I investigate a practical route for internal meshing spiral bevel gear cutting in nutation reducers used as robot joint reducers. The central problem is not the outer bevel gear but the internal bevel gear, whose cone angle is larger than \(90^\circ\). Conventional spiral bevel gear cutting theories are mainly developed for external bevel gears, and they cannot be transferred directly to the internal member without a systematic treatment of curvature interference, secondary cutting, tool tilt, and machine motion. My work therefore combines the generating gear principle, the meshing theory of conjugate surfaces, cutter-tilting kinematics, finite element analysis, multibody kinematic simulation, and virtual five-axis gear cutting verification. The final aim is to obtain a manufacturable tooth surface model for an internal meshing double-circular-arc spiral bevel gear and to show that the proposed gear cutting scheme is feasible before a physical trial cut is attempted.
Industrial robot joint reducers require high reduction ratio, compact axial size, low backlash, and stable transmission. Cycloidal-pinwheel reducers and harmonic reducers are widely used, but their core patents and manufacturing know-how are concentrated in a small number of suppliers. A nutation drive is attractive because it can achieve a large ratio in a single stage and can be arranged coaxially. In a nutation reducer, the external bevel gear performs a nutating motion with a fixed nutation angle, while the internal bevel gear is fixed or used as the reaction member. The relative motion is similar to a coin rolling on a table with a fixed inclination. This motion creates a slow output rotation, and the reduction ratio is governed by the difference between the numbers of teeth of the two bevel gears. For the internal meshing configuration, the reduction ratio can be written as
$$ i=\frac{1}{1-\frac{Z_1}{Z_2}}=\frac{Z_2}{Z_2-Z_1}, $$
where \(Z_1\) is the number of teeth of the external bevel gear and \(Z_2\) is the number of teeth of the internal bevel gear. A small tooth-number difference produces a large reduction ratio. This is the main kinematic advantage of the nutation drive. However, the same small tooth-number difference also makes the internal gear cutting process sensitive to tooth surface errors. Any error in the internal tooth surface is amplified in the contact pattern and in the transmission error.
| Symbol | Meaning | Typical role in gear cutting |
|---|---|---|
| \(Z_1\) | External bevel gear tooth number | Determines cutter position and generating motion |
| \(Z_2\) | Internal bevel gear tooth number | Determines internal gear cutting interference risk |
| \(\delta_1\) | External pitch cone angle | Controls machine setup and generating roll |
| \(\delta_2\) | Internal pitch cone angle | Larger than \(90^\circ\) for nutation drive |
| \(\beta\) | Spiral angle | Controls tooth trace curvature and contact orientation |
| \(\alpha_n\) | Normal pressure angle | Defines double-circular-arc profile |
| \(m_n\) | Normal module | Scales tooth height and cutter geometry |
| \(b\) | Face width | Affects cutter radius and interference depth |
| \(u\) | Radial cutter position | Sets cutter center relative to generating gear |
| \(q\) | Angular cutter position | Sets angular location of cutter center |
| \(I\) | Theoretical cutter tilt angle | Avoids secondary cutting in internal gear cutting |
I base the tooth surface design on the generating gear principle. The generating gear is a virtual gear that is conjugate to both the external bevel gear and the internal bevel gear. During gear cutting, the cutter and the workpiece perform a relative motion that is equivalent to the meshing motion between the generating gear and the target gear. The cutting edges of the cutter sweep a family of surfaces, and the envelope of this family is the tooth surface. For nutation drives with double-circular-arc spiral bevel gears, the generating gear is not a simple plane gear. I use a conical generating gear for the internal member because a plane generating gear can cause curvature interference. The geometric relations between the generating gear and the two bevel gears are summarized as follows:
$$ Z_c=\frac{Z_1}{\sin\delta_1}=\frac{Z_2}{\sin\delta_2}, $$
$$ r_c=\frac{r_1}{\sin\delta_1}=\frac{r_2}{\sin\delta_2}, $$
$$ \frac{\varphi_c}{\sin\delta_1}=\frac{\varphi_1}{1}, \qquad \frac{\varphi_c}{\sin\delta_2}=\frac{\varphi_2}{1}, $$
$$ I_{c1}=\frac{\omega_c}{\omega_1}=\frac{Z_1}{Z_c}=\sin\delta_1, \qquad I_{c2}=\frac{\omega_c}{\omega_2}=\frac{Z_2}{Z_c}=\sin\delta_2. $$
These relations allow me to describe the external and internal bevel gears through a common generating gear. This is important for gear cutting because the same cutter geometry and the same generating motion can be adapted to both members by changing the coordinate transformation and the machine settings. The double-circular-arc profile is used because it provides a convex arc and a concave arc in one tooth, which improves load capacity and bending strength compared with a single arc profile. The basic profile consists of eight arc segments. In the normal profile coordinate system \(S_n\), a point on the \(i\)-th arc can be expressed as
$$ x_{ni}=r_{ni}\sin\alpha_i+E_i, \qquad y_{ni}=r_{ni}\cos\alpha_i+F_i, \qquad z_{ni}=0, $$
where \(i=1,2,\ldots,8\), \(r_{ni}\) is the arc radius, \(\alpha_i\) is the arc angle parameter, and \((E_i,F_i)\) is the arc center. The eight segments include the working convex arc, the working concave arc, and the transition and root arcs. In gear cutting, only the working arcs should participate in the meshing contact under normal load. The transition arcs should not become load-bearing surfaces, although they can come into contact under edge loading if the tooth surface is not properly modified.
| Arc segment | Function | Desired role in meshing |
|---|---|---|
| 1 | Convex working arc | Main load-bearing surface |
| 2 | Convex transition arc | No normal contact |
| 3 | Convex root arc | Clearance and fillet |
| 4 | Convex tip arc | Tip relief and clearance |
| 5 | Concave tip arc | Tip relief and clearance |
| 6 | Concave root arc | Clearance and fillet |
| 7 | Concave transition arc | No normal contact |
| 8 | Concave working arc | Main load-bearing surface |
For the tooth trace, I use a circular arc generated by the cutter head. The mid-point spiral angle is the design spiral angle \(\beta\). The cutter position relative to the generating gear is defined by the radial cutter position \(u\) and the angular cutter position \(q\). The cutter center \(O_t\) is located in the generating gear coordinate system, and the tooth trace arc is the path of the cutter edge. For a mid-point spiral angle \(\beta\), the cutter position can be calculated from the triangle formed by the generating gear center, the cutter center, and the mid-point of the tooth trace. In general form,
$$ q=\arctan\left(\frac{R_t\cos\beta}{R_c-R_t\sin\beta}\right), \qquad u=\frac{R_t\cos\beta}{\sin q}, $$
where \(R_t\) is the cutter radius and \(R_c\) is the generating gear radius at the mid-point. When single-side cutting is used, the concave and convex flanks are cut separately. The angular cutter position is then shifted by half of the tooth width angle to maintain the same design spiral angle on both flanks. This is important for gear cutting because double-side cutting can produce different mid-point spiral angles on the two flanks, which leads to uneven tooth thickness and a poor contact pattern.
The generating gear tooth surface is obtained by transforming the cutter rotation surface into the generating gear coordinate system. The transformation from the normal profile coordinate system \(S_n\) to the cutter coordinate system \(S_t\) is
$$ \mathbf{r}_t=\mathbf{M}_{tn}\mathbf{r}_n, $$
where
$$ \mathbf{M}_{tn}=
\begin{bmatrix}
1 & 0 & 0 & 0\\
0 & \cos\theta & -\sin\theta & R_t\cos\theta\\
0 & \sin\theta & \cos\theta & R_t\sin\theta\\
0 & 0 & 0 & 1
\end{bmatrix}. $$
Here \(\theta\) is the cutter rotation angle. The transformation from \(S_t\) to the generating gear coordinate system \(S_c\) is
$$ \mathbf{M}_{ct}=
\begin{bmatrix}
\cos q & -\sin q & 0 & -u\sin q\\
\sin q & \cos q & 0 & u\cos q\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{bmatrix}. $$
Therefore, the generating gear tooth surface is
$$ \mathbf{r}_c=\mathbf{M}_{ct}\mathbf{M}_{tn}\mathbf{r}_n. $$
This surface is the envelope of the cutter family in the generating gear coordinate system. It is the theoretical surface from which the external and internal bevel gear tooth surfaces are derived. In gear cutting, the generating gear is not a physical part. It is a kinematic reference that allows the machine axes to be set so that the cutter removes the correct material from the workpiece.
To obtain the external bevel gear tooth surface, I introduce the fixed reference coordinate system \(S_0\), the external gear coordinate system \(S_1\), and the generating gear coordinate system \(S_c\). The transformation from \(S_c\) to \(S_1\) is
$$ \mathbf{r}_1=\mathbf{M}_{1c}\mathbf{r}_c, \qquad \mathbf{M}_{1c}=\mathbf{M}_{1m}\mathbf{M}_{m0}\mathbf{M}_{0c}. $$
The individual matrices are
$$ \mathbf{M}_{0c}=
\begin{bmatrix}
\cos\varphi_c & \sin\varphi_c & 0 & 0\\
-\sin\varphi_c & \cos\varphi_c & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{bmatrix}, $$
$$ \mathbf{M}_{m0}=
\begin{bmatrix}
1 & 0 & 0 & 0\\
0 & \sin\delta_1 & \cos\delta_1 & 0\\
0 & -\cos\delta_1 & \sin\delta_1 & 0\\
0 & 0 & 0 & 1
\end{bmatrix}, $$
$$ \mathbf{M}_{1m}=
\begin{bmatrix}
\cos\varphi_1 & -\sin\varphi_1 & 0 & 0\\
\sin\varphi_1 & \cos\varphi_1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{bmatrix}. $$
For the internal bevel gear, the transformation is
$$ \mathbf{r}_2=\mathbf{M}_{2c}\mathbf{r}_c, \qquad \mathbf{M}_{2c}=\mathbf{M}_{2m}\mathbf{M}_{m0}\mathbf{M}_{0c}, $$
$$ \mathbf{M}_{m0}=
\begin{bmatrix}
1 & 0 & 0 & 0\\
0 & -\sin\delta_2 & \cos\delta_2 & 0\\
0 & \cos\delta_2 & \sin\delta_2 & 0\\
0 & 0 & 0 & 1
\end{bmatrix}, $$
$$ \mathbf{M}_{2m}=
\begin{bmatrix}
\cos\varphi_2 & -\sin\varphi_2 & 0 & 0\\
\sin\varphi_2 & \cos\varphi_2 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{bmatrix}. $$
The meshing equation is obtained from the condition that the relative velocity at the contact point is perpendicular to the common normal. This is the fundamental condition for conjugate tooth surfaces. It can be written as
$$ \Phi=\mathbf{n}_c\cdot\mathbf{v}_c^{(i)}=0, $$
where \(\mathbf{n}_c\) is the unit normal of the generating gear tooth surface and \(\mathbf{v}_c^{(i)}\) is the relative velocity between the generating gear and the target gear \(i\). The unit normal is
$$ \mathbf{n}_c=\frac{\mathbf{r}_{c,\alpha}\times\mathbf{r}_{c,\theta}}{|\mathbf{r}_{c,\alpha}\times\mathbf{r}_{c,\theta}|}. $$
The relative velocity for the external gear is
$$ \mathbf{v}_c^{(1)}=\boldsymbol{\omega}_c\times\mathbf{r}_c-\boldsymbol{\omega}_1\times\mathbf{r}_1, $$
and for the internal gear it is
$$ \mathbf{v}_c^{(2)}=\boldsymbol{\omega}_c\times\mathbf{r}_c-\boldsymbol{\omega}_2\times\mathbf{r}_2. $$
Because the generating gear and each target gear are conjugate, there is no relative rolling in the normal direction. Substituting the coordinate transformations and the angular velocity relations into the meshing equation gives an equation for the profile parameter \(\alpha\) as a function of the cutter rotation angle \(\theta\) and the generating gear rotation angle. In a compact form,
$$ \tan\alpha=\frac{\cos\varphi\sin(\theta-q)+u\sin\varphi}{\sin(\theta-q)+u\cos\varphi}. $$
This equation is used to determine the contact line on the generating gear surface. By combining the meshing equation with the generating gear surface equation, I obtain the theoretical tooth surface equations for the external and internal bevel gears. The external gear surface can be written as
$$ \mathbf{r}_1=\mathbf{M}_{1c}\mathbf{r}_c(\alpha,\theta), \qquad \Phi(\alpha,\theta,\varphi_c)=0, $$
and the internal gear surface can be written as
$$ \mathbf{r}_2=\mathbf{M}_{2c}\mathbf{r}_c(\alpha,\theta), \qquad \Phi(\alpha,\theta,\varphi_c)=0. $$
These equations describe the ideal conjugate surfaces. They are the starting point for the gear cutting analysis, but they are not yet a machining solution. The internal bevel gear has a cone angle larger than \(90^\circ\), and the tooth space is enclosed by the gear body. A cutter that is large enough to generate the theoretical surface may collide with the opposite flank or with the root region. This is the secondary cutting problem. In addition, the relative curvature between the generating gear and the internal gear can become negative or positive in a way that causes curvature interference. Therefore, I analyze the internal gear cutting process in two steps: first, the curvature interference condition, and second, the secondary cutting condition.
For curvature interference, I consider a conical generating gear. The internal gear pitch cone angle is \(\delta_2\), and the generating gear cone angle is \(\delta_c\). The shaft angle is
$$ \Sigma=\delta_2+\delta_c. $$
The induced normal curvature along the contact line direction can be expressed as
$$ K_c^{(2)}=\frac{p^2}{A_2\left(p\cdot v-n\cdot q\right)}, $$
where
$$ p=\boldsymbol{\omega}_c\times\mathbf{n}, \qquad q=\left(\boldsymbol{\omega}_c-\boldsymbol{\omega}_2\right)\times\mathbf{R}, \qquad v=\frac{\boldsymbol{\omega}_c-\boldsymbol{\omega}_2}{|\boldsymbol{\omega}_c-\boldsymbol{\omega}_2|}. $$
After substituting the angular velocity components and the pressure angle \(\phi_2\), the induced curvature takes the form
$$ K_c^{(2)}=
\frac{i_{c2}^2+2i_{c2}\cos\phi_2+\sin^2\phi_2}{A_2\sin^2\phi_2}
-\frac{\sin^2\Sigma}{A_2\sin^2\phi_2}\cos\phi_2. $$
The sign of this curvature determines whether curvature interference occurs. For the convex flank, interference is avoided when \(K_c^{(2)}<0\). For the concave flank, interference is avoided when \(K_c^{(2)}>0\). My analysis shows that when the shaft angle satisfies \(0<\Sigma<180^\circ\), the conical generating gear can avoid curvature interference for both flanks. Since \(\Sigma=\delta_2+\delta_c\), the cone angle of the conical generating gear only needs to satisfy \(\delta_2+\delta_c<180^\circ\). In the nutation drive, the external bevel gear already satisfies this condition when paired with the internal gear. Therefore, the external bevel gear can be used as a conical generating gear for the internal gear. This is the dual generating gear method. It means that the internal gear can be cut by a generating motion that is conjugate to the external gear. This is a key conclusion for internal gear cutting because it avoids the need for a new generating gear geometry.
For secondary cutting, the cutter head rotates around its own axis and is fed into the internal gear tooth space. Because the internal gear tooth surface grows inward, the cutter rotation surface may intersect the workpiece at two locations: the intended tooth flank and an unintended nearby region. The unintended intersection is the secondary cutting region. The depth of secondary cutting increases with the cutter radius. To suppress this effect, I tilt the cutter head in the normal plane of the tooth trace. The tilt angle is called the theoretical cutter tilt angle \(I\). Tilting the cutter changes the effective cutting section and moves the cutter away from the unintended region. However, a large tilt angle introduces asymmetry between the convex and concave flanks. Therefore, I choose the smallest tilt angle that clears the interference. The geometric relation is
$$ i_0=\arcsin\left(\frac{\Delta H+h}{2R_t}\right), $$
$$ \Delta H=(x_1-x_2)\sin\delta_a, $$
$$ I=i_0+\Delta i, $$
where \(\Delta H\) is the maximum interference depth, \(h\) is the tooth height, \(\delta_a\) is the installation angle, \(x_1\) and \(x_2\) are the coordinates of the cutter tip circle intersections with the internal gear outer circle, and \(\Delta i\) is a small safety allowance. This tilt angle is applied in the normal plane of the tooth trace. In actual machine setup, it is more convenient to decompose the theoretical tilt angle into a vertical tilt component and a horizontal component. The vertical component is produced by a tool tilt mechanism, and the horizontal component is absorbed by the installation angle. The decomposition is
$$ \tan i=\frac{\sin I}{\cos\beta}, \qquad \tan\Delta=\frac{\sin I}{\sin\beta}. $$
With this decomposition, the machine does not need a full normal-plane tilt mechanism. Instead, a simpler vertical tilt head and an adjusted installation angle can realize the required cutter orientation. This simplification is important for practical gear cutting because it reduces the number of machine axes and the complexity of the setup.
After introducing the cutter tilt, the generating gear surface is no longer the same as the un-tilted surface. I rebuild the cutter coordinate system with the tilt angle. The tilted cutter coordinate system \(S_i\) is obtained from the cutter coordinate system \(S_t\) by a rotation about the tooth trace tangent. The transformation is
$$ \mathbf{M}_{it}=
\begin{bmatrix}
1 & 0 & 0 & 0\\
0 & \cos i & -\sin i & R_t\cos i\\
0 & \sin i & \cos i & R_t\sin i\\
0 & 0 & 0 & 1
\end{bmatrix}. $$
The transformation from \(S_i\) to the generating gear coordinate system \(S_c\) is
$$ \mathbf{M}_{ci}=
\begin{bmatrix}
1 & 0 & 0 & -u\sin q\\
0 & \cos q & -\sin q & u\cos q\cos i\\
0 & \sin q & \cos q & u\cos q\sin i\\
0 & 0 & 0 & 1
\end{bmatrix}. $$
The tilted cutter rotation surface in the generating gear coordinate system is
$$ \mathbf{r}_c=\mathbf{M}_{ci}\mathbf{M}_{it}\mathbf{r}_n. $$
Expanding this expression gives the tilted generating gear surface. For the double-circular-arc profile, the normal profile point is
$$ \mathbf{r}_n=
\begin{bmatrix}
r_{ni}\sin\alpha_i+E_i\\
r_{ni}\cos\alpha_i+F_i\\
0\\
1
\end{bmatrix}. $$
Substituting into the transformation yields the tilted generating gear surface. The external gear surface is then
$$ \mathbf{r}_{1z}=\mathbf{M}_{1c}\mathbf{r}_c, $$
and the internal gear surface is
$$ \mathbf{r}_{2z}=\mathbf{M}_{2c}\mathbf{r}_c. $$
These surfaces include the effect of the cutter tilt. They are the surfaces that can be produced by the proposed gear cutting method. The theoretical un-tilted surface is still useful as a reference, but the tilted surface is the one that must be evaluated for contact performance. In my numerical example, the theoretical tilt angle is about \(29^\circ\), and the vertical tilt component is about \(26.67^\circ\). The horizontal component is absorbed into the installation angle. The resulting internal gear surface is slightly asymmetric, but the asymmetry is small enough to maintain stable meshing when the gears are paired in the nutation drive.
For the forming method, the relative motion between the cutter and the workpiece is simpler than in the generating method. The cutter is fed along its own axis or along the machine \(Z\)-axis, and the workpiece is indexed after each tooth space. There is no generating roll. The forming method is attractive for internal gear cutting because it reduces the number of simultaneous machine axes and avoids the complex roll motion that is difficult to synchronize for a small internal gear. The forming method coordinate transformation from the generating gear to the workpiece is
$$ \mathbf{r}_{2m}=\mathbf{M}_{m c}\mathbf{r}_c, $$
where
$$ \mathbf{M}_{m c}=
\begin{bmatrix}
\cos\delta_a & 0 & -\sin\delta_a & -X_{2t}\cos\delta_a\\
0 & 1 & 0 & 0\\
\sin\delta_a & 0 & \cos\delta_a & X_{2t}\sin\delta_a\\
0 & 0 & 0 & 1
\end{bmatrix}. $$
Here \(\delta_a\) is the workpiece installation angle and \(X_{2t}\) is the horizontal correction of the workpiece. The forming method tooth surface equation is obtained by substituting the tilted generating gear surface into this transformation. The result is a surface that can be cut by a single plunge and index cycle. This is the basis of the proposed efficient gear cutting scheme for the internal bevel gear.
| Parameter | Value | Unit |
|---|---|---|
| Nutation angle | 10 | deg |
| Spiral angle | 35 | deg |
| Normal pressure angle | 24 | deg |
| Normal module | 2.5 | mm |
| External gear tooth number | 20 | — |
| Internal gear tooth number | 22 | — |
| Gear ratio | 1.1 | — |
| External pitch diameter | 61.0387 | mm |
| Internal pitch diameter | 67.1426 | mm |
| Shaft angle | 170 | deg |
| External pitch cone angle | 56.4411 | deg |
| Internal pitch cone angle | 113.5589 | deg |
| Face width | 15.2709 | mm |
| Outer cone distance | 51.8836 | mm |
| Addendum | 2.25 | mm |
| Dedendum | 2.75 | mm |
| Whole depth | 5 | mm |
I validate the proposed gear cutting surfaces by building a single-stage nutation reducer model. The external gear, internal gear, input shaft, nutation sleeve, and output mechanism are assembled with the correct cone apex coincidence. The cone apexes of the two bevel gears must coincide to ensure correct meshing. In the finite element analysis, I simplify the model by retaining only the two gears and the contact pair. The input shaft and nutation sleeve are replaced by appropriate constraints. The material is assumed to be \(38\)CrMoAl with elastic modulus \(210\) GPa, Poisson ratio \(0.3\), density \(7850\) kg/m\(^3\), and yield strength \(835\) MPa. The contact is defined as a no-penetration surface-to-surface contact. The external gear is loaded with a torque through its inner bore, and the internal gear is fixed. A hinge constraint on the external gear prevents rigid body motion except rotation. The mesh is refined near the contact zone.
| Component | Material | Elastic modulus (GPa) | Poisson ratio | Density (kg/m\(^3\)) | Yield strength (MPa) |
|---|---|---|---|---|---|
| External bevel gear | 38CrMoAl | 210 | 0.3 | 7850 | 835 |
| Internal bevel gear | 38CrMoAl | 210 | 0.3 | 7850 | 835 |
The finite element results show that both the convex working arc and the concave working arc participate in load transfer. The maximum contact stress on the external gear appears near the root region, while the maximum stress on the internal gear appears near the tip region. This indicates that the load distribution is not perfectly uniform. The asymmetry comes from the cutter tilt and from the internal gear geometry. The maximum contact stress is about \(223\) MPa, which is well below the yield strength of \(835\) MPa. Therefore, the gear pair has sufficient strength under the assumed load. However, the non-uniform load distribution suggests that tooth surface modification may be needed to improve the contact pattern and to avoid edge contact under higher loads.
| Result | External gear | Internal gear |
|---|---|---|
| Maximum contact stress | Near root region | Near tip region |
| Working arcs in contact | Convex and concave | Convex and concave |
| Load distribution | Non-uniform | Non-uniform |
| Maximum stress value | About 223 MPa | About 223 MPa |
| Yield strength | 835 MPa | 835 MPa |
| Safety margin | Sufficient | Sufficient |
For kinematic validation, I use a multibody simulation environment. The input shaft is driven with a smooth speed profile that reaches \(600\) deg/s in one second and then remains constant. Gravity is set in the negative \(Y\) direction. Solid contact is defined between the gear flanks. The simulated input speed, external gear speed, and output speed are recorded. The theoretical reduction ratio of the single-stage nutation reducer is \(10\). The output speed should be \(60\) deg/s when the input speed is \(600\) deg/s. The simulation results show that the output speed settles to approximately \(60.27\) deg/s for the theoretical gear surface and approximately \(60.50\) deg/s for the cutter-tilted gear surface. The error is less than one percent in both cases. The cutter-tilted surface produces a slightly larger fluctuation because the convex and concave flanks are not perfectly symmetric. Nevertheless, the transmission remains stable, and the reduction ratio is satisfied. The external gear also exhibits a periodic angular velocity component about the \(Z\)-axis, which is consistent with the nutation motion.
| Case | Theoretical amplitude (deg/s) | Simulated amplitude (deg/s) | Error |
|---|---|---|---|
| Un-tilted theoretical gear | 60 | 60.27 | 0.45% |
| Cutter-tilted gear | 60 | 60.50 | 0.85% |
The kinematic results confirm that the cutter-tilted internal gear can mesh correctly with the external gear in the nutation drive. The output fluctuation is small enough for a joint reducer application, although further tooth surface optimization would be beneficial. The simulation also shows that the acceleration phase produces larger fluctuations than the steady phase. This is expected because the contact ratio and the dynamic load change during acceleration. In the steady phase, the output speed is stable, and the transmission error is within acceptable limits. From the viewpoint of gear cutting, this means that the proposed tool tilt and forming method do not destroy the fundamental meshing behavior of the nutation drive.
To verify the machining feasibility, I build a virtual five-axis gear cutting machine. The machine structure includes three linear axes \(X\), \(Y\), and \(Z\), a workpiece rotary axis \(C\), and a workpiece tilting axis \(B\). The cutter spindle rotates about its own axis. The \(X\) and \(Y\) axes together replace the cradle motion of a traditional mechanical gear cutting machine. The \(Z\) axis provides the feed motion. The \(B\) axis sets the installation angle of the workpiece. The \(C\) axis indexes the workpiece after each tooth space. The kinematic chain is arranged so that the cutter is mounted on the \(Y\) axis, the \(Y\) axis is mounted on the \(X\) axis, and the workpiece is mounted on the \(B\) axis, which is mounted on the \(Z\) axis. This arrangement allows the machine to reproduce the required relative motion between the cutter and the workpiece.
| Machine axis | Function in gear cutting | Relation to generating motion |
|---|---|---|
| \(X\) | Horizontal cutter position | Radial cutter position component |
| \(Y\) | Vertical cutter position | Angular cutter position component |
| \(Z\) | Feed and retract | Cutting depth and safety clearance |
| \(B\) | Workpiece installation angle | Pitch cone angle and tilt correction |
| \(C\) | Workpiece indexing | Tooth space division |
| Spindle | Cutter rotation | Cutting speed and cutting edge sweep |
The cutter is not a standard straight-sided disk cutter. Because the internal gear cone angle is large and the required tilt angle is significant, I modify the cutter into a truncated conical cutter. The cutting edge profile remains the double-circular-arc profile in the normal section. The cutter axis is tilted so that the effective cutting edge is normal to the desired tooth surface at the cutting point. The cutter radius is chosen so that the projected cutter position in the machine plane matches the theoretical radial and angular cutter positions. This design reduces the risk of secondary cutting while preserving the desired spiral angle. The cutter is modeled as a rotation surface for the virtual gear cutting simulation. The normal section is generated from the eight-arc double-circular-arc profile.
For the forming method, the machine settings are derived from the generating gear geometry. The horizontal cutter position \(X_C\), vertical cutter position \(Y_C\), installation angle \(B_C\), theoretical tilt angle \(I\), vertical tilt angle \(i\), indexing angle \(C_C\), cutter radius \(R_t\), horizontal wheel position \(X_L\), and bed position \(Z_L\) are calculated from the gear parameters. The relations are
$$ X_C=u\cos q, \qquad Y_C=u\sin q\cos i, $$
$$ B_C=\frac{\pi}{2}-\delta_2-\Delta, $$
$$ C_C=\frac{\pi}{Z_2}, $$
$$ X_L=L\cos B_C, \qquad Z_L=L\sin B_C. $$
The resulting machine settings for the example internal gear are listed in the following table. These settings are used to configure the virtual machine and to generate the numerical control program. The program follows a simple cycle: the cutter and workpiece move to the prepared position, the spindle starts, the workpiece feeds along \(Z\), the tooth space is cut to the required depth, the workpiece retracts, and the \(C\) axis indexes to the next tooth space. This cycle is repeated until all internal teeth are cut.
| Parameter | Internal flank cutting | External flank cutting |
|---|---|---|
| Horizontal cutter position \(X_C\) (mm) | 33.98 | 37.95 |
| Vertical cutter position \(Y_C\) (mm) | 26.01 | 19.75 |
| Installation angle \(B_C\) (deg) | 114.625 | 114.625 |
| Theoretical tilt angle \(I\) (deg) | 29.000 | 29.000 |
| Vertical tilt angle \(i\) (deg) | 26.673 | 26.673 |
| Indexing angle \(C_C\) (deg) | 16.364 | 16.364 |
| Cutter radius \(R_t\) (mm) | 28.584 | 28.584 |
| Horizontal wheel position \(X_L\) (mm) | 41.500 | 41.500 |
| Bed position \(Z_L\) (mm) | 32.074 | 32.074 |
The virtual gear cutting simulation is carried out in a numerical control simulation environment. The machine model is assembled from the kinematic chain, the control system is configured, the cutter is loaded as a rotation surface, and the workpiece blank is loaded as a stock model. The simulation shows the cutter entering the internal tooth space, cutting the flanks, and retracting without collision. The final internal gear shape matches the design intent. The simulation also confirms that the vertical tilt angle and the adjusted installation angle can be realized by the five-axis machine. The secondary cutting region is eliminated by the cutter tilt, and the remaining tooth surface is smooth enough for the subsequent lapping or grinding operation if required. This virtual gear cutting verification supports the feasibility of the proposed method.
From the results, I draw several conclusions. First, the double-circular-arc spiral bevel gear can be used in a nutation drive, and its tooth surface can be derived from a common generating gear. The external and internal gear surfaces are conjugate to the same generating gear, which simplifies the design and the gear cutting setup. Second, the internal gear cannot be cut reliably with an un-tilted cutter because of curvature interference and secondary cutting. A conical generating gear is needed to avoid curvature interference, and a cutter tilt is needed to avoid secondary cutting. The theoretical tilt angle can be decomposed into a vertical tilt angle and a horizontal installation angle correction. This makes the gear cutting process compatible with a five-axis machine. Third, the cutter-tilted tooth surface produces stable meshing in the nutation reducer. The finite element analysis shows that both working arcs participate in load transfer, although the load distribution is not perfectly uniform. The maximum contact stress is well below the yield strength. Fourth, the multibody simulation shows that the output speed error is less than one percent, which is acceptable for a joint reducer. Fifth, the virtual gear cutting simulation demonstrates that the proposed forming method can produce the internal gear with a simple plunge-and-index cycle. This is a significant improvement over the complex five-axis milling of the entire tooth surface as a free-form surface.
The main innovation of this work is the combination of the double-circular-arc spiral bevel gear with the nutation drive and the development of a cutter-tilted forming method for the internal gear. The method reduces the number of required machine axes, avoids the most serious interference problems, and provides a practical path for gear cutting of internal meshing nutation reducers. The second innovation is the derivation of the tilted generating gear surface and the corresponding tooth surface equations. These equations allow the designer to evaluate the effect of the cutter tilt before machining. The third innovation is the virtual gear cutting verification, which shows that the proposed machine settings and cutter design can produce the internal gear without collision.
Further work is still needed. The current study assumes ideal machine geometry and ideal cutter geometry. In actual gear cutting, axis errors, cutter wear, thermal deformation, and workpiece clamping errors will affect the tooth surface. These errors should be introduced into the tooth surface model and into the contact analysis. The tooth surface optimization should also be studied. The double-circular-arc profile has two working arcs, but the load distribution between them is not equal. A proper modification of the cutter profile and the spiral angle may improve the contact pattern and reduce the sensitivity to misalignment. In addition, a physical trial cut should be performed to measure the actual tooth surface and to compare it with the virtual gear cutting result. The measurement data can then be used to correct the machine settings and to refine the gear cutting process. With these improvements, the internal meshing nutation reducer can become a competitive candidate for robot joint applications, and the proposed gear cutting method can be extended to other internal bevel gear systems with large cone angles.