Spiral bevel gears transmit torque between intersecting shafts in a compact envelope, which is why they are used in aero-engine central drives. Their curved tooth trace produces a contact path that moves across the tooth as the gears rotate, and thin webs are used to save weight. Both features make mesh stiffness a three-dimensional contact and body-compliance problem rather than a simple two-dimensional tooth-bending problem. The time-varying meshing stiffness (TVMS) is the ratio of normal contact force to the elastic deformation of the contacting flanks. It sets the parametric excitation that drives gearbox vibration and also controls load sharing, root stress, and noise. At high speed and alternating load, a root crack changes local compliance; at the same time, load torque changes the contact patch and the number of teeth in contact. A useful calculation therefore has to separate tooth and body compliance from rigid-body motion and support deflection, and it has to resolve both load and crack length.
Potential-energy and analytical slice models are efficient, but they rely on section approximations and are difficult to apply to a thin-web bevel gear with a moving contact path. Loaded tooth contact analysis captures the contact pattern but needs a stiffness model for the gear body. Finite-element contact analysis represents the blank, web, and contact directly, but the result is sensitive to how the hub and supports are constrained. The calculation described here uses the finite-element route with a separate single-gear model for each member of the pair.
Separating tooth compliance from rigid-body motion
For the i-th contacting tooth pair, TVMS is written as the normal contact force divided by the normal elastic deformation of the pair:
$$k_i=\frac{F_i}{u_i}$$
For each gear member, the same ratio can be defined from its own loaded-flank displacement. The pair stiffness is the series combination of the two single-gear stiffnesses:
$$k_i=\left(\frac{1}{k_{i1}}+\frac{1}{k_{i2}}\right)^{-1}$$
When several pairs contact simultaneously, their stiffnesses act in parallel and the total stiffness is the sum over the contacting pairs:
$$k_{\mathrm{m}}=\sum_{i=1}^{n}k_i$$
In a two-gear finite-element model, the relative displacement at the mesh includes the rigid-body rotation of the driven gear and the deflection of both hubs and supports. If that relative displacement is used as the elastic deformation, the calculated stiffness is not a tooth-property. The equivalent model avoids this by computing each gear separately. For the 47-tooth gear, its hub inner surface is fixed and the 35-tooth gear is free to rotate under torque. For the 35-tooth gear, the reverse is done. The free gear’s rigid-body rotation is allowed and is not part of the fixed gear’s compliance. The shaft is omitted in the sector models, and the hub inner surface is fixed, so shaft bending and support compliance do not enter the tooth and body compliance.
To make the two single-gear calculations mechanically equivalent, the normal contact force must be the same in both. The normal force is the torque divided by the pitch radius. For a standard pair the pitch radii scale with the tooth numbers, so equal normal force requires
$$T_{47}=\frac{47}{35}T_{35}$$
Thus a torque of 205.0 N·m on the 35-tooth gear corresponds to 275.3 N·m on the 47-tooth gear. The same scaling gives 26.9 N·m and 1 342.9 N·m for the low- and high-torque cases.
Gear pair, finite-element models, and crack states
The pair is an aero-engine central-drive spiral bevel set with 47 driving teeth and 35 driven teeth. The material is 16Cr3NiWMoVNbE carburized gear steel. A full three-dimensional model was built from the design tooth count, cutter geometry, and flank modification. For the contact calculation, the shaft was removed and the non-analyzed gear was reduced to a five-tooth sector. This keeps five possible contacting teeth while reducing the model size. In the driving-gear calculation, the driven gear is the five-tooth sector; in the driven-gear calculation, the driving gear is the sector.
Cracks were introduced in the 35-tooth driven gear by removing elements along the measured crack paths from the maximum-stress location at the small-end convex fillet. Two paths are represented: one turns from the small end toward the web and lightening holes, and the other runs along the tooth groove from the small end to the large end. The four modeled states are summarized in Table 1. JD1 and JD2 are early-stage models; JD3f and JD4b are the final modeled states from three-nodal-diameter forward and four-nodal-diameter backward traveling-wave fatigue tests, respectively.
| Crack state | Along rim (mm) | Along tooth groove (mm) | Total crack length (mm) | Modeled condition |
|---|---|---|---|---|
| JD1 | 0.35 | 1.02 | 1.37 | Early-stage model |
| JD2 | 3.52 | 3.07 | 6.59 | Stage-2 model |
| JD3f | 5.91 | 4.09 | 10.00 | Final state, three-nodal-diameter forward traveling-wave test |
| JD4b | 10.12 | 8.18 | 18.30 | Final state, four-nodal-diameter backward traveling-wave test |
The total crack length in the comparison is the sum of the along-rim and along-groove components. Static contact used 200 °C properties: elastic modulus 210 GPa and Poisson ratio 0.41. The mesh used C3D8R reduced-integration hexahedral elements, surface-to-surface contact with a friction coefficient of 0.15, and multipoint constraint beam couplings from the loaded flanks and web to a rotation control point. The hub inner surface was fixed to a separate control point. The analysis had three steps: close the backlash with a small rotation; release that small rotation from the free gear and apply the torque; then rotate the 35-tooth gear in 1° increments, with the 47-tooth gear following the tooth ratio, for 27 rotational positions. The torque ramps linearly from zero to its final value, and each analysis step has a time span of 0.10 s.
Modal check of the structural model
Five driven gears from the same production batch were suspended by elastic cords to approximate free-free conditions. An accelerometer at the large-end rim recorded the response to hammer impacts at multiple points. The finite-element free-modal solution used 20 °C properties. Table 2 compares the simulated frequencies with the measurements.
| Mode family | FE (Hz) | A1 (Hz) | A2 (Hz) | A3 (Hz) | A4 (Hz) | A5 (Hz) |
|---|---|---|---|---|---|---|
| Three nodal diameters | 8 040.1 | 7 882.0 | 7 932.0 | 7 940.0 | 7 964.0 | 7 908.0 |
| Four nodal diameters | 13 380.0 | 13 322.0 | 13 380.0 | 13 390.0 | 13 420.0 | 13 310.0 |
The three-nodal-diameter simulation gives two split modes at 8 026.4 Hz and 8 053.7 Hz; the table uses their average, 8 040.1 Hz. The split occurs because the three nodal diameters can either avoid or pass through the lightening holes, which changes the axial bending stiffness. At four nodal diameters, the two symmetric shapes have the same frequency, 13 380.0 Hz. The measured ranges are 7 882.0-7 964.0 Hz and 13 310.0-13 420.0 Hz. The simulated values are within about 2% for the three-nodal-diameter family and within 0.53% for the four-nodal-diameter family. The batch itself shows spreads of 82.0 Hz and 110.0 Hz, so the correlation is at the level of the batch scatter. This supports the geometry, mesh, and material representation used in the contact models.
Quasi-static contact calculation and reference stiffness
The reference load is 205.0 N·m on the 35-tooth gear, corresponding to 275.3 N·m on the 47-tooth gear. This reference corresponds to the three-nodal-diameter forward traveling-wave load used for the crack models. In the five-tooth sector, tooth pairs are numbered 1 to 5. At the initial position only pair 2 carries load. The normal force stays near zero while the initial clearance is closed, then rises with torque and reaches its maximum at the end of the load step. The total deformation of all nodes on the loaded flank of the driven tooth is averaged. Because the driven hub is fixed in that calculation, this average is the driven-gear elastic deformation. The same procedure gives the driving-gear deformation. Dividing the normal force at full load by each deformation gives the single-gear stiffness, and the pair stiffness follows from the series relation.
At 205.0 N·m, the total TVMS is periodic with the 10.3° pitch of the 35-tooth gear. Two-pair contact occurs near 8° and 18°, with maxima of 2.08 × 108 N/m and 2.09 × 108 N/m. One-pair contact is about 1.80 × 108 N/m. The mean over the cycle is 1.83 × 108 N/m. For tooth pair 3, contact lasts 14° of driven-gear rotation and adjacent pairs are about 10° apart, so the contact ratio is
$$\varepsilon_\alpha=\frac{T}{\Delta t}=\frac{14^\circ}{10^\circ}=1.40$$
This is a contact-duration ratio extracted from the finite-element contact history, not a geometric transverse contact ratio.
Effect of load torque
Table 3 lists the mean stiffness and contact ratio at 20.0 N·m, 205.0 N·m, and 1 000.0 N·m on the 35-tooth gear. The mean total stiffness rises from 1.63 × 108 N/m to 1.83 × 108 N/m and then to 1.93 × 108 N/m. The contact ratio rises from 1.10 to 1.40 and then to 1.64.
| Torque on 35-tooth gear (N·m) | Equivalent torque on 47-tooth gear (N·m) | Mean total TVMS (108 N/m) | Contact ratio |
|---|---|---|---|
| 20.0 | 26.9 | 1.63 | 1.10 |
| 205.0 | 275.3 | 1.83 | 1.40 |
| 1 000.0 | 1 342.9 | 1.93 | 1.64 |
The gain is not proportional. A 50-fold torque increase from 20.0 N·m to 1 000.0 N·m raises the mean stiffness by about 18%, while the contact ratio rises by about 49%. The turning points in the stiffness curve also become less pronounced at high torque. Higher torque enlarges the contact patch and extends contact over a larger part of the pitch, so more of the transition is shared by two pairs. The result is a smoother stiffness curve and a higher mean. At low torque, contact is more concentrated and the stiffness is more sensitive to the instantaneous contact state.
Effect of root-crack length
Table 4 compares the undamaged gear with JD3f and JD4b at 205.0 N·m. The mean total stiffness falls from 1.83 × 108 N/m to 1.64 × 108 N/m and 1.61 × 108 N/m, reductions of 10.4% and 12.1%.
| Crack state | Total crack length (mm) | Mean total TVMS (108 N/m) | Reduction from undamaged (%) |
|---|---|---|---|
| Undamaged | 0 | 1.83 | 0 |
| JD3f | 10.00 | 1.64 | 10.4 |
| JD4b | 18.30 | 1.61 | 12.1 |
The local stiffness of the cracked tooth is lower. The total is an average over the mesh cycle, so the reduction is smaller than the local loss. The trend is monotonic but sublinear between the two crack lengths: an 8.3 mm increase in total crack length adds only 1.7 percentage points of mean stiffness loss. This is consistent with the total compliance being shared among the contact, the uncracked portion of the root, and the gear body. A mean-stiffness threshold alone would therefore have limited resolution for crack length in this range.
Implications for design and condition monitoring
Three practical points follow. First, TVMS should be specified with its load. A dynamic model that uses a single stiffness at all torques will misplace the mean stiffness and the contact-ratio transitions. Second, the contact-ratio change from 1.10 to 1.64 is large enough to change the harmonic content of the mesh force; low-torque operation is not simply a scaled version of high-torque operation. Third, root cracks reduce the mean stiffness, but the average change is modest. Condition monitoring should normalize the stiffness estimate by torque and examine the angular variation around the mesh cycle, not only a scalar mean.
The equivalent single-gear model is most useful where support compliance is comparable to tooth compliance, as in thin-web bevel gears. It requires the pitch-radius ratio, which for a standard pair follows the tooth-number ratio. The model does not require the two gears to have the same support stiffness, because each is calculated with its own hub boundary.
Scope and limitations
The numbers above apply to the modeled 47/35 pair, 16Cr3NiWMoVNbE steel, 200 °C elastic properties, the 20.0-1 000.0 N·m torque range, and the four idealized crack states. The contact solution is quasi-static; it does not resolve impact, lubricant film dynamics, or thermal gradients. Cracks are represented by element removal, which gives a traction-free gap rather than a sharp crack tip. The modal check was at 20 °C and used the full gear, while the stiffness calculation used the no-shaft sector at 200 °C. These choices are appropriate for comparing load and crack effects within one geometry, but absolute stiffness values should be revalidated before transfer to another tooth count, web design, or support. Within that scope, the calculations show a consistent trend: total TVMS and contact ratio increase with load torque, and a longer root crack lowers total TVMS.