Spiral bevel gears carry motion and torque between intersecting axes in helicopter rotor drives, turboprop reduction gearboxes and vehicle axles. Because the teeth are cut by a rotating cutter head on a generating cradle or a multi-axis CNC machine, the two flanks of a pair are never exactly conjugate. A controlled amount of geometric mismatch, known as the ease-off, is introduced deliberately: it localises the contact pattern, keeps the pattern away from the tooth edges under deflection, and shapes the transmission error that governs mesh excitation and noise.
The classical way to specify that mismatch is to fix a small set of geometric properties at one reference point, usually the midpoint of the gear flank. Modern flank-modification practice instead specifies the whole ease-off surface as a topological map and obtains the contact pattern and transmission error from a tooth contact analysis (TCA) on that map. The step that remains open is the description of contact quality at orders higher than the second, because the running behaviour over a complete mesh cycle is governed not only by curvature at the reference point but by how the contact point drifts, and how fast the instantaneous ratio changes, as it sweeps along the contact trace.
What Second-Order Contact Parameters Leave Undescribed
Second-order contact description rests on the principal curvatures and principal directions at the instantaneous contact point, together with three derived quantities: the direction of the contact trace on each member, the first derivative of the transmission ratio, and the ratio of the contact-ellipse semi-major axis to the face width. The Gleason local-conjugate method builds a gear flank by choosing a reference point on the wheel flank and computing the first- and second-order contact parameters of the mating pinion from conjugacy at that point. Litvin’s local synthesis inverts the procedure, prescribing the second-order parameters and circumventing the overdetermined system that a full conjugacy condition would produce. Both are pointwise constructions: they guarantee a well-controlled contact at the reference point and say nothing quantitative about the rest of the path.
Third-order quantities close that gap. Differentiating the principal curvatures with respect to position along the path gives the rate at which the contact-ellipse geometry changes; the short-range (geodesic) curvature of the contact trace measures how strongly the trace bends on the flank; and the higher derivatives of the transmission error describe the short-period motion error that is felt as vibration rather than as a steady ratio offset. Earlier work on the third-order problem includes Wang Xiaochun’s 1983 analysis of three-point contact of point-contact surfaces, which established the theoretical framework without the higher-order tangency conditions or a design procedure; Liu Huran and co-workers’ use of high-order Taylor expansions of the surfaces to obtain higher-order tangency conditions, without the associated contact parameters; and Zhu Limin and Ding Han’s third-order point-contact theory for five-axis machining of sculptured surfaces with non-ball-end tools, developed as a tool-envelope approximation and positioning strategy rather than as a gear design tool. Wu Baolin and Zheng Gang examined relations among higher-order contact parameters and applied them in other contexts. On the manufacturing side, ease-off-based flank modification for face-hobbing and face-milling, ease-off surface synthesis with osculating surfaces, cutter-error compensation, supplemental kinematic flank-correction motions on universal hypoid generators, and higher-order flank-form error correction have all been formulated on second-order contact theory.
Constructing the Discrete Ease-Off Surface
The geometric model uses two moving frames, S1(O1–x1y1z1) fixed to the pinion and rotating about x1 through angle φ1, and S2(O2–x2y2z2) fixed to the wheel and rotating about x2 through φ2, together with two stationary auxiliary frames that coincide with them at zero rotation. The shaft angle is Σ.
The wheel flank r2(u2, θ2) and its unit normal n2(u2, θ2) follow from the cutting parameters and the machine kinematic chain. Transformed into the stationary frame Sh, the generating condition is the meshing equation
$$\mathbf{n}_h\cdot\mathbf{v}_h^{(21)}=f(u_2,\theta_2,\phi_2)=0$$
which is solved for u2 = u2(θ2, φ2); the relative velocity vh(21) is that between the two members. Mapping the resulting point and normal back into S1 gives the flank exactly conjugate to the wheel,
$$\mathbf{r}_{1c}=M_{1h}\mathbf{r}_h,\qquad \mathbf{n}_{1c}=L_{1h}\mathbf{n}_h$$
which serves as the datum for modification. The pinion design flank r1(θ1, φ1) with normal n1 comes from the cutter-to-flank coordinate chain. Because the conjugate flank and the design flank are generated independently, they must first be located angularly so that flank points correspond one-to-one. Rotating both about x1 by angles λ1 and λ1c chosen to satisfy
$$y_1^{(C)}\cos\lambda_1+z_1^{(C)}\sin\lambda_1=0,\qquad y_{1c}^{(C)}\cos\lambda_{1c}+z_{1c}^{(C)}\sin\lambda_{1c}=0$$
brings them into tangency at the reference point C at the midpoint of the pitch cone. The ease-off is then the signed normal separation of the two located surfaces,
$$\Delta\delta(\theta_1,\phi_1)=\left[\tilde{\mathbf{r}}_1(\theta_1,\phi_1)-\tilde{\mathbf{r}}_{1c}(\theta_2,\phi_2)\right]\cdot\tilde{\mathbf{n}}_1(\theta_1,\phi_1)$$
To obtain a surface over the flank rather than a pointwise error, the design flank is projected onto a plane by rotating it about x1, giving a regular grid with coordinates
$$X_m=x_1(\theta_1,\phi_1),\qquad Y_m=\sqrt{\left[y_1(\theta_1,\phi_1)\right]^2+\left[z_1(\theta_1,\phi_1)\right]^2}$$
which is subsequently rotated into a frame X–Y aligned with the face width and tooth height, using the pitch cone angle β and the origin offset (XC, YC). Within that frame the modification is represented by a fourth-order polynomial,
$$\Delta\delta=a_{11}X^2Y^2+a_{12}X^3Y+a_{13}XY^3+a_{14}X^4+a_{15}Y^4$$
in which a11 acts together with a14 and a15 to shape a higher-order torsional mismatch, while a14 and a15 carry the higher-order crowning along the face width and tooth height respectively. Sampling the polynomial on the projected grid produces the discrete mismatch matrix
$$M=\begin{bmatrix}R^e_{11}&R^e_{12}&\cdots&R^e_{1n}\\ R^e_{21}&R^e_{22}&\cdots&R^e_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ R^e_{m1}&R^e_{m2}&\cdots&R^e_{mn}\end{bmatrix},\qquad R^e_{ij}=\left[X_{ij},Y_{ij},F(X_{ij},Y_{ij})\right]$$
with m and n the number of grid divisions along tooth height and face width and F the local mismatch depth.
Two Higher-Order Contact Parameters
The first parameter follows from the geometric meaning of transmission error, i.e. the departure of the wheel’s actual rotation from its theoretical value. With N1 and N2 the tooth numbers and K = N2/N1,
$$\Delta\phi_2=\frac{N_1\Delta\delta}{N_2}=\frac{F(X,Y)}{K}$$
As the pinion rotates, the contact point travels along the trace, so X and Y are functions of φ1 and the mismatch is evaluated along that trajectory, F = F(X(φ1), Y(φ1)). Differentiating twice gives the rate of change of the instantaneous ratio,
$$\frac{d^2\phi_2}{d\phi_1^2}=\frac{1}{K}\left[\frac{\partial^2F}{\partial X^2}\left(\frac{dX}{d\phi_1}\right)^2+2\frac{\partial^2F}{\partial X\partial Y}\frac{dX}{d\phi_1}\frac{dY}{d\phi_1}+\frac{\partial^2F}{\partial Y^2}\left(\frac{dY}{d\phi_1}\right)^2+\frac{\partial F}{\partial X}\frac{d^2X}{d\phi_1^2}+\frac{\partial F}{\partial Y}\frac{d^2Y}{d\phi_1^2}\right]$$
The second parameter is the short-range curvature of the contact trace. At the contact point C the design flank Σ1, the conjugate flank Σ1c and the ease-off surface Σe share a common tangent plane π, which is also the plane of the projected grid. The contact trace Γ1 on Σ1, projected along the normal n1 onto Σ1c and Σe, yields traces Γ1c and Γe. Because the three surfaces are tangent at C, the curvature of Γe measured within π equals in absolute value the short-range curvatures of Γ1 and Γ1c. Flattening the contact point to R̃e = [X(φ1), Y(φ1), 0] makes the quantity directly computable from the planar trace:
$$\kappa_g=\frac{\left|\dfrac{dX}{d\phi_1}\cdot\dfrac{d^2Y}{d\phi_1^2}-\dfrac{dY}{d\phi_1}\cdot\dfrac{d^2X}{d\phi_1^2}\right|}{\left[\left(\dfrac{dX}{d\phi_1}\right)^2+\left(\dfrac{dY}{d\phi_1}\right)^2\right]^{3/2}}$$
Equations for d2φ2/dφ12 and κg together constitute the higher-order contact description of the discrete flank.
Finite-Difference Evaluation on the Grid
Both expressions require first and second derivatives of the projected coordinates with respect to pinion rotation. Rather than solving the full third-order tangency system, the derivatives are obtained numerically on the same grid that carries the mismatch data. The discrete points are indexed as a mesh in the tangent plane, and the step is tied to the number of teeth and the number of contact points N along the trace:
$$h=\frac{2\pi}{NN_1}$$
so that, for example, 34 pinion teeth and 40 contact points give h ≈ 0.004 6 rad, about 0.26° of pinion rotation. Central differences are taken along the grid diagonal, which is the direction in which the contact trace crosses the mesh:
$$\left(\frac{dX}{d\phi_1}\right)_{ij}=\frac{X_{i+1,j+1}-X_{i-1,j-1}}{2h},\qquad \left(\frac{d^2X}{d\phi_1^2}\right)_{ij}=\frac{X_{i+1,j+1}-2X_{ij}+X_{i-1,j-1}}{h^2}$$
and analogously for Y. The partial derivatives of F are evaluated analytically from the polynomial coefficients at each node,
$$\frac{\partial^2F}{\partial X^2}=A_{ij},\quad \frac{\partial^2F}{\partial Y^2}=B_{ij},\quad \frac{\partial^2F}{\partial X\partial Y}=C_{ij},\quad \frac{\partial F}{\partial X}=D_{ij},\quad \frac{\partial F}{\partial Y}=E_{ij}$$
$$A_{ij}=2a_{11}Y_{ij}^2+6a_{12}X_{ij}Y_{ij}+12a_{14}X_{ij}^2,\qquad B_{ij}=2a_{11}X_{ij}^2+6a_{13}X_{ij}Y_{ij}+12a_{15}Y_{ij}^2$$
$$C_{ij}=4a_{11}X_{ij}Y_{ij}+3a_{12}X_{ij}^2+3a_{13}Y_{ij}^2$$
Substitution into the two derivative expressions yields the second derivative of the output rotation and, via the same finite differences, the short-range curvature of the trace. The computational burden reduces to arithmetic on a stored matrix: the difficult part of higher-order contact theory, the construction and solution of high-order tangency conditions, is replaced by differentiation of a known polynomial along a known path.
Case Study: A Helicopter Transmission Gear Pair
The procedure was applied to the working flanks of a spiral bevel gear pair from a helicopter transmission. The basic data are given in Table 1 and the pinion cutting settings in Table 2.
| Parameter | Pinion | Wheel |
|---|---|---|
| Number of teeth | 34 | 43 |
| Outer transverse module / mm | 2.0 | |
| Face width / mm | 15 | |
| Pressure angle / (°) | 20 | |
| Spiral angle / (°) | 20 | |
| Shaft angle / (°) | 108 | |
| Pitch cone angle / (°) | 44.860 8 | 63.139 2 |
| Face cone angle / (°) | 49.463 7 | 66.035 5 |
| Root cone angle / (°) | 41.964 5 | 58.536 2 |
| Hand of spiral | Right | Left |
| Parameter | Value |
|---|---|
| Cutter tip radius Rc / mm | 44.471 3 |
| Angular cutter position q0 / rad | 1.082 7 |
| Radial cutter position Sk / mm | 51.569 7 |
| Roll ratio Ra | 1.439 1 |
| Vertical offset E / mm | −3.283 9 |
| Axial setting Xp / mm | −0.238 9 |
| Sliding base Xb / mm | 0.159 8 |
| Mounting angle δm / rad | 0.732 4 |
The initial flank is a nominally second-order surface with a peak modification of 11.21 μm. Its transmission error curve looks like a parabola, yet the second derivative of the output rotation is far from constant, varying over a band of about 0.000 9 in the normalised ordinate (plotted at 10−4 scale). The contact trace, likewise, is not a smooth curve: the short-range curvature, of magnitude near 0.010 7 in the millimetre-based grid, fluctuates by up to 0.000 2. A flank designed to second-order specifications therefore does not run in a second-order manner.
Three sets of ease-off coefficients were then prescribed to test whether the analysis discriminates between deliberately higher-order surfaces. Their topology and the resulting contact behaviour are summarised in Table 3, where the peak modification depths are of the order of 10–13 μm and the contact traces are rotated relative to the initial trace.
| Surface | a11 | a12 | a13 | a14 | a15 | Fluctuation of d2φ2/dφ12 | Fluctuation of κg | Trace rotation / (°) |
|---|---|---|---|---|---|---|---|---|
| Initial | 0 | 0 | 0 | 0 | 0 | 0.000 9 | 0.000 2 | — |
| Set 1 | 0 | 0.000 005 | 0 | 0.000 001 2 | 0.000 206 3 | 0.003 1 | 0.000 076 9 | 2.40 |
| Set 2 | 0 | −0.000 005 | 0 | 0.000 001 4 | 0.000 206 3 | 0.001 9 | 0.000 586 | 5.63 |
| Set 3 | 0.000 01 | 0 | 0.000 006 | 0.000 001 4 | 0.000 206 3 | 0.001 | 0.000 127 | 23.76 |
For all three modified surfaces the second derivative of the output rotation peaks at zero, because the flanks contact fully in the mid-tooth region, and its fluctuation (0.003 1, 0.001 9 and 0.001 respectively) is concentrated near the tooth edges, where the modification is heaviest. The parameter thus reflects the global morphology of the flank rather than a local feature at the reference point. The short-range curvature of the contact trace, with fluctuations of 0.000 076 9, 0.000 586 and 0.000 127, stays numerically small but varies non-monotonically along the path in every case, which is a direct indication that the contact process along the trace is not smooth. The trace rotation separates the three designs clearly — 2.40°, 5.63° and 23.76° — and shows that coefficients which differ only in the third and fourth orders can reposition the contact pattern substantially without changing the second-order data at the reference point.
Implications for Flank Design
Three conclusions follow from the formulation and the example. First, an analytical link exists between the ease-off polynomial and the higher-order contact parameters of a discrete flank: the second derivative of the output rotation and the short-range curvature of the contact trace can both be read off the ease-off surface along the computed contact path, with no additional geometric construction. Second, the finite-difference treatment removes the practical obstacle that has kept higher-order contact theory out of design loops. Solving high-order tangency conditions directly is overdetermined and numerically awkward; differentiating a stored polynomial on a grid of a few dozen points is not, and it returns the same parameters. Third, the parameters are global rather than pointwise diagnostics, so they can be used as objective functions: the coefficients a11–a15 can be optimised against smoothness of the ratio derivative and of the trace curvature over the entire path, which is the natural formulation of a global flank design problem.
The scope of the analysis should be stated plainly. It is a geometric, unloaded evaluation in the tradition of classical TCA: mesh deflection, misalignment under torque and manufacturing scatter enter only through whatever flank geometry they produce, and the fourth-order polynomial limits the description to mismatch shapes that this basis can represent. Within those limits the method gives a quantitative, low-cost answer to a question that second-order parameter sets cannot answer — how the contact actually evolves along the tooth, and how much of that evolution a given modification leaves uncontrolled.