The spiral face gear is a planar gear whose modulus and helix angle vary continuously along the tooth width. Because it meshes with a helical involute cylindrical pinion in a staggered-axis arrangement, it offers a compact structure, a high contact ratio, steady transmission, low noise, a wide range of transmission ratios and a strong load-carrying capacity. In my research I focused on one central problem: the conventional generating methods that are widely used for straight face gears, namely gear shaping, gear hobbing and gear grinding, cannot reproduce the large helix angle of a spiral face gear with sufficient accuracy. Hot die casting is generally used at present, but the resulting tooth surface hardness and pitch accuracy are too low for precision transmission in high-end machinery. Therefore I proposed and investigated a two-degree-of-freedom gear cutting method, in which a straight-toothed conical skiving cutter removes material from the spiral face gear blank while the two members perform a forced generating motion combined with a continuous radial feed. In the following I describe the theoretical tooth surface model, the kinematic model of the gear cutting process, the error analysis, the principal curvature model, the cutter structure and a numerical control gear cutting simulation.
1. Modeling of the theoretical tooth surface
I began by deriving the tooth surface of a right-hand involute helical cylindrical pinion, since this pinion defines the theoretical conjugate surface of the spiral face gear. Let \(r_{bt}\) be the base radius, \(\lambda_{bt}\) the lead angle on the base cylinder, \(p_t\) the helical parameter and \(\theta_t\), \(\mu_t\) the two surface parameters. The position vector of a point on flank I of the pinion, expressed in the coordinate system \(S_t\) rigidly attached to the pinion, is
$$\mathbf{r}_t^{I}(\theta_t,\mu_t)=\begin{bmatrix} r_{bt}\cos\theta_t+\mu_t\cos\lambda_{bt}\sin\theta_t\\[2pt] r_{bt}\sin\theta_t-\mu_t\cos\lambda_{bt}\cos\theta_t\\[2pt] -\mu_t\sin\lambda_{bt}-p_t\theta_t\end{bmatrix}$$
and the corresponding unit normal is
$$\mathbf{n}_t^{I}=\begin{bmatrix}-\sin\lambda_{bt}\sin\theta_t\\[2pt] \sin\lambda_{bt}\cos\theta_t\\[2pt] \cos\lambda_{bt}\end{bmatrix}$$
For flank II the sign of the parameter \(\mu_t\) is reversed, giving
$$\mathbf{r}_t^{II}(\theta_t,\mu_t)=\begin{bmatrix} r_{bt}\cos\theta_t-\mu_t\cos\lambda_{bt}\sin\theta_t\\[2pt] r_{bt}\sin\theta_t+\mu_t\cos\lambda_{bt}\cos\theta_t\\[2pt] \mu_t\sin\lambda_{bt}-p_t\theta_t\end{bmatrix},\qquad \mathbf{n}_t^{II}=\begin{bmatrix}\sin\lambda_{bt}\sin\theta_t\\[2pt] -\sin\lambda_{bt}\cos\theta_t\\[2pt] \cos\lambda_{bt}\end{bmatrix}$$
The helical parameter is related to the base radius and the base lead angle by \(p_t = r_{bt}\tan\lambda_{bt}\), and the transverse pressure angle \(\alpha_t\) at an arbitrary radius satisfies
$$\cos\alpha_t=\frac{r_{bt}}{r_{pt}},\qquad \tan\lambda_{bt}=\frac{\tan\lambda_{pt}}{\cos\alpha_t}$$
The angular offset of the involute at the base circle is
$$\theta_{t0}=\frac{\pi}{2N_t}-\operatorname{inv}\alpha_t,\qquad \operatorname{inv}\alpha_t=\tan\alpha_t-\alpha_t$$
where \(N_t\) is the number of teeth of the imaginary pinion.
2. Gear shaping coordinate system and meshing equation
To obtain the theoretical tooth surface of the spiral face gear I constructed the installation layout of the imaginary pinion and the spiral face gear. The pinion rotates about its own axis with angular velocity \(\omega_t\) and the face gear rotates with \(\omega_c\), the two being linked by the ratio
$$i_{ct}=\frac{\omega_t}{\omega_c}=\frac{N_c}{N_t}$$
Four coordinate systems were defined: two moving systems \(S_t\) and \(S_c\) rigidly connected to the pinion and the face gear, and two fixed systems \(S_p\) and \(S_q\). The centre distance between the two origins is taken as
$$L_0=\frac{L_1+L_2}{2}$$
where \(L_1\) and \(L_2\) are the inner and outer radii of the spiral face gear. The overall transformation from \(S_t\) to \(S_c\) is the product
$$\mathbf{M}_{ct}=\mathbf{M}_{cf}\,\mathbf{M}_{fe}\,\mathbf{M}_{et}$$
with the individual matrices
$$\mathbf{M}_{et}=\begin{bmatrix}\cos\varphi_t & \sin\varphi_t & 0 & 0\\ -\sin\varphi_t & \cos\varphi_t & 0 & 0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix},\quad \mathbf{M}_{fe}=\begin{bmatrix}1&0&0&-L_0\\ 0&0&-1&0\\ 0&1&0&0\\ 0&0&0&1\end{bmatrix},\quad \mathbf{M}_{cf}=\begin{bmatrix}\cos\varphi_c & -\sin\varphi_c & 0 & 0\\ \sin\varphi_c & \cos\varphi_c & 0 & 0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}$$
Because the two conjugate surfaces are tangent at every instant of contact and the relative velocity must be perpendicular to the common normal, the meshing equation of the shaping process is
$$f(\theta_t,\mu_t,\varphi_t)=\mathbf{n}_t\cdot\mathbf{v}_t^{(tc)}=0$$
The relative velocity of the pinion with respect to the face gear, written in \(S_t\), is
$$\mathbf{v}_t^{(tc)}=\mathbf{v}_t^{(t)}-\mathbf{v}_t^{(c)}=\bigl(\boldsymbol{\omega}_t^{(t)}-\boldsymbol{\omega}_t^{(c)}\bigr)\times\mathbf{r}_t-\mathbf{R}_t\times\boldsymbol{\omega}_t^{(c)}$$
where \(\mathbf{R}_t=[-L_0,\,0,\,0]^{\mathsf T}\) and the angular velocity of the face gear expressed in \(S_t\) is \(\boldsymbol{\omega}_t^{(c)}=\omega_c[\cos\varphi_t,\,-\sin\varphi_t,\,0]^{\mathsf T}\). Substitution gives the explicit form
$$f=\pm i_{ct}\bigl[\sin\lambda_{bt}\sin\theta_t\,(x_t\cos\varphi_t-y_t\sin\varphi_t)+\cos\lambda_{bt}\,z_t\bigr]+\sin\lambda_{bt}\bigl[x_t\sin\theta_t-y_t\cos\theta_t-L_0\sin\theta_t\bigr]=0$$
Combining the meshing equation with the family of surfaces swept by the pinion tooth surface yields the working tooth surface of the spiral face gear:
$$\begin{cases}\mathbf{r}_c(\theta_t,\mu_t,\varphi_t)=\mathbf{M}_{ct}(\varphi_t)\,\mathbf{r}_t(\theta_t,\mu_t)\\[4pt] f(\theta_t,\mu_t,\varphi_t)=0\end{cases}$$
The fillet, or transition surface, is generated by the tip of the imaginary pinion. At the tip, \(\theta_t\) takes the fixed value
$$\theta_t^{*}=\frac{\sqrt{r_{at}^2-r_{bt}^2}}{r_{bt}}$$
and therefore
$$\mathbf{r}_c^{*}(\theta_t^{*},\mu_t,\varphi_t)=\mathbf{M}_{ct}(\varphi_t)\,\mathbf{r}_t(\theta_t^{*},\mu_t)$$
3. Tooth width design
The tooth width of a spiral face gear is controlled by two limits: undercutting at the inner radius and pointing at the outer radius. The undercutting limit function is obtained from the theory of envelope singularities,
$$F(\theta_t,\mu_t,\varphi_t)=\begin{vmatrix} \dot{x}_t & \dot{y}_t & \dot{z}_t\\[2pt] \partial x_t/\partial\mu_t & \partial y_t/\partial\mu_t & \partial z_t/\partial\mu_t\\[2pt] \partial f/\partial\mu_t & \partial f/\partial\theta_t & \partial f/\partial\varphi_t\end{vmatrix}=0$$
Solving this equation together with the meshing equation at the tip value \(\theta_t^{*}\) gives the coordinates of the singular point, and hence the minimum inner radius
$$L_1=\sqrt{x_t^{*2}+y_t^{*2}}$$
Pointing occurs where the two flanks of the same tooth intersect at the tip, so that
$$x_{cI}=x_{cII},\qquad y_{cI}=y_{cII}$$
Solving this pair of equations gives the coordinate of the pointing point and the maximum outer radius
$$L_2=\sqrt{x_t’^{2}+y_t’^{2}}$$
The tooth width is then
$$H=L_2-L_1$$
The two flanks do not give identical values of \(L_1\) because the concave flank undercuts earlier than the convex flank; I therefore adopted the larger of the two values so that neither flank is undercut.
4. Digital model of the theoretical spiral face gear
To build an accurate geometric model I discretized the tooth surface. Each spatial point of the tooth surface corresponds to one point on the axial section obtained by rotation projection. If a grid node in the projection plane has coordinates \((R_{hij},Z_{hij})\), the mapping is
$$R_{hij}=L_1+\frac{j-1}{8}(L_2-L_1),\qquad Z_{hij}=r_{ft}-\frac{i-1}{4}\bigl(r_{ft}-r_{at}\bigr)$$
and the correspondence between the spatial point and its projection is
$$R_{hij}=\sqrt{x_t^2+y_t^2},\qquad Z_{hij}=z_t$$
These two relations form a system of nonlinear equations in the surface parameters. I solved it by the Newton iteration method, using the converged solution of each node as the initial guess for the neighbouring node in order to guarantee convergence. Twenty-one columns along the tooth width and nine rows along the tooth height were used for each flank, giving one hundred and eighty-nine discrete points per flank. The computed point clouds were exported and imported into a three-dimensional modelling environment, where they were fitted into curves, blended into surfaces and finally arrayed to produce the complete digital model of the spiral face gear.
5. Principle of the two-degree-of-freedom gear cutting method
The existing generating methods cannot handle a large helix angle, so I proposed a gear cutting method in which a straight-toothed conical cutter performs a two-degree-of-freedom generating motion. Three members participate in the process: the imaginary generating gear, the gear cutting cutter, and the spiral face gear. At every instant the three members are in contact at a single common point, which means that the cutter and the face gear are conjugate through the imaginary generating gear. This instantaneous point-contact relationship is the basis of the whole gear cutting principle.
5.1 Installation layout
The axis of the imaginary generating gear is orthogonal to the axis of the spiral face gear. The axis of the gear cutting cutter is skew with respect to the axis of the face gear, and the skew angle between the cutter axis and the generating gear axis is denoted by \(\gamma\). When the cutter is a straight-toothed cutter, this angle equals the helix angle of the imaginary generating gear:
$$\gamma=\beta$$
The offset distance between the cutter axis and the generating gear axis is
$$E_{gs}=r_{ag}-r_{as}$$
where \(r_{ag}\) and \(r_{as}\) are the tip radii of the generating gear and of the cutter respectively.
5.2 Motion analysis
During gear cutting the spiral face gear rotates about its axis with angular velocity \(\omega_2\) while the cutter rotates about its own axis with \(\omega_s\). These two rotations form the generating motion. In addition the cutter translates continuously along the radial direction of the face gear with velocity \(v\), and this translation generates the whole tooth width. The translation can be decomposed into a component along the cutter axis and a component in the direction of the cutter rotation, and the second component disturbs the correct meshing relation. To compensate, an additional rotation is superimposed either on the cutter or on the face gear. I chose to apply it to the face gear, so that the relation between the three motion parameters becomes
$$\varphi_2=i_{s2}\varphi_s+p_s s,\qquad p_s=\frac{N_s}{2\sin\beta},\qquad i_{s2}=\frac{N_s}{N_2}$$
with the additional rotation
$$\Delta\omega_2=\frac{2v\sin\beta}{N_2 m}$$
If the compensation is applied to the cutter instead, the sign is reversed and
$$\Delta\omega_s=-\frac{2v\sin\beta}{N_s m}$$
The resulting relation between the rotational speeds is
$$\omega_2=\frac{\omega_s}{i_{s2}}+\frac{2v\sin\beta}{N_2 m}$$
6. Cutter geometry and the skiving tooth surface
6.1 Cutter tooth profile
The normal profile of the imaginary generating gear is very close to an involute profile, so I replaced it by a straight involute profile and used it as the cutting edge of the gear cutting cutter. The number of teeth of the cutter is taken as the equivalent number of teeth of the generating gear:
$$N_s=\frac{N_g}{\cos^3\beta}$$
For flank I of the cutter the position vector is
$$\mathbf{r}_s^{I}(\theta_s)=\begin{bmatrix} r_{bs}\bigl[\sin(\theta_{s0}+\theta_s)-\theta_s\cos(\theta_{s0}+\theta_s)\bigr]\\[2pt] r_{bs}\bigl[\cos(\theta_{s0}+\theta_s)+\theta_s\sin(\theta_{s0}+\theta_s)\bigr]\\[2pt] 0\end{bmatrix}$$
and the unit normal is
$$\mathbf{n}_s^{I}=\begin{bmatrix}\sin(\theta_{s0}+\theta_s)\\[2pt] \cos(\theta_{s0}+\theta_s)\\[2pt] 0\end{bmatrix}$$
For flank II the signs of the second term of the first component and of the first component of the normal are reversed. The angular offset at the base circle is
$$\theta_{s0}=\frac{\pi}{2N_s}-\operatorname{inv}\alpha$$
6.2 Coordinate systems for gear cutting
I established six coordinate systems: three moving systems \(S_s\), \(S_g\) and \(S_2\) rigidly attached to the cutter, the imaginary generating gear and the spiral face gear, two fixed systems \(S_c\) and \(S_p\), and one auxiliary system \(S_b\). After the cutter, the generating gear and the face gear have rotated through \(\varphi_s\), \(\varphi_g\) and \(\varphi_2\) respectively, and after the cutter has translated by the distance \(s\), the overall transformation from \(S_s\) to \(S_2\) is
$$\mathbf{M}_{s2}=\mathbf{M}_{2m}\,\mathbf{M}_{mp}\,\mathbf{M}_{pb}\,\mathbf{M}_{bc}\,\mathbf{M}_{cs}$$
and the individual matrices are
$$\mathbf{M}_{cs}=\begin{bmatrix}\cos\varphi_s & \sin\varphi_s & 0 & 0\\ -\sin\varphi_s & \cos\varphi_s & 0 & 0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix},\quad \mathbf{M}_{bc}=\begin{bmatrix}1&0&0&0\\ 0&\cos\gamma & -\sin\gamma & 0\\ 0&\sin\gamma & \cos\gamma & 0\\ 0&0&0&1\end{bmatrix}$$
$$\mathbf{M}_{pb}=\begin{bmatrix}1&0&0&-E_{gs}\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix},\quad \mathbf{M}_{mp}=\begin{bmatrix}1&0&0&-L_1\\ 0&0&-1&0\\ 0&1&0&0\\ 0&0&0&1\end{bmatrix},\quad \mathbf{M}_{2m}=\begin{bmatrix}\cos\varphi_2 & -\sin\varphi_2 & 0 & 0\\ \sin\varphi_2 & \cos\varphi_2 & 0 & 0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}$$
6.3 Enveloping equation of the gear cutting process
Because the two motion parameters \(\varphi_s\) and \(s\) are linked by the relation \(\varphi_2=i_{s2}\varphi_s+p_s s\), the family of surfaces swept by the cutter edge is a single-parameter family. The family, expressed in the coordinate system of the face gear, is
$$\mathbf{r}_2(\theta_s,\varphi_s)=\mathbf{M}_{s2}(\varphi_s)\,\mathbf{r}_s(\theta_s)$$
and the enveloping condition is
$$\left(\frac{\partial \mathbf{r}_2}{\partial \theta_s}\times\frac{\partial \mathbf{r}_2}{\partial \varphi_s}\right)\cdot\frac{\partial \mathbf{r}_2}{\partial s}=0$$
Expanding this determinant gives the meshing equation of the gear cutting process, which takes the compact form
$$\sum_{k=1}^{9} t_k=0$$
where the nine terms \(t_1\) to \(t_9\) collect the coefficients arising from the involute parameters, the offset distance, the axial setting distance and the rotational angles. Eliminating \(\varphi_s\) between the family of surfaces and the meshing equation gives the working tooth surface of the spiral face gear produced by gear cutting:
$$\begin{cases}\mathbf{r}_2(\theta_s,\varphi_s)=\mathbf{M}_{s2}(\varphi_s)\,\mathbf{r}_s(\theta_s)\\[4pt] f(\theta_s,\varphi_s)=0\end{cases}$$
The transition surface follows from the tip value of the cutter parameter,
$$\theta_s^{*}=\frac{\sqrt{r_{as}^2-r_{bs}^2}}{r_{bs}},\qquad \mathbf{r}_2^{*}=\mathbf{M}_{s2}(\varphi_s)\,\mathbf{r}_s(\theta_s^{*})$$
7. Error analysis of the skiving tooth surface
Let the tooth surface produced by the imaginary pinion be called the theoretical surface and the surface produced by the gear cutting cutter be called the skiving surface. I defined the tooth surface error as the normal distance between the two surfaces at corresponding grid points:
$$\delta=(\mathbf{r}_{2s}-\mathbf{r}_{2g})\cdot\mathbf{n}_t$$
where \(\mathbf{n}_t\) is the inward unit normal of the theoretical surface.
Two numerical examples were computed. In both examples the helical pinion had twenty-four teeth, a normal modulus of \(3.25\) mm and a normal pressure angle of \(20^{\circ}\), while the spiral face gear had seventy-two teeth. The helix angle and the shaft angle were both \(15^{\circ}\) in the first example and both \(25^{\circ}\) in the second. The cutter had twenty-seven teeth in the first example and thirty-two teeth in the second.
| Object | Parameter | Example 1 | Example 2 |
|---|---|---|---|
| Gear cutting cutter | Number of teeth \(N_s\) | 27 | 32 |
| Module \(m\) (mm) | 3.25 | 3.25 | |
| Pressure angle \(\alpha\) (deg) | 20 | 20 | |
| Hand | straight | straight | |
| Imaginary generating gear | Number of teeth \(N_g\) | 24 | 24 |
| Normal modulus \(m_n\) (mm) | 3.25 | 3.25 | |
| Normal pressure angle (deg) | 20 | 20 | |
| Helix angle \(\beta\) (deg) | 15 | 25 | |
| Hand | right | right | |
| Spiral face gear | Number of teeth \(N_2\) | 72 | 72 |
| Inner radius \(L_1\) (mm) | 117 | 125 | |
| Outer radius \(L_2\) (mm) | 137 | 147 | |
| Tooth width \(H\) (mm) | 20 | 22 | |
| Hand | left | left | |
| Shaft angle | Magnitude (deg) | 15 | 25 |
| Direction | left offset | left offset |
The computed tooth surface errors are summarized below. On the concave flank the error of the working part of the surface is essentially zero, while the transition part shows a larger deviation because of undercutting. On the convex flank the error remains within a narrow band.
| Case | Flank | Working surface error | Transition surface behaviour |
|---|---|---|---|
| \(\gamma=\beta=15^{\circ}\) | concave | approximately \(0\) | large deviation caused by undercutting |
| \(\gamma=\beta=15^{\circ}\) | convex | \(-0.011\) to \(-0.017\) mm | gradual deviation |
| \(\gamma=\beta=25^{\circ}\) | concave | approximately \(0\) | large deviation caused by undercutting |
| \(\gamma=\beta=25^{\circ}\) | convex | \(-0.040\) to \(0.001\) mm | gradual deviation |
I also examined the tooth profile error by cutting cylindrical sections at three radii and projecting the resulting three-dimensional profiles onto the principal planes. In the first example the radii were \(127\) mm, \(118.5\) mm and \(117\) mm, and in the second example they were \(136\) mm, \(127.2\) mm and \(125\) mm. On the convex flank the skiving profile and the theoretical profile coincided almost completely at every radius. On the concave flank the two profiles also coincided except near the inner end, where the undercutting produced a visible deviation. These results confirmed that the working part of the spiral face gear produced by gear cutting is geometrically consistent with the theoretical surface, and they provided the first verification of the correctness of the proposed gear cutting method.
8. Principal curvature and principal directions
To characterize the local shape of the skiving tooth surface I derived the principal curvatures and principal directions. The unit normal of the theoretical surface is
$$\mathbf{n}=\frac{\mathbf{r}_{\theta_2}\times\mathbf{r}_{\varphi_2}}{\lvert \mathbf{r}_{\theta_2}\times\mathbf{r}_{\varphi_2}\rvert}$$
The first fundamental quantities are
$$E=\mathbf{r}_{\theta_2}\cdot\mathbf{r}_{\theta_2},\qquad F=\mathbf{r}_{\theta_2}\cdot\mathbf{r}_{\varphi_2},\qquad G=\mathbf{r}_{\varphi_2}\cdot\mathbf{r}_{\varphi_2}$$
and the second fundamental quantities are
$$L=\mathbf{r}_{\theta_2\theta_2}\cdot\mathbf{n},\qquad M=\mathbf{r}_{\theta_2\varphi_2}\cdot\mathbf{n},\qquad N=\mathbf{r}_{\varphi_2\varphi_2}\cdot\mathbf{n}$$
Applying the Rodrigues formula and eliminating the surface parameter increments gives the quadratic equation for the principal curvatures:
$$(EG-F^2)K^2+(2FM-EN-GL)K+(LN-M^2)=0$$
The discriminant of this equation is
$$\Delta=(2FM-EN-GL)^2-4(EG-F^2)(LN-M^2)\geq 0$$
so that two real distinct principal curvatures \(K_I\) and \(K_{II}\) always exist unless the surface is umbilically flat. The corresponding principal directions are obtained from the ratio \(\mathrm{d}\theta_2/\mathrm{d}\varphi_2\) determined by the same equation, and the unit vectors along these directions are
$$\mathbf{e}_i=\frac{\mathbf{r}_{\theta_2}+\dfrac{\mathrm{d}\theta_2}{\mathrm{d}\varphi_2}\mathbf{r}_{\varphi_2}}{\left\lvert \mathbf{r}_{\theta_2}+\dfrac{\mathrm{d}\theta_2}{\mathrm{d}\varphi_2}\mathbf{r}_{\varphi_2}\right\rvert},\qquad i=I,II$$
This model provides a quantitative description of the flatness and the bending of the skiving surface at every point and forms the basis for subsequent tooth surface modification.
9. Design of the gear cutting cutter
The two main issues in cutter design are chip removal and interference. During gear cutting the chip is formed by the front cutting edge and is carried on the rake face until the cutter separates from the blank. Because the interference problem dominates, I designed the cutter as a conical straight-toothed cutter. A relief angle of \(24.5^{\circ}\) was found to eliminate interference completely. The initial basic parameters are listed below.
| Parameter | Symbol | Value |
|---|---|---|
| Module | \(m\) | 3.25 mm |
| Number of teeth | \(N_s\) | 27 |
| Pressure angle | \(\alpha\) | \(20^{\circ}\) |
| Relief angle | \(\eta\) | \(24.5^{\circ}\) |
| Addendum coefficient | \(h_a^{*}\) | 1.0 |
| Dedendum coefficient | \(c^{*}\) | 0.25 |
| Transverse pitch | \(H_s\) | 10 mm |
The geometric dimensions of the front cutting edge follow the standard involute relations:
$$d=N_s m,\qquad d_b=d\cos\alpha,\qquad h_a=h_a^{*}m,\qquad h_f=c^{*}m$$
$$d_a=d+2h_a,\qquad d_f=d-2h_f$$
A three-dimensional model of the cutter was built from these dimensions and used in the subsequent simulation.
10. Numerical control simulation of the gear cutting process
To verify the gear cutting method at the level of the machine tool I carried out a virtual machining experiment. The kinematic model of the gear cutting process contains two rotations and one continuous translation, and I mapped these onto a five-axis machine tool with three linear axes and two rotary axes.
10.1 Machine tool model
In the machine coordinate system the transformation from the cutter to the face gear has the form
$$\mathbf{M}_{s2}^{C}=\begin{bmatrix}\cos D_2 & -\sin D_2 & 0 & 0\\ \sin D_2 & \cos D_2 & 0 & 0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}\begin{bmatrix}1&0&0&-D_3\\ 0&1&0&-D_4\\ 0&0&1&-D_5\\ 0&0&0&1\end{bmatrix}\begin{bmatrix}\cos D_1 & \sin D_1 & 0 & 0\\ -\sin D_1 & \cos D_1 & 0 & 0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}$$
Identifying this matrix with the theoretical transformation gives the two rotary motion parameters
$$D_1=\varphi_s,\qquad D_2=i_{s2}\varphi_s+p_s s$$
and the three linear motion parameters
$$D_3=-L_1\sin\gamma,\qquad D_4=E_{gs},\qquad D_5=-L_1\cos\gamma$$
together with the two feed components
$$D_6=s\sin\gamma,\qquad D_7=s\cos\gamma$$
10.2 Simulation workflow
The simulation was carried out in the following steps: building the machine tool model, selecting the numerical control system, building the cutter and blank models, writing the numerical control program, and running the comparison analysis. A sinusoidal numerical control system was selected because of its support for the macro used to switch the broaching mode on and off. The cutter and blank models were first created in a three-dimensional modelling environment and then transferred to the simulation environment through the standard interface. The blank was clamped on the workpiece spindle and the cutter was mounted on the tool spindle with automatic clamping and automatic tool setting.
The numerical control program was written according to the kinematic relations derived above. The radial feed rate was chosen as \(0.0166666\) mm/s, corresponding to a feed displacement of \(0.0001\) mm per step and a time interval of \(0.006\) s. Decomposed along the machine axes this becomes \(0.000008586161733\) mm per step along the \(Y\) axis and \(0.00009366721892\) mm per step along the \(Z\) axis. The rotational parameters were updated at every step from the relation between \(\varphi_s\) and \(\varphi_2\).
| Step | Quantity | Expression or value |
|---|---|---|
| 1 | Radial feed speed \(v\) | 0.0166666 mm/s |
| 2 | Feed per step \(\Delta s\) | 0.0001 mm |
| 3 | Time interval \(\Delta t\) | 0.006 s |
| 4 | \(Y\) component per step | 0.000008586161733 mm |
| 5 | \(Z\) component per step | 0.00009366721892 mm |
| 6 | Cutter rotation \(D_1\) | updated from \(\varphi_s\) |
| 7 | Face gear rotation \(D_2\) | \(i_{s2}\varphi_s+p_s s\) |
| 8 | Termination criterion | \(Z\) coordinate threshold |
10.3 Simulation results
The virtual gear cutting experiment produced a complete three-dimensional model of the spiral face gear. The characteristic undercutting near the root and pointing near the tip appeared in the simulated geometry, as expected. I then compared the simulated model with the theoretical model using the automatic comparison function of the simulation environment. Undercutting and overcutting were displayed with different colours, and a tolerance band was set for the comparison.
The comparison showed no overcutting on the working tooth surface. Only a small amount of residual material remained on the transition surface, which does not participate in meshing and can therefore be neglected. The maximum deviation on the working tooth surface between the simulated model and the theoretical model was less than \(22.6\ \mu\mathrm{m}\). This value is consistent with the theoretical error analysis presented earlier, especially when the modelling accuracy of the machine tool, the cutter and the blank, the import and export tolerances, the rounding errors and the internal tolerances of the simulation environment are taken into account. The virtual gear cutting experiment therefore provides a second and independent verification of the correctness of the proposed method.
| Comparison item | Result | Remark |
|---|---|---|
| Overcutting on working surface | none detected | geometry is not violated |
| Undercutting on working surface | none detected | full tooth height generated |
| Residual material on transition surface | small | not a meshing surface |
| Maximum working surface deviation | less than \(22.6\ \mu\mathrm{m}\) | consistent with theoretical prediction |
| Undercut and pointing | reproduced | typical face gear behaviour |
11. Concluding remarks
I proposed and investigated a two-degree-of-freedom gear cutting method for spiral face gears with large helix angles. The main outcomes of my work can be summarized as follows. The tooth surface of a right-hand involute helical pinion was derived from the spatial curve generation principle, and the installation layout and coordinate systems of the shaping process were established. The meshing equation of the shaping process was obtained from the theory of conjugate surfaces, and the working tooth surface and the transition tooth surface of the spiral face gear were derived from the envelope of the pinion surface family. The tooth width was determined from the undercutting limit at the inner radius and the pointing limit at the outer radius.
Based on the instantaneous point-contact relation among the imaginary generating gear, the gear cutting cutter and the spiral face gear, I built the installation layout, the kinematic model and the coordinate systems of the gear cutting process. A straight-toothed conical cutter whose tooth profile approximates the normal profile of the imaginary generating gear was adopted. The meshing equation of the gear cutting process was derived by the enveloping method, and the skiving tooth surface was obtained. The theoretical error analysis showed that the working part of the skiving surface coincides with the theoretical surface, whereas the transition part deviates because of undercutting. The principal curvature and principal direction model was established from the first and second fundamental quantities of the surface.
Finally, a five-axis numerical control gear cutting simulation was constructed. The rotary and linear motion parameters were derived from the one-to-one correspondence between the theoretical coordinate systems and the machine coordinate systems. The virtual experiment produced a complete spiral face gear model, and the comparison with the theoretical model showed a maximum working surface deviation below \(22.6\ \mu\mathrm{m}\). Both the theoretical error analysis and the virtual gear cutting experiment therefore confirm that the proposed method is correct and feasible for spiral face gears with large helix angles. The method combines the advantages of shaping and hobbing in a single continuous gear cutting operation, avoids the idle strokes of shaping, and requires only a relatively simple straight-toothed cutter, which makes it a promising candidate for the precision manufacturing of spiral face gears.