BIZOMAT MAKEOS · Gears
Helical Gear Set Design (Geometry, Forces and Bending)
What are the sizes, tooth forces and bearing thrust of this helical gear pair, and is the tooth bending stress within the allowable?
Design check of a parallel-axis helical gear pair from power and speed: transverse module, pitch diameters, centre distance, transverse pressure angle, virtual teeth and overlap ratio (as the helical gear tool), the tangential, axial and separating forces the bearings must carry, pitch-line velocity and the Lewis bending stress in the normal plane with the velocity factor (as the Lewis bending tool), using the form factor for the virtual number of teeth.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Power | kW | Power through the gear pair. |
| Pinion speed | rpm | Speed of the pinion. |
| Normal module | mm | Normal module (the hob or cutter module). |
| Helix angle | ° | Helix angle at the pitch diameter. |
| Normal pressure angle | ° | Normal pressure angle. |
| Pinion teeth | Number of teeth on the pinion. | |
| Gear teeth | Number of teeth on the gear. | |
| Face width | mm | Face width in mesh. |
| Lewis form factor at the virtual teeth | Lewis form factor read for the pinion’s virtual number of teeth (z ÷ cos³β), shown in the results. | |
| Tooth finish | choice | Manufacturing quality, which sets the velocity factor. |
| Allowable bending stress | MPa | Allowable bending stress of the pinion material (your design value). |
| Least overlap ratio wanted | Least overlap (face contact) ratio you want for smooth running (often at least 1). |
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 the pinion passes to the gear, normally the rated output of the motor or the power the driven machine absorbs at this mesh.
- 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 7.5 kW as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Pinion torque, all three tooth forces and the bending stress rise in proportion.
- Get it right
- Use the power at this gear pair, after any losses in stages before it; kW, not W or hp.
What this means: Pinion speed
- What it is
- How fast the pinion (the smaller gear) turns. With the power it sets the pinion torque, and with the pitch diameter it sets the pitch-line velocity.
- 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; the two differ by the tooth ratio.
What this means: Normal module
- What it is
- The tooth size measured square to the teeth (in the normal plane). It is the module of the hob or cutter that cuts the gear, and it sets tooth thickness and height.
- 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 3 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 both gears and the centre distance and lowers the bending stress markedly.
- Get it right
- Do not enter the transverse module (measured across the face); the tool works that out as mₙ ÷ cos β.
What this means: Helix angle
- What it is
- The angle the teeth make with the gear axis, measured at the pitch diameter. Both gears of an external pair share it with opposite hands.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0° to 45°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 15° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger helix angle raises the axial thrust (Fₐ = Fₜ tan β), enlarges the pitch diameters and raises the overlap ratio.
- Get it right
- A spur gear has 0°; with 0° the overlap ratio is zero and there is no axial thrust.
What this means: Normal pressure angle
- What it is
- The pressure angle of the tooth profile measured square to the teeth, as given by the hob or cutter. The tool converts it to the transverse pressure angle.
- 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 separating (radial) force on the bearings.
- Get it right
- Enter the normal value from the cutter or drawing; the transverse angle shown in the results is larger.
What this means: Pinion teeth
- What it is
- The number of teeth on the pinion, the smaller and usually driving gear. With the module it sets the pinion pitch diameter.
- Units and format
- Enter a plain number with no unit. The form accepts 8 to 300. Use a whole number. 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
- More pinion teeth enlarge its pitch diameter, which lowers the tangential force for the same torque.
What this means: Gear teeth
- What it is
- The number of teeth on the larger gear of the pair. With the pinion teeth it sets the ratio, the gear pitch diameter and the centre distance.
- Units and format
- Enter a plain number with no unit. The form accepts 8 to 1000. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 45 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Only the gear diameter and centre distance change; the pinion forces and bending stress do not.
What this means: Face width
- What it is
- The axial width over which the two gears’ teeth are actually in contact. It carries the tooth load and sets the overlap ratio.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 2000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 35 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Bending stress falls in proportion to face width, and the overlap ratio rises.
- Get it right
- If the two gears differ in width, use the narrower one, since only the common width is in mesh.
What this means: Lewis form factor at the virtual teeth
- What it is
- A dimensionless factor for the strength of the tooth shape in bending, read from a Lewis form factor table for the pressure angle and the pinion’s virtual number of teeth.
- 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 0.33 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Bending stress is inversely proportional to the form factor.
- Get it right
- Read it for the virtual teeth z ÷ cos³β shown in the results, not the actual tooth count; run once to see that number.
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 so a lower bending stress.
- Get it right
- Choose the finish the gears will actually have, not the best one available.
What this means: Allowable bending stress
- What it is
- The bending stress you allow at the pinion tooth root, from the material data or the gear standard your design follows, including your own margin.
- 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 (root) allowable, not a surface contact or pitting value; the tool checks bending only.
What this means: Least overlap ratio wanted
- What it is
- The smallest overlap (face contact) ratio you accept: how many axial pitches the face width spans, which keeps more than one tooth pair in contact for smooth running.
- Units and format
- Enter a plain number with no unit. The form accepts 0 to 5. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- It only triggers a warning; it does not change the forces or the stress.
Method and formulas
- m_t = m_n ÷ cos β; d = z m_t; a = (d₁ + d₂) ÷ 2; tan α_t = tan α_n ÷ cos β; virtual teeth z_v = z ÷ cos³β; overlap ratio ε_β = b sin β ÷ (π m_n).
- T₁ = P ÷ ω₁; tangential force F_t = 2T₁ ÷ d₁; axial F_a = F_t tan β; separating F_r = F_t tan α_t.
- Lewis bending σ = K_v F_t ÷ (b m_n Y) with K_v from the pitch-line velocity (as the Lewis bending tool).
- Example: 7.5 kW at 1450 rpm gives a 1,590.3 N tangential force and 426.13 N of thrust; the bending stress is 73.895 MPa.
Limitations
- Lewis method for a first sizing; full ratings use AGMA or ISO 6336 factors and pitting resistance.
- Form factor must be read for the virtual number of teeth.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Size a helical gear pair for a reducer
- Find the thrust load for bearing selection
- Compare helix angles for smoothness against thrust
Who it is for
- Design engineers
- Maintenance engineers
- Power transmission engineers
What you get
- Gear geometry and centre distance
- Tangential, axial and separating forces
- Bending stress and overlap checks
Interpreting the result
- A larger helix angle runs smoother but raises the thrust the bearings must take.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2195–2210, 2131–2160. Topic reference; no source tables or recommended settings are reproduced.
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Open the Helical Gear Set Design (Geometry, Forces and Bending) calculator