BIZOMAT MAKEOS · Engineering
Signed Three-Term Tolerance Stack
What are the worst-case limits of this signed assembly dimension?
Calculate exact interval bounds for a three-term linear dimension chain with asymmetric tolerances and signed coefficients.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Term 1: nominal dimension | mm | Nominal dimension in the signed assembly equation, measured from a consistent datum. |
| Term 1: lower deviation magnitude | mm | Positive distance from nominal down to the permitted minimum; 0 means no lower deviation. |
| Term 1: upper deviation magnitude | mm | Positive distance from nominal up to the permitted maximum; 0 means no upper deviation. |
| Term 1: signed coefficient | × | Coefficient in the linear equation: +1 adds, −1 subtracts and 0 excludes this term. |
| Term 2: nominal dimension | mm | Nominal dimension in the signed assembly equation, measured from a consistent datum. |
| Term 2: lower deviation magnitude | mm | Positive distance from nominal down to the permitted minimum; 0 means no lower deviation. |
| Term 2: upper deviation magnitude | mm | Positive distance from nominal up to the permitted maximum; 0 means no upper deviation. |
| Term 2: signed coefficient | × | Coefficient in the linear equation: +1 adds, −1 subtracts and 0 excludes this term. |
| Term 3: nominal dimension | mm | Nominal dimension in the signed assembly equation, measured from a consistent datum. |
| Term 3: lower deviation magnitude | mm | Positive distance from nominal down to the permitted minimum; 0 means no lower deviation. |
| Term 3: upper deviation magnitude | mm | Positive distance from nominal up to the permitted maximum; 0 means no upper deviation. |
| Term 3: signed coefficient | × | Coefficient in the linear equation: +1 adds, −1 subtracts and 0 excludes this term. |
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: Term 1: nominal dimension
- What it is
- The basic size of this dimension in the chain, as drawn, measured from the same datum scheme as the other terms. It is multiplied by the term’s coefficient before the terms are added.
- Units and format
- Enter a number in mm: millimetres. The form accepts -1000000 mm to 1000000 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
- Each nominal moves the result by its value times the coefficient.
- Get it right
- Enter the nominal as a positive size and let the coefficient carry the sign; entering a negative nominal and −1 adds it back.
What this means: Term 1: lower deviation magnitude
- What it is
- How far below the nominal this dimension is allowed to go, written as a positive distance. For a drawing callout of 60 −0.05, this is the 0.05.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.1 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 lower deviation lowers the minimum result for a positive coefficient and raises the maximum for a negative one.
- Get it right
- Type the size of the deviation without its minus sign; the tool applies the direction itself.
What this means: Term 1: upper deviation magnitude
- What it is
- How far above the nominal this dimension is allowed to go, written as a positive distance. For a callout of 60 +0.1, this is the 0.1.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.2 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 upper deviation raises the maximum result for a positive coefficient and lowers the minimum for a negative one.
- Get it right
- Enter a positive magnitude even when the drawing shows a plus sign; 0 means the dimension may not exceed nominal.
What this means: Term 1: signed coefficient
- What it is
- The multiplier that places this dimension in the assembly equation: +1 when it adds to the gap or result, −1 when it subtracts, 0 to leave it out, or another value where the equation scales the dimension.
- Units and format
- Enter a number in ×: a multiplier (type 1.5 for 1.5 times). The form accepts -100 × to 100 ×. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 × as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A negative coefficient swaps the lower and upper contributions of the term before they are added.
- Get it right
- Draw the loop first: dimensions that push the result larger are positive, those that make it smaller are negative.
What this means: Term 2: nominal dimension
- What it is
- The basic size of this dimension in the chain, as drawn, measured from the same datum scheme as the other terms. It is multiplied by the term’s coefficient before the terms are added.
- Units and format
- Enter a number in mm: millimetres. The form accepts -1000000 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 60 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Each nominal moves the result by its value times the coefficient.
- Get it right
- Enter the nominal as a positive size and let the coefficient carry the sign; entering a negative nominal and −1 adds it back.
What this means: Term 2: lower deviation magnitude
- What it is
- How far below the nominal this dimension is allowed to go, written as a positive distance. For a drawing callout of 60 −0.05, this is the 0.05.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 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.
- Effect on the result
- A larger lower deviation lowers the minimum result for a positive coefficient and raises the maximum for a negative one.
- Get it right
- Type the size of the deviation without its minus sign; the tool applies the direction itself.
What this means: Term 2: upper deviation magnitude
- What it is
- How far above the nominal this dimension is allowed to go, written as a positive distance. For a callout of 60 +0.1, this is the 0.1.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.1 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 upper deviation raises the maximum result for a positive coefficient and lowers the minimum for a negative one.
- Get it right
- Enter a positive magnitude even when the drawing shows a plus sign; 0 means the dimension may not exceed nominal.
What this means: Term 2: signed coefficient
- What it is
- The multiplier that places this dimension in the assembly equation: +1 when it adds to the gap or result, −1 when it subtracts, 0 to leave it out, or another value where the equation scales the dimension.
- Units and format
- Enter a number in ×: a multiplier (type 1.5 for 1.5 times). The form accepts -100 × to 100 ×. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -1 × as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A negative coefficient swaps the lower and upper contributions of the term before they are added.
- Get it right
- Draw the loop first: dimensions that push the result larger are positive, those that make it smaller are negative.
What this means: Term 3: nominal dimension
- What it is
- The basic size of this dimension in the chain, as drawn, measured from the same datum scheme as the other terms. It is multiplied by the term’s coefficient before the terms are added.
- Units and format
- Enter a number in mm: millimetres. The form accepts -1000000 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Each nominal moves the result by its value times the coefficient.
- Get it right
- Enter the nominal as a positive size and let the coefficient carry the sign; entering a negative nominal and −1 adds it back.
What this means: Term 3: lower deviation magnitude
- What it is
- How far below the nominal this dimension is allowed to go, written as a positive distance. For a drawing callout of 60 −0.05, this is the 0.05.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.02 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 lower deviation lowers the minimum result for a positive coefficient and raises the maximum for a negative one.
- Get it right
- Type the size of the deviation without its minus sign; the tool applies the direction itself.
What this means: Term 3: upper deviation magnitude
- What it is
- How far above the nominal this dimension is allowed to go, written as a positive distance. For a callout of 60 +0.1, this is the 0.1.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.03 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 upper deviation raises the maximum result for a positive coefficient and lowers the minimum for a negative one.
- Get it right
- Enter a positive magnitude even when the drawing shows a plus sign; 0 means the dimension may not exceed nominal.
What this means: Term 3: signed coefficient
- What it is
- The multiplier that places this dimension in the assembly equation: +1 when it adds to the gap or result, −1 when it subtracts, 0 to leave it out, or another value where the equation scales the dimension.
- Units and format
- Enter a number in ×: a multiplier (type 1.5 for 1.5 times). The form accepts -100 × to 100 ×. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -1 × as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A negative coefficient swaps the lower and upper contributions of the term before they are added.
- Get it right
- Draw the loop first: dimensions that push the result larger are positive, those that make it smaller are negative.
Method and formulas
- Define y = c₁n₁ + c₂n₂ + c₃n₃. Each dimension lies between n − lower deviation and n + upper deviation.
- Multiply both limits by the signed coefficient, take the smaller and larger products, then sum the three lower products and three upper products.
- Example: 100 (−0.1/+0.2) − 60 (−0.05/+0.1) − 30 (−0.02/+0.03) mm gives 10 mm nominal, with limits 9.77 to 10.27 mm.
Limitations
- Linear size-only worst-case calculation; geometry, deformation and temperature require separate treatment.
- Coefficients may combine repeated terms for worst-case arithmetic. Correlated constraints can make the interval conservative. There is no RSS or probability claim.
- Three terms are supported. Carry validated limits when combining a longer sub-chain into one term.
- Default values are worked examples, not recommended settings or material properties.
Typical uses
- assembly clearance
- worst-case stack
- asymmetric tolerance
Who it is for
- Design engineers
- Drawing checkers
- Toolmakers
What you get
- Nominal result
- Minimum and maximum result
- Contribution of each term
Interpreting the result
- Write the physical equation first and check subtraction directions against the drawing.
- Compare both limits with the functional requirement; a positive nominal gap alone does not establish clearance.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 608–636. Topic reference; implementation and examples are original, and no source tables are reproduced.
- Method: direct interval arithmetic for a linear dimension equation; no statistical tolerancing standard is assumed.
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