Spiral bevel gears transmit motion and torque between intersecting or offset shafts in aerospace drives, electric-vehicle axles, precision machine tools, marine propulsion, and other high-end equipment. Their flanks are spatial surfaces generated by a coordinated five-axis motion between a cup-shaped grinding wheel and the workpiece. After carburizing and hardening, generating grinding is the final operation that determines flank accuracy, surface integrity, and contact pattern. The process is geometrically nonstationary: as the cradle angle sweeps, the wheel enters different regions of the tooth, the instantaneous contact line changes length and orientation, and the undeformed chip volume per increment can vary by an order of magnitude. With a fixed generating speed, this geometric variation is converted directly into a time-varying grinding force. The resulting force spikes excite the machine structure, produce chatter marks and profile errors, concentrate heat, and accelerate uneven wheel wear.
Conventional force control relies on dynamometers, spindle-power feedback, adaptive feed drives, or data-driven models. These approaches can reduce mean force or reject low-frequency disturbances, but they either add hardware, introduce servo lag, or depend on black-box models that are difficult to certify in a real-time CNC. They also act after the force has appeared. A different route is feedforward motion design: pre-schedule the generating speed so that the geometric source of the force is spread over time. The control layer then needs no additional force sensor or actuator; the existing CNC axes execute the schedule.
The Geometric Forcing Function
Classical grinding-force models separate the force into a cutting term associated with chip formation and a friction or ploughing term associated with contact area and rubbing. Werner’s 1978 model linked force to chip thickness and active grain density; Malkin’s 1989 two-component formulation separated material-removal and interface-friction contributions; Hecker and co-workers added a probabilistic chip-thickness distribution in 2007. For a generating grind, the practical consequence is that the cutting term is controlled primarily by the local material-removal rate, while the friction term is controlled primarily by the wheel–flank contact area. At fixed depth of cut and wheel speed, a useful control-oriented representation is
$$F_t \approx F_{\mathrm{cut}}\!\left(v(q)\,\omega\right)+F_{\mathrm{fric}}\!\left(l_c(q)\right),$$
where \(q\) is the cradle angle, \(\omega=\dot q\) is the generating angular speed, \(v(q)\) is the local undeformed chip volume per unit cradle angle, and \(l_c(q)\) is the instantaneous contact-line length. Neither \(v(q)\) nor \(l_c(q)\) can be changed at a given cradle angle without changing the gear design, wheel profile, or machine setting. What can be changed is the time mapping \(q(t)\).
For a fixed cradle angle, the contact line is obtained by solving the meshing equation over a set of wheel rotation angles. Its length is the sum of Euclidean distances between adjacent contact points:
$$l_c(q)=\sum_{i=1}^{n-1}\left\|\mathbf{P}_{i+1}(q)-\mathbf{P}_{i}(q)\right\|.$$
For the representative Gleason-type gear in Table 1, \(l_c(q)\) rises from near zero at entry to a broad maximum near the middle of the stroke and falls at exit. The chip-volume sequence is sharper. It has a dominant peak near \(60^\circ\) cradle angle, where the wheel tip radius passes through the root fillet and the flank curvature changes. At a fixed generating speed, the time axis is linear in \(q\), so every geometric peak appears at the same relative time in the force signal. This is why the measured spindle current in the constant-speed trial has a single sharp pulse at about 65% of the stroke.
| Parameter | Value |
|---|---|
| Number of teeth | 33 |
| Face width | 38 mm |
| Mid-spiral angle | 35° |
| Outer diameter | 217.63 mm |
| Outer cone distance | 119.19 mm |
| Wheel diameter | 190.5 mm |
| Wheel specification | 32A80J8VS ceramic-bond form wheel |
| Workpiece material | 20CrMnTi, carburized and tempered, 58–62 HRC |
| Finishing depth | 0.1 mm |
| Wheel spindle speed | 3000 rpm |
| Single-slot cycle | 7 s |
| Generating speed range | 200–1000 mm/min |
To obtain \(v(q)\), a two-step CAD simulation was used instead of a single Boolean subtraction. A rough wheel cut the full slot; a finishing wheel, reduced radially by 0.1 mm, repeated the generating motion; at each cradle increment the intersection volume was recorded. The sum over the stroke was 28.42 mm³, compared with 28.50 mm³ from the 0.1 mm depth times the computed flank area, a 0.28% difference. The sequence reaches about 0.92 mm³ per 0.5° increment near the middle of the stroke and falls to near zero at entry and exit. The volume check and the phase agreement with the contact-line curve support the geometric extraction.
Two Feedforward Laws
Let \(q\) be the cradle angle, \(\omega=\dot q\), \(v(q)\) the local chip volume per unit angle, and \(l_c(q)\) the contact-line length. Both functions are precomputed and represented by cubic splines, so they are continuous and differentiable for optimization.
Constant material-removal rate (Strategy 1, S1). The target is \(v(q)\omega=C\), which gives \(\omega(q)=C/v(q)\). Where the local chip volume is large, the cradle slows; where it is small, the cradle speeds up. This is a feedforward peak-shaving law. It does not reduce the total material removed or the total stroke; it redistributes removal over time. The instantaneous cutting-force excitation is lowered at the price of a longer dwell in the heavy-cut region.
Equal contact-line exposure (Strategy 2, S2). An exact constant \(l_c(t)\) is not attainable over the full stroke because \(l_c(q)\) is non-monotonic. The practical target equalizes contact-line exposure in each segment:
$$\int_{q_i}^{q_{i+1}} l_c(q)\,dq=\frac{1}{N}\int_{q_{\mathrm{start}}}^{q_{\mathrm{end}}} l_c(q)\,dq.$$
Within a segment, the speed is set inversely proportional to the mean \(l_c\), and the proportionality constant is fixed by the 7 s cycle. This keeps the contact area per unit time nearly uniform rather than the material-removal rate.
Both laws are discretized into seven constant-speed segments. The segment count is a compromise: more segments approximate continuous modulation better but lengthen the NC program and increase acceleration events. Speed is constrained to 200–1000 mm/min, total time to 7 s, and segment-to-segment speed changes are kept moderate. The constant-speed reference is 900 mm/min, equivalent to 0.0960 rad/s. Table 2 lists the schedules used in the grinding trials.
| Segment | S1 speed (mm/min) | S2 speed (mm/min) |
|---|---|---|
| 1 | 721 | 681 |
| 2 | 277 | 289 |
| 3 | 216 | 226 |
| 4 | 191 | 209 |
| 5 | 215 | 237 |
| 6 | 289 | 309 |
| 7 | 980 | 814 |
Both schedules slow the cradle in the middle of the stroke, where the wheel is fully engaged. Strategy 1 is more aggressive at the ends, reaching 980 mm/min in the last segment; Strategy 2 is more conservative, ending at 814 mm/min. This difference is consistent with their objectives: Strategy 1 responds to the sharp chip-volume function, while Strategy 2 responds to the broader contact-line function.
Experimental Evidence
The three strategies were compared on an H350GH five-axis CNC spiral bevel gear grinder. The workpiece was a Gleason-type 33-tooth gear in 20CrMnTi, carburized and tempered to 58–62 HRC. The wheel was a 32A80J8VS ceramic-bond form wheel of 190.5 mm diameter. Finishing depth was 0.1 mm, wheel spindle speed 3000 rpm, and the single-slot cycle was 7 s. All parameters except the generating-speed program were held fixed. Spindle U-phase current was logged at 50 Hz as a proxy for tangential grinding force; an ICP accelerometer on the spindle housing captured vibration at 100 kHz. Flank error was measured on a gear measuring center with a 25 × 25 grid, 625 points per flank. Roughness was measured with a stylus instrument using a 0.8 mm cutoff and a 4.0 mm evaluation length, with three repeats per flank. At least three slots were ground per strategy on the same gear.
| Strategy | Peak current (A) | Current fluctuation ΔIpp (A) | ΔIpp reduction (%) | Vibration RMS (m/s²) | Vibration reduction (%) | Profile error RMS (µm) | Profile improvement (%) | Ra (µm) | Ra improvement (%) |
|---|---|---|---|---|---|---|---|---|---|
| Constant speed | 2.55 ± 0.08 | 1.23 ± 0.10 | — | 0.0481 ± 0.0025 | — | 1.276 ± 0.105 | — | 0.642 ± 0.032 | — |
| S1 | 2.39 ± 0.06 | 0.58 ± 0.08 | 58.12 | 0.0184 ± 0.0012 | 61.75 | 1.071 ± 0.087 | 16.06 | 0.588 ± 0.028 | 8.41 |
| S2 | 2.28 ± 0.05 | 0.51 ± 0.06 | 66.31 | 0.0175 ± 0.0010 | 63.62 | 0.849 ± 0.065 | 33.46 | 0.585 ± 0.027 | 8.79 |
The constant-speed current trace has a single sharp peak of 2.55 A at about 65% of the stroke, with a peak-to-peak fluctuation of 1.23 A. The timing matches the predicted chip-volume peak. Under Strategy 1, the peak falls to 2.39 A and the fluctuation to 0.58 A; the sharp pulse becomes a broader plateau. Under Strategy 2, the peak falls to 2.28 A and the fluctuation to 0.51 A; the trace is flattest over most of the cycle. Relative to constant speed, the current-fluctuation amplitude is reduced by 58.12% for Strategy 1 and 66.31% for Strategy 2. The current RMS follows the same order, falling from 0.31 A to 0.11 A and 0.09 A, respectively.
The spindle-housing acceleration shows the same ranking. Constant-speed grinding produces a strong mid-stroke oscillation with a peak above 0.25 m/s² and an RMS of 0.0481 m/s². Strategy 1 reduces the RMS to 0.0184 m/s², a 61.75% reduction; Strategy 2 reduces it to 0.0175 m/s², a 63.62% reduction. Strategy 2 also decays faster after the main excitation. This is consistent with a lower force impulse and a more stable contact area, although the vibration measurement alone does not separate the two contributions.
Profile error is measured over 625 points per flank. The RMS error falls from 1.276 µm for constant speed to 1.071 µm for Strategy 1 and 0.849 µm for Strategy 2, improvements of 16.06% and 33.46%. The maximum peak-to-valley error falls from 5.82 to 3.21 µm. The variable-speed flanks show no groove-like or ridge-like discontinuities that can appear when speed is switched abruptly. Ra falls from 0.642 µm to 0.588 and 0.585 µm, improvements of 8.41% and 8.79%. The roughness gain is smaller than the profile gain, which is expected because Ra is less sensitive to low-frequency dynamic error than the flank form is.
The ranking Strategy 2 > Strategy 1 > constant speed can be read as a coupling effect. Strategy 1 directly targets the cutting-force source by flattening the chip-volume rate; it should be most effective at reducing the sharpest force pulses. Strategy 2 targets the contact-area source and the contact stiffness; it should be more effective at reducing the structural response that produces profile error. Because \(v(q)\) and \(l_c(q)\) are both fixed by the same meshing geometry, each strategy also partially improves the other variable. The experiments are consistent with this: Strategy 1 reduces vibration almost as much as Strategy 2, while Strategy 2 gives a much larger profile improvement. A plausible interpretation is that the contact-line schedule keeps the wheel–flank engagement more uniform, reducing time-varying elastic deflection and chatter excitation, so the CNC path is replicated more faithfully. This interpretation is supported by the correlation between lower vibration and lower profile error, but it is not a direct measurement of contact stiffness.
Engineering Scope and Limits
At the control layer, the method uses only the standard CNC axes. Once \(v(q)\) and \(l_c(q)\) are known for a gear, wheel, and machine set, the schedule can be generated offline and written as a seven-segment NC program. The experimental spindle current and accelerometer were used for measurement, not for feedback control. This makes the approach attractive for existing grinders that lack force dynamometers or fast force-feedback loops.
The validation covers one Gleason-type 33-tooth gear, one 20CrMnTi workpiece at 58–62 HRC, one 32A80J8VS wheel, one H350GH five-axis grinder, one 0.1 mm finishing depth, one 3000 rpm wheel speed, and a 7 s cycle. The force signal is inferred from spindle current, not measured with a calibrated dynamometer. The speed law is open-loop and precomputed; it does not adapt to wheel wear, thermal drift, or material batch variation. The seven-segment approximation cannot cancel high-frequency force components. The model is geometric and kinematic; it does not include grinding heat, residual stress, or wheel topology. Extrapolation to other gear sizes, modified flanks, or form grinding requires recalculation and new validation.
Three extensions follow naturally. First, the geometric schedule can be coupled with thermal and wear models so that speed is chosen to balance force, temperature, and wheel life, not force alone. Second, slow online adaptation can be added using already available spindle current or vibration signals, keeping the feedforward schedule as the baseline and correcting for drift. Third, the same motion-design logic can be transferred to form grinding, profile-modified gears, and other gear-cutting operations where a kinematic degree of freedom can be used to reshape a time-varying load. The broader point is that generating speed is not only a productivity parameter; in a geometrically nonstationary process such as spiral bevel gear grinding, it is a control variable.