KEVOS® MAKEOS · Gears
Crossed Helical Gears at an Exact Centre Distance (Any Shaft Angle, Minimum Centre Distance, Cutter Numbers)
For a pair of crossed helical gears on non-parallel shafts at an exact centre distance, what helix angles make the gears fit, what are their pitch and outside diameters and leads, what is the least centre distance for the ratio and which formed cutters do I use?
Helical gears for shafts at any angle and an exact centre distance by the handbook procedure (same hand, helix angles adding to the shaft angle Σ): pinion teeth nearest n = 2 C Pn cos αa cos βa ÷ (R cos βa + cos αa) for approximate helix angles, exact helix angles found by trial from R sec α + sec β = 2 C Pn ÷ n with β = Σ − α, pitch diameters D = N ÷ (Pn cos α) and d = n ÷ (Pn cos β), outside diameters D + 2 ÷ Pn and d + 2 ÷ Pn, leads L = π D cot α and l = π d cot β, the helix angle for the least centre distance (cot α = R^⅓ at 90°), and formed cutter numbers from N′ = N sec³ α + Pn Dc tan² α.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Shaft angle Σ | ° | Angle between the two shafts; 90° for shafts at right angles. |
| Exact centre distance C | mm | Exact centre distance the gears must fit. |
| Normal module mn (25.4 ÷ Pn) | mm | Normal module of the teeth and cutter; for a normal diametral pitch Pn it is 25.4 ÷ Pn. |
| Teeth in gear N | teeth | Number of teeth in the larger gear. |
| Teeth in pinion n | teeth | Number of teeth in the pinion, the whole number nearest the estimate shown. |
| Approximate helix angle of gear αa | ° | Approximate helix angle you want on the gear; the pinion takes the shaft angle less this. |
| Pitch diameter of the formed cutter Dc | mm | Pitch diameter of the formed milling cutter, from the cutter maker or the handbook’s cutter table. |
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: Shaft angle Σ
- What it is
- Angle between the two shafts the crossed helical gears connect.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 1° to 179°. 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.
- Effect on the result
- The helix angles of gear and pinion add to it.
- Get it right
- 90° for shafts at right angles.
What this means: Exact centre distance C
- What it is
- Exact centre distance the gear pair must fit between the shafts.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 254 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the helix angles found by trial.
- Get it right
- Measure square to both shafts along their common perpendicular.
What this means: Normal module mn (25.4 ÷ Pn)
- What it is
- Normal module of the gear teeth, the size of tooth in the normal plane.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3.175 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Scales every diameter, lead and tooth dimension.
- Get it right
- For an inch normal diametral pitch Pn, enter 25.4 ÷ Pn.
What this means: Teeth in gear N
- What it is
- Number of teeth in the larger gear of the crossed helical pair.
- Units and format
- Enter a number in teeth: a whole number of teeth. The form accepts 1 teeth to 10000 teeth. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 84 teeth as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the ratio and the gear diameter.
- Get it right
- Gear teeth ÷ pinion teeth gives the speed ratio.
What this means: Teeth in pinion n
- What it is
- Number of teeth in the smaller gear (pinion) of the pair.
- Units and format
- Enter a number in teeth: a whole number of teeth. The form accepts 1 teeth to 10000 teeth. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 28 teeth as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the ratio and the pinion diameter.
- Get it right
- Take the whole number nearest the estimate shown.
What this means: Approximate helix angle of gear αa
- What it is
- Approximate helix angle you want on the larger gear.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0.1° to 179°. 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
- Picks which of the two fitting helix angles is used.
- Get it right
- The pinion takes the shaft angle less this; 45° each at 90° is common.
What this means: Pitch diameter of the formed cutter Dc
- What it is
- Pitch diameter of the formed milling cutter used for the teeth.
- 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 66.294 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 cutter raises the cutter tooth number N′.
- Get it right
- From the cutter maker or the handbook’s table of standard cutters.
Method and formulas
- R = N ÷ n; nest = 2 C cos αa cos βa ÷ (mn (R cos βa + cos αa)) with βa = Σ − αa (2 C Pn ÷ n = 2 C ÷ (n mn) in SI).
- Solve R sec α + sec (Σ − α) = 2 C ÷ (n mn) for the root nearest αa; β = Σ − α.
- D = N mn ÷ cos α; d = n mn ÷ cos β; O = D + 2 mn; o = d + 2 mn; L = π D cot α; l = π d cot β.
- Least centre distance where R sec α + sec β is smallest (cot α = R^⅓ at Σ = 90°).
- N′ = N sec³ α + (Dc ÷ mn) tan² α and n′ = n sec³ β + (Dc ÷ mn) tan² β, rounded and looked up in the cutter series; N ÷ cos³ α shown for comparison.
- Example: 84 and 28 teeth at 254 mm take helix angles of 46.099° and 43.901°; the least centre distance for the ratio is 240.28 mm.
Limitations
- Covers gears of the same hand whose helix angles add to the shaft angle; for opposite-hand pairs the difference of the angles equals the shaft angle.
- Crossed helical gears have point contact and suit light loads; rate them separately.
- Choose the hand of each helix from the rotation and the direction the thrust must be taken.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find the helix angles that fit crossed helical gears to a fixed centre distance
- Find the least centre distance for a crossed helical ratio
- Work out leads and formed cutter numbers for milling crossed helical gears
Who it is for
- Gear designers
- Machinists and gear cutters
- Students
What you get
- Pinion teeth estimated for the approximate helix angles
- Exact helix angles of gear and pinion
- Pitch and outside diameters
- Leads of both helices
- Helix angle and centre distance for the least centre distance
- Formed cutter numbers by the preferred and approximate formulas
Interpreting the result
- There are usually two helix angles that fit; the tool takes the one nearest your approximate angle.
- If the centre distance is below the least centre distance, no helix angles fit: increase it or change the teeth.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2199–2205, 2150. Topic reference; no source tables or recommended settings are reproduced.
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