_2} f_1$ | $f_5 = \frac{z_1 z_4}{(z_3 – z_4)(z_1 + z_2)} f_1$ |
| Pin Wheel | $f_6 = -\frac{(z_4 – z_3) z_1}{z_2 z_4} f_1$ | $f_6 = 0$ |
| Support Disc (Output) | $f_7 = 0$ | $f_7 = \frac{z_1}{z_1 + z_2} f_1$ |
| 1st Stage Meshing | $f_{1c} = z_1 f_1$ | $f_{1c} = \frac{z_1 z_2}{z_1 + z_2} f_1$ |
| 2nd Stage Meshing | $f_{2c} = z_3 |f_5| = \frac{z_1 z_3}{z_2} f_1$ | $f_{2c} = z_3 |f_4| = \frac{z_1 z_3}{z_1 + z_2} f_1$ |
Frequencies of Rolling Bearing Components
A rotary vector reducer typically incorporates three sets of critical bearings: the crankshaft support bearings, the crankpin bearings, and the main output bearings. Each bearing’s internal components—inner ring, outer ring, rolling elements, and cage—have distinct rotational frequencies. Assuming pure rolling contact and no gross slip, these frequencies can be derived from kinematic relations. The key parameters are: pitch diameter $D_j$, rolling element diameter $d_j$, contact angle $\alpha_j$, and number of rolling elements $Z_j$, where $j=1,2,3$ denotes the support, crankpin, and main bearing respectively.
The fundamental kinematic relationship for a bearing is that the cage (rolling element carrier) speed is the average of the inner and outer ring raceway velocities at the pitch circle. For a bearing with inner ring rotation frequency $f_{i}$ and outer ring frequency $f_{o}$, the cage rotation frequency $f_c$ (which is also the rolling element revolution frequency) is given by:
$$
f_c = \frac{1}{2} \left[ f_i \left(1 + \frac{d}{D} \cos \alpha \right) + f_o \left(1 – \frac{d}{D} \cos \alpha \right) \right]
$$
The rolling element spin frequency $f_b$ relative to the cage can be found by considering the relative velocity at the contact point with the inner ring. The frequency at which a single rolling element passes a point on the inner raceway is $f_i – f_c$, and the ratio of rolling element surface velocity to this passing frequency relates to the geometry. The derived formula is:
$$
f_b = \frac{D f_i}{2d} \left[ 1 – \left( \frac{d}{D} \right)^2 \cos^2 \alpha \right] – \frac{D f_o}{2d} \left[ 1 – \left( \frac{d}{D} \right)^2 \cos^2 \alpha \right]
$$
Applying these general formulas to each bearing in the two configurations of the rotary vector reducer requires substituting the appropriate inner and outer ring speeds. For the Support Disc Fixed case:
- Crankshaft Support Bearing: Inner ring spins with the crankshaft ($f_{i1}=f_2$), outer ring is fixed in the support disc ($f_{o1}=0$).
- Crankpin Bearing: Inner ring spins with the crankshaft ($f_{i2}=f_2$), outer ring is housed in the cycloidal disc. Since the cycloidal disc only orbits without spinning in this configuration, its absolute rotation is zero, but the bearing outer ring experiences a translational motion. For kinematic purposes relative to the crankshaft axis, its effective rotational speed $f_{o2}=0$.
- Main Bearing: Inner ring is fixed to the stationary support disc ($f_{i3}=0$), outer ring rotates with the output pin wheel ($f_{o3}=f_6$).
For the Pin Wheel Fixed case:
- Crankshaft Support Bearing: Inner ring has combined spin and revolution ($f_{i1}=f_2 + f_3$), outer ring rotates with the output support disc ($f_{o1}=f_7$).
- Crankpin Bearing: Inner ring has combined motion ($f_{i2}=f_2 + f_3$), outer ring rotates with the cycloidal disc, which spins at $f_4 = f_7$ ($f_{o2}=f_7$).
- Main Bearing: Inner ring rotates with the output support disc ($f_{i3}=f_7$), outer ring is fixed in the stationary pin wheel housing ($f_{o3}=0$).
The complete set of derived formulas for the bearing component frequencies in a rotary vector reducer is presented in the following comprehensive table. Note that for deep groove ball bearings or cylindrical roller bearings often used in the crank locations, the contact angle $\alpha = 0^\circ$ simplifies the expressions significantly.
| Bearing & Component | Support Disc Fixed Configuration | Pin Wheel Fixed Configuration |
|---|---|---|
| Crankshaft Support Bearing (j=1) | ||
| Inner Ring Raceway Frequency ($f_{i1}$) | $f_2$ | $f_2 + f_3$ |
| Outer Ring Raceway Frequency ($f_{o1}$) | $0$ | $f_7$ |
| Cage Rotation / Roller Revolution ($f_{c1}$) | $\frac{1}{2} f_2 \left(1 – \frac{d_1}{D_1} \cos \alpha_1 \right)$ | $\frac{1}{2} \left[ (f_2+f_3)\left(1 – \frac{d_1}{D_1} \cos \alpha_1 \right) + f_7\left(1 + \frac{d_1}{D_1} \cos \alpha_1 \right) \right]$ |
| Roller Spin ($f_{b1}$) | $\frac{D_1 f_2}{2d_1} \left[ 1 – \left( \frac{d_1}{D_1} \right)^2 \cos^2 \alpha_1 \right]$ | $\frac{D_1 (f_2+f_3)}{2d_1} \left[ 1 – \left( \frac{d_1}{D_1} \right)^2 \cos^2 \alpha_1 \right]$ |
| Crankpin Bearing (j=2) | ||
| Inner Ring Raceway Frequency ($f_{i2}$) | $f_2$ | $f_2 + f_3$ |
| Outer Ring Raceway Frequency ($f_{o2}$) | $0$ | $f_7$ |
| Cage Rotation / Roller Revolution ($f_{c2}$) | $\frac{1}{2} f_2 \left(1 – \frac{d_2}{D_2} \cos \alpha_2 \right)$ | $\frac{1}{2} \left[ (f_2+f_3)\left(1 – \frac{d_2}{D_2} \cos \alpha_2 \right) + f_7\left(1 + \frac{d_2}{D_2} \cos \alpha_2 \right) \right]$ |
| Roller Spin ($f_{b2}$) | $\frac{D_2 f_2}{2d_2} \left[ 1 – \left( \frac{d_2}{D_2} \right)^2 \cos^2 \alpha_2 \right]$ | $\frac{D_2 (f_2+f_3)}{2d_2} \left[ 1 – \left( \frac{d_2}{D_2} \right)^2 \cos^2 \alpha_2 \right]$ |
| Main Output Bearing (j=3) | ||
| Inner Ring Raceway Frequency ($f_{i3}$) | $0$ | $f_7$ |
| Outer Ring Raceway Frequency ($f_{o3}$) | $f_6$ | $0$ |
| Cage Rotation / Roller Revolution ($f_{c3}$) | $\frac{1}{2} f_6 \left(1 + \frac{d_3}{D_3} \cos \alpha_3 \right)$ | $\frac{1}{2} f_7 \left(1 – \frac{d_3}{D_3} \cos \alpha_3 \right)$ |
| Roller Spin ($f_{b3}$) | $\frac{D_3 f_6}{2d_3} \left[ 1 – \left( \frac{d_3}{D_3} \right)^2 \cos^2 \alpha_3 \right]$ | $\frac{D_3 f_7}{2d_3} \left[ 1 – \left( \frac{d_3}{D_3} \right)^2 \cos^2 \alpha_3 \right]$ |
Numerical Example and Validation via Multibody Dynamics Simulation
To demonstrate the application of the derived formulas and verify their accuracy, a specific rotary vector reducer model, the RV-40E, is analyzed. The input parameters are as follows: Sun gear speed $n_1 = 1500 \text{ rpm}$ ($f_1 = 25 \text{ Hz}$), tooth numbers $z_1=16$, $z_2=32$, $z_3=39$, $z_4=40$. The bearing parameters are:
- Crankshaft Support Bearing (Deep Groove Ball): $D_1=25\text{mm}$, $d_1=6.38\text{mm}$, $\alpha_1=0^\circ$, $Z_1=10$.
- Crankpin Bearing (Needle Roller): $D_2=27\text{mm}$, $d_2=2\text{mm}$, $\alpha_2=0^\circ$, $Z_2=24$.
- Main Bearing (Angular Contact Ball): $D_3=130\text{mm}$, $d_3=8\text{mm}$, $\alpha_3=34^\circ$, $Z_3=36$.
Applying the theoretical formulas from the tables above yields the calculated operating frequencies for the RV-40E rotary vector reducer.
To validate these theoretical results, a detailed multi-body dynamics simulation was performed using ADAMS software. A three-dimensional model of the RV-40E rotary vector reducer was created, incorporating necessary simplifications such as modeling a single cycloidal disc and merging non-critical components. Appropriate joints (revolute, fixed) and contacts were defined between gear teeth and bearing components. A step function was applied to the sun gear to bring it to 1500 rpm. After simulation, the absolute angular velocities of all components were extracted from the post-processor and converted to frequencies. For components with compound motion, the spin frequency was obtained by subtracting the revolution frequency from the absolute angular velocity. The comparison between the theoretical calculations and the ADAMS simulation results for the primary frequencies is presented below. The percentage error is calculated based on the absolute difference.
| Frequency Component | Theoretical Value (Hz) | ADAMS Simulation (Hz) | Absolute Error (%) |
|---|---|---|---|
| Support Disc Fixed Configuration | |||
| Sun Gear, $f_1$ | 25.00 | 25.00 | 0.00 |
| Crankshaft Spin, $f_2$ | -12.50 | -12.50 | 0.00 |
| Pin Wheel Output, $f_6$ | -0.31 | -0.31 | 0.96 |
| 1st Stage Meshing, $f_{1c}$ | 400.00 | 400.00 | 0.00 |
| 2nd Stage Meshing, $f_{2c}$ | 487.50 | 487.50 | 0.00 |
| Crankpin Bearing Roller Spin, $f_{b2}$ | -83.91 | -83.94 | 0.04 |
| Main Bearing Roller Spin, $f_{b3}$ | 2.53 | 2.61 | 2.99 |
| Pin Wheel Fixed Configuration | |||
| Sun Gear, $f_1$ | 25.00 | 25.00 | 0.00 |
| Crankshaft Spin, $f_2$ | -12.35 | -12.35 | 0.03 |
| Crankshaft Revolution, $f_3$ | 0.31 | 0.31 | 0.32 |
| Support Disc Output, $f_7$ | 0.31 | 0.31 | 0.32 |
| 1st Stage Meshing, $f_{1c}$ | 395.06 | 395.06 | 0.00 |
| 2nd Stage Meshing, $f_{2c}$ | 481.48 | 481.48 | 0.00 |
| Crankpin Bearing Roller Spin, $f_{b2}$ | -82.88 | -82.76 | 0.15 |
| Main Bearing Roller Spin, $f_{b3}$ | 2.50 | 2.61 | 4.19 |
The results show excellent agreement between the theoretical predictions and the multi-body dynamics simulation. The maximum error observed across all calculated frequencies for this rotary vector reducer is 4.19%, which falls within an acceptable range for engineering analysis and validates the derived theoretical framework.
Discussion and Design Implications
The analysis reveals crucial insights into the dynamics of the rotary vector reducer. A particularly significant finding is the exceptionally high spin speed of the crankpin bearing rollers. In the provided example, with an input of 1500 rpm, the crankpin roller spin frequency exceeds 80 Hz, corresponding to a rotational speed near 5000 rpm. Within the confined, grease-lubricated environment of a rotary vector reducer, this high-speed spinning of small-diameter rollers generates significant friction and heat, which is a primary contributor to bearing wear and potential failure. This highlights why the crankpin bearings are often the life-limiting component in a rotary vector reducer.
The derived formula for crankpin roller spin frequency, $f_{b2} = \frac{D_2 (f_2+f_3)}{2d_2} \left[ 1 – \left( \frac{d_2}{D_2} \right)^2 \cos^2 \alpha_2 \right]$, provides a direct path for design optimization. For a given pitch diameter $D_2$ and kinematic input $(f_2+f_3)$, the roller spin frequency $f_{b2}$ is inversely proportional to the roller diameter $d_2$. Therefore, a key design strategy to enhance the durability and thermal performance of a rotary vector reducer is to maximize the diameter of the crankpin bearing rollers within the spatial constraints. This effectively reduces their surface velocity and spin rate, leading to lower operating temperatures and improved fatigue life.
Conclusion
This study establishes a complete and generalized theoretical framework for calculating the full spectrum of operating frequencies in a rotary vector reducer. The formulas cover both common structural configurations (support disc fixed and pin wheel fixed) and account for the frequencies of all major transmission components as well as the inner components of the critical bearing sets. The methodology, based on planetary gear kinematics and rolling bearing pure-roll kinematics, has been rigorously validated against multi-body dynamics simulation for a specific rotary vector reducer model (RV-40E), showing a high degree of accuracy with a maximum error of 4.19%.
The derived formulas provide a powerful, efficient, and generic analytical tool. Unlike simulation methods which require building a unique model for each specific rotary vector reducer design, these closed-form equations allow for rapid calculation and sensitivity analysis during the design and troubleshooting phases. The identification of the high spin speed of crankpin bearing rollers as a critical factor offers clear guidance for design optimization, emphasizing the importance of maximizing roller diameter to improve the reliability and service life of the entire rotary vector reducer assembly. This work forms a essential foundation for advanced dynamic modeling, vibration analysis, and predictive maintenance strategies for these precision speed reducers.