KEVOS® MAKEOS · Gears
Worm Profile Change with Cutter or Grinding Wheel Size (Profile Deviation and Pressure Angle Correction)
If one worm is milled with a small cutter and another is ground with a large wheel, how far does each normal profile depart from a straight side, how much does each change the pressure angle, and what correction makes the two worms interchangeable?
Profile deviation and pressure angle change produced in a worm thread by the diameter of the milling cutter or grinding wheel, by the formulas the handbook gives from ANSI B6.9: the radius of curvature of the normal profile of an involute thread and of the thread actually produced, the difference between the pressure angle of the thread and of the tool in minutes, the slant height of the addendum, the departure of the profile from a straight side and the effect of the pressure angle change along the slant height, for two tools of different diameter, with the profile curve and pressure angle correction that make their worms alike, compared with your profile tolerance.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Worm | ||
| Pitch diameter of the worm | mm | Pitch diameter of the worm; the formulas use the pitch radius r = d ÷ 2. |
| Lead angle | ° | Lead angle of the worm thread, measured from a plane square to the worm axis. |
| Normal pressure angle of the cutter or wheel | ° | Normal pressure angle of the milling cutter or grinding wheel, normally the pressure angle the worm is specified with. |
| Threads (starts) on the worm | Number of threads or starts on the worm, as the formula for the pressure angle change uses it. | |
| Addendum of the worm | mm | Addendum of the worm thread, from the worm proportions. |
| Tools | ||
| First tool diameter (milling cutter) | mm | Diameter of the first (normally smaller) tool, for example a milling cutter; the handbook works with 2 in (50.8 mm). |
| Second tool diameter (grinding wheel) | mm | Diameter of the second (normally larger) tool, for example a grinding wheel; the handbook works with 20 in (508 mm). |
| Tolerance | ||
| Profile tolerance accepted (curve departure) | mm | Largest difference in profile curvature between the two worms that your profile tolerance allows, measured from a straight side. |
| Tolerance accepted along the slant height (pressure angle) | mm | Largest difference along the slant height of the thread, from the pressure angle change, that your tolerance allows. |
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.
Worm
What this means: Pitch diameter of the worm
- What it is
- Pitch diameter of the worm, where the thread thickness equals the space.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.1 mm to 2000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 22.405 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 worm changes the profile curvature and the pressure angle change of every tool.
- Get it right
- Take it from the worm drawing; the formulas use half of it, the pitch radius.
What this means: Lead angle
- What it is
- Lead angle of the worm thread, measured from a plane square to the worm axis.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0.5° to 45°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Steeper leads give much larger profile changes: the pressure angle change grows with the cube of sin λ.
- Get it right
- Use the lead angle at the pitch diameter, not the helix angle measured from the axis.
What this means: Normal pressure angle of the cutter or wheel
- What it is
- Normal pressure angle of the milling cutter or grinding wheel.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 5° to 40°. 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
- Sets the involute profile radius and the slant height of the addendum.
- Get it right
- Normally the specified worm pressure angle, such as 20°.
What this means: Threads (starts) on the worm
- What it is
- Number of threads or starts on the worm.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 20. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The pressure angle change in the formula falls as the number of threads rises.
- Get it right
- Count the thread starts at the end of the worm.
What this means: Addendum of the worm
- What it is
- Addendum of the worm thread, from the pitch line to the tip.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 200 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.12 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The profile departure grows with the square of the addendum’s slant height.
- Get it right
- Take it from the worm proportions; for the fine-pitch standard it is 0.3183 × normal pitch.
Tools
What this means: First tool diameter (milling cutter)
- What it is
- Diameter of the first, smaller tool, usually a disc milling cutter.
- Units and format
- Enter a number in mm: millimetres. The form accepts 1 mm to 5000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50.8 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A small cutter curves the profile and changes the pressure angle the most.
- Get it right
- Use the cutter’s actual diameter; it shrinks a little with resharpening.
What this means: Second tool diameter (grinding wheel)
- What it is
- Diameter of the second, larger tool, usually a grinding wheel.
- Units and format
- Enter a number in mm: millimetres. The form accepts 1 mm to 5000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 508 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A large wheel comes closest to the involute profile.
- Get it right
- Use the wheel diameter you will work at; it falls as the wheel wears.
Tolerance
What this means: Profile tolerance accepted (curve departure)
- What it is
- Profile tolerance you allow between the two worms, as a departure from a straight side.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.005 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Only moves the profile check.
- Get it right
- Take it from the worm drawing or the wormgear maker.
What this means: Tolerance accepted along the slant height (pres…
- What it is
- Tolerance you allow along the slant height of the thread from the pressure angle change.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.005 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Only moves the pressure angle check.
- Get it right
- Take it from the pressure angle tolerance on the worm drawing.
Method and formulas
- Pitch radius r = d ÷ 2; tool radius R = tool diameter ÷ 2 for each tool.
- ρ_ni = r sin φn ÷ (sin λ)² — radius of curvature of the normal profile of an involute thread (1).
- ρ_n = ρ_ni + r ρ_ni ÷ (R (cos λ)²) — radius of curvature of the normal profile the tool produces (2).
- Δφ = 5400 r (sin λ)³ ÷ (n (R (cos λ)² + r)) minutes — difference between the thread and tool normal pressure angles (3).
- q = a sec φn (4); y = q² ÷ (2 ρ_n) (5); s = 0.000582 q Δφ (6).
- Δy = y_w − y_c (7); Δs = s_c − s_w (8), where c is the first tool and w the second.
- Example: a 50.8 mm cutter changes the pressure angle by 24.995′ and a 508 mm wheel by 3.7489′; making the worms alike needs a profile correction of 0.014588 mm and 0.014738 mm along the slant height.
Limitations
- The formulas are those the handbook gives in slightly modified form from ANSI B6.9-1977 for straight-sided cutters and wheels on cylindrical worms.
- The handbook prints the profile departure of the 2 in cutter in Example 3 with the wheel’s radius of curvature; the tool uses formula (5) for each tool, which gives the example’s final Δy = 0.00057 in.
- Wormgear hob design and the tolerances themselves are not set here; enter the tolerance your drawing allows.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Decide whether a milled worm and a ground worm can run with the same wormgears
- Find the pressure angle and profile correction for a cutter or grinding wheel
- Set a worm profile tolerance that suits the tools in the shop
Who it is for
- Gear designers
- Worm and hob makers
- Production engineers
What you get
- Radius of curvature of the involute normal profile
- Radius of curvature of the profile each tool produces
- Pressure angle change of each tool in minutes
- Slant height of the addendum
- Profile departure y and slant-height effect s for each tool
- Profile and pressure angle correction Δy and Δs between the tools
Interpreting the result
- A small cutter changes the pressure angle and curves the profile much more than a large wheel, so worms made by the two methods are not interchangeable unless one tool is corrected.
- The correction applies to either tool: change its pressure angle by the amount Δs represents and give it a profile curved by Δy.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 3329–3330. Topic reference; no source tables or recommended settings are reproduced.
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