Gear drives are among the most widely used mechanisms and transmission devices. Due to their numerous advantages, they remain the primary form of transmission in many systems and are considered a symbol of a nation’s industrial technological level. This paper focuses on the spiral face gear drive, a novel type of spatial crossed-axis gearing composed of an involute spiral pinion and a spiral face gear (spiroid face gear). This transmission offers significant advantages such as a large contact ratio, compact structure, high transmission ratio, low noise, mature manufacturing processes for the pinion, strong load-bearing capacity, and smooth operation, leading to increasingly broad applications across various engineering fields.
The spiral face gear studied in this paper is a core component used internally in fishing reels, typically manufactured by die-casting. However, influenced by mold machining accuracy and thermal deformation, die-cast spiral face gears suffer from poor surface quality. To improve this, the method of gear shaving for spiral face gears is proposed. The main research content includes the following aspects.
Tooth Surface Generation of the Spiroid Face Gear
The spiral face gear drive can be categorized into orthogonal, non-orthogonal, or offset configurations based on the spatial relationship between its axis and the pinion axis. This research concerns the case of a left-hand involute spiral pinion meshing with the face gear. Based on the generating principle, the tooth surface of the spiral face gear can be enveloped by an imaginary pinion cutter identical to the meshing pinion.
First, the tooth surface equation of the involute spiral pinion is derived. For a right-hand pinion, the position vector for flank I is derived from the generation principle of an involute helicoid. The position vector $\mathbf{O_1N}$ for a point on flank I is given by:
$$ \mathbf{O_1N} = \mathbf{O_1K} + \mathbf{KM} + \mathbf{MN} $$
where:
$$ \mathbf{O_1K} = r_b (\cos\theta \mathbf{i} + \sin\theta \mathbf{j}) $$
$$ \mathbf{KM} = \mu \mathbf{k} $$
$$ \mathbf{MN} = \mu \tan\lambda_b (-\sin\theta \mathbf{i} + \cos\theta \mathbf{j}) $$
Here, $r_b$ is the base circle radius, $\lambda_b$ is the lead angle on the base cylinder, and $p = r_b / \tan\lambda_b$ is the spiral parameter. The variables $\mu$ and $\theta$ are the surface parameters. The resulting coordinates in coordinate system $S_1$ are:
$$ \begin{cases}
x_1 = r_b \cos\theta + \mu \sin\lambda_b \sin\theta \\
y_1 = r_b \sin\theta – \mu \sin\lambda_b \cos\theta \\
z_1 = \mu \cos\lambda_b – p\theta
\end{cases} $$
The unit normal vector for flank I is:
$$ \mathbf{n}_1 = \frac{\partial \mathbf{r}_1/\partial \mu \times \partial \mathbf{r}_1/\partial \theta}{|\partial \mathbf{r}_1/\partial \mu \times \partial \mathbf{r}_1/\partial \theta|} = [\sin\lambda_b \sin\theta, \; -\sin\lambda_b \cos\theta, \; \cos\lambda_b]^T $$
Similarly, the equations for flank II can be derived. For the left-hand pinion used in this study, the tooth surface equations and unit normals for both flanks (I and II) are adjusted accordingly, introducing an initial angle parameter $\theta_{01}$ which accounts for the semi-tooth space angle on the pitch circle minus the involute spread angle.
Assuming the imaginary pinion cutter has the same tooth surface $\Sigma_c$ as the actual pinion, the tooth surface generation of the spiral face gear $\Sigma_2$ is analyzed. The coordinate systems for the generation process are established, as shown in the derived kinematic model. The homogeneous transformation matrix from the cutter system $S_c$ to the gear system $S_2$, $\mathbf{M}_{2c}$, is derived through intermediate coordinate systems $S_m$, $S_p$, $S_a$, and $S_n$:
$$ \mathbf{M}_{2c} = \mathbf{M}_{2n} \mathbf{M}_{na} \mathbf{M}_{ap} \mathbf{M}_{pm} \mathbf{M}_{mc} $$
Where $\gamma_m$ is the shaft angle (90° for orthogonal), $E$ is the offset, $L$ is the axial distance defining the cutter origin relative to the gear, and $\phi_c$ and $\phi_2$ are rotation angles of the cutter and gear, related by $\phi_2 = m_{2c} \phi_c$, with $m_{2c} = N_2 / N_c$ being the gear ratio.
The meshing equation is established based on the condition that the relative velocity at the contact point is perpendicular to the common normal. The relative velocity $\mathbf{v}_c^{(2c)}$ of the cutter point relative to the gear in $S_c$ is:
$$ \mathbf{v}_c^{(2c)} = \mathbf{v}_c^{(c)} – \mathbf{v}_c^{(2)} = (\boldsymbol{\omega}^{(c)} – \boldsymbol{\omega}^{(2)}) \times \mathbf{r}_c – \boldsymbol{\omega}^{(2)} \times \mathbf{R} $$
Here, $\boldsymbol{\omega}^{(c)}$ and $\boldsymbol{\omega}^{(2)}$ are the angular velocity vectors, and $\mathbf{R} = \mathbf{o_c o_2}$ is the position vector between origins. The meshing function $f(\mu_c, \theta_c, \phi_c)$ is:
$$ f(\mu_c, \theta_c, \phi_c) = \mathbf{n}_c \cdot \mathbf{v}_c^{(2c)} = 0 $$
Solving this equation yields a relationship between the surface parameters and the motion parameter $\phi_c$.
The generated spiral face gear tooth surface $\Sigma_2$ consists of the working flank and the fillet (transition surface). The working flank equation is obtained by combining the coordinate transformation with the meshing equation:
$$ \begin{cases}
\mathbf{r}_2(\mu_c, \theta_c, \phi_c) = \mathbf{M}_{2c}(\phi_c) \cdot \mathbf{r}_c(\mu_c, \theta_c) \\
f(\mu_c, \theta_c, \phi_c) = 0
\end{cases} $$
The fillet surface is generated by the tip edge (or tip arc) of the imaginary pinion cutter and can be represented similarly with $\theta_c$ fixed at the value corresponding to the cutter tip.
The face width of the spiral face gear is limited by undercutting at the inner radius and pointing (tip sharpening) at the outer radius. The condition for no undercutting is derived by finding the envelope singularity. The minimum inner radius $L_1$ is determined by solving for the point on the gear surface corresponding to the intersection of the cutter’s limit curve with its tip circle. The condition for no pointing is found by solving for the intersection point of the left and right flank edge lines on the gear tooth at the tip. The maximum outer radius $L_2$ is the radial distance of this intersection point. The design values are determined by calculations for both flanks.
To construct an accurate 3D geometric model, the tooth surface is discretized. A grid of points (e.g., 5 points along the tooth height and 9 points along the tooth length) is planned in a rotational projection plane ($S_g: O_g-Z_g-R_g$). The coordinates $(R_{gij}, Z_{gij})$ of grid nodes are calculated based on $L_1$, $L_2$, and the tooth height $H$. For each grid node, a system of nonlinear equations derived from the gear surface equations and the projection relationship ($R_{gij}^2 = x_2^2 + y_2^2$, $Z_{gij} = z_2$) is solved numerically (e.g., using MATLAB’s `fsolve` function) to obtain the corresponding surface parameters $(\mu_c, \theta_c, \phi_c)$. These parameters are then substituted into the gear surface equation to get the 3D point cloud $(x_2, y_2, z_2)$. This point cloud data is imported into CAD software (e.g., Pro/E) to generate curves, surfaces, and finally the solid model of the spiral face gear and the involute pinion through operations like boundary blending, merging, solidifying, and patterning.
Principle of Spiroid Face Gear Shaving
Given the poor surface quality of die-cast gears, a finishing process is required. Existing methods like hobbing or grinding for face gears involve complex tool design and machine modifications. This paper proposes a gear shaving process for spiral face gears, inspired by the shaving of cylindrical gears. The principle is based on the crossed-axis meshing of the spiral face gear and the involute pinion. By cutting chip flutes (gashes) into the tooth flanks of the pinion, a shaving cutter is created. When this cutter is meshed under pressure with the die-cast gear blank in a CNC machine, the cutting edges remove a thin layer of material, refining the tooth surface.
The kinematic model for gear shaving is established. The shaving cutter rotates with angular velocity $\omega_s$, and the workpiece (spiral face gear) is driven by the cutter with angular velocity $\omega_g$, satisfying $\omega_g / \omega_s = N_s / N_g = m_{gs}$. To shave the entire face width, the cutter must perform a radial feed motion $v_{s0}$ along the gear’s radius direction. This radial feed disturbs the correct kinematic relationship. To maintain the correct generated tooth profile, an additional rotational motion must be superimposed on the gear’s rotation. The modified angular velocity of the gear $\dot{\omega}_g$ is given by:
$$ \dot{\omega}_g = m_{gs} \left( \omega_s – \frac{v_{s0}}{p_s} \right) $$
where $p_s$ is the spiral parameter of the shaving cutter.
The corresponding rotation angles are related by $\phi_g = m_{gs} \phi_s – (N_s / N_g) \cdot (\Delta l / p_s)$, where $\Delta l$ is the radial feed displacement.
The coordinate systems for the gear shaving process are defined, including fixed systems $S_t$, $S_h$, moving systems $S_s$ (cutter), $S_g$ (workpiece), and auxiliary systems $S_f$, $S_q$. The homogeneous transformation matrix from $S_s$ to $S_g$, $\mathbf{M}_{gs}$, is derived considering the radial feed $\Delta l$ and the modified gear rotation $\phi_g$.
The tooth surface of the shaving cutter $\Sigma_s$ is identical to the pinion tooth surface, obtained by replacing subscript ‘1’ with ‘s’ in the pinion equations. The relative velocity $\mathbf{v}_{sg}$ at a potential contact point in $S_s$ is calculated as:
$$ \mathbf{v}_{sg} = \mathbf{v}_s^{(s)} – \mathbf{v}_s^{(g)} = (\boldsymbol{\omega}^{(s)} + \mathbf{v}_{s0}) – \boldsymbol{\omega}^{(g)} \times \mathbf{r}_s^{(g)} $$
After substituting the kinematic relations, the meshing equation for the shaving process becomes:
$$ f_{gs}(\mu_s, \theta_s, \phi_s) = \mathbf{n}_s \cdot \mathbf{v}_{sg} = 0 $$
The tooth surface of the spiral face gear generated by gear shaving is then given by:
$$ \begin{cases}
\mathbf{r}_g(\mu_s, \theta_s, \phi_s) = \mathbf{M}_{gs}(\phi_s) \cdot \mathbf{r}_s(\mu_s, \theta_s) \\
f_{gs}(\mu_s, \theta_s, \phi_s) = 0
\end{cases} $$
When $\Delta l = 0$ and $v_{s0} = 0$, the transformation matrix $\mathbf{M}_{gs}$ and relative velocity $\mathbf{v}_{sg}$ become identical to those in the theoretical generation by the imaginary cutter ($\mathbf{M}_{2c}$ and $\mathbf{v}_c^{(2c)}$), validating the correctness of the proposed shaving kinematics.
Design of the Shaving Cutter
The shaving cutter is designed based on the involute spiral pinion. The workpiece material is zinc alloy (ZZnAl4Cu3Y) with low hardness (~120-130 HBS). High-speed steel W6Mo5Cr4V2 is chosen for the cutter material due to its good toughness, wear resistance, and hot plasticity.
The basic parameters of the shaving cutter are the same as those of the pinion, as shown in the table below:
| Parameter | Symbol | Value |
|---|---|---|
| Normal Module | $m_n$ | 0.65 mm |
| Number of Teeth | $N_s$ | 7 |
| Pressure Angle | $\alpha_n$ | 20° |
| Helix Angle | $\beta$ | 52° (LH) |
| Normal Addendum Coefficient | $h_{an}^*$ | 1 |
| Normal Dedendum Coefficient | $c_n^*$ | 0.25 |
| Normal Profile Shift Coefficient | $x_n$ | 0.277 |
The geometrical dimensions of the cutter, such as pitch diameter $d$, tip diameter $d_a$, root diameter $d_f$, and transverse tooth thickness $s_t$, are calculated using standard formulas for helical gears.
The chip flute design is crucial. For this small module cutter, axial straight flutes (grooves) are chosen. The flute width is designed as 1.0 mm, and the depth is set to the full tooth depth plus 0.5 mm, rounded to 2.0 mm. The flutes are evenly distributed along the face width of the cutter. The 3D model is created in CAD software. The actual manufactured axial-grooved shaving cutter is shown in the provided figure. Technical requirements for the cutter, such as profile error, lead deviation, pitch deviation, and runout, are specified according to precision tool standards.
NC Machining Simulation of Gear Shaving
To verify the proposed gear shaving method, a machining simulation is conducted using VERICUT software. A five-axis machine tool model is conceptualized with three linear axes (X, Y, Z) and two rotary axes (A, C). The A-axis rotates the workpiece, the C-axis rotates the shaving cutter, and the Z-axis provides the radial feed motion. The simulation workflow involves:
- Creating the machine tool model, cutter model, and gear blank model in Pro/E and importing them into VERICUT.
- Configuring the kinematic tree in VERICUT to represent the machine structure.
- Selecting a control system (e.g., Siemens Sinumerik 840D).
- Programming the NC code based on the shaving kinematics.
The NC code controls the synchronized motion of the C, Z, and A axes. The relationship between the axes is derived from the shaving kinematics. For a given radial feed velocity $v_{s0}=0.0166666 \text{ mm/ms}$ and feed increment of 0.0001 mm per step, the time increment is $\Delta t=0.006$ s. If the cutter rotates by $\Delta \phi_s = \omega_s \Delta t$, the corresponding gear rotation and Z-axis displacement are calculated using $\phi_g = m_{gs} \phi_s – (N_s / N_g) \cdot (\Delta l / p_s)$. This logic is implemented in the NC program using parametric cycles.
The simulation is run in VERICUT, which performs Boolean subtraction of the cutter model from the blank. The simulation successfully generates the spiral face gear tooth form, including the characteristic undercut at the inner radius and pointed tip at the outer radius. The simulated gear model is compared with the theoretical CAD model within VERICUT. The comparison shows no gouging, and the maximum deviation on the working flank surfaces is within 9.8 μm, which can be attributed to modeling and discretization errors. This result validates the correctness of the shaving kinematics and the NC program.
Two improved cutter designs were simulated: one with an increased addendum coefficient ($h_{an}^* = 1.25$) and another with an increased profile shift coefficient ($x_n = 0.45$). The first alteration significantly changed the tooth form, while the second produced a similar form but removed slightly more material from the concave flank, proving to be a feasible modification.
Experimental Study on Gear Shaving
An experimental gear shaving test was conducted on a CNC machining center using the designed and manufactured axial-groove shaving cutter. A batch of die-cast zinc alloy spiral face gears was used as workpieces. The key parameters for the experiment are summarized below:
| Category | Parameter | Symbol | Value |
|---|---|---|---|
| Workpiece (Gear) | Number of Teeth | $N_g$ | 36 |
| Inner Radius | $R_1$ | 11.7 mm | |
| Outer Radius | $R_2$ | 14.5 mm | |
| Shaving Cutter | Number of Teeth | $N_s$ | 7 |
| Helix Angle | $\beta$ | 52° LH | |
| Normal Module | $m_n$ | 0.65 mm | |
| Profile Shift Coefficient | $x_n$ | 0.277 | |
| Machine Setup | Shaft Angle | $\gamma_m$ | 90° |
| Offset | $E$ | 7 mm | |
| Spindle Speed / Radial Feed | $n_s$ / $v_{s0}$ | 2000 rpm / 50 mm/min |
The cutter was mounted on the machine spindle, and the workpiece was clamped on the rotary table. After alignment, the shaving process was executed. The gear shaving operation successfully removed material from the die-cast gear teeth, resulting in a visibly smoother surface finish compared to the as-cast state.
To quantitatively evaluate the effect of gear shaving, the surface roughness of the gear teeth before and after shaving was measured using an Infinite Focus optical 3D measurement system. The measured arithmetic mean roughness ($R_a$) values are as follows:
| Condition | Convex Flank ($R_a$) | Concave Flank ($R_a$) |
|---|---|---|
| Before Shaving (As-Cast) | 0.8527 μm | 0.8365 μm |
| After Shaving | 0.6419 μm | 0.6311 μm |
The results show a significant reduction in surface roughness on both flanks after the shaving process. The concave flank, which is the primary driving surface in this application, achieved a roughness of 0.6311 μm. This experiment successfully demonstrates the practical feasibility and effectiveness of the proposed spiral face gear gear shaving method for improving the surface quality of die-cast gears.
Conclusions and Future Work
This research proposed and investigated a gear shaving method for finishing die-cast spiral face gears. The main conclusions are:
- The mathematical model for the tooth surface generation of the spiroid face gear was established based on the meshing theory with an imaginary pinion cutter. The conditions for avoiding undercut and pointing were derived to determine the feasible inner and outer radii.
- The principle of gear shaving for this gear type was formulated. The kinematic model incorporating the necessary radial feed and the compensating rotational motion was developed and validated mathematically.
- An axial-groove shaving cutter was designed and manufactured. A five-axis NC machining simulation was successfully implemented in VERICUT. The deviation between the simulated gear model and the theoretical model was within an acceptable range, confirming the kinematic correctness.
- Practical gear shaving experiments were conducted on a CNC machining center. Measurement results confirmed that the surface roughness of the gear teeth was significantly improved after shaving, proving the method’s practical feasibility and effectiveness.
The innovation of this work lies in the application of the gear shaving process to spiroid face gears, featuring a simple cutter design based on the mating pinion and implementable on commercially available multi-axis CNC machines.
Future work may focus on:
- Cutter Modification Design: Introducing deliberate modifications (e.g., crowning) to the shaving cutter tooth surface to achieve a localized bearing contact between the shaved gear and the original pinion, potentially improving meshing performance and reducing noise.
- Dynamic Analysis and Vibration Testing: Investigating the dynamic characteristics, tooth stresses, and vibration of the spiral face gear drive to assess its service life and operational behavior.
- Grooves on Gear Teeth: Exploring the effect of machining grooves along the contact path on the gear’s concave flank to alter the meshing frequency and potentially reduce vibration perception.