Spiral bevel gears transmit motion between intersecting axes in differentials, helicopter drives, and machine-tool spindles. Their final tooth-surface quality is set by lapping, a free-abrasive finishing process that corrects the contact pattern and reduces transmission error. In conventional lapping, the pinion and wheel run in mesh with an abrasive slurry; local sliding velocity, contact pressure, and abrasive entrapment determine how much material is removed. Because the contact is not uniform along the tooth, the toe, heel, and tooth edges can receive different finishing action.
Superimposing a longitudinal ultrasonic vibration on the pinion changes the local kinematics at the tooth surface. The vibrating flank repeatedly strikes and releases abrasive particles, adding hammering, impact, friction, and cavitation effects to the sliding motion. Ultrasonic-assisted lapping of spiral bevel gears has been associated with roughness values near 0.2 μm, noise reductions of 3–8 dB, and material removal rates roughly three times those of conventional lapping. The practical bottleneck is the acoustic hardware: a piezoelectric transducer produces only a few micrometres of displacement, while the pinion has a fixed geometry and substantial mass. A stepped horn can amplify the displacement, but the pinion cannot be treated as a passive attachment. It becomes the last section of the resonant system, so the horn and pinion must be tuned together at the target frequency.
Acoustic architecture and design constraints
The vibration system combines a half-wave piezoelectric transducer, a stepped titanium horn, and the pinion. The transducer converts electrical excitation into longitudinal motion. Its output face is designed as a displacement antinode with near-zero strain, so the ideal axial force transmitted into the horn is small. The horn has three cylindrical sections: section I at the transducer side, section II in the middle, and section III at the pinion side. Section III ends in a threaded connection to the pinion. The thread is treated as rigid and as part of the horn, so the connection is not modelled as a separate elastic joint.
The pinion is represented as a truncated cone, matching the pitch-cone envelope of the spiral bevel gear. Its large-end diameter is 33.2963 mm, and its small-end diameter follows from the pitch-cone angle and face width. This conical section is not a horn in the usual sense; its dimensions are set by the gear design. The horn is therefore designed around the pinion rather than forcing the pinion to a separate resonance.
Two hardware constraints shape the dimensions. The transducer output diameter sets the large end of the horn, and the machine-spindle envelope limits its length. In the design example, section I has a diameter of 54 mm and a length of 68 mm. Sections II and III have diameters of 40 mm and 30 mm. These moderate area reductions provide displacement amplification while keeping the one-dimensional force-continuity approximation reasonable. A flange and locating disk hold the assembly in the machine; they are placed at a displacement node and isolated with rubber gaskets to reduce acoustic leakage into the machine frame.
| Parameter | Symbol | Value |
|---|---|---|
| Horn material | TC4 titanium | Density 4450 kg/m3; elastic modulus 118 GPa; Poisson ratio 0.34 |
| Pinion material | 20CrMnTi | Density 7800 kg/m3; elastic modulus 207 GPa; Poisson ratio 0.25 |
| Section I | d1, l1 | 54 mm, 68 mm |
| Section II | d2 | 40 mm |
| Section III | d3, l3 | 30 mm; l3 = 28.55 mm theoretical, 28 mm machined |
| Pinion cone large end | d4 | 33.2963 mm |
| Pinion cone small end | d5 | Derived from pitch-cone angle and face width |
| Design frequency | f | 20 kHz nominal; search range 15–25 kHz |
Longitudinal-wave model and frequency equation
The acoustic model assumes homogeneous, isotropic materials; lateral dimensions much smaller than the longitudinal wavelength; uniform axial displacement over each cross-section; negligible mechanical loss; and a rigid threaded joint. Under these conditions, the system is treated as a one-dimensional longitudinal resonator. The governing wave equation for a variable cross-section is
$$\frac{\partial^2 \xi}{\partial x^2}+\frac{1}{S(x)}\frac{\partial S}{\partial x}\frac{\partial \xi}{\partial x}+k^2\xi=0,\qquad k=\frac{\omega}{c},\quad c=\sqrt{\frac{E}{\rho}},$$
where \(\xi\) is the axial displacement, \(S(x)\) is the cross-sectional area, \(\omega\) is the angular frequency, \(E\) is the elastic modulus, and \(\rho\) is the density.
For the three constant-area cylindrical sections, the displacement and strain are
$$\xi_n(x_n)=A_n\cos(k_nx_n)+B_n\sin(k_nx_n),$$
$$\frac{\partial \xi_n}{\partial x_n}=-A_nk_n\sin(k_nx_n)+B_nk_n\cos(k_nx_n),\qquad n=1,2,3.$$
For the conical pinion section, the area varies linearly with the axial coordinate. The displacement solution is
$$\xi_4(x_4)=\frac{A_4\cos(k_4x_4)+B_4\sin(k_4x_4)}{x_4-1/\alpha_4},\qquad \alpha_4=\frac{2\tan\delta}{d_4},$$
where \(d_4\) is the cone large-end diameter and \(\delta\) is the cone half-angle.
The boundary condition at the transducer side follows from the half-wave transducer output face acting as a displacement antinode with zero strain:
$$E_1S_1\left.\frac{\partial \xi_1}{\partial x}\right|_{x=-l_1}=0.$$
At the free pinion end, the axial force is zero:
$$E_4S_5\left.\frac{\partial \xi_4}{\partial x}\right|_{x=l_2+l_3+l_4}=0.$$
At each interface between adjacent sections, displacement and axial force are continuous:
$$\xi_j=\xi_{j+1},\qquad S_jE_j\frac{\partial \xi_j}{\partial x}=S_{j+1}E_{j+1}\frac{\partial \xi_{j+1}}{\partial x}.$$
Substituting the four displacement fields into these conditions gives eight linear homogeneous equations in the eight unknown coefficients \(A_1,B_1,\ldots,A_4,B_4\). A nontrivial solution requires the determinant of the coefficient matrix to vanish:
$$\Delta(f,l_3)=\det\left[D_{ij}(f,l_3)\right]_{8\times8}=0.$$
This is the frequency equation of the stepped system. If the geometry is known, its roots give the resonant frequencies. If the operating frequency is fixed, the same equation can be solved for an unknown geometric parameter. The model treats the pinion end as free; during actual lapping, the mesh force acts on the tooth flanks. The free-end condition is therefore a design-stage approximation, not a full loaded-contact model.
Tuning the stepped system
Section III is the tuning element. Its diameter is fixed by the horn design, but its length can be machined to shift the resonance without altering the transducer or the pinion. The determinant was evaluated over a working-frequency range of 15–25 kHz and a section-III length range of 0–43 mm. The resulting \(\Delta(f,l_3)\) surface is smooth and continuous. Multiple frequency–length pairs satisfy \(\Delta=0\), so the root search must be constrained by the available spindle space, the target frequency, and practical machining limits.
With the frequency fixed at 20 kHz, the determinant crosses zero at a single length within the search interval: \(l_3=28.55\) mm. Rounding to 28 mm for machining gives a predicted resonance of 20.042 kHz. The shift is 42 Hz, or 0.21% of the nominal frequency. That is small enough to be accepted at the design stage, and it is much smaller than the frequency deviation observed after assembly.
The calculated longitudinal displacement rises from the transducer end to the pinion small end. A single displacement minimum occurs near 0.02 m from the input face; this is the mounting node. The flange and locating disk are placed at that plane to minimize acoustic coupling to the machine. Over the pinion region, approximately 111–137 mm from the input face, the displacement is close to its maximum. The free small end reaches 13.79 μm, giving a displacement amplification of 2.758 relative to the input. The near-plateau over the tooth region means that the face width is driven at high amplitude, not just the tooth tip.
Experimental validation
A prototype was assembled according to the design and tested with a TJS-3000 intelligent numerical-control ultrasonic generator, SA-PE50W piezoelectric accelerometers, an SAPE08 charge amplifier, and an SA1804A data acquisition unit. Three accelerometers were used: one at the front output face, one at the rear end, and one on the flange connected to the machine. The acceleration signals were Fourier transformed to obtain the amplitude–frequency response.
At 19.50 kHz, both the front and rear faces showed maximum acceleration, while the response at nearby frequencies remained close to zero. The measured resonant frequency was therefore 19.50 kHz, which is 0.50 kHz or 2.5% below the 20 kHz nominal target and 2.7% below the 20.042 kHz prediction obtained with the rounded length. The front and rear acceleration amplitudes were 254.347 m/s2 and 96.972 m/s2, respectively. Their ratio was 2.623, about 4.9% below the calculated displacement ratio of 2.758. The acceleration ratio is not a direct displacement measurement, but it is a practical proxy for the relative motion of the two ends when both faces oscillate at the same frequency.
| Quantity | Design or prediction | Measurement | Comparison |
|---|---|---|---|
| Resonant frequency | 20.000 kHz nominal; 20.042 kHz with l3 = 28 mm | 19.50 kHz | −2.5% versus nominal; −2.7% versus rounded-length prediction |
| Amplification ratio | 2.758 | 2.623 | −0.135; −4.9% |
| Flange vibration | Designed at displacement node | Acceleration near zero | Consistent with node placement and isolation |
The flange accelerometer measured almost no vibration. This confirms that the mounting plane is close to the calculated node and that the rubber isolation limits mechanical coupling between the resonator and the machine. The measured frequency shift is larger than the 0.21% effect of rounding the tuning length, so it is dominated by assembly-level factors. Elastic-property tolerances, machining tolerances, thread compliance, and clamp load are all plausible contributors. A lower measured resonance is consistent with a small increase in effective compliance or mass relative to the ideal model.
Design implications and limits
The main design value of this approach is system-level integration. The pinion is not a passive workpiece attached to a separately tuned horn; it is the last acoustic section of the resonator. Section III provides a single machining variable for frequency tuning, and the calculated mode shape determines the mounting plane. Placing the flange at the displacement node reduces energy loss to the machine, while the stepped area reduction and the tapered pinion produce an output displacement of 13.79 μm at the gear end. The near-maximum displacement over the tooth region is favourable for activating abrasive particles along the full face width.
The model is a design tool rather than a complete representation of loaded lapping. It is linear, one-dimensional, and lossless; it neglects transverse and torsional modes, nonlinear contact, thermal effects, and slurry loading. The threaded joint is treated as rigid, although real thread compliance can shift the resonance. Most importantly, the pinion end is modelled as free, while lapping introduces dynamic mesh forces on the tooth flanks. The experimental validation reported here is a vibration test of the assembled system, not a fully loaded lapping test. Under operating load, the resonance, mode shape, and output amplitude may differ from the free-end prediction.
For production use, the resonator should be characterized after assembly, and the generator should either track the actual resonance or the tuning length should be adjusted to compensate for joint compliance and material tolerances. A loaded test with a dummy mesh force or an instrumented lapping run would establish how much the tooth contact shifts the resonance and whether the amplitude remains sufficient at the tooth surface. Within those limits, the stepped-horn design method provides a practical way to tune a spiral bevel gear pinion as an ultrasonic lapping tool while keeping the mounting node and the high-amplitude tooth region in the intended locations.