In modern industrial applications, integrated planetary gear reduction motors have gained prominence due to their compact design, high reliability, and substantial torque output. These systems are widely used in fields such as new energy vehicles, intelligent robotics, and high-end manufacturing equipment. The planetary reducer, as the core component for output speed and torque, significantly influences overall performance through its transmission stability and load distribution uniformity. However, the torsional deformation of double planetary gear shafts under load often leads to uneven load sharing and increased meshing impact, which can compromise system efficiency and longevity. This paper addresses these issues by proposing a topological modification method that explicitly considers the torsional deformation of double planetary gear shafts. We analyze the causes and effects of this deformation, employ multi-body gear load tooth contact analysis (PLTCA) and gear load tooth contact analysis (LTCA) to determine load distribution and transmission error, and utilize optimization algorithms to derive optimal modification parameters. Simulation and experimental results validate the effectiveness of our approach in reducing unit length loads and transmission error amplitudes, thereby enhancing gear performance.
The double planetary gear shaft is a critical element in NGWN-type planetary gear trains, where it connects primary and secondary planetary gears. Under operational loads, these gear shafts experience torsional deformation due to the circumferential forces acting on the gears. This deformation induces misalignment in gear meshing, leading to biased load distribution and increased stress concentrations. The torsional angle of the gear shaft can be expressed as:
$$ \theta_t = \frac{M_t b}{G I} $$
where \( M_t \) is the torque applied to the gear shaft, \( b \) is the gear width, \( G \) is the shear modulus of the shaft material, and \( I \) is the polar moment of inertia of the equivalent shaft diameter. The resulting misalignment in meshing is given by:
$$ b_w = \frac{M_t b d_n}{2 G I} $$
Here, \( d_n \) represents the pitch circle diameter of the planetary gears (e.g., primary and secondary). For instance, in a typical NGWN configuration, the misalignment values for primary and secondary planetary gears might be calculated as -0.913 µm and 1.336 µm, respectively. This misalignment causes the gear teeth to contact偏向ly at one end, resulting in uneven load distribution across the tooth face and exacerbating wear and tear.
To mitigate these issues, we first analyze the meshing state of the gear system using PLTCA and LTCA. The PLTCA model accounts for the coupling between multiple gear pairs in the planetary system. The normal compliance matrix of the tooth surface, incorporating the additional compliance due to gear shaft torsional deformation, is defined as:
$$ \mathbf{F}_k = \mathbf{F}_{ij} + \mathbf{F}_{i’j} $$
where \( \mathbf{F}_{ij} \) is the normal compliance matrix of the tooth surface mesh, and \( \mathbf{F}_{i’j} \) is the additional compliance matrix from the torsional deformation of the planetary gear shaft. The PLTCA model for the planetary gear train is formulated as a set of equations representing deformation compatibility and force equilibrium:
$$ \begin{cases}
\mathbf{F}_k \mathbf{p}_{nk} + \boldsymbol{\omega}_{nk} = \mathbf{Z}_{r} \mathbf{e}_k + \mathbf{d}_{nk} & \text{(for internal meshing)} \\
\mathbf{F}_k \mathbf{p}_{nk} + \boldsymbol{\omega}_{nk} = \mathbf{Z}_{s} \mathbf{e}_k + \mathbf{d}_{nk} & \text{(for external meshing)} \\
\sum \mathbf{p}_{I1} + \sum \mathbf{p}_{II1} + \sum \mathbf{p}_{I2} + \sum \mathbf{p}_{II2} = P & \text{(internal force balance)} \\
\sum \mathbf{p}_{III1} + \sum \mathbf{p}_{IV1} + \sum \mathbf{p}_{III2} + \sum \mathbf{p}_{IV2} = P & \text{(external force balance)} \\
\text{subject to } d_{jk} = 0 \text{ if } p_{jk} > 0, \text{ and } d_{jk} > 0 \text{ if } p_{jk} = 0, \mathbf{Z}_r \geq 0, \mathbf{Z}_s \geq 0
\end{cases} $$
In these equations, \( \boldsymbol{\omega}_{nk} \), \( \mathbf{p}_{nk} \), and \( \mathbf{d}_{nk} \) represent the initial gap, normal load, and deformed tooth surface gap at discrete points along the contact ellipse, respectively. \( \mathbf{Z}_r \) and \( \mathbf{Z}_s \) denote the normal displacements for internal and external meshing pairs. The load distribution coefficient for planetary gear \( m \) on tooth pair \( k \) is calculated as:
$$ L_{k1} = \frac{\sum_{j=1}^{N_k} p_{jk}}{P} $$
where \( N_k \) is the number of discrete points on the contact line. The loaded transmission error is given by:
$$ T_{e1} = \frac{3600 \times 180 Z_l}{\pi r_b} $$
with \( r_b \) being the base circle radius of the driven gear. For the secondary planetary gear train, which involves only internal meshing, the LTCA model is used:
$$ \begin{cases}
\min \sum_{i=1}^{2n+1} X_i – \mathbf{F}_k \mathbf{p}_k + Z_k \mathbf{E}_k + \mathbf{d}_k + \mathbf{X} = \boldsymbol{\omega} \\
P = \mathbf{e}^T \mathbf{p} + X_{2n+1} \\
\text{subject to } d_{ik} = 0 \text{ if } p_{jk} > 0, \text{ and } d_{ik} > 0 \text{ if } p_{jk} = 0, Z \geq 0, X_j \geq 0
\end{cases} $$
Here, \( \mathbf{p}_k \) and \( \mathbf{d}_k \) are the normal load and deformed gap vectors, \( Z_k \) is the normal displacement, and \( \mathbf{X} \) is a vector of custom variables. The load distribution coefficient and transmission error for the secondary system are derived similarly.
Based on the meshing state analysis, we propose a topological modification method that includes both profile and lead modifications. The profile modification curve combines two parabolic segments and one linear segment, as illustrated in the design. The modification parameters include amounts \( y_1 \) and \( y_2 \), and lengths \( y_3 \) and \( y_4 \), with a radial constant \( H \). The ranges for these parameters are determined empirically, as summarized in Table 1.
| Parameter | Lower Bound | Upper Bound |
|---|---|---|
| \( y_1 \) | 0.005 mm | 0.01 mm |
| \( y_2 \) | 0.005 mm | 0.015 mm |
| \( y_3 \) | 0.1 mm | 0.6 mm |
| \( y_4 \) | 0.15 mm | 0.8 mm |
For lead modification, a barrel-shaped profile is adopted to compensate for the torsional deformation of the gear shaft. The maximum torsional deformation量 is calculated as:
$$ \delta_t = \frac{4 \phi_d^2 K_i W_t}{\pi G} $$
where \( \phi_d \) is the width-to-diameter ratio, \( K_i \) is the gear bore influence coefficient, and \( W_t \) is the load per unit tooth width. This deformation informs the drum amount \( y_5 \), with optimization ranges provided in Table 2.
| Parameter | Lower Bound | Upper Bound |
|---|---|---|
| \( y_5 \) | 0.001 mm | 0.05 mm |
To optimize the modification parameters, we define an objective function that minimizes the load distribution non-uniformity and transmission error amplitude. The function is formulated as:
$$ F(y_1, y_2, y_3, y_4, y_5) = \min \left\{ \lambda_1 f_1 + \lambda_2 f_2 + \lambda_3 f_3 + \lambda_4 f_4 \right\} $$
where \( f_1 = L_{k1} / L_{k’1} \), \( f_2 = L_{k2} / L_{k’2} \), \( f_3 = \Delta T_{e1} / \Delta T_{e’1} \), and \( f_4 = \Delta T_{e2} / \Delta T_{e’2} \), with \( \lambda_i \) as weighting coefficients. The parameters are constrained within their respective bounds. Using an intelligent optimization algorithm (e.g., genetic algorithm) with a population size of 50, iteration count of 20, and mutation rate of 0.03, we obtain the optimal modification parameters for the left tooth surfaces of the primary and secondary planetary gears, as listed in Table 3.
| Parameter | Primary Planetary Gear | Secondary Planetary Gear |
|---|---|---|
| \( y_1 \) | 0.0053 mm | 0.0084 mm |
| \( y_2 \) | 0.007 mm | 0.011 mm |
| \( y_3 \) | 0.15 mm | 0.2 mm |
| \( y_4 \) | 0.2 mm | 0.423 mm |
| \( y_5 \) | 0.0015 mm | 0.0035 mm |
We conduct simulation analyses to evaluate the performance of the modified gear system. The unit length load on the tooth surface, which reflects load magnitude and distribution uniformity, is compared before and after modification. For the primary planetary gear, the unit length load decreases from 10.81 N/mm to 7.18 N/mm, a reduction of 33.58%. For the secondary planetary gear, it drops from 43.1 N/mm to 33.9 N/mm, a 21.35% improvement. The load distribution becomes more uniform, with no stress concentrations or edge contacts, as evidenced by the cloud diagrams of unit length load.
The loaded transmission error is another critical indicator of vibrational performance. Before modification, the transmission error amplitude is 0.83102 µm, with maximum and minimum values of 1.76 µm and 0.92992 µm, respectively. After modification, the amplitude reduces to 0.185 µm, a 77.74% decrease, although the symmetry of the error curve is slightly affected due to the optimization trade-offs. This reduction signifies enhanced transmission stability and reduced noise.
Experimental validation is performed on a reducer load test bench over 7,200 hours of operation. The three-dimensional topography of the tooth surfaces is measured using a VR-3000 micro-topography instrument. Prior to modification, the primary planetary gear exhibits severe pitting on the right end and edge contact at the tooth tip, while the secondary gear shows overload spalling that decreases diagonally from left to right. After modification, the tooth surfaces display even wear patterns centered along the face width, confirming the simulation results. The elimination of biased loading and improved contact distribution demonstrate the efficacy of our topological modification method.
In conclusion, our study presents a comprehensive approach to addressing torsional deformation in double planetary gear shafts through topological modification. By integrating PLTCA and LTCA analyses with optimization algorithms, we derive optimal modification parameters that significantly reduce unit length loads and transmission error amplitudes. The method effectively resolves load distribution issues in NGWN-type planetary gear trains, enhancing transmission accuracy and service life. This approach can be extended to other planetary gear systems with similar transmission principles, offering broad applicability in industrial contexts.
The torsional behavior of the gear shaft is a recurring theme throughout this work, underscoring the importance of considering shaft dynamics in gear design. Future research could explore the effects of other deformation modes, such as bending or thermal expansion, on gear performance. Additionally, real-time monitoring of gear shaft conditions could further optimize modification strategies in operational environments.