Spiral bevel gears are used in aerospace, automotive, and precision machine-tool drives where high torque, low noise, and long service life are required. After heat treatment, their flanks are usually finished by CNC generating grinding to correct distortion and restore tooth geometry. In double-flank generating grinding, the wheel engages the concave and convex flanks at the same time, and the contact geometry changes continuously as the cradle angle sweeps through the generating stroke. The instantaneous contact line length and the undeformed chip volume are not constant; they rise and fall with the cradle angle. At a fixed generating speed, this geometric variation becomes a variation in material removal rate, which appears as grinding-force fluctuation, spindle vibration, and tooth-surface error. A model-based way to suppress this chain at its source is to vary the generating speed so that the material removal rate stays approximately constant.
Force Fluctuation in Double-Flank Generating Grinding
A two-term grinding-force model separates the cutting contribution from the friction contribution. For this process, the tangential force can be written as
$$
F_T=\frac{K_c}{\alpha+1}V\left(\frac{v_w}{v_s}\right)^{\alpha}\left(\frac{a_p}{D_e}\right)^{\alpha/2}
+\frac{K_f}{\alpha+1}l_c\left(\frac{v_w}{v_s}\right)^{\alpha}\left(\frac{a_p}{D_e}\right)^{\alpha/2}
$$
where \(F_T\) is the tangential grinding force, \(V\) is the instantaneous undeformed chip volume, \(l_c\) is the instantaneous contact line length, \(v_w\) is the generating speed of the workpiece, \(v_s\) is the wheel peripheral speed, \(a_p\) is the depth of cut, \(D_e\) is the equivalent wheel diameter, \(K_c\) and \(K_f\) are model coefficients, and \(\alpha\) is an exponent determined by the abrasive-edge distribution, typically in the range \(0 \le \alpha \le 2/3\). The first term represents chip formation; the second represents friction. The model is not used here to predict an absolute force. It identifies the geometric terms that dominate the fluctuation when the process parameters are otherwise fixed.
In finish grinding, \(v_s\), \(a_p\), and \(D_e\) are held constant, and \(K_c\), \(K_f\), and \(\alpha\) are treated as process coefficients. The instantaneous contact line length \(l_c(q_1)\) and the undeformed chip volume \(V(q_1)\) are functions of the cradle angle \(q_1\), fixed by the tooth-surface geometry and the generating path. Their nonlinearity is the geometric source of the force oscillation. A variable generating speed does not change these functions. It changes the rate at which \(q_1\) advances, and therefore changes the time evolution of the removal load. Reducing the cradle speed where the chip load is large and increasing it where the load is small flattens the time-averaged material removal rate and reduces the forcing amplitude.
Contact Line and Chip Volume in a Representative Gear
The tooth surface can be described by a two-parameter mathematical model. For a fixed cradle angle \(q_1\), varying the wheel rotation angle \(\theta\) traces the instantaneous contact line; its length is obtained by summing the Euclidean distances between adjacent points on that line. In double-flank grinding, the total contact line is the sum of the concave-flank and convex-flank lines. The calculation below uses a 33-tooth spiral bevel gear with a 38 mm face width and a 35° spiral angle. The generating stroke runs from \(q_1=81.72^\circ\) to \(q_1=37.72^\circ\), a span of 44°, with the center at \(59.72^\circ\).
| Parameter | Value |
|---|---|
| Tooth count | 33 |
| Face width | 38 mm |
| Spiral angle | 35° |
| Outer diameter | 217.63 mm |
| Outer cone distance | 119.19 mm |
| Generating stroke | 37.72°–81.72° (span 44°) |
| Center cradle angle | 59.72° |
| Roll ratio | 1.108401 |
| Wheel diameter | 190.5 mm |
| Finish depth of cut | 0.1 mm |
| Wheel speed | 20 m/s |
The total contact line does not vary linearly with \(q_1\). It rises from the start of the generating stroke, reaches a maximum near the middle, and then falls toward the end. Because both flanks are ground together, this total line is the relevant quantity for the overall contact area and friction load. A constant cradle speed therefore produces a contact-area history that is largest in the middle of the stroke and smaller at the ends.
The undeformed chip volume was extracted with an AutoCAD/VBA simulation of the generating motion. The model used a rough-slotting followed by finish-grinding sequence, advanced the cradle angle in 0.5° steps, changed the relative wheel–workpiece position according to the generating motion, and used Boolean subtraction to isolate the undeformed chip. The resulting volume is strongly nonlinear. In the illustrated interval, it rises from 0.0007 mm3 at \(q_1=43.22^\circ\) to 0.0416 mm3 at \(q_1=48.72^\circ\), and the full-stroke curve continues to vary strongly. At a constant cradle speed, this chip-volume history is directly converted into a removal-rate history, which is the origin of the force swing.
Equal-Integral Speed Schedule
Let \(V(q_1)\) denote the local chip-volume function and \(\omega_c=dq_1/dt\) the cradle angular speed. The instantaneous material removal rate can be expressed through the local chip-volume derivative and the cradle angular speed:
$$
R(t)=\frac{dV}{dt}=\frac{dV}{dq_1}\frac{dq_1}{dt}
$$
At constant \(\omega_c\), \(R(t)\) follows \(dV/dq_1\). To flatten \(R(t)\), the cradle speed should be inversely proportional to that derivative: the motion should slow down where the chip volume changes rapidly and speed up where it changes slowly. The implementation divides the generating stroke into \(N\) contiguous intervals. The interval boundaries are chosen so that each interval removes the same volume, \(V_{\text{total}}/N\). If the total stroke time \(T_0\) is divided equally among the intervals, the constant speed in interval \(i\) is
$$
\omega_c^{(i)}=\frac{\Delta q_1^{(i)}N}{T_0}
$$
where \(\Delta q_1^{(i)}\) is the angular span of interval \(i\). For the reported schedule, \(N=7\) was selected as a compromise between force smoothing, cycle time, and CNC program complexity. The result is a piecewise-constant approximation to constant material removal rate, not a continuous ideal.
Two weighting functions were used. Scheme 1 equalizes the integral of the chip-volume function \(V(q_1)\); scheme 2 equalizes the integral of the contact-line function \(l_c(q_1)\). The first targets the chip-formation term of the force model; the second targets the friction term through the contact area. Both schedules are slow in the middle of the stroke, where the geometric load is largest, and fast near the ends. Scheme 1 spans 191–980 mm/min across its seven intervals; scheme 2 spans 209–814 mm/min. The constant-speed baseline was an empirical production value of 900 mm/min.
The control is a feedforward motion strategy. It does not measure force or vibration in real time and does not close a feedback loop. The schedule is precomputed from the tooth-surface and stock-removal model and stored as a segmented feed-rate table in the CNC program. This avoids the latency, hardware cost, and interference sensitivity of sensor-based adaptive force control, but it also means that the schedule is only as good as the model and the assumed stock distribution.
Experimental Comparison
Tests were run on an H350GH CNC spiral bevel gear grinding machine. The gear material was 20CrMnTi; the wheel was a 32A80J8VS microcrystalline ceramic alumina wheel; the finish depth of cut was 0.1 mm; the wheel speed was 20 m/s. Three schedules were compared: a constant 900 mm/min baseline, scheme 1, and scheme 2. Each schedule was repeated three times. Spindle current was acquired by the CNC system and used as an indirect force indicator through the current–force relationship. Spindle acceleration was measured with a PCB352C23 vibration sensor. Tooth-surface error was measured on an L65G gear measuring center with a 25 × 25 point grid per flank. The control itself used no additional force or vibration sensor; the sensors were used for evaluation.
| Metric | Constant 900 mm/min | Scheme 1: chip volume | Scheme 2: contact line |
|---|---|---|---|
| Current fluctuation coefficient | 0.6924 | 0.3189 (−53.94%) | 0.2820 (−59.27%) |
| Spindle vibration RMS (m/s2) | 0.0481 | 0.0184 (−61.75%) | 0.0175 (−63.62%) |
| Tooth-surface error RMS (µm) | 1.276 | 1.071 (−16.06%) | 0.849 (−33.46%) |
The baseline current trace rose and fell strongly over the stroke. Its worst run had a fluctuation coefficient of 0.7446, and the three-run average was 0.6924. Scheme 1 reduced the average to 0.3189, a 53.94% reduction. Its individual values were 0.3117, 0.3215, and 0.3236. Scheme 2 reduced the average to 0.2820, a 59.27% reduction; its individual values were 0.2895, 0.2508, and 0.3057. The improvement was therefore consistent across repeats, not the result of a single outlier.
Spindle vibration followed the same trend. The baseline amplitude was about ±0.13 m/s2 with an RMS of 0.0481 m/s2. Scheme 1 reduced the amplitude to about ±0.04 m/s2 and the RMS to 0.0184 m/s2, a 61.75% reduction. Scheme 2 gave an RMS of 0.0175 m/s2, a 63.62% reduction. Tooth-surface error also decreased: from 1.276 µm at constant speed to 1.071 µm with scheme 1 (16.06%) and 0.849 µm with scheme 2 (33.46%). The three metrics improved together, which is consistent with reduced excitation of the wheel–spindle–workpiece system.
Scheme 2 performed better than scheme 1. In finish grinding, the friction component is a substantial part of the total force, and the contact line length directly sets the wheel–flank contact area and the frictional load. Stabilizing that area also reduces swings in thermal load and the associated changes in contact stiffness. Equalizing chip volume alone mainly addresses the chip-formation term, so it leaves more of the friction-driven fluctuation in place. This explanation is an interpretation of the force model and the observed ranking; the experiments did not separate the cutting and friction contributions by direct force measurement.
Engineering Implications and Limits
The practical value of the method is that it reduces the geometric forcing at the source without adding hardware to the control loop. It can be implemented as a seven-segment feed-rate table in an existing CNC program, and it is compatible with the generating motion already used on spiral bevel gear grinders. The speed profile is low in the middle of the stroke, where the chip volume and contact line are largest, and high near the ends, where the load is smaller. This is the opposite of a constant-speed schedule, which imposes the largest removal rate exactly where the contact geometry is most demanding.
The evidence also has clear boundaries. The comparison used one gear geometry, one material, one wheel specification, and one machine. The percentage reductions are relative to a single empirical baseline of 900 mm/min, not to a globally optimized constant speed. The current signal is an indirect force indicator, not a direct dynamometer measurement. The 25 × 25 grid RMS is an aggregate surface metric and does not by itself identify localized defects. Three repeats per schedule provide a preliminary repeatability check rather than a full statistical distribution. The experiments show that the variable-speed schedules improve all three measured outcomes under the tested conditions; they do not establish universal gains for other spiral bevel gear geometries or process parameters.
The method is model-limited by design. Its performance depends on the accuracy of the tooth-surface model, the assumed stock distribution, and the equal-integral partition. If the actual removal distribution differs from the nominal model, the schedule will not be optimal. An adaptive extension could update the schedule from spindle current or other process signals, but that would reintroduce sensor hardware and response latency. Within the open-loop framework, the main engineering lever is the accuracy of the precomputed geometric load model.
For spiral bevel gear finishing, the results support a straightforward conclusion: when the contact geometry is known in advance, a segmented variable generating speed can flatten the material removal rate enough to reduce force fluctuation, spindle vibration, and tooth-surface error without additional sensors. The contact-line-based schedule is the stronger of the two tested variants because it targets the contact area that governs the friction term. The approach is therefore a practical feedforward complement to, rather than a replacement for, sensor-based adaptive control in high-precision generating grinding.