Why spiral bevel gears are hard to inspect
Spiral bevel gears transmit motion and torque between intersecting shafts. The curved, gradually engaged teeth keep several pairs in contact at once, which gives the pair high load capacity, smooth running and a large contact ratio; the same properties make it a standard choice for locomotive drives, petroleum machinery and aerospace transmissions. Because the flanks are generated helicoids whose curvature changes along the tooth, the tolerances that decide whether a pair will run quietly — single pitch deviation, cumulative pitch deviation, runout and profile deviation — must be evaluated on a circle of a few tens of millimetres at the mid-face width, as specified in GB/T 11365—2019. Every measurement problem therefore reduces to locating the mid-face plane of an individual tooth accurately and comparing the real flank with the theoretical one there.
Inspection practice falls into four families: single-flank rolling tests that record the composite transmission error of a meshing pair; measurement of individual geometric errors on a gear measuring centre; on-machine measurement carried out on the cutting or grinding machine; and non-contact measurement using optical or laser probes. CNC gear measuring centres built by Harbin Measuring & Cutting Tool Plant, Chengdu Tool Research Institute and Harbin Jingda Measuring Instruments successively implemented automatic bevel gear measurement with error evaluation, a MECT three-coordinate machine developed at Xi’an Jiaotong University was used for spiral bevel gears, and five-axis on-machine measurement has also been demonstrated on machining centres. The remaining gap relative to imported instruments is usually described in terms of stability, functional range and achievable accuracy.
Contact probing is the bottleneck. A stylus of finite radius cannot reach the root of a strongly curved flank, so the measured profile is truncated exactly where the stress is highest; the probing path has to be planned in advance from the gear parameters, with the tangential and normal corrections recomputed at every point; and tip wear adds a slow drift that is difficult to separate from the gear error itself. On a complex helicoid this produces long cycles and an accuracy that is limited by the probe rather than by the machine. A laser displacement sensor removes the stylus from the loop: it measures distance along a fixed beam, it needs no radius compensation, and a single revolution of the worktable yields enough data for several error items at once.
Instrument architecture and measurement procedure
The measuring device combines three linear slides (x, y, z), a rotary worktable (c axis) and a laser displacement sensor. The y slide is mounted on the worktable base, the x slide on the y slide, and the z slide carries the bracket of the sensor; the sensor head itself can be rotated about a horizontal axis so that the emitted beam stays perpendicular to the pitch cone of the gear. Linear gratings on the slides and a circular grating on the rotary axis provide position feedback, and the motion of the three slides together with the worktable is programmed through the control and servo system. The coordinates of a measured point follow from the machine position of the probe and the distance reading of the sensor.
Orientation of the probe is handled by a separate adjustment unit built from two freely rotating brackets, corresponding to axes I and II, and a rotating platform, so the sensor can be pointed in any direction in space. The setting matters for gears with a large spiral angle: the plane containing the emitted and the received beam must be tilted until it coincides with the flank plane at the mid-face width, otherwise the reflected light misses the receiving window and the spot image is incomplete. With this adjustment the same hardware covers the full range of bevel gear parameters.
For a measurement the sensor is first rotated about axis II by an angle close to the spiral angle of the gear, and the gear is clamped on a high-precision arbor on the rotary table. After the gear data are entered, the sensor is rotated about axis I until the beam is perpendicular to the pitch cone generatrix, and the slides bring the beam to the mid-face width at the mid-point pitch radius. The automatic scan then starts: the table turns at constant angular velocity while the system records, every 100 ms, the sensor reading and the angle reported by the circular grating, and acquisition stops after one full revolution. The head is retracted to a safe position and the data are passed to the analysis module, which produces the deviation report. Sampling interval and table speed together fix the angular resolution of the sampled flank.
Tooth surface model of the Gleason generating process
Gleason spiral bevel gears are cut by a generating process in which an imaginary plane gear — the cradle, or generating gear — meshes with the blank. The cutter disk is mounted eccentrically on the cradle and rotates about its own axis while the cradle turns, so the cutting edges of the disk trace one tooth of the generating gear; the blank rotates at a fixed ratio to the cradle, a tooth slot is generated, the blank is indexed, and the cycle repeats. The tooth surface of the workpiece is therefore the envelope of the blade positions that satisfy the meshing condition, and it can be written analytically once the cutter geometry and the machine settings are expressed in a common coordinate system.
The machine frame $O_1x_1y_1z_1$ is fixed to the machine, with its origin at the machine centre, its $x_1O_1y_1$ plane in the machine plane through the blade tips perpendicular to the cradle axis, and $z_1$ along the cradle axis pointing outwards. The cradle frame $O_2x_2y_2z_2$ rotates with the cradle by the current cradle angle $\beta_1$; the cutter frame $O_3x_3y_3z_3$ rotates with the disk by the cutter angle $\theta$; the auxiliary frame $O_{f2}x_{f2}y_{f2}z_{f2}$ is obtained from the machine frame by translating $T+P\sin\delta$ along $z_1$ and $P\cos\delta$ along $x_1$ and then rotating by the blank mounting angle $\delta$ about $y$; the workpiece frame $O_4x_4y_4z_4$ rotates with the blank by the angle $\beta_2$. The cutter position is given by the angular setting $q$ (initial value $q_0$) and the radial setting $S$; $T$ is the axial blank offset and $P$ the vertical offset of the bed.
A point on the blade surface is described in the cutter frame by
$$\mathbf{r}_p(\theta,u)=\begin{pmatrix}(r_{p0}-u\sin\alpha)\cos\theta\\ (r_{p0}-u\sin\alpha)\sin\theta\\ -u\cos\alpha\end{pmatrix}$$
where $\alpha$ is the blade pressure angle, $u$ the parameter along the cutting edge measured from the tip $P_0$, $r_d$ the nominal cutter radius and $w$ the point width. For the inner blade $r_{p0}=r_d-w/2$ with $\alpha=\alpha_1$; for the outer blade $r_{p0}=r_d+w/2$ with $\alpha=-\alpha_2$. Seen from the machine frame the same point is
$$\mathbf{r}_p=\begin{pmatrix}S\cos q+r_{p0}\sin(q-\theta)\\ S\sin q-r_{p0}\cos(q-\theta)\\ 0\end{pmatrix}$$
with the unit normal and the tangent to the blade given by
$$\mathbf{n}=\begin{pmatrix}\cos\alpha\sin(q-\theta)\\ -\cos\alpha\cos(q-\theta)\\ -\sin\alpha\end{pmatrix},\qquad \mathbf{t}=\begin{pmatrix}\sin\alpha\sin(q-\theta)\\ -\sin\alpha\cos(q-\theta)\\ \cos\alpha\end{pmatrix}$$
Sweeping the blade along $\mathbf{t}$ produces the tooth surface of the generating gear,
$$\mathbf{r}_1=\mathbf{r}_{p0}-u\,\mathbf{t}=\begin{pmatrix}S\cos q+r_{p0}\sin(q-\theta)-u\sin\alpha\sin(q-\theta)\\ S\sin q-r_{p0}\cos(q-\theta)+u\sin\alpha\cos(q-\theta)\\ -u\cos\alpha\end{pmatrix}$$
The generating gear and the blank are conjugate, so at the contact point the relative velocity $\mathbf{v}_{12}$ satisfies $\mathbf{n}\cdot\mathbf{v}_{12}=0$. Solving this condition for $u$ gives the position of the contact point on the blade:
$$u(q,\theta)=\frac{-a\cos\alpha\,(b+c)+d\sin\alpha\,e}{d\cos(q-\theta)}$$
with
$$a=i\sin\delta-1,\qquad b=\left[S\sin q-r_{p0}\cos(q-\theta)\right]\sin(q-\theta)$$
$$c=\left[S\cos q+r_{p0}\sin(q-\theta)\right]\cos(q-\theta),\qquad d=i\cos\delta,\qquad e=S\sin q-r_{p0}\cos(q-\theta)$$
where $i$ is the ratio between cradle rotation and blank rotation. The surface and its normal are then carried into the workpiece frame,
$$\mathbf{r}_4=M_{4f2}M_{f21}\mathbf{r}_2,\qquad \mathbf{n}_4=L_{4f2}L_{f21}\mathbf{n}$$
where $M_{f21}$ is the homogeneous transformation from the machine frame to the auxiliary frame, $L_{f21}$ its $3\times3$ rotation part, and $M_{4f2}$ the transformation from the auxiliary frame to the workpiece frame with $L_{4f2}$ its rotation part. Because the workpiece rotation is $\beta_2=i(q-q_0)$, the transformed surface reads
$$\mathbf{r}_4=\begin{pmatrix}r_{2x}\cos\delta+r_{2z}\sin\delta\\ r_{2y}\cos\beta_2+\sin\beta_2\,(r_{2z}\cos\delta-r_{2x}\sin\delta)\\ -r_{2y}\sin\beta_2+\cos\beta_2\,(r_{2z}\cos\delta-r_{2x}\sin\delta)\end{pmatrix}$$
and the normal transforms with the same rotation, with $n_x$, $n_y$, $n_z$ replacing $r_{2x}$, $r_{2y}$, $r_{2z}$. Equations of this form define the theoretical flank against which the sampled points are compared, and they depend only on the gear and cutter data — no measured data enter the model.
Mid-face profile as an involute
At the mid-face width the tooth behaves as a spur gear with module $m_m$ and equivalent tooth count $z_v$,
$$m_m=m_e\left(1-\frac{\phi_R}{2}\right),\qquad z_v=\frac{z}{\cos\delta},\qquad \phi_R=\frac{b}{R_e}$$
so the base circle radius of the equivalent gear is
$$r_b=\frac{m_m z_v\cos\alpha}{2}=\frac{m_e\left(1-\phi_R/2\right)z\cos\alpha}{2\cos\delta}$$
and the involute profile in rectangular coordinates centred on the gear axis is
$$\begin{cases}x_p=r_b\left(\sin u_k-u_k\cos u_k\right)\\ y_p=r_b\left(\cos u_k+u_k\sin u_k\right)\end{cases},\qquad u_k=\alpha_k+\theta_k$$
where $\alpha_k$ is the pressure angle and $\theta_k$ the roll angle at the point. The normal to the profile at a point $P(x_p,y_p)$ has the slope
$$k=\tan\left(\arctan\frac{y_p}{x_p}+\frac{\pi}{2}+\alpha_H\right)$$
with $\alpha_H$ the pressure angle at that point, and the normal line itself is written through the reference point, $y=k(x-x_D)+y_D$.
Extracting pitch and profile deviations from the sampled points
After the sensor readings and grating angles are converted into coordinates in the transverse mid-face plane, each tooth contributes a set of sampled points $N_i(x_i,y_i)$ along one flank. Only the points in a window around the pitch circle are used, because the comparison is defined there. A cubic
$$y=a_0+a_1x+a_2x^2+a_3x^3$$
is fitted to $n=8$ of these points in the least-squares sense, which requires solving the normal equations
$$\begin{bmatrix}n & \sum x_i & \sum x_i^2 & \sum x_i^3\\ \sum x_i & \sum x_i^2 & \sum x_i^3 & \sum x_i^4\\ \sum x_i^2 & \sum x_i^3 & \sum x_i^4 & \sum x_i^5\\ \sum x_i^3 & \sum x_i^4 & \sum x_i^5 & \sum x_i^6\end{bmatrix}\begin{bmatrix}a_0\\a_1\\a_2\\a_3\end{bmatrix}=\begin{bmatrix}\sum y_i\\ \sum x_i y_i\\ \sum x_i^2 y_i\\ \sum x_i^3 y_i\end{bmatrix}$$
with all sums running over $i=1\ldots n$ and the coordinate origin taken at the gear axis. The intersection of the fitted curve with the pitch circle of radius $r_m=m_m z/2$ is accepted as the actual crossing point $C_i$ of that flank. The theoretical involute crosses the same circle at $D_i$, where the pressure angle is $\alpha_D=20°$, and the profile relative deviation of the tooth is the distance between $D_i$ and the point $E_i$ where the fitted curve meets the normal to the theoretical involute erected at $D_i$:
$$\Delta f_{ci}=\pm\sqrt{\left(x_{D_i}-x_{E_i}\right)^2+\left(y_{D_i}-y_{E_i}\right)^2}$$
The sign is positive when $E_i$ lies outside the theoretical flank and negative when it lies inside, which corresponds to the difference in rotational angle, expressed as a linear value at the mid-face width, between the real and the reference flank. Summing the contributions of all teeth gives the figure compared with the tolerance:
$$f_c=\sum_{i=1}^{n}\Delta f_{ci}$$
The same crossing points carry the pitch information, since the pitch is the arc between the crossings of two adjacent teeth on the same flank. Given the chord, the arc follows from
$$\widehat{C_iC_{i+2}}=m_m z\arcsin\frac{\sqrt{\left(x_{i+2}-x_i\right)^2+\left(y_{i+2}-y_i\right)^2}}{m_m z}$$
where $m_m z=2r_m$ is the diameter of the mid-face pitch circle. With the nominal pitch $p=\pi m_m=\pi m_e\left(1-\phi_R/2\right)$, the single pitch deviation, the cumulative deviation over $K$ pitches and the total cumulative deviation are
$$\Delta f_{pt}=\widehat{C_iC_{i+2}}-p,\qquad F_{pK}=\widehat{C_iC_k}-pK,\qquad F_p=\widehat{C_iC_n}-pn$$
Once the crossing points are known, single tooth thickness, backlash and slot width can be computed from the same data without another scan.
Validation on an IT8 gear pair
The procedure was applied to a Gleason pair declared as IT8 accuracy with a 1:1 ratio and the parameters listed in Table 1. The gears were measured tooth by tooth with a single revolution of the worktable per gear, and the deviations were evaluated at the mid-face width.
| Parameter | Symbol | Value |
|---|---|---|
| Accuracy grade | — | IT8 |
| Gear ratio | — | 1:1 |
| Outer transverse module | $m_e$ | 3 mm |
| Number of teeth | $z_1=z_2$ | 30 |
| Outer cone distance | $R_e$ | 63.64 mm |
| Face width | $b$ | 20.00 mm |
| Pitch cone angle | $\delta$ | 45° |
| Spiral angle | $\beta$ | 35° |
The face width ratio of this pair is $\phi_R=b/R_e=0.314$, close to the value of 0.3 usually assumed when the mid-face module is calculated. The measured deviations are collected in Table 2.
| Quantity | Value (mm) |
|---|---|
| Mean single pitch deviation $f_{pt}$ | 0.000829 |
| Largest single pitch deviation $\max \Delta f_{pt}$ | −0.02788 |
| Smallest single pitch deviation $\min \Delta f_{pt}$ | 0.00097 |
| Maximum total cumulative pitch deviation $F_p$ | −0.05557 |
| Tooth profile relative deviation $f_c$ | 0.01093 |
Judged against the tolerance bands of GB/T 11365—2019, the single pitch deviation, the total cumulative pitch deviation and the profile relative deviation all fall inside the grade 8 interval, so the gear is classified as grade 8 — the accuracy that had been declared for it before measurement. The agreement between the declared and the measured grade is the practical test of the whole chain: probe orientation, coordinate conversion, involute model and curve fitting.
What the method changes, and where its limits lie
The gain over contact measurement comes from three features of the arrangement rather than from the sensor alone. First, no probing path has to be planned from the tooth geometry: the beam stays fixed in orientation while the c axis sweeps the gear, so the flank is sampled as a polar curve and the transformation from stylus centre to contact point disappears together with its numerical error. Second, there is no tip radius compensation and no tip wear, and the lateral resolution is set by the spot size rather than by the size of a stylus. Third, one acquisition serves several error items — pitch, cumulative pitch and profile deviation — because all of them are derived from the same crossing points of the fitted flanks with the pitch circle.
The limits follow from the same design. Evaluation is confined to the mid-face plane, which is what the standard requires but is not a substitute for flank topography; describing the whole surface would need measurements at several axial positions and a correspondingly larger data set. The fitted crossing point is sensitive to the local density of sampled points and to the choice of the fitting window, and although a cubic is adequate near the pitch circle, where the involute is nearly straight, the fit must not be extrapolated towards the tip or the root. Optical access is the other constraint: deep slots and large spiral angles demand the tilt adjustment of the sensor head, and the flank has to return enough light, so ground or lapped teeth are measured far more reliably than rough-cut ones. Within these bounds the approach converts a path-planning problem into a data-processing problem, and the same processing chain — sampling along a swept coordinate, cubic least-squares fitting, comparison with an analytically generated reference — is transferable to other complex helical surfaces.