BIZOMAT MAKEOS · Quality
Machine Axis Error Budget (Positioning, Abbe, Thermal, Fixture)
Can this machine axis hold my tolerance once positioning error, repeatability, Abbe error, thermal growth of the screw and fixture error are added up?
Error budget for a dimension made or measured along one machine axis: positioning accuracy and repeatability from the machine test, the Abbe error from the offset between the work and the scale and the axis tilt (as the Abbe error tool), the uncompensated part of the screw or scale thermal growth (as the thermal growth tool), and the fixture or set-up error, combined by worst-case sum and by root-sum-square, against the share of the tolerance you allow the machine.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Tolerance band on the dimension | mm | Total tolerance band (upper minus lower limit). |
| Share of the tolerance allowed for the machine | % | Part of the tolerance you allow for machine errors, leaving the rest for tool wear, deflection and measurement. |
| Positioning accuracy (from the axis test) | µm | Positioning accuracy over the travel from an ISO 230-2 or similar test. |
| Repeatability | µm | Repeatability from the same test. |
| Abbe offset | mm | Distance between the work point and the axis scale or screw line. |
| Axis pitch or yaw over the travel | arcsec | Angular error (pitch or yaw) of the axis over the travel. |
| Screw or scale length in use | mm | Length of screw between the fixed bearing and the working position (or scale length). |
| Screw expansion coefficient | µm/m·K | Expansion coefficient of the screw or scale. |
| Screw temperature rise | K | Temperature rise of the screw in running. |
| Share of thermal growth compensated | % | Share of the thermal growth the control compensates (0 if none). |
| Fixture and set-up error | µm | Error from locating and clamping the part. |
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: Tolerance band on the dimension
- What it is
- The total width of the tolerance on the dimension, upper limit minus lower limit, from the drawing.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.0005 mm to 100 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.05 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- The whole band: a tolerance of ±0.02 mm is a band of 0.04 mm.
What this means: Share of the tolerance allowed for the machine
- What it is
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 5% to 100%. 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: Positioning accuracy (from the axis test)
- What it is
- The positioning accuracy of the axis over its travel, as reported by an acceptance test (ISO 230-2 or similar) or the builder’s certificate.
- Units and format
- Enter a number in µm: micrometres (µm, microns). The form accepts 0 µm to 1000 µm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 8 µm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Micrometres, not millimetres.
What this means: Repeatability
- What it is
- The repeatability of the axis: the scatter when the same position is approached again and again, from the same axis test.
- Units and format
- Enter a number in µm: micrometres (µm, microns). The form accepts 0 µm to 1000 µm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3 µm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Abbe offset
- What it is
- The perpendicular distance between the line where the work is cut or measured and the line of the axis scale or ballscrew.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 5000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 100 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The Abbe error rises in direct proportion to the offset.
What this means: Axis pitch or yaw over the travel
- What it is
- The angular error of the axis (pitch or yaw) over the travel, from a laser or level test. It turns the offset into an Abbe error.
- Units and format
- Enter a number in arcsec: arcsec. The form accepts 0 arcsec to 3600 arcsec. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5 arcsec as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Arc-seconds: 3600 arcsec = 1°.
What this means: Screw or scale length in use
- What it is
- The length of screw between its fixed bearing and the working position, or the length of scale in use, over which heat growth adds up.
- 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 500 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The thermal error rises in direct proportion.
What this means: Screw expansion coefficient
- What it is
- The linear thermal expansion coefficient of the screw or scale material, from the material or scale data.
- Units and format
- Enter a number in µm/m·K: micrometres per metre per kelvin. The form accepts 0 µm/m·K to 40 µm/m·K. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 11.5 µm/m·K as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- µm/m·K is the same as 10⁻⁶ per kelvin; enter the number without the ×10⁻⁶.
What this means: Screw temperature rise
- What it is
- How much the screw or scale warms up in running, above the temperature at which the axis was calibrated.
- Units and format
- Enter a number in K: kelvin (a difference of 1 K equals 1 °C). The form accepts 0 K to 100 K. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5 K as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Share of thermal growth compensated
- What it is
- The part of the thermal growth that the control corrects, for example through a thermal compensation table or temperature-compensated scale.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 0% to 100%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 80% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The thermal error is proportional to the uncompensated share, 100 % minus this value.
- Get it right
- 0 = no compensation: the full growth counts.
What this means: Fixture and set-up error
- What it is
- The error from locating and clamping the part in its fixture, from a set-up study or fixture inspection.
- Units and format
- Enter a number in µm: micrometres (µm, microns). The form accepts 0 µm to 1000 µm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5 µm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Method and formulas
- Abbe error = offset × tan(tilt) (as the Abbe error tool).
- Thermal error = L α ΔT × (1 − compensation) (as the thermal growth tool).
- Worst case = sum of all contributions; RSS = √(sum of squares); compare with the allowed share of the tolerance band.
- Example: worst case 24.174 µm, RSS 11.702 µm, against 25 µm allowed; the largest source is Positioning accuracy.
Limitations
- Treats contributions as independent for RSS; correlated thermal errors add directly.
- One axis only; multi-axis features need volumetric errors.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Decide whether a machine can hold a tight tolerance
- Find the largest error source on an axis
- Justify thermal compensation or a better fixture
Who it is for
- Manufacturing engineers
- Quality engineers
- Machine tool engineers
What you get
- Each error contribution
- Worst-case and root-sum-square totals
- Total against the allowed share of tolerance
Interpreting the result
- Thermal growth is often the largest error on long screws: compensation or a scale feedback removes most of it.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 709–735, 375–376. Topic reference; no source tables or recommended settings are reproduced.
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