KEVOS® MAKEOS · Gears
Milled Bevel Gear Dimensions (Cone Angles, Blank, Cutting Angle, Vertex Distances)
For a pair of bevel gears to be cut with formed milling cutters, what are the pitch cone, addendum, dedendum, face and cutting angles, the outside diameters and vertex distances to turn the blanks to, the small-end tooth sizes, the widest face allowed and the tooth numbers for selecting the cutters?
Dimensions of a pair of milled bevel gears with shafts at any angle by the handbook’s rules for milled bevel gears, where pinion and gear have equal tooth thickness, addendum and dedendum: pitch cone angles tan γP = sin Σ ÷ (NG ÷ NP + cos Σ) and γG = Σ − γP, pitch diameters N ÷ P, addendum 1 ÷ P, dedendum 1.157 ÷ P, whole depth 2.157 ÷ P, tooth thickness 1.5708 ÷ P, pitch cone radius C = D ÷ 2 sin γ, the widest face (C ÷ 3 or 8 ÷ P, whichever is smaller), small-end addendum and thickness in the ratio (C − F) ÷ C, addendum and dedendum angles tan α = S ÷ C and tan δ = (S + A) ÷ C, face angle γ + α, compound rest angle 90° − γ − α, cutting angle γ − δ (or γ − α for uniform clearance), angular addendum S cos γ, outside diameter D + 2K, vertex distances J = (O ÷ 2) cot (γ + α) and j = J (C − F) ÷ C, and the number of teeth N ÷ cos γ for which to select each formed cutter.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Module m (25.4 ÷ P) | mm | Module of the gears; for a diametral pitch P it is 25.4 ÷ P. |
| Teeth in pinion NP | teeth | Number of teeth in the pinion. |
| Teeth in gear NG | teeth | Number of teeth in the gear. |
| Shaft angle Σ | ° | Angle between the pinion and gear shafts; 90° for shafts at right angles. |
| Face width F | mm | Width of face (tooth length) you intend for both gears. |
| Cutting (root) angle | choice | Whether to cut the root with the dedendum angle as in the handbook table, or with the addendum angle so the clearance is uniform as the handbook recommends. |
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: Module m (25.4 ÷ P)
- What it is
- Module of the pinion and gear, the pitch diameter per tooth in millimetres.
- 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 6.35 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, depth and tooth thickness.
- Get it right
- For an inch diametral pitch P, enter 25.4 ÷ P.
What this means: Teeth in pinion NP
- What it is
- Number of teeth in the pinion, the smaller bevel gear.
- Units and format
- Enter a number in teeth: a whole number of teeth. The form accepts 5 teeth to 1000 teeth. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 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 pinion pitch cone angle and diameters.
- Get it right
- Count the teeth on the drawing or the part.
What this means: Teeth in gear NG
- What it is
- Number of teeth in the gear, the larger bevel gear.
- Units and format
- Enter a number in teeth: a whole number of teeth. The form accepts 5 teeth to 1000 teeth. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 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 gear pitch cone angle and diameters.
- Get it right
- Gear teeth ÷ pinion teeth gives the speed ratio.
What this means: Shaft angle Σ
- What it is
- Angle between the pinion and gear shafts.
- 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 70° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The two pitch cone angles add up to it.
- Get it right
- 90° for shafts at right angles; acute or obtuse angles also work.
What this means: Face width F
- What it is
- Width of face, the length of the teeth along the pitch cone.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 38.1 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A wider face makes the small-end teeth smaller.
- Get it right
- Keep it to a third of the pitch cone radius or less.
What this means: Cutting (root) angle
- What it is
- How the cutting (root) angle is worked out.
- Units and format
- Choose one of 2 options from the list: Pitch cone angle − dedendum angle (handbook table); Pitch cone angle − addendum angle (uniform clearance).
- Example
- The form opens with “Pitch cone angle − dedendum angle (handbook table)” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Changes the dividing head angle by the difference of the two angles.
- Get it right
- Uniform clearance (addendum angle) is what the handbook recommends.
Method and formulas
- tan γP = sin Σ ÷ (NG ÷ NP + cos Σ); γG = Σ − γP; D = N m.
- S = m; S + A = 1.157 m; W = 2.157 m; T = 1.5708 m; C = D ÷ (2 sin γ).
- F (max) = the smaller of C ÷ 3 and 8 m; s = S (C − F) ÷ C; t = T (C − F) ÷ C.
- tan α = S ÷ C; tan δ = (S + A) ÷ C; face angle γ + α; compound rest angle 90° − γ − α; cutting angle γ − δ, or γ − α for uniform clearance.
- K = S cos γ; O = D + 2K; J = (O ÷ 2) cot (γ + α); j = J (C − F) ÷ C.
- Number of teeth for selecting the cutter N′ = N ÷ cos γ.
- Example: the pinion’s pitch cone angle is 18.296° and its cutter is selected for 21.065 teeth; the gear’s for 80.681 teeth.
Limitations
- These are the proportions for milled bevel gears, with equal addendums and tooth thicknesses on pinion and gear; generated bevel gears use long and short addendums.
- Milled bevel gears are approximate in form and are not suitable for high speeds or accurate motion.
- Pitch cone angles of 90° or more (crown and internal bevel gears) are not covered.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Turn bevel gear blanks to the right outside diameter, face angle and vertex distance
- Set the dividing head to the cutting angle for milling bevel gear teeth
- Find the tooth numbers for selecting formed bevel gear cutters
Who it is for
- Machinists and gear cutters
- Gear designers
- Students
What you get
- Pitch cone angles and pitch diameters
- Addendum, dedendum, whole depth and tooth thickness
- Pitch cone radius and the widest face allowed
- Small-end addendum and tooth thickness
- Addendum, dedendum, face, compound rest and cutting angles for each gear
- Angular addendum, outside diameter and vertex distances for each gear
- Tooth numbers for selecting the formed cutters
Interpreting the result
- The face width should not exceed the maximum, or the teeth become too small at the inner end.
- Select the cutters from the spur gear cutter series for the tooth numbers N′ shown, using the gear cutter number tool.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2181–2184, 2187. Topic reference; no source tables or recommended settings are reproduced.
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