The Central Idea
Engineering design often begins with a nominal value and then specifies an allowable range around it.
In compact form,
\[ \boxed{ \text{nominal geometry} + \text{allowable variation} = \text{acceptable geometry} }. \]The goal is not to make every physical object mathematically exact. The goal is to control variation closely enough that the design still works.
Nominal Dimensions
A nominal dimension is the stated or intended size of a feature.
For example, a shaft may have nominal diameter
\[ 20.00\text{ mm}. \]The actual manufactured diameter may differ slightly from this value while remaining acceptable.
Tolerance
A tolerance specifies the allowed variation in a measured quantity.
A symmetric tolerance may be written
\[ \boxed{20.00\pm0.05\text{ mm}}. \]This means the actual acceptable diameter ranges from
\[ 19.95\text{ mm} \]to
\[ 20.05\text{ mm}. \]Upper and Lower Limits
If a nominal dimension is \(N\) with symmetric tolerance \(\pm t\), then the lower limit is
\[ \boxed{L=N-t}, \]and the upper limit is
\[ \boxed{U=N+t}. \]Therefore, an acceptable measurement \(x\) satisfies
\[ \boxed{N-t\le x\le N+t}. \]Example: Symmetric Tolerance
A component length is specified as
\[ 75.0\pm0.2\text{ mm}. \]The lower limit is
\[ 75.0-0.2=74.8\text{ mm}, \]and the upper limit is
\[ 75.0+0.2=75.2\text{ mm}. \]Therefore,
\[ \boxed{74.8\text{ mm}\le L\le75.2\text{ mm}}. \]Unilateral Tolerances
Tolerances do not have to extend equally in both directions.
For example,
\[ 40.00^{+0.10}_{-0.00}\text{ mm} \]allows the dimension to be as large as
\[ 40.10\text{ mm} \]but not smaller than
\[ 40.00\text{ mm}. \]This is a unilateral tolerance.
Bilateral Tolerances
A bilateral tolerance allows variation in both directions.
It may be symmetric:
\[ 25.00\pm0.03\text{ mm}, \]or asymmetric:
\[ 25.00^{+0.04}_{-0.02}\text{ mm}. \]In the asymmetric case, the acceptable range is
\[ \boxed{24.98\text{ mm}\le x\le25.04\text{ mm}}. \]Total Tolerance Width
The total width of an allowable interval is
\[ \boxed{T=U-L}. \]For
\[ 20.00\pm0.05\text{ mm}, \]the total tolerance width is
\[ 20.05-19.95=0.10\text{ mm}. \]The \(\pm0.05\) tolerance therefore creates a total acceptable span of \(0.10\) millimeter.
Deviation from Nominal
If the nominal value is \(N\) and the measured value is \(x\), then the signed deviation is
\[ \boxed{d=x-N}. \]A positive deviation means the measured value is above nominal.
A negative deviation means it is below nominal.
Example: Checking a Measured Dimension
A dimension is specified as
\[ 50.00\pm0.08\text{ mm}. \]A measured part is
\[ 50.06\text{ mm}. \]Its deviation from nominal is
\[ 50.06-50.00=0.06\text{ mm}. \]Since
\[ |0.06|\le0.08, \]the measured dimension lies within tolerance.
Out-of-Tolerance Measurements
Suppose instead that the measured value is
\[ 50.11\text{ mm}. \]The upper allowable limit is
\[ 50.08\text{ mm}. \]Since
\[ 50.11>50.08, \]the feature is outside the specified dimensional tolerance.
Why Tolerances Exist
Perfect geometric exactness is not physically attainable.
Variation can arise from:
- machine resolution;
- tool wear;
- material variation;
- temperature;
- measurement uncertainty;
- assembly conditions;
- manufacturing process variation.
Tolerances define how much variation the design can accept.
Tighter Is Not Automatically Better
A smaller tolerance may require more precise tools, slower processes, additional inspection, or more expensive manufacturing.
Engineering therefore balances:
\[ \boxed{ \text{function} + \text{manufacturability} + \text{cost} + \text{verification} }. \]A useful tolerance is tight enough to protect function but not tighter than necessary.
Fit Between Parts
Many design problems depend on the relationship between two dimensions rather than either dimension alone.
Suppose a shaft with diameter \(d_s\) fits inside a hole with diameter \(d_h\).
The diametral clearance is
\[ \boxed{C=d_h-d_s}. \]Clearance Fit
A clearance fit occurs when the hole remains larger than the shaft across the intended allowable ranges.
For guaranteed clearance,
\[ \boxed{ d_{h,\min}>d_{s,\max} }. \]This condition ensures that even the smallest acceptable hole is larger than the largest acceptable shaft.
Example: Guaranteed Clearance
A hole may range from
\[ 20.10\text{ mm} \]to
\[ 20.20\text{ mm}, \]while a shaft may range from
\[ 19.90\text{ mm} \]to
\[ 20.00\text{ mm}. \]Minimum clearance occurs with the smallest hole and largest shaft:
\[ C_{\min}=20.10-20.00=0.10\text{ mm}. \]Maximum clearance occurs with the largest hole and smallest shaft:
\[ C_{\max}=20.20-19.90=0.30\text{ mm}. \]Therefore,
\[ \boxed{0.10\text{ mm}\le C\le0.30\text{ mm}}. \]Interference Fit
An interference fit occurs when the shaft is larger than the hole over the intended ranges.
In a guaranteed interference condition,
\[ \boxed{ d_{s,\min}>d_{h,\max} }. \]Assembly then requires deformation, force, thermal methods, or another appropriate process.
Transition Fit
A transition fit occurs when the tolerance ranges overlap so that some combinations produce clearance while others produce interference.
The resulting assembly condition depends on the actual manufactured sizes.
Worst-Case Analysis
Worst-case analysis examines the combination of allowable dimensions that produces the most extreme result.
For clearance,
\[ \boxed{ C_{\min}=H_{\min}-S_{\max} } \]and
\[ \boxed{ C_{\max}=H_{\max}-S_{\min} }. \]This method is deliberately conservative because it assumes extreme limits can occur together.
Tolerance Stack-Up
When several dimensions combine in a chain, their variations can accumulate.
Suppose
\[ L=A+B+C. \]If each dimension has its own tolerance, then the final length \(L\) also varies.
In a simple worst-case model, the maximum total variation is the sum of the individual absolute tolerances.
Worst-Case Stack-Up
Suppose
\[ A=20.0\pm0.1\text{ mm}, \] \[ B=15.0\pm0.2\text{ mm}, \] \[ C=10.0\pm0.1\text{ mm}. \]The nominal total is
\[ L=20+15+10=45\text{ mm}. \]The worst-case tolerance is
\[ 0.1+0.2+0.1=0.4\text{ mm}. \]Therefore,
\[ \boxed{L=45.0\pm0.4\text{ mm}} \]under this simple worst-case model.
Stack-Up with Subtraction
Suppose a clearance is defined by
\[ C=A-B. \]Even though one dimension is subtracted, uncertainty in both dimensions can still widen the possible clearance range.
Worst-case limits should be found directly from the extreme combinations:
\[ \boxed{C_{\min}=A_{\min}-B_{\max}}, \] \[ \boxed{C_{\max}=A_{\max}-B_{\min}}. \]Example: Clearance Stack-Up
Let
\[ A=50.0\pm0.2\text{ mm} \]and
\[ B=47.0\pm0.1\text{ mm}. \]Then
\[ A_{\min}=49.8,\qquad A_{\max}=50.2, \] \[ B_{\min}=46.9,\qquad B_{\max}=47.1. \]Minimum clearance:
\[ C_{\min}=49.8-47.1=2.7\text{ mm}. \]Maximum clearance:
\[ C_{\max}=50.2-46.9=3.3\text{ mm}. \]Therefore,
\[ \boxed{2.7\text{ mm}\le C\le3.3\text{ mm}}. \]Geometric Tolerances
Size alone does not completely describe a physical feature.
Engineering may also need to control:
- straightness;
- flatness;
- circularity;
- parallelism;
- perpendicularity;
- angular orientation;
- position;
- profile.
These are geometric properties of shape, orientation, and location.
Size Tolerance Versus Geometric Tolerance
A rectangular plate can have the correct length and width while still being warped or out of square.
Likewise, a hole can have the correct diameter while its center is misplaced.
Dimensional tolerance answers
\[ \text{How large?} \]Geometric tolerance may answer
\[ \text{How straight? How flat? How oriented? Where located?} \]Flatness
A perfectly flat surface lies in a plane.
Real surfaces deviate slightly.
A flatness tolerance limits how far the actual surface may depart from an ideal planar condition.
Conceptually, the acceptable surface must fit between two parallel planes separated by a specified tolerance distance.
Straightness
A perfectly straight line has no lateral deviation from its ideal direction.
A straightness tolerance limits how far a real line element or axis may wander from an ideal straight condition.
The same geometric idea appears in rails, shafts, beams, edges, and linear guides.
Parallelism
Two ideal parallel lines or planes maintain constant direction.
If two surfaces are intended to be parallel, the design may limit their allowable angular or positional departure from that condition.
In a planar analytic model, parallel nonvertical lines satisfy
\[ m_1=m_2. \]Physical parts approximate this ideal within specified limits.
Perpendicularity
Ideal perpendicular features meet at
\[ 90^\circ. \]A perpendicularity tolerance limits departure from this right-angle relationship.
This matters when mounting faces, holes, shafts, rails, or structural components must align correctly.
Angular Tolerance
An angle may be specified with allowable variation.
For example,
\[ 45.0^\circ\pm0.5^\circ \]permits angles from
\[ 44.5^\circ \]through
\[ 45.5^\circ. \]Small Angular Errors Can Create Large Linear Errors
A small angular deviation can create a larger positional offset over a long distance.
If a line of length \(L\) deviates by angle \(\theta\), its transverse offset is approximately related by
\[ \boxed{e=L\sin\theta}. \]For small angles measured in radians,
\[ \sin\theta\approx\theta, \]so
\[ \boxed{e\approx L\theta}. \]Example: Angular Error over Distance
A \(2\)-meter member is misaligned by
\[ 1^\circ. \]The lateral offset at the end is approximately
\[ e=2\sin1^\circ. \]Therefore,
\[ e\approx0.0349\text{ m}. \]Thus,
\[ \boxed{e\approx34.9\text{ mm}}. \]A seemingly small angular error can therefore become significant over distance.
Positional Tolerance
A feature may have an intended coordinate position
\[ (x_0,y_0). \]If the actual position is
\[ (x,y), \]then the positional error in the plane is
\[ \boxed{ e= \sqrt{(x-x_0)^2+(y-y_0)^2} }. \]A positional tolerance can limit how far the actual feature center may move from its intended location.
Example: Hole Position Error
A hole center is intended at
\[ (40.00,25.00)\text{ mm}. \]The measured center is
\[ (40.06,24.92)\text{ mm}. \]The coordinate differences are
\[ \Delta x=0.06\text{ mm}, \] \[ \Delta y=-0.08\text{ mm}. \]Therefore,
\[ e=\sqrt{0.06^2+(-0.08)^2} =0.10\text{ mm}. \]Thus,
\[ \boxed{e=0.10\text{ mm}}. \]Radial and Diametral Clearance
When a circular shaft is centered inside a circular hole, diametral clearance is
\[ C_d=d_h-d_s. \]The radial clearance on one side is half of that:
\[ \boxed{ C_r=\frac{d_h-d_s}{2} }. \]Distinguishing radial from diametral clearance prevents a common factor-of-two error.
Example: Radial Clearance
A hole diameter is
\[ 30.20\text{ mm}, \]and the shaft diameter is
\[ 30.00\text{ mm}. \]Diametral clearance is
\[ 30.20-30.00=0.20\text{ mm}. \]Radial clearance is
\[ \frac{0.20}{2}=0.10\text{ mm}. \]Therefore,
\[ \boxed{C_r=0.10\text{ mm}}. \]Thermal Expansion and Dimensional Change
Dimensions can change with temperature.
A simplified linear thermal-expansion model is
\[ \boxed{ \Delta L=\alpha L_0\Delta T }, \]where \(L_0\) is the original length, \(\Delta T\) is the temperature change, and \(\alpha\) is the material's coefficient of linear expansion.
This matters when clearances or alignments are sensitive to operating temperature.
Design Conditions Can Change Geometry
A component that fits at one temperature, load, or operating condition may behave differently at another.
Engineering design may therefore need to consider geometry under:
- temperature change;
- elastic deformation;
- wear;
- assembly preload;
- motion;
- environmental conditions.
Nominal dimensions alone may not fully describe operating geometry.
Measurement Resolution and Tolerance
A measuring instrument must have enough resolution to evaluate the required tolerance meaningfully.
If a feature must be controlled within hundredths of a millimeter, an instrument that only reads to the nearest millimeter is insufficient.
Measurement capability must be appropriate for the scale of the design requirement.
Accuracy, Precision, and Repeatability
Accuracy concerns closeness to the correct value.
Precision concerns measurement detail or spread.
Repeatability concerns how consistently a measurement system reproduces similar results under similar conditions.
Tolerance verification depends on all three ideas.
Pass-Fail Inspection
A simple inspection asks whether a measured value lies inside the allowed interval.
If
\[ L\le x\le U, \]the dimension passes the stated limit check.
If
\[ xit fails that check.
Go/No-Go Thinking
Some inspection systems are designed around limits rather than exact numerical measurement.
Conceptually, one boundary verifies that a feature does not exceed one limit, while another verifies that it does not cross the opposite limit.
This reflects the same interval logic:
\[ \boxed{L\le x\le U}. \]Design Margin
A design margin separates a predicted operating condition from a limiting condition.
If a system requires minimum clearance \(C_{\text{req}}\) and the worst-case predicted clearance is \(C_{\min}\), one simple margin is
\[ \boxed{ M=C_{\min}-C_{\text{req}} }. \]A positive value means the predicted worst case still exceeds the stated minimum requirement.
Example: Clearance Margin
Suppose a mechanism requires at least
\[ 0.20\text{ mm} \]clearance, and worst-case analysis predicts
\[ 0.28\text{ mm}. \]Then
\[ M=0.28-0.20=0.08\text{ mm}. \]Therefore,
\[ \boxed{M=0.08\text{ mm}}. \]Constraint-Based Design
An engineering design often must satisfy several spatial constraints at the same time.
For example:
\[ L_{\min}\le L\le L_{\max}, \] \[ C\ge C_{\text{required}}, \] \[ \theta_{\min}\le\theta\le\theta_{\max}, \] \[ e\le e_{\max}. \]The feasible design region consists of choices satisfying all applicable constraints simultaneously.
Tolerance Zones as Geometric Regions
A tolerance can often be visualized as a region in which an actual feature is allowed to lie.
Examples include:
- an interval for a length;
- a band between parallel lines;
- a region between parallel planes;
- a circular region around a nominal point;
- an angular wedge around a nominal direction.
This geometric viewpoint makes tolerances easier to connect to spatial reasoning.
Common Error: Treating Nominal as Exact
A nominal value such as
\[ 25.00\text{ mm} \]describes the target or stated size.
The actual acceptable size depends on the associated tolerance.
Nominal dimensions should not be confused with infinitely exact physical dimensions.
Common Error: Using Only One Side of a Tolerance
A specification
\[ 30.0\pm0.2 \]has both a lower and an upper limit:
\[ 29.8\le x\le30.2. \]Checking only whether the part is below \(30.2\) ignores the lower limit.
Common Error: Adding Tolerances Without Understanding the Chain
For a simple sum,
\[ L=A+B, \]worst-case absolute tolerances add.
But for a difference such as
\[ C=A-B, \]minimum and maximum results should be formed from the appropriate opposing extremes.
The geometry and algebra of the chain determine the correct limit combinations.
Common Error: Confusing Radial and Diametral Clearance
If a centered hole is \(0.20\) millimeter larger in diameter than a shaft, the clearance on one side is not \(0.20\) millimeter.
It is
\[ \frac{0.20}{2}=0.10\text{ mm}. \]Always identify whether a clearance is being stated across the full diameter or from one surface to the other on one side.
Common Error: Ignoring Geometry While Checking Size
A hole can have the correct diameter and still be misplaced.
A plate can have the correct thickness and still be warped.
A shaft can have the correct diameter and still be bent.
Size conformity does not guarantee geometric conformity.
Common Error: Over-Specifying Precision
A tolerance much tighter than functional requirements can increase cost and complexity without improving the design meaningfully.
Engineering specifications should be connected to actual performance requirements.
A Reliable Procedure for Dimensional Tolerance Problems
- Identify the nominal dimension.
- Identify the upper and lower allowed deviations.
- Compute the lower and upper limits.
- Compare the measured value with the allowable interval.
- For fits, identify the extreme size combinations.
- Calculate minimum and maximum clearance or interference.
- For dimension chains, write the geometric relationship first.
- Evaluate worst-case combinations consistently.
- Check units and signs.
- Confirm that the result addresses the intended functional requirement.
Example: Full Fit Analysis
A hole is specified as
\[ 25.00^{+0.08}_{+0.02}\text{ mm}, \]and a shaft is specified as
\[ 24.90^{+0.05}_{-0.03}\text{ mm}. \]The hole limits are
\[ H_{\min}=25.02\text{ mm}, \] \[ H_{\max}=25.08\text{ mm}. \]The shaft limits are
\[ S_{\min}=24.87\text{ mm}, \] \[ S_{\max}=24.95\text{ mm}. \]Minimum clearance:
\[ C_{\min}=25.02-24.95=0.07\text{ mm}. \]Maximum clearance:
\[ C_{\max}=25.08-24.87=0.21\text{ mm}. \]Therefore,
\[ \boxed{0.07\text{ mm}\le C\le0.21\text{ mm}}. \]Because the minimum clearance remains positive, this simplified dimensional analysis guarantees clearance across the stated limits.
Practice 1: Symmetric Limits
Find the lower and upper limits of
\[ 80.0\pm0.3\text{ mm}. \]Practice 2: Asymmetric Limits
Find the allowable range for
\[ 40.00^{+0.06}_{-0.02}\text{ mm}. \]Practice 3: Pass or Fail?
A dimension is specified as
\[ 12.50\pm0.04\text{ mm}. \]A measured part is
\[ 12.55\text{ mm}. \]Is it within the stated tolerance?
Practice 4: Minimum Clearance
A hole ranges from
\[ 15.10\text{ mm} \]to
\[ 15.18\text{ mm}, \]and a shaft ranges from
\[ 14.96\text{ mm} \]to
\[ 15.02\text{ mm}. \]Find the minimum clearance.
Practice 5: Maximum Clearance
Using the same hole and shaft ranges from Practice 4, find the maximum clearance.
Practice 6: Worst-Case Stack-Up
Three dimensions are
\[ 10.0\pm0.1\text{ mm}, \] \[ 15.0\pm0.2\text{ mm}, \] \[ 5.0\pm0.1\text{ mm}. \]They are added to form a total length. Find the nominal total and the simple worst-case tolerance.
Practice 7: Radial Clearance
A centered hole has diameter
\[ 40.30\text{ mm} \]and a shaft has diameter
\[ 40.10\text{ mm}. \]Find the radial clearance.
Practice 8: Positional Error
A point should be located at
\[ (10.00,20.00)\text{ mm}, \]but is measured at
\[ (10.03,20.04)\text{ mm}. \]Find the planar positional error.
Practice 9: Angular Offset
A \(1.5\)-meter member is misaligned by
\[ 2^\circ. \]Approximate the transverse offset at its end using
\[ e=L\sin\theta. \]Practice 10: Identify the Error
A student says that a shaft and hole have guaranteed clearance because their nominal hole diameter is larger than their nominal shaft diameter.
Explain why nominal dimensions alone are insufficient.
Practice Answers
Practice 1
\[ L=80.0-0.3=79.7\text{ mm}, \] \[ U=80.0+0.3=80.3\text{ mm}. \]Therefore,
\[ \boxed{79.7\text{ mm}\le x\le80.3\text{ mm}}. \]Practice 2
\[ L=40.00-0.02=39.98\text{ mm}, \] \[ U=40.00+0.06=40.06\text{ mm}. \]Therefore,
\[ \boxed{39.98\text{ mm}\le x\le40.06\text{ mm}}. \]Practice 3
The acceptable interval is
\[ 12.46\text{ mm}\le x\le12.54\text{ mm}. \]Since
\[ 12.55>12.54, \]the measurement is
\[ \boxed{\text{outside tolerance}}. \]Practice 4
\[ C_{\min}=15.10-15.02=0.08\text{ mm}. \]Therefore,
\[ \boxed{C_{\min}=0.08\text{ mm}}. \]Practice 5
\[ C_{\max}=15.18-14.96=0.22\text{ mm}. \]Therefore,
\[ \boxed{C_{\max}=0.22\text{ mm}}. \]Practice 6
Nominal total:
\[ 10+15+5=30\text{ mm}. \]Worst-case tolerance:
\[ 0.1+0.2+0.1=0.4\text{ mm}. \]Therefore,
\[ \boxed{30.0\pm0.4\text{ mm}}. \]Practice 7
Diametral clearance is
\[ 40.30-40.10=0.20\text{ mm}. \]Radial clearance is
\[ \frac{0.20}{2}=0.10\text{ mm}. \]Therefore,
\[ \boxed{0.10\text{ mm}}. \]Practice 8
\[ e= \sqrt{(10.03-10.00)^2+(20.04-20.00)^2}. \] \[ e= \sqrt{0.03^2+0.04^2} = \sqrt{0.0025}. \]Therefore,
\[ \boxed{e=0.05\text{ mm}}. \]Practice 9
\[ e=1.5\sin2^\circ. \]Therefore,
\[ e\approx0.0524\text{ m}. \]Thus,
\[ \boxed{e\approx52.4\text{ mm}}. \]Practice 10
Nominal dimensions describe only the intended central values. Guaranteed clearance depends on the extreme allowable sizes.
The relevant worst-case condition is
\[ \boxed{H_{\min}>S_{\max}}. \]If the tolerance ranges overlap, a nominally larger hole may still produce interference for some manufactured combinations.
What to Remember
- Nominal dimensions describe intended size; tolerances describe allowable variation.
- Symmetric, bilateral, and unilateral tolerances produce measurable upper and lower limits.
- A measured value passes a simple dimensional check only when it lies inside the allowed interval.
- Fits depend on the relationship between tolerance ranges, not nominal dimensions alone.
- Worst-case analysis combines extreme allowable dimensions to find limiting conditions.
- Tolerance stack-up describes how variation accumulates through a dimension chain.
- Size, form, orientation, and position are different geometric properties.
- Small angular errors can produce significant linear offsets over long distances.
- Radial clearance is half of diametral clearance for centered circular parts.
- Measurement resolution and capability must match the required tolerance scale.
- Tighter tolerances can increase manufacturing and inspection difficulty.
- A useful engineering tolerance protects function without demanding unnecessary precision.
The central design relationship is:
\[ \boxed{ \text{intended geometry} + \text{controlled variation} \longrightarrow \text{functional physical system} }. \]Where We Go Next
Tolerances control the allowable geometry of parts and assemblies. Many engineered systems must also move through space while maintaining position, direction, timing, and clearance.
The next lesson, Motion, Trajectories, and Robotics, applies spatial mathematics to paths, moving coordinate systems, robot motion, collision avoidance, and controlled trajectories.