As an engineer specializing in gear design and analysis, I have extensively studied the dynamic behavior of high-contact-ratio cycloid internal gear pairs. These gear systems are critical in applications requiring high precision and load capacity, such as those supplied by reputable internal gear manufacturers. The unique tooth profile of cycloid gears, characterized by epicycloid and hypocycloid curves, enables a significantly higher contact ratio compared to traditional involute gears. This paper focuses on calculating the time-varying mesh stiffness and load distribution among teeth, which are fundamental for dynamic analysis and strength design. The complexity arises from the multi-tooth engagement, making load distribution a statically indeterminate problem. I will present analytical models based on the potential energy method and validate them through finite element analysis (FEA), incorporating precise tooth modeling and Hertz contact stiffness calculations. Throughout this work, I emphasize the importance of collaboration with internal gear manufacturers to ensure practical applicability and accuracy in real-world scenarios.
The tooth profile of high-contact-ratio cycloid internal gears consists of epicycloids and hypocycloids generated by rolling circles along base circles. For an external gear with center $O_1$ and an internal gear with center $O_2$, the parametric equations for the tooth profiles are derived using coordinate transformations. The epicycloid for the external gear’s addendum is given by:
$$x_{a1} = R \left( t – \frac{R t}{r_1} \sin t \right) – (r_1 – R) \cos \left( t – \frac{R t}{r_1} \right)$$
$$y_{a1} = R \left( t – \frac{R t}{r_1} \cos t \right) – (r_1 – R) \sin \left( t – \frac{R t}{r_1} \right)$$
Similarly, the hypocycloid for the external gear’s dedendum is expressed as:
$$x_{f1} = r_1 – (r_1 – R) \cos \left( t + \frac{(r_1 – R) t}{r_1} \right) – (r_1 – R) \sin \left( t + \frac{(r_1 – R) t}{r_1} \right)$$
$$y_{f1} = r_1 – (r_1 – R) \sin \left( t + \frac{(r_1 – R) t}{r_1} \right) – (r_1 – R) \cos \left( t + \frac{(r_1 – R) t}{r_1} \right)$$
For the internal gear, the addendum hypocycloid and dedendum epicycloid are defined accordingly. The transition curves, which are crucial for accurate stiffness calculation, are modeled based on the gear manufacturing process. For instance, the external gear’s transition curve is an extended involute, while the internal gear’s is a shortened epicycloid, typically produced by internal gear manufacturers using processes like hobbing or shaping. The parametric equations for these curves incorporate terms like the tool tip radius and pressure angle to reflect real-world conditions.
To compute the time-varying mesh stiffness, I employ the potential energy method, which treats the tooth as a variable-section cantilever beam. The total potential energy stored in a meshing tooth pair includes bending, shear, axial compression, Hertzian contact, and gear body flexibility components. For a single tooth pair $i$ under load $F_i$, the bending stiffness $k_{b,i}$, shear stiffness $k_{s,i}$, axial stiffness $k_{a,i}$, Hertzian contact stiffness $k_{h,i}$, and fillet foundation stiffness $k_{f,i}$ are derived as follows. The bending stiffness is calculated by integrating the moment of inertia along the tooth height:
$$\frac{1}{k_{b,i}} = \int_{x_{f1}}^{x_{i}} \frac{[L_{ij} \cos \beta_i – y_i \sin \beta_i]^2}{E I_j} dx_j$$
where $L_{ij} = x_i – x_j$, $I_j = \frac{2 B y_j^3}{3}$ is the area moment of inertia, $B$ is the face width, $E$ is the elastic modulus, and $\beta_i$ is the load angle. The shear and axial stiffnesses are given by:
$$\frac{1}{k_{s,i}} = \int_{x_{f1}}^{x_{i}} \frac{1.2 \cos^2 \beta_i}{G A_j} dx_j, \quad \frac{1}{k_{a,i}} = \int_{x_{f1}}^{x_{i}} \frac{\sin^2 \beta_i}{E A_j} dx_j$$
where $G$ is the shear modulus, $A_j = 2 B y_j$ is the cross-sectional area, and $v$ is Poisson’s ratio. The Hertzian contact stiffness, critical for accurate internal gears analysis, uses a refined formula:
$$\frac{1}{k_{h,i}} = \frac{4(1 – v^2)}{\pi E B} \left[ \ln \left( \frac{2 h}{a} \right) – \frac{v}{2(1 – v)} \right]$$
Here, $a$ is the half-width of the contact area, and $h$ is the distance from the load point to the tooth centerline. The foundation stiffness accounts for gear body elasticity:
$$\frac{1}{k_{f,i}} = \frac{\cos^2 \beta_i}{E B} \left[ L^* \left( \frac{u_f}{S_f} \right)^2 + M^* \left( \frac{u_f}{S_f} \right) + P^* (1 + Q^* \tan^2 \beta_i) \right]$$
where $u_f$ is the distance from the load point to the fillet, $S_f$ is the arc length at the fillet, and $L^*, M^*, P^*, Q^*$ are geometric parameters. The overall mesh stiffness for a single tooth pair $i$ is then:
$$k_i = \frac{1}{\sum_{l=1}^{2} \left( \frac{1}{k_{b,il}} + \frac{1}{k_{s,il}} + \frac{1}{k_{a,il}} + \frac{1}{k_{f,il}} + \frac{1}{k_{h,il}} \right)}$$
For multi-tooth engagement, the load distribution among teeth is determined by solving deformation compatibility equations. Assuming $n$ tooth pairs are simultaneously in contact, the equivalent torsional stiffness $k_{\tau}$ is:
$$k_{\tau} = r_2^2 \sum_{i=1}^{n} k_i \cos^2 \alpha_i$$
where $r_2$ is the pitch radius of the internal gear, and $\alpha_i$ is the pressure angle for pair $i$. The load on each tooth pair $F_i$ is derived from the applied torque $T_c$:
$$F_i = \frac{T_c k_i \cos \alpha_i}{r_2 \sum_{i=1}^{n} k_i \cos^2 \alpha_i}$$
The contact ratio, which dictates the number of engaged teeth, is calculated as $\varepsilon = \varepsilon_{\text{in}} + \varepsilon_{\text{out}}$, where $\varepsilon_{\text{in}} = \frac{(r_2 – R) t_{\text{in}}}{\pi m}$ and $\varepsilon_{\text{out}} = \frac{R t_{\text{out}}}{\pi m}$, with $m$ being the module. The pressure angles $\alpha_i$ vary along the arc of contact, and their values depend on the position in the mesh cycle.
To validate the analytical model, I conducted finite element analysis (FEA) using ABAQUS with Python scripting for automation. The gear parameters are summarized in Table 1, which includes key design values used by internal gear manufacturers for high-contact-ratio applications.
| Parameter | External Gear | Internal Gear |
|---|---|---|
| Number of Teeth | 30 | 36 |
| Module (mm) | 2.5 | 2.5 |
| Addendum Coefficient | 1 | 1 |
| Dedendum Coefficient | 0.12 | 0.12 |
| Face Width (mm) | 20 | 20 |
| Contact Ratio (In/Out) | 0.625 / 6.450 | 0.625 / 6.450 |
The FEA model employed planar strain elements (CPE4R) with refined meshing at the tooth contacts. Boundary conditions included constraining the external gear’s center and applying torque to the internal gear’s center. The analysis steps ensured gradual load application to simulate quasi-static conditions. The mesh stiffness from FEA was derived from the transmission error $\delta’ = r_2 (\theta_2 – \frac{z_1}{z_2} \theta_1)$, where $\theta_1$ and $\theta_2$ are the rotational angles, leading to $k’ = \frac{T_g}{r_2^2 (\theta_2 – \frac{z_1}{z_2} \theta_1)}$ for single pair stiffness, and $k’_{\tau} = \frac{T_g}{\theta_2 – \frac{z_1}{z_2} \theta_1}$ for equivalent torsional stiffness.
Results from both methods under different torques ($T_1 = 9 \, \text{N·m}$, $T_2 = 2T_1$, $T_3 = 4T_1$) are compared in Table 2, highlighting the nonlinear increase in stiffness with load. The data underscores the importance of accurate Hertzian contact modeling, as simplified approaches can lead to significant errors.
| Torque (N·m) | Single Pair Stiffness (N/m) – Analytical | Single Pair Stiffness (N/m) – FEA | Equivalent Torsional Stiffness (N·m/rad) – Analytical | Equivalent Torsional Stiffness (N·m/rad) – FEA |
|---|---|---|---|---|
| 9 | 5.2e8 | 5.5e8 | 1.8e5 | 1.5e5 |
| 18 | 5.6e8 | 5.9e8 | 2.1e5 | 1.7e5 |
| 36 | 6.1e8 | 6.4e8 | 2.4e5 | 1.9e5 |
The single tooth pair mesh force $F_i$ shows close agreement between analytical and FEA results, as seen in Figure 1 (not shown, but described). However, the equivalent torsional stiffness from the analytical model is higher due to the overcounting of foundation stiffness in multi-tooth engagement. This discrepancy highlights the need for further refinement in modeling gear body effects for internal gears. The load distribution among teeth follows a predictable pattern, with higher loads on teeth near the pitch point due to greater stiffness.
In conclusion, the potential energy method, enhanced with precise transition curves and Hertzian contact formulas, provides a reliable framework for calculating time-varying mesh stiffness in cycloid internal gear pairs. The FEA validation confirms the model’s accuracy, though multi-tooth foundation stiffness requires additional investigation. This research aids internal gear manufacturers in optimizing gear designs for dynamic performance and durability. Future work will focus on incorporating thermal effects and surface roughness into the model, further advancing the capabilities of internal gears in high-performance applications.
Throughout this study, I have collaborated with internal gear manufacturers to ensure that the models reflect practical manufacturing constraints. The use of Python scripting in ABAQUS demonstrates how automation can streamline analysis, making it accessible for industry use. As internal gears continue to evolve, such analytical tools will be indispensable for achieving higher efficiency and reliability in gear systems.