Helicopter main gearboxes must transfer engine power to the rotor across a large speed reduction while remaining light, reliable and quiet. The spiral bevel gear stage is a key part of that chain: it transmits torque between intersecting shafts, and its curved teeth engage gradually across the face width. In the civil helicopter main gearbox considered here, the pinion and wheel do not run on perpendicular axes. The shaft angle is 95.5°, so the pair is non-orthogonal. The mesh is also nonlinear in service: designed backlash prevents tooth seizure and direct flank contact, while the number of loaded tooth pairs changes as the gears rotate. The coupling of alternating engagement with backlash, time-varying mesh stiffness and transmission error turns the gear pair into a source of vibration that affects gearbox life and rotor-system behavior.
For civil helicopters, safety, operating cost and passenger comfort are primary design goals. Reducing gearbox vibration at the tooth surface is attractive because it acts on the mesh itself rather than adding mass or external damping. Spiral bevel gears are not fully conjugate, so the tooth surface near the reference point can be deliberately modified. Local synthesis describes that modification with three second-order parameters: the angle between the contact trace on the driven gear and the tooth-height direction, the first derivative of the transmission ratio, and the ratio of the contact-ellipse semi-major axis to the face width. These parameters shape the contact path, the local motion error and the contact-ellipse size, and therefore the mesh stiffness and dynamic response.
Mesh Geometry and Design Variables
The gear pair has 11 pinion teeth and 49 wheel teeth. The module is 6.285 mm, the pressure angle is 20°, the mid-face spiral angle is 30°, and the face width is 46.584 mm. The pinion is right-hand and the wheel is left-hand. The shaft angle is 95.5°, which makes the axes non-perpendicular. The transmission input is 106.647 kW at a pinion speed of 2097 r/min.
| Parameter | Value |
|---|---|
| Number of teeth | 11 pinion / 49 wheel |
| Module | 6.285 mm |
| Pressure angle | 20° |
| Spiral angle | 30° |
| Face width | 46.584 mm |
| Shaft angle | 95.5° |
| Hand | Right pinion / left wheel |
| Input power | 106.647 kW |
| Pinion speed | 2097 r/min |
In the local-synthesis formulation, the tooth surface is controlled near a reference point by three parameters. The angle η is measured on the driven wheel between the contact trace and the tooth-height direction. The derivative m′pg describes the local slope of the transmission ratio; on the working flank it is negative. The dimensionless ratio KD is the semi-major axis of the contact ellipse divided by the face width. The first two parameters influence the direction and motion character of the contact path, while KD controls the contact-ellipse footprint and, through the contact pattern, the contact ratio.
Eight-Degree-of-Freedom Dynamic Model
The gear pair is represented by a lumped-parameter model. The pinion and wheel are treated as rigid disks, and the shaft segments and rolling bearings are replaced by equivalent stiffness and damping acting at the mid-face. Tooth friction is neglected. The coordinates are three translations and one rotation for each gear: Xp, Yp, Zp, θp for the pinion and Xg, Yg, Zg, θg for the wheel, giving eight degrees of freedom. The two local frames share an origin at the axis intersection, with their X axes along the respective gear axes and the angle between them equal to the shaft angle.
The normal mesh force at the mid-face is:
$$
F_n=c_m\dot{\lambda}_n+k_h(t)f(\lambda_n)
$$
where cm is mesh damping, kh(t) is time-varying mesh stiffness, λn is the normal relative displacement, and f(λn) is the backlash function. The force is resolved into the local x, y and z directions through direction cosines that depend on the normal pressure angle, the mid-face spiral angle and the pitch-cone angles of the two gears. The backlash function has a dead zone:
$$
f(\lambda_n)=
\begin{cases}
\lambda_n-b_n, & \lambda_n>b_n,\\
0, & |\lambda_n|\le b_n,\\
\lambda_n+b_n, & \lambda_n<-b_n,
\end{cases}
$$
where bn is half the normal backlash. This piecewise form allows the mesh force to vanish inside the clearance and to act only when the flanks are in contact.
The normal relative displacement combines the translational motions of the two gears along the mesh line with the torsional contributions and the unloaded transmission error:
$$
\lambda_n=-a_1X_p-a_2Y_p-a_3Z_p-a_4X_g-a_5Y_g-a_6Z_g+a_3r_p\theta_p+a_6r_g\theta_g-e_n(t)
$$
Here a1–a6 are the direction cosines of the normal force in the two local frames, rp and rg are the mesh-point radii, and en(t) is the static transmission error. The latter is represented as a harmonic series at the mesh frequency and its harmonics, so the model is excited by both the time-varying stiffness and the transmission error.
The translational equations balance inertia, support damping, support stiffness and the corresponding force component for each coordinate. The rotational equations include the input torque on the pinion and the load torque on the wheel. Because the two rotations contain a rigid-body component, the system is recast using λn; the resulting coupled equations retain the six translations and the normal relative displacement. The equations are non-dimensionalized before integration because the physical quantities differ by several orders of magnitude. A Runge-Kutta method solves the non-dimensionalized system, and the initial transient is discarded so that the steady-state dynamic response can be evaluated.
Optimization Workflow
The optimization objective is the root-mean-square (RMS) vibration displacement of the transmission system. The three tooth-surface parameters are the design variables, with the search limited to the working flank:
$$
\min_{\eta,\,m’_{pg},\,K_D} V(\eta,m’_{pg},K_D)
$$
$$
\text{subject to}\quad 0^\circ<\eta<90^\circ,\quad -0.05<m'_{pg}<0,\quad $$="" 0.15<k_d<0.25=""
where V is the RMS vibration displacement obtained from the steady-state solution. The macro geometry and operating point are fixed; only the second-order tooth-surface parameters are varied.
The three parameters do not enter the dynamic equations directly. Their effect is carried through the contact geometry and mesh stiffness. Tooth contact analysis converts a candidate tooth surface into contact-pattern and contact-ratio data. A three-dimensional finite element model in ANSYS supplies discrete mesh-stiffness values, which are fitted by a Fourier series to form kh(t). The contact ratio, single-pair mesh stiffness and combined mesh stiffness therefore provide the link between tooth-surface micro-geometry and the dynamic excitation. For each candidate set, the dynamic equations are solved and the RMS response is returned to the genetic algorithm as the fitness value.
The genetic algorithm uses binary encoding, single-point crossover and a mutation probability of 0.07. The population size is 10 and the run is limited to 10 iterations. The crossover probability is 0.8. The parameter ranges and algorithm settings are summarized in Table 2.
| Item | Setting |
|---|---|
| Search range of η | 0°–90° |
| Search range of m′pg | −0.05 to 0 |
| Search range of KD | 0.15–0.25 |
| Population size | 10 |
| Iterations | 10 |
| Crossover probability | 0.8 |
| Mutation probability | 0.07 |
| Encoding | Binary |
| Crossover | Single-point |
Optimized Tooth Surface and Vibration Response
The population-mean RMS changed little after about eight iterations and the run was treated as converged. Table 3 compares the first and final iterations. The first iteration used η = 1.7°, m′pg = −0.0175 and KD = 0.1738. The final design used η = 71°, m′pg = −0.0195 and KD = 0.1512. All values in Table 3 lie inside the prescribed ranges.
| Quantity | First iteration | Final iteration | Change |
|---|---|---|---|
| η | 1.7° | 71° | — |
| m′pg | −0.0175 | −0.0195 | — |
| KD | 0.1738 | 0.1512 | — |
| RMS vibration displacement | 153.86 μm | 139.35 μm | −9.43% |
The RMS vibration displacement fell from 153.86 μm to 139.35 μm:
$$
\frac{153.86-139.35}{153.86}\times100\%=9.43\%
$$
This is a 9.43% reduction for the modeled pair at the specified operating point. The steady-state displacement–time traces for the first and final iterations show the same trend: the optimized tooth surface produces a lower vibration envelope after the initial transient has decayed.
Design Implications and Scope
The result shows that the three local-synthesis parameters can be used as system-level vibration design variables, not only as contact-pattern descriptors. In this case, the optimized set combines a much larger contact-trace angle on the driven wheel (71° instead of 1.7°), a slightly more negative transmission-ratio derivative (−0.0195 instead of −0.0175) and a smaller contact-ellipse ratio (0.1512 instead of 0.1738). Within the model, the parameter set acts on the RMS response through the contact ratio and the mesh-stiffness waveform; the optimization finds a combination that lowers that response. The 9.43% reduction applies to the parameter set as a whole.
The optimum is specific to the 11/49-tooth, 95.5° shaft-angle pair, the 106.647 kW input and the 2097 r/min pinion speed. It is also conditional on the chosen search ranges: KD lies close to its lower bound of 0.15, so the optimum is bounded by that interval. The 9.43% reduction is a model-based estimate from the lumped-parameter dynamic solution at the specified operating point. Practical implementation would need to account for manufacturing scatter, load sharing and other speed and torque conditions, with test confirmation of the predicted response. Within those limits, the workflow provides a practical way to screen spiral bevel gear tooth surfaces for vibration before detailed hardware testing.
</m'_{pg}