The finishing problem that ultrasonic lapping addresses
Spiral bevel gears transmit torque between intersecting shafts, and their final quality is fixed by lapping, the last mechanical step that corrects the contact pattern and, with it, the running noise of the pair. Conventional lapping relies on relative rotation of the two gears in an abrasive compound. Its removal rate is limited by contact pressure and by the rate at which fresh abrasive reaches the mesh, and the tooth flanks near the heel and toe are the hardest regions to finish uniformly.
Superimposing a high-frequency longitudinal vibration on the pinion changes the cutting mechanism. The abrasive grains are no longer only dragged across the flank; they are repeatedly hammered into it, and the alternating pressure field adds friction heating and cavitation effects in the compound film. Material is removed by micro-chipping at the tooth surface rather than by bulk sliding contact, so the removal rate can rise without a proportional increase in static load. Comparative lapping trials on spiral bevel gear pairs have reported surface roughness reduced to about $0.2\ \mu\text{m}$, running noise lowered by 3–8 dB, and a material removal rate roughly three times that of the conventional process. These values describe the process target that motivates the hardware design; they are outcomes of specific gear pairs and operating conditions rather than guaranteed figures.
Architecture of the vibration system
The functional core of the process is a longitudinal resonator that carries the pinion at its output face. It has three active parts: a piezoelectric transducer driven by an ultrasonic generator, a two-stage stepped horn, and the spiral bevel pinion itself. The transducer is built as a half-wave resonator from the inverse piezoelectric effect, so its output face oscillates at maximum displacement and essentially zero strain. The horn amplifies that displacement and concentrates the acoustic energy onto a smaller output cross-section. The pinion is threaded onto the end of the horn and treated, acoustically, as the final section of the resonator.
The gear geometry cannot be altered to suit the acoustics: its module, cone angle and face width are set by the transmission design. The pinion is therefore designed on a non-resonant basis and accepted as a given termination, while the horn carries the tuning freedom. A flange and a locating disk position the assembly in the machine spindle and clamp it rigidly, with rubber gaskets providing vibration isolation between the resonator and the machine frame.
Reducing the gear to a tractable acoustic element
For the one-dimensional longitudinal model, the pinion is replaced by an equivalent truncated cone. The tooth cone is bounded by a large-end diameter $d_4$, a small-end diameter derived from the pitch cone angle $\delta$ and the gear thickness $l_4$, and a cone half-angle $\delta$ between the generatrix and the axis. Together with three cylindrical sections of the horn, this gives four regions along the axis: sections I, II and III of the stepped horn and section IV, the gear cone. The threaded joint is treated as rigid and its compliance is not modelled separately.
The formulation assumes homogeneous, isotropic, loss-free material, transverse dimensions much smaller than the acoustic wavelength so that only longitudinal motion need be retained, and a free output face. The last assumption is a design simplification: during lapping the pinion meshes with the wheel and carries a dynamic lapping force, which is neglected when the resonator is dimensioned. The resulting geometry and material data are collected in Table 1.
| Element | Parameter | Value |
|---|---|---|
| Horn, TC4 titanium alloy | Density $\rho$ / elastic modulus $E$ / Poisson ratio $\nu$ | 4450 kg/m³ / 118 GPa / 0.34 |
| Horn, section I | Diameter $d_1$ / length $l_1$ | 54 mm / 68 mm |
| Horn, section II | Diameter $d_2$ | 40 mm |
| Horn, section III | Diameter $d_3$ / length $l_3$ | 30 mm / 0–43 mm (tuning range) |
| Pinion, 20CrMnTi | Density $\rho$ / elastic modulus $E$ / Poisson ratio $\nu$ | 7800 kg/m³ / 207 GPa / 0.25 |
| Pinion cone, section IV | Large-end diameter $d_4$ / thickness $l_4$ | 33.30 mm / set by gear design |
| Design target | Resonant frequency / search range | 20 kHz / 15–25 kHz |
Wave equation and frequency equation
With $x$ measured along the axis from the interface between sections I and II, the longitudinal displacement $\xi$ in a section of cross-sectional area $S(x)$ satisfies the Webster-type wave equation
$$\frac{\partial^2 \xi}{\partial x^2} + \frac{1}{S}\frac{\partial S}{\partial x}\frac{\partial \xi}{\partial x} + k^2 \xi = 0, \qquad k = \frac{\omega}{c}, \quad c = \sqrt{\frac{E}{\rho}}$$
where $\omega$ is the angular frequency and $c$ the longitudinal wave speed. For the three constant-area sections ($\partial S/\partial x = 0$) the solution and its strain are harmonic:
$$\xi_n(x_n) = A_n \cos(k_n x_n) + B_n \sin(k_n x_n), \qquad n = 1,2,3$$
For the conical section, the area varies as the square of the distance from the cone apex, and the solution acquires a $1/(x_4 – 1/\alpha_4)$ amplitude factor:
$$\xi_4(x_4) = \frac{A_4 \cos(k_4 x_4) + B_4 \sin(k_4 x_4)}{x_4 – 1/\alpha_4}, \qquad \alpha_4 = \frac{2\tan\delta}{d_4}$$
The boundary conditions follow from the architecture. The transducer output face is a displacement antinode, so the internal force there vanishes; the output face of the gear is treated as free:
$$\left.\frac{\partial \xi_1}{\partial x}\right|_{x=-l_1} = 0, \qquad \left.\frac{\partial \xi_4}{\partial x}\right|_{x=l_2+l_3+l_4} = 0$$
At each of the three internal interfaces, displacement and force are continuous:
$$\xi_j = \xi_{j+1}, \qquad S_j E_j \frac{\partial \xi_j}{\partial x} = S_{j+1} E_{j+1} \frac{\partial \xi_{j+1}}{\partial x}, \qquad j = 1,2,3$$
Substituting the section solutions into these eight conditions yields a homogeneous linear system in the eight coefficients $A_n$, $B_n$. A non-trivial solution exists only when the coefficient determinant vanishes:
$$\Delta(\omega, l_3) = \det \mathbf{D}(\omega, l_3) = 0$$
This is the frequency equation of the stepped resonator. With all dimensions fixed it gives the resonant frequencies; with the target frequency fixed it gives the unknown horn dimension, which in this design is the length $l_3$ of section III. Because $c \approx 5.15$ km/s in both TC4 and 20CrMnTi, the half-wavelength at 20 kHz is about 129 mm, consistent with the 137 mm overall length of the finished assembly.
Numerical design: locating a 20 kHz resonance
The determinant is a strongly transcendental function, so it was evaluated numerically with the frequency $f$ and the length $l_3$ as independent variables and $\Delta$ as the response. The resulting surface is smooth and continuous over the 15–25 kHz search band, and $\Delta = 0$ is crossed at several combinations of length and frequency, reflecting the higher-order longitudinal modes of the same stack.
Fixing the working frequency at 20 kHz reduces the problem to a single-variable root search. The error curve rises from positive values, crosses zero once and continues into negative values, so exactly one length in the 0–43 mm range satisfies the frequency equation: $l_3 = 28.55$ mm. Rounding to $l_3 = 28$ mm for manufacturability shifts the computed resonance to 20.042 kHz, an increase of 42 Hz, or $2.1 \times 10^{-3}$ relative. A deviation of this size is negligible for the process, which means that the tuning length can be specified as an integer millimetre dimension without compromising the design.
Predicted vibration behaviour
Solving the non-homogeneous system obtained by replacing one boundary condition with a prescribed input displacement gives the coefficients $A_n$, $B_n$ and hence the displacement distribution along the axis. The computed amplitude rises from the input face to the output face and passes through a single low-amplitude location close to the input end, at about 0.02 m from the datum, where the displacement falls to $0.47\ \mu\text{m}$. The flange and locating disk are placed at this position: clamping the resonator where its motion is smallest minimizes acoustic coupling to the machine and the associated mechanical loss.
Over the last 111–137 mm of the assembly, which is exactly the axial extent occupied by the gear teeth, the displacement curve flattens near its maximum. Every point of the tooth flank from heel to toe therefore receives a strong, nearly uniform vibration output, which favours the hammering, impact, friction and cavitation action of the abrasive. At the gear end face, 137 mm from the datum, the displacement reaches $13.79\ \mu\text{m}$; relative to the input, this is an amplitude gain of 2.758, corresponding to an input-face amplitude of about $5\ \mu\text{m}$.
Experimental validation
A resonator was manufactured to the rounded dimensions and tested with an intelligent CNC ultrasonic generator (TJS-3000), three piezoelectric accelerometers (SA-PE50W), a charge amplifier (SAPE08) and a data acquisition unit (SA1804A). Accelerometer 1 was mounted at the front face to measure forward-radiated output, accelerometer 3 at the rear face for backward-radiated output, and accelerometer 2 on the mounting flange.
Fourier analysis of the three signals shows a single dominant peak near 19.50 kHz, with the front and rear faces both at maximum acceleration and all other frequencies oscillating around zero. The measured resonant frequency is thus 19.50 kHz, 2.5% below the 20 kHz design target. At resonance the front-face acceleration is $254.347\ \text{m/s}^2$ and the rear-face value $96.972\ \text{m/s}^2$, a ratio of 2.623, close to the predicted amplitude gain of 2.758. Because both values are taken at the same frequency, their ratio is a valid proxy for the displacement ratio and confirms that most of the vibration energy is radiated forward through the gear. Accelerometer 2 on the flange recorded an acceleration close to zero, consistent with the choice of a low-amplitude clamping position and with effective isolation of the resonator from the machine.
| Quantity | Model prediction | Measured | Deviation |
|---|---|---|---|
| Resonant frequency | 20.042 kHz (rounded geometry) | 19.50 kHz | −2.5% |
| Amplitude gain, output/input | 2.758 | 2.623 (acceleration ratio) | −5% |
| Vibration at the flange | Low-amplitude location by design | Acceleration near zero | Consistent |
| Displacement at gear end face | 13.79 μm | Not measured directly | — |
Design rules and remaining uncertainty
Three conclusions follow for the engineering of such systems. First, the workpiece has to be part of the acoustic model: because the gear cone contributes mass and stiffness at the output, treating the horn and the gear separately and then attaching one to the other would place the resonance at the wrong frequency. The determinant formulation handles the loaded horn directly and returns the length that restores the target frequency. Second, the tuning dimension is forgiving in discrete steps but sensitive in absolute terms: a 0.55 mm truncation of section III costs only 42 Hz, yet the 2.5% gap between prediction and measurement shows that manufacturing tolerance, thread-joint compliance and material property scatter together move the resonance by several hundred hertz. A final frequency trim on the assembled resonator, typically by adjusting $l_3$, is therefore advisable before the system is released to production. Third, clamping belongs at the computed low-amplitude location, not at a structurally convenient one; the near-zero flange acceleration is the measurable signature that this rule has been met.
The model is deliberately one-dimensional. It neglects transverse and torsional modes, the compliance and possible micro-slip of the threaded connection, the temperature dependence of the elastic modulus under continuous ultrasonic excitation, and the dynamic lapping force that appears once the pinion meshes with the wheel. These omissions do not affect the validity of the design method, but they explain the direction and size of the observed frequency offset and set the limit within which the predicted gain of 2.758 should be read. Within that envelope, the measured 19.50 kHz resonance, 2.623 gain and quiescent flange together confirm that the stepped resonator delivers the longitudinal vibration amplitude and the front-to-back energy concentration that ultrasonic lapping of spiral bevel gears requires.