BIZOMAT MAKEOS · Geometry
Rollers Around a Shaft and Circles in Circles
What size rollers fit around this shaft, how big a circle holds N equal circles, or how many circles fit in a rectangle?
Diameter of equal rollers fitting around a shaft (or the shaft for given rollers), the enclosing circle for up to ten equal circles, and the number of equal circles in a rectangle by square and hexagonal packing.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Problem | choice | Which packing problem to solve. |
| Number of rollers or circles | items | Number of rollers around the shaft, or circles inside the circle (rectangle mode ignores it). |
| Shaft diameter, or circle diameter | mm | Shaft diameter (rollers around a shaft) or diameter of each small circle (other modes). |
| Rectangle width | mm | Rectangle width (rectangle mode). |
| Rectangle height | mm | Rectangle height (rectangle mode). |
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: Problem
- What it is
- Which packing problem to solve: rollers fitted round a central shaft, equal circles inside the smallest possible circle, or equal circles nested in a rectangle.
- Units and format
- Choose one of 3 options from the list: N rollers around a shaft; N equal circles inside a circle (N ≤ 10); Equal circles in a rectangle.
- Example
- The form opens with “N rollers around a shaft” selected as a worked example. Choose the option that matches your case.
- Get it right
- Each mode reads different inputs: rectangle mode ignores the count; the other modes ignore the rectangle size.
What this means: Number of rollers or circles
- What it is
- How many rollers sit around the shaft, all touching their neighbours and the shaft, or how many equal circles must fit inside the enclosing circle.
- Units and format
- Enter a number in items: a count of items. The form accepts 1 items to 10000 items. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 8 items as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Rollers around a shaft need at least three; the circle-in-circle mode handles up to ten. Not used in rectangle mode.
What this means: Shaft diameter, or circle diameter
- What it is
- In roller mode, the diameter of the central shaft the rollers sit round. In the other modes, the diameter of each equal small circle being packed.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- In the roller and circle-in-circle modes the result is directly proportional to this diameter.
- Get it right
- Its meaning changes with the mode: shaft diameter for rollers, small circle diameter otherwise.
What this means: Rectangle width
- What it is
- The width of the rectangular area or sheet the circles must fit in, along the direction the rows of circles run.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 500 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Deduct edge margins and cutting gaps before entering it. Used only in rectangle mode.
What this means: Rectangle height
- What it is
- The height of the rectangular area or sheet, across the rows of circles.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 300 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Deduct edge margins before entering it. Used only in rectangle mode.
Method and formulas
- Rollers around a shaft of radius R: roller diameter = 2R sin(π/n) ÷ (1 − sin(π/n)); outside diameter = shaft + 2 × roller.
- Smallest circle around N equal circles of diameter d: d × k, with k = 1, 2, 2.1547, 2.4142, 2.7013, 3, 3, 3.3048, 3.6131, 3.8130 for N = 1…10 (proven optimal packings).
- Rectangle: square grid ⌊W/d⌋ × ⌊H/d⌋; hexagonal rows of alternating ⌊W/d⌋ and ⌊(W − d/2)/d⌋ at a pitch of 0.866d.
- Example: 8 rollers around a 40 mm shaft are 24.797 mm each.
Limitations
- Rollers around a shaft touch each other and the shaft (no clearance); add running clearance in practice.
- Rectangle counts use simple regular patterns; better irregular packings may exist.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Design roller cages and needle-roller layouts
- Size a sleeve or tube for a bundle of bars
- Estimate parts per sheet or tubes per box
Who it is for
- Design engineers
- Toolmakers
- Estimators
What you get
- Roller diameter and ring outside diameter
- Enclosing circle diameter
- Circles per rectangle by square and hexagonal packing
Interpreting the result
- Allow clearance between rollers and for cage bars when designing bearings.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 88–94. Topic reference; no source tables or recommended settings are reproduced.
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Open the Rollers Around a Shaft and Circles in Circles calculator