A spiral bevel gear pair transmits torque between intersecting shafts through curved teeth that engage gradually along an elliptical contact patch. This load path makes the gear efficient and quiet, but it also concentrates cyclic bending stress at the tooth-root fillet. Under heavy or alternating load, a root crack can initiate, alter the local compliance of the tooth, redistribute contact load, and eventually lead to tooth fracture. The important technical question is not simply whether a crack exists, but how its geometry interacts with the moving contact ellipse. A crack that remains outside the loaded region may have little immediate effect; the same crack can produce a sharp stiffness trough once the contact patch crosses its trace.
For a spiral bevel gear, mesh stiffness and tooth stress are therefore position-dependent quantities. They depend on the instantaneous contact ellipse, the crack location, the crack depth, and the direction in which the crack intersects the tooth section. Three crack families are relevant to this problem: a plane-type root crack with controlled width and depth, a spatial crack that extends at an angle toward the tooth tip, and a broken-tooth condition in which the remaining section no longer carries load. The finite element results below separate these families and show how each one changes the meshing stiffness, tooth-surface contact stress, and root bending stress.
Tooth contact, mesh stiffness, and the finite element load path
The tooth flanks of a spiral bevel gear are generated from the envelope of the cutter surface in a machine-tool coordinate system. Tooth contact analysis provides the corresponding flank coordinates, which can then be imported into a finite element model. In the static contact model used for the crack study, the gear body is discretized with six-node-tooth sectors and hexahedral Solid185 elements. Six contact pairs are defined with Conta174 contact elements and Targe170 target elements. Rigid reference points are coupled to the inner surfaces of the pinion and gear; torque is applied at the pinion reference point, while the gear reference point is fully constrained. The pinion reference point is released only for rotation about its axis.
A six-tooth sector is sufficient for this class of analysis because the contact ratio of the spiral bevel gear pair is not greater than 3, and six teeth capture three mesh cycles for one tooth without introducing significant far-field coupling from teeth outside the contact zone. The model is static: it resolves the loaded contact state at a given angular position, not the transient vibration of the complete transmission.
The normal mesh stiffness is defined as the normal load per unit normal deflection at the contact:
$$
k=\frac{F}{E},\qquad
F=\frac{T}{R\sin\delta_{A1}\cos\alpha_n\cos\beta},\qquad
E=\theta R\sin\delta_{A1}\cos\alpha_t\cos\beta_b.
$$
Here, T is the driving torque, R is the mid-cone distance, δA1 is the pinion pitch-cone angle, αn is the transverse pressure angle, β is the workpiece spiral angle, βb is the spiral angle in the meshing plane, αt is the tangential pressure angle, and θ is the equivalent static rotation of the master node along the pinion axis. The stiffness is thus obtained from the normal load and the total normal deformation, not from a local contact stiffness alone.
The model was checked against loaded tooth contact analysis at driving torques of 50, 100, 150, and 200 N·m. The stiffness curves followed the same variation and the maximum difference was 4.66%. Stress contours for the cracked cases were evaluated at an input torque of 120 N·m. These conditions define the numerical scope: the trends are relevant to the specified geometry, boundary conditions, and torque range, but they do not by themselves establish fatigue life or measured vibration spectra.
How the crack geometries are parameterized
Root cracks are introduced at the pinion tooth-root fillet, where the bending stress is highest. The crack is implanted by a node-replacement technique: new nodes are generated at the same coordinates as the original nodes along the intended crack boundary, the elements on the crack edge are deleted, and new elements are built from the replacement nodes. Originally connected elements are thus separated without prescribing a pre-existing crack-opening displacement. This approach allows the crack trace and depth to be varied while keeping the surrounding mesh and contact definition consistent.
For the plane-type crack, the crack width is measured as an arc length W across the face width, and the crack depth is measured as an arc length P through the tooth thickness. Three width levels, W/3, 2W/3, and W, are combined with five depth levels from P/6 to 5P/6. For the spatial crack, the crack extends toward the tooth tip over angular extents α1, α2, and α3, with depth S divided into five levels from S/6 to 5S/6. The broken-tooth cases correspond to complete through-thickness separation at the three angular extents.
| Crack family | Geometric control | Levels | Case numbers | Largest stiffness reduction |
|---|---|---|---|---|
| Plane-type root crack | Width along face width: W/3, 2W/3, W; depth through tooth thickness: P/6, P/3, P/2, 2P/3, 5P/6 | 3 widths × 5 depths | 1–15 | 27.58% (case 15) |
| Spatial root crack | Angular extent toward tip: α1, α2, α3; depth: S/6, S/3, S/2, 2S/3, 5S/6 | 3 angles × 5 depths | 1–15 | 14.12% (case 15) |
| Broken tooth | Through-thickness fracture at α1, α2, or α3 | 3 angular extents | 16–18 | 32.82% (case 18) |
Stiffness response across crack types
The stiffness reduction is localized in the mesh cycle. When the contact ellipse is away from the crack, the cracked tooth can still behave nearly like a healthy tooth. As the ellipse moves into the cracked region, the effective load-bearing section is reduced, and the mesh stiffness falls. The reduction increases with crack severity, but it is not a simple monotonic function of depth at every instant: the instantaneous overlap between the contact ellipse and the crack trace controls the magnitude.
For the plane-type cracks, the largest drop reaches 27.58% in case 15, where the crack spans the full face width and extends to 5P/6 through the tooth thickness. For the spatial cracks, the largest drop is 14.12% in case 15. For the broken-tooth cases, the largest drop is 32.82% in case 18. In the smallest width or angular-extent group, cases 1–5, the stiffness reduction during initial engagement is comparatively small even when the depth is increased. The cracked tooth then exits the contact zone, the next healthy tooth picks up load, and the stiffness returns toward its healthy variation.
The timing of the stiffness trough differs between the two crack geometries. Because the spatial crack trace is inclined toward the tooth tip, and because the contact ellipse has a finite size and a specific direction of travel, the ellipse reaches the affected region later than it does for the plane crack. The resulting affected interval within the cracked-tooth mesh cycle is therefore shorter for the spatial crack. Under the same model and load conditions, the plane crack family produces the larger peak stiffness reduction: 27.58% versus 14.12%. The plane crack also has a longer interval of influence, whereas the spatial crack produces a more localized disturbance whose stress consequences are concentrated around the crack line.
When the spatial crack depth becomes through-thickness, the tooth no longer behaves as a cracked but continuous body. The remaining broken tooth may carry little or no load at the corresponding position, and the original double-tooth contact can become single-tooth contact. This produces the largest stiffness drop, 32.82%, together with pronounced fluctuation as load is transferred to the neighboring tooth.
Contact and root bending stress redistribution
Stiffness alone does not describe the failure state. The same crack can lower the contact stress on one part of the tooth while raising it on another. Stress was therefore extracted at three positions within the cracked-tooth mesh region: position 1, before the contact ellipse reaches the crack; position 2, when the ellipse is on or near the crack line; and position 3, after the ellipse has passed the crack line.
For the plane-type crack, when the contact ellipse is relatively far from the root crack at positions 1 and 2, the tooth-surface contact stress is almost unchanged as the crack depth increases. The root fillet is a different matter. Because the plane crack lies at the root transition, the stress concentration at the crack tip grows with depth even while the contact patch remains elsewhere. Once the ellipse moves over the crack at position 3, the load-bearing section is controlled by the reduced section inertia. The tooth-surface contact stress then decreases with increasing crack depth, while the stress concentration at the root crack tip continues to rise. The plane crack therefore combines a broad stiffness reduction with a persistent root-tip stress concentration.
For the spatial crack, the response is more clearly divided by the crack line. Before the contact ellipse reaches the crack line at position 1, the loaded section does not include the crack, and the changes in contact stress and root bending stress are very small. When the ellipse crosses the crack line at position 2, the contact patch is split between the tooth material outside the crack and the material inside it. The outer body loses resistance to deformation and its contact stress falls; the inner body takes a larger share of the load and its contact stress rises. For case 5, the inner-body contact stress increases by 15.26%, while the root bending stress increases by 2.3%. This split across the crack line is a characteristic spatial-crack signature: one side of the crack unloads while the other side loads up.
After the ellipse has passed the crack line at position 3, the loaded portion lies mainly outside the crack. The reduced effective section changes the compliance of the tooth, and the contact stress decreases with increasing crack depth; for case 5 the reduction is 8.91%. The root-fillet bending stress also decreases, by 47.11% in case 5, as the load path shifts away from the crack-controlled section. This behavior differs from the plane crack, where the root crack-tip concentration continues to increase after the ellipse passes over the crack.
| Contact position relative to crack line | Tooth-surface contact stress | Root bending stress | Mechanical interpretation |
|---|---|---|---|
| Before the crack line | Nearly unchanged | Nearly unchanged | The loaded section does not include the crack. |
| On the crack line | Outer body decreases; inner body increases; maximum increase 15.26% in case 5 | Increase of 2.3% in case 5 | Load is redistributed across the crack; the outer body is less resistant to deformation. |
| After the crack line | Decrease of 8.91% in case 5 | Decrease of 47.11% in case 5 | The loaded region lies mainly outside the crack, and the effective load path is altered by the reduced section. |
These stress trends explain why a crack cannot be characterized by depth alone. A plane crack can leave the contact stress almost unchanged at an early mesh position while raising the root-tip stress; a spatial crack can produce a local contact-stress increase on the inner side of the crack line and a decrease on the outer side at the same instant. The broken-tooth condition goes further: once the broken region is large enough, the damaged tooth stops participating in load sharing, and the neighboring tooth must carry the load that would otherwise have been shared.
Diagnostic meaning and modeling scope
For model-based fault diagnosis, the useful indicators are the amplitude and timing of the stiffness trough within the mesh cycle, together with the spatial redistribution of contact stress. The plane-type crack produces a relatively long interval of stiffness influence and a large peak reduction, with sustained stress concentration at the root crack tip. The spatial crack produces a later-onset, shorter stiffness disturbance and a contact-stress split across the crack line. A broken tooth produces the largest stiffness loss and the strongest fluctuation because load sharing is lost at the damaged position. These signatures are more specific than a single “crack depth” value, but they are not direct measurements of an installed gear: they are predictions of a static contact model under prescribed geometry and torque.
The reported trends apply to the specified spiral bevel gear pair, the six-tooth sector model, the stated boundary conditions, and the 50–200 N·m stiffness validation range. The model does not include transient dynamics, lubrication, friction, misalignment, thermal effects, or crack propagation. Those factors can shift the contact ellipse and change the local compliance, so the stiffness trough and stress split should be treated as mechanism-level indicators rather than as a complete diagnostic rule. Within that scope, the central result is consistent: the meshing response of a cracked spiral bevel gear is controlled by the moving contact ellipse, the crack trace, and the remaining load-bearing section, not by crack depth alone.