Timing-Belt Service Interval Estimator

How to use

1. This calculator has one normal-life-distribution mode; use it only for a reasonably normal population of comparable timing-belt lives.
2. Enter positive mean life μ and standard deviation σ in 10³ km. Enter baseline R₁ and a higher conservative R₂ as probabilities between 0 and 1.
3. Select Calculate or edit an input to update both intervals, z-values, and their ratio. Select Reset to restore the worked example.
4. Interpret t₂ as the more conservative statistical distance. Also apply manufacturer limits and account for calendar age, heat, contamination, overload, installation, and inspection findings.
5. Correct any warning before use. Do not apply this normal model to mixed populations, sparse or censored data without distribution fitting, non-normal wear-out, safety-critical release decisions, or nonpositive predicted intervals.

Worked example

For μ = 150 × 10³ km, σ = 20 × 10³ km, R₁ = 0.90, and R₂ = 0.99, the results are k₁ = 1.280, t₁ = 124 × 10³ km, k₂ = 2.328, t₂ = 103 × 10³ km, and t₂/t₁ = 0.8316. The 99% target shortens the screening distance to about 83% of the 90% target.

Equations used

For each target i ∈ {1,2}, convert survival reliability to lower-tail failure probability:

Fᵢ = 1 − Rᵢ
qᵢ = 0.5 − |Fᵢ − 0.5|
wᵢ = √[−2 ln(qᵢ)]

The implemented inverse-normal approximation is:

zᵢ = sgn(Fᵢ − 0.5){wᵢ − (2.30753 + 0.27061wᵢ) / [1 + wᵢ(0.99229 + 0.04481wᵢ)]}
tᵢ = μ + σzᵢ
kᵢ = −round(zᵢ, 3)
a = t₂ / t₁

Here Φ⁻¹ is the inverse standard-normal cumulative distribution, zᵢ ≈ Φ⁻¹(Fᵢ), μ and σ are in 10³ km, tᵢ is the unrounded interval, and kᵢ is the displayed positive reliability z-value. Displayed intervals are rounded to the nearest 10³ km; a is rounded to four decimals from unrounded t₁ and t₂.