Spiral bevel gears transmit torque between intersecting shafts in applications where smoothness, load capacity, and compact packaging matter, including helicopter drive trains, aeroengine accessories, vehicle axles, and heavy machinery. Their curved, oblique teeth enter mesh gradually, giving a high contact ratio and a more gradual load transfer than straight bevel gears. The same geometry makes each tooth a short cantilever rooted in a thin fillet. Under heavy or alternating torque, the root fillet is therefore a natural site for fatigue crack initiation. Once a crack forms, it changes the load path: the effective load-bearing section shrinks, mesh stiffness falls locally, and contact pressure and root bending stress redistribute. If the crack grows through the tooth thickness, the tooth can break and the pair loses part of its load-sharing capacity.
Why root cracks require a contact-mechanics description
A spiral bevel tooth does not engage along a simple line. The contact ellipse moves across a curved flank while the tooth rotates, so the loaded section changes continuously through the mesh cycle. A root crack introduces a second geometric discontinuity: it may lie behind the contact ellipse, intersect it, or pass beyond it. The resulting changes in mesh stiffness and stress are therefore not determined by crack depth alone. Crack type, crack location along the root, and the instantaneous contact position all matter.
A finite element contact study of a spiral bevel gear pair with controlled root cracks examined this coupling by comparing plane cracks, spatial cracks, and broken teeth under the same input torque. The analysis focused on three quantities: time-varying mesh stiffness, tooth-surface contact stress, and root bending stress.
Contact model and mesh stiffness definition
Tooth flanks were generated by tooth contact analysis (TCA) from the cutter-head geometry and machine-tool settings. The resulting surface nodes were imported into ANSYS, and the tooth body was built with hexahedral Solid185 elements. A six-tooth sector model was used for the pair. Because the contact ratio of the spiral bevel pair does not exceed 3, a six-tooth sector retains enough teeth to cover consecutive mesh cycles while keeping the far-field boundary away from the contact zone.
Six contact pairs were defined with Conta174 and Targe170 elements. Rigid reference points at the pinion and gear axes were coupled to the inner rings. Torque was applied at the pinion reference point, the gear reference point was fully constrained, and the pinion was allowed to rotate only about its axis. This arrangement represents a driven gear held in a steady loaded state while the pinion rotates under torque.
Mesh stiffness is defined as the normal force per unit normal deflection at the contact:
$$k=\frac{F}{E}$$
where \(F\) is the normal contact force derived from the applied torque and \(E\) is the total normal deformation obtained from the equivalent static rotation. The resulting stiffness curve varies with contact position and therefore with the mesh cycle.
The model was checked against loaded tooth contact analysis (LTCA) at pinion torques of 50, 100, 150, and 200 N·m. The stiffness curves followed the same variation, with a maximum difference of 4.66%. This agreement supports the use of the contact model for the cracked cases.
Crack geometries and finite element insertion
Two crack families were introduced at the root fillet, where bending stress concentrates. A plane crack was defined by its arc length along the face width \(W\) and its depth along the tooth thickness \(P\). Fifteen cases combined three widths (\(W/3\), \(2W/3\), \(W\)) with five depths (\(P/6\), \(P/3\), \(P/2\), \(2P/3\), \(5P/6\)). A spatial crack extended from the fillet toward the tooth tip at one of three angles \(\alpha_1\), \(\alpha_2\), or \(\alpha_3\), with depth \(S\). Fifteen spatial cases combined the three angles with depths from \(S/6\) to \(5S/6\). Three additional broken-tooth cases represented spatial cracks that had passed through the tooth thickness.
To insert a crack without remeshing the entire tooth, an element-node replacement method was used. Nodes were duplicated at the intended crack boundary, elements along the crack path were deleted, and new elements were generated from the duplicated nodes. This separated the originally connected tooth material and produced a traction-free crack surface. The cracked-case simulations applied a pinion torque of 120 N·m.
| Fault family | Geometric control | Severity levels | Cases | Mechanical interpretation |
|---|---|---|---|---|
| Plane crack | Width along face \(W\); depth along tooth thickness \(P\) | \(W/3\), \(2W/3\), \(W\); \(P/6\) to \(5P/6\) | 15 | Root crack reduces the load-bearing section in a plane normal to the tooth |
| Spatial crack | Inclination angles \(\alpha_1\), \(\alpha_2\), \(\alpha_3\); depth \(S\) | \(S/6\) to \(5S/6\) | 15 | Inclined crack splits the load path between inner and outer tooth-body regions |
| Broken tooth | Spatial crack through the tooth thickness | Three angle cases | 3 | Part of the tooth no longer carries contact load; load shifts to remaining teeth |
Mesh stiffness under crack fault
Stiffness response depended on where the contact ellipse was relative to the crack. For small cracks, the drop during early engagement was small. As the contact ellipse reached the cracked region, stiffness decreased progressively with crack depth. The maximum reductions were 27.58% for the plane crack, 14.12% for the spatial crack, and 32.82% for the broken tooth. After the cracked tooth left the contact and the next healthy tooth entered, stiffness recovered. The reduction was therefore confined to the cracked tooth’s engagement interval rather than appearing as a uniform loss across the whole revolution.
The plane and spatial cracks did not affect the same portion of the mesh cycle. Because the spatial crack trace and the moving contact ellipse intersect later and over a shorter arc, the spatial crack influenced a shorter interval of the cracked tooth’s mesh. In the tested cases, the plane crack produced the larger peak stiffness reduction, while the broken tooth produced the largest reduction and the strongest fluctuation. The broken tooth also changed the contact state: when the broken region was large enough to stop carrying load, the original two-tooth contact became single-tooth contact at that mesh position.
| Fault case | Maximum stiffness reduction | Observed mesh behavior |
|---|---|---|
| Plane crack, condition 15 | 27.58% | Reduction spread over a longer portion of the cracked tooth engagement |
| Spatial crack, condition 15 | 14.12% | Shorter affected interval; local load split across the crack line |
| Broken tooth, condition 18 | 32.82% | Sharp stiffness fluctuation and loss of contact on the broken region |
Contact and bending stress redistribution
Stress extraction focused on three positions in the cracked tooth’s mesh zone: position ① before the contact ellipse reached the crack, position ② on the crack line, and position ③ after the ellipse passed the crack. The response differed between plane and spatial cracks because the crack orientation controlled which part of the tooth carried the contact load.
For the plane crack, when the contact ellipse was away from the root crack, the tooth-surface contact stress was almost independent of crack depth. The root bending stress at the crack tip, however, rose with depth because the crack concentrated stress at the fillet. When the ellipse moved over the plane crack, the load-bearing section was reduced. Contact stress then decreased with increasing depth, while the crack-tip stress concentration at the root fillet increased.
For the spatial crack, before the ellipse reached the crack line, the crack lay outside the main load path, so neither contact stress nor root bending stress changed much. On the crack line, the contact ellipse straddled two tooth-body regions. The outer region lost stiffness as the crack deepened, so its contact stress fell. The inner region carried more load, and its contact stress rose. In condition 5, the inner-side contact stress was 15.26% higher than in the healthy case, and the root bending stress increased by 2.30%. After the ellipse passed the crack line, the loaded region lay outside the crack. Contact stress decreased with depth, reaching 8.91% below the healthy case in condition 5, and the root bending stress decreased by 47.11%.
For broken teeth, the missing region no longer carried contact load. Load shifted to the remaining teeth, and the mesh changed from double-tooth to single-tooth contact when the broken area was large enough. This produces a sharp stiffness fluctuation rather than a gradual local reduction.
| Fault type and contact position | Tooth-surface contact stress | Root bending stress |
|---|---|---|
| Plane crack, ellipse away from crack | Nearly unchanged | Crack-tip concentration rises with depth |
| Plane crack, ellipse over crack | Decreases with depth | Crack-tip concentration at root fillet rises |
| Spatial crack, before crack line | Negligible change | Negligible change |
| Spatial crack, on crack line | Outer side decreases; inner side increases by up to 15.26% | Slight increase, 2.30% in condition 5 |
| Spatial crack, after crack line | Decreases, 8.91% in condition 5 | Decreases, 47.11% in condition 5 |
| Broken tooth | Contact lost on broken portion; load shifts to remaining teeth | Single-tooth contact and stronger fluctuation |
Implications for fault diagnosis and design
The results show that a root crack is not a single scalar fault. Its type, depth, and position relative to the contact ellipse determine whether contact stress rises or falls and whether root bending stress concentrates or relaxes. A plane crack can produce a larger peak stiffness reduction over a longer mesh interval, while a spatial crack can split the contact load and raise stress on the inner side of the crack. Broken teeth produce the strongest stiffness variation and a clear change in contact ratio.
For vibration-based condition monitoring, these localized stiffness changes imply that the fault signature will be angularly localized to the cracked tooth’s engagement and will depend on load level and contact position. A healthy-baseline comparison over one pinion revolution is therefore more informative than a single global stiffness value. For design and inspection, the root fillet remains the critical region. Detecting a crack before it crosses the tooth thickness is important because the transition from a local stiffness reduction to broken-tooth behavior is abrupt.
Conclusions
1. A TCA-based finite element contact model with a six-tooth sector reproduced LTCA stiffness trends within 4.66% over pinion torques from 50 to 200 N·m, providing a basis for the cracked-case analysis.
2. When the contact ellipse moved into the crack region, mesh stiffness decreased with crack severity. The maximum reductions were 27.58% for the plane crack, 14.12% for the spatial crack, and 32.82% for the broken tooth.
3. Plane and spatial cracks affected different portions of the mesh cycle. The plane crack influenced a longer interval and produced a larger peak stiffness drop in the tested cases; the spatial crack affected a shorter interval but caused a local split in contact load and a stress increase on the inner side of the crack line.
4. Contact stress and root bending stress depended primarily on the instantaneous tooth rotation position and secondarily on the geometric relation between the contact ellipse and the crack line. Spatial crack condition 5 increased inner-side contact stress by 15.26% and root bending stress by 2.30% on the crack line, while after the crack line it reduced contact stress by 8.91% and root bending stress by 47.11%. Broken teeth transferred load to remaining teeth and changed double-tooth contact into single-tooth contact.