In an aero-engine central drive, spiral bevel gears carry power between intersecting shafts at high speed and under alternating load, and the mesh stiffness of the tooth pair is the parameter that couples the transmitted torque to the vibration the casing actually sees. Because the teeth are curved in space and the contact path runs obliquely across the flank, stiffness is not a single number: it varies with rotation, with the number of teeth simultaneously in contact, and with the elastic compliance of everything behind the tooth — the web, the shaft and the bearing support. Root cracks, which start in the fillet of the small end where the alternating bending stress concentrates, remove local material and therefore local stiffness, and the resulting asymmetry distorts the stiffness waveform that diagnostics rely on.
A finite-element study of an aero-engine central drive pair — 47 teeth on the driving member, 35 on the driven member, both cut from the carburized gear steel 16Cr3NiWMoVNbE — quantified these dependencies. The work addressed two questions that are easy to state and awkward to answer: how much of the measured compliance belongs to the tooth and how much to the support structure, and how large a root crack has to be before the mesh stiffness signature becomes usable as a damage indicator.
Meshing Stiffness as a Series Compliance
The stiffness of a single contacting tooth pair is defined through the linear elastic relation between the normal contact force and the elastic deformation at the contact, evaluated separately for the two members:
$$k_{i1}=\frac{F_{i1}}{u_{i1}},\qquad k_{i2}=\frac{F_{i2}}{u_{i2}}$$
where the subscript i identifies the tooth pair, 1 and 2 denote the driving and driven members, F is the normal contact force and u the comprehensive elastic deformation. For the pair in contact, Newton’s third law makes the two forces equal, so the pair stiffness is
$$k_i=\frac{F_{in}}{u_{i1}+u_{i2}}$$
which, on dividing and multiplying by the individual stiffnesses, reduces to the series combination
$$k_i=\frac{k_{i1}k_{i2}}{k_{i1}+k_{i2}}=\frac{1}{\dfrac{1}{k_{i1}}+\dfrac{1}{k_{i2}}}$$
Two consequences follow. First, the softer member dominates: a thin web or a flexible shaft on one side of the mesh pulls down the pair stiffness regardless of how stiff the tooth flanks are. Second, u must contain only deformation — Hertzian contact compression, bending and shear deflection along the tooth, and the displacement of the contact point caused by web, shaft and support compliance — and must exclude rigid-body rotation, which carries no stiffness information but which a naive finite-element setup will happily include.
When several pairs share the load, the contributions add in parallel:
$$k_m=\sum_{i=1}^{n}k_i$$
Eliminating Rigid-Body Displacement with an Equivalent Driving/Driven Model
The usual way to load a gear pair in a static solve — fix one shaft, apply torque to the other — lets the loaded member rotate freely as a rigid body. Any support deflection in that load path is then absorbed into the measured deformation, and the calculated stiffness is contaminated by an amount that depends on the support model rather than on the gear. The remedy adopted here is to split the calculation into two mirror-image models. For the driving-member stiffness, the inner surface of the driving shaft bore is fixed, the rotational degree of freedom of the driven shaft bore is released, and a torque T1 is applied; for the driven-member stiffness, the roles are reversed and T2 is applied. In each case the free member can rotate without contributing spurious elastic deflection to the fixed one.
The two models are only equivalent if the normal force at the contact is the same in both. Assuming the mesh force acts at the pitch point, force equality requires the applied torques to scale with the pitch radii, hence with the tooth numbers:
$$T_{47}=\frac{Z_{47}}{Z_{35}}\,T_{35}=\frac{47}{35}\,T_{35}$$
At the reference operating point the 35-tooth member carried 205.0 N·m, so 275.3 N·m was applied to the 47-tooth member. The same ratio converts the other load cases: 20.0 N·m becomes 26.9 N·m, and 1 000.0 N·m becomes 1 342.9 N·m. Pair stiffness is then obtained from the series formula and summed over the contacting pairs.
Finite Element Models: Intact, Cracked, and Reduced Tooth Count
Three families of model were built. The intact assembly used the full geometry of both members, generated from the design parameters, the actual cutter geometry and the flank modification. For the cracked cases, the mesh of the driven member was edited by deleting elements along the two propagation directions observed in earlier traveling-wave fatigue work: from the small end toward the web, through the shaft bore and lightening holes, and then turning to the adjacent tooth small end and running along the root to the large end; and directly along the tooth slot from the root small end to the large end. Four crack states were represented, defined by the two projections of the crack front:
- JD1 — 0.35 mm along the rim, 1.02 mm along the tooth slot (early stage).
- JD2 — 3.52 mm along the rim, 3.07 mm along the tooth slot (second stage).
- JD3f — 5.91 mm along the rim, 4.09 mm along the tooth slot, a total of 10.00 mm; the terminal state of the three-nodal-diameter forward traveling-wave fatigue test.
- JD4b — 10.12 mm along the rim, 8.18 mm along the tooth slot, a total of 18.30 mm; the terminal state of the four-nodal-diameter backward traveling-wave test.
For the stiffness sweeps, computational cost was cut by replacing the intact counterpart with a shaftless five-tooth sector: when the driving member was evaluated, its shaft was simplified and the driven member reduced to five teeth, and vice versa. This keeps five consecutive contacts inside the model, enough to capture single/double-tooth alternation, while removing the shaft length that carries no useful deformation information.
Modal Verification Against Hammer Testing
Model fidelity was checked on the driven member, whose thin web makes it the more flexible and more error-prone part. Five gears from the same production batch were suspended by elastic cord to approximate free boundary conditions, an accelerometer was placed on the rim at the large end, and a multi-point excitation, single-point response hammer test was performed. The simulation used material properties at 20 °C.
The nodal-diameter modes behave as the geometry dictates. For three nodal diameters, two limiting configurations exist — nodal lines that miss the lightening holes and nodal lines that pass through them — with slightly different axial bending stiffness, and the simulated frequencies differ accordingly (8 026.4 Hz and 8 053.7 Hz, averaged to 8 040.1 Hz for comparison). For four nodal diameters the four lines cannot form two distinct limiting configurations, so the two symmetric mode shapes are degenerate at 13 380.0 Hz.
| Mode | Simulation | Measured range (A1–A5) | Spread within batch | Relative error |
|---|---|---|---|---|
| Three nodal diameters | 8 040.1 | 7 882.0–7 964.0 | 82.0 | < 1.97% |
| Four nodal diameters | 13 380.0 | 13 310.0–13 420.0 | 110.0 | < 0.53% |
The batch scatter is itself informative: nominally identical gears differ by up to 82 Hz in the three-diameter mode and 110 Hz in the four-diameter mode, so an experimental validation that quotes a single specimen cannot claim better than a few tenths of a percent. Within that resolution the finite-element model is accurate, which justifies using it to separate the stiffness contributions that cannot be measured directly.
Static Contact Analysis at 205 N·m
Stiffness sweeps used a general static analysis with material data at 200 °C: elastic modulus 210 GPa and Poisson’s ratio 0.41 for the carburized steel, hexahedral first-order reduced-integration elements (C3D8R), and a step length of 0.10 s. Because C3D8R elements carry no rotational degrees of freedom, the tooth flanks, the tooth-to-web junction and the shaft bore were coupled to control points with MPC beam constraints; the five engaging flank pairs were defined as surface-to-surface contacts with a tangential friction coefficient of 0.15. Each load case ran in three steps: a small rotation to close the backlash and establish initial contact, removal of that imposed rotation with a torque applied about the axis of the free member, and then a sweep in which the driven member was indexed by 1° per model — 27 positions, covering the full meshing cycle — with the driving member rotated by the corresponding ratio. Torque was ramped linearly from zero to its full value within each step.
At a driven-member torque of 205.0 N·m and zero rotation angle, only one of the five candidate pairs carried load. Its resultant normal force stayed at zero until 0.04 s, rose as the flanks closed the gap, and reached its maximum at 0.20 s when the torque first attained its full value; the surface-averaged elastic deformation of the same flank followed the identical ramp. Dividing force by deformation at full load, and repeating the procedure in the mirror model, gave the two member stiffnesses and, through the series relation, the pair stiffness.
The assembled result reproduces the expected meshing behaviour. The comprehensive stiffness is periodic with 10.3°, the pitch of the 35-tooth member (360°/35). Double-tooth contact produces peaks of 2.08×108 N/m and 2.09×108 N/m at driven-member rotation angles of 8° and 18°, single-tooth contact gives roughly 1.80×108 N/m, and the cycle mean is 1.83×108 N/m. The transition between the two regimes appears as a sharp corner in the waveform rather than a smooth curve, which is the feature that most strongly excites the system.
Contact ratio was read directly from the contact duration. For the third pair, contact persisted from 6° to 20° of driven-member rotation, a duration of 14°, while successive pairs entered mesh 10° apart:
$$\varepsilon=\frac{T}{\Delta t}=\frac{14°}{10°}=1.40$$
The value slightly exceeds unity, so the drive alternates between one and two loaded pairs, and the 40% overshoot corresponds to the double-contact intervals visible as peaks in the stiffness curve.
Load Torque: Stiffening and Contact Ratio Growth
Repeating the sweep at 20.0 N·m and 1 000.0 N·m on the driven member shows a monotonic trend in both the mean level and the shape of the stiffness waveform.
| Torque on 35-tooth member (N·m) | Torque on 47-tooth member (N·m) | Mean stiffness (×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 |
Two mechanisms operate together. Higher torque flattens the contact ellipse and extends the loaded region along the flank, so more of the curved tooth shares the force and the effective stiffness rises; simultaneously the longer loaded path delays the exit of each pair, raising the contact ratio from 1.10 to 1.64. The mean stiffness increases by about 18% between the lightest and heaviest cases, which is a larger change than the crack-induced losses described below. Equally important for vibration, the corners of the waveform flatten as torque rises: the step between single- and double-contact branches shrinks, so the periodic excitation becomes smoother and the drive runs more steadily at high load. Any stiffness model calibrated at a single torque should therefore be treated as valid only near that torque.
Crack Length: Local Compliance and Global Stiffness Loss
Cracks act on the stiffness through asymmetry. Removing material at the fillet of one tooth lowers the local bending stiffness of that tooth, breaks the cyclic symmetry that makes nodal-diameter modes degenerate, and shows up as mode splitting in the modal response — which is why stiffness is the natural quantity to track when a crack is suspected. At the reference torque of 205.0 N·m the comprehensive stiffness waveform retains its period but drops in level as the crack lengthens, and the drop is concentrated in the portion of the cycle where the cracked tooth is loaded.
| Case | Crack projection along rim (mm) | Along tooth slot (mm) | Total crack length (mm) | Mean stiffness (×108 N/m) | Change vs intact |
|---|---|---|---|---|---|
| Intact | – | – | – | 1.83 | – |
| JD3f | 5.91 | 4.09 | 10.00 | 1.64 | −10.4% |
| JD4b | 10.12 | 8.18 | 18.30 | 1.61 | −12.1% |
Normalizing the mean stiffness by the intact value gives ratios of approximately 0.90 at 10.00 mm and 0.88 at 18.30 mm. The relation is strongly nonlinear: nearly doubling the crack length from 10.00 mm to 18.30 mm removes only about 1.7 additional percentage points of stiffness. Two factors limit the growth. The crack front advances along the root while the load path through the remaining ligament stays comparatively short, and the lost stiffness on the cracked tooth is partly compensated by the neighbouring pairs, which carry more of the total force once the damaged tooth deflects further. The engineering consequence is that stiffness loss saturates early in relative terms, and the last increment of crack growth before fracture produces only a small additional change in the mean stiffness — so a monitoring threshold set on mean stiffness alone will not distinguish a long crack from a very long one.
What the Numbers Imply for Diagnostics and Design
The magnitudes deserve emphasis. A root crack 10 mm long, in a gear whose tooth height is a few millimetres, reduces the mean comprehensive stiffness by about a tenth; a load change from light cruise to maximum torque raises it by nearly a fifth. A diagnostic indicator built on absolute stiffness, or on the raw amplitude of a mesh-frequency component, can therefore mistake a load change for damage and vice versa. What survives is the shape information: the crack suppresses the local peak associated with the damaged tooth while leaving the period intact, so an order-tracking scheme that normalizes by measured torque and inspects the angular position of the stiffness trough is considerably more robust than one that watches the mean.
From the design side, the load dependence is the more consequential finding. Because the contact ratio climbs from 1.10 to 1.64 across the torque range, the number of load-sharing teeth is itself a function of operating point, which makes the mesh behave as a stiffening spring at high load and a softening one at low load. Dynamic models that linearize the mesh at a nominal stiffness will misplace resonance speeds at part load, precisely the regime where a thin-web bevel gear is most likely to encounter a nodal-diameter excitation. The measured sensitivity of the web and support to the pair stiffness also argues for treating the housing and shaft train as part of the mesh model rather than as rigid boundaries, which is exactly the error the split driving/driven formulation was designed to avoid.
Scope and Limitations
The results apply to one geometry and one material: a 47/35-tooth pair in 16Cr3NiWMoVNbE, evaluated at a fixed 200 °C with linear elastic properties and a Coulomb friction coefficient of 0.15. The analysis is quasi-static — the mesh is stepped through 27 positions and inertia, gyroscopic effects and time-dependent contact are excluded — so the calculated stiffness is the input to a dynamic model, not a measurement of the running gear. Cracks are represented by element deletion along two prescribed fronts rather than by a propagating crack with a stress-intensity criterion, so the progression between states is geometric rather than physically simulated. Contact ratio is derived from contact duration in the static solution and does not account for elastic flattening beyond the modelled load step. Within these bounds the calculation is internally consistent and experimentally anchored at the modal level: the finite-element models reproduce measured natural frequencies to better than 2% in the three-nodal-diameter mode and better than 0.6% in the four-nodal-diameter mode, and the batch spread of the physical parts is of the same order as the modelling error.
The practical takeaway is a two-variable picture of spiral bevel gear mesh stiffness: torque sets the level and the smoothness, crack length sets a localized deficit that grows quickly at first and then slowly, and both must be known before a stiffness measurement can be interpreted as condition information.