BIZOMAT MAKEOS · Mathematics
Geometric Speed and Size Series
What geometric series of speeds (or sizes) runs between these limits in this many steps?
Step ratio and every value of a geometric progression between a minimum and maximum, as used for machine spindle speeds, feeds and preferred-number size ranges, with the nearest Renard (R10/R20/R40) ratio.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Smallest value | First value of the series (e.g. lowest spindle speed). | |
| Largest value | Last value of the series (e.g. highest spindle speed). | |
| Number of values | values | Number of values including both ends. |
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: Smallest value
- What it is
- The first and smallest value of the series, such as the lowest spindle speed or the smallest size in a range.
- Units and format
- Enter a plain number with no unit. The form accepts 0.000001 or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Largest value
- What it is
- The last and largest value of the series, such as the highest spindle speed or the largest size in a range.
- Units and format
- Enter a plain number with no unit. The form accepts 0.000001 or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2000 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- It must be larger than the smallest value; use the same unit for both.
What this means: Number of values
- What it is
- How many values the series contains, counting both the smallest and the largest.
- Units and format
- Enter a number in values: values. The form accepts 2 values to 100 values. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 12 values as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- More values give a smaller step ratio between neighbours.
- Get it right
- Count the end values too: n values have n − 1 steps between them.
Method and formulas
- φ = (max ÷ min)^(1/(n − 1)).
- Values: min × φ^k for k = 0 … n − 1.
- Renard ratios: R5 1.585, R10 1.259, R20 1.122, R40 1.059.
- Example: 12 speeds from 50 to 2000 rpm need a step ratio of 1.3984.
Limitations
- Geometric series only; arithmetic series have equal differences instead of equal ratios.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Design spindle speed or feed steps for a gearbox
- Build a size range of preferred values
- Check a machine’s speed steps
Who it is for
- Machine designers
- Product designers
- Engineers
What you get
- Step ratio φ
- Every value of the series
- Nearest Renard series
Interpreting the result
- A ratio near 1.26 (R10) is common for machine tool speeds; smaller ratios give finer choice but more gears.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 37–39. Topic reference; no source tables or recommended settings are reproduced.
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