In modern mechanical systems, helical gears are widely used due to their high load-bearing capacity, large overlap ratio, and smooth transmission characteristics. However, factors such as installation errors and manufacturing inaccuracies can lead to meshing impacts, resulting in vibration and noise. Diagonal modification has been proven as an effective method to reduce these issues. This study focuses on developing a high-precision reverse modeling approach for helical gears with diagonal modification, utilizing MATLAB for coordinate solving and CATIA for 3D model reconstruction. We address the problem of unsmooth connections at the tooth root by employing Hermite interpolation, ensuring geometric continuity and accuracy.
The foundation of our work lies in establishing the mathematical models for the tooth surfaces of helical gears. We begin by considering a hypothetical generating rack to derive the standard working tooth surface and the root transition surface. The coordinate systems involved in the generation process are defined, and the equations for the tooth surfaces are derived based on gear meshing theory. For the standard involute helical gear, the position vector and unit normal vector are expressed as functions of parameters from the rack cutter. The transformation matrices facilitate the conversion from the rack coordinate system to the gear coordinate system, ensuring accurate representation of the tooth geometry.
The meshing equation plays a crucial role in defining the contact conditions between the rack and the gear. For the standard involute surface, the position vector \(\mathbf{r}_1(u_t, l_t, \theta_1)\) and unit normal vector \(\mathbf{n}_1(u_t, l_t, \theta_1)\) are given by:
$$\mathbf{r}_1(u_t, l_t, \theta_1) = \mathbf{M}_{1t} \mathbf{r}_t(u_t, l_t)$$
$$\mathbf{n}_1(u_t, l_t, \theta_1) = \mathbf{L}_{1t} \mathbf{n}_t(u_t, l_t)$$
$$f = \mathbf{n}_1(u_t, l_t, \theta_1) \cdot \frac{\partial \mathbf{r}_1(u_t, l_t, \theta_1)}{\partial \theta_1} = 0$$
Here, \(u_t\) and \(l_t\) are parameters along the rack cutter’s surface, \(\theta_1\) is the gear rotation angle, \(\mathbf{M}_{1t}\) is the transformation matrix from the rack to the gear, and \(\mathbf{L}_{1t}\) is the corresponding submatrix. The meshing equation ensures that the normal vector is perpendicular to the relative velocity at the contact point, defining the gear tooth surface accurately.
For the root transition curve, which is generated by the rack’s tip fillet, we define the position vector \(\mathbf{r}’_t(\phi_t, l_t)\) and unit normal vector \(\mathbf{n}’_t(\phi_t, l_t)\) based on the circular arc profile of the rack. The transition surface on the gear is then derived as:
$$\mathbf{r}’_1(\phi_t, l_t, \theta_1) = \mathbf{M}_{1t} \mathbf{r}’_t(\phi_t, l_t)$$
$$\mathbf{n}’_1(\phi_t, l_t, \theta_1) = \mathbf{L}_{1t} \mathbf{n}’_t(\phi_t, l_t)$$
$$f = \mathbf{n}’_1(\phi_t, l_t, \theta_1) \cdot \frac{\partial \mathbf{r}’_1(\phi_t, l_t, \theta_1)}{\partial \theta_1} = 0$$
The parameter \(\phi_t\) ranges from 0 to \(\pi/2 – \alpha_n\), where \(\alpha_n\) is the normal pressure angle. This ensures a smooth transition from the involute profile to the root fillet, critical for stress distribution and fatigue life.
Diagonal modification involves altering the tooth profile at the meshing-in and meshing-out ends while keeping the central portion unchanged or lightly modified. This approach reduces impact forces and noise. For a right-hand helical gear’s left tooth surface, the modification zones are defined based on the tooth top and root regions. The modification termination points are determined using geometric constraints, such as the base circle radius and meshing conditions. The radius at the root modification start point \(r_{k1}\) is calculated as:
$$r_{k1} = \sqrt{r_{b1}^2 + \left[ (r_{p1} + r_{p2}) \sin \alpha_t – \sqrt{r_{a2}^2 – r_{b2}^2} \right]^2}$$
Here, \(r_{b1}\) and \(r_{b2}\) are the base circle radii of the driving and driven gears, \(r_{p1}\) and \(r_{p2}\) are the pitch circle radii, and \(\alpha_t\) is the transverse pressure angle. The modification height and length are specified, and the modification amount \(\delta(z_2, x_2)\) at any point in the modification zone is defined using linear, quadratic, or quartic functions. For instance, a quadratic modification can be expressed as:
$$\delta(z_2, x_2) = \begin{cases}
\left( \frac{l_G}{l_a} \right)^2 C_a & \text{if } G \in \triangle ABC \\
\left( \frac{l_G}{l_d} \right)^2 C_d & \text{if } G \in \triangle DEF \\
0 & \text{otherwise}
\end{cases}$$
where \(C_a\) and \(C_d\) are the maximum modification amounts at the tip and root, \(l_a\) and \(l_d\) are the distances from the modification start lines, and \(l_G\) is the distance from point \(G\) to the start line. The modified tooth surface is then obtained by superimposing the modification surface in the normal direction of the standard involute surface:
$$\mathbf{r}_{1m}(u_t, l_t, \theta_1) = \mathbf{r}_1(u_t, l_t, \theta_1) + \delta(z_2, x_2) \mathbf{n}_1(u_t, l_t, \theta_1)$$
$$\mathbf{n}_{1m}(u_t, l_t, \theta_1) = \left( \frac{\partial \mathbf{r}_1}{\partial u_t} + \frac{\partial \delta}{\partial u_t} \mathbf{n}_1 + \frac{\partial \mathbf{n}_1}{\partial u_t} \delta \right) \times \left( \frac{\partial \mathbf{r}_1}{\partial l_t} + \frac{\partial \delta}{\partial l_t} \mathbf{n}_1 + \frac{\partial \mathbf{n}_1}{\partial l_t} \delta \right)$$
To address discontinuities at the boundary between the modified surface and the root transition surface, we employ Hermite interpolation. This method ensures that the interpolated curve matches the position and tangent vectors at the endpoints, providing a smooth connection. Given position vectors \(\mathbf{P}_0\) and \(\mathbf{P}_1\), and tangent vectors \(\mathbf{T}_0\) and \(\mathbf{T}_1\), the Hermite curve is defined as:
$$\mathbf{r}(t) = (2t^3 – 3t^2 + 1) \mathbf{P}_0 + (-2t^3 + 3t^2) \mathbf{P}_1 + (t^3 – 2t^2 + t) \mathbf{T}_0 + (t^3 – t^2) \mathbf{T}_1$$
where \(t \in [0, 1]\). The blending functions \(b_1, b_2, b_3, b_4\) are used to weight the contributions of the endpoints and tangents. For gear modeling, the tangent vectors are scaled by the normal module \(m_n\) and a parameter \(t_h\) to control the curve’s shape, typically ranging from 0.5 to 1.5.
In our implementation, we apply this to the tooth root transition, ensuring that the Hermite surface closely approximates the original transition surface. The coordinates of any point on the Hermite curve are given by:
$$x_P(t) = b_1 x_{P0} + b_2 x_{P1} + b_3 x_{T0} + b_4 x_{T1}$$
$$y_P(t) = b_1 y_{P0} + b_2 y_{P1} + b_3 y_{T0} + b_4 y_{T1}$$
$$z_P(t) = b_1 z_{P0} + b_2 z_{P1} + b_3 z_{T0} + b_4 z_{T1}$$
This approach effectively eliminates sharp transitions, enhancing the model’s accuracy and suitability for finite element analysis.
For the reverse modeling process, we use MATLAB to compute the discrete coordinates of the tooth surface, including the modification zones. The data points are then imported into CATIA V5 R20 to generate a 3D model. The Digital Shape Editor (DSE) module is used to create point clouds, which are fitted into surfaces using the Power Fit command. The deviation between the modified surface and the standard involute surface is analyzed to ensure precision. Key parameters for the helical gear pair and diagonal modification are summarized in the following tables:
| Parameter | Pinion | Gear |
|---|---|---|
| Handedness | Right | Left |
| Number of Teeth | 30 | 72 |
| Module (mm) | 5.000 | |
| Pressure Angle (°) | 20.000 | |
| Helix Angle (°) | 33.273 | |
| Tip Radius (mm) | 94.706 | 220.294 |
| Root Radius (mm) | 83.456 | 209.044 |
| Face Width (mm) | 40.000 | |
| Parameter | Tip | Root |
|---|---|---|
| Modification Order | 2 | 2 |
| Modification Height (mm) | 5.000 | 5.000 |
| Modification Length (mm) | 18.519 | 23.597 |
| Max Modification (μm) | 30.000 | 30.000 |
| Start Line Helix Angle (°) | 15.109 | 11.964 |
The discrete points for the quadratic diagonal modification surface are generated in MATLAB, considering both the tooth height and width directions. The Hermite interpolation is applied to the root transition, and the resulting curve profile is validated for smoothness. The following figure illustrates the 3D model of the modified helical gear, showcasing the seamless integration of the modification zones.
To evaluate the modeling accuracy, we perform deviation analysis in CATIA. The standard involute surface is compared with the modified surface at specific cross-sections along the tooth profile and tooth direction. For the profile direction, cross-sections at \(z_1 = -20\) mm, \(0\) mm, and \(20\) mm are analyzed, with 50 points each. The maximum fitting deviation after subtracting the modification amount is found to be 1.103 μm on the left tooth surface at \(z_1 = 0\) mm. For the tooth direction, cross-sections at radii \(r = 85.666\) mm (root modification end), \(89.706\) mm (pitch circle), and \(94.706\) mm (tip circle) are examined, with 100 points each. The maximum deviation in this direction is 0.691 μm. Overall, the maximum fitting deviation across the tooth surfaces is approximately 1.1 μm, meeting high-precision design requirements.
The complete 3D model of the helical gear with quadratic diagonal modification is constructed in CATIA using operations such as extrusion, trimming, closing, rotation, and patterning. This model can be directly used for further engineering analyses, including contact stress evaluation and dynamic simulation.
In conclusion, we have developed a comprehensive methodology for high-precision reverse modeling of helical gears with diagonal modification. The mathematical models for standard and modified tooth surfaces are derived, and Hermite interpolation is effectively applied to ensure smooth transitions at the root. The integration of MATLAB and CATIA facilitates accurate 3D reconstruction, with deviations within acceptable limits. This approach enhances the design and analysis of helical gears in various applications, contributing to improved performance and reduced noise. Future work may focus on optimizing the modification parameters for specific operational conditions and extending the method to other gear types.