Spiral bevel gears transmit torque between intersecting shafts through teeth whose flanks are curved along the face width. Because contact begins at one end of the tooth and migrates diagonally across the flank, the mesh is gradual, giving a high contact ratio, smooth running, low noise and high load capacity. Axle drives, helicopter tail-rotor transmissions and machine-tool spindle trains all depend on this behaviour, and in each case the flank geometry produced by the last finishing operation controls it.
After carburising and hardening, the tooth flanks must be re-finished to remove distortion and to restore the intended contact pattern. Generating grinding performs this task with a cup-shaped or dish-shaped wheel whose positions are coordinated with a virtual generating gear. The operation therefore fixes three properties at once: the arithmetic mean roughness Ra of the flank, the residual stress state in the surface layer, and the dynamic response of the machine spindle that carries the wheel. Roughness and compressive stress act on different failure modes—roughness on film breakdown, contact fatigue and noise; compressive stress on crack initiation and tooth-root bending fatigue—while spindle vibration feeds back into the surface by displacing the wheel relative to the workpiece, and simultaneously wears the spindle bearings that hold the machine’s accuracy over years of production.
The three responses do not improve together. Wheel speed raises the number of cutting grits per unit time and refines the surface, but it also amplifies the unbalance forces of the wheel; generating speed trades productivity against heat generation; grinding depth trades mechanical burnishing against cutting force. Parameter selection is therefore a constrained multi-objective problem rather than a single-response tuning exercise.
Generating Kinematics on a Five-Axis Grinding Machine
The generating method reproduces the meshing of the workpiece with a virtual crown gear. The grinding wheel represents one tooth of the cutter head, and the flank is the envelope of the wheel positions swept as the workpiece rolls in mesh with the virtual gear. On a CNC gear grinder this motion is synthesised by three linear axes (X, Y, Z) and two rotary axes (A for the workpiece, B for the wheel head), whose interpolation over the tooth surface is defined by the machine settings: radial cutter position, cutter tilt and swivel, vertical and horizontal wheel offsets, workpiece mounting angle, roll ratio, and the roll positions at the toe, centre and heel of the tooth.
Machine settings determine the path; the process parameters—wheel linear speed vs, generating speed vw and grinding depth ap—determine the local interaction along that path. The path fixes where the wheel goes, the parameters fix how much material each grit removes, how much heat enters the flank, and how hard the spindle is driven.
Table 1. Gear and grinding wheel specifications.
| Component | Parameter | Value |
|---|---|---|
| Gear | Number of teeth | 33 |
| Gear | Module | 6.5 mm |
| Gear | Pressure angle | 20° |
| Gear | Spiral angle | 35° |
| Gear | Outer diameter | 217.6 mm |
| Gear | Outer cone distance | 119.19 mm |
| Gear | Crown-to-apex distance | 48.77 mm |
| Gear | Pitch cone angle | 64.1333° |
| Gear | Face cone angle | 66.45° |
| Gear | Root cone angle | 59.9667° |
| Wheel | Diameter | 190.5 mm |
| Wheel | Grinding edge pressure angle | 20° |
| Wheel | Tip distance | 3.3 mm |
| Wheel | Grit size | 80 |
| Wheel | Maximum linear speed | 35 m/s |
| Wheel | Tip radius | 2.2 mm |
Table 2. Machine settings used for the gear pair.
| Setting | Value | Setting | Value |
|---|---|---|---|
| Radial cutter position | 90.1 mm | Bed position | 0 mm |
| Cutter tilt | 0° | Roll ratio | 1.108401 |
| Cutter swivel | 204.72° | Centre roll position | 59.72° |
| Vertical wheel offset | 0 mm | Heel roll position | 42.07° |
| Horizontal wheel offset | 0 mm | Toe roll position | 75.97° |
| Workpiece mounting angle | 59.97° | — | — |
Experimental Design and Measurement Chain
The grinding trials were run on a domestic H350GH CNC spiral bevel gear grinder. The workpiece was a 33-tooth gear in 20CrMnTi, and the wheel was a 32A80J8VS ceramic alumina wheel of 190.5 mm diameter. Fixture and workpiece were installed and the wheel dressed before the first cut; the wheel type and coolant flow were held constant, and the wheel was dressed again after every tooth slot to suppress the drift that wear would otherwise introduce between runs.
Vibration was measured radially on the wheel spindle with a CT9104 four-channel USB acquisition unit and a CT1000 piezoelectric accelerometer connected by BNC-M5 cable. One record was taken per ground slot, giving 23 signal files at a sampling frequency of 9600 Hz. From each record the grinding stage was extracted and reduced to a root-mean-square value, used as the scalar measure of vibration intensity:
$$G_{\mathrm{rms}}=\sqrt{\frac{1}{n}\sum_{i’=1}^{n}x_{i’}^{2}}$$
where n is the number of samples in the analysed segment and xi′ the acceleration at sample i′.
Because the flank of a spiral bevel gear cannot be measured in place, each tooth was wire-cut into a single-tooth specimen before inspection. Roughness was obtained with a FORMTRACER SV-C3200 profilometer, using the arithmetic mean deviation Ra, which captures both the height and the fine peak-and-valley texture of the ground surface. Residual stress was measured with a Pulstec µ-X360S portable X-ray diffractometer having a stated accuracy of ±10 MPa; all recorded values are compressive and are reported with a negative sign. For every parameter set, Ra and stress were averaged over the concave and convex flanks.
The parameter study used a central composite design (CCD) with three factors and five levels, three centre points and 23 runs in total, with factor ranges bounded by the machine’s capability and by shop-floor practice. The 33 tooth slots of the workpiece were numbered, and each parameter set was assigned to one slot; the concave flank of the first tooth and the convex flank of the second tooth were ground with the same setting, so that both flanks of a parameter set could be characterised.
Table 3. Central composite design: factors and levels.
| Level | Wheel speed vs (m/s) | Generating speed vw (mm/min) | Grinding depth ap (mm) |
|---|---|---|---|
| −2 | 16 | 400 | 0.05 |
| −1 | 18 | 600 | 0.09 |
| 0 | 20 | 800 | 0.13 |
| +1 | 22 | 1000 | 0.17 |
| +2 | 24 | 1200 | 0.21 |
Table 4. Design matrix and measured responses (roughness and stress averaged over concave and convex flanks).
| Run | vs (m/s) | vw (mm/min) | ap (mm) | Ra (μm) | σ (MPa) | Grms (m/s2) |
|---|---|---|---|---|---|---|
| 1 | 16 | 800 | 0.13 | 0.5863 | −459.5 | 15.8 |
| 2 | 20 | 800 | 0.21 | 0.4638 | −454.5 | 18.1 |
| 3 | 18 | 600 | 0.17 | 0.5057 | −586.0 | 17.5 |
| 4 | 16 | 800 | 0.13 | 0.5197 | −486.0 | 15.6 |
| 5 | 20 | 400 | 0.13 | 0.4715 | −622.5 | 17.2 |
| 6 | 20 | 1200 | 0.13 | 0.5371 | −405.0 | 17.5 |
| 7 | 20 | 1200 | 0.13 | 0.5212 | −407.5 | 17.8 |
| 8 | 20 | 800 | 0.13 | 0.2860 | −447.5 | 17.0 |
| 9 | 18 | 1000 | 0.09 | 0.5144 | −506.0 | 16.3 |
| 10 | 20 | 800 | 0.05 | 0.5405 | −461.5 | 17.2 |
| 11 | 20 | 800 | 0.13 | 0.3257 | −418.5 | 17.6 |
| 12 | 24 | 800 | 0.13 | 0.5139 | −429.5 | 20.1 |
| 13 | 18 | 1000 | 0.17 | 0.5371 | −415.0 | 17.1 |
| 14 | 20 | 800 | 0.05 | 0.5093 | −440.5 | 17.3 |
| 15 | 22 | 1000 | 0.09 | 0.4436 | −413.5 | 19.9 |
| 16 | 20 | 800 | 0.21 | 0.5276 | −494.5 | 17.9 |
| 17 | 22 | 1000 | 0.17 | 0.4704 | −438.0 | 18.3 |
| 18 | 18 | 600 | 0.09 | 0.5283 | −518.0 | 16.8 |
| 19 | 22 | 600 | 0.09 | 0.4896 | −474.0 | 17.3 |
| 20 | 22 | 600 | 0.17 | 0.4372 | −569.0 | 16.9 |
| 21 | 24 | 800 | 0.13 | 0.4627 | −453.0 | 18.8 |
| 22 | 20 | 800 | 0.13 | 0.3180 | −421.0 | 17.7 |
| 23 | 20 | 400 | 0.13 | 0.4868 | −586.0 | 17.0 |
Response Surface Models
A second-order polynomial was fitted to each response, since the underlying relationships are not expected to be monotonic over the whole parameter window:
$$Y=\beta_0+\sum_{i=1}^{k}\beta_i x_i+\sum_{i=1}^{k}\beta_{ii}x_i^{2}+\sum_{1\le i<j\le
with Y the response, k the number of factors, xi, xj the coded inputs, β the regression coefficients for the constant, linear, quadratic and interaction terms, and ε the random error. The three fitted functions, in the units of Table 3, are:
$$f_1=16.57224-0.103646\,v_s-0.027286\,v_w+125.72917\,a_p+0.001531\,v_sv_w-5.46875\,v_sa_p-0.017187\,v_wa_p$$
$$f_2=6.93792-0.516586\,v_s-0.001931\,v_w-8.91288\,a_p-9.46875\times10^{-6}\,v_sv_w-0.040156\,v_sa_p+0.001945\,v_wa_p+0.012972\,v_s^{2}+1.1941\times10^{-6}\,v_w^{2}+30.81356\,a_p^{2}$$
$$f_3=-1883.08474+95.01879\,v_s+0.457643\,v_w+2640.92061\,a_p+0.002656\,v_sv_w-222.65625\,v_sa_p+3.58594\,v_wa_p-1.57137\,v_s^{2}-0.000459\,v_w^{2}-4826.85811\,a_p^{2}$$
Table 5. Analysis of variance for the three response models.
| Response | Model | F | p | R2 | Signal-to-noise ratio |
|---|---|---|---|---|---|
| Grms (f1) | Linear with interactions | 20.26 | <0.0001 | 0.8837 | 15.9462 |
| Ra (f2) | Full quadratic | 11.62 | <0.0001 | 0.8894 | 11.6117 |
| σ (f3) | Full quadratic | 14.36 | <0.0001 | 0.9086 | 13.0119 |
For all three models the difference between R2 and the adjusted R2 stayed below 0.2, and the signal-to-noise ratios exceeded the value of 4 usually taken as adequate, so the fits are usable as predictors inside the experimental window. Measured responses spanned 0.286–0.5863 μm for Ra, −405 to −622.5 MPa for residual stress, and 15.6–20.1 m/s2 for spindle vibration, giving the models a working range that covers typical production settings for this gear size.
Spindle Vibration Response
Vibration rises monotonically with wheel speed across the tested range, and the effect is the strongest of the three factors. A grinding wheel is not a homogeneous body: abrasive grit distribution is uneven, porosity varies locally, and wear flats develop during use, so the wheel carries a residual unbalance that excites the spindle once per revolution. The resulting force grows with the square of rotational speed, which is why the response surface is steep in the vs direction and why vibration rose from 15.6–15.8 m/s2 at 16 m/s to 20.1 m/s2 at 24 m/s.
Generating speed and grinding depth also increase vibration, but far more gently: faster table motion requires larger axis accelerations, and a deeper cut raises the normal grinding force, yet neither effect compares with the unbalance excitation. The interaction surface between generating speed and grinding depth is almost planar, indicating that their combined effect on spindle vibration is negligible. The practical consequence is direct: keeping spindle load low means limiting wheel speed rather than limiting feed.
Surface Roughness Response
Roughness behaves nonlinearly in all three parameters, and the response contours are elliptical, which is the signature of an interior optimum. Increasing wheel speed raises the number of active grits per unit time and reduces the undeformed chip thickness per grit, so the residual ridge height left on the flank falls. Beyond roughly 22 m/s the trend reverses: spindle vibration grows large enough to imprint kinematic error on the surface, and roughness deteriorates even though the cutting conditions are nominally finer.
Generating speed acts through the scratch length. A higher vw lengthens the track traced by each grit, so successive tracks overlap more closely and the ridge between them becomes shallower. Past a certain value, however, the volume removed per unit time and the grinding force increase, and the surface degrades again. Grinding depth has a similar two-sided character specific to bevel gear grinding: at very small depths the flank near the root is not cut cleanly because wheel-dressing imperfections are not fully removed, leaving a poor surface, while at large depths the undeformed chip thickness of individual grits grows and the scratches deepen. Within the studied window, the roughness optimum lies toward the higher end of the wheel-speed range and the lower half of the generating-speed range.
Residual Stress Response
All measured stresses are compressive, between −405 and −622.5 MPa, and the response is governed by the competition between mechanical burnishing and grinding heat. Generating speed has the largest influence and correlates positively with the signed stress, meaning that compressive magnitude falls as the table speeds up: removal rate, force and heat all increase together, and the thermal tensile contribution outweighs the compressive contribution of the ploughing action.
Wheel speed reduces compressive magnitude for a different reason. Although the force per grit falls as speed rises, the proportion of grits that only rub or plough increases, so more energy is converted to heat and the thermal stress dominates the superposition. Grinding depth works in the opposite direction: a deeper cut raises the undeformed chip thickness and the normal force, mechanical stress becomes dominant, and compressive magnitude increases. The effect is weaker than that of generating speed but stronger than that of wheel speed. The practical rule for this gear type is therefore to reduce generating speed and increase depth when a high compressive stress state is wanted.
Optimization Model and Constraints
The three fitted functions were combined into a single constrained problem in which vibration and roughness are minimised and the signed residual stress is minimised, which is equivalent to maximising the compressive magnitude:
$$\min\left\{f_1(v_s,v_w,a_p),\ f_2(v_s,v_w,a_p),\ f_3(v_s,v_w,a_p)\right\}$$
subject to the parameter box and two process constraints:
$$16\le v_s\le 24\ \mathrm{m/s},\qquad 400\le v_w\le 1200\ \mathrm{mm/min},\qquad 0.05\le a_p\le 0.21\ \mathrm{mm}$$
$$f_2<0.6\ \mu\mathrm{m},\qquad 60\,v_s\sqrt{a_p\,d_s}\le C_b=8348\ \mathrm{m\cdot mm/min}$$
The first constraint caps roughness at a level consistent with the finishing requirement. The second is a burn limit of the form used for grinding-process charts, with ds the wheel diameter and Cb a critical coefficient fixed by the workpiece material and wheel specification; the factor 60 converts the wheel speed from m/s to m/min so that the product carries the units of the coefficient. In production trials this gear showed slight burn at 22 m/s and 0.21 mm depth, which sets Cb = 8348 m·mm/min; higher combinations of speed and depth are therefore excluded regardless of their predicted surface quality.
Pareto Front and AHP-TOPSIS Selection
The constrained problem was solved with a multi-objective whale optimization algorithm (MOWOA), which models the encircling, bubble-net and random-search behaviour of humpback whales and maintains a non-dominated archive through the search. The run used a population of 200, 500 iterations, three decision variables and a spiral constant of 1; solutions violating either constraint were discarded after the run, leaving the Pareto front shown by the trade-off between roughness, compressive stress and vibration.
Selecting one point from that front requires an explicit statement of shop-floor priorities. Analytic hierarchy process (AHP) weighting was applied with surface integrity first: because roughness is already capped as a constraint, the largest weight goes to residual stress, the second to roughness, and the smallest to spindle vibration. The resulting weight vector over (Ra, σ, Grms) is (0.2976, 0.4938, 0.2086), and the pairwise comparison passed the consistency check.
The weighted matrix was then ranked by similarity to an ideal solution (TOPSIS). Each criterion is normalised by its Euclidean column norm and multiplied by its weight,
$$y_{ab}=\frac{x_{ab}}{\sqrt{\sum_{k’=1}^{M}x_{k’b}^{2}}},\qquad v_{ab}=y_{ab}\,w_b$$
positive and negative ideal solutions are formed from the best and worst weighted values in each criterion, and each candidate is scored by its relative closeness:
$$D_a^{+}=\sqrt{\sum_{b=1}^{N}\left(v_{ab}-v_b^{+}\right)^{2}},\qquad D_a^{-}=\sqrt{\sum_{b=1}^{N}\left(v_{ab}-v_b^{-}\right)^{2}},\qquad S_a=\frac{D_a^{-}}{D_a^{+}+D_a^{-}}$$
Table 6. Highest-ranked parameter sets from the Pareto front.
| Rank | vs (m/s) | vw (mm/min) | ap (mm) | Closeness Sa |
|---|---|---|---|---|
| 1 | 20.9300 | 599.06 | 0.1606 | 0.7050 |
| 2 | 21.1542 | 576.35 | 0.1513 | 0.6945 |
| 3 | 20.8417 | 618.77 | 0.1539 | 0.6786 |
| 4 | 19.6033 | 456.73 | 0.1600 | 0.6679 |
| 5 | 21.7169 | 519.11 | 0.1459 | 0.6674 |
The top-ranked solutions cluster in a narrow band: wheel speed slightly above 21 m/s, generating speed between 520 and 620 mm/min, and depth between 0.15 and 0.16 mm. This is a compromise region rather than an extreme of any single objective, which is what a front with three competing responses should produce. The solutions ranked second and third differ from the first by less than 0.03 in closeness, so the ranking is shallow at the top and a machine-command rounding step does not move the choice off the front.
Experimental Validation
The highest-ranked set was rounded to commandable values—21 m/s, 600 mm/min and 0.16 mm—and used to grind a complete 33-tooth gear on the same H350GH machine. Vibration was recorded and the flank roughness and residual stress measured as in the design runs, with the whole tooth slot averaged. The comparison baseline was the existing production setting of 20 m/s, 1000 mm/min and 0.07 mm. The rounded optimum satisfies both constraints: predicted roughness remains under 0.6 μm, and 60 × 21 × √(0.16 × 190.5) ≈ 6957 m·mm/min lies well below the burn threshold of 8348 m·mm/min.
Table 7. Production setting versus validated optimum.
| Parameter set | vs (m/s) | vw (mm/min) | ap (mm) | Ra (μm) | σ (MPa) | Grms (m/s2) |
|---|---|---|---|---|---|---|
| Production | 20 | 1000 | 0.07 | 0.5816 | −476 | 17.5 |
| Optimized | 21 | 600 | 0.16 | 0.4713 | −538 | 16.2 |
| Change | +5.0% | −40.0% | +128.6% | −18.96% | +13.02% compressive magnitude | −7.43% |
The validated gain is a simultaneous improvement in all three responses: flank roughness fell by 18.96%, compressive stress rose by 13.02%, and spindle vibration fell by 7.43%. The direction of the change follows the individual response surfaces. The moderate increase in wheel speed is more than offset, in vibration terms, by the large reduction in generating speed, while the deeper cut supplies the mechanical component that raises compressive stress. The improvement in roughness comes from operating near the interior minimum of the Ra surface rather than at the edge of the feasible box.
Scope and Practical Reading
Three limitations bound how far these results can be transferred. The models are valid only inside the design box of 16–24 m/s, 400–1200 mm/min and 0.05–0.21 mm for this gear geometry, material and wheel specification; the strong curvature of the roughness and stress surfaces makes extrapolation unreliable. The burn coefficient Cb = 8348 m·mm/min was calibrated for 20CrMnTi with a 32A80J8VS wheel, and a change of either material or wheel grade shifts that limit and therefore the feasible region. Finally, the responses were characterised by three scalars—Ra, a single X-ray stress value with ±10 MPa accuracy, and a spindle-mounted RMS acceleration—which do not capture waviness, subsurface stress gradients, or the vibration at the contact zone itself.
Within those bounds, the procedure offers a transferable template for hard finishing of spiral bevel gears: design a CCD over the parameters the control program can actually change, fit second-order surfaces to roughness, residual stress and spindle vibration, impose the surface-integrity and burn constraints explicitly, generate a Pareto front, and use AHP weights that encode the shop’s priorities to select a single setting. The reported outcome—roughness below 0.48 μm with compressive stress above 500 MPa while spindle load decreases—shows that the three responses are not as deeply conflicted as their individual trends suggest, provided the parameters are moved together rather than one at a time. Parameter selection at the last grinding pass is thus a constrained compromise that can be computed, validated and then held as a fixed recipe for a given gear family.