Bevel gear & bevel gear drive calculator
Calculate the geometry of a straight bevel gear pair: pitch cone angles, outer and mean pitch diameters, outer and mean cone distance, mean module and the virtual tooth number of the equivalent spur gearing. The pinion torque gives the tangential force. The tool provides geometry and preliminary sizing with a traffic-light rating for undercut and face width, live with every input.
Calculation
Model: straight bevel gear, geometry and preliminary sizing (pitch cone angles, diameters, cone distances, virtual tooth number, tangential force). Basic rack αn = 20° for the limiting tooth number. No full load capacity check to ISO 10300 (tooth root, flank, scuffing), no spiral bevel or hypoid gearing.
Results
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Formulas and fundamentals
Pitch cone angle and ratio
In a bevel gear drive the axes of the two gears intersect at the shaft angle Σ (usually 90°). From the tooth-number ratio and the shaft angle the pinion pitch cone angle follows as:
The gear pitch cone angle is:
For the special case Σ = 90° this simplifies to δ1 = atan(z1/z2). The ratio is i = z2/z1 = tan δ2/tan δ1 and depends solely on the pitch cone angles.
Diameters and cone distance
The module, and thus the tooth size, decreases linearly from the outer to the inner end of the teeth. Referred to the outer module m, the outer pitch diameters are:
The outer cone distance, measured from the common cone apex to the outer end of the teeth, is:
With the face width b:
The mean pitch diameters are:
Virtual tooth number and tangential force
The load capacity of a bevel gear is estimated via a notional spur gear with the virtual tooth number zv = z/cos δ, whose radius equals the back cone radius. This lets the known form factors and limiting tooth numbers of spur gearing be transferred. The pinion of a bevel pair always has the smaller virtual tooth number due to its smaller pitch cone angle and therefore governs undercut: if zv falls below the limiting tooth number (about 17 theoretically, roughly 14 in practice for a 20° basic rack), profile shift is required. The pinion torque T1 gives the tangential force at the mean pitch circle:
It is the starting point for tooth root and flank checks.
Scope of the calculator
The calculator deliberately covers only the geometry and preliminary sizing: pitch cone angles, diameters, cone distances, virtual tooth numbers and tangential force. The full load capacity calculation of bevel gears to ISO 10300 with its tooth root, flank and scuffing checks and the numerous load, geometry and material factors is not included. As guide values the calculator checks undercut (virtual tooth number) and face width (b ≤ Re/3, typically b ≈ 0.25…0.3·Re).
Worked example
Reference example: a straight bevel gear pair with shaft angle Σ = 90°, z1 = 20, z2 = 40 and outer module m = 4 mm. The ratio is i = 40/20 = 2. The pitch cone angles are δ1 = atan(20/40) = 26.565° and δ2 = 63.435°; their sum equals the shaft angle of 90°.
The outer pitch diameters are d1 = 4·20 = 80 mm and d2 = 4·40 = 160 mm. The outer cone distance is Re = 0.5·4·√(20² + 40²) = 89.44 mm. For a face width b = 25 mm the mean cone distance is Rm = 76.94 mm, the mean module mm = 3.44 mm and the mean diameters dm1 = 68.81 mm and dm2 = 137.64 mm.
The virtual tooth numbers are zv1 = 20/cos 26.565° = 22.36 and zv2 = 40/cos 63.435° = 89.44. Since zv1 exceeds the limiting tooth number of about 17, no undercut occurs. For a pinion torque of T1 = 100 Nm the tangential force at the mean pitch circle is Ft = 2·100,000/68.81 = 2906 N. With b/Re = 0.28 the face width is within the guide range - the preliminary sizing is plausible.
Frequently asked questions
What is the pitch cone angle and how does it relate to the ratio?
The pitch cone angle δ is the half apex angle of the pitch cone on which the two gears roll. It follows from δ1 = atan(sin Σ/(z2/z1 + cos Σ)) and δ2 = Σ − δ1. The ratio is i = z2/z1 = tan δ2/tan δ1; for Σ = 90° simply i = tan δ2. A larger tooth-number ratio therefore increases the pitch cone angle of the gear.
Why calculate with the virtual tooth number zv?
The teeth of a bevel gear approximately correspond to those of a notional spur gear with the virtual tooth number zv = z/cos δ, whose radius equals the back cone radius. This allows the familiar form factors and limiting tooth numbers of spur gearing to be applied. Because zv is always larger than z, a bevel gear behaves in terms of load capacity like a spur gear with more teeth.
When does undercut occur in bevel gears?
Undercut occurs when the virtual tooth number zv of the pinion falls below the limiting tooth number - about 17 theoretically and roughly 14 in practice for a 20° basic rack. The pinion with the smaller pitch cone angle always governs. A positive profile shift helps; bevel gears are often designed as V-zero drives with x1 = −x2, the pinion shifted positively.
How large may the face width of a bevel gear be?
As a guide value b ≤ Re/3, typically b ≈ 0.25 to 0.3·Re. An excessive face width leads to very small teeth at the inner end and an unfavourable contact pattern, because module and tooth height decrease towards the cone apex. The calculator rates the ratio b/Re with a traffic light.
Does the calculator replace a load capacity check to ISO 10300?
No. The calculator provides the geometry and a preliminary sizing with tangential force, virtual tooth number and the guide-value checks for undercut and face width. The full verification of tooth root, flank and scuffing load capacity of bevel gears is carried out to ISO 10300 with the load, geometry and material factors defined there and is not included here.
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