The finishing bottleneck after carburizing
Spiral bevel gears transmit motion between intersecting shafts through a curved tooth trace that rolls rather than slides through most of the mesh cycle. High contact ratio, load-sharing across several teeth and quiet running make them standard components in aircraft drives, precision machine tools, heavy transport and mining equipment. The manufacturing chain that produces them has three stages: blank cutting, carburizing and hardening, and finish machining of the tooth flanks. Carburizing dominates the second stage — it changes chemistry, microstructure and physical properties so that the flank reaches 58–60 HRC while the core stays tough. The same thermal and transformation strains, however, distort the part, typically costing two to three accuracy grades and displacing the contact pattern, which reappears later as transmission error, noise and accelerated wear. Finish machining after hardening is therefore not optional.
Grinding is the default response. Above roughly 38 HRC, cutting becomes impractical and CNC form-wheel grinding takes over, correcting distortion and lifting accuracy to grade 6 or better. The economics are awkward: imported gear grinding machines cost on the order of CNY 10 million, single-part and small-batch work is dominated by set-up time, and stock allowance becomes a process risk in both directions. Remove too much and the effective case depth is cut away; remove too little and regions that have already reached size still carry unground “black” patches, scrapping the gear.
Lapping avoids the wheel altogether. A mating pair is run together with an abrasive slurry — white corundum is common — and material is removed by the relative sliding of the flanks, producing a fine finish and a controlled contact pattern. Its weakness is geometric correction, and on spiral bevel gears the weakness is structural rather than procedural: near the pitch line the flanks roll on each other with almost no sliding, so removal there is negligible and lapping becomes uneven along the tooth height.
What ultrasonic assistance adds to lapping
Superimposing a high-frequency vibration on the tool gear changes the kinematics of the abrasive. The transducer output is amplified by a horn and drives the tool gear at above 20 kHz with micron-scale amplitude, so each abrasive particle receives a periodic impact acceleration on top of the steady sliding motion. Three effects follow. Micro-impact raises the cutting efficiency of fine abrasive, lowering roughness. Cavitation in the slurry improves its penetration into the contact and homogenises the contact pattern. Intermittent cutting reduces the dwell of heat in the contact zone. Reported results for ultrasonic-assisted lapping of spiral bevel gears give surface roughness Ra down to 0.2 μm against 0.33 μm for the conventional process, with material removal rate roughly three times higher — the removal mechanisms usually cited being abrasive hammering, particle bouncing and slurry cavitation.
The relevant engineering question is not whether the vibration helps, but how to build a resonator that keeps a well-defined amplitude at the tooth flank once the gear itself becomes part of the acoustic stack.
System architecture and non-resonant design
The chain consists of an ultrasonic power supply, a piezoelectric transducer, a horn and the tool gear. The gear is not a passive workpiece holder: it is the output end of the resonator, and its mass and stiffness shift the resonant frequency of the assembly. A design method that first tunes a free-standing horn and then attaches the gear will miss the target frequency.
Non-resonant design reverses the order. The gear geometry is fixed by the process and cannot be adjusted, so the gear and horn are treated as one composite horn, and a dedicated adjusting section is used to move the composite resonance onto the generator frequency of 20 kHz. For a stepped and conical stack the axial motion follows the one-dimensional wave equation for a variable cross-section,
$$\frac{\partial^{2}\xi_n}{\partial x_n^{2}}+\frac{1}{S_n}\frac{\partial S_n}{\partial x_n}\frac{\partial \xi_n}{\partial x_n}+k^{2}\xi_n=0,\qquad k=\frac{\omega}{c},\quad c=\sqrt{E/\rho},$$
where $\xi_n$ is the axial displacement of segment $n$, $S_n$ its cross-section, $E$ the elastic modulus and $\rho$ the density. Cylindrical segments take the harmonic solution $\xi_n=A_n\cos(kx_n)+B_n\sin(kx_n)$; the conical load section, with taper parameter $\alpha$, takes the form
$$\xi_4=\frac{1}{x_4-1/\alpha}\left(A_4\cos kx_4+B_4\sin kx_4\right).$$
Displacement continuity and force equilibrium at the three internal interfaces, together with free-end conditions at the transducer face and the tooth end, produce eight homogeneous equations in the coefficients $A_n$, $B_n$. A non-trivial solution exists only when the coefficient determinant vanishes,
$$\left|C_{ij}(f,l_3)\right|=0,$$
which is the frequency equation of the composite system, implicit in the resonant frequency $f$ and the length of the adjusting section $l_3$. Once the coefficients are known, the amplitude gain follows as
$$M_p=\frac{\xi_4\!\left(l_1+l_2+l_3+l_4\right)}{\xi_1\!\left(0\right)}.$$
In the hardware described here the first three segments have diameters of 54, 40 and 30 mm and lengths of 68, 7 and 26 mm. The first segment length is fixed at 68 mm by the bore of the lapping machine spindle, and $l_3$ is the free variable, allowed to range over 0–50 mm. The conical load section is the gear, with $l_4$ equal to the face width. Flanges and threaded joints are ignored in this analytical stage; they are introduced later at the displacement node.
Replacing the toothed gear with an equivalent cone
The tooth form is not axisymmetric, so the analytical model needs an equivalent truncated cone. Three candidates were compared: sections taken at the addendum circle, at the pitch circle and at the dedendum circle, all with the same 26 mm face width. Their radii and cone half-angles differ substantially, and so does the composite resonator that results.
| Equivalent section | $R_4$ (mm) | Cone half-angle (°) | $l_3$ for $f$ = 20 kHz (mm) | $f$ theory (kHz) | $f$ FEM (kHz) | $f$ measured (kHz) | $\xi_o$ FEM (μm) | $\xi_o$ measured (μm) |
|---|---|---|---|---|---|---|---|---|
| Addendum circle | 21.66 | 14.775 | 30.96 | 18.3 | 19.005 | 18.154 | 12.1 | 13.312 |
| Pitch circle | 16.648 | 11.041 | 34.88 | 21.1 | 21.645 | 21.433 | 7.7 | 6.827 |
| Dedendum circle | 13.94 | 9.387 | 47.32 | 21.8 | 23.169 | 23.894 | 4.7 | 5.628 |
Solving the determinant for $l_3$ at the design frequency gives three distinctly different adjusting lengths, spanning almost 17 mm, which already indicates that the choice of equivalent section is a design decision rather than a formality. Re-substituting these lengths and solving for frequency gives 18.3, 21.1 and 21.8 kHz, that is, deviations of 8.88%, 5.35% and 8.61% from the 20 kHz target. Modal analysis of the three simplified assemblies over 15–25 kHz returns longitudinal modes at 19.005, 21.645 and 23.169 kHz, within 3.78%, 2.55% and 6.09% of their own analytical values. A full toothed model of the same stack resonates at 20.306 kHz, so the three simplifications deviate from the actual gear by 6.62%, 6.38% and 13.17% respectively.
Harmonic analysis with a 5 μm axial excitation at the input end shows the expected energy concentration at the tooth end: 12.1, 7.7 and 4.7 μm for the three models against 9.3 μm for the full toothed model, deviations of 26.1%, 18.8% and 65.7%. The displacement nodes fall at 69.863, 68.833 and 67.338 mm from the transducer face — the positions that later carry the mounting flanges. Machined prototypes were measured with an impedance analyzer (1 kHz–1 MHz, ±10 ppm frequency accuracy, 0.1 Hz step) and a laser Doppler vibrometer (0.01 nm displacement resolution). Measured resonances were 18.154, 21.433 and 23.894 kHz, within 4.58%, 0.98% and 3.08% of simulation, and measured axial amplitudes were 13.312, 6.827 and 5.628 μm, within 9.54%, 12.02% and 17.97% of simulation.
Across all three comparisons the pitch-circle equivalent is the closest match: 21.433 kHz measured against 21.259 kHz for the fully toothed system (0.82% apart) and 6.827 μm against 7.964 μm (15.4% apart). The addendum model, with the largest equivalent radius, overestimates the amplitude delivered to the tooth by roughly a factor of 1.7; the dedendum model, with the smallest radius, underestimates it by about 30%. Frequency prediction is evidently the robust output of the analytical model, while amplitude prediction carries an error of the order of 15–25% even for the best case and should be treated as an estimate to be confirmed by impedance measurement of the assembled stack.
Horn material and the damped double-flange mount
Horn material enters the design through both the wave speed and the acoustic impedance. Three candidates were evaluated against the 20CrMnTi gear: TC4 titanium alloy, 45 carbon steel and an aluminium alloy. Computed longitudinal wave speeds and acoustic impedances are listed below; the wavelength at 20 kHz is 245–258 mm in the metals, an order of magnitude above the segment lengths, which justifies the one-dimensional treatment.
| Component | Material | $E$ (GPa) | $\rho$ (kg/m³) | $\nu$ | $c$ (m/s) | $Z=\rho c$ (MRayl) |
|---|---|---|---|---|---|---|
| Horn | TC4 titanium alloy | 111 | 4429 | 0.338 | 5006 | 22.2 |
| Horn | 45 carbon steel | 209 | 7850 | 0.269 | 5160 | 40.5 |
| Horn | Aluminium alloy | 68 | 2840 | 0.330 | 4893 | 13.9 |
| Tool gear | 20CrMnTi carburized steel | 207 | 7800 | 0.250 | 5152 | 40.2 |
| Damping washer | Nylon 66 | 3.5 | 1150 | 0.390 | 1745 | 2.0 |
Solving the frequency equation at 20 kHz for each horn material gives adjusting lengths of 35.18 mm (45 steel), 43.86 mm (TC4) and 41.80 mm (aluminium), and the resulting resonant frequencies are 21.9, 21.4 and 19.28 kHz, deviations of 9.6%, 6.8% and 3.7%. The material choice therefore moves the achievable frequency only modestly, but it changes the required length substantially: 45 steel shortens the adjusting section by 6.6–8.7 mm relative to the other two. In a spindle bore of fixed internal envelope this is decisive, and 45 steel was selected, with $l_3$ rounded to 35 mm for machining. A second argument points the same way: the acoustic impedance of 45 steel (40.5 MRayl) is within 1% of the 20CrMnTi gear (40.2 MRayl), against 22.2 MRayl for TC4 and 13.9 MRayl for aluminium, so the interface between horn and tool reflects far less of the incident wave. Where a larger horn envelope is available, TC4 or aluminium become the better options because they produce a longer adjusting section at the same frequency.
The mounting is the other place where energy is lost. The vibration system is clamped into the lapping machine spindle through flanges, and a conventional rigid flange at a non-node position couples the machine frame into the resonator: interfacial friction converts mechanical energy into heat, contact faces yield plastically and distort the waveform, and spindle vibration can drive the stack into unwanted resonances. The design used here places a double-flange module at the displacement node — one flange mating with the horn, one with the spindle — and inserts nylon 66 washers as interface damping elements at both junctions, clamped by high-strength bolts. The steel flange body supplies the stiffness needed for location; the polymer layer suppresses thermal-elastic deformation at the contact and decouples the machine from the resonator.
Finite element analysis of the complete assembly (169,648 nodes, 94,481 elements, Block Lanczos extraction over 15–25 kHz) identifies the longitudinal mode at 21.964 kHz, 9.4% above the 20 kHz target, with the displacement increasing from the large end to the small end and from root to tip. Harmonic analysis with fixed supports at the flanges and a 5 μm input gives an output of 8.3 μm and a node at 65.167 mm. Impedance measurement of the assembled system gives 21.259 kHz, 3.26% below simulation, with a single closed admittance circle and a single phase peak, which excludes parasitic modes in the working band; the laser vibrometer gives 7.964 μm, 4.13% below the simulated value.
Sensitivity to tooth width and helix angle
Gear geometry is not held to the micrometre. Tooth form is produced by turning and CNC milling, then modified by heat treatment, so face width and helix angle vary within the tolerance chain and shift the dynamics of the resonator that carries them. Two series of parametric models were run: face width $b$ from 24 to 28 mm in 0.5 mm steps, and helix angle $\beta$ from 31° to 39° in 1° steps, each with all other parameters frozen.
In simulation, resonant frequency falls monotonically with face width, by 51–119 Hz per 0.5 mm increment, and the axial output displacement falls with it. Helix angle is a much weaker lever on frequency: over 31–39° the resonance moves within 20.115–21.395 kHz, with a downward trend, while the axial output drops by 0.05–0.41 μm per degree. Local coordinate systems set on the concave and convex flanks, with one axis normal to the flank surface, show the tooth-surface vibration components remaining small and non-monotonic — 0.66–1.34 μm across the face-width series and 0.7–1.25 μm across the helix-angle series. The physical reading is that face width adds axial length and mass to the resonator and therefore lowers its stiffness-to-mass ratio, whereas helix angle mainly redistributes a fixed longitudinal excitation into components normal to the flank, which is why the axial output barely responds and the flank-normal component does.
Preload of the threaded joint
The gear is fixed to the horn by a threaded connection at its hub, and the preload determines how much of the vibration crosses that joint. Too little preload and the friction at the contact faces is insufficient to prevent micro-slip, so the joint loses self-locking and dissipates energy; too much and stress concentration at the thread root accelerates wear and can initiate fatigue cracks. Reports on threaded joints under vibration attribute roughly 72% of the transmitted vibratory torque to the rear flank of the thread, where fretting wear is the main source of progressive slip. The gear hub was given an external hexagon so that a preset torque wrench (5–60 N·m range, 0.5 N·m resolution) can apply a controlled torque.
| Preload torque (N·m) | Resonant frequency $f$ (kHz) | Mechanical quality factor $Q_m$ | Output amplitude $\xi_o$ (μm) |
|---|---|---|---|
| 20 | 21.336 | 165.4 | 7.643 |
| 25 | 21.434 | 253.5 | 7.752 |
| 30 | 21.412 | 344.6 | 7.913 |
| 35 | 21.508 | 462.7 | 7.994 |
| 40 | 21.574 | 471.1 | 7.991 |
| 45 | 21.596 | 480.9 | 8.018 |
Frequency, quality factor and amplitude all rise with torque and then flatten: between 20 and 35 N·m the quality factor nearly triples and the amplitude gains 0.35 μm, whereas between 35 and 45 N·m the quality factor gains less than 4% and the amplitude less than 0.03 μm. A preload of 35 N·m was adopted for all subsequent measurements, since the marginal gain beyond that point does not justify the additional thread-root stress. The quality factor is a useful side benefit of this experiment: it can be read directly from an impedance sweep and serves as a cheap acceptance indicator for joint tightness on the shop floor.
Measured behaviour of machined gears
Two series of samples were cut on a CNC spiral tooth milling machine — five face widths and five helix angles — and measured with the impedance analyzer and the laser vibrometer, the latter aimed at the axial end face and at both flanks. All other conditions were held fixed at the 35 N·m preload.
| Parameter | $f$ (kHz) | $Q_m$ | Axial $\xi_o$ (μm) | Concave flank (μm) | Convex flank (μm) |
|---|---|---|---|---|---|
| $b$ = 24 mm | 21.422 | 380.5 | 14.942 | 1.134 | 1.210 |
| $b$ = 25 mm | 21.381 | 412.2 | 10.540 | 1.382 | 1.428 |
| $b$ = 26 mm | 21.259 | 462.7 | 7.964 | 1.360 | 1.259 |
| $b$ = 27 mm | 21.226 | 391.3 | 5.627 | 0.962 | 1.093 |
| $b$ = 28 mm | 21.184 | 411.8 | 4.315 | 0.834 | 0.786 |
| $\beta$ = 31° | 21.344 | 421.3 | 7.854 | 1.362 | 1.268 |
| $\beta$ = 33° | 21.284 | 460.4 | 7.239 | 1.155 | 1.233 |
| $\beta$ = 35° | 21.259 | 462.7 | 7.964 | 1.204 | 1.106 |
| $\beta$ = 37° | 21.248 | 442.0 | 8.362 | 1.137 | 1.164 |
| $\beta$ = 39° | 21.226 | 356.2 | 7.153 | 0.956 | 0.862 |
The face-width series reproduces the simulated trends and exaggerates them. Resonance falls from 21.422 to 21.184 kHz over the 4 mm range, about 60 Hz/mm, and the axial amplitude collapses from 14.942 to 4.315 μm, an average of about 2.7 μm/mm. The response is markedly asymmetric about the 26 mm nominal: reducing the face width by 2 mm raises the amplitude by 6.98 μm, while increasing it by 2 mm lowers it by 3.65 μm, so a thin gear is a far larger perturbation than a thick one. Step-normalised ranges quoted for the series are 33–122 Hz per mm of face width for frequency and 1.31–4.40 μm per mm for amplitude.
The helix-angle series is flatter and partly non-monotonic. Frequency falls by only 118 Hz across 31–39°, about 15 Hz per degree, and the axial amplitude wanders between 7.153 and 8.362 μm without a clear trend, while the two flank components decrease steadily as the angle grows. Step-normalised ranges are 11–60 Hz per 2° for frequency and 0.62–1.21 μm per 2° for amplitude. The quality factor peaks near the 35° nominal and drops to 356 at 39°, which is consistent with the flank geometry moving the load away from the purely longitudinal mode.
All amplitude values on the flanks are an order of magnitude below the axial values and show scatter of 20–30% between nominally identical parts, which is the same order as the parameter effect being measured. Simulation and experiment agree in trend for both parameters but differ in magnitude, and the reported agreement for the fully toothed system itself varies between 0.02% and 3.3% depending on which comparison is quoted. Frequency agreement should therefore be read as “within a few percent”, not as a validated prediction.
What the numbers imply for design
Three engineering conclusions follow from the data. First, the equivalent-section choice must be justified by measurement, not assumed: the three obvious candidates differ by 17 mm in required horn length and by a factor of 2.6 in predicted output amplitude, and only the pitch-circle model stays within 1% of the real toothed system in frequency. Second, the horn material is selected mainly by envelope and acoustic matching, not by resonance: all three materials land within 10% of the 20 kHz target, but 45 steel needs 6.6–8.7 mm less axial length and reflects almost nothing at the gear interface because its impedance matches 20CrMnTi to within 1%. Third, the load parameters set a process window that is tight in one direction and loose in the other. Because the measured axial amplitude changes by about 2.7 μm/mm of face width, holding the delivered amplitude within ±10% of 8 μm implies a face-width tolerance of roughly ±0.3 mm — an estimate derived from the measured slope rather than a validated specification, but one that places face width among the critical tolerances of the tool gear. Helix angle, by contrast, can be allowed a wider band if only the axial amplitude matters, although it does control the flank-normal component that governs local lapping action.
The preload experiment adds a maintenance dimension. Quality factor read from an impedance sweep responds strongly to joint condition between 20 and 35 N·m and is essentially flat above it, so a single sweep after assembly can confirm that a joint has been tightened into the plateau without instrumenting the machine for vibration measurement.
Open issues
The reported evidence covers the resonator, not the process outcome. Frequency, quality factor and axial amplitude are characterised as functions of load geometry and preload, but the corresponding lapping results — roughness, contact-pattern shift, removal rate and tooth-form error — are not mapped onto the same parameter grid, so the amplitude sensitivity measured here cannot yet be converted into a quality or productivity specification. The flank-normal amplitudes are small and scattered, and whether that scatter produces local over-cutting or, under some conditions, beneficial micro-finishing remains unresolved.
The modelling is linear, one-dimensional and axisymmetric after simplification, with interfaces bonded in the finite element model and flanges omitted from the analytical stage; transverse and torsional modes, thermal drift during long runs and the nonlinear contact of the threaded joint are outside its scope. Extending the parameter study to a full factorial design that includes lapping response, and replacing the bonded interface with a contact model calibrated against measured quality factors, would be the natural next steps before the load-parameter sensitivities reported here can be used as production tolerances.