Spiral bevel gears transmit motion and torque between intersecting or offset shafts, and they are used wherever a drive has to combine high load capacity with smooth, quiet running: aerospace actuation and accessory drives, vehicle differentials, marine propulsion, machine-tool indexing heads. The curved tooth flank is a free-form spatial surface generated by a conjugate motion between a rotating cup-shaped grinding wheel and the workpiece, and the finishing grind after heat treatment is what finally fixes flank geometry and surface integrity. Because the contact between wheel and flank changes continuously during the generating roll, the finishing pass is also where the process is least stable.
Where the force fluctuation comes from
Grinding force is generated by many thousands of discrete abrasive grains acting simultaneously. At the scale of a single grain, material response passes through three regimes: elastic rubbing, where the tip deforms the surface recoverably and generates friction and heat without removing material; ploughing, where the grain pushes material plastically to the sides and forms a raised ridge; and chip formation, where the shear stress exceeds the flow stress and material is removed. In a real wheel all three occur at once on different grains, and the measured force is the statistical sum of those events.
For engineering purposes the force is resolved into a tangential component, parallel to the wheel velocity and proportional to the power drawn by the spindle, and a normal component, directed into the workpiece and responsible for elastic deflection of the machine–tool–workpiece loop, wheel wear and chatter onset. The two are linked by a force ratio typically between 1.5 and 3, depending on workpiece material, wheel sharpness and cooling condition.
The practical consequence for a generated bevel flank is that, with depth of cut and wheel speed held fixed, the cutting contribution tracks the volume of material removed per unit time, while the rubbing and ploughing contribution tracks the real area of wheel–workpiece contact. Both quantities vary strongly with the generating roll position on a spiral bevel gear. Written as a two-term model,
$$F_t = \underbrace{k_1\,\dot{V}^{\,c}}_{\text{chip formation}} + \underbrace{k_2\,L_c}_{\text{rubbing and ploughing}}, \qquad 0 \le c \le 2/3$$
where $V$ is the instantaneous undeformed chip volume, $L_c$ the instantaneous contact-line length, and the exponent $c$ is set by the distribution of active cutting edges on the wheel face; $k_1$ and $k_2$ absorb wheel speed, depth of cut, equivalent wheel diameter and the wheel–workpiece pair, and are identified per speed level. The structure echoes the classic Werner, Malkin and Hecker formulations, and a bevel-gear-specific variant fitted to measured data uses the same two geometric inputs: undeformed chip volume and contact-line length.
The important reading of this model is that force fluctuation on a generated flank is geometric in origin. Wheel speed and depth of cut are constant across a finishing stroke; what changes is how much material sits under the wheel at each instant and how long the contact line is. Neither quantity can be altered by changing the tooth geometry, which is fixed by design. They can, however, be redistributed in time — which is the basis of the control approach described below.
Computing the two geometric inputs
Tooth surface and contact-line length
The flank model is built from spatial meshing theory coupled to the kinematics of a five-axis generating grinder: three linear axes (X, Y, Z) and two rotary axes (A for the workpiece spindle, B for blank mounting), with the wheel spindle as C. The cradle angle $q$ drives the generating roll; the workpiece rotation follows from the roll ratio plus higher-order motion coefficients that produce the required flank modification and contact pattern. Solving the meshing equation yields the surface position vector and, for each cradle angle, a set of discrete points along the instantaneous contact line. Its length is obtained by summing Euclidean distances between adjacent, ordered points:
$$L_c(q) = \sum_{j=1}^{m-1}\left|\mathbf{P}_{j+1}(q)-\mathbf{P}_{j}(q)\right|$$
For a representative case — 33 teeth, face width 38 mm, mean spiral angle 35°, outer cone distance 119.19 mm, outer diameter 217.63 mm, addendum 3.58 mm, dedendum 8.69 mm, pitch cone angle 64.13°, root cone angle 59.98° — the total contact-line length starts at zero at the beginning of the sweep, rises steeply, peaks near mid-roll at roughly 88 mm, and returns to zero at exit. Concave and convex flanks are computed separately and summed for double-sided grinding. This single curve already shows why a constant generating speed cannot produce a constant force: the same angular increment of the cradle removes several times more contact length near mid-roll than near the ends.
Undeformed chip volume by two-step simulation
Chip volume is harder to obtain analytically because it depends on the instantaneous boolean intersection between the wheel solid and the material left in the tooth slot. A one-pass simulation of the whole removal is not representative: the finishing grind removes a skin of about 0.1 mm from a slot that already has its macroscopic shape. The simulation therefore reproduces the process sequence. A first step generates the complete slot with a full-size wheel solid through the entire generating motion, producing a geometrically accurate tooth space. A second step re-runs the same motion with a wheel solid reduced radially by the finishing depth (0.1 mm), and at each discrete cradle position records the volume of the intersection between the wheel and the previously generated slot. The routine is implemented as an AutoCAD secondary development in VBA, with the machine kinematics applied as homogeneous transformations (Rotate3D and Move) so that every relative pose matches the theoretical model, and the intersection volume read out through the Volume command.
Two independent checks were applied to the extracted data. Volume conservation compares the summed chip volumes against the product of finishing depth and slot surface area, the latter obtained by numerical integration on the tooth surface model. For the case above, the simulated total is 28.42 mm³ against a theoretical 28.50 mm³ from 0.1 mm × (14.3764 + 14.1316) mm², a relative error of 0.28%, well inside the 1.5% band attributable to step discretisation and CAD boolean tolerance. Trend consistency compares the chip-volume sequence with the independently computed contact-line curve: peak and trough positions coincide, and both follow the same rise–plateau–fall pattern corresponding to root entry, mid-flank contact and tip exit. Since the two quantities are derived by different routes but from the same meshing geometry, their agreement supports the physical validity of the simulated sequence, not merely its internal arithmetic.
Both discrete sequences are finally converted into continuous, differentiable functions of cradle angle by cubic spline interpolation. A spline is chosen over polynomial fitting because the data are noise-free but strongly non-linear with sharp local peaks; the interpolant passes exactly through every computed point, is twice continuously differentiable — needed later for the material-removal-rate term — and preserves local shape without the oscillation of high-order polynomials.
Two feed-forward control strategies
The control variable is the generating speed itself, that is, the rate at which the cradle angle advances. Since flank geometry and depth of cut are fixed, changing $\omega_c(t) = dq/dt$ rescales the time axis of the process and thereby redistributes removal load and contact state over the stroke. No sensor and no additional actuator are involved: the strategy is embedded in the motion program. Two targets follow directly from the two terms of the force model.
Constant material removal rate. If the chip-formation term is to be held constant, the volumetric removal rate must be constant:
$$\dot{V}(t) = \frac{dV}{dq}\,\omega_c(t) = \bar{R} \quad\Longrightarrow\quad \omega_c(q) = \frac{\bar{R}}{V'(q)}$$
With $\bar{R} = V_{total}/T_0$ set by the total removed volume and the allotted cycle time, the generating speed is slowed where the chip-volume derivative is large and accelerated where it is small — a direct feed-forward of the geometric load, applied before the force peak develops rather than in response to it.
Constant instantaneous contact line. Holding $L_c$ strictly constant with respect to time is not achievable, because $L_c(q)$ has extrema where its derivative vanishes. The engineering substitute is to make the contact length uniform in time: divide the cradle sweep into $N$ segments of equal contact-length integral,
$$\int_{q_i}^{q_{i+1}} L_c(q)\,dq = \frac{1}{N}\int_{q_0}^{q_N} L_c(q)\,dq, \qquad i = 0,\ldots,N-1$$
and set each segment speed inversely proportional to the mean contact length in that segment, with the proportionality constant fixed by the total time constraint
$$\sum_{i=1}^{N}\frac{\Delta q_i}{\omega_i} = T_0$$
Both strategies are subject to the same constraints: generating speed inside the servo-feasible and process-acceptable band (200–1000 mm/min in this case), total single-slot time equal to the conventional cycle (7 s), and speed continuity between segments to avoid jerk. Segment count was set to seven, a compromise between fidelity to the sharp features of $V(q)$ and $L_c(q)$ and the block-processing capability of the CNC. The resulting schedules are listed below; the conventional reference is a constant 900 mm/min (0.0960 rad/s) over the whole sweep.
| Segment | Strategy 1 — constant MRR | Strategy 2 — constant contact line | ||
|---|---|---|---|---|
| Cradle span (°) | Speed (mm/min) | Cradle span (°) | Speed (mm/min) | |
| 1 | 37.72–49.45 | 721 | 37.72–50.97 | 681 |
| 2 | 49.45–52.81 | 277 | 50.97–54.54 | 289 |
| 3 | 52.81–55.43 | 216 | 54.54–57.21 | 226 |
| 4 | 55.43–57.74 | 191 | 57.21–59.74 | 209 |
| 5 | 57.74–60.34 | 215 | 59.74–62.61 | 237 |
| 6 | 60.34–63.84 | 289 | 62.61–66.35 | 309 |
| 7 | 63.84–81.72 | 980 | 66.35–81.72 | 814 |
The two schedules differ in character. Strategy 1 is aggressive: it drops to about 190–220 mm/min through the mid-flank region where the chip-volume derivative peaks — roughly a quarter of the conventional speed — and recovers to 721 and 980 mm/min at the two ends, sitting at the upper bound in the final segment. Strategy 2 is smoother, with a minimum of about 210 mm/min and a final-segment speed of 814 mm/min, reflecting that it equalises contact length rather than suppressing the chip-volume pulse. Feeding either schedule back into the force model predicts the intended reshaping: the sharp single peak of the constant-speed case becomes a broad, low plateau, with strategy 2 giving the flattest force trace and strategy 1 removing the highest-frequency content of the pulse.
Experimental verification
Tests were run on an H350GH CNC spiral bevel gear grinder with a 32A80J8VS ceramic-bonded CBN cup wheel (diameter 190.5 mm, inner and outer blade angles 19.5°, point width 3.3 mm; radial cutter position 90.1 mm, cutter swivel 204.72°, roll ratio 1.108401, blank mounting angle 59.97°). Workpieces were Gleason-generated gear members in 20CrMnTi, carburised, quenched and low-temperature tempered to 58–62 HRC. Fixed conditions across all trials were: depth of cut 0.1 mm, wheel spindle speed 3000 r/min, 7 s per tooth slot, dedicated grinding oil for cooling and lubrication, and diamond-roll dressing before each trial to standardise wheel topography. Only the generating-speed schedule varied, giving three groups: constant velocity (CV, 900 mm/min), strategy 1 (S1) and strategy 2 (S2). Each strategy was applied to at least three slots on the same gear to allow scatter estimates.
Two in-process signals were recorded. Tangential force was inferred from the U-phase current of the wheel-spindle servo motor (C1 axis), captured through the CNC trace function at 50 Hz, which resolves the low-frequency, generating-motion-dominated content below about 10 Hz. Vibration was measured with a PCB 352C23 ICP accelerometer (100 mV/g, 0.5 Hz–10 kHz) magnetically mounted on the spindle housing near the bearing seat, with its sensitive axis normal to the grinding plane, sampled at 100 kHz by an NI cDAQ-9174 with an NI 9234 module. A programmable digital output from the CNC issued a trigger pulse at the start of the grinding cycle so that both records share a common time origin. Off-line, flank topography was measured on an L65G gear measuring centre using a 25 × 25 grid (625 points per flank) rather than the conventional 45-point pattern, specifically to detect any step, furrow or ridge that a segmented speed schedule might imprint at segment boundaries. Surface roughness was measured with a Taylor Hobson Surtronic S-100 stylus instrument along the tooth-height direction, 0.8 mm cut-off and 4.0 mm evaluation length, three repeats per flank.
| Strategy | Peak current (A) | Current ripple $\Delta I_{pp}$ (A) | Ripple reduction | Current RMS (A) | Vibration RMS (m/s²) | Vibration reduction |
|---|---|---|---|---|---|---|
| CV | 2.55 ± 0.08 | 1.23 ± 0.10 | — | 0.31 ± 0.04 | 0.0481 ± 0.0025 | — |
| S1 | 2.39 ± 0.06 | 0.58 ± 0.08 | 58.12% | 0.11 ± 0.03 | 0.0184 ± 0.0012 | 61.75% |
| S2 | 2.28 ± 0.05 | 0.51 ± 0.06 | 66.31% | 0.09 ± 0.02 | 0.0175 ± 0.0010 | 63.62% |
| Strategy | Profile error RMS (µm) | Accuracy improvement | Max peak-to-valley (µm) | Ra (µm) | Roughness improvement |
|---|---|---|---|---|---|
| CV | 1.276 ± 0.105 | — | 5.82 | 0.642 ± 0.032 | — |
| S1 | 1.071 ± 0.087 | 16.06% | — | 0.588 ± 0.028 | 8.41% |
| S2 | 0.849 ± 0.065 | 33.46% | 3.21 | 0.585 ± 0.027 | 8.79% |
Reading the results
Under constant generating speed the current trace shows a single, sharp pulse peaking at about 2.55 A and located at roughly 65% of the stroke. That position coincides with the predicted maximum of the undeformed chip volume and with the peak of the contact-line curve. This coincidence is the most direct experimental evidence available here that the geometric time-variation of the tooth space — not random wheel behaviour — is the dominant excitation source, and it justifies treating the force problem as a motion-planning problem.
Under strategy 1 the peak is cut and, more tellingly, the narrow pulse is replaced by a broad hump, exactly as the constant-MRR formulation predicts: load that was concentrated in a short cradle interval is spread over a longer time window. Under strategy 2 the trace approaches a flat band for most of the cycle, with the lowest peak (2.28 A) and the lowest RMS current; the corresponding vibration record shows both the smallest main peak (0.088 m/s² against 0.263 m/s² for CV) and the fastest decay of residual oscillation. The ordering S2 > S1 > CV is consistent across current ripple, vibration RMS and flank accuracy, which is what a causal chain from excitation to response would require: smaller force modulation means less forced vibration, less dynamic deflection of the machine loop, and therefore a closer reproduction of the commanded tool path on the workpiece.
The accuracy gain is the largest single effect: 33.46% reduction in profile error RMS and a fall in maximum peak-to-valley deviation from 5.82 µm to 3.21 µm. Because the 625-point grid would expose any discontinuity at a segment boundary, the absence of furrows or ridges is itself a result — it confirms that the seven-segment approximation with continuity constraints does not leave a geometric signature. Roughness improves by a much smaller margin, 8.79%, which is physically reasonable: roughness is set by the individual grain interaction, and a smoother force history mainly removes the anomalous grain fracture and plastic ploughing events caused by load spikes rather than changing the baseline cutting mechanism.
The comparison between the two strategies is instructive. Strategy 2 targets contact length and yet outperforms strategy 1 on every indicator, including the current ripple that strategy 1 was designed to suppress. The likely explanation is that the contact-line term governs the dynamic stiffness of the contact zone and the friction contribution simultaneously, so equalising it stabilises both the force amplitude and the structural response, whereas equalising removal rate leaves the contact condition still varying. For practical purposes, controlling the contact-length history appears to be the more effective single lever.
Scope, limitations and next steps
The results above were obtained on one gear design, one wheel specification and one machine, with three slots per strategy; they establish feasibility rather than generality. The current signal is a proxy for tangential force calibrated through the motor torque relationship, and the 50 Hz sampling deliberately excludes high-frequency force components, so the vibration channel carries the high-frequency evidence. The speed schedules are computed offline from nominal geometry and a fixed depth of cut, which means the method is feed-forward by construction and has no mechanism to compensate for wheel wear, thermal drift of the machine or batch-to-batch variation in blank hardness — the disturbances that dominate over a production shift.
Three extensions follow naturally. First, coupling the geometric model with thermal, residual-stress and wheel-wear models would allow the speed schedule to be optimised against several objectives rather than force alone, since a force-optimal schedule is not automatically burn-optimal. Second, combining the offline schedule with online monitoring — vibration or acoustic emission — would give an adaptive or robust outer loop that corrects the nominal plan when the process drifts, at the cost of reintroducing sensors. Third, the same motion–geometry–physics reasoning can be transferred to other generating operations and to formed grinding with cup wheels, and to modified flanks and higher accuracy grades, where the contact-line variation is different in shape but not in nature.
The engineering value of the approach lies in what it does not need: no dynamometer, no accelerometer in the control loop, no additional actuator, no change to tooth geometry or to the grinding cycle time. The whole intervention is a rewriting of the generating-motion program, which makes it deployable on existing five-axis grinders through their standard numerical control. For a process where the dominant disturbance is geometric and fully known in advance, that is arguably the appropriate place to intervene.