Unbalance is not a one-dimensional measurement, and tolerancing it as though it were causes real problems. A diameter specified as Ø20 mm ± 0.10 mm has a tolerance you can draw on a number line. Unbalance always carries both a magnitude and a direction, so its tolerance is an area in a plane, not an interval on a line.
Drawn from Unbalance Tolerance in Vector Space by Joel M. Book and John Haidler.
Unbalance Is a Vector
Every unbalance reading has a magnitude and an angle. It can be written in polar form (R, Θ) or resolved into Cartesian components (X, Y), but either way it occupies a point in a plane rather than a position on a scale.
Adding or resolving those vectors by hand is tedious and easy to get wrong; our vector calculator converts between polar and Cartesian form and adds unbalance vectors together directly. For multi-step vector expressions, such as influence coefficient work, use the vector expression calculator.
That single fact is what makes the four specification styles below behave so differently, even when the numbers in them look similar.
Case 1 — Maximum Unbalance of T
The most common form: maximum residual unbalance T g·mm, with no angle specified.
The tolerance zone is a circle of radius T centred on zero. Any result falling inside that circle passes, at any angle.
| Property | Value |
|---|---|
| Zone shape | Circle centred at the origin |
| Radius | T g·mm |
| Upper limit | +T g·mm |
| Lower limit | −T g·mm |
| Total width | 2T g·mm |
Note what the lower limit means here. There is no minimum acceptable unbalance — zero is the ideal and sits at the centre of the zone. The tolerance is genuinely two-sided in each Cartesian direction while being one-sided in magnitude, which is exactly where one-dimensional intuition starts to fail.
Case 2 — Unbalance of U ± T
Here a non-zero unbalance is the target. The tolerance zone becomes an annulus — a ring — with inner radius U − T and outer radius U + T.
| Property | Value |
|---|---|
| Zone shape | Annulus (ring) centred at the origin |
| Nominal | U g·mm |
| Upper limit | U + T g·mm |
| Lower limit | U − T g·mm |
| Total width | 2T g·mm |
This form arises when a part is deliberately built with a known unbalance, most often to compensate for something it will be assembled to. Angle is still unconstrained, so the acceptable region is the full ring.
Case 3 — Unbalance of U ± T at a Defined Angle
Adding an angular requirement — ±G degrees from a reference axis A — reduces the annulus to a segment of a ring.
| Property | Value |
|---|---|
| Zone shape | Annular sector |
| Magnitude limits | U + T and U − T g·mm |
| Angular limits | ±G° from axis A |
| Total magnitude tolerance | 2T g·mm |
| Total angular tolerance | 2G° |
One caution worth flagging. Where T exceeds U, the inner radius U − T goes negative and the geometry stops behaving as a simple sector. The zone becomes a pair of unequal circular sectors on opposite sides of the origin, which is almost never what the person writing the specification intended. If T > U on a drawing, the specification is worth revisiting before it reaches production.
Case 4 — Maximum Unbalance with a Bobweight
A bobweight is a known mass fitted at a known radius to represent a component the part will later carry — a connecting rod on a crankshaft, for instance. The part is balanced with the bobweight installed, so the specification has to account for the bobweight’s own variation.
| Variable | Meaning | Units |
|---|---|---|
| W | Bobweight mass | g |
| V | Weight variation | g |
| L | Location radius | mm |
| Φ | Angular position | degrees |
| P | Positional tolerance on Φ | — |
Where the positional tolerance is small relative to the radius and the weight variation small relative to the weight — P « L and V « W — the zone approximates a circle:
| Property | Value |
|---|---|
| Nominal | W × L g·mm |
| Upper limit | L × (W + V) g·mm |
| Lower limit | L × (W − V) g·mm |
| Total width | L × 2V g·mm |
Those simplifications are worth checking rather than assuming. Where the bobweight is light or its position loosely toleranced, the approximation stops holding and the real zone has to be worked out properly.
Why the Zone Shape Matters
Two specifications can carry identical numbers and describe quite different acceptable regions. A circular zone centred at zero, an annulus, a sector and a bobweight-offset circle are four different shapes, and a part accepted under one can fail another.
When comparing a supplier specification against a customer drawing, or translating between measurement systems, the shape of the zone is the thing to compare — not the magnitude alone.
For where those planes should be defined in the first place, see specifying unbalance and tolerance plane location.
Frequently Asked Questions
Why can’t unbalance be toleranced like a diameter?
Because unbalance has direction as well as magnitude. A diameter tolerance is an interval on a number line; an unbalance tolerance is an area in a plane.
What shape is a maximum-unbalance tolerance zone?
A circle centred on zero, with radius equal to the stated maximum. Any magnitude within it passes at any angle.
What happens if the tolerance is larger than the nominal unbalance?
The zone stops being a simple ring or sector and becomes a pair of unequal circular sectors on opposite sides of the origin. This is rarely intended and usually indicates the specification needs revisiting.
What is a bobweight?
A known mass fitted at a known radius to simulate a component the part will carry in service, so the part can be balanced as it will actually run.
Source
Adapted from Unbalance Tolerance in Vector Space by Joel M. Book and John Haidler, Balance Technology Inc.
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