Spiral bevel gears transmit motion between intersecting shafts and are widely used in automotive, aerospace, and industrial drivetrains because they combine high load capacity with smooth, low-noise operation. Their tooth flanks are complex conjugate surfaces, and the final geometry is usually produced after heat treatment by generating grinding. This finishing pass removes distortion, corrects pitch and profile errors, and establishes the surface state that governs contact fatigue, wear, and transmission noise. The grinding wheel is not a simple form tool: it represents a tooth of an imaginary generating gear, and the workpiece and wheel move in a timed conjugate relationship so that the final flank is the envelope of a moving contact line.
Three machine settings dominate the adjustable process window: grinding-wheel linear speed, generating speed, and grinding depth. They influence surface roughness, residual stress, and spindle vibration through coupled mechanical and thermal mechanisms. Reducing one defect often shifts the process toward another: a higher wheel speed can improve grit overlap and surface finish, but it also raises heat input and spindle vibration; a higher generating speed can increase productivity, but it can intensify heat generation and reduce compressive residual stress; a larger depth can strengthen the mechanical burnishing effect, but it also raises force and burn risk. The practical problem is therefore not to optimize one indicator in isolation, but to find a parameter combination that balances surface quality, fatigue-beneficial stress, and process stability.
Process window and competing objectives
In generating grinding of spiral bevel gears, the grinding wheel replaces the cutter of the generating machine. The wheel and workpiece rotate according to a predetermined ratio, and the instantaneous contact line sweeps across the flank. In double-flank grinding, the concave and convex flanks are machined in the same pass, which makes the process efficient but also couples the thermal and mechanical loads on both sides of the tooth slot.
The three adjustable parameters have different physical roles. Wheel linear speed controls the number of active cutting events per unit time and the rate of heat input. Generating speed controls how quickly the contact line travels along the flank and therefore affects removal rate, contact time, and heat generation. Grinding depth controls the undeformed chip thickness, normal force, and contact width. The experimental window examined here covered wheel speeds of 16–24 m/s, generating speeds of 400–1200 mm/min, and depths of 0.05–0.21 mm. These ranges are representative of a production finishing operation, but they are not universal: the appropriate window depends on wheel specification, coolant condition, machine stiffness, and gear geometry.
The target quantities also have different natures. Surface roughness is a geometric measure of the residual height left by overlapping grit scratches. Residual stress is a mechanical-thermal state variable: compressive stress is generally beneficial for fatigue resistance, while tensile stress can reduce fatigue life and increase cracking risk. Spindle vibration is a dynamic indicator of process stability and wheel life. The three responses are not independent. A parameter change that improves roughness may worsen the stress state, and a change that reduces vibration may reduce productivity or increase burn risk.
Surface topography from grit kinematics
The ground surface is formed by many abrasive grits cutting at different protrusion heights. A single grit path in workpiece coordinates is approximately a trochoidal scratch, and the final surface at any point is the lower envelope of all scratches that pass through that point. If the height of the wheel surface at a discretized point is represented by a matrix, the final tooth-surface height can be written as
$$z(m,n)=\min_{k\in K(m,n)} z_k(m,n)$$
where \(K(m,n)\) is the set of grit trajectories crossing the surface point \((m,n)\), and \(z_k\) is the height left by the \(k\)-th trajectory. The wheel surface is not deterministic: grit size and protrusion height are statistically distributed. For the 80-grit wheel examined here, a normal distribution of grit protrusion was used to generate the wheel topography, and a 1 mm × 1 mm patch of the tooth surface was discretized for the envelope calculation.
The simulation gives a clear mechanism for each parameter. Increasing wheel speed raises the number of active grits per unit time and increases the number of overlapping scratches, so the residual height tends to fall. Increasing generating speed lengthens the scratch path of each grit along the flank, which also reduces the residual height when the process remains stable. Reducing grinding depth lowers the undeformed chip thickness and the depth of individual scratches, which improves the geometric finish. Under ideal assumptions—rigid spindle, no built-up edge, and complete removal by the cutting edges—all three trends are monotonic within the tested range.
The experiments show a more restrictive behavior: roughness first decreases and then increases as each parameter is raised. The reason is that the ideal geometric model captures only the cutting geometry. At high wheel speed, spindle vibration and wheel imbalance become large enough to imprint additional irregularities. At high generating speed, the removal rate and grinding force rise, and the resulting dynamic and thermal effects can degrade the surface. At very small depth, incomplete grinding near the root and local wheel-dressing defects can leave a poorer finish. The practical optimum therefore lies inside the parameter window, not at its boundary.
Contact geometry, force, and heat input
The thermal-mechanical state of the flank is controlled by the contact-zone geometry. For the 33-tooth spiral bevel gear examined here, the instantaneous contact line at the mid-roll position reached about 41.09 mm. As the grinding depth increased from 0.05 to 0.21 mm, the computed short semi-axis of the contact zone increased from 0.9585 to 1.1517 mm. This widening matters because it spreads the heat input over a larger area and changes the balance between mechanical compression and thermal expansion.
Representative outputs from the contact-zone model are summarized below. The values are taken at the mid-roll position and are intended to show trends, not to serve as a universal calibration for other gear or wheel combinations.
| Parameter sweep | Tangential grinding force | Heat flux | Contact-zone short semi-axis |
|---|---|---|---|
| Wheel speed 16 → 24 m/s (generating speed 800 mm/min, depth 0.13 mm) | ≈20.0 N, nearly constant | 8.29 × 106 → 1.24 × 107 W/m² | — |
| Generating speed 400 → 1200 mm/min (wheel speed 20 m/s, depth 0.13 mm) | 12.07 → 30.53 N | 6.26 × 106 → 1.59 × 107 W/m² | — |
| Depth 0.05 → 0.21 mm (wheel speed 20 m/s, generating speed 800 mm/min) | 17.67 → 22.46 N | 1.01 × 107 → 1.07 × 107 W/m² | 0.9585 → 1.1517 mm |
The table reveals the central coupling. Wheel speed raises heat flux strongly while the tangential force remains nearly unchanged, so the thermal effect becomes more important as speed increases. Generating speed raises both force and heat flux, but heat flux rises more steeply. Depth raises force and contact width, while heat flux increases only modestly because the contact area grows at the same time. These different ratios explain why the three parameters do not have interchangeable effects on residual stress.
A sequential thermal-mechanical finite-element model was used to resolve the transient temperature field and the final stress state. The gear material was 20CrMnTi, with temperature-dependent elastic modulus decreasing from about 210 GPa at 20 °C to 191 GPa at 300 °C and Poisson’s ratio varying from 0.300 to 0.303. A 0.126 s heat pulse was applied in five load steps, followed by a 10 s cooling stage. The temperature field was then imported into a static structural analysis with the mechanical grinding load, and the residual stress was evaluated after unloading and cooling.
The computed surface stress with zero initial stress was tensile, in the range of about 100–350 MPa. The mechanism is consistent with the heat-flux and force trends. Increasing wheel speed raises heat input without a significant force increase, so the tensile contribution grows. Increasing generating speed raises both heat and force, but heat generation dominates, so the tensile stress again increases. Increasing depth raises the mechanical burnishing effect and widens the contact zone; the heat flux does not rise in proportion, so the compressive mechanical contribution becomes more prominent and the net tensile stress decreases.
The measured initial state of the heat-treated tooth surface was already compressive, approximately −626 to −713 MPa. After grinding, the measured residual stress remained compressive but its magnitude decreased, with values from about −405 to −622.5 MPa. The finite-element trend agrees with the experiment, but the absolute values are not a simple linear superposition because the model did not include phase transformation, initial stress, and all thermal boundary details. The agreement is therefore strongest in the direction of change and in the ranking of parameter effects, which is what the optimization model requires.
Experimental design and measurement
The grinding trials used a 23-run central composite design. The gear had 33 teeth, a face width of 38 mm, a mean spiral angle of 35°, an outer diameter of 217.63 mm, an outer cone distance of 119.19 mm, and a working depth of 11 mm. The workpiece material was 20CrMnTi alloy carburizing steel. The wheel was a 32A80J8VS ceramic-bonded CBN wheel with a diameter of 190.5 mm, a pressure angle of 20°, a point width of 3.3 mm, 80 grit, and a tip radius of 2.2 mm. Grinding was performed on an H350GH CNC spiral bevel gear grinder.
| Level | Wheel speed (m/s) | Generating speed (mm/min) | Grinding depth (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 |
The design combined factorial points, repeated axial points, and three center points. Each parameter set was used to grind one tooth slot; 23 of the 33 slots were used. The wheel was dressed after each slot to reduce the effect of wheel wear. Wheel type, coolant flow, and machine configuration were held constant. Because the process was double-flank generating grinding, the concave and convex flanks of a slot were produced in the same pass, and the roughness and stress results were averaged over the two flanks.
Spindle vibration was recorded with a piezoelectric accelerometer mounted on the horizontal section of the grinding spindle, measuring the radial vibration signal. The sampling frequency was 9600 Hz, and the root-mean-square value of the acceleration during grinding was used as the vibration indicator. Tooth-surface roughness was measured with a white-light interferometer after wire-cutting the teeth to expose the flanks. The arithmetic mean deviation \(R_a\) was used as the quantitative roughness measure. Residual stress was measured by X-ray diffraction using the \(\sin^2\psi\) method, with a portable diffractometer having a stated deviation within about ±10 MPa. Measurements were taken near the mid-face along the tooth height and averaged over the concave and convex flanks.
Regression models and parameter effects
The experimental data were fitted with a second-order response-surface model of the form
$$y=\beta_0+\sum_{i=1}^{3}\beta_i x_i+\sum_{i<j}\beta_{ij}x_i
where \(x_1\), \(x_2\), and \(x_3\) represent wheel speed, generating speed, and grinding depth, respectively. Three separate models were built for spindle vibration \(G_{\mathrm{rms}}\), tooth-surface roughness \(R_a\), and residual stress \(\sigma\). The model-quality indicators are given below.
| Response | \(R^2\) | Adjusted \(R^2\) | Adequate precision |
|---|---|---|---|
| Spindle vibration \(G_{\mathrm{rms}}\) | 0.8837 | 0.8401 | 15.95 |
| Tooth-surface roughness \(R_a\) | 0.8894 | 0.8128 | 11.61 |
| Residual stress \(\sigma\) | 0.9086 | 0.8453 | 13.01 |
All three models were significant at the 0.0001 level, with model F-values of 20.26, 11.62, and 14.36. The lack-of-fit tests were not significant, with p-values of 0.3361, 0.2969, and 0.0841, all above 0.05. Adequate precision above 4 is generally considered sufficient for navigation of the design space; the values here are well above that threshold. The adjusted and predicted coefficients remained reasonably close, which indicates that the models were not strongly overfitted.
Spindle vibration was dominated by wheel speed. The linear wheel-speed term was highly significant, while generating speed had a smaller but still significant effect. Grinding depth did not show a significant linear effect on vibration within this design, although the response surface indicated a mild positive trend. The interactions between wheel speed and generating speed, and between wheel speed and depth, were significant. The practical interpretation is that wheel imbalance and high-speed dynamic effects are the main vibration source; increasing generating speed or depth changes the load, but not as strongly as increasing wheel speed.
Roughness behaved nonlinearly. The linear wheel-speed term was significant, but the linear effects of generating speed and depth were not; all three quadratic terms were highly significant. This is the statistical signature of a response surface with an interior minimum. At low parameter values, material removal is incomplete or grit overlap is insufficient. At high values, spindle vibration, grinding force, or thermal effects begin to dominate, and the surface deteriorates again. The optimum roughness therefore lies away from the extremes of the design box.
Residual stress was most strongly controlled by generating speed. Increasing generating speed reduced the compressive residual stress, consistent with the heat-dominated mechanism. Grinding depth entered through a significant interaction with generating speed and through a quadratic term; the overall trend was toward a larger compressive stress as depth increased. Wheel speed had a weaker effect than generating speed. The response surface therefore shows a clear trade-off: the parameter that most improves productivity, generating speed, is also the parameter that most strongly erodes the beneficial compressive stress state.
Multi-objective optimization and decision
The optimization problem was formulated with three objectives: minimize spindle vibration, minimize tooth-surface roughness, and maximize the magnitude of compressive residual stress. Because the measured stress is negative, maximizing compression is equivalent to driving the algebraic stress more negative. The constraints included the tested parameter ranges, a roughness limit of \(R_a \le 0.6\ \mu\mathrm{m}\), and a burn-limit condition calibrated from a slight-burn case at 22 m/s and 0.21 mm.
$$
\begin{aligned}
\min \quad & G_{\mathrm{rms}}(v_s,v_w,a_p),\\
\min \quad & R_a(v_s,v_w,a_p),\\
\min \quad & \sigma(v_s,v_w,a_p) \quad \text{(more negative means more compressive)},\\
\text{s.t.}\quad & 16 \le v_s \le 24\ \mathrm{m/s},\\
& 400 \le v_w \le 1200\ \mathrm{mm/min},\\
& 0.05 \le a_p \le 0.21\ \mathrm{mm},\\
& R_a \le 0.6\ \mu\mathrm{m},\\
& \text{burn limit satisfied}.
\end{aligned}
$$
The Pareto set was generated with the multi-objective whale optimization algorithm (MOWOA), using a population of 150 and 500 iterations. NSGA-II was run as a comparison. The hypervolume (HV) and inverted generational distance (IGD) are reported below; larger HV and smaller IGD indicate a better approximation of the Pareto front.
| Algorithm | HV | IGD |
|---|---|---|
| MOWOA | 0.581 | 0.042 |
| NSGA-II | 0.555 | 0.060 |
MOWOA produced a wider and more evenly distributed Pareto set in this problem, and its convergence curve reached a lower final fitness value. The difference is not a claim of general algorithm superiority; it is specific to the response surfaces, constraints, and population settings used here. The result is still useful because the optimization problem is small enough to compare the two algorithms directly and because the Pareto front contains several distinct compromise regions.
Final selection from the Pareto set used an AHP-TOPSIS procedure. The priority weights were 0.2976 for roughness, 0.4938 for residual stress, and 0.2086 for vibration, with a consistency ratio of 0.021, below the 0.1 threshold. This weighting reflects a quality-first strategy in which fatigue-beneficial residual stress receives the highest priority, surface finish the next, and dynamic stability the lowest. The highest-closeness solution was Pareto point 60, with a closeness of 0.7050. After rounding, the selected parameters were a wheel speed of 21 m/s, a generating speed of 600 mm/min, and a grinding depth of 0.16 mm.
| Item | Empirical baseline | Optimized | Change |
|---|---|---|---|
| Wheel speed (m/s) | 20 | 21 | — |
| Generating speed (mm/min) | 1000 | 600 | — |
| Grinding depth (mm) | 0.07 | 0.16 | — |
| Tooth-surface roughness \(R_a\) (µm) | 0.582 | 0.471 | −19.07% |
| Residual stress (MPa) | −476 | −538 | 13.02% larger compressive magnitude |
| Spindle vibration RMS (mm/s²) | 17.5 | 16.2 | −7.43% |
The optimized point is a compromise rather than the extreme of any single objective. It reduces generating speed from 1000 to 600 mm/min, which lowers the heat input and vibration associated with a fast-moving contact line. It increases depth from 0.07 to 0.16 mm, which strengthens the mechanical burnishing effect and helps preserve compressive stress. It raises wheel speed modestly to 21 m/s, improving grit overlap without driving spindle vibration to the level expected at 24 m/s. The residual stress remains compressive; the 13.02% improvement refers to the magnitude of compression, from 476 to 538 MPa.
The Pareto set also contains alternative operating points for different priorities. A roughness-first solution was found near 20.18 m/s, 705 mm/min, and 0.14 mm. A compression-first solution was found near 19.6 m/s, 456 mm/min, and 0.16 mm. The second option illustrates the cost of stressing residual stress too heavily: lowering generating speed raises compressive stress but reduces productivity. In production, the weighting should therefore reflect the actual duty cycle, the cost of finishing time, and the fatigue requirement of the gear.
Implementation boundaries
The result is valid within the tested window and for the specific combination of gear geometry, 20CrMnTi material, CBN wheel, coolant condition, and machine tool. The regression models are empirical surrogates; they should not be extrapolated beyond the central composite design range without new trials. The thermal-mechanical model simplifies the grinding contact, omits phase transformation, and starts from a zero-stress state, so its absolute stress values are less reliable than its trends. X-ray residual-stress measurement also carries uncertainty from surface preparation, crystallographic texture, and the assumed elastic constants.
Within those limits, the case shows a useful design principle for spiral bevel gear grinding: the best parameter set is rarely the one that maximizes wheel speed or generating speed. Heat input, mechanical burnishing, and dynamic stability must be balanced together. A moderate wheel speed improves grit overlap; a lower generating speed reduces thermal loading; a larger depth restores the mechanical compression that would otherwise be lost. When these effects are represented by response surfaces and searched as a multi-objective problem, the result is a transparent trade-off between surface finish, residual stress, vibration, and productivity.
Further development would benefit from online signals—spindle vibration, acoustic emission, and spindle power—linked to a real-time model, so that the process can adapt to wheel wear and thermal drift rather than relying on a fixed parameter table. Extending the same framework to other gear materials, wheel specifications, and high-speed or dry-grinding conditions would also require new calibration, but the underlying logic remains the same: identify the dominant thermal and mechanical mechanisms, quantify their competing effects, and select the operating point against an explicit production priority.