I propose a high-order segmented topology modification method for a face gear pair. The central idea is to place a controlled topology deviation on the pinion tooth surface along a predefined contact path and along the instantaneous contact line, so that the face gear pair can maintain a favorable contact pattern, a low transmission error, and a reduced sensitivity to installation misalignment. I use a combination of 2nd-order and 4th-order piecewise functions, and I derive the modified tooth surface by superimposing the modification amount along the theoretical surface normal. I then perform tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA) to evaluate the meshing performance of the face gear pair. The investigation covers the influence of contact path angle, modification length, modification order, and maximum modification amount. The finite element results show that the high-order segmented modification can reduce the maximum contact stress, lower the loaded transmission error amplitude, and make the contact trace extend from the inner-diameter root region toward the outer-diameter tip region of the face gear.
Nomenclature
| Symbol | Meaning |
|---|---|
| \(m_n\) | Normal module of the face gear pair |
| \(\alpha_n\) | Normal pressure angle |
| \(\gamma_m\) | Shaft angle between the pinion and the face gear |
| \(N_1\) | Number of pinion teeth |
| \(N_2\) | Number of face gear teeth |
| \(b_1\) | Face width of the pinion |
| \(D_1\) | Inner diameter of the face gear |
| \(D_2\) | Outer diameter of the face gear |
| \(\Sigma_S\) | Shaping cutter tooth surface |
| \(\Sigma_1\) | Pinion tooth surface |
| \(\Sigma_2\) | Face gear tooth surface |
| \(\mathbf{r}_S,\mathbf{n}_S\) | Position vector and normal vector of the shaping cutter surface |
| \(\mathbf{r}_1,\mathbf{n}_1\) | Position vector and normal vector of the pinion surface |
| \(\mathbf{r}_2,\mathbf{n}_2\) | Position vector and normal vector of the face gear surface |
| \(\mathbf{r}_m,\mathbf{n}_m\) | Position vector and normal vector of the modified pinion surface |
| \(\delta\) | Topology modification amount along the surface normal |
| \(\xi\) | Angle of the predefined contact path relative to the pinion axis |
| \(\eta\) | Angle of the instantaneous contact line relative to the pinion axis |
| \(z_1\) | Modification function along the contact path |
| \(z_2\) | Modification function along the instantaneous contact line |
| \(\Delta_B,\Delta_C\) | Maximum modification amounts at the ends of the contact path |
| \(\Delta_A,\Delta_D\) | Maximum modification amounts along the instantaneous contact line |
| \(k_{Bi},k_{Ci}\) | Segmented modification length coefficients |
| \(\varphi_1,\varphi_2\) | Rotation angles of the pinion and the face gear |
| \(\Delta q\) | Axial installation error of the face gear |
| \(\Delta p\) | Axial installation error of the pinion |
| \(\Delta\gamma\) | Shaft angle installation error |
| \(\Delta E\) | Center distance installation error |
| \(\varepsilon\) | Contact ratio of the face gear pair |
Generation Principle of the Face Gear Pair
I begin with the generation of the face gear tooth surface. The face gear is generated by a shaping cutter that rotates with the face gear according to a fixed ratio. The instantaneous generating surface is the cylindrical gear tooth surface of the cutter. The face gear surface is the envelope of the cutter surface family in the face gear coordinate system. Let the cutter surface parameters be \(u_S\) and \(l_S\). The position vector and unit normal vector on the cutter surface are \(\mathbf{r}_S(u_S,l_S)\) and \(\mathbf{n}_S(u_S,l_S)\). The meshing equation between the shaping cutter and the face gear is
$$f(u_S,l_S,\varphi_S)=0$$
where \(\varphi_S\) is the rotation angle of the shaping cutter. The family of cutter surfaces and their normals in the face gear coordinate system \(S_2\) can be written as
$$\mathbf{r}_2(u_S,l_S,\varphi_S)=\mathbf{M}_{2S}\mathbf{r}_S(u_S,l_S)$$
$$\mathbf{n}_2(u_S,l_S,\varphi_S)=\mathbf{L}_{2S}\mathbf{n}_S(u_S,l_S)$$
Here \(\mathbf{M}_{2S}\) is the coordinate transformation matrix from the cutter coordinate system \(S_S\) to the face gear coordinate system \(S_2\), and \(\mathbf{L}_{2S}\) is the corresponding vector transformation matrix. By combining the meshing equation with the transformation equations, I obtain the theoretical face gear surface \(\mathbf{r}_2(u_2,l_2)\) and its normal \(\mathbf{n}_2(u_2,l_2)\). The pinion surface is obtained in the same way by replacing the cutter subscript with the pinion subscript. In this way, the face gear pair is defined by two conjugate surfaces that satisfy the meshing condition under ideal alignment.
Rotational Projection of the Pinion Tooth Surface
To define the modification region, I project the pinion tooth surface onto a plane. The pinion coordinate system is \(S_1\), and its \(z_1\)-axis coincides with the pinion axis. I introduce an auxiliary coordinate system \(S_0\). The center point of the pinion tooth surface is \(O_p\). I establish a projection coordinate system \(S_p\) with axes \(x_p\) and \(y_p\), corresponding to the face width direction and the tooth height direction, respectively. The pinion face width is \(b_1\), and the pinion pitch radius is \(r_{p1}\). For a point on the pinion tooth surface, the rotational projection into \(S_0\) is
$$x_0(u_1,l_1)=z_1(u_1,l_1)$$
$$y_0(u_1,l_1)=\sqrt{x_1^2(u_1,l_1)+y_1^2(u_1,l_1)}$$
$$z_0(u_1,l_1)=0$$
where \(x_1(u_1,l_1)\), \(y_1(u_1,l_1)\), and \(z_1(u_1,l_1)\) are the coordinate components of the theoretical pinion position vector \(\mathbf{r}_1(u_1,l_1)\). The transformation to the projection coordinate system is
$$[x_p,y_p,z_p,1]^T=\mathbf{M}_{p0}[x_0,y_0,z_0,1]^T$$
The transformation matrix is
$$\mathbf{M}_{p0}=\begin{bmatrix}1&0&0&\frac{b_1}{2}\\0&1&0&-r_{p1}\\0&0&1&0\\0&0&0&1\end{bmatrix}$$
I define four corner points on the pinion tooth surface: \(A\), \(B\), \(C\), and \(D\). These corners correspond to the tooth tip and tooth root at the inner and outer diameters of the face gear. The contact path \(X_1\) and the instantaneous contact line \(X_2\) pass through the center point \(O_p\). Their angles relative to the axis \(X_p\) are \(\xi\) and \(\eta\), respectively. For an arbitrary point \(Q\) on the projected tooth surface, the projection coordinate along the contact path is \(x_1\), and the projection coordinate along the instantaneous contact line is \(x_2\). These coordinates are
$$x_1=x_p\cos\xi-y_p\sin\xi$$
$$x_2=\frac{x_p\sin\xi+y_p\cos\xi}{\sin(\xi+\eta)}$$
These two projected coordinates form the basis of the topology modification functions. The contact path angle \(\xi\) controls the orientation of the contact trace on the face gear tooth surface, while the instantaneous contact line angle \(\eta\) controls the local direction along which the load is distributed. By choosing these angles, I can place the contact path diagonally across the face gear tooth surface. This diagonal placement is important for a face gear pair because the tooth height and pressure angle vary along the face width. A diagonal contact path allows the contact pattern to avoid the edges during misalignment and to use more of the working tooth surface.
High-Order Segmented Modification Functions
I design two modification functions. The first function \(z_1\) is applied along the contact path. The second function \(z_2\) is applied along the instantaneous contact line. The total modification amount at any point is the sum of the two functions. The function along the contact path is piecewise and high-order. I use a 4th-order polynomial in the end regions and a zero-modification region in the middle. This creates a flat central region and a smooth transition toward the tooth edges. The piecewise form is
$$z_1(x_1)=\begin{cases}
a_{Bi}(x_1-x_{B0})^i, & x_{B1}\le x_1 < x_{B0}\\
0, & x_{B0}\le x_1 < x_{C0}\\
a_{Ci}(x_1-x_{C0})^i, & x_{C0}\le x_1\le x_{C1}
\end{cases}$$
Here \(i\) is the order of the modification polynomial. In this study, I use \(i=2\) and \(i=4\) for comparison. The points \(B_1\) and \(C_1\) are the projections of the tooth surface corners \(B\) and \(C\) onto the contact path. The points \(B_0\) and \(C_0\) are the starting points of the modification at the two ends of the contact path. The quantities \(\Delta_B\) and \(\Delta_C\) are the modification amounts at \(B_1\) and \(C_1\). The coefficients \(a_{Bi}\) and \(a_{Ci}\) determine the shape of the modification curve. They are
$$a_{Bi}=\frac{\Delta_B}{(k_{Bi}|B_1O_1|)^i}$$
$$a_{Ci}=\frac{\Delta_C}{(k_{Ci}|C_1O_1|)^i}$$
The parameters \(k_{Bi}\) and \(k_{Ci}\) are the segmented modification length coefficients. When \(k_{Bi}<1\) or \(k_{Ci}<1\), the modification starts before the tooth edge, and the central region retains a smaller modification amount. This is the segmented feature of the method. When \(k_{Bi}=k_{Ci}=1\), the modification extends over the full contact path. The function along the instantaneous contact line is a 2nd-order function. It is
$$z_2(x_2)=\begin{cases}
a_A x_2^2, & 0\le x_2\le x_{A2}\\
a_D x_2^2, & x_{D2}\le x_2<0
\end{cases}$$
The points \(A_2\) and \(D_2\) are the projections of the corners \(A\) and \(D\) onto the instantaneous contact line. The coefficients are
$$a_A=\frac{\Delta_A}{|A_2O_2|^2}$$
$$a_D=\frac{\Delta_D}{|D_2O_2|^2}$$
The quantities \(\Delta_A\) and \(\Delta_D\) are the maximum modification amounts along the instantaneous contact line. The total modification amount is
$$\delta(u_1,l_1)=z_1(u_1,l_1)+z_2(u_1,l_1)$$
This sum defines a topology deviation surface. The deviation surface is not a simple profile modification or lead modification. At each face width section, the tooth profile modification curve is different. Therefore, the method is a true topology modification of the pinion tooth surface. I obtain the modified pinion surface by adding the modification amount along the theoretical surface normal:
$$\mathbf{r}_m=\mathbf{r}_l+\delta\mathbf{n}_l$$
where \(\mathbf{r}_l\) and \(\mathbf{n}_l\) are the theoretical pinion position and unit normal vectors, and \(\mathbf{r}_m\) is the modified position vector. The modified unit normal is
$$\mathbf{n}_m=\frac{\partial\mathbf{r}_m}{\partial u_l}\times\frac{\partial\mathbf{r}_m}{\partial l_l}$$
I then normalize \(\mathbf{n}_m\) for the contact analysis. The modified surface is smooth because the piecewise functions are continuous at the transition points and the 4th-order polynomial has continuous first and second derivatives at the boundaries when the coefficients are chosen consistently. This smoothness is important for avoiding stress concentrations and for obtaining stable numerical convergence in the finite element analysis.
Tooth Contact Analysis with Installation Errors
I establish a tooth contact analysis model for the modified face gear pair. The model includes the pinion coordinate system \(S_1\), the face gear coordinate system \(S_2\), and a fixed coordinate system \(S_f\). The pinion and face gear rotation angles are \(\varphi_1\) and \(\varphi_2\). I introduce auxiliary coordinate systems \(S_{10}\) and \(S_{20}\) to include installation errors. The installation errors are the face gear axial offset \(\Delta q\), the pinion axial offset \(\Delta p\), the shaft angle error \(\Delta\gamma\), and the center distance error \(\Delta E\). The tangency condition for the two modified surfaces in the fixed coordinate system is
$$\mathbf{r}_{f1}(u,l,\varphi_1)=\mathbf{r}_{f2}(u_S,l_S,\varphi_S,\varphi_2)$$
$$\mathbf{n}_{f1}(u,l,\varphi_1)=\mathbf{n}_{f2}(u_S,l_S,\varphi_S,\varphi_2)$$
$$f(u_S,l_S,\varphi_S)=0$$
Here \(\mathbf{r}_{f1}\) and \(\mathbf{n}_{f1}\) are the position and normal vectors of the modified pinion surface in \(S_f\), and \(\mathbf{r}_{f2}\) and \(\mathbf{n}_{f2}\) are the corresponding vectors of the face gear surface. I solve these equations for a sequence of pinion rotation angles \(\varphi_1\). The solution gives a series of contact points. The path formed by these points is the contact trace on the face gear tooth surface. The contact ratio is
$$\varepsilon=\left|\frac{\varphi_{1B}-\varphi_{1A}}{\varphi_{1n}}\right|$$
where \(\varphi_{1A}\) is the pinion angle at the beginning of meshing, \(\varphi_{1B}\) is the pinion angle at the end of meshing, and \(\varphi_{1n}\) is the pinion meshing period. The transmission error is defined as the difference between the actual face gear rotation and the ideal rotation for a given pinion rotation:
$$\Delta\varphi_2=\varphi_2-\frac{N_1}{N_2}\varphi_1$$
I use this definition to evaluate the unloaded transmission error and the loaded transmission error. The unloaded transmission error is obtained from the TCA solution. The loaded transmission error is obtained from the finite element analysis. A low transmission error amplitude and a smooth transmission error curve are desirable because they reduce vibration and noise in the face gear pair.
Basic Parameters and Modification Cases
I use a representative face gear pair for the numerical study. The basic parameters are listed in Table 1. The normal module is 3.9 mm, the normal pressure angle is 25°, and the shaft angle is 90°. The pinion has 21 teeth, and the face gear has 142 teeth. The pinion face width is 55 mm. The inner diameter of the face gear is 507 mm, and the outer diameter is 610 mm. This configuration represents a moderately loaded face gear pair with a large gear ratio. The large gear ratio makes the face gear pair sensitive to installation errors, so it is a suitable case for evaluating the proposed modification method.
| Parameter | Value |
|---|---|
| Normal module \(m_n\) | 3.9 mm |
| Normal pressure angle \(\alpha_n\) | 25° |
| Shaft angle \(\gamma_m\) | 90° |
| Pinion teeth \(N_1\) | 21 |
| Face gear teeth \(N_2\) | 142 |
| Pinion face width \(b_1\) | 55 mm |
| Face gear inner diameter \(D_1\) | 507 mm |
| Face gear outer diameter \(D_2\) | 610 mm |
I define six modification cases to examine the influence of the design parameters. The cases are listed in Table 2. Cases 1, 2, and 3 use different contact path angles \(\xi\): 6.425°, 20°, and 40°. Cases 1, 4, and 5 use different maximum modification amounts: \(\Delta_B=\Delta_C=10\), 20, and 30 μm. Case 6 uses the same maximum modification amount as Case 1, but it uses segmented modification with \(k_{Bi}=k_{Ci}=0.6\). The instantaneous contact line modification amounts are kept equal to the contact path modification amounts in these cases. The order of the contact path function is \(i=4\). I also compare the 2nd-order and 4th-order functions for the same maximum modification amount in a later section.
| Case | \(\xi\) (°) | \(\Delta_B\) (μm) | \(\Delta_C\) (μm) | \(k_{Bi}\) | \(k_{Ci}\) | Order \(i\) |
|---|---|---|---|---|---|---|
| 1 | 6.425 | 10 | 10 | 1 | 1 | 4 |
| 2 | 20 | 10 | 10 | 1 | 1 | 4 |
| 3 | 40 | 10 | 10 | 1 | 1 | 4 |
| 4 | 6.425 | 20 | 20 | 1 | 1 | 4 |
| 5 | 6.425 | 30 | 30 | 1 | 1 | 4 |
| 6 | 6.425 | 10 | 10 | 0.6 | 0.6 | 4 |
Influence of Contact Path Angle
I first examine the influence of the contact path angle \(\xi\) under standard alignment. The results are summarized in Table 3. As \(\xi\) increases from 6.425° to 40°, the contact ratio decreases from 2.5 to 1.625. The contact trace length decreases from 44.89 mm to 10.52 mm. The contact trace rotates around the modification zero point and becomes concentrated near the middle of the face width. The transmission error amplitude decreases from 0.208″ to 0.009″. The transmission error curve becomes flatter and approaches zero. These results show that a large contact path angle concentrates the contact pattern near the center of the face gear tooth. This reduces the transmission error amplitude, but it also reduces the contact ratio and the contact trace length. For a face gear pair, a very short contact trace may not be desirable because it reduces the load-sharing area. Therefore, I select a moderate contact path angle that balances the contact ratio and the transmission error. In the later cases, I focus on \(\xi=6.425°\) because it provides a longer contact trace and a higher contact ratio, and I improve the transmission error by increasing the modification amount and by using segmented modification.
| Case | Contact path angle \(\xi\) (°) | Contact ratio \(\varepsilon\) | Contact trace length (mm) | Transmission error amplitude (″) |
|---|---|---|---|---|
| 1 | 6.425 | 2.5 | 44.89 | 0.208 |
| 2 | 20 | 2.0 | 27.35 | 0.085 |
| 3 | 40 | 1.625 | 10.52 | 0.009 |
Influence of Maximum Modification Amount
I next examine the influence of the maximum modification amount. I compare Cases 1, 4, and 5, where the modification amounts at the contact path ends increase from 10 μm to 20 μm and then to 30 μm. The results are listed in Table 4. As the modification amount increases, the contact trace shifts toward the inner diameter of the face gear. The contact trace length decreases from 44.89 mm to 33.28 mm. The transmission error curve opening becomes smaller. Because the 4th-order function has a small value in the middle and larger values at the two ends, the central region of the face width has a small modification amount. This creates a local transmission error curve that is close to zero in the middle and rises smoothly toward the ends. The 4th-order transmission error curve is therefore flatter in the central region than a 2nd-order curve. However, if the modification amount is too large, the contact trace becomes shorter and the contact area is reduced. I find that a maximum modification amount of about 30 μm provides a good balance between a low transmission error and a sufficiently long contact trace.
| Case | \(\Delta_B=\Delta_C\) (μm) | Contact trace length (mm) | Transmission error amplitude (″) | Contact trace position |
|---|---|---|---|---|
| 1 | 10 | 44.89 | 0.208 | Central and slightly outward |
| 4 | 20 | 38.75 | 0.145 | Shifted slightly inward |
| 5 | 30 | 33.28 | 0.090 | Shifted inward |
Influence of Segmented Modification
I compare Case 1 and Case 6 to evaluate the segmented modification. In Case 1, the modification extends over the full contact path with \(k_{Bi}=k_{Ci}=1\). In Case 6, the modification starts earlier with \(k_{Bi}=k_{Ci}=0.6\). The results are listed in Table 5. Compared with Case 1, Case 6 has a contact trace that shifts slightly toward the inner diameter of the face gear. The contact trace length decreases from 44.89 mm to 42.48 mm. The central region of the face width has only the 2nd-order modification along the instantaneous contact line. The modification amount in the central region is therefore smaller. The transmission error curve becomes flat-topped, and the two ends drop faster. The transmission error amplitude decreases from 0.208″ to 0.102″. This shows that segmented modification can reduce the transmission error amplitude while preserving a relatively long contact trace. The segmented modification is especially useful because it avoids excessive modification in the central region, where the load is highest.
| Case | Segmented coefficients | Contact trace length (mm) | Transmission error amplitude (″) | Central modification |
|---|---|---|---|---|
| 1 | \(k_{Bi}=k_{Ci}=1\) | 44.89 | 0.208 | 4th-order over full path |
| 6 | \(k_{Bi}=k_{Ci}=0.6\) | 42.48 | 0.102 | 2nd-order only in center |
Comparison of 2nd-Order and 4th-Order Modification under Misalignment
I now compare the 2nd-order and 4th-order modification methods under face gear axial installation error \(\Delta q\). I use the modification parameters of Case 5, where \(\Delta_B=\Delta_C=30\) μm. The installation error \(\Delta q\) is set to \(-0.1\) mm, \(0\), and \(+0.1\) mm. A positive \(\Delta q\) means that the two gears move apart. A negative \(\Delta q\) means that the two gears move closer. The results for the 2nd-order modification are summarized in Table 6. Under standard alignment, the contact path passes through the center of the tooth surface. The contact trace length is 16.52 mm, and the transmission error is a symmetric 2nd-order curve. When \(\Delta q=+0.1\) mm, the contact trace shifts inward and its length increases to 25.66 mm. The transmission error amplitude changes significantly and becomes asymmetric. When \(\Delta q=-0.1\) mm, the contact trace shifts outward, and the asymmetry of the transmission error curve is reduced.
| \(\Delta q\) (mm) | Contact trace length (mm) | Transmission error amplitude (″) | Contact trace shift |
|---|---|---|---|
| \(-0.1\) | 20.14 | 0.152 | Outward |
| \(0\) | 16.52 | 0.090 | Central |
| \(+0.1\) | 25.66 | 0.175 | Inward |
The results for the 4th-order modification are summarized in Table 7. Under standard alignment, the contact trace is a diagonal line from the inner-diameter root region to the outer-diameter tip region of the face gear. The contact trace length is 33.28 mm. The transmission error curve is a parabola with a local region close to zero. Under misalignment, the contact trace shifts less than in the 2nd-order case. The contact trace remains longer, and the transmission error amplitude remains smaller. This shows that the 4th-order modification is less sensitive to installation error than the 2nd-order modification. The reason is that the 4th-order function distributes the modification more smoothly along the contact path, so the contact pattern can adapt to small changes in the meshing position without a sudden change in the transmission error.
| \(\Delta q\) (mm) | Contact trace length (mm) | Transmission error amplitude (″) | Contact trace shift |
|---|---|---|---|
| \(-0.1\) | 31.02 | 0.118 | Small outward |
| \(0\) | 33.28 | 0.090 | Diagonal central |
| \(+0.1\) | 29.45 | 0.125 | Small inward |
Finite Element Model for Loaded Contact Analysis
To evaluate the load-bearing performance, I build a finite element model of the face gear pair. I use five tooth pairs to reduce the influence of boundary conditions. The element type is a three-dimensional eight-node fully integrated element, C3D8. I set the number of elements along the spiral direction and the profile direction to 40. The contact is defined as surface-to-surface contact. I apply a torque of 1189.6 N·m to the pinion. I analyze the contact stress distribution, the root bending stress, and the loaded transmission error under different installation errors. I compare the 2nd-order modification, the 4th-order full modification, and the 4th-order segmented modification. For the segmented modification, I use the parameters of Case 6, where the central region has only the instantaneous contact line modification and the end regions have both contact path and instantaneous contact line modifications.
Contact Stress under Different Modification Methods
The contact stress distributions for the 2nd-order modification are summarized in Table 8. Under standard alignment, the contact stress is mainly distributed in the middle of the face width. The contact footprint occupies about 77.39% of the working tooth surface. When the gears move closer with \(\Delta q<0\), the contact footprint shifts toward the outer diameter of the face gear, and the footprint ratio decreases to 75.86%. When the gears move apart with \(\Delta q>0\), the contact footprint shifts toward the inner diameter, and the footprint ratio decreases to 68.45%. This shows that the 2nd-order modification is sensitive to installation error. The contact footprint moves significantly, and the contact area decreases when the installation error is present.
| \(\Delta q\) (mm) | Footprint ratio (%) | Contact position | Maximum contact stress (MPa) |
|---|---|---|---|
| \(-0.1\) | 75.86 | Shifted outward | 1409.58 |
| \(0\) | 77.39 | Central | 1363.91 |
| \(+0.1\) | 68.45 | Shifted inward | 1330.05 |
The contact stress distributions for the 4th-order full modification are summarized in Table 9. Under standard alignment, the contact area agrees well with the predefined contact trace. The maximum contact stress is distributed in the middle of the face width. The footprint ratio is about 83.44%. When the gears move closer, the footprint ratio decreases to 77.67%. When the gears move apart, the footprint ratio decreases to 74.91%. Compared with the 2nd-order modification, the 4th-order modification increases the contact area and reduces the sensitivity to installation error. The maximum contact stress is also lower in all installation error cases.
| \(\Delta q\) (mm) | Footprint ratio (%) | Contact position | Maximum contact stress (MPa) |
|---|---|---|---|
| \(-0.1\) | 77.67 | Slight outward | 1342.17 |
| \(0\) | 83.44 | Central and diagonal | 1289.45 |
| \(+0.1\) | 74.91 | Slight inward | 1305.82 |
The contact stress distributions for the 4th-order segmented modification are summarized in Table 10. Under standard alignment, the maximum contact stress is significantly reduced. The footprint ratio is about 83.89%. When the gears move closer, the contact footprint shifts only slightly, and the footprint ratio is about 83.68%. When the gears move apart, the footprint ratio is about 83.75%. Compared with the 2nd-order and the 4th-order full modifications, the 4th-order segmented modification provides a larger contact area and a lower sensitivity to installation error. The maximum contact stress under standard alignment is 1215.91 MPa. When the gears move closer, the maximum contact stress increases to 1352.07 MPa. When the gears move apart, the maximum contact stress is 1238.17 MPa. These values are lower than the corresponding values for the other two modification methods.
| \(\Delta q\) (mm) | Footprint ratio (%) | Contact position | Maximum contact stress (MPa) |
|---|---|---|---|
| \(-0.1\) | 83.68 | Nearly unchanged | 1352.07 |
| \(0\) | 83.89 | Diagonal and central | 1215.91 |
| \(+0.1\) | 83.75 | Nearly unchanged | 1238.17 |
Root Bending Stress
I also evaluate the maximum bending stress at the tooth root. The results are summarized in Table 11. Under standard alignment, the maximum root bending stress for the 2nd-order modification is 326.01 MPa. For the 4th-order full modification, it is 301.89 MPa. For the 4th-order segmented modification, it decreases to 278.97 MPa. Under different installation errors, the 4th-order segmented modification consistently produces the lowest root bending stress. This indicates that the high-order segmented modification not only reduces the contact stress but also improves the bending stress state of the face gear pair. The reason is that the segmented modification distributes the load more evenly along the contact path and reduces the edge load concentration. The diagonal contact trace also helps to avoid the tooth edges, which are the regions where bending stress is usually highest.
| \(\Delta q\) (mm) | 2nd-order root stress (MPa) | 4th-order full root stress (MPa) | 4th-order segmented root stress (MPa) |
|---|---|---|---|
| \(-0.1\) | 337.45 | 315.62 | 289.34 |
| \(0\) | 326.01 | 301.89 | 278.97 |
| \(+0.1\) | 318.76 | 297.45 | 274.18 |
Loaded Transmission Error
The loaded transmission error is a key indicator of the dynamic performance of the face gear pair. I compare the loaded transmission error under different modification methods and installation errors. The results for the 2nd-order modification are listed in Table 12. Under standard alignment, the loaded transmission error amplitude is 3.4″. When \(\Delta q=-0.1\) mm, the amplitude increases to 3.9″. When \(\Delta q=+0.1\) mm, the amplitude decreases to 3.2″. The variation is relatively large, which indicates that the 2nd-order modification is sensitive to installation error.
| \(\Delta q\) (mm) | Loaded transmission error amplitude (″) | Variation relative to standard alignment (″) |
|---|---|---|
| \(-0.1\) | 3.9 | +0.5 |
| \(0\) | 3.4 | 0.0 |
| \(+0.1\) | 3.2 | -0.2 |
The loaded transmission error for the 4th-order full modification is listed in Table 13. Under standard alignment, the amplitude is about 3.3″. When \(\Delta q=-0.1\) mm, the amplitude increases to 3.6″. When \(\Delta q=+0.1\) mm, the amplitude decreases to 3.1″. The variation is smaller than that of the 2nd-order modification. This confirms that the 4th-order modification reduces the sensitivity to installation error.
| \(\Delta q\) (mm) | Loaded transmission error amplitude (″) | Variation relative to standard alignment (″) |
|---|---|---|
| \(-0.1\) | 3.6 | +0.3 |
| \(0\) | 3.3 | 0.0 |
| \(+0.1\) | 3.1 | -0.2 |
The loaded transmission error for the 4th-order segmented modification is listed in Table 14. Under standard alignment, the amplitude is about 3.1″. When \(\Delta q=-0.1\) mm, the amplitude increases to 3.3″. When \(\Delta q=+0.1\) mm, the amplitude decreases to 2.9″. Under all installation error conditions, the loaded transmission error amplitude is lower than that of the other two modification methods. The transmission error curve also has a wider opening and a smaller fluctuation range. This shows that the 4th-order segmented modification provides the best dynamic performance among the three methods.
| \(\Delta q\) (mm) | Loaded transmission error amplitude (″) | Variation relative to standard alignment (″) |
|---|---|---|
| \(-0.1\) | 3.3 | +0.2 |
| \(0\) | 3.1 | 0.0 |
| \(+0.1\) | 2.9 | -0.2 |
Discussion of the Modification Mechanism
The high-order segmented modification works because it separates the modification into two directions. Along the contact path, the modification controls the transmission error and the position of the contact trace. Along the instantaneous contact line, the modification controls the local load distribution and the contact area. The 4th-order function along the contact path creates a flat central region and a smooth transition at the ends. The flat central region has a small modification amount, so the transmission error is close to zero in the middle of the meshing cycle. The ends have a larger modification amount, so the contact trace can avoid the tooth edges when installation error occurs. The segmented feature further reduces the modification amount in the central region. This is important because the central region carries the highest load. By keeping the central modification small, the contact area remains large, and the contact stress is reduced.
In contrast, the 2nd-order modification has a parabolic shape over the entire contact path. This means that the modification amount is not zero in the central region, and the transmission error curve is more rounded. Under installation error, the contact trace shifts more easily, and the transmission error amplitude changes significantly. The 4th-order modification has a flatter central region, so it is less sensitive to installation error. The segmented modification further improves this by allowing the central region to have only the instantaneous contact line modification. The result is a contact pattern that extends from the inner-diameter root to the outer-diameter tip of the face gear, which is the preferred contact direction for a face gear pair.
The finite element results confirm the TCA predictions. The maximum contact stress and the root bending stress are lower for the 4th-order segmented modification. The footprint ratio is higher under all installation error conditions. The loaded transmission error amplitude is also lower. These improvements are achieved without increasing the complexity of the manufacturing process significantly. The modification function can be applied to the pinion tooth surface by a computer-controlled grinding or milling process. The piecewise polynomial form is simple and can be evaluated efficiently. The coefficients can be adjusted to meet different load and misalignment requirements. Therefore, the proposed method is practical for engineering applications.
Parametric Guidelines
Based on the numerical results, I can provide some guidelines for selecting the modification parameters. The contact path angle \(\xi\) should be chosen to balance the contact ratio and the transmission error. A small angle gives a long contact trace and a high contact ratio, but the transmission error amplitude may be larger. A large angle gives a lower transmission error, but the contact trace becomes short. I recommend a moderate angle between about 6° and 20° for a face gear pair with a large gear ratio. The maximum modification amount should be large enough to avoid edge contact under the expected installation error, but not so large that the contact area becomes too small. I recommend a maximum modification amount between 20 μm and 30 μm for the case studied here. The order of the modification function should be 4 or higher. The 4th-order function provides a flat central region and a smooth transition, which reduces the transmission error and the sensitivity to installation error. The segmented coefficients \(k_{Bi}\) and \(k_{Ci}\) should be less than 1. A value between 0.5 and 0.8 is effective. This keeps the central region lightly modified and preserves a large contact area.
Comparison of Modification Strategies
I summarize the comparison of the three modification strategies in Table 15. The 2nd-order modification has the simplest form, but it is the most sensitive to installation error. The 4th-order full modification improves the transmission error and reduces the contact stress, but the central region still has a non-negligible modification amount. The 4th-order segmented modification gives the best overall performance. It has the lowest contact stress, the lowest root bending stress, and the lowest loaded transmission error amplitude. It also has the highest footprint ratio under installation error. The contact trace extends from the inner-diameter root to the outer-diameter tip of the face gear, which is the desired contact pattern for a face gear pair. Therefore, I recommend the 4th-order segmented topology modification for high-performance face gear pairs.
| Performance indicator | 2nd-order | 4th-order full | 4th-order segmented |
|---|---|---|---|
| Maximum contact stress under standard alignment (MPa) | 1363.91 | 1289.45 | 1215.91 |
| Maximum root bending stress under standard alignment (MPa) | 326.01 | 301.89 | 278.97 |
| Loaded transmission error amplitude under standard alignment (″) | 3.4 | 3.3 | 3.1 |
| Footprint ratio under standard alignment (%) | 77.39 | 83.44 | 83.89 |
| Footprint ratio under \(\Delta q=+0.1\) mm (%) | 68.45 | 74.91 | 83.75 |
| Sensitivity to installation error | High | Medium | Low |
Mathematical Summary of the Method
The complete mathematical procedure of the proposed method can be summarized as follows. First, I generate the theoretical face gear surface and the pinion surface from the shaping cutter geometry. The face gear surface is
$$\mathbf{r}_2(u_S,l_S,\varphi_S)=\mathbf{M}_{2S}\mathbf{r}_S(u_S,l_S)$$
$$\mathbf{n}_2(u_S,l_S,\varphi_S)=\mathbf{L}_{2S}\mathbf{n}_S(u_S,l_S)$$
with the meshing equation
$$f(u_S,l_S,\varphi_S)=0$$
Second, I project the pinion surface onto the projection plane and compute the contact path coordinate \(x_1\) and the instantaneous contact line coordinate \(x_2\):
$$x_1=x_p\cos\xi-y_p\sin\xi$$
$$x_2=\frac{x_p\sin\xi+y_p\cos\xi}{\sin(\xi+\eta)}$$
Third, I define the high-order segmented modification function along the contact path:
$$z_1(x_1)=\begin{cases}
a_{Bi}(x_1-x_{B0})^i, & x_{B1}\le x_1 < x_{B0}\\
0, & x_{B0}\le x_1 < x_{C0}\\
a_{Ci}(x_1-x_{C0})^i, & x_{C0}\le x_1\le x_{C1}
\end{cases}$$
and the 2nd-order modification function along the instantaneous contact line:
$$z_2(x_2)=\begin{cases}
a_A x_2^2, & 0\le x_2\le x_{A2}\\
a_D x_2^2, & x_{D2}\le x_2<0
\end{cases}$$
Fourth, I obtain the total modification amount:
$$\delta(u_1,l_1)=z_1(u_1,l_1)+z_2(u_1,l_1)$$
and the modified pinion surface:
$$\mathbf{r}_m=\mathbf{r}_l+\delta\mathbf{n}_l$$
$$\mathbf{n}_m=\frac{\partial\mathbf{r}_m}{\partial u_l}\times\frac{\partial\mathbf{r}_m}{\partial l_l}$$
Fifth, I solve the tooth contact analysis equations with installation errors:
$$\mathbf{r}_{f1}(u,l,\varphi_1)=\mathbf{r}_{f2}(u_S,l_S,\varphi_S,\varphi_2)$$
$$\mathbf{n}_{f1}(u,l,\varphi_1)=\mathbf{n}_{f2}(u_S,l_S,\varphi_S,\varphi_2)$$
$$f(u_S,l_S,\varphi_S)=0$$
and compute the contact ratio and transmission error:
$$\varepsilon=\left|\frac{\varphi_{1B}-\varphi_{1A}}{\varphi_{1n}}\right|$$
$$\Delta\varphi_2=\varphi_2-\frac{N_1}{N_2}\varphi_1$$
Finally, I perform the finite element analysis to obtain the contact stress, root bending stress, and loaded transmission error. The results show that the 4th-order segmented modification provides the best meshing performance for the face gear pair.
Conclusion
I have proposed a high-order segmented topology modification method for a face gear pair. The method uses a predefined contact path and a combination of 4th-order and 2nd-order modification functions. The modification is applied along the pinion tooth surface normal, and the modified surface is used in both TCA and finite element analysis. The main conclusions are as follows. The contact path angle affects the contact ratio and the meshing position. A diagonal contact path allows the contact pattern to use more of the working tooth surface and to avoid edge contact. The 4th-order modification produces a flat central region with a small modification amount, so the transmission error is close to zero in the middle of the meshing cycle. The 4th-order modification reduces the maximum contact stress and the loaded transmission error compared with the 2nd-order modification. The 4th-order segmented modification further improves the performance by keeping the central region lightly modified. Under installation error, the 4th-order segmented modification has the highest footprint ratio, the lowest contact stress, the lowest root bending stress, and the lowest loaded transmission error amplitude. The contact trace extends from the inner-diameter root to the outer-diameter tip of the face gear, which is the preferred contact direction. Therefore, the high-order segmented topology modification is an effective method for improving the meshing performance of a face gear pair and reducing its sensitivity to installation errors.
The method can be extended to other types of face gear pairs, including helical face gears and face gears with different shaft angles. The modification parameters can be optimized for specific load cases and misalignment conditions. Future work can include experimental validation and the integration of the modification method with manufacturing processes. The mathematical formulation is general and can be implemented in a computer-aided design environment. The use of a piecewise polynomial function makes the method computationally efficient and suitable for engineering applications. The face gear pair remains a key component in helicopter transmissions, automotive differentials, and industrial machinery. Improving its meshing performance is important for increasing load capacity, reducing vibration, and extending service life. The high-order segmented topology modification provides a practical way to achieve these goals.