In modern mechanical transmission systems, spur gears play a critical role in parallel-axis power transmission due to their simplicity and efficiency. However, traditional involute spur gears exhibit inherent sliding between tooth surfaces, leading to friction, wear, and noise. To address these issues, we propose a novel design for pure rolling spur gears with a circular arc tooth profile in the normal section. This design aims to eliminate relative sliding at the meshing point, thereby reducing energy losses and improving durability. Our approach is based on actively designing the meshing line equation, ensuring pure rolling contact throughout the engagement process. This paper details the geometric design, mathematical modeling, and tooth contact analysis (TCA) of these gears, emphasizing their performance under ideal and misaligned conditions.
The fundamental concept involves deriving the transverse active tooth profiles from a circular arc defined in the normal section. The entire tooth profile combines this active segment with a root fillet constructed using a Hermite curve, ensuring smooth continuity at the junction point. The three-dimensional tooth surfaces are generated by sweeping the transverse profiles along the pure rolling contact curves on both the pinion and gear. We establish coordinate systems to describe the meshing kinematics and derive parametric equations for the tooth surfaces. Key design parameters, such as module, pressure angle, and spiral angle, are defined to facilitate the construction of the gear model. The mathematical formulation includes transformations between coordinate systems and the application of the meshing line function to achieve pure rolling motion.
To define the meshing line, we consider a fixed coordinate system where the meshing point moves uniformly along the z-axis. The motion of the meshing point is described by the parameter equation: $$ x_k = 0, \quad y_k = 0, \quad z_k = c_1 t, $$ where $c_1$ is a coefficient and $t$ is the motion parameter. The rotation angles of the pinion and gear are related by: $$ \phi_1 = k_\phi t, \quad \phi_2 = \frac{\phi_1}{i_{12}}, $$ with $k_\phi$ as the rotational coefficient and $i_{12}$ as the transmission ratio. The circular arc tooth profile in the normal section is defined in local coordinates, with radii $\rho_a$ and $\rho_b$ for the pinion and gear, respectively. The profile equations are: $$ \mathbf{r}_{a} = \begin{bmatrix} \rho_a \sin \xi_a \\ \rho_a – \rho_a \cos \xi_a \\ 0 \end{bmatrix}, \quad \mathbf{r}_{b} = \begin{bmatrix} \rho_b \sin \xi_b \\ \rho_b – \rho_b \cos \xi_b \\ 0 \end{bmatrix}, $$ where $\xi_a$ and $\xi_b$ are angular parameters.
Coordinate transformations are essential to relate the normal section profiles to the transverse planes. The transformation matrix from the normal coordinate system $S_a$ to the fixed system $S_k$ is: $$ \mathbf{M}_{ka} = \begin{bmatrix} \cos \alpha_n & \sin \alpha_n & 0 & 0 \\ -\sin \alpha_n \cos \beta & \cos \alpha_n \cos \beta & \sin \beta & 0 \\ \sin \alpha_n \sin \beta & -\cos \alpha_n \sin \beta & \cos \beta & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}, $$ where $\alpha_n$ is the normal pressure angle and $\beta$ is the spiral angle. For the gear, the transformation from $S_b$ to $S_k$ is: $$ \mathbf{M}_{kb} = \mathbf{M}_{ka} \mathbf{M}_{ab}, $$ with $\mathbf{M}_{ab}$ being the identity matrix with negative signs for x and y axes. The tooth surfaces are then generated by sweeping the transverse profiles along the contact lines, resulting in parametric equations for the pinion and gear surfaces.
The transverse tooth profiles are derived by setting the z-coordinate to zero in the surface equations. This yields the relationship between the angular parameters and the motion parameter: $$ \sin \alpha_n \sin \beta \, x_a – \cos \alpha_n \sin \beta \, y_a + c_1 t = 0, $$ for the pinion, and a similar equation for the gear. The entire transverse profile includes the active segment and the Hermite fillet curve. The Hermite curve connects the active profile at point $P_S$ to the root circle at point $P_E$, with tangent weights $T_{Hp}$ and $T_{Hg}$ controlling the curvature. The parametric equations for the Hermite curve are based on cubic interpolation, ensuring first-order continuity at the junction.
Key design parameters for spur gears are summarized in the table below. These parameters are used to compute geometric dimensions such as pitch radius, addendum, and dedendum.
| Parameter | Symbol | Value |
|---|---|---|
| Number of teeth (pinion) | $z_1$ | 30 |
| Transmission ratio | $i_{12}$ | 2.0 |
| Normal module | $m_n$ | 2.0 mm |
| Normal pressure angle | $\alpha_n$ | 20° |
| Spiral angle | $\beta$ | 20° |
| Addendum coefficient | $h_{an}^*$ | 0.8 |
| Dedendum coefficient | $c_n^*$ | 0.3 |
| Face width | $b$ | 30 mm |
The geometric dimensions are calculated using the following formulas: $$ m_t = \frac{m_n}{\cos \beta}, \quad \alpha_t = \arctan \left( \frac{\tan \alpha_n}{\cos \beta} \right), \quad R_i = z_i m_t, $$ where $m_t$ is the transverse module, $\alpha_t$ is the transverse pressure angle, and $R_i$ is the pitch radius for gear $i$. The addendum and dedendum are: $$ h_{ai} = h_{an}^* m_n, \quad h_{fi} = (h_{an}^* + c_n^*) m_n. $$ The radii for the addendum, root, and Hermite start points are: $$ R_{ai} = R_i + h_{ai}, \quad R_{fi} = R_i – h_{fi}, \quad R_{Hi} = R_i – h_{ai}. $$ The angular parameters for tooth thickness and fillet placement are: $$ \lambda_i = \frac{\pi}{z_i}, \quad \eta_i = 0.1 \left( \frac{2\pi}{z_i} \right), \quad \delta_{li} = \frac{2\pi}{5z_i}. $$
For tooth contact analysis, we employ an algorithm that solves for the contact patterns and unloaded transmission error. The analysis considers multiple contact positions as the pinion rotates through two pitch angles, divided into 21 steps. Under ideal alignment conditions, the contact patterns for unmodified spur gears consist of ellipses centered on the pitch line, confirming pure rolling motion. However, edge contact occurs at the tooth ends. To mitigate this, we apply a parabolic lead crowning modification to the pinion tooth surface with a magnitude of 8 μm. This modification results in contact ellipses that vary in size along the tooth width, maintaining the pure rolling characteristic while eliminating edge contact.
The unloaded transmission error for unmodified spur gears is zero, represented by a horizontal line. With lead crowning, the transmission error curve becomes parabolic, with a peak-to-peak value of 10 arcseconds. This parabolic shape helps absorb vibrations caused by misalignments. The table below summarizes the TCA results under different conditions.
| Condition | Contact Pattern | Transmission Error |
|---|---|---|
| Ideal alignment, unmodified | Ellipses on pitch line, edge contact | 0 arcseconds |
| Ideal alignment, modified | Ellipses varying in size, no edge contact | 10 arcseconds (peak-to-peak) |
| Center distance error ($\Delta C = 1$ mm) | Ellipses shifted toward tip | Similar to ideal |
| Crossed axis error ($\Delta V = 0.05°$) | Ellipses shifted to one end | Parabolic, increased magnitude |
| Intersecting axis error ($\Delta H = 0.05°$) | Ellipses shifted significantly | Parabolic, highest magnitude |
Misalignment errors include center distance error $\Delta C$, crossed axis error $\Delta V$, and intersecting axis error $\Delta H$. Under center distance error, the contact patterns shift toward the tooth tip without changing shape, risking tip edge contact. For crossed axis error, the patterns shift laterally, causing partial loading and stress concentration. Intersecting axis error results in the most severe shift, leading to pronounced edge contact and higher transmission error. The transmission error curves for these cases remain parabolic but with increased peak-to-peak values, indicating higher sensitivity to axis errors than center distance variations.
The mathematical model for TCA involves solving the equation of meshing between the pinion and gear surfaces. The position vectors and normal vectors are derived for both gears, and the contact condition is expressed as: $$ \mathbf{r}_1(u_1, v_1) = \mathbf{r}_2(u_2, v_2), \quad \mathbf{n}_1(u_1, v_1) = \mathbf{n}_2(u_2, v_2), $$ where $u_i$ and $v_i$ are surface parameters. The transmission error is computed as the difference between the actual and theoretical rotation angles: $$ \Delta \phi = \phi_2 – \frac{\phi_1}{i_{12}}. $$ For modified spur gears, the surface equation includes a crowning term, such as: $$ z_{\text{modified}} = z + k_c (y – y_0)^2, $$ where $k_c$ is the crowning coefficient and $y_0$ is the reference point.
In conclusion, our design of pure rolling spur gears with circular arc tooth profile successfully achieves pure rolling contact, as verified by TCA. The lead crowning modification enables parabolic transmission error, which enhances noise and vibration resistance. These spur gears exhibit center distance insensitivity but are highly sensitive to axis misalignments. Future work will focus on stress analysis and loaded transmission error evaluation to further validate the design for practical applications. The use of spur gears in this context highlights their adaptability to advanced geometric modifications for improved performance.
The derivation of the tooth surface equations begins with the meshing line function. For pure rolling spur gears, the meshing point moves along a straight line in the fixed coordinate system. The position vector of the meshing point is transformed into the gear coordinate systems using rotation matrices. The pinion tooth surface is given by: $$ \mathbf{r}_1 = \mathbf{M}_{1k} \mathbf{r}_k, $$ where $\mathbf{M}_{1k}$ is the transformation matrix from $S_k$ to $S_1$. Similarly, for the gear: $$ \mathbf{r}_2 = \mathbf{M}_{2k} \mathbf{r}_k. $$ The surface parameters are related through the meshing equation, which ensures continuous contact.
The effectiveness of the design is demonstrated through the contact patterns and transmission error curves. For spur gears, the pure rolling condition minimizes sliding velocities, reducing wear and friction. The mathematical model ensures that the relative velocity at the contact point is zero, i.e., $$ \mathbf{v}_{12} = \mathbf{v}_1 – \mathbf{v}_2 = \mathbf{0}, $$ where $\mathbf{v}_1$ and $\mathbf{v}_2$ are the velocities of the contact point on the pinion and gear, respectively. This condition is satisfied by the active design of the meshing line function.
In summary, the proposed spur gears offer a significant improvement over traditional designs by eliminating sliding motion. The geometric design process, supported by mathematical modeling and TCA, provides a robust framework for developing high-performance transmission systems. The integration of circular arc profiles and lead crowning makes these spur gears suitable for applications requiring low noise and high efficiency. Further research will explore the manufacturing aspects and experimental validation of these gears.