Spiral bevel gears transmit motion between intersecting shafts and are the default choice in automotive drivelines, aircraft accessory and rotor systems, and machine-tool spindle trains, where smooth meshing, high load capacity, low noise, and high transmission efficiency are required simultaneously. The flank geometry that governs those properties is created in the last manufacturing operation, generating grinding, so the three parameters an operator can change at the machine panel—wheel linear speed, generating speed, and grinding depth—set the final surface roughness, the residual stress state of the tooth flank, and the dynamic load the spindle carries during the pass.
Vibration is usually treated as a machine-health indicator rather than a quality variable, yet it couples directly to the result: relative displacement between wheel and workpiece is superimposed on the commanded generating motion, imprinting waviness on the flank and accelerating wear of spindle bearings, which erodes the long-term accuracy of the machine and shortens wheel life. A parameter set chosen purely for minimum roughness can therefore be unacceptable in production if it drives the spindle hard. The optimization problem addressed here combines the three responses—flank roughness, flank residual stress, and radial spindle vibration—in a single model, calibrates them from a central composite design on a production gear, and ranks the resulting Pareto front to obtain one set of shop-floor parameters.
Generating Kinematics on a Five-Axis Grinding Machine
Generating grinding reproduces the meshing action of a virtual crown gear (the generating gear) with the workpiece. The grinding wheel represents one tooth of that virtual gear; as the cradle rolls, the wheel sweeps through the tooth space and the envelope of successive wheel positions forms the flank. On the H350GH architecture used here, the motion is distributed over three linear axes (X, Y, Z) and two rotary axes (A for the workpiece, B for the wheel cradle), with the wheel spindle as an additional servo axis. The set of machine settings—radial cutter position, cutter swivel angle, cutter tilt, vertical and horizontal offset, workpiece mounting angle, roll ratio, and the roll positions at the toe, center, and heel—fixes the generating motion and hence the unloaded flank form. These settings were held constant throughout the trials so that the parameter effects isolated below belong to the process variables alone.
Experimental Platform, Workpiece, and Measurement Chain
The trials ran on a domestically produced H350GH CNC spiral bevel gear grinder. The workpiece was a 33-tooth spiral bevel gear in 20CrMnTi (a case-carburizing gear steel), ground with a 32A80J8VS ceramic alumina wheel of 190.5 mm diameter and 80 grit. Gear, wheel, and machine-setting data are collected in Tables 1–3. The gear geometry is a typical automotive-hypoid-adjacent configuration: module 6.5 mm, 20° pressure angle, 35° spiral angle, 217.6 mm outer diameter, and a 119.19 mm outer cone distance.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Number of teeth | 33 | Outer cone distance | 119.19 mm |
| Module | 6.5 mm | Crown-to-apex distance | 48.77 mm |
| Pressure angle | 20° | Pitch cone angle | 64.1333° |
| Spiral angle | 35° | Face cone angle | 66.45° |
| Outer diameter | 217.6 mm | Root cone angle | 59.9667° |
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Wheel diameter | 190.5 mm | Grit size | 80 |
| Blade pressure angle | 20° | Maximum linear speed | 35 m/s |
| Blade tip distance | 3.3 mm | Tip fillet radius | 2.2 mm |
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Radial cutter position | 90.1 mm | Bed position | 0 mm |
| Cutter tilt angle | 0° | Roll ratio | 1.108401 |
| Cutter swivel angle | 204.72° | Center roll position | 59.72° |
| Vertical offset | 0 mm | Heel roll position | 42.07° |
| Horizontal offset | 0 mm | Toe roll position | 75.97° |
| Workpiece mounting angle | 59.97° |
Vibration was acquired with a CT9104 four-channel USB signal conditioner and a CT1000 piezoelectric accelerometer mounted radially on the wheel spindle, connected through BNC-M5 leads. One record was taken per ground tooth slot at a 9,600 Hz sampling rate, giving 23 records. Because the raw waveform mixes transient impacts with steady grinding, the root-mean-square value of the acceleration over the grinding interval was used as the scalar feature:
$$G_{\mathrm{rms}}=\sqrt{\frac{1}{n}\sum_{i’=1}^{n}x_{i’}^{2}}$$
where $n$ is the number of samples in the record and $x_{i’}$ the acceleration at sample $i’$.
Flank quality was measured after wire-EDM sectioning of individual teeth. Roughness was taken on a FORMTRACER SV-C3200 profilometer and residual stress on a Pulstec µ-X360S portable X-ray diffractometer with a stated accuracy of ±10 MPa. The arithmetic mean deviation $R_a$ was adopted as the roughness index because it captures both peak-valley height and overall finish. Each reported $R_a$ and $\sigma$ value is the mean of the concave and convex flanks ground under the same parameter set.
Design of Experiments and Measured Responses
The three process variables were wheel linear speed $v_s$, generating speed $v_w$, and grinding depth $a_p$. Ranges were fixed from machine capability and shop experience, and a three-factor, five-level central composite design (CCD) was generated with three center points. The response surface approach rests on the standard second-order model
$$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 $k$ factors, first-order, quadratic, and interaction coefficients $\beta_i$, $\beta_{ii}$, $\beta_{ij}$, constant $\beta_0$, and random error $\varepsilon$. Factor levels are listed in Table 4; the resulting 23 parameter combinations were ground one per tooth slot on the numbered 33-tooth gear, and the wheel was dressed after each slot to keep wear from contaminating the comparison. Table 5 gives the full data set.
| Level | $v_s$ (m/s) | $v_w$ (mm/min) | $a_p$ (mm) |
|---|---|---|---|
| −2 | 16 | 400 | 0.05 |
| −1 | 18 | 600 | 0.09 |
| 0 | 20 | 800 | 0.13 |
| +1 | 22 | 1,000 | 0.17 |
| +2 | 24 | 1,200 | 0.21 |
| Run | $v_s$ (m/s) | $v_w$ (mm/min) | $a_p$ (mm) | $R_a$ (µm) | $\sigma$ (MPa) | $G_{\mathrm{rms}}$ (mm/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 | 1,200 | 0.13 | 0.5371 | −405.0 | 17.5 |
| 7 | 20 | 1,200 | 0.13 | 0.5212 | −407.5 | 17.8 |
| 8 | 20 | 800 | 0.13 | 0.2860 | −447.5 | 17.0 |
| 9 | 18 | 1,000 | 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 | 1,000 | 0.17 | 0.5371 | −415.0 | 17.1 |
| 14 | 20 | 800 | 0.05 | 0.5093 | −440.5 | 17.3 |
| 15 | 22 | 1,000 | 0.09 | 0.4436 | −413.5 | 19.9 |
| 16 | 20 | 800 | 0.21 | 0.5276 | −494.5 | 17.9 |
| 17 | 22 | 1,000 | 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
Regression on the 23 runs produced one model per response. Vibration is described by a model without quadratic terms, because the interaction and curvature terms for $G_{\mathrm{rms}}$ were not significant over this window:
$$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$$
Roughness and residual stress both required full quadratic forms:
$$\begin{aligned}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}\end{aligned}$$
$$\begin{aligned}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}\end{aligned}$$
Here $v_s$ is in m/s, $v_w$ in mm/min, $a_p$ in mm, $f_1$ in mm/s², $f_2$ in µm, and $f_3$ in MPa, with compressive stress carrying a negative sign. Table 6 summarizes the analysis of variance. All three models are significant at $p<0.0001$, the coefficients of determination exceed 0.88, the difference between $R^2$ and the adjusted $R^2$ stays below 0.2, and adequate precision is well above the value of 4 regarded as the threshold for a usable model. Prediction-versus-measurement plots scatter closely about the 45° line over the whole measured span, from 15.6 to 20.1 mm/s² for vibration, 0.286 to 0.586 µm for roughness, and −622.5 to −368.5 MPa for residual stress.
| Model | Response | $F$ value | $p$ value | $R^2$ | Adequate precision |
|---|---|---|---|---|---|
| $f_1$ | Spindle vibration $G_{\mathrm{rms}}$ | 20.26 | < 0.0001 | 0.8837 | 15.95 |
| $f_2$ | Surface roughness $R_a$ | 11.62 | < 0.0001 | 0.8894 | 11.61 |
| $f_3$ | Residual stress $\sigma$ | 14.36 | < 0.0001 | 0.9086 | 13.01 |
How the Three Parameters Act
Spindle vibration rises almost linearly with wheel speed. Vibration is positively correlated with $v_s$ and $v_w$, and $v_s$ dominates: at a fixed generating speed of 800 mm/min and depth of 0.13 mm, raising wheel speed from 16 to 24 m/s lifts $G_{\mathrm{rms}}$ from about 15.7 to 20.1 mm/s². Depth of cut also increases vibration, though more weakly, and the $a_p$–$v_w$ surface is nearly planar, indicating a negligible interaction. The mechanism is wheel imbalance: non-uniform abrasive distribution, porosity, and wear-induced surface irregularity create a rotating unbalance that scales with the square of rotational speed, and the resulting excitation is transmitted to the spindle. The practical consequence is that a low-vibration setting should not use an aggressive wheel speed, however favourable that speed may be for finish.
Roughness is non-monotonic in all three parameters. The $R_a$ surfaces are elliptical over the design window, so an interior optimum exists. Increasing $v_s$ raises the number of active grits per unit time, reducing the residual height left by overlapping cuts; beyond a certain speed, however, the vibration described above dominates and the finish degrades again. Increasing $v_w$ lengthens the scratch produced by each grit, which lowers the ridge height between neighbouring tracks, but past a threshold the removal volume per unit time and the grinding force grow enough to worsen the surface. Depth of cut has a two-sided effect specific to bevel gear grinding: at very small depths, dressing imperfections prevent complete grinding near the tooth root and roughness suffers, while at large depths the undeformed chip thickness per grit rises and the scratches deepen. The lowest roughness in the data set, 0.286 µm, occurred at the center-point setting of 20 m/s, 800 mm/min, and 0.13 mm; the replicate center-point runs gave 0.318 and 0.326 µm, which indicates the level of run-to-run scatter that any optimum must be judged against.
Residual compressive stress is governed mainly by generating speed, in the opposite direction to what the roughness trend would suggest. Over the tested range, $\sigma$ moves from about −622 MPa at 400 mm/min to about −405 MPa at 1,200 mm/min, so the compressive magnitude falls as the generating speed increases. Higher $v_s$ also reduces compressive stress, by two mechanisms acting together: the thinner undeformed chip per grit lowers the mechanical stress, while the larger share of grits in the rubbing and ploughing regimes generates more heat and hence more thermal tensile stress. Depth of cut works the other way—a larger $a_p$ thickens the undeformed chip, the mechanical pressure on the flank outweighs the additional heat, and the compressive stress grows. Since a compressive field on the flank retards crack initiation and improves fatigue life, the process window therefore contains a genuine conflict: the combination that produces a fine finish is close to the one that produces an unfavourable stress state.
Multi-Objective Formulation and Constraints
The three regression models were combined into a constrained minimization problem. Because compressive stress is stored as a negative number, maximizing the compressive magnitude is equivalent to minimizing the signed value:
$$\min\;\{f_1(v_s,v_w,a_p),\;f_2(v_s,v_w,a_p),\;f_3(v_s,v_w,a_p)\}$$
subject to the process window and two process constraints. The first caps roughness at 0.6 µm, which corresponds to the upper bound of the measured range and to normal acceptance for this gear class. The second guards against grinding burn, which appears when the specific grinding energy per unit contact area exceeds what the coolant can remove; it is expressed through the empirical burn criterion
$$v_s\sqrt{a_p d_s}\le C_b=139\ \mathrm{m\cdot mm/s}\;\;(C_b=8348\ \mathrm{m\cdot mm/min})$$
with wheel diameter $d_s=190.5$ mm. The critical coefficient was identified on the shop floor: a slight burn appeared at 22 m/s and 0.21 mm depth, which fixes $C_b$ at that boundary after unit conversion. The complete feasible region is thus
$$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 v_s\sqrt{a_p d_s}\le 139\ \mathrm{m\cdot mm/s}$$
Pareto Optimization and AHP–TOPSIS Selection
The multi-objective whale optimization algorithm (MOWOA) was used to search this region. The algorithm mimics three humpback-whale behaviours—encircling prey, bubble-net attack, and random search—and switches between them through the coefficient vector $A$ and the convergence factor $\alpha$, which falls linearly from 2 to 0 over the run; non-dominated sorting maintains the archive of compromise solutions. Settings were a population of 200 whales, 500 iterations, three decision variables, and a spiral constant of 1. Solutions violating the roughness or burn constraints were removed, leaving the Pareto set that spans roughly 15.6–19.5 mm/s² in vibration, 0.35–0.55 µm in roughness, and −620 to −420 MPa in stress.
Selecting a single point from that front requires a preference structure. The analytic hierarchy process (AHP) supplied the weights, and the technique for order preference by similarity to an ideal solution (TOPSIS) ranked the candidates by their Euclidean distance to the positive and negative ideal solutions after vector normalization and weighting. Since roughness is already bounded by a constraint, surface integrity was expressed mainly through residual stress: the weights assigned to roughness, residual stress, and vibration were $w_b=[0.2976,\;0.4938,\;0.2086]$, which passed the consistency check. The closeness score
$$S_a=\frac{D_a^{-}}{D_a^{+}+D_a^{-}},\qquad 0\le S_a\le 1$$
then orders the front, with larger $S_a$ meaning closer to the ideal. Table 7 lists the five best-ranked parameter sets.
| Rank | $v_s$ (m/s) | $v_w$ (mm/min) | $a_p$ (mm) | Closeness $S_a$ |
|---|---|---|---|---|
| 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 ranking converges on a characteristic compromise: a wheel speed near the upper part of the window, a generating speed near the lower-middle of the range, and a depth close to 0.15–0.16 mm. The high wheel speed preserves the grit-density benefit for roughness, the low generating speed protects the compressive stress state, and the intermediate depth keeps the chip thickness high enough for mechanical compressive stress without approaching the burn boundary.
Verification Against the Production Setting
The top-ranked solution was rounded to machine-settable values (21 m/s, 600 mm/min, 0.16 mm), and one complete 33-tooth gear was ground with those settings on the same H350GH machine. Vibration was recorded and roughness and residual stress measured as before, averaging over the full slot flank. Table 8 compares the outcome with the production setting previously used for this part (20 m/s, 1,000 mm/min, 0.07 mm).
| Parameter set | $v_s$ (m/s) | $v_w$ (mm/min) | $a_p$ (mm) | $R_a$ (µm) | $\sigma$ (MPa) | $G_{\mathrm{rms}}$ (mm/s²) |
|---|---|---|---|---|---|---|
| Production setting | 20 | 1,000 | 0.07 | 0.5816 | −476 | 17.5 |
| Optimized setting | 21 | 600 | 0.16 | 0.4713 | −538 | 16.2 |
| Change | — | — | — | −18.96% | +13.02% (compressive magnitude) | −7.43% |
The optimized parameters reduce flank roughness by 18.96%, raise the compressive residual stress by 13.02%, and lower the radial spindle vibration acceleration by 7.43%. The gain is thus not a single-response improvement traded against the others: all three responses move in the desired direction, which is possible because the production setting was far from the roughness optimum in wheel speed and too shallow in depth to develop compressive stress, while the higher generating speed it used was costing stress without compensating vibration benefit.
Scope and Engineering Reading
Three points determine how far these results transfer. First, the regression models are valid inside the box 16–24 m/s, 400–1,200 mm/min, and 0.05–0.21 mm, for this gear geometry, this 20CrMnTi material state, and this wheel specification; extrapolation outside that region, in particular to other modules or to carburized surfaces with different hardness, is not supported by the data. Second, the burn coefficient $C_b=8348$ m·mm/min is a machine- and coolant-specific calibration derived from a single observed burn boundary, and it should be re-identified for other coolant delivery conditions. Third, the weighting vector, not the optimization algorithm, decides which Pareto point is called optimal: giving vibration a larger weight shifts the choice toward lower wheel speed at the cost of roughness, and giving stress the dominant weight drives the solution toward the slowest acceptable generating speed. The reported 18.96%, 13.02%, and 7.43% improvements therefore describe one defensible preference among many, not an absolute optimum.
The broader engineering message concerns the role of the machine structure in process planning. Including spindle vibration in the objective set changes the recommended parameters in a way that surface-integrity-only optimization would miss, because the two objectives are driven by different physics: roughness responds to grit density and scratch geometry, stress to the balance between mechanical and thermal loading, and vibration to rotating unbalance that grows with wheel speed. For production, this argues for treating vibration as a measured process output on gear grinders, alongside roughness and residual stress, and for re-running the Pareto ranking whenever wheel specification, coolant strategy, or spindle condition changes, since all three shift the underlying response surfaces.