Level 4 — ApplicationSpacecraft Structures & Systems Engineering

Spacecraft Structures & Systems Engineering

⏱ 60 minutes60 marksprintable — key stays hidden on paper

Level 4 — Application Examination

Time limit: 60 minutes Total marks: 60 Instructions: Answer all questions. Show all working. Use $g = 9.81\ \text{m/s}^2$ where needed. Symbols carry their usual meaning.

Question 1 — Launch load path & buckling (14 marks)

A cylindrical satellite adapter is a thin-walled aluminium tube of outer radius $R = 0.30\ \text{m}$ , wall thickness $t = 3.0\ \text{mm}$ , and length $L = 1.2\ \text{m}$ . It supports a spacecraft of mass $m = 850\ \text{kg}$ during a launch phase with steady axial acceleration of $6.0\,g$ plus a quasi-static bending moment from wind shear of $M = 4.0\ \text{kN·m}$ . Aluminium: $E = 70\ \text{GPa}$ , yield stress $\sigma_Y = 280\ \text{MPa}$ .

(a) Compute the axial compressive load and the axial membrane stress in the wall. (3)

(b) Compute the peak bending stress at the tube surface. (Treat as thin ring: $I \approx \pi R^3 t$ .) (3)

(c) Combine (a) and (b) to get the peak compressive stress, and state the margin of safety against yield if the required factor of safety is $\text{FOS} = 1.4$ . (3)

(d) Estimate the classical shell buckling stress $\sigma_{cr} = \dfrac{E t}{R\sqrt{3(1-\nu^2)}}$ with $\nu = 0.33$ , and comment on whether yield or buckling governs. (5)

A component is mounted on a bracket whose fundamental mode is at $f_n = 220\ \text{Hz}$ with quality factor $Q = 12$ . The base random vibration input is a flat PSD of $W_0 = 0.04\ \text{g}^2/\text{Hz}$ over the band containing $f_n$ .

(a) Using the Miles' equation $\text{GRMS} = \sqrt{\dfrac{\pi}{2} f_n\, Q\, W_0}$ , compute the RMS acceleration response of the component. (4)

(b) State the "3-sigma" peak acceleration used for design and convert it to a force on a component of mass $0.65\ \text{kg}$ . (4)

(c) The bracket natural frequency requirement is " $f_n \geq 200\ \text{Hz}$ ." Rewrite this as a SMART requirement and state which verification method (analysis/test/inspection/demonstration) is most appropriate and why. (4)

Question 3 — Fatigue and fracture (12 marks)

A structural fitting experiences, per mission, the following cyclic loading blocks. Its S-N behaviour is $N = 10^{12}\,\sigma^{-3}$ (with $\sigma$ in MPa, $N$ cycles to failure).

Block	Stress amplitude $\sigma$ (MPa)	Cycles per mission $n_i$
A	200	3000
B	100	40000

(a) Using Miner's rule, compute the cumulative damage per mission and the number of missions to failure. (6)

(b) The fitting contains a surface crack of length $a = 1.5\ \text{mm}$ . The stress intensity factor is $K = Y\sigma\sqrt{\pi a}$ with $Y = 1.12$ . Fracture toughness $K_{IC} = 30\ \text{MPa}\sqrt{\text{m}}$ . Find the critical stress for fast fracture at this crack length, and state whether the 200 MPa block is safe against fracture. (6)

Question 4 — Power, thermal & mass budget (12 marks)

A LEO smallsat requires an orbit-average electrical load of $180\ \text{W}$ . Orbit period $95\ \text{min}$ , eclipse fraction $35\%$ .

(a) The battery must supply the load through eclipse. Compute the energy (Wh) drawn per eclipse, and the required battery capacity if the allowable depth of discharge is $\text{DoD} = 25\%$ and battery-to-load efficiency is $0.90$ . (4)

(b) Solar array must power the load AND recharge the battery during sunlight. If charging efficiency is $0.85$ , compute the minimum solar array output power needed (assume load runs continuously). (4)

(c) The spacecraft dry mass is estimated at $210\ \text{kg}$ with a $20\%$ system mass margin. Propellant is $18\ \text{kg}$ . Compute the design dry mass (with margin) and wet mass. (4)

Question 5 — Reliability & redundancy (10 marks)

A subsystem uses two identical units, each with constant failure rate $\lambda = 8\times10^{-6}\ \text{h}^{-1}$ .

(a) Compute the MTTF of a single unit, and its reliability at $t = 3$ years (use $8766$ h/yr). (4)

(b) The two units are in active (parallel) redundancy (subsystem works if at least one works). Write the subsystem reliability $R_{sys}(t)$ and evaluate it at $t = 3$ years. (4)

Answer keyMark scheme & solutions

Question 1

(a) Axial load: $F = m\,a = 850 \times 6.0 \times 9.81 = 50{,}013\ \text{N} \approx 50.0\ \text{kN}$ . (1) Wall area: $A = 2\pi R t = 2\pi (0.30)(0.003) = 5.655\times10^{-3}\ \text{m}^2$ . (1) $\sigma_{axial} = F/A = 50013 / 5.655\times10^{-3} = 8.84\times10^{6}\ \text{Pa} = 8.84\ \text{MPa}$ . (1)

(b) $I = \pi R^3 t = \pi (0.30)^3 (0.003) = 2.545\times10^{-4}\ \text{m}^4$ . (1) $\sigma_{bend} = M R / I = (4000)(0.30)/2.545\times10^{-4} = 4.71\times10^{6}\ \text{Pa} = 4.71\ \text{MPa}$ . (2)

(c) Peak compressive stress $= 8.84 + 4.71 = 13.56\ \text{MPa}$ . (1) Allowable with FOS: $\sigma_{allow} = \sigma_Y/\text{FOS} = 280/1.4 = 200\ \text{MPa}$ . (1) Margin of Safety $= \dfrac{\sigma_{allow}}{\sigma_{peak}} - 1 = \dfrac{200}{13.56} - 1 = 13.7$ — very large positive margin (yield not a concern). (1)

(d) $\sigma_{cr} = \dfrac{E t}{R\sqrt{3(1-\nu^2)}} = \dfrac{70\times10^9 \times 0.003}{0.30\sqrt{3(1-0.33^2)}}$ . (2) Denominator: $\sqrt{3(1-0.1089)} = \sqrt{2.673} = 1.635$ ; $\times 0.30 = 0.4905$ . (1) $\sigma_{cr} = 2.1\times10^{8} / 0.4905 = 4.28\times10^{8}\ \text{Pa} = 428\ \text{MPa}$ (classical). (1) Comparison: applied peak $13.6\ \text{MPa} \ll \sigma_{cr}$ classical, and even with a knockdown factor of ~0.3 (real shells) $\sim128\ \text{MPa}$ still far above applied stress. Both yield and buckling have huge margins; the driver is elsewhere (or design is heavily over-conservative). (1)

Question 2

(a) $\text{GRMS} = \sqrt{\dfrac{\pi}{2} \times 220 \times 12 \times 0.04} = \sqrt{\dfrac{\pi}{2}\times 105.6} = \sqrt{165.9} = 12.88\ \text{g}$ . (4)

(b) 3-sigma peak $= 3 \times 12.88 = 38.6\ \text{g}$ . (2) Force $= m \times (3\sigma)\, g = 0.65 \times 38.6 \times 9.81 = 246\ \text{N}$ . (2)

(c) SMART version, e.g.: "The bracket's first natural frequency shall be $\geq 200\ \text{Hz}$ , verified by modal test/analysis at ambient conditions." — Specific (frequency), Measurable (Hz value), Achievable, Relevant (avoids low-frequency coupling with launcher), Testable. (2) Verification: Test (modal/sine-sweep) is most appropriate — natural frequency is directly measurable on hardware; analysis (FEM) may support but a test provides authoritative verification of the as-built structure. (2)

Question 3

(a) Cycles to failure: $N_A = 10^{12}\times 200^{-3} = 10^{12}/8\times10^{6} = 1.25\times10^{5}$ . (1) $N_B = 10^{12}\times 100^{-3} = 10^{12}/10^{6} = 1.0\times10^{6}$ . (1) Damage per mission: $D = \dfrac{n_A}{N_A} + \dfrac{n_B}{N_B} = \dfrac{3000}{125000} + \dfrac{40000}{1000000} = 0.024 + 0.040 = 0.064$ . (3) Missions to failure $= 1/D = 1/0.064 = 15.6 \Rightarrow 15$ complete missions. (1)

(b) Critical stress: $\sigma_c = \dfrac{K_{IC}}{Y\sqrt{\pi a}}$ . (2) $\sqrt{\pi a} = \sqrt{\pi \times 0.0015} = \sqrt{4.712\times10^{-3}} = 0.06865$ . (1) $\sigma_c = \dfrac{30}{1.12\times 0.06865} = \dfrac{30}{0.07689} = 390\ \text{MPa}$ . (2) Since $390\ \text{MPa} > 200\ \text{MPa}$ applied, the 200 MPa block is safe against fast fracture (margin ~1.95). (1)

Question 4

(a) Eclipse duration $= 0.35 \times 95 = 33.25\ \text{min} = 0.5542\ \text{h}$ . (1) Energy per eclipse $= 180 \times 0.5542 = 99.75\ \text{Wh}$ . (1) Accounting for efficiency, energy drawn from battery $= 99.75 / 0.90 = 110.8\ \text{Wh}$ . (1) Required capacity $= 110.8 / 0.25 = 443.3\ \text{Wh}$ . (1)

(b) Sunlight duration $= 0.65 \times 95 = 61.75\ \text{min} = 1.0292\ \text{h}$ . (1) Recharge energy needed (into battery) $= 110.8\ \text{Wh}$ ; energy the array must generate for charging $= 110.8/0.85 = 130.4\ \text{Wh}$ . (1) Array power during sunlight $=$ (load) $+$ (charge power) $= 180 + 130.4/1.0292 = 180 + 126.7 = 306.7\ \text{W}$ . (2)

(c) Design dry mass $= 210 \times 1.20 = 252\ \text{kg}$ . (2) Wet mass $= 252 + 18 = 270\ \text{kg}$ . (2)

Question 5

(a) Single-unit MTTF $= 1/\lambda = 1/(8\times10^{-6}) = 1.25\times10^{5}\ \text{h}$ . (2) $t = 3\times 8766 = 26298\ \text{h}$ ; $\lambda t = 8\times10^{-6}\times 26298 = 0.2104$ . $R(t) = e^{-0.2104} = 0.8103$ . (2)

(b) Active parallel: $R_{sys} = 1 - (1-R)^2 = 2R - R^2$ . (2) $R_{sys} = 2(0.8103) - (0.8103)^2 = 1.6206 - 0.6566 = 0.9640$ . (2)

(c) Cold standby: backup unit is unpowered/off until the primary fails, then switched in (lower wear, needs fault detection & switch). Hot standby: backup runs continuously (powered) alongside primary, enabling instant takeover but accumulating wear/aging. (2)

[
  {"claim":"Q1a axial stress ~8.84 MPa","code":"F=850*6.0*9.81; A=2*pi*Rational(3,10)*0.003; s=F/A; result = abs(float(s)-8.84e6) < 5e4"},
  {"claim":"Q1d shell buckling ~428 MPa","code":"scr=(70e9*0.003)/(0.30*sqrt(3*(1-0.33**2))); result = abs(float(scr)-4.28e8) < 5e6"},
  {"claim":"Q2a GRMS ~12.88 g","code":"g=sqrt((pi/2)*220*12*0.04); result = abs(float(g)-12.88) < 0.05"},
  {"claim":"Q3a missions to failure ~15.6","code":"D=3000/125000 + 40000/1000000; result = abs(1/D-15.625) < 0.1"},
  {"claim":"Q3b critical stress ~390 MPa","code":"sc=30/(1.12*sqrt(pi*0.0015)); result = abs(float(sc)-390) < 3"},
  {"claim":"Q5b redundant reliability ~0.964","code":"R=exp(-8e-6*3*8766); Rsys=2*R-R**2; result = abs(float(Rsys)-0.9640) < 0.002"}
]