BIZOMAT MAKEOS · Gears
Straight Bevel Gear Set Design (Geometry, Forces and Bending)
What are the cone angles, tooth forces and bearing loads of this straight bevel gear pair, and is the bending stress within the allowable?
Design check of a straight bevel gear pair from power and speed: pitch cone angles, pitch diameters and outer cone distance (as the bevel gear tool), face width against your limit as a share of the cone distance, the mean pitch diameter and the tangential, axial and radial forces on pinion and gear at the mean diameter, and the Lewis bending stress for bevel teeth with the face-width correction (L − b) ÷ L and the velocity factor.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Power | kW | Power through the bevel pair. |
| Pinion speed | rpm | Speed of the pinion. |
| Module at the large end | mm | Module at the outer (large) end. |
| Pinion teeth | Number of teeth on the pinion. | |
| Gear teeth | Number of teeth on the gear. | |
| Shaft angle | ° | Angle between the shafts (90° for a right-angle drive). |
| Pressure angle | ° | Pressure angle of the teeth. |
| Face width | mm | Face width along the cone. |
| Largest face width ÷ cone distance | Largest face width as a share of the outer cone distance you design to (a third is common). | |
| Lewis form factor (for the formative teeth) | Lewis form factor for the formative (equivalent spur) number of teeth. | |
| Tooth finish | choice | Manufacturing quality, which sets the velocity factor. |
| Allowable bending stress | MPa | Allowable bending stress of the pinion material (your design value). |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Power
- What it is
- The mechanical power passed through the bevel pair, usually the motor output or the power the driven shaft absorbs.
- Units and format
- Enter a number in kW: kilowatts. The form accepts 0.000001 kW to 100000 kW. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3 kW as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Torque, tooth forces, bearing loads and bending stress rise in proportion.
- Get it right
- Use the power at this pair after losses upstream; kW, not W or hp.
What this means: Pinion speed
- What it is
- How fast the bevel pinion (the smaller gear) turns. With the power it sets the pinion torque.
- Units and format
- Enter a number in rpm: revolutions per minute. The form accepts 0.000001 rpm to 100000 rpm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1450 rpm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- For the same power, a higher speed lowers torque and tooth forces but raises the velocity factor.
- Get it right
- Enter the pinion speed, not the gear speed.
What this means: Module at the large end
- What it is
- The tooth size at the outer (large) end of the bevel teeth, where the pitch diameters d = m z are measured. Bevel teeth taper, so the size falls towards the apex.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.5 mm to 50 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 4 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger module enlarges the gears and lowers the tooth forces and bending stress.
- Get it right
- Use the outer-end module from the drawing, not the mean module at mid-face.
What this means: Pinion teeth
- What it is
- The number of teeth on the bevel pinion. With the gear teeth and shaft angle it sets the pitch cone angles.
- Units and format
- Enter a plain number with no unit. The form accepts 8 to 200. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 18 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Gear teeth
- What it is
- The number of teeth on the larger bevel gear. It sets the ratio, the gear pitch diameter and, with the shaft angle, both cone angles and the cone distance.
- Units and format
- Enter a plain number with no unit. The form accepts 8 to 500. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 36 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Shaft angle
- What it is
- The angle between the centre lines of the pinion shaft and the gear shaft where they meet at the cone apex.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 30° to 150°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 90° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Measure between the shaft axes, not between a shaft and the mounting face.
What this means: Pressure angle
- What it is
- The pressure angle of the tooth profile. It decides how much of the tooth force pushes the shafts axially and radially.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 14.5° to 25°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger pressure angle raises the axial and radial bearing forces; it does not enter the bending stress here.
What this means: Face width
- What it is
- The length of the teeth measured along the pitch cone, from the outer end towards the apex. It carries the load and is compared with the outer cone distance.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 22 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Measure along the slant of the cone, not parallel to the shaft axis.
What this means: Largest face width ÷ cone distance
- What it is
- Units and format
- Enter a plain number with no unit. The form accepts 0.1 to 0.5. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.33 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter a fraction (a third is 0.33), not a percentage.
What this means: Lewis form factor (for the formative teeth)
- What it is
- The Lewis form factor for the tooth shape, read from a table for the formative (equivalent spur) number of teeth z ÷ cos δ and the pressure angle.
- Units and format
- Enter a plain number with no unit. The form accepts 0.1 to 0.6. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.3 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Read it for the formative tooth count of the pinion, not its actual tooth count; the cone angle δ₁ is shown in the results.
What this means: Tooth finish
- What it is
- How the teeth were made or finished, which sets the velocity factor that allows for dynamic loading from tooth errors at speed.
- Units and format
- Choose one of 4 options from the list: Cast teeth; Cut teeth; Hobbed or shaped teeth; Shaved or ground teeth.
- Example
- The form opens with “Hobbed or shaped teeth” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- A finer finish gives a smaller velocity factor and a lower bending stress.
- Get it right
- Choose the finish the gears will actually have.
What this means: Allowable bending stress
- What it is
- The bending stress you allow at the pinion tooth root, from material data or the bevel gear standard your design follows, with your margin included.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 2000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 150 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use a bending allowable, not a surface contact value; the tool checks bending only.
Method and formulas
- tan δ₁ = sin Σ ÷ (z₂ ÷ z₁ + cos Σ); δ₂ = Σ − δ₁; d = m z; outer cone distance L = d₂ ÷ (2 sin δ₂) (as the bevel gear tool).
- Mean pitch diameter d_m1 = d₁ (1 − 0.5 b ÷ L); F_t = 2T₁ ÷ d_m1; pinion axial F_a1 = F_t tan α sin δ₁ and radial F_r1 = F_t tan α cos δ₁ (the gear’s axial and radial forces swap these).
- Lewis bending σ = K_v F_to L ÷ [b m Y (L − b)], with F_to = 2T₁ ÷ d₁ at the large end.
- Example: 3 kW at 1450 rpm puts 635.67 N tangential load at the mean diameter; the bending stress is 47.393 MPa.
Limitations
- Lewis method for a first sizing; AGMA 2003 or ISO 10300 give full ratings.
- Straight bevel teeth; spiral bevels have different force directions that depend on hand and rotation.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Size a bevel pair for a right-angle drive
- Find the axial and radial loads for the bearings
- Check a face width against the cone distance
Who it is for
- Design engineers
- Maintenance engineers
- Power transmission engineers
What you get
- Cone angles, diameters and cone distance
- Forces on pinion and gear at the mean diameter
- Bending stress and face-width checks
Interpreting the result
- Face width beyond about a third of the cone distance adds little strength because the inner teeth are small.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2181–2190, 2131–2160. Topic reference; no source tables or recommended settings are reproduced.
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