Transmission error as a noise mechanism
Transmission error is the difference between the rotation that a gear pair would have under rigid, conjugate motion and the rotation it actually transmits. In a spiral bevel gear, the quasi-conjugate flank form and the periodic variation of mesh stiffness make this error both geometric and load-dependent. The unloaded geometric transmission error (GTE) is fixed by flank topology and machine settings; the loaded transmission error (LTE) adds the elastic deflection of the contacting teeth. High-order harmonics of LTE are a key excitation source in high-speed helicopter main reducers and electrified commercial-vehicle drivetrains, where mesh tonal noise is an NVH concern.
Second-order parabolic GTE has been widely used because it absorbs linear mounting and manufacturing errors and reduces sensitivity to misalignment. Its weakness is at the mesh transfer: the acceleration curve changes abruptly when contact hands over between tooth pairs, producing an impact that can dominate the high-frequency response. Higher-order motion curves increase the included angle at the transfer point, but a single high-order polynomial can be difficult to realize or may not control the loaded spectrum well. A hybrid approach keeps parabolic curves in the outer mesh regions and replaces the single-tooth-contact region with a sinusoid.
Design targets: symmetry and load fluctuation
Two metrics guide the flank and machine-setting optimization. The first is symmetry of the GTE about the design reference point \(C\): the entry and exit halves should have equal amplitude and mesh time, so edge contact does not begin early. The second is peak-to-peak fluctuation of the LTE at the working load, which should be small to reduce stiffness excitation. The optimization objective is
$$F(\Delta y,\eta_2,m’_{21})=\min\left\{\omega f_1+(1-\omega)f_2\right\}$$
$$f_1=\frac{|\phi_{1A}-\phi_{1C}|}{|\phi_{1E}-\phi_{1C}|},\qquad f_2=\max(T_e)-\min(T_e)$$
subject to
$$\Delta y_{\min}\le \Delta y\le \Delta y_{\max},\quad
\eta_{2\min}\le \eta_2\le \eta_{2\max},\quad
m’_{21\min}\le m’_{21}\le m’_{21\max}.$$
Here \(A\) and \(E\) are the entry and exit points of the current tooth pair, \(C\) is the design reference point, \(T_e\) is the loaded transmission error, and \(\omega\) is a weighting factor. The design variables are the tooth-height position \(\Delta y\) of the reference point, the contact-path angle \(\eta_2\) relative to the root cone, and the first derivative \(m’_{21}\) of the transmission ratio. The bounds on \(\Delta y\) come from the wheel addendum and dedendum; the bounds on \(\eta_2\) come from tooth strength and mounting accuracy; the bounds on \(m’_{21}\) are set by the working load.
The objective cannot be written as a closed-form function of the machine settings, and multiple local minima exist. Each candidate solution is therefore passed through local synthesis, tooth contact analysis (TCA), and loaded tooth contact analysis (LTCA). A genetic algorithm with a population of 50, 100 generations, crossover probability 0.5, and mutation probability 0.1 is used to search this loop.
For the example discussed below, the optimized local synthesis parameters were \(\eta_2=15.713^\circ\), \(m’_{21}=-0.008512\), ellipse major axis \(2a=11.1\) mm, elastic deformation \(\delta=0.00635\) mm, pitch-cone offset \(\Delta x=0.05\) mm, tooth-height offset \(\Delta y=0.5422\) mm, and design contact ratio \(\varepsilon_r=2.2\). These parameters produced the machine settings used for the hybrid curve.
A three-segment hybrid transmission error
Five points define one complete mesh cycle: \(A\) is entry of the current tooth pair, \(B\) is the entry transfer to the next pair, \(C\) is the design reference, \(D\) is the exit transfer, and \(E\) is exit. At zero load, \(B\) and \(D\) are separated from \(C\) by half a mesh period:
$$\phi_{1B}=\phi_{1C}-\frac{T_m}{2},\qquad
\phi_{1D}=\phi_{1C}+\frac{T_m}{2}.$$
TCA gives the GTE amplitudes \(\delta_{2B}\) and \(\delta_{2D}\) at the transfer points. In a symmetric design these amplitudes are nearly equal. The hybrid curve uses a sinusoid between \(B\) and \(D\) and parabolas between \(A\)-\(B\) and \(D\)-\(E\):
$$
\delta_2(\phi_1)=
\begin{cases}
a_{2B}(\phi_1-\phi_{1B})^k+\delta_{2B}, & \phi_{1A}<\phi_1\le\phi_{1B},\\[4pt]
A_m\sin\left[\dfrac{2\pi}{T_m}(\phi_1-\phi_{1B})+\dfrac{\pi}{2}\right]+A_m, & \phi_{1B}<\phi_1\le\phi_{1D},\\[8pt]
a_{2C}(\phi_1-\phi_{1D})^k+\delta_{2D}, & \phi_{1D}<\phi_1\le\phi_{1E},
\end{cases}
\qquad k=2,4,\ldots
$$
The parabolic coefficients follow from the end-point amplitudes:
$$a_{2B}=\frac{\delta_{2A}-\delta_{2B}}{(\phi_{1A}-\phi_{1B})^k},\qquad
a_{2C}=\frac{\delta_{2E}-\delta_{2D}}{(\phi_{1E}-\phi_{1D})^k}.$$
The sinusoid is phased so that it reaches an extremum at \(B\) and \(D\) and passes through the reference value at \(C\). For the symmetric case, its amplitude is \(A_m=\delta_{2B}/2\), and its circular frequency is \(\omega_m=2\pi/T_m\). At \(B\) and \(D\), the sinusoidal and parabolic pieces have the same value and zero slope for \(k\ge2\). The transfer tangents are collinear, corresponding to an included angle of \(180^\circ\) between the transmission-error curves of adjacent tooth pairs. This removes the abrupt slope change associated with a second-order parabola and reduces the mesh impact at handover.
A higher parabolic order makes the curve steeper near the transfer point. It has little additional effect on mesh performance but is harder to realize through modified roll, so the example uses \(k=2\); both parabolic segments have a coefficient of \(-0.0115\). The sinusoid uses \(A_m=-7.5\) arcsec and \(\omega_m=23\). Under light load, contact remains in the single-tooth sinusoid region and the LTE follows the same harmonic shape. Under working load, load sharing adds elastic deformation, but the hybrid curve is intended to keep the LTE fluctuation low.
Converting the curve into machine settings
Once \(\delta_2(\phi_1)\) is defined, the corresponding wheel rotation follows from the transmission-error relation:
$$\phi_2=\delta_2+\frac{Z_1}{Z_2}(\phi_1-\phi_{1C})+\phi_{2C},$$
where \(Z_1\) and \(Z_2\) are the pinion and wheel tooth numbers, and \(\phi_{1C}\), \(\phi_{2C}\) are the reference rotations. Here \(\delta_2\) is used in radians when evaluating the rotation relation; the tabulated values below are in arcseconds.
Each of the three intervals is sampled at 100 points. Inverse TCA determines the instantaneous roll ratio and cradle rotation that generate the required rotation relation. A fifth-order polynomial fit for each interval gives a maximum fitting error below \(1.0\times10^{-5}\). The fit is converted into a basic roll ratio and second- through sixth-order modified-roll coefficients. The pinion settings for the three segments are listed in Table 1.
| Segment | Basic roll ratio | 2nd order | 3rd order | 4th order | 5th order | 6th order |
|---|---|---|---|---|---|---|
| Left parabolic | 2.81868 | −0.75340 | 15.38470 | −154.50197 | 814.93680 | −1765.30910 |
| Sinusoidal | 2.86216 | 0.07416 | 0.01231 | −25.60220 | 37.06650 | 274.51730 |
| Right parabolic | 2.90589 | −0.73830 | −15.60000 | −153.45930 | −818.28210 | −1792.60890 |
The basic roll ratio rises from 2.81868 in the left parabolic segment to 2.86216 in the sinusoidal segment and 2.90589 in the right parabolic segment. These are not coefficients of one global polynomial; the controller switches among three sets during grinding. Commercial computer numerical control (CNC) spiral bevel gear grinders, such as the Gleason Phoenix and Klingelnberg C series, accept machine-setting polynomials up to sixth order. The three-segment coefficients therefore fit within existing modified-roll control and do not require a new machine concept.
Example: contact pattern, deviation, and spectrum
The design was evaluated on a 23/65-tooth spiral bevel gear pair. The working flanks were the wheel convex flank and the pinion concave flank. Table 2 lists the main geometry.
| Parameter | Pinion | Wheel |
|---|---|---|
| Number of teeth | 23 | 65 |
| Module (mm) | 3.9 | 3.9 |
| Mean spiral angle (°) | 25 | 25 |
| Hand of spiral | Right | Left |
| Shaft angle (°) | 90 | 90 |
| Pitch cone angle (°) | 70.5138 | 19.4861 |
| Face cone angle (°) | 71.6718 | 21.7770 |
| Root cone angle (°) | 68.2229 | 18.3281 |
| Mean cone distance (mm) | 115.951 | 115.951 |
| Face width (mm) | 37 | 37 |
| Addendum (mm) | 1.9844 | 4.6455 |
| Dedendum (mm) | 5.3787 | 2.7176 |
TCA at the five key points gave the pinion rotations and TE amplitudes in Table 3. The entry and exit rotations relative to the reference point differ by less than \(0.001^\circ\), and the corresponding TE amplitude mismatch is below 0.1 arcsec, so the symmetry target is met.
| Point | Role | Pinion rotation φ1 (°) | TE amplitude δ2 (arcsec) |
|---|---|---|---|
| A | Entry | −15.5498 | −80.0192 |
| B | Entry transfer | −6.0743 | −15.0 |
| C | Reference | −1.7517 | 0 |
| D | Exit transfer | 9.5778 | −15.0 |
| E | Exit | 18.9627 | −79.2132 |
Compared with the parabolic design, the hybrid design preserves the design contact ratio, the transfer-point amplitude, and the entry and exit amplitudes within close limits. In the TCA contact pattern, the hybrid design gives the \(180^\circ\) included angle at \(B\) and \(D\), while the parabolic design does not. This is the geometric mechanism for smoother tooth-pair handover.
The tooth-surface deviation was evaluated with the parabolic design as the reference surface. The hybrid pinion flank deviates by at most \(+81\) μm, at the large-end root. Positive deviation appears at the small-end tip and large-end root because the hybrid curve applies less modification in the entry-to-\(B\) and \(D\)-to-exit regions; the sign reverses near the mid-reference region. This is a relative flank change, not an absolute machining error.
The load sweep covered wheel torques from 5 to 1000 N·m, with a working load of 400 N·m. At 400 N·m the actual contact changes from two tooth pairs to three tooth pairs. From low load to the working load, both designs keep a similar LTE shape and decreasing amplitude. Above 400 N·m the LTE shape changes and the amplitude grows for both designs. The hybrid design has lower peak-to-peak LTE across the sweep.
At 400 N·m, a fast Fourier transform of the LTE gave the first six harmonic amplitudes. All six amplitudes were lower for the hybrid design. The mean amplitude was 0.527 arcsec for the hybrid design and 0.6582 arcsec for the parabolic design, a reduction of 19.86%. These are transmission-error amplitudes, not sound-pressure levels; they indicate reduced high-order excitation rather than a directly measured acoustic benefit.
Design and manufacturing implications
The hybrid curve addresses the mesh-transfer problem in three ways. Under light load, when the loaded curve is close to the geometric curve, the single-tooth region is a sinusoid with continuous derivatives and a \(180^\circ\) transfer angle. This smooths the handover and reduces the acceleration impact that appears with a second-order parabola. Across the load sweep, the peak-to-peak LTE fluctuation is smaller, which lowers the excitation associated with time-varying mesh stiffness. At the working load, the first six LTE harmonic amplitudes are lower, which is the relevant signature for high-order mesh noise.
The manufacturing path is also practical. The hybrid curve is generated by three sets of basic roll ratio and higher-order modified coefficients, and these settings fit within the polynomial capacity of existing CNC spiral bevel gear grinders. The method therefore does not depend on a special machine tool, although it does require accurate setup and a contact-pattern check because the hybrid flank differs from the parabolic reference by up to 81 μm.
The evidence is numerical: TCA, LTCA, tooth-surface deviation, and Fourier analysis of a specific 23/65-tooth pair. It does not include acoustic measurements or durability testing. The benefit is also tied to the working-load range. Above 400 N·m the LTE shape changes for both designs, so a duty cycle that spends substantial time at higher torque would require re-optimization of the local synthesis parameters and the three-segment curve. Within the intended light-to-working-load range, the hybrid transmission-error design provides a lower high-order excitation while remaining compatible with modified-roll grinding of spiral bevel gears.