Probabilistic Fit and Dimensional Tolerance Analysis

How to use

1. Select Interval bounds, Threshold probability, or Radial-clearance stack.
2. Enter minimum and maximum deviations in µm. Every minimum must be below its matching maximum.
3. For interval or stack bounds, enter central coverage R as a fraction strictly between 0 and 1. For threshold mode, also enter δ₁ in µm.
4. Select Calculate. Reset restores the mode’s example values.
5. Read positive fit results as interference and negative fit results as clearance. In the stack mode, negative radial clearance indicates possible interference.
6. Review warnings and compare probabilistic bounds with deterministic worst-case limits.
7. Do not use this model when distributions are not approximately normal and independent, process tolerances do not represent ±3σ, or thermal, elastic, roughness, form, or safety effects dominate.

Worked example

1. 20H7/r6-style interval: enter hole deviations 0 to 21 µm, shaft deviations 28 to 41 µm, and R = 0.98. The result is a mean interference of 24.00 µm, deterministic range 7.00–41.00 µm, and central probabilistic range 14.42–33.58 µm. This suggests interference throughout the entered worst-case and probabilistic ranges, before roughness, temperature, and deformation corrections.

Equations used

For hole limits H_min, H_max and shaft limits S_min, S_max, all in µm:

µ_H = (H_min + H_max)/2;   µ_S = (S_min + S_max)/2

σ_H = (H_max − H_min) / 6;   σ_S = (S_max − S_min) / 6

µ_I = µ_S − µ_H;   σ_I = √(σ_H² + σ_S²)

I_min = S_min − H_max;   I_max = S_max − H_min

I_p,min = µ_I − z_Rσ_I;   I_p,max = µ_I + z_Rσ_I

R is the selected central probability (0 < R < 1), p = (1 + R)/2, and z_R approximates Φ⁻¹(p): q = 0.5 − |p − 0.5|, w = √[−2 ln(q)], z = w − (2.30753 + 0.27061w)/(1 + w(0.99229 + 0.04481w)). Here Φ is the standard-normal cumulative distribution.

Threshold probability

For threshold δ₁, t₁ = (δ₁ − µ_I)/σ_I and F(δ₁) ≈ Σ φ(t_i)Δt from t = −5 to t₁, with Δt = 0.001 and φ(t) = exp(−t²/2)/√(2π). The displayed percentage preserves the implemented two-decimal numerical-integration convention.

Radial-clearance stack

With housing Dh, bushing outside diameter D, bushing inside diameter d, shaft ds, and bushing wall deviation s:

Theory and method

The model converts each entered tolerance interval to a normal distribution by treating the midpoint as its mean and the full width as six standard deviations. Independent dimensional contributions are combined by root-sum-square propagation. Deterministic bounds combine adverse limits directly; probabilistic bounds use a central normal interval. This distinction is useful for screening manufacturing fits, but it does not replace process-capability evidence or ISO fit verification.

All values are deviations from compatible basic sizes. Surface roughness can reduce effective interference; temperature and elastic deformation can alter assembly behavior. Correlated machining errors make the independence assumption optimistic or conservative depending on covariance.

Assumptions and limits

• Normal, independent dimensions; each tolerance span represents ±3σ.
• One-dimensional deviations in µm with compatible basic sizes.
• No thermal expansion, elastic deformation, roughness, waviness, form error, or assembly damage.
• DIN ISO 286:1990 context is retained, but current design release should use the applicable current standard and drawing requirements.
• Not suitable for safety-critical acceptance without measured capability data and independent engineering review.

References

• DIN ISO 286:1990, ISO system of limits and fits.
• ISO 286-1:2010, Geometrical product specifications — ISO code system for tolerances on linear sizes — Part 1.
• Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions, normal-distribution approximations.

Continue in TriboSolver to refine contact pressure, deformation, thermal effects, and interface behavior where a tolerance-only screen is insufficient.