A single bevel stage with a reduction ratio near 10:1 removes an entire gear pair from a drive train, and with it a large part of the mass, the housing volume and the losses. The obstacle is not the ratio itself but the tooth geometry: to obtain it in one mesh the pinion must be reduced to a handful of teeth, which concentrates sliding, weakens the pinion root and pushes the wheel flank towards unfavourable contact. The case examined here is a 4:41 pair — pinion pitch diameter 18 mm against a wheel of 184.5 mm, 4.5 mm outer module, 90° shaft angle, 20° normal pressure angle and 26 mm face width — realized as spiral bevel gears with arc-shaped, constant-depth teeth. The design combines two ideas: composite addendum modification, which redistributes material between the two members, and meshing outside the pitch cone, which displaces the path of contact away from the pitch point so that the sliding velocity along the path keeps one sign and the friction force does not reverse during a mesh cycle.
Geometric envelope of a 4:41 pair
With a pitch cone angle of only 5.5722° on the pinion and 84.4278° on the wheel, conventional proportions are not viable. Four constraints define the admissible region, and all of them are evaluated on the equivalent cylindrical (virtual) gear rather than on the cone itself, because the modification design is naturally expressed in the transverse plane of that virtual gear.
The first constraint enforces meshing outside the pitch cone. Written for the wheel, the coefficient must be negative:
$$\lambda=\frac{0.5\,d_{va2}-0.5\,d_{v2}}{m_{mt}}<0$$
where dva2 is the tip diameter and dv2 the pitch diameter of the wheel’s virtual gear, and mmt the mid-face transverse module. A negative value means the wheel tip circle lies inside its pitch circle, so no contact occurs at the pitch point. The second constraint keeps the tips from sharpening, applied identically to both members:
$$s_{a1}=s_{a2}\ge 0.25\,m_{mt}$$
The third prevents undercut in a tooth count as low as four, giving a lower bound on the profile shift coefficients:
$$x_1\ge h^{*}_{at}\left(1-\frac{z_{v1}}{z_{v\min}}\right),\qquad x_2\ge h^{*}_{at}\left(1-\frac{z_{v2}}{z_{v\min}}\right),\qquad z_{v\min}=\frac{2h^{*}_{at}}{\sin^{2}\alpha_{vt}}$$
and the fourth preserves smooth transfer of load between teeth by demanding a total contact ratio of at least 1.5:
$$\varepsilon_{o}=\sqrt{\varepsilon_{\alpha}^{2}+\varepsilon_{\beta}^{2}}\ge 1.5$$
where εα and εβ are the transverse and overlap contact ratios. The configuration finally adopted is listed in Table 1; the total contact ratio of 1.6961, obtained mainly from a 36° spiral angle over a 26 mm face width, satisfies the fourth condition with margin.
| Parameter | Pinion | Wheel |
|---|---|---|
| Spiral angle β (°) | 36 | 36 |
| Outer cone distance Re (mm) | 92.688 | |
| Tooth depth coefficient kd | 1.66 | |
| Clearance coefficient c* | 0.125 | |
| Addendum coefficient ha* | 0.02 | −0.02 |
| Tangential modification xt | 0.27 | −0.27 |
| Pitch cone angle δ (°) | 5.5722 | 84.4278 |
| Tip diameter da (mm) | 28.5492 | 184.4798 |
| Pitch diameter d (mm) | 18 | 184.5 |
| Addendum ha (mm) | 5.2997 | −0.1039 |
| Dedendum hf (mm) | 0.5456 | 5.9491 |
| Total contact ratio εo | 1.6961 | |
The negative wheel addendum (−0.1039 mm, tip diameter 184.4798 mm against a pitch diameter of 184.5 mm) is the geometric signature of off-pitch-cone meshing, and the tangential shift of ±0.27 mm balances tooth thickness between a four-tooth pinion and a forty-one-tooth wheel. The penalty is paid on the pinion side, where the addendum grows to 5.2997 mm on a 18 mm pitch diameter; the tip-thickness inequality is what keeps that tooth from becoming a knife edge.
Cutting kinematics and tooth surface equations
The two members are machined by different processes, and the choice drives the rest of the design. The wheel, with a pitch cone angle of 84.4278°, is cut by the formate method: the cutter head is a rotating conical surface whose axis keeps a fixed orientation relative to the workpiece, and no generating roll is applied. Four settings define the process — radial setting Sr2, angular setting q2, sliding base Xb2 and axial wheel position correction Xd2. In the cutter coordinate system the blade surface and its unit normal are
$$r_{c}=\begin{bmatrix}\left(r_{0}-u_{2}\sin\alpha_{2}\right)\cos\theta_{2}\\ \left(r_{0}-u_{2}\sin\alpha_{2}\right)\sin\theta_{2}\\ -u_{2}\cos\alpha_{2}\\ 1\end{bmatrix},\qquad n_{c}=\begin{bmatrix}-\cos\alpha_{2}\cos\theta_{2}\\ -\cos\alpha_{2}\sin\theta_{2}\\ \sin\alpha_{2}\end{bmatrix}$$
with r0 the nominal cutter radius, α2 the blade profile angle (positive for the inner blade, negative for the outer) and u2, θ2 the surface parameters. Transforming into the wheel frame by the homogeneous matrices of the machine kinematics gives the wheel flank and normal:
$$r_{2}=M_{2f’}M_{f’m2}M_{m2c}\,r_{c},\qquad n_{2}=L_{2f’}L_{f’m2}L_{m2c}\,n_{c}$$
The conjugate pinion flank follows from the meshing condition nm·vm(12) = 0 between the wheel and its generating gear, which reduces to a closed-form relation for the wheel rotation angle:
$$\cos\left(\varphi_{2}+c_{2}\right)=\frac{W_{2}}{\sqrt{U_{2}^{2}+V_{2}^{2}}}$$
with c2 = arctan(V2/U2) and the intermediate quantities U2 = n2yz2 − n2zy2, V2 = n2zx2 − n2xz2, W2 = (n2xy2 − n2yx2)z2/z1, where z2/z1 = 41/4 is the tooth ratio. The pinion rotation follows from φ1 = φ2z2/z1, and the conjugate pinion surface Σ1 is obtained by the corresponding frame transformation.
The pinion itself is generated on the machine tool with a cutter head that is not left with a straight profile. Because the wheel is cut without generating motion, its flank curvature is essentially that of the cutter blade and is much too flat for a 4.5 mm module; matched against the conjugate pinion, the pair would touch along an excessively long line and would be intolerant of mounting error. The pinion blades are therefore ground to a circular arc of radius R, bending inwards on the inner blade and outwards on the outer blade. The blade surface becomes
$$r_{t}=\begin{bmatrix}\left(r_{c1}+u_{0}\sin\alpha_{1}+R\cos\alpha_{1}-R\cos\varphi\right)\cos\theta_{1}\\ \left(r_{c1}+u_{0}\sin\alpha_{1}+R\cos\alpha_{1}-R\cos\varphi\right)\sin\theta_{1}\\ -u_{0}\cos\alpha_{1}+R\cos\alpha_{1}-R\sin\varphi\end{bmatrix}$$
$$n_{t}=\begin{bmatrix}-\cos\varphi\cos\theta_{1}\\ -\cos\varphi\sin\theta_{1}\\ \sin\varphi\end{bmatrix},\qquad \sin\varphi=\frac{-u_{0}\cos\alpha_{1}+R\sin\alpha_{1}+u_{1}\cos\alpha_{1}}{R}$$
where rc1 is the theoretical tip radius, α1 the blade profile angle (positive for the outer, negative for the inner blade), u0 the modification datum and u1, θ1 the surface parameters. The generating motion adds the cradle rotation φp, again fixed by a meshing equation of the same form, with machine settings Sr1, q1, vertical offset Em1, axial position Xd1 and roll ratio m21. The resulting pinion flank Σ1′ is the surface whose curvature has to be brought into agreement with a prescribed contact pattern.
Curvature synthesis: from an ease-off target to machine settings
The design target is not a flank shape but a gap. Three surfaces are considered to touch at a single point M: the conjugate pinion Σ1, the modified pinion Σ1′ and the difference (ease-off) surface Σd between them. Σd is prescribed as an elliptical paraboloid in the tangent plane at M:
$$z_{d}=\tfrac{1}{2}k_{a}x_{d}^{2}+\tfrac{1}{2}k_{b}y_{d}^{2}$$
where zd is measured along the common normal, and the principal curvatures ka, kb fix the size, shape and orientation of the contact zone as well as the amplitude of the transmission error. The depth of the paraboloid at the boundary of one mesh cycle, zT0, was set to 0.012 mm: 0.006 mm to match the thickness of a marking-compound film used to read contact patterns, plus 0.006 mm assigned to the transmission error at the fundamental conversion point. The corresponding elliptical contour has a semi-major axis of 0.4 times the face width, a semi-minor axis of 0.5 times the tooth depth and a 20° tilt in the tangent plane, which gives
$$k_{a}=\frac{8z_{d}}{a^{2}},\qquad k_{b}=\frac{8z_{d}}{b^{2}}$$
Rotating this curvature tensor into the longitudinal (x) and transverse (y) directions of the flank yields the normal curvatures and the geodesic torsion contributed by the modification:
$$A_{x}=k_{a}\cos^{2}\lambda_{x}+k_{b}\sin^{2}\lambda_{x},\qquad A_{y}=k_{a}\cos^{2}\lambda_{y}+k_{b}\sin^{2}\lambda_{y},\qquad C_{x}=\tfrac{1}{2}\left(k_{b}-k_{a}\right)\sin 2\lambda_{x}$$
with λy = λx + π/2. Subtracting these from the curvatures of the conjugate flank gives the required curvatures of the modified pinion,
$$k_{i}=k_{x}-A_{x},\qquad k_{j}=k_{y}-A_{y},\qquad \tau_{x}=C_{x}$$
and the principal directions σ1,2 and principal curvatures k1,2 of that flank follow from the Euler–Bertrand relations:
$$\tan 2\sigma_{1,2}=\frac{2\tau_{x}}{k_{j}-k_{i}},\qquad k_{1,2}=k_{i}\cos^{2}\sigma_{1,2}+\tau_{x}\sin 2\sigma_{1,2}+k_{j}\sin^{2}\sigma_{1,2}$$
Since the prescribed surface and the machined surface must coincide at M in position, normal, principal direction and both principal curvatures, nine scalar equations are obtained:
$$r_{1}-r_{1}’=0,\quad n_{1}-n_{1}’=0,\quad e_{1}-e_{\mathrm{I}}=0,\quad k_{1}-k_{\mathrm{I}}=0,\quad k_{2}-k_{\mathrm{II}}=0$$
each vector equation counting for three components, which makes nine equations in the nine unknowns Sr1, q1, Em1, Xd1, rc1, R, m21, u1, θ1. Two further parameters are fixed in advance: the blank mounting angle γm1 and the blade profile angle α1, the latter being −16.0° for the convex and 24.0° for the concave flank; the sliding base is tied to the axial position through Xb1 = −Xd1sin γ1 − hfm1, where hfm1 is the mid-face dedendum of the pinion. A constrained nonlinear optimisation gives the two sets of settings in Table 2.
| Parameter | Convex flank | Concave flank |
|---|---|---|
| Blade profile angle α1 (°) | −16.0 | 24.0 |
| Radial setting Sr1 (mm) | 42.9147 | 46.7355 |
| Angular setting q1 (°) | 65.2822 | 64.6142 |
| Sliding base Xb1 (mm) | −0.4497 | −0.5839 |
| Axial position Xd1 (mm) | −0.9876 | 0.3946 |
| Roll ratio m21 | 10.6656 | 9.9004 |
| Cutter tip radius rc1 (mm) | 58.5734 | 55.3288 |
| Blade arc radius R (mm) | 32.0 | 80.0 |
| Vertical offset Em1 (mm) | 0 | 0 |
The asymmetry between the two flanks is substantial: the concave side needs a blade arc of 80 mm radius, the convex side only 32 mm, and the roll ratios differ by 0.77 from the nominal 10.25 tooth ratio. These are the degrees of freedom that the curvature equations actually use, and they explain why a single cutter head setting cannot serve both flanks of so small a pinion.
Simulated meshing behaviour
Reconstructing the difference surface from the solved settings closes the loop between the prescribed paraboloid and the manufactured flank. The computed ease-off maps of the two flanks are of the order of tens of micrometres over the active surface, with the prescribed gradient and a single tangency point, which confirms that the optimisation reproduced the intended modification rather than merely satisfying the equations locally.
Contact path and contact lines are extracted from the same surface: along each instantaneous contact line the point of minimum ease-off is a contact trace point, and the locus of those points over a mesh cycle is the path. On the concave flank the path runs from the heel at the root to the toe at the tip; on the convex flank it runs from the toe at the root to the heel at the tip. Both are internal-diagonal paths, the intended layout for a pair with this much cone-angle asymmetry, since an internal diagonal keeps the contact zone away from the tooth ends under deflection.
Plotting the ease-off depth at the trace points against pinion rotation gives a transmission error curve of parabolic form, with the 0.006 mm amplitude built into zT0 and a single extremum per mesh cycle. A solid model of the pinion built from the same settings and assembled with the wheel in a CAD environment shows an instantaneous contact ellipse of the expected proportions, entering at the root and leaving at the tip without crossing the tip boundary — the practical criterion for avoiding edge contact and the stress concentration that comes with it.
Cutting trial and roll inspection
The parameters of Table 2 were converted into the adjustment data of a GH-35 mechanical cutter, listed in Table 3, and the corresponding numerical control programs were written separately for the formate-cut wheel and the generated pinion. After cutting, the pair was run in rolling contact with marking compound and the pattern was read on the wheel.
| Setting | Wheel | Pinion, convex | Pinion, concave |
|---|---|---|---|
| Cradle eccentric angle (°) | 51.6030 | 51.5979 | 51.0318 |
| Cradle angle (°) | 109.1121 | 197.116 | 201.220 |
The first observation concerns the tooth tips: after cutting, neither member showed distortion or sharpening at the tip, which is the direct check on the tip-thickness constraint of the design stage. Several iterations of matching and contact-zone adjustment were needed before the pattern settled, after which the marking on the wheel appeared as an ellipse lying along the internal diagonal, positioned clear of the tip and the ends. This agrees with the simulated contact ellipse in both orientation and location, and it is the main evidence that the geometric design, the arc-modified cutter and the solved machine settings are mutually consistent.
What the trial establishes, and what it does not
The result is a demonstration of geometric feasibility rather than of transmission performance. Three things are supported by the evidence: a 4:41 spiral bevel pair can be cut with a wheel whose tip circle lies inside its pitch circle, so that meshing occurs entirely outside the pitch cone; the curvature deficit left by formate cutting of the wheel can be compensated on the pinion side by a circular-arc blade, with the arc radius acting as a direct control on the contact ellipse; and the nine curvature-matching equations can be solved for settings that a mechanical cutter accepts, since the conversion to cradle eccentric and cradle angles is a straightforward remapping.
What remains open is the behaviour under load. The verification here rests on unloaded rolling contact and on a geometric simulation of the unloaded mesh, so it says nothing directly about the transmission error spectrum at torque, about contact-zone migration under deflection, or about the effect of case hardening on the thin, heavily shifted pinion tooth. The friction argument for off-pitch-cone meshing — a sliding velocity of constant sign along the path and hence a friction force that does not reverse — is a kinematic property that favours scoring resistance but does not by itself quantify it. For a four-tooth pinion with a 5.2997 mm addendum and a 0.5456 mm dedendum, bending fatigue of the pinion root and distortion during heat treatment are the two items most likely to decide whether the design reaches production, and both require loaded tooth contact analysis and bench testing on the actual pair.
Within those limits, the configuration points to a practical route for compact final reductions: accept a very small pinion, compensate the resulting asymmetry through composite modification and off-pitch-cone contact, and recover the lost flank curvature through a deliberately modified cutter rather than through a generating motion the wheel cannot use. The same sequence — geometry, curvature synthesis, ease-off simulation, cutting trial — transfers to other high-ratio spiral bevel and hypoid pairs wherever the wheel is cut by formate and the pinion carries the burden of contact localisation.