A gear pair whose tooth flanks are defined by machine settings
Spiral bevel gears transmit rotation between intersecting shafts through teeth whose trace on the pitch cone is a circular arc. Because the teeth engage gradually from one end of the face to the other and several pairs can share the load, the format carries high torque in a compact envelope, runs with low excitation and is comparatively tolerant of small assembly errors. These properties explain its position in aircraft accessory drives, vehicle axles, marine reduction units and general industrial gearboxes.
The manufacturing chain separates into two families. In generating (interrupted cutting), the cutter head and the blank rotate about their own axes at a fixed ratio so that the straight-sided blades envelope the flank; in forming (plunge cutting), the head is fed directly into the blank and the flank reproduces the blade profile, which is faster but inflexible. Roughing is normally done with a duplex head that cuts both flanks of a slot at once. Finishing offers single-sided, duplex and double-duplex variants; the double-duplex route cuts both flanks of both members in one pass and is the productive choice for volume production.
What matters for the present subject is that the finished flank is not described on the blank drawing. It is the envelope of the blade surfaces swept past the blank under a particular set of machine settings — cradle angle, radial and angular cutter position, roll ratio, vertical offset, axial setting, bed position and blank mounting angle. Altering any one of them changes the curvature of the flank, and the change appears not as a measurable dimension but as a displacement of the contact pattern and a modification of the transmission error. The design problem is therefore inverse: the useful output is not a flank equation but a set of machine settings that produces a prescribed meshing behaviour.
From local conjugate design to local synthesis
Olivier described an enveloping-surface construction for bevel tooth surfaces in 1820, distinguishing what are now recognised as line-contact and point-contact solutions. The first machine tool for cutting a circular-arc tooth trace was built by James Gleason in 1913, and the format subsequently spread from automobile axles into general manufacture.
Gleason’s local conjugate method was the first systematic answer to the inverse problem. The gear member is cut first; a reference point is chosen on its flank; the pinion surface that is conjugate to the gear at that point is computed; and the pinion settings are adjusted until its generating surface matches. The weakness is that the second-order data at the reference point — the curvatures that decide how quickly the contact ellipse grows and how the pattern moves under load — are not known in advance. The inducing curvature of the pinion must be corrected iteratively, which is slow and gives no guarantee of the final pattern.
The situation changed when the meshing behaviour itself became computable. Baxter, at Gleason, published the formulation of tooth contact analysis (TCA) in 1961: given the two flank equations and the mounting, the contact path and the transmission error of a misaligned pair can be solved numerically before any metal is cut. Combined with blank selection and cutting calculation, TCA became the core of computer-aided design for these gears. Edge contact was added in the 1970s as EDGE-TCA, and in 1981 Krenzer extended the analysis to loaded contact (LTCA), where the contact pattern is computed with elastic deflection and mounting error included.
The local synthesis method formulated by Litvin in the late 1980s attacked the inverse problem directly. Instead of accepting whatever second-order geometry the gear flank happens to have, it prescribes at the reference point the direction of the contact path, the length of the instantaneous contact ellipse and the rate of change of the gear ratio, then solves for the pinion curvatures that satisfy these conditions. It is a refinement of the Gleason procedure, and it removed most of the trial-and-error. Its limitation is equally clear: only second-order quantities are controlled, so third-order behaviour — including the risk of tip contact or edge contact at the ends of the cycle — is left to the subsequent TCA check.
Later work addressed that gap from several directions. Stadtfeld replaced the parabolic transmission error with higher-order functions to shape the dynamic excitation of the mesh. Fuentes and Litvin designed a nearly straight contact path to reduce sensitivity to misalignment, and optimised second- and third-order modification coefficients to control the amplitude of the transmission error and the shape of the pattern. Barone and co-workers used finite element simulation to evaluate contact stress and contact ratio and to assess the effect of modification.
Chinese development followed a parallel track. Machines were copied with Soviet assistance at the Tianjin No. 1 Machine Tool Plant in the 1950s, and in 1972 the Ministry of Machine Building organised a programme on the Gleason technology system that involved Nankai University and several institutes. Yan Zhida and Wu Daren established the underlying meshing theory; Zheng Changqi derived TCA equations and described how contact path and transmission error follow from local conjugate conditions; Wu Xutang analysed the generating principles of the different cutting methods and the corresponding machine adjustments; Zeng Tao reproduced the Gleason calculation cards through pitch-surface analysis. Local synthesis was introduced into this body of work by Fang Zongde, who combined it with global optimisation to remove third-order contact defects, while Deng Xiaozhong and Wei Bingyang raised the contact ratio by tilting the contact path, Tian Xingbin applied global optimisation to the reference-point pattern, and Jiang Yugang built accurate solid models from discrete flank points. On the machine side, the YA2150 mechanical generator appeared in 1977, the seven-axis five-linkage CNC grinder YK2045 in 2002, and the six-axis CNC generator YK2275 in 2010, the last reaching grade 6 accuracy.
The second-order contact problem
Two flanks $\Sigma_1$ (pinion) and $\Sigma_2$ (gear) are tangent at a common point $M$ and share a common normal there. Their relative velocity is
$$\mathbf{v}^{(12)}=\boldsymbol{\omega}^{(12)}\times\mathbf{r}^{(1)}-\boldsymbol{\omega}^{(2)}\times\mathbf{R},\qquad \boldsymbol{\omega}^{(12)}=\boldsymbol{\omega}^{(1)}-\boldsymbol{\omega}^{(2)}$$
where $\mathbf{r}^{(1)}$ locates the contact point from a point on the pinion axis and $\mathbf{R}$ locates the gear axis relative to the pinion axis. Tangency requires the meshing equation
$$\mathbf{n}^{(2)}\cdot\mathbf{v}^{(12)}=0$$
and, since contact is maintained along the path, its time derivative vanishes as well:
$$\dot{\mathbf{n}}^{(2)}\cdot\mathbf{v}^{(12)}+\mathbf{n}^{(2)}\cdot\dot{\mathbf{v}}^{(12)}=0$$
Applying the Rodrigues relation $\dot{\mathbf{n}}^{(i)}=-K_i\mathbf{v}^{(i)}$ to both surfaces and projecting onto the common tangent plane, spanned by the gear’s principal directions $\mathbf{e}_s,\mathbf{e}_q$ with curvatures $k_s,k_q$, gives a linear system for the velocity components of the pinion surface at $M$:
$$
\begin{bmatrix}
a_{11} & a_{12}\\
a_{21} & a_{22}\\
a_{31} & a_{32}
\end{bmatrix}
\begin{bmatrix} v_s^{(1)}\\ v_q^{(1)}\end{bmatrix}
=
\begin{bmatrix} a_{13}\\ a_{23}\\ a_{33}\end{bmatrix}
$$
with
$$a_{11}=k_f\cos^2\sigma_{12}+k_h\sin^2\sigma_{12}-k_s,\qquad a_{12}=a_{21}=\tfrac{1}{2}\left(k_f-k_h\right)\sin 2\sigma_{12}$$
$$a_{22}=k_f\sin^2\sigma_{12}+k_h\cos^2\sigma_{12}-k_q$$
Here $k_f,k_h$ are the pinion curvatures along its principal directions $\mathbf{e}_f,\mathbf{e}_h$, and $\sigma_{12}$ is the angle from $\mathbf{e}_f$ to $\mathbf{e}_s$ measured in the tangent plane. The coefficients $a_{13},a_{23},a_{33}$ contain the relative velocity and angular velocity terms and are not reproduced here; what matters is the structure of the system.
Between the cutter and the blank the contact is linear, the relative velocity direction at $M$ is not determined, the solution is not unique, and the coefficient matrix degenerates:
$$a_{12}^2=a_{11}a_{22},\qquad a_{11}a_{23}=a_{12}a_{13},\qquad a_{12}a_{33}=a_{13}a_{23}$$
These three relations determine $k_f,k_h$ and $\sigma_{12}$ from the known cutter geometry, and the pinion principal directions follow by rotating $\mathbf{e}_s,\mathbf{e}_q$ through $\sigma_{12}$. This is the step that converts cutter data into flank curvature.
Between the two finished members the contact is a point contact and the system has a unique solution, so the augmented matrix must be singular. The vanishing determinant supplies one scalar relation among $k_f,k_h,\sigma_{12}$; two further conditions are required, and local synthesis supplies them by prescription at the reference point:
- $\eta_2$, the direction of the contact path tangent on the gear flank, which fixes how the pattern migrates across the face as the pair rolls and therefore how it behaves under deflection;
- $a$, the semi-major axis of the instantaneous contact ellipse, which sets the pattern length and, through the ellipse area, the contact stress level;
- $m’_{21}$, the derivative of the gear ratio at $M$, which is the slope of the transmission error curve at the reference point.
Once the three are fixed, $k_f,k_h$ and $\sigma_{12}$ follow from the compatibility condition together with the ellipse geometry. Writing $K_\Sigma^{(1)}=k_f+k_h$, $K_\Sigma^{(2)}=k_s+k_q$, $g_1=k_f-k_h$ and $g_2=k_s-k_q$, the ellipse coefficients are
$$A=\frac{1}{4}\left[K_\Sigma^{(1)}-K_\Sigma^{(2)}-\sqrt{g_1^2-2g_1g_2\cos 2\sigma_{12}+g_2^2}\right]$$
$$B=\frac{1}{4}\left[K_\Sigma^{(1)}-K_\Sigma^{(2)}+\sqrt{g_1^2-2g_1g_2\cos 2\sigma_{12}+g_2^2}\right]$$
$$a=\sqrt{\varepsilon/A},\qquad b=\sqrt{\varepsilon/B}$$
where $\varepsilon=0.00635$ mm is the elastic approach of the two flanks under light load and $b$ is the semi-minor axis. The prescribed semi-major axis therefore acts as one of the equations that determine the pinion curvature, closing the system.
The transmission error itself follows from the same expansion. With $\varphi_1,\varphi_2$ the rotation angles and $N_1,N_2$ the tooth numbers,
$$\delta\varphi_2=\left(\varphi_2-\varphi_2^{0}\right)-\frac{N_1}{N_2}\left(\varphi_1-\varphi_1^{0}\right)$$
and, for a constant mean roll ratio, the second-order expansion about the reference position gives a parabola whose curvature is the prescribed derivative:
$$\delta\varphi_2=\tfrac{1}{2}m’_{21}\left(\varphi_1-\varphi_1^{0}\right)^2$$
A negative $m’_{21}$ on the working flank produces a convex error curve in which the driven member lags the theoretical position, which is the usual way of pre-loading the mesh and postponing edge contact. The ranges adopted in the present design are collected in Table 1.
| Quantity | Symbol | Range adopted | Effect |
|---|---|---|---|
| Derivative of gear ratio | $m’_{21}$ | −0.008 to 0.008; negative on the working flank, positive on the non-working flank | Curvature and sign of the transmission error parabola |
| Contact path direction on gear flank | $\eta_2$ | 0°–90° working flank; 90°–180° non-working flank | Orientation of the pattern, behaviour under deflection |
| Semi-major axis of contact ellipse | $a$ | 0.15–0.20 of face width | Pattern length, contact stress level |
Blank geometry and gear member settings
The working example is a 9/39 pair with a 5.69 mm module, 35 mm face width, 22.5° pressure angle and 29.43° mid-face spiral angle, left-hand pinion and right-hand gear. The cutting scheme is the Gleason SGM arrangement: the gear is generated with a duplex cutter head and the pinion by the modified-roll method, in which the roll ratio varies during the generating motion. Blank data are listed in Table 2.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 9 | 39 |
| Module / mm | 5.69 | |
| Face width / mm | 35 | |
| Pressure angle / ° | 22.5 | |
| Pitch cone angle / ° | 12.99 | 77.01 |
| Face cone angle / ° | 16.57 | 79.72 |
| Root cone angle / ° | 10.28 | 73.43 |
| Mid-face spiral angle / ° | 29.43 | |
| Whole tooth depth / mm | 11.38 | |
| Addendum / mm | 5.98 | 4.27 |
The gear member is cut with the settings in Table 3, obtained from the Gleason calculation card. With bed position, axial setting and vertical offset all zero, only the cutter position and the mounting angle remain to be adjusted. The cutter head diameter of 190.50 mm and the point width of 1.50 mm fix the slot width at the root, while the two blade profile angles of 22° and −18° act on the convex and concave flanks respectively; the asymmetry reflects the different curvature demands of the two sides rather than a symmetric pressure-angle correction.
| Parameter | Value |
|---|---|
| Blade profile angle, convex flank / ° | 22 |
| Blade profile angle, concave flank / ° | −18 |
| Cutter point width / mm | 1.50 |
| Cutter head diameter / mm | 190.50 |
| Blank mounting angle / ° | 73.43 |
| Bed position / mm | 0 |
| Radial cutter position / mm | 96.64 |
| Angular cutter position / ° | 59.21 |
Converting preset contact conditions into pinion settings
The gear flank equation is obtained by transforming the blade cone from the cutter frame through the cradle and machine frames into the gear frame, eliminating the surface parameter through the meshing equation between blade and generating wheel. A reference point $M$ is then placed at the centre of the intended pattern, its position controlled by two coordinates along and across the face, and located by solving the resulting nonlinear system with initial values $\theta_g^{(0)}=270°+\beta$ and $\varphi_g^{(0)}=0$ for a left-hand pinion. At that point the cutter principal directions and curvatures are known in closed form, and the line-contact relations of the previous section convert them into the gear flank curvatures $k_I^{(2)},k_{II}^{(2)}$ and directions $\mathbf{e}_I^{(2)},\mathbf{e}_{II}^{(2)}$.
Local synthesis is then applied in the fixed frame of the gearbox, in which the pinion axis and the gear axis intersect at the pitch apex with the shaft angle between them. The gear flank data are rotated into that frame, the three quantities of Table 1 are prescribed, and the corresponding pinion curvatures $k_I^{(1)},k_{II}^{(1)}$ and directions follow.
The remaining step converts pinion flank curvature into machine settings, using the line-contact condition between the pinion and its generating wheel. The cutter tip radius follows from matching the normal curvature of the blade cone at the reference point to the synthesised value, with the surface coordinate fixed by the cone apex distance. The roll ratio is obtained by projecting the relative velocity onto the two principal directions and solving the resulting system. The vertical offset and axial setting follow from the components of the same velocity field, the bed position from the cone geometry,
$$X_{B1}=\left(Z_{R1}-X_{G1}\right)\sin\gamma_1$$
and the radial and angular cutter positions from the coordinates of the reference point in the generating frame,
$$S_p=\sqrt{a_1^2+a_2^2},\qquad q_1=\arctan\frac{a_1}{a_2}$$
Because the three presets are applied separately to each flank, the two sides of the pinion receive markedly different settings. The computed values are given in Table 4; the concave flank requires a much smaller radial cutter position and a much larger roll ratio than the convex flank, which is the expected consequence of imposing different curvature targets rather than accepting a single generating motion for both sides.
| Parameter | Concave flank | Convex flank |
|---|---|---|
| Cutter tip diameter / mm | 188.46 | 187.90 |
| Blade profile angle / ° | 18 | −21 |
| Blank mounting angle / ° | 10.28 | 10.28 |
| Axial setting / mm | −7.89 | 11.57 |
| Bed position / mm | 1.41 | −2.28 |
| Vertical offset / mm | −9.44 | 2.26 |
| Radial cutter position / mm | 83.80 | 142.41 |
| Angular cutter position / ° | 60.56 | 68.23 |
| Roll ratio | 5.46 | 3.28 |
Tooth contact analysis of the computed settings
Before cutting, the two flank equations and their normals are assembled in the fixed frame. Contact at any instant requires equality of position vectors and of unit normals,
$$\mathbf{r}_h^{(1)}\left(\theta_p,\phi_p,\varphi_1\right)=\mathbf{r}_h^{(2)}\left(\theta_g,\phi_g,\varphi_2\right),\qquad \mathbf{n}_h^{(1)}\left(\theta_p,\phi_p,\varphi_1\right)=\mathbf{n}_h^{(2)}\left(\theta_g,\phi_g,\varphi_2\right)$$
which projects into five independent scalar equations in six unknowns. Fixing the pinion rotation angle and solving for the rest yields one contact point; sweeping the angle produces the contact path, and the corresponding gear rotations produce the transmission error through the relation given earlier.
For the settings in Table 4 the computed contact path is close to a straight line running across the middle of the face, and the contact ellipses at successive positions have similar major axes on the gear convex flank. The transmission error curve is approximately symmetric about the reference position and remains continuous through the cycle, without the abrupt change of slope that signals the transfer of load to the tooth edges. Both features match the intent of the presets: a straight path keeps the pattern away from the ends of the face, and a symmetric error curve keeps the mesh stiffness variation small in both directions of rotation. The settings were accepted on this basis and carried forward to modelling and cutting.
Building the solid model from the calculated flanks
The flank equations are not convenient input for a CAD kernel, so the surfaces were discretised by rotating projection. A point $(x,y,z)$ on the flank maps to $(X,Y)$ in the axial section through
$$x=X,\qquad y^2+z^2=Y^2$$
so a grid laid out in the projection plane corresponds one-to-one with a grid on the flank. The boundary vertices were computed from the cone angles, addendum and dedendum at the inner and outer ends, the interior nodes from the intersections of the two families of straight lines joining opposite boundary points, and the corresponding three-dimensional coordinates obtained by solving the two nonlinear equations above together with the flank equation of the member.
A 9 × 5 grid, four divisions across the tooth height and eight along the face, was used. The resulting point sets were exported as DAT files and imported into UG NX 12.0, where they were fitted with splines, lofted into convex and concave patches, translated by $360/Z$ about the axis to create the opposite flank of the slot, closed at the root with ruled surfaces, extended tangentially and sewn into a solid tooth slot. The blank was revolved from a sketch built on the pitch, face and root cone lines, trimmed with the slot, and patterned around the axis. The pinion was built the same way. The assembled pair was constrained on the two axes with a centre distance and an axial offset, and the two working flanks were separated by 0.00635 mm — the same elastic approach used in the ellipse calculation — so that the surfaces touch rather than interpenetrate. A kinematic simulation of the assembly confirmed that the contact zone falls in the middle of the face before any finite element work was started.
Contact stress under load
Stress analysis used three teeth from each member, since at most three pairs share the load at any instant and a full wheel model is wasteful. The trimmed assemblies were imported into ANSYS Workbench. The material is 20CrMnTi carburising steel, with density $7800\ \mathrm{kg/m^3}$, Poisson ratio 0.25 and elastic modulus $2.1\times10^{5}\ \mathrm{MPa}$.
The contact is treated as flexible-to-flexible, surface-to-surface, with the pinion flanks as the target surface and the gear flanks as the contact surface. A frictional contact formulation was chosen with a coefficient of 0.1, since sliding accompanies the rolling motion, and the augmented Lagrangian algorithm was used to enforce contact compatibility while limiting penetration sensitivity. Free tetrahedral meshing gave a global element size of 0.6 mm and a minimum size of 0.013 mm, refined on the flanks; the mesh contained 1,341,875 elements and 1,953,178 nodes, and the quality metric was close to unity over most of the model.
The two parameters that govern convergence were selected by numerical experiment rather than by default. For the normal stiffness factor $FKN$, a torque of 15 N·m was applied and the factor varied over 0.01 to 10; the maximum contact stress became insensitive to further increase beyond a value of 3, which was therefore adopted. The penetration tolerance was set in the same way, starting from a loose value and reducing it until the solution remained convergent, giving 0.02.
The pinion was allowed to rotate about its axis only, all other degrees of freedom being suppressed, and the gear was fixed at the hub and on both sides. A driving torque of 200 N·m was applied to the pinion. The resulting stress field is concentrated in an elliptical zone whose intensity decays outward from the centre, with the peak located where the gear addendum meets the pinion dedendum. Because bending at the root contributes to the stress state there, the peak on the pinion flank is lower than on the gear flank, which is favourable for avoiding local concentration. The maximum contact stress and the strength check are summarised in Table 5.
| Item | Value |
|---|---|
| Driving torque on pinion / N·m | 200 |
| Maximum contact stress / MPa | 1227.9 |
| Allowable contact stress / MPa | 1280 |
| Margin below allowable / % | 4.07 |
The computed stress lies below the allowable value, but the margin is small enough that manufacturing scatter in hardness or in the pattern position would consume it. For a production part this result would justify either a lower rated torque or a further iteration on the presets.
Stress over a mesh cycle
Because the contact position moves continuously across the face during meshing, the quasi-static result describes only one instant. A transient analysis was therefore run with the pinion at 1000 r/min and a resisting torque of 200 N·m on the gear, applied through joint loads on the two rotational joints. Over one mesh cycle the contact zone travels from the heel toward the toe and from the tip toward the root, following the path predicted by the contact analysis and by the kinematic simulation of the assembly. The maximum contact stress varies through the cycle as single and then double tooth pairs carry the load, and the peak value remains consistent with the static result, which confirms that the quasi-static solution is representative of the loaded state.
Cutting and roll inspection
Cutting trials were run on a GH-35 mechanical gear milling machine of Japanese origin that has been retrofitted with servo drives in place of the mechanical feed train, with the settings of Tables 3 and 4 entered through the numerical control interface, including roll ratio and cradle angle. Both members were cut from the same blank stock, and the resulting tooth form matched the solid model.
The pair was then mounted on a rolling tester with the mounting distance set from the blank data, the flanks coated with red lead, and the drive run for a short period. On the gear convex flank the contact pattern fell in the middle region of the face, with a size consistent with normal running; the pinion flank showed the corresponding pattern. Both are within the range expected for a correctly set pair, which confirms that the settings derived from the prescribed second-order quantities are realisable on the machine and produce the intended meshing behaviour. It should be noted that this check is qualitative: the rolling test establishes the position and approximate size of the pattern, not the transmission error, which would require single-flank measurement.
What second-order control can and cannot guarantee
The value of the procedure lies in the direction of the calculation. Instead of cutting a gear, measuring the pattern, and correcting settings by judgement, the designer states three quantities that have direct mechanical meaning — contact path orientation, ellipse size and the curvature of the transmission error — and obtains the pinion curvatures and machine settings that realise them. The TCA step then verifies that the resulting pair behaves as intended before any material is removed, and the finite element and cutting stages confirm that the settings are physically achievable and that the stress remains within the allowable range at the rated torque.
The boundaries are equally clear. Only second-order data are prescribed, so the behaviour away from the reference point — where third-order terms dominate and where tip contact or edge contact would appear — is not controlled by construction and must be checked afterwards. The preset ranges in Table 1 are empirical, and the resulting settings differ substantially between the two flanks, which places demands on machine repeatability when the pair is cut in separate passes. The stress margin of 4.07% was established for a single torque level and a single mounting condition, without the misalignment that LTCA would introduce. Extending the work in three directions would address most of these limits: a sensitivity study of pattern and transmission error against the three presets, so that the ranges can be chosen rather than assumed; a direct data path from the parameter calculation to the solid model and the finite element model, so that iterations do not require manual transfer; and experimental measurement of flank stress or transmission error, which would test the computed results rather than only the contact pattern.