Spiral bevel gears transmit torque between intersecting axes through curved teeth that provide gradual contact, high load capacity and smooth operation. Their contact pattern is sensitive to support deflection, and the mesh is not a rigid constraint: backlash, time-varying mesh stiffness, static transmission error and tooth-surface friction act together. Friction is distinctive among these excitations because its magnitude scales with the normal load, its direction reverses as contact passes the pitch point, and it applies both a force and a moment to each gear. The lubrication state therefore changes not only efficiency but also the internal excitation of the transmission.
The analysis here examines a non-orthogonal spiral bevel gear pair from a helicopter main gearbox: 29 and 76 teeth, 3.75 mm module, 20° pressure angle, 30° spiral angle, 40 mm face width and a 75° shaft angle, transmitting 1,200 kW at 20,000 r/min. The resulting mesh frequency is approximately 9.67 kHz, so the dynamic response is relevant to gear whine and high-cycle tooth loading, not only to quasi-static strength. The central question is how strongly the steady-state vibration and dynamic mesh force depend on the friction coefficient and on the contact ratio.
Eight-degree-of-freedom model and mesh excitation
The transmission is represented as two rigid disks on elastic supports. The origin is at the intersection of the two axes. The driving gear defines the reference frame, and the driven gear frame is rotated by the shaft angle of 75°. Each gear has three translational coordinates and one rotation, giving eight degrees of freedom. Tilt and wobble are neglected, and the shaft and bearing compliances are lumped at the mid-face. The mesh point is also taken at the mid-face, where the normal force, friction force and friction moment are applied.
The normal mesh force is written as
$$F_n = c_m \dot{\lambda}_n + k_h(t) f(\lambda_n)$$
where cm is the mesh damping, kh(t) is the time-varying mesh stiffness, λn is the normal relative displacement and f(λn) is the backlash function. With bn denoting half the normal backlash,
$$f(\lambda_n)=\left\{\begin{matrix}\lambda_n-b_n,\ \lambda_n \gt b_n\\ 0,\ |\lambda_n|\le b_n\\ \lambda_n+b_n,\ \lambda_n \lt -b_n\end{matrix}\right.$$
The force is projected onto the six translational coordinates through direction cosines that depend on the normal pressure angle, the mid-face spiral angle and the pitch cone angles. The two torsional coordinates are not independent because the gear pair can rotate as a rigid body. To remove that rigid-body mode, the normal relative displacement is used as a generalized coordinate:
$$\lambda_n=-a_1X_p-a_2Y_p-a_3Z_p-a_4X_g-a_5Y_g-a_6Z_g+a_3r_p\theta_p+a_6r_g\theta_g-e_n(t)$$
Here rp and rg are the mesh radii, θp and θg are the torsional displacements, and en(t) is the static transmission error. The sign convention is that a displacement projection along the positive normal is negative. The transmission error is represented by a mean plus harmonics:
$$e_n(t)=e_0+\sum_{i=1}^{\infty}e_i\cos(i\omega_e t+\phi_i)$$
In the case considered, the mean is zero, the first harmonic dominates and the phase is zero. The input and load torques are likewise split into mean and alternating components, so torque ripple is included alongside mesh stiffness and transmission error.
Friction is applied at the same mesh point. The friction force is proportional to the normal force, and the friction moment is the force times the moment arm:
$$\mathbf{F}_{fi}=n_{fi}F_f,\qquad \mathbf{M}_{fi}=r_{fi}F_f,\qquad F_f=\mu F_n$$
The direction cosines nfi and radii rfi are obtained from the tooth-surface geometry and the sliding velocity at the mid-face. Friction on the driving gear opposes the relative sliding; friction on the driven gear acts with the relative sliding. Because the sliding velocity reverses at the pitch point, the friction excitation changes sign within each mesh cycle, producing an internal excitation rather than a simple additional damping term.
Mesh stiffness and lubrication-dependent friction
The mesh stiffness is not constant. A three-tooth loaded tooth contact analysis provides the torsional stiffness and the mesh radius over a mesh cycle; the result is converted to normal mesh stiffness and fitted by a Fourier series. Three harmonics are retained to include sub- and super-harmonic components:
$$k_h(t)=k_m+\sum_{i=1}^{N}\left[k_{ai}\cos(i\omega_e t+\phi_{ki})+k_{bi}\sin(i\omega_e t+\phi_{ki})\right]$$
where km is the mean stiffness and kai, kbi are harmonic amplitudes. The mesh frequency follows from the input speed and tooth count, ωe = n1z1/60.
Friction is treated through a mixed-lubrication model. The regime is set by the film thickness ratio Λ = hmin/σ, where hmin is the minimum film thickness and σ is the composite surface roughness. Λ > 2 corresponds to full-film lubrication, Λ < 0.2 to boundary lubrication, and 0.2 ≤ Λ ≤ 2 to mixed lubrication. The mixed friction coefficient combines full-film and boundary values:
$$\mu_m=f^{1.2}\mu_{ml}+(1-f)\mu_{bl}$$
$$f=\frac{1.21\Lambda^{0.64}}{1+0.37\Lambda^{1.26}},\qquad \Lambda=\frac{h_{\min}}{\sigma}$$
Here μbl is the boundary friction coefficient, typically 0.1–0.2, and μml is the full-film coefficient:
$$\mu_{ml}=0.0607\left(\frac{q_0}{v_s}\right)^{0.2}\eta^{-0.05}\left(\frac{\sigma}{d_1}\right)^{0.25}$$
In this expression q0 is the normal load per unit face width, vs is the sliding speed, η is the lubricant dynamic viscosity, d1 is the driving gear pitch diameter and R is the composite curvature radius. Under starved conditions, the load is shared between the elastohydrodynamic film and asperity contacts:
$$\mu_f=\mu_{ml}k_c+\mu_{bl}k_b,\qquad k_c+k_b=1$$
Because the load per unit width and the sliding speed vary along the contact path, the friction coefficient also varies within a mesh cycle. The values used to characterize the three lubrication states are therefore representative levels rather than constants across the entire contact.
| Parameter | Driving gear | Driven gear |
|---|---|---|
| Number of teeth | 29 | 76 |
| Module (mm) | 3.75 | 3.75 |
| Pressure angle (°) | 20 | 20 |
| Spiral angle (°) | 30 | 30 |
| Face width (mm) | 40 | 40 |
| Shaft angle (°) | 75 | |
| Hand | Left | Right |
Baseline response
The eight equations are non-dimensionalised and integrated with a variable-step Runge–Kutta method. The first few hundred mesh cycles are discarded to remove the start-up transient. In the normal mixed-lubrication case, the normal relative displacement oscillates between 80.0458 and 85.4185 μm. The phase portrait is a closed periodic orbit, the frequency spectrum is dominated by the mesh frequency and a few harmonics, and the dynamic mesh force remains positive, so contact is single-sided without reverse impact. This is the reference state for the parameter studies.
Friction coefficient: normal, starved and dry running
Three lubrication states are compared: normal mixed lubrication with μ = 0.0244, starved lubrication with μ = 0.050 and dry running with μ = 0.52. The displacement bands are 80.0458–85.4185 μm, 80.0461–85.4210 μm and 80.0457–85.4722 μm, respectively. The root-mean-square (RMS) values are collected in Table 2.
| Lubrication state | Friction coefficient, μ | RMS displacement (μm) | RMS normal mesh force (N) |
|---|---|---|---|
| Normal mixed | 0.0244 | 82.937 | 15,070 |
| Starved | 0.050 | 82.939 | 15,071 |
| Dry | 0.52 | 82.977 | 15,092 |
The trend is monotonic but weak. A twenty-one-fold increase in the friction coefficient, from 0.0244 to 0.52, raises the RMS displacement from 82.937 to 82.977 μm, less than 0.05%, and the RMS normal mesh force from 15,070 to 15,092 N, about 0.15%. The peak-to-peak displacement band widens from 5.373 to 5.427 μm, about 1.0%.
The friction force itself rises in proportion to μ, so friction power loss and heat generation are affected far more strongly than the global vibration. In this model the normal mesh force is set mainly by the transmitted torque and the mesh stiffness; the friction term enters through direction cosines and moment arms, and its sign reversal at the pitch point reduces its net contribution to the line-of-action response. The practical consequence is that global vibration RMS is a poor discriminator of lubrication state for this gear pair, whereas friction torque, temperature or efficiency would respond directly.
Contact ratio: contact trace angle and load sharing
The contact ratio is changed without altering the gear macro geometry. The contact trace on the driven gear tooth surface is rotated relative to the root cone by 15°, 45° and 75° through active tooth-surface design. These angles correspond to design contact ratios of 2.125, 1.417 and 1.250. The loaded tooth contact analysis shows that the mesh stiffness decreases as the angle increases and the contact ratio falls. Fewer tooth pairs share the load, so the mean stiffness drops and the load per contacting tooth rises.
The dynamic response changes more strongly than in the friction study. The displacement bands are 80.0112–87.2067 μm at 15°, 86.5481–91.8493 μm at 45° and 88.7469–95.3988 μm at 75°. The RMS values and normal mesh forces are given in Table 3.
| Contact trace angle (°) | Contact ratio, ε | RMS displacement (μm) | RMS normal mesh force (N) |
|---|---|---|---|
| 15 | 2.125 | 83.612 | 14,735 |
| 45 | 1.417 | 89.263 | 14,829 |
| 75 | 1.250 | 92.085 | 14,865 |
As the contact ratio falls from 2.125 to 1.250, a reduction of about 41%, the RMS displacement rises from 83.612 to 92.085 μm, an increase of about 10.1%, and the RMS normal mesh force rises from 14,735 to 14,865 N, about 0.9%. The peak-to-peak spread is not monotonic: 7.20 μm at 15°, 5.30 μm at 45° and 6.65 μm at 75°. The dominant effect is an upward shift of the mean level and an increase in the upper envelope as the contact ratio decreases.
The comparison with the friction study is instructive. A twenty-one-fold change in friction coefficient moved the RMS displacement by less than 0.1%, while a 41% reduction in contact ratio moved it by about 10%. For this transmission, contact ratio is a first-order design lever for vibration, and friction is a second-order lever for global vibration but a first-order lever for heat and efficiency.
Implications for design and monitoring
For a high-speed helicopter main gearbox, the results point to a clear hierarchy. The mesh stiffness variation associated with contact ratio, together with the backlash nonlinearity, sets the global vibration level. Friction shifts the operating point and adds a high-frequency internal excitation, but its effect on RMS displacement and mesh force is small in the cases examined. Designs that maintain a high contact ratio, for example by keeping the contact trace angle low relative to the root cone, would be expected to reduce the dynamic displacement and the normal load per tooth. The trade-off is that the contact pattern, alignment sensitivity and manufacturability must be checked; the present model does not include misalignment or thermal distortion.
Lubrication still matters. A dry-running friction coefficient of 0.52 is more than twenty times the normal mixed-lubrication value, and the friction force scales directly with it. The resulting heat, power loss and surface distress can be severe even when the global vibration signature changes little. Monitoring should therefore use friction-sensitive quantities such as oil temperature, friction torque or efficiency rather than relying on vibration RMS alone.
The model is a lumped-parameter sensitivity analysis: the gears are rigid disks, tilt and wobble are neglected, the supports are lumped at the mid-face, and the friction coefficient is supplied by mixed-lubrication relations rather than updated from a coupled thermal solution. Within that scope, the conclusions are consistent: increasing lubrication severity raises the friction coefficient and slightly raises vibration and mesh force; reducing contact ratio raises vibration and mesh force more strongly.
Key findings
- Friction coefficient is a weak global-vibration parameter in this case: a twenty-one-fold rise from 0.0244 to 0.52 changed RMS displacement by less than 0.05% and RMS normal mesh force by about 0.15%.
- Contact ratio is a strong parameter: reducing it from 2.125 to 1.250 raised RMS displacement by about 10.1% and RMS normal mesh force by about 0.9%.
- The baseline response is periodic with single-sided contact, and the frequency content is concentrated near the mesh frequency and its first harmonics.
- Lubrication state strongly affects friction force, heat and efficiency even when vibration RMS is insensitive, so condition monitoring should not rely on vibration RMS alone.