Chapter 32 — Key Takeaways (Reliability, Testing, and Why Rockets Fail)

A one-page reference. Reread this before an exam, or before you sign off on a risk assessment.

The four equations of reliability

Form Use it to find Notes
$R_{\text{series}} = \prod_i R_i$ reliability when all components are required always below the weakest link
$R_{\text{parallel}} = 1 - \prod_i (1 - R_i)$ reliability of redundant (any-one-works) units multiply the unreliabilities
$R = 1 - (1-R)^n$ reliability of $n$ identical parallel units each unit adds a "nine" — if independent
$R(t) = e^{-\lambda t}$ reliability over time (constant hazard) $\text{MTBF} = 1/\lambda$

The one number to remember

$$ 0.9999^{10{,}000} \approx e^{-1} \approx 0.37 $$

Ten thousand excellent parts in series give a system that fails almost two flights in three. "Everything must work" is arithmetic, not attitude. Attack it by cutting part count, breaking series with redundancy, and pushing each $R_i$ toward 1 with margin and testing.

The worked example (series vs. one redundant unit)

Step Chain System $R$ Mission-loss
No redundancy $0.98 \times 0.90 \times 0.97$ 0.8555 14.4%
Dual computer (weak link 0.90 → $1-0.1^2$ = 0.99) $0.98 \times 0.99 \times 0.97$ 0.9411 5.9%

One backup on the weakest link cut mission-loss by ~60%. Redundancy pays least on already-strong components — always harden the weakest string first.

Definitions worth memorizing

Term One-line meaning
Reliability Probability of the function, under stated conditions, for a stated duration. A number in [0,1].
Redundancy Two+ independent means to a function; parallel — fails only if all fail. Active vs. standby.
Single point of failure (SPOF) A no-backup series element whose failure alone loses the mission.
Common-cause failure One fault disabling several "redundant" units at once (shared software/power/environment). Breaks independence.
FMEA Bottom-up: for each part → failure mode → cause → effect → detection → mitigation; score S × L.
Qualification testing Over-stress a design beyond flight to prove margin; article usually not flown.
Acceptance testing Screen each flight unit at flight levels for workmanship (infant mortality).
"Test like you fly" Reproduce flight hardware/software/interfaces/sequence/environment; every gap is an untested assumption.
Root-cause analysis Trace symptom → proximate cause → root cause (technical and organizational).

Decision aid — "which move do I use?"

Situation Move
Function needs several parts, all must work series: multiply, then attack the weakest link
A block is too unreliable and can be duplicated add parallel redundancy: $1-(1-R)^n$
Redundant units share software/power/sensor STOP — common-cause; use dissimilar, separated backups
Item cannot be duplicated (structure, heat shield, main engine) margin + derating + qualification testing, and accept the SPOF
Need to rank many risks with limited budget FMEA severity × likelihood; fix high-risk (esp. catastrophic) first
Limited mass for redundancy rank by reliability gain per kilogram, add greedily

Case studies — root cause, not symptom

Failure Symptom Root cause Lesson
Ariane 5 Flight 501 (1996) nozzles hard over, breakup reused Ariane-4 software assumption never re-validated; both SRIs identical → common-cause re-validate heritage code; redundancy needs dissimilar implementations
Mars Climate Orbiter (1999) entered atmosphere too low unmanaged units interface (lbf·s vs. N·s); warnings not escalated manage interfaces; detection without response is worthless
Soviet N-1 (1969–72) four launch failures 30-engine stage never static-fired as a unit; non-benign engine failures cascaded "test like you fly"; redundancy needs contained failures

Common pitfalls

Pitfall Reality
"Add a backup and you're safe." Only if failures are independent; common-cause defeats the $1-(1-R)^n$ math.
"More engines = less reliable." True if all must work ($0.99^9=0.91$); false with engine-out ($\ge 8$ of 9 = 0.9966).
"Reliability is the average of the parts." It's the product — far below the average for long series chains.
"Test until it passes." You can't test reliability in; testing verifies a design and reveals defects.
"Cut the margin to save mass." Margin is spent by the physics you didn't model — not by the schedule.

Numbers worth memorizing

  • Industry orbital-launch success ≈ 90–95%; new vehicles do worse (infant mortality).
  • Structural factor of safety ≈ 1.25–1.5 (higher for pressure vessels / crew) — Ch. 23.
  • Crewed loss-of-crew targets are orders of magnitude stricter than cargo (e.g., ~1-in-270 ascent/entry, Tier 2).
  • Redundancy adds a "nine" per independent unit; the cost of reliability rises steeply toward 1.

Mission / astrotools additions this chapter

  • MDR: added the risk assessment — reliability block diagram, FMEA-lite (S × L), reliability allocation by gain-per-kg, and documented residual SPOFs.
  • reliability.py: series(rs), parallel(rs), redundant(r, n), k_of_n(r, n, k).