Verification methods — analysis, test, inspection, demonstration: Exercises

This page is a self-test. Read each problem, try it with pen and paper, then open the collapsible solution. Problems climb from L1 (just recognise the idea) to L5 (put everything together). Every symbol used here was built in the parent note — if you feel lost, re-read that first.

Level 1 — Recognition

Exercise 1.1 (L1)

For each requirement, name the single best verification method (analysis, test, inspection, or demonstration):

(a) "Total spacecraft dry mass shall not exceed 180 kg."
(b) "Structure shall survive 8 g RMS random vibration during launch."
(c) "Reaction wheels shall despin the spacecraft from 10 °/s to <0.1 °/s within 120 s on orbit."
(d) "Peak stress in the primary strut shall stay below yield under an 8 g quasi-static launch load."

Recall Solution

(a) Inspection — mass is a static property. You put it on a scale and read it. No physics prediction, no environment, no operation needed.
(b) Test — vibration survival is actual dynamic behaviour in a hard-to-model environment. You put it on a shaker and see if it lives. See Vibration Testing.
(c) Demonstration — it is an operational capability: run the actual control loop and watch it perform the task. You are not measuring a physical quantity to high precision, you are showing "it does the job."
(d) Analysis — stress under launch loads is predicted behaviour from deterministic physics ( $σ = M c / I$ ). You compute it before ever building flight hardware. See Finite Element Analysis.

Exercise 1.2 (L1)

Fill the blank: qualification testing applies a level higher than flight, while acceptance testing applies a level equal to flight limit. State the standard factors.

Recall Solution

Qualification level $= Flight Limit \times == 1.25 ==$
Acceptance level $= Flight Limit \times == 1.0 ==$

The qualification unit is stressed harder to prove the design; the flight unit is only stressed to flight level so you don't consume its fatigue life. See Acceptance Testing.

Level 2 — Application

Exercise 2.1 (L2)

A rectangular aluminium strut carries a tip mass $m = 4 kg$ at $L = 0.30 m$ from its fixed root. Cross-section: width $b = 0.020 m$ , height $h = 0.010 m$ . Launch load $n = 8$ g, $g = 9.81 m/s^{2}$ .

Using $σ = \frac{6 m n g L}{b h ^{2}}$ (from the parent note), find the peak bending stress.

Figure — Verification methods — analysis, test, inspection, demonstration

Recall Solution

What we do: plug into the derived formula. Why: all quantities are given, so analysis is a direct calculation. $σ = \frac{6 \cdot 4 \cdot 8 \cdot 9.81 \cdot 0.30}{0.020 \cdot ( 0.010 ) ^{2}}$ Numerator: $6 \times 4 = 24$ ; $\times 8 = 192$ ; $\times 9.81 = 1883.52$ ; $\times 0.30 = 565.056$ . Denominator: $0.020 \times 0.0001 = 2 \times 1 0^{- 6}$ . $σ = \frac{565.056}{2 \times 1 0 ^{- 6}} = 2.83 \times 1 0^{8} Pa \approx== 282.5 MPa ==$

Exercise 2.2 (L2)

Aluminium 6061-T6 has yield strength $σ_{y} = 276 MPa$ . Using the stress from 2.1, compute the Margin of Safety and state whether the requirement "MoS > 0" is met.

Recall Solution

$MoS = \frac{σ _{y}}{σ _{applied}} - 1 = \frac{276}{282.5} - 1 === - 0.023 ==$ MoS is negative → requirement NOT met by analysis. The strut yields under load. Fix: increase $h$ (stress $\propto 1/ h^{2}$ , so it pays off fast) or choose a stronger alloy. Feed this back to Requirements Development.

Exercise 2.3 (L2)

Flight limit random vibration is $8 g RMS$ . Compute the qualification level and the acceptance level.

Recall Solution

Qualification: $8 \times 1.25 === 10 g RMS ==$
Acceptance: $8 \times 1.0 === 8 g RMS ==$

Level 3 — Analysis

Exercise 3.1 (L3)

Manufacturing introduces $\pm 12%$ variation in a component's natural frequency; the flight environment adds $\pm 9%$ variation. Assuming independence, find the combined 1-sigma uncertainty $σ_{total}$ , and the 3-sigma test level (as a multiple of the mean). Compare to the industry 1.25× factor.

Recall Solution

Why root-sum-square: independent uncertainties add in quadrature, not linearly — variances add, so standard deviations combine as $σ_{1}^{2} + σ_{2}^{2}$ . $σ_{total} = 0.1 2^{2} + 0.0 9^{2} = 0.0144 + 0.0081 = 0.0225 === 0.15 ==$ 3-sigma test level: $Mean + 3 σ = 1.0 + 3 (0.15) === 1.45 ==$ Strict 3-sigma coverage demands 1.45×, higher than the industry 1.25×. The 1.25× factor is a cost/risk compromise — accepting slightly less than 3-sigma coverage to avoid over-designing every unit.

Exercise 3.2 (L3)

A thermal analysis predicts a battery cold-case minimum of $5^{\circ} C$ and hot-case maximum of $38^{\circ} C$ . The requirement is $0$ – $40^{\circ} C$ . Compute the temperature margins on each end. Are both satisfied? Which end is the design driver?

Recall Solution

Cold margin: $5 - 0 === 5^{\circ} C ==$ above the lower limit ✓
Hot margin: $40 - 38 === 2^{\circ} C ==$ below the upper limit ✓ Both pass. The hot case is the design driver — only 2 °C of slack, so it is most vulnerable to model error. See Margin Philosophy and Thermal Math Modeling.

Level 4 — Synthesis

Exercise 4.1 (L4)

A requirement reads: "The deployable antenna shall deploy fully within 30 s after command, on orbit, at $- 40^{\circ} C$ ." Design a combined verification approach using more than one method, and justify each choice.

Recall Solution

No single method covers this cleanly — synthesise:

Analysis — model the deployment kinematics and spring torque vs. hinge friction at $- 40^{\circ} C$ to predict deploy time. Cheap, done early, informs design.
Test — thermal-vacuum test of the hinge mechanism at $- 40^{\circ} C$ to measure actual friction, which analysis can only estimate. This validates the model — see Model Validation.
Demonstration — a full deploy of the flight-like article (gravity-offloaded) to show the end-to-end operational capability: command → full deployment.
Inspection — verify the as-built latch geometry and cable routing match the design that was analysed/tested; ensures the tested config equals flight config. Track via Configuration Management.

Every method answers a different sub-question: analysis predicts, test measures the uncertain input, demonstration proves the operation, inspection confirms the article is the right one.

Exercise 4.2 (L4)

Redo Exercise 2.1's strut so the requirement passes with a 1.5× safety factor (i.e. $σ \leq σ_{y} /1.5$ ). Keep $b = 0.020 m$ ; solve for the minimum height $h$ . Use $σ_{y} = 276 MPa$ .

Recall Solution

Target stress: $σ_{allow} = 276/1.5 = 184 MPa = 1.84 \times 1 0^{8} Pa$ . Rearrange $σ = \frac{6 mn g L}{b h ^{2}}$ for $h$ : $h = \frac{6 mn g L}{b σ _{allow}} = \frac{565.056}{0.020 \times 1.84 \times 1 0 ^{8}}$ Denominator: $0.020 \times 1.84 \times 1 0^{8} = 3.68 \times 1 0^{6}$ . $h = \frac{565.056}{3.68 \times 1 0 ^{6}} = 1.5355 \times 1 0^{- 4} === 0.01239 m ==\approx 12.4 mm$ So growing $h$ from 10 mm to ≈12.4 mm buys the full 1.5× factor — because stress falls as $1/ h^{2}$ , a 24% height increase gives a 54% stress drop.

Level 5 — Mastery

Exercise 5.1 (L5)

You are the systems engineer closing out a Traceability Matrix. A requirement states: "Primary structure shall withstand 8 g launch load with MoS > 0 AND survive 8 g RMS random vibration."

Your team reports:

FEA analysis: peak stress 282.5 MPa vs. yield 276 MPa.
Vibration test at qualification level: no failures, natural frequency shifted 6% after test.

Acceptance criterion for the vibration test was "frequency shift < 5%". Decide the verification status of the full requirement and state your action. Cite the numbers.

Recall Solution

Static (analysis) part: MoS $= 276/282.5 - 1 = - 0.023 < 0$ . FAIL. The structure yields under 8 g quasi-static load. Dynamic (test) part: frequency shift $= 6% > 5%$ acceptance limit. A >5% shift signals stiffness degradation (a crack, delamination, or loosened joint softens the structure and lowers its resonance). FAIL.

Overall status: NOT VERIFIED. Both halves fail independently. Action: the two failures likely share a root cause — an under-sized structure. Redesign (e.g. Ex 4.2's taller strut), re-run FEA to restore MoS > 0, then re-qualify on the shaker. Update the Traceability Matrix status to "open" and log the change under Configuration Management. Do not ship on the basis of "it didn't physically break during the test" — the 6% shift is direct evidence of hidden damage.

Recall Quick self-check

Which verifies a mass requirement? ::: Inspection Qualification factor over flight? ::: 1.25× How do independent uncertainties combine? ::: Root-sum-square (quadrature) Stress in a rectangular beam scales as? ::: $1/ h^{2}$ A 6% frequency shift after vibe means? ::: Stiffness degradation — a failure