As an engineer specializing in mechanical dynamics, I have extensively studied the torsional vibration characteristics of rotary vector reducers, which are critical components in robotics and precision machinery. The rotary vector reducer, often abbreviated as RV reducer, is a high-performance gear system derived from cycloidal drive technology. It offers compact size, high torque capacity, and large reduction ratios, making it ideal for applications like robotic arms where dynamic loads are common. However, torsional vibrations induced by rapid stops or oscillatory motions can lead to performance degradation and fatigue failures. In this article, I will delve into the natural frequency of torsional vibration in rotary vector reducers, using a lumped-parameter approach to model the system, and identify key influencing factors through analytical and experimental insights. Throughout this discussion, the term ‘rotary vector reducer’ will be emphasized to underscore its relevance in dynamic analysis.
The rotary vector reducer combines a planetary gear stage with a cycloidal pin-wheel stage, resulting in a two-stage transmission. The first stage involves a sun gear, planetary gears, and a carrier, while the second stage uses crankshafts with eccentric bearings to drive cycloid discs against a fixed pin ring. This unique design allows for high reduction ratios, often exceeding 100:1, but it also introduces complex dynamic behaviors. Torsional vibration arises from elastic deformations in components like gears, shafts, and bearings, especially under transient loads. Understanding the natural frequency, particularly the first-order mode, is essential for avoiding resonance and ensuring reliable operation. Previous studies on rotary vector reducers have modeled torsional vibrations using methods like transfer matrices and multi-degree-of-freedom systems, yet there is ongoing debate about the primary factors affecting the fundamental frequency. Some researchers emphasize the torsional stiffness of the input shaft, while others highlight the crankshaft or bearing stiffness. In my analysis, I will show that bearing stiffness, particularly at the carrier, plays a dominant role, a finding supported by experimental data.
To begin, let’s review the transmission ratio of the rotary vector reducer, as it underpins the dynamic model. The overall reduction ratio is derived from the kinematics of both stages. For the planetary stage, with the carrier fixed, the gear ratio between the sun gear and planetary gear is given by:
$$ i^{H}_{sp} = \frac{\omega_s – \omega_H}{\omega_p – \omega_H} = -\frac{z_p}{z_s} $$
where \( \omega_s \), \( \omega_p \), and \( \omega_H \) are the angular velocities of the sun gear, planetary gear, and carrier, respectively, and \( z_s \) and \( z_p \) are the tooth numbers of the sun and planetary gears. For the cycloidal stage, with the crankshaft fixed (i.e., the planetary gear fixed relative to the carrier), the ratio between the cycloid disc and pin ring is:
$$ i^{p}_{cr} = \frac{\omega_c – \omega_p}{\omega_r – \omega_p} = \frac{z_r}{z_c} $$
Here, \( \omega_c \) and \( \omega_r \) are the angular velocities of the cycloid disc and pin ring, with \( z_c \) and \( z_r \) as their tooth numbers. The cycloid disc is connected to the carrier, so \( \omega_c = \omega_H \), and the pin ring is typically fixed (\( \omega_r = 0 \)). The tooth difference is \( z_c = z_r – 1 \). From these relations, the speed ratio between the planetary gear and sun gear is:
$$ i_{ps} = \frac{\omega_p}{\omega_s} = -\frac{z_s z_c}{z_s + z_r z_p} $$
and the carrier-to-sun gear ratio is:
$$ i_{Hs} = \frac{\omega_H}{\omega_s} = \frac{1}{1 + (z_c + 1) \frac{z_p}{z_s}} $$
These ratios are crucial for converting parameters in the dynamic model. For instance, a typical rotary vector reducer might have \( z_s = 10 \), \( z_p = 34 \), \( z_c = 29 \), and \( z_r = 30 \), yielding \( i_{Hs} \approx 1/103 \) and \( i_{ps} \approx -29/103 \). This high reduction ratio amplifies torques and stresses, making dynamic analysis vital.
Next, I develop a torsional vibration model using the lumped-parameter method, which simplifies the rotary vector reducer into discrete inertias and springs. This approach assumes pure torsion, ignoring bending and axial deformations, which is reasonable for first-order vibration modes. The system is symmetric, with two crankshafts and cycloid discs, allowing me to model them as a single equivalent component. Key assumptions include: identical motion for both crankshafts, negligible bending stiffness of crankshafts due to short spans, and average values for time-varying meshing stiffnesses. The model comprises five degrees of freedom: sun gear, planetary gear, crankshaft, cycloid disc, and carrier. Each component is represented by an equivalent inertia disk and torsional spring, with parameters折算 to the input shaft for consistency.
The equivalent moments of inertia are derived from energy equivalence. For a component \( i \) with rotational inertia \( J_i \) and mass \( m_i \), the equivalent inertia \( J^e_i \) referred to the sun gear shaft is:
$$ J^e_i = J_i \left( \frac{\omega_i}{\omega_s} \right)^2 + m_i \left( \frac{v_i}{\omega_s} \right)^2 $$
where \( v_i \) is the translational velocity of the center of mass. For symmetric parts like planetary gears, crankshafts, and cycloid discs, I double the values to account for both sides. Specific expressions are:
For planetary gears (\( p \)): \( J^e_p = 2J_p i_{ps}^2 + 2m_p r_b^2 i_{Hs}^2 \), where \( r_b \) is the radius to the crankshaft axis on the carrier.
For crankshafts (\( b \)): \( J^e_b = 2J_b i_{ps}^2 + 2m_b r_b^2 i_{Hs}^2 \).
For cycloid discs (\( c \)): \( J^e_c = 2J_c i_{Hs}^2 + 2m_c a^2 i_{ps}^2 \), with \( a \) as the crankshaft eccentricity.
For the carrier (\( H \)): \( J^e_H = J_H i_{Hs}^2 \), including any attached load inertia.
The equivalent torsional stiffnesses are similarly derived from potential energy. For a stiffness element \( k_i \) associated with deformation \( \theta_i \), the equivalent stiffness \( k^e_i \) is:
$$ k^e_i = k_i \left( \frac{\theta_i}{\theta_s} \right)^2 $$
where \( \theta_s \) is the sun gear’s angular deflection. Key stiffness elements include:
- Sun-planet meshing stiffness: \( k^e_{sp} = 2k_{sp} r_s^2 \), with \( k_{sp} \) as the mesh stiffness and \( r_s \) the sun gear base radius.
- Crankshaft torsional stiffness: \( k^e_b = 2k_b i_{ps}^2 \), where \( k_b \) is the crankshaft’s torsional stiffness.
- Cycloid disc bearing stiffness: \( k^e_{cb} = 2k_{cb} a^2 i_{ps}^2 \), with \( k_{cb} \) as the radial bearing stiffness.
- Cycloid-pin meshing stiffness: This has two parts—rotational and translational. The equivalent stiffness is \( k^e_{cr} = 2k_{cr} i_{Hs}^2 + 2k’_{cr} a^2 i_{ps}^2 \), where \( k_{cr} \) is the torsional mesh stiffness and \( k’_{cr} = \frac{k_{cr}}{r_c} \cos \alpha \) is the tangential component, with \( r_c \) as the cycloid disc’s fixed radius and \( \alpha \) the equivalent pressure angle.
- Carrier bearing stiffness: \( k^e_{Hb} = k_{Hb} r_b^2 i_{Hs}^2 \), where \( k_{Hb} \) is the radial stiffness of the carrier bearings.
These derivations lead to a 5-DOF lumped-parameter model, as shown in the equivalent system diagram. The equations of motion for free torsional vibration are:
$$ \mathbf{J} \ddot{\boldsymbol{\theta}} + \mathbf{K} \boldsymbol{\theta} = \mathbf{0} $$
where \( \mathbf{J} \) is the diagonal inertia matrix, \( \mathbf{K} \) is the stiffness matrix, and \( \boldsymbol{\theta} = [\theta_s, \theta^e_p, \theta^e_b, \theta^e_c, \theta^e_H]^T \) is the vector of angular displacements. The stiffness matrix \( \mathbf{K} \) is symmetric and includes all equivalent stiffnesses. For computational simplicity, I average time-varying meshing stiffnesses over one cycle. The matrices are:
$$ \mathbf{J} = \begin{bmatrix}
J_s & 0 & 0 & 0 & 0 \\
0 & J^e_p & 0 & 0 & 0 \\
0 & 0 & J^e_b & 0 & 0 \\
0 & 0 & 0 & J^e_c & 0 \\
0 & 0 & 0 & 0 & J^e_H
\end{bmatrix} $$
$$ \mathbf{K} = \begin{bmatrix}
k_I + k^e_{sp} & -k^e_{sp} & 0 & 0 & 0 \\
-k^e_{sp} & k^e_{sp} + k^e_b & -k^e_b & 0 & 0 \\
0 & -k^e_b & k^e_b + k^e_{cb} + k^e_{Hb} & -k^e_{cb} & -k^e_{Hb} \\
0 & 0 & -k^e_{cb} & k^e_{cr} + k^e_{cb} & 0 \\
0 & 0 & -k^e_{Hb} & 0 & k^e_{Hb}
\end{bmatrix} $$
Here, \( k_I \) is the torsional stiffness of the input shaft. The natural frequencies are found by solving the eigenvalue problem \( (\mathbf{K} – \omega_n^2 \mathbf{J}) \boldsymbol{\phi} = \mathbf{0} \), where \( \omega_n \) are the natural frequencies in rad/s, and \( \boldsymbol{\phi} \) are the mode shapes.
To illustrate, I perform an instance calculation for an RV-6A-type rotary vector reducer. Parameters are based on typical values: reduction ratio \( i_{Hs} = 1/103 \), sun gear inertia \( J_s = 105 \times 10^{-7} \, \text{kg} \cdot \text{m}^2 \), and load inertia included in the carrier. The equivalent inertias are computed as shown in Table 1.
| Component | Symbol | Value (kg·m²) |
|---|---|---|
| Sun gear | \( J_s \) | \( 105 \times 10^{-7} \) |
| Planetary gears (equivalent) | \( J^e_p \) | \( 4.44 \times 10^{-7} \) |
| Crankshafts (equivalent) | \( J^e_b \) | \( 1.35 \times 10^{-7} \) |
| Cycloid discs (equivalent) | \( J^e_c \) | \( 0.446 \times 10^{-7} \) |
| Carrier (with load) | \( J^e_H \) | \( 123 \times 10^{-7} \) |
Stiffness values are estimated from literature: sun-planet mesh stiffness \( k_{sp} = 4.08 \times 10^7 \, \text{N/m} \), crankshaft torsional stiffness \( k_b = 7800 \, \text{N} \cdot \text{m/rad} \), cycloid-pin mesh stiffness \( k_{cr} = 2.05 \times 10^5 \, \text{N} \cdot \text{m/rad} \), cycloid disc bearing stiffness \( k_{cb} = 10 \times 10^7 \, \text{N/m} \), carrier bearing stiffness \( k_{Hb} = 20 \times 10^7 \, \text{N/m} \), and input shaft stiffness \( k_I = 6088 \, \text{N} \cdot \text{m/rad} \). The equivalent stiffnesses are summarized in Table 2.
| Stiffness Element | Symbol | Value (N·m/rad) |
|---|---|---|
| Sun-planet mesh | \( k^e_{sp} \) | 1800 |
| Crankshaft torsion | \( k^e_b \) | 1236.6 |
| Cycloid disc bearing | \( k^e_{cb} \) | 13.36 |
| Cycloid-pin mesh | \( k^e_{cr} \) | 46.7038 |
| Carrier bearing | \( k^e_{Hb} \) | 9.5438 |
Solving the eigenvalue problem yields the first natural frequency \( f_1 = \omega_{n1} / (2\pi) = 132.4 \, \text{Hz} \). The corresponding mode shape is \( \boldsymbol{\phi}_1 = [0.0003, 0.0012, 0.0025, 0.1247, 0.9722]^T \), indicating that the carrier has the largest amplitude, which aligns with its high inertia and low bearing stiffness. This suggests that the carrier bearings are critical in governing the first-order torsional vibration.
To identify key influencing factors, I conduct a parametric study by varying individual stiffnesses while keeping others constant. The results are compiled in Table 3, showing the sensitivity of the first natural frequency to changes in input shaft stiffness, crankshaft stiffness, cycloid-pin mesh stiffness, cycloid disc bearing stiffness, and carrier bearing stiffness.
| Stiffness Parameter | Base Value | Varied Value | First Natural Frequency (Hz) | Change (%) |
|---|---|---|---|---|
| Input shaft torsion | 6088 N·m/rad | 10000 N·m/rad | 132.430 | +0.02 |
| 8000 N·m/rad | 132.429 | +0.02 | ||
| 5000 N·m/rad | 132.427 | +0.02 | ||
| Crankshaft torsion | 7800 N·m/rad | 12000 N·m/rad | 132.44 | +0.03 |
| 8000 N·m/rad | 132.43 | +0.02 | ||
| 4000 N·m/rad | 132.40 | -0.03 | ||
| Cycloid-pin mesh | 2.05×10⁵ N·m/rad | 3.0×10⁵ N·m/rad | 134.97 | +1.94 |
| 2.5×10⁵ N·m/rad | 133.86 | +1.10 | ||
| 1.6×10⁵ N·m/rad | 130.72 | -1.27 | ||
| Cycloid disc bearing | 10×10⁷ N/m | 14×10⁷ N/m | 133.14 | +0.56 |
| 12×10⁷ N/m | 132.80 | +0.30 | ||
| 8×10⁷ N/m | 132.03 | -0.28 | ||
| Carrier bearing | 20×10⁷ N/m | 30×10⁷ N/m | 154.51 | +16.70 |
| 20×10⁷ N/m | 134.20 | +1.36 | ||
| 16×10⁷ N/m | 118.30 | -10.65 |
As evident from Table 3, the carrier bearing stiffness has the most pronounced effect on the first natural frequency. A 50% increase from the base value raises the frequency by about 16.7%, while a 20% decrease lowers it by 10.65%. In contrast, changes in input shaft or crankshaft stiffness result in negligible variations (less than 0.1%). This underscores that in rotary vector reducers, the carrier bearings are the weakest link due to high radial loads and relatively low stiffness. The radial load on each carrier bearing can be estimated as \( F_{Hb} = T_o / (n r_b) \), where \( T_o \) is the output torque (e.g., 50 N·m), \( n \) is the number of bearings, and \( r_b \) is the radius. For typical values, \( F_{Hb} \approx 556 \, \text{N} \), leading to significant deflections if bearing stiffness is low (e.g., \( \delta \approx F_{Hb} / k_{Hb} \approx 11 \, \mu \text{m} \)). This deformation translates into torsional motion via the crankshafts, amplifying vibration.
To validate the model, I compare the calculated natural frequency with experimental data. Using an impact hammer test on an RV-6A rotary vector reducer, the first natural frequency was measured at 126.8 Hz. The discrepancy of about 4.2% from the theoretical 132.4 Hz is acceptable, considering simplifications like ignoring damping and assuming average stiffnesses. This confirms that the lumped-parameter model effectively captures the dominant dynamics of the rotary vector reducer. Further refinements could include damping effects or nonlinearities, but for first-order analysis, this model suffices.
Beyond the first mode, higher-order torsional vibrations in rotary vector reducers may involve complex interactions, such as gear mesh harmonics or housing resonances. However, the first mode is often most critical for operational stability. My analysis highlights that designers should prioritize carrier bearing selection to enhance dynamic performance. Increasing bearing stiffness, perhaps by using more bearings or optimized designs, can raise the natural frequency away from excitation ranges. For instance, in robotic applications where start-stop cycles occur at frequencies below 100 Hz, ensuring the first natural frequency above 150 Hz might prevent resonance. Alternatively, adding damping to the carrier assembly could mitigate vibration amplitudes.
In summary, the rotary vector reducer’s torsional vibration is governed by a system of inertias and stiffnesses, with the carrier bearings playing a pivotal role. The lumped-parameter model, despite its simplifications, provides valuable insights and aligns with experimental findings. Future work could explore asymmetric effects or thermal influences, but the core takeaway remains: optimizing bearing stiffness is key to improving the dynamic response of rotary vector reducers. This knowledge aids in the reliable integration of these reducers in high-precision machinery, ensuring longevity and performance.