Bevel pairs

The Bevel module rates bevel gear pairs with intersecting axes: straight (or skew), spiral and Zerol teeth, without hypoid offset. It computes the mean cone geometry, the forces on both shafts, the virtual cylindrical gears and load factors of ISO 10300-1, and the pitting safety by ISO 10300-2 method B1. The tooth root is rated on the virtual cylindrical pair by ISO 6336-3, because ISO 10300-3 is not implemented.

Inputs

Choose + Bevel pair above the list (or Add bevel pair in an empty module). The inspector groups:

Geometry

A pinion with few teeth needs a positive shift. The template's 15-tooth pinion has x_hm1 = 0.41, the long-addendum rule of straight bevel gears $x_{hm1}=0.46\cdot (1-1/u^2)$.

Load

Pinion torque (N·m), Pinion speed (rpm), Application factor KA (Annex C: 1.0–2.25) and Life (h).

Load factors (ISO 10300-1)

• Mounting: Both straddle-mounted, One member overhung (default) or Both members overhung.
• Contact pattern checked: Each set in its housing, full load, Each set, light test load (default) or Sample set, estimated for full load. With the mounting this gives the mounting factor KHβ-be from Table 4 of ISO 10300-1; Mounting factor KHβ-be overrides it (1.0 to 1.5).
• Loaded contact pattern b2,eff/b2: 0.85 when unknown; a satisfactory pattern covers 80 to 85 % of the face.
• Accuracy grade (ISO 17485) (7 by default) and Single pitch deviation f_pt (from the grade when empty).
• Dynamic factor: Method B (resonance model) or Method C (empirical curves) (ISO 10300-1, 7.7).
• Profile crowning (load sharing Z_LS): High (industrial) or Low (automotive).

Materials and requirements

σHlim and σFlim for pinion and wheel, Softer flank hardness (Z_W) in HB for a through-hardened gear running with a surface-hardened one (Z_W = 1 when empty), Min pitting safety (1.0) and Min root safety (1.4).

Results

The view card draws the pitch cones meeting at their apex and shows Safeties per gear (pitting and tooth root, pinion and wheel). The key figures are Pitting safety with σH and Root safety with the cone angles. Details lists the forces, the mean normal module, the virtual contact ratios, the load factors, the pitting factors and the permissible contact stress.

In 3D the two gears stand on their pitch cones at the result's cone angles, with the face width and the spiral angle; hovering turns them in mesh at the ratio. The 2D and 3D tabs above the drawing switch to a 3D model; the choice is kept for the module. In 3D, drag to orbit and scroll to zoom; the model stands still and plays its motion once when the pointer moves onto it or it gets the keyboard focus (never with reduced motion). Without WebGL the drawing stays, with a note.

The Spiral bevel pair template (15/45, m_et 4 mm, Σ 90°, β_m 35°, b 30 mm, x_hm1 0.41, 150 N·m at 1500 rpm, KA 1.25, 16MnCr5 case hardened) gives:

Pinion and wheel share one contact stress; their pitting safeties are equal here because both gears are of the same material and both life factors Z_NT are 1. The forces act at the mean diameter; on a 90° pair the pinion's axial force equals the wheel's radial force and the other way round.

What the factors mean

• $K_{H\beta }=1.5\cdot K_{H\beta \text{-}be}$: the face load factor of method C comes from the mounting and how the contact pattern is verified. Straddle-mounting both members and checking the pattern under full load in the housing lowers it: on the template K_Hβ-be falls to 1.0, K_Hβ to 1.5 and the pitting safety rises from 1.178 to 1.218.
• $K_{F\beta }=K_{H\beta }/K_{F0}$: the lengthwise curvature factor K_F0 from the cutter radius relieves the root of spiral teeth.
• Z_LS spreads the load over the tip, middle and root contact lines; Z_M-B takes the curvature at the governing point of the path of contact.

When a pair is refused

As for cylindrical pairs, the contact must stay on the involutes of both virtual cylindrical gears. The template with straight teeth and no shift is refused:

involute interference on the virtual cylindrical gears (z_v = 15.81, 142.30): the wheel's tip meets the pinion 0.273 mm (roll length) inside its base circle Ø50.03 mm, where it has no involute; use a pinion profile shift x_hm1 of at least 0.05 (profileShift; the wheel takes −x_hm1) or more pinion teeth

Enter at least the suggested Profile shift x_hm1 (wheel −x), or add pinion teeth. With 35° spiral and no shift the template solves (SH 1.12, SF 2.222), because the spiral angle raises the transverse pressure angle of the virtual gears.

Fixing a failing check

• Pitting: larger outer module or face width (within the usual $b\le R_e/3$), a more favourable mounting or contact check (lower K_Hβ), a finer accuracy grade, better oil viscosity or flank finish.
• Tooth root: larger module, more pinion shift, a cutter radius for spiral teeth (K_F0).

Standards and scope

ISO 10300-1:2014 method B1 geometry with the virtual cylindrical gears of Annex A, K_A (Annex C guide values), K_v by method B or C, K_Hβ and K_Fβ by method C, K_Hα = K_Fα by method B; pitting by ISO 10300-2:2014 method B1 (clause 6) with Z_L, Z_v, Z_R, Z_W and Z_NT of clause 8.

Not covered:

• ISO 10300-3 (tooth root) is not implemented. The root safety is that of the virtual cylindrical pair rated by ISO 6336-3 with the ISO 10300-1 load factors; part 3's own factors are not applied.
• Hypoid offsets are refused (the hypoid geometry of ISO 23509 is not implemented); Z_Hyp is 1.
• The bevel slip factor of ISO 10300-2's informative Annex A, scuffing and mounting on the project's shafts.

From the command line

axm new bevel.axm.json --template bevel-pair
axm bevel update bevel.axm.json BV1 --patch '{"mounting":"neitherCantilevered","contactCheck":"fullLoad"}'
axm solve bevel.axm.json

Related

• Gear pairs
• ISO 6336 gear rating
• Shaft for the shafts that carry the forces
• Open the spiral bevel pair example