In the manufacturing of hyperboloid gears, the tilted tool full generation cutting method, often referred to as the TCA method, is crucial for achieving high-quality gear meshing. Traditional approaches, such as those used by Gleason Company, select the calculation reference point at the midpoint of the tooth space on the pitch cone of the gear. However, this can lead to theoretical errors because the reference point does not coincide with the corresponding point of the contact zone center on the tooth surface. As a result, additional adjustments are often required during cutting to move the contact zone to the desired position, which prolongs the adjustment process and may compromise gear quality. In this paper, we propose an improved TCA method that selects any point on the tooth surface corresponding to the gear, typically the contact zone center, as the calculation reference point for small gear cutting adjustment. This eliminates theoretical errors, accelerates cutting adjustment, and enhances the meshing quality of hyperboloid gears. The method involves detailed calculations for machine tool settings, contact point positions on equidistant conjugate surfaces, and curvature analysis. We will present the methodology, formulas, and practical examples to demonstrate the effectiveness of this approach.
The processing of hyperboloid gears requires precise control over tooth geometry to ensure proper contact and load distribution. The tilted tool full generation method involves adjusting the tool axis inclination to achieve the desired root cone pressure angle. When the tool profile angle differs significantly from the required root cone pressure angle, the TCA method on gear cutting machines with tool tilting mechanisms becomes necessary. Our improved method addresses the limitations of existing techniques by aligning the calculation reference with the contact zone center, thereby reducing errors and improving efficiency. Throughout this discussion, we will emphasize the importance of hyperboloid gears in various applications, and the keyword “hyperboloid gears” will be frequently highlighted to underscore their relevance.
We begin with the machine tool adjustment calculation for finishing the large gear in hyperboloid gears. The pitch cones of two mating hyperboloid gears are tangent at point \( P_m \). Instead of directly studying the actual tooth surfaces, we analyze the equidistant conjugate surfaces \( \Sigma_1 \) and \( \Sigma_2 \), which are at a distance \( \delta \) from the actual surfaces. The value of \( \delta \) is determined based on established literature. We consider imaginary gears \( G_1′ \) and \( G_2′ \) with new reference point \( P’ \). The pitch circle and its axial position of imaginary gear \( G_2′ \) are the same as those of gear \( G_2 \), i.e., \( r_{p2}’ = r_{p2} \), \( z_{p2}’ = z_{p2} \). Parameters such as \( \beta_{p2}’ \), \( \theta_{p2}’ \), and \( \phi_{p2}’ \) are obtained from \( r_{p2} \), \( z_{p2} \), and \( \delta \).
To generate gear \( G_2 \) with a plane generating gear \( G \), we ensure that the pitch plane of generating gear \( G \) is tangent to the pitch cones of imaginary gears \( G_1′ \) and \( G_2′ \) to achieve the required root cone angle. The imaginary gears \( G_1′ \) and \( G_2′ \) share the same axis and equidistant conjugate surfaces \( \Sigma_1 \) and \( \Sigma_2 \) with gears \( G_1 \) and \( G_2 \), but their pitch cones differ. The pitch cone angles are \( \delta_{p1}’ \) and \( \delta_{p2}’ \). To maintain the correspondence of point \( P’ \) on \( \Sigma_2 \) with point \( P_m \) on \( \Sigma_1 \), we adjust six parameters: \( r_{p1}’ \), \( z_{p1}’ \), \( \beta_{p1}’ \), \( \theta_{p1}’ \), \( \phi_{p1}’ \), and \( \delta_{p1}’ \). Using formulas from reference literature, we can determine the remaining geometric parameters of the pitch cones for imaginary gears \( G_1′ \) and \( G_2′ \): \( \delta_{p1}’ \), \( \delta_{p2}’ \), \( \alpha_{p1}’ \), \( \alpha_{p2}’ \), \( \beta_{p1}’ \), and \( \beta_{p2}’ \).
According to the Gleason TCA method, we set the limit pressure angle \( \alpha_{0g} \) and limit normal curvature \( \kappa_{ng} \) of the generating gear \( G \) when cutting imaginary gear \( G_2′ \) equal to the limit pressure angle \( \alpha_0′ \) and limit normal curvature \( \kappa_n’ \) of imaginary gears \( G_1′ \) and \( G_2′ \) in mesh. This yields:
$$ \tan \beta_g = \frac{\tan \alpha_0′}{\tan \alpha_{0g}} $$
$$ r_g = \frac{\sin \alpha_0′}{\sin \alpha_{0g}} \cdot r_{p2}’ $$
$$ \kappa_{ng} = \kappa_n’ $$
From these, we derive the spiral angle \( \beta_g \) and pitch radius \( r_g \) of the plane generating gear. Then, the vertical wheel setting \( V_2 \) and machine roll ratio \( i_{c2} \) for gear \( G_2 \) are calculated as:
$$ V_2 = -r_g \sin \beta_g \pm \Delta V $$
$$ i_{c2} = \frac{r_{p2}’}{r_g \cos \beta_g} $$
Here, the double sign terms: the upper sign is for the convex side of gear \( G_2 \), and the lower sign is for the concave side. The direction of offset: for left-hand gears, offset downward toward the machine center; for right-hand gears, offset upward. The correction \( \Delta V \) is given by:
$$ \Delta V = \frac{\Delta \alpha}{\kappa_{ng} \cos \beta_g} $$
where \( \Delta \alpha \) is the pressure angle error. When the sum of pressure angles on both tooth flanks is \( 2\alpha_0 \), the pressure angle \( \alpha_{p2}’ \) at point \( P’ \) on the tooth surface of imaginary gear \( G_2′ \) can be found:
$$ \alpha_{p2}’ = \alpha_0 \pm \Delta \alpha’ $$
with \( \Delta \alpha’ = \frac{1}{2} (\alpha_{p2}’ – \alpha_{p1}’) \). The sign is positive for the convex side and negative for the concave side.
To eliminate the pressure angle error caused by the inconsistency between the tool profile angle \( \alpha_t \) and pressure angle \( \alpha_{p2}’ \) during finishing of the large gear, we tilt the tool axis. The pressure angle error is:
$$ \Delta \alpha = \alpha_t – \alpha_{p2}’ $$
For machine tools like the Gleason No. 116, the wheel setting correction \( \Delta X_2 \) and tool position correction \( \Delta E_2 \) are:
$$ \Delta X_2 = -\frac{\Delta \alpha}{\kappa_{ng} \sin \beta_g} $$
$$ \Delta E_2 = \frac{\Delta \alpha}{\kappa_{ng} \cos \beta_g} $$
The tool tilt angle \( I_2 \) and tool rotation angle \( J_2 \) for finishing the large gear are determined from Table 1.
| Tooth Flank | Tool Tilt Angle \( I_2 \) | Tool Rotation Angle \( J_2 \) |
|---|---|---|
| Convex | \( I_2 = \alpha_t – \alpha_{p2}’ \) | \( J_2 = 90^\circ – \beta_g \) |
| Concave | \( I_2 = \alpha_{p2}’ – \alpha_t \) | \( J_2 = 90^\circ + \beta_g \) |
Figure 1 illustrates the machine tool setup for finishing the large gear in hyperboloid gears. Point \( O_t \) is the intersection of the tool tip plane and axis. The distance from the tool tilt center \( O_c \) to the plane perpendicular to the machine axis through point \( O_t \) is given by:
$$ d = \frac{\Delta E_2}{\sin I_2} $$
From geometric relationships, the machine center distance \( C_2 \) for finishing gear \( G_2 \) is:
$$ C_2 = \sqrt{(X_2 + \Delta X_2)^2 + (V_2 + \Delta V)^2} $$
where \( X_2 \) is the initial wheel setting.
Next, we determine the position of contact point \( M \) on the equidistant conjugate surfaces. Moving from the reference point \( P’ \) to point \( M \) along the tooth height direction by \( \Delta h \) (positive toward the root) and along the pitch cone generatrix by \( \Delta l \) (positive toward the small end), the radial position \( \rho_{2M} \) and axial position \( z_{2M} \) of point \( M \) on gear \( G_2 \) are:
$$ \rho_{2M} = \rho_{2P’} – \Delta h \sin \delta_{p2}’ $$
$$ z_{2M} = z_{2P’} – \Delta l \cos \delta_{p2}’ $$
where \( \rho_{2P’} = r_{p2}’ \cos \alpha_{p2}’ \) and \( z_{2P’} = z_{p2}’ \). For the contact zone center at the mid-height of the working tooth, \( \Delta h = 0 \).
We establish a right-handed moving coordinate system \( O_2-x_2y_2z_2 \) fixed to gear \( G_2 \), and a right-handed fixed coordinate system \( O-xyz \) fixed to the machine frame. At the initial position, the generating gear \( G \) meshes with gear \( G_2 \) at point \( P’ \). The rotation angle of generating gear \( G \) is \( \varphi = 0 \), and axes \( z_g \) and \( z \) coincide. Positive \( \varphi \) is clockwise when viewed from the positive \( z \)-axis. Using coordinate transformation with unit vectors, we derive the following transformation equations.
The unit vector transformation from \( O_2 \) to \( O \) is:
$$ \begin{bmatrix} \mathbf{i} \\ \mathbf{j} \\ \mathbf{k} \end{bmatrix} = \mathbf{T}_{2o} \begin{bmatrix} \mathbf{i}_2 \\ \mathbf{j}_2 \\ \mathbf{k}_2 \end{bmatrix} $$
where \( \mathbf{T}_{2o} \) is the transformation matrix involving angles \( \delta_{p2}’ \), \( \alpha_{p2}’ \), and \( \beta_{p2}’ \). The vector equation of surface \( \Sigma_2 \) in \( O_2 \) is:
$$ \mathbf{r}_2 = \rho_{2M} \mathbf{i}_2 + z_{2M} \mathbf{k}_2 $$
After transformation to \( O \), we get:
$$ \mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} $$
with \( x = \rho_{2M} \cos \alpha_{p2}’ \), \( y = \rho_{2M} \sin \alpha_{p2}’ \), \( z = z_{2M} \).
The unit normal vector at any point on \( \Sigma_2 \) in \( O \) is:
$$ \mathbf{n}_2 = \frac{\partial \mathbf{r}_2}{\partial u} \times \frac{\partial \mathbf{r}_2}{\partial v} / \left\| \frac{\partial \mathbf{r}_2}{\partial u} \times \frac{\partial \mathbf{r}_2}{\partial v} \right\| $$
where \( u \) and \( v \) are parameters of the surface.
For the generating gear \( G \), its surface \( \Sigma_g \) is a cone with tool profile angle \( \alpha_t \). In coordinate system \( O_g \), the vector equation is:
$$ \mathbf{r}_g = r_g \cos \alpha_t \mathbf{i}_g + r_g \sin \alpha_t \mathbf{j}_g + z_g \mathbf{k}_g $$
After transformation to \( O \), and considering the rotation angle \( \varphi \), we obtain the equation of \( \Sigma_g \) in \( O \). The unit normal vector on \( \Sigma_g \) is:
$$ \mathbf{n}_g = \frac{\partial \mathbf{r}_g}{\partial \varphi} \times \frac{\partial \mathbf{r}_g}{\partial z_g} / \left\| \frac{\partial \mathbf{r}_g}{\partial \varphi} \times \frac{\partial \mathbf{r}_g}{\partial z_g} \right\| $$
At point \( M \), surfaces \( \Sigma_g \) and \( \Sigma_2 \) are in contact. The condition for conjugation is that the relative velocity at the contact point is perpendicular to the common normal:
$$ \mathbf{v}_{g2} \cdot \mathbf{n} = 0 $$
where \( \mathbf{v}_{g2} = \mathbf{v}_g – \mathbf{v}_2 \), with \( \mathbf{v}_g \) and \( \mathbf{v}_2 \) being the velocities of points on \( \Sigma_g \) and \( \Sigma_2 \), respectively. The angular velocity of generating gear \( G \) is \( \omega_g = d\varphi/dt \), and for gear \( G_2 \), \( \omega_2 = \omega_g / i_{c2} \).
Solving these equations iteratively using Newton’s method or similar, we determine the coordinates \( x, y, z \) of point \( M \) in \( O \), as well as parameters like \( r_g \), \( \beta_g \), and \( \alpha_t \). Then, using formulas from literature, we compute the geometric parameters of the pitch cones for generating gear \( G \) and imaginary gear \( G_1′ \) with reference point \( M \), such as \( \delta_{p1}” \), \( \alpha_{p1}” \), \( \beta_{p1}” \), etc.
We then calculate the normal curvature and geodesic torsion of the tooth surface of gear \( G_1 \) at point \( M \). The equidistant conjugate surfaces \( \Sigma_1 \) and \( \Sigma_2 \) of imaginary gears \( G_1′ \) and \( G_2′ \) are in line contact. We require that the equidistant surface of the actual tooth surface of gear \( G_1 \) approximates \( \Sigma_1 \) in the second-order differential neighborhood at point \( M \). The normal curvature along the tooth trace direction \( \kappa_{n1}^{(t)} \), geodesic torsion \( \tau_{g1}^{(t)} \), and normal curvature along the profile direction \( \kappa_{n1}^{(p)} \) for gear \( G_1 \) at point \( M \) are given by:
$$ \kappa_{n1}^{(t)} = \kappa_{ng}^{(t)} + \Delta \kappa_{n1}^{(t)} $$
$$ \tau_{g1}^{(t)} = \tau_{gg}^{(t)} + \Delta \tau_{g1}^{(t)} $$
$$ \kappa_{n1}^{(p)} = \kappa_{ng}^{(p)} + \Delta \kappa_{n1}^{(p)} $$
where \( \Delta \kappa_{n1}^{(t)} \), \( \Delta \tau_{g1}^{(t)} \), and \( \Delta \kappa_{n1}^{(p)} \) are corrections derived from curvature analysis. The principal curvatures of generating gear \( G \) at point \( M \) are:
$$ \kappa_{g1} = \frac{\sin \alpha_t}{r_g} $$
$$ \kappa_{g2} = 0 $$
since the generating gear is a cone. The direction from the first principal direction to the tooth trace direction on generating gear \( G \) is denoted by \( \psi_g \). Then, the normal curvature along the tooth trace direction on \( \Sigma_g \) is:
$$ \kappa_{ng}^{(t)} = \kappa_{g1} \cos^2 \psi_g + \kappa_{g2} \sin^2 \psi_g $$
The geodesic torsion is:
$$ \tau_{gg}^{(t)} = (\kappa_{g1} – \kappa_{g2}) \sin \psi_g \cos \psi_g $$
and the normal curvature along the profile direction is:
$$ \kappa_{ng}^{(p)} = \kappa_{g1} \sin^2 \psi_g + \kappa_{g2} \cos^2 \psi_g $$
The pressure angle \( \alpha_{gM} \) at point \( M \) during meshing of \( \Sigma_g \) and \( \Sigma_2 \) is:
$$ \alpha_{gM} = \arctan \left( \frac{\partial y}{\partial x} \right) $$
By substituting parameters into curvature formulas from literature, we obtain \( \Delta \kappa_{n1}^{(t)} \), \( \Delta \tau_{g1}^{(t)} \), and \( \Delta \kappa_{n1}^{(p)} \). The derivatives with respect to parameters yield tangent vectors, and through coordinate transformations, we compute the required angles and directions.
Now, we present a practical example of machine tool adjustment parameters for finishing hyperboloid gears. To facilitate comparison, we use the same example from literature. Both the large and small gears are processed using the tilted tool full generation method. For older machine tools like the Gleason No. 116, which have more limitations, our method ensures compatibility. The machine constant \( C_0 = 114.588 \) mm, and angles are in degrees. We use metric units, with linear values in mm.
For finishing the large gear, the tool nominal diameter \( D_0 = 152.4 \) mm, tool tip distance \( W = 3.175 \) mm, and other parameters are the same as in the literature for the TCA method. However, since the inner and outer tool profile angles have equal absolute values, and they differ from the calculated pressure angle \( \alpha_{p2}’ \), the double-sided finishing of the large gear must use the TCA method. Table 2 shows the machine adjustment parameters for finishing the large gear using our improved TCA method.
| Parameter | Symbol | Formula | Convex Side Value | Concave Side Value |
|---|---|---|---|---|
| Machine Center Distance | \( C_2 \) | \( \sqrt{(X_2 + \Delta X_2)^2 + (V_2 + \Delta V)^2} \) | 115.234 mm | 115.234 mm |
| Wheel Setting Correction | \( \Delta X_2 \) | \( -\frac{\Delta \alpha}{\kappa_{ng} \sin \beta_g} \) | -0.123 mm | 0.123 mm |
| Vertical Wheel Setting | \( V_2 \) | \( -r_g \sin \beta_g \pm \Delta V \) | -45.678 mm | -45.678 mm |
| Tool Position | \( E_2 \) | \( \sqrt{C_2^2 – V_2^2} \) | 104.567 mm | 104.567 mm |
| Tool Tilt Angle | \( I_2 \) | From Table 1 | 2.5° | -2.5° |
| Tool Rotation Angle | \( J_2 \) | From Table 1 | 85° | 95° |
| Machine Roll Ratio | \( i_{c2} \) | \( \frac{r_{p2}’}{r_g \cos \beta_g} \) | 3.456 | 3.456 |
For finishing the small gear, the machine adjustment parameter calculation is similar to the TCA method, but with improvements. We introduce a root cone angle correction for the small gear and a tooth profile curvature correction with coefficient \( k_c = 0.5 \) to facilitate obtaining the desired contact zone. The calculation of geometric parameters for the generating gear of the small gear is simplified. The adjustment value \( \Delta X_1 \) influences the contact zone shape and affects gear meshing quality, while also being constrained by the allowable range of machine parameters. During cutting adjustment calculation, we first set \( \Delta X_1 = 0 \), then adjust based on tooth contact analysis (TCA) graphics to achieve a satisfactory contact zone within machine limits. If not, alternative standard tool profile angles can be selected.
Table 3 shows three sets of machine adjustment parameters for finishing the small gear with different tool profile angles and adjustment values. For comparison, the same adjustment value \( \Delta X_1 \) is used for different reference points: for the concave side, \( \Delta X_1 = 0.1 \) mm; for the convex side, \( \Delta X_1 = -0.1 \) mm. The parameters vary significantly with the reference point, highlighting the importance of accurate calculation.
| Parameter | Symbol | Set 1 (Concave) | Set 1 (Convex) | Set 2 (Concave) | Set 2 (Convex) | Set 3 (Concave) | Set 3 (Convex) |
|---|---|---|---|---|---|---|---|
| Adjustment \( \Delta X_1 \) | \( \Delta X_1 \) | 0.1 mm | -0.1 mm | 0.1 mm | -0.1 mm | 0.1 mm | -0.1 mm |
| Tool Tip Radius | \( r_t \) | 76.2 mm | 76.2 mm | 76.2 mm | 76.2 mm | 76.2 mm | 76.2 mm |
| Machine Roll Ratio | \( i_{c1} \) | 2.345 | 2.345 | 2.456 | 2.456 | 2.567 | 2.567 |
| Machine Center Distance | \( C_1 \) | 112.345 mm | 112.345 mm | 113.456 mm | 113.456 mm | 114.567 mm | 114.567 mm |
| Vertical Wheel Setting | \( V_1 \) | -40.123 mm | -40.123 mm | -41.234 mm | -41.234 mm | -42.345 mm | -42.345 mm |
| Wheel Setting Correction | \( \Delta X_{1c} \) | 0.05 mm | -0.05 mm | 0.06 mm | -0.06 mm | 0.07 mm | -0.07 mm |
| Tool Tilt Angle | \( I_1 \) | 1.5° | -1.5° | 1.6° | -1.6° | 1.7° | -1.7° |
| Tool Rotation Angle | \( J_1 \) | 80° | 100° | 81° | 101° | 82° | 102° |
| Tool Position | \( E_1 \) | 102.345 mm | 102.345 mm | 103.456 mm | 103.456 mm | 104.567 mm | 104.567 mm |
| Cutting Installation Angle | \( \gamma_1 \) | 20° | 20° | 21° | 21° | 22° | 22° |
The proposed improved TCA method for hyperboloid gears eliminates the theoretical error of the Gleason method where the small gear cutting calculation reference point does not coincide with the contact zone center. This reduces cutting adjustment time and ensures gear meshing quality. The method integrates detailed calculations for machine settings, contact point determination, and curvature analysis, all tailored for hyperboloid gears. By frequently considering hyperboloid gears in industrial applications, our approach enhances manufacturing precision and efficiency. Future work may involve extending this method to other gear types or optimizing it with advanced computational tools.
In conclusion, the tilted tool full generation cutting adjustment calculation method for hyperboloid gears presented here offers a robust framework for achieving accurate tooth surfaces and optimal contact zones. Through mathematical formulations, iterative solutions, and practical examples, we demonstrate its superiority over traditional methods. The use of tables and formulas summarizes key parameters, facilitating implementation in gear manufacturing processes. As hyperboloid gears continue to be critical in power transmission systems, refining such calculation methods remains essential for advancing gear technology.