In modern mechanical engineering, gears serve as fundamental transmission components, with cycloid gears gaining renewed interest due to advancements in manufacturing technologies. As an internal gear manufacturer, I recognize that precise root fillet curves are critical for evaluating bending strength in gear systems. Unlike external gears, internal gears, particularly cycloid types, present unique challenges in machining and strength analysis. Internal gears are commonly produced using gear shaping tools via the generating method, which relies on the principle of conjugate tooth profiles as envelopes during meshing. This paper explores the root fillet curves of internal cycloid gear pairs, focusing on parametric equations derived from gear meshing theory. By analyzing common machining techniques and tool profiles, I establish accurate models for transition curves, which are essential for reliable bending strength calculations in applications involving internal gears.
The root fillet curve significantly influences stress concentration and fatigue life in gears. Various machining methods yield different transition curve forms, such as extended epicycloids or their equidistant curves. For internal gears, gear shaping is predominant, and the fillet curve typically consists of shortened hypocycloid equidistant curves or segments with root arcs. As an internal gear manufacturer, I emphasize that the choice of tool—whether rack-type or gear-type with single or double rounded tips—directly affects the fillet geometry. In this study, I select representative machining schemes: rack-type tools for external cycloid gears and gear-type tools for internal cycloid gears, both with double-rounded tips to generalize the analysis. The objective is to derive parametric equations for these curves and validate them through computational examples, thereby enhancing the design accuracy for internal gears.
Forms of Root Fillet Curves and Tool Profiles
Root fillet curves vary based on the machining method and tool geometry. Common types include equidistant curves of extended involutes for rack-type tools and extended epicycloids for gear-type tools. For internal gears, gear shaping produces transition curves as equidistant curves of shortened hypocycloids, often with intermediate root arcs. The generating method, which simulates gear meshing, is preferred for its precision over forming methods like milling. I focus on double-rounded tip tools to cover a broad range of scenarios, as single-rounded tips are a special case. Below, I summarize the transition curve forms in a table to clarify the relationships.
| Machining Method | Tool Type | Tip Geometry | Transition Curve Form |
|---|---|---|---|
| Generating (Rack-type) | Rack | Single Round | Equidistant curve of extended involute |
| Generating (Rack-type) | Rack | Double Round | Equidistant curve of extended involute with root arc |
| Generating (Gear-type) | Gear | Single Round | Equidistant curve of extended epicycloid |
| Generating (Gear-type) | Gear | Double Round | Equidistant curve of extended epicycloid with root arc |
| Forming | Milling Cutter | N/A | Single circular arc |
For internal gears, gear-type tools are standard, and the transition curve is an equidistant curve of a shortened hypocycloid. As an internal gear manufacturer, I model the tool profiles mathematically to derive the fillet curves. The rack-type tool for external gears and gear-type tool for internal gears are detailed in the following sections.
Rack-Type Tool Profile Parameters
The rack-type tool with double-rounded tips is used for generating external cycloid gears. Its tooth profile includes a cycloidal segment at the tip and rounded corners. Let me define the key parameters: the distance \( h \) from the round center \( C_\rho \) to the pitch line, the distance \( l \) from \( C_\rho \) to the tooth space centerline, the radius \( r_\rho \) of the round, the addendum coefficient \( h_a^* \), the clearance coefficient \( c^* \), and the module \( m \). The relationships are given by:
$$ h = h_a^* m + c^* m – r_\rho $$
$$ l = \frac{\pi m}{4} + x_{c0} + r_\rho \cos \alpha_\rho $$
$$ r_\rho = \frac{c^* m}{1 – \sin \alpha_\rho} $$
Here, \( x_{c0} \) is the x-coordinate of the point where the cycloidal segment meets the round, and \( \alpha_\rho \) is the angle between the normal at this point and the pitch line. The parametric equations for the cycloidal segment of the rack-type tool are:
$$ x_c = r_{ca} (t – \sin t) $$
$$ y_c = -r_{ca} (1 – \cos t) $$
where \( r_{ca} \) is the rolling circle radius and \( t \) is the parameter. At the point of tangency \( A \) with coordinates \( (x_{c0}, y_{c0}) \), we have \( |y_{c0}| = h_a^* m \), leading to:
$$ t_0 = \arccos \left( \frac{r_{ca} – h_a^* m}{r_{ca}} \right) $$
Thus, \( x_{c0} = r_{ca} (t_0 – \sin t_0) \), and the normal angle is \( \alpha_\rho = \frac{t_0}{2} \). These equations form the basis for the rack-type tool profile used in generating external gears, which indirectly influences internal gears when considering full gear pairs.
Gear-Type Tool Profile Parameters
For internal gears, a gear-type tool with double-rounded tips is employed in shaping processes. The tool profile features an epicycloidal segment at the tip. The parametric equations for this epicycloid are:
$$ x’_c = -r_{ca} \sin \left( \frac{r_{ca} t}{r_c} + t \right) + (r_{ca} + r_c) \sin \left( \frac{r_{ca} t}{r_c} \right) $$
$$ y’_c = r_{ca} \cos \left( \frac{r_{ca} t}{r_c} + t \right) – (r_{ca} + r_c) \cos \left( \frac{r_{ca} t}{r_c} \right) $$
where \( r_c \) is the tool pitch radius, \( r_{ca} \) is the rolling circle radius, and \( t \) is the parameter. The point of tangency \( T \) between the epicycloid and the round has coordinates \( (x’_{c0}, y’_{c0}) \), and since \( |O_c T| = r_c + h_a^* m \), we derive:
$$ t’_0 = \arccos \left( \frac{(r_{ca} + r_c)^2 + r_{ca}^2 – (r_c + h_a^* m)^2}{2 r_{ca} (r_{ca} + r_c)} \right) $$
The normal angle at this point is \( \alpha’_\rho = \frac{r_{ca} t’_0}{r_c} + \frac{t’_0}{2} \). Let \( (x_{C_\rho}, y_{C_\rho}) \) be the coordinates of the round center \( C_\rho \), and \( r’_\rho \) be the round radius. Then:
$$ r’_\rho = |T C_\rho| $$
$$ |O_c C_\rho| = r_c + h_a^* m + c^* m – r’_\rho $$
Using the slope condition from the normal:
$$ \tan \alpha’_\rho = \frac{y_{C_\rho} – y’_{c0}}{x_{C_\rho} – x’_{c0}} $$
Solving these equations yields \( (x_{C_\rho}, y_{C_\rho}) \) and \( r’_\rho \). The angle \( \beta’ \) is calculated as:
$$ \beta’ = \frac{\pi}{2 z_c} + \angle T O_c C_\rho $$
where \( z_c \) is the number of teeth on the tool, and \( \angle T O_c C_\rho \) is derived from the cosine rule in triangle \( \Delta T O_c C_\rho \). This comprehensive model allows internal gear manufacturers to accurately describe the tool profile for generating internal gears.
Parametric Equations for Root Fillet Curves in Internal Cycloid Gear Pairs
Based on gear meshing theory, I derive the parametric equations for the root fillet curves when using rack-type tools for external gears and gear-type tools for internal gears. The generating method involves relative rolling between the tool and gear pitch circles, with the fillet curve formed as an envelope of the tool profile.
Fillet Curve for External Gears with Rack-Type Tool
When a rack-type tool generates an external gear, the pitch line of the tool rolls without slipping on the pitch circle of the gear. The transition curve is an equidistant curve of an extended involute. In the coordinate system \( Oxy \) with origin at the gear center, the parametric equations are:
$$ x_1 = r_1 \sin \phi_1 – \left( \frac{h}{\sin \alpha’} + r_\rho \right) \cos (\alpha’ – \phi_1) $$
$$ y_1 = r_1 \cos \phi_1 – \left( \frac{h}{\sin \alpha’} + r_\rho \right) \sin (\alpha’ – \phi_1) $$
Here, \( r_1 \) is the pitch radius of the external gear, \( \phi_1 \) is the rotation angle, and \( \alpha’ \) is the parameter representing the angle between the common normal and the pitch line, with \( \alpha’ \in [\alpha_\rho, \frac{\pi}{2}] \). The relationship between \( \alpha’ \) and \( \phi_1 \) is:
$$ r_1 \phi_1 = h \tan \alpha’ + l $$
By substituting tool parameters, I can compute coordinates for points on the transition curve. This approach is vital for internal gear manufacturers when analyzing meshing pairs involving external gears.
Fillet Curve for Internal Gears with Gear-Type Tool
For internal gears generated with a gear-type tool, the tool pitch circle rolls inside the gear pitch circle. The transition curve is an equidistant curve of a shortened hypocycloid. In the coordinate system \( Oxy \) centered on the internal gear, the parametric equations are:
$$ x_2 = r_2 \sin \phi_2 – \left[ \frac{r_c \sin \beta}{\cos (\alpha’ + \beta)} + r’_\rho \right] \cos (\alpha’ + \phi_2) $$
$$ y_2 = r_2 \cos \phi_2 + \left[ \frac{r_c \sin \beta}{\cos (\alpha’ + \beta)} + r’_\rho \right] \sin (\alpha’ + \phi_2) $$
where \( r_2 \) is the pitch radius of the internal gear, \( \phi_2 \) is the rotation angle, \( \alpha’ \) is the parameter (angle between common normal and common tangent), with \( \alpha’ \in [\alpha_\rho, \frac{\pi}{2}] \), and \( \beta \) is an auxiliary angle. The angle \( \phi_2 \) relates to the tool rotation \( \phi_c \) by \( \phi_2 = \frac{z_c}{z_2} \phi_c \), where \( z_2 \) is the number of teeth on the internal gear, and \( \phi_c = \beta’ + \beta \). The lower limit \( \alpha_\rho \) is the engagement angle at the tangency point, given by \( \alpha_\rho = \frac{\pi}{2 z_c} + \frac{t_0}{2} – \frac{\pi r_2}{2 r_c z_2} \). When \( \alpha’ = \frac{\pi}{2} \) and \( \beta = 0 \), the curve transitions smoothly to the root circle at point \( (r_{f2} \sin \phi, r_{f2} \cos \phi) \), where \( r_{f2} \) is the root radius. These equations enable precise modeling of fillet curves for internal gears.
Computational Example and Results
To validate the derived equations, I consider an internal cycloid gear pair with parameters listed in the table below. As an internal gear manufacturer, I use these values to compute the root fillet curves and full tooth profiles for both internal and external gears.
| Parameter | Value |
|---|---|
| Number of teeth on external gear, \( z_1 \) | 44 |
| Number of teeth on internal gear, \( z_2 \) | 51 |
| Module, \( m \) (mm) | 3.25 |
| Rolling circle radius, \( r_{ca} \) (mm) | 5.5 |
| Addendum coefficient, \( h_a^* \) | 1 |
| Clearance coefficient, \( c^* \) | 0.2 |
Using MATLAB, I implement the parametric equations to plot the root fillet curves and complete tooth profiles. For the external gear, the transition curve blends smoothly with the root cycloid, while for the internal gear, the fillet curve integrates with the root epicycloid. The results confirm the accuracy of the equations, as the curves exhibit continuous transitions without discontinuities. This computational approach is essential for internal gear manufacturers to optimize gear design and ensure high bending strength.
Discussion on Implications for Gear Design
The accurate modeling of root fillet curves has profound implications for gear performance, particularly in bending strength assessment. Stress concentrations often occur at the fillet region, and a larger transition radius can reduce peak stresses, enhancing fatigue life. For internal gears, the generating method with gear-type tools provides control over the fillet geometry, allowing internal gear manufacturers to tailor curves for specific loads. The parametric equations derived here facilitate finite element analysis and optimization algorithms. Moreover, the use of double-rounded tip tools offers flexibility; for instance, adjusting \( r_\rho \) or \( \alpha_\rho \) can modify the curve to mitigate stress. In high-load applications, such as those involving internal gears in planetary systems, precise fillet curves are crucial for reliability. This study underscores the importance of integrating manufacturing considerations into design phases, enabling better predictions of gear behavior under operational conditions.
Conclusion
In this analysis, I have examined the root fillet curves of internal cycloid gear pairs generated via the generating method. By selecting representative machining schemes and modeling tool profiles, I derived parametric equations for the transition curves using gear meshing theory. The computational examples validate the equations, showing smooth transitions between fillet curves and root profiles. This work highlights that the machining method and tool geometry directly influence the fillet shape, and accurate modeling is essential for bending strength evaluation. For internal gear manufacturers, these findings provide a foundation for designing robust internal gears with optimized transition curves, ultimately improving performance in demanding mechanical systems. Future work could explore dynamic loading effects and material variations on fillet stress distributions.