Intuition The One Core IdeaBefore a spacecraft ever flies, we must prove that every part will do its job — and there are exactly four honest ways to get that proof: calculate it, test it, look at it, or run it. Everything in this topic is about matching the right kind of proof to the right kind of promise (a "requirement").
This page assumes you know nothing . We build every word, symbol, and picture the parent note leaned on, one at a time, so that when you read about stress formulas or vibration margins, no symbol is a stranger.
A requirement is a written promise about what the hardware must do, in numbers. It has three parts: a quantity (what we measure), a limit (the allowed range), and a condition (when it must hold).
Example: "Battery temperature (quantity) shall stay between 0 ∘ C 4 0 ∘ C (limit) throughout every orbit, during operation (condition) ." Without the condition, "between 0 and 40" is meaningless — 40°C in a factory is fine; 40°C during discharge in eclipse is a different promise.
The picture (figure s01): a gate with a fixed opening stands on the right, its gap marked in red as the allowed range. On the left a blue box labelled "hardware" slides toward it, and a green arrow reading "verification = check it fits" shows the whole job of this topic — proving the box passes before launch day. See Requirements Development for where these promises come from.
Verification is the act of producing evidence that a requirement is met. Not an opinion — evidence . The four ways to make that evidence are the whole topic: analysis, test, inspection, demonstration .
Intuition Why four methods and not oneA promise about mass you weigh (inspection). A promise about launch shaking you shake (test). A promise about stress you cannot safely apply you calculate (analysis). A promise about "it can point at a star" you run (demonstration). One tool cannot cover all four — like you cannot measure temperature with a ruler.
Every formula in the parent note is built from these atoms. We define each and draw it.
Definition Variable and the lettersm , g , n variable is a box that holds a number we haven't fixed yet.
m mass : how much stuff is in an object, measured in kilograms (kg g gravitational acceleration near Earth, g ≈ 9.81 m/s 2 n plain multiplier ("n n = 8
F Force F N F = m ⋅ a a acceleration (how fast the speed changes, in m/s 2
Why the topic needs it: during launch the rocket accelerates hard, so every bolt feels a force F = m ( n g )
Intuition Why acceleration "feels like" extra gravitySitting in a car that floors it, you feel pressed back into the seat. Your body can't tell the difference between "being pushed" and "gravity got stronger." A spacecraft under 8 g quasi-static load .
The parent's stress formula needs a picture of a beam. Here it is.
Figure s02 draws a blue cantilever bar clamped into a gray wall on the left. A red arrow labelled F = mn g neutral axis (defined just below). Below the bar a double-headed arrow marks the length L b h c = h /2
When a beam bends, the material on the outside of the curve stretches and the material on the inside squashes . Somewhere in between is a line that does neither — it keeps its original length. That line is the neutral axis . For a symmetric bar it runs straight through the geometric middle (the green dashed line in s02).
Definition The beam and its lettersL , b , h , c , y beam is a bar fixed at one end (like a diving board). Every length here is measured in metres (m
L length from the fixed wall to where the load hangs.b width of the cross-section (how wide, looking at the cut end).h height of the cross-section (how tall).y up or down from the neutral axis (now defined above). Positive y y c largest such distance, i.e. the outer edge, c = h /2
Definition Bending momentM bending moment M M = F ⋅ L end of a long spanner bends it more than pushing near the pivot. Longer arm L M N⋅m
σ Stress σ spread over an area :
σ = area force Pa = N/m 2 Note on the letter: in this section σ mechanical stress . Much later, in Level 6, the same Greek letter σ
Intuition Why the stress in a bent beam varies withy σ = M c / I y y stress at height y y — it grows in a straight line from zero at the axis to a maximum at the edge:
σ ( y ) = k ⋅ y k c I
Definition Second moment of areaI ∫ ∫ "add up infinitely many tiny slices." The second moment of area is
I = ∫ y 2 d A d A y 2 far-flung the material is from the middle — far material resists bending far more. For a rectangle this sum works out to I = 12 b h 3
y 2 y more and has a longer lever arm to push back — two effects, each proportional to distance, so their product goes as distance-squared. That is why an I-beam puts its metal far out top and bottom: I
Common mistake Which edge is in tension? (sign convention)σ ( y ) = k y positive on one side and negative on the other . For the downward load in s02, the top fibres stretch (positive σ tension ) and the bottom fibres squash (negative σ compression ). If the moment reverses sign (load pushes up), tension and compression swap sides. The magnitude ∣ σ ∣ = M c / I which edge cracks first depends on the sign of M M c / I
Deeper structural modelling of exactly this kind is Finite Element Analysis .
Definition Yield strengthσ y Yield strength σ y stops springing back and stays bent — the danger line for that material.
Definition Margin of Safety (MoS)MoS = σ applied σ y − 1
If applied stress equals the limit, ratio = 1 = 0
If MoS > 0 safe .
If MoS < 0 fails .
Figure s03 is a number line for this idea: a black tick in the middle marks "applied = yield, MoS = 0 (edge)". The green band to the left (low applied stress) is labelled "MoS > 0 SAFE"; the red band to the right (high applied stress) is labelled "MoS < 0 FAILS". Reading the line left-to-right shows margin shrinking to zero and then going negative as the load grows.
The philosophy of how big that gap should be is Margin Philosophy .
The thermal example needs a new family of symbols.
T d t d T T temperature (in ∘ C K d t d T "how fast T — a rate . If it's positive, the thing is heating up; negative, cooling.
Q Q heat energy flowing per second (watts, W stored energy change = energy in − energy out. For a battery node:
m b c p d t d T b = Q in − Q out
m b c p 1 kg 1 K If Q in = Q out d t d T b = 0
Common mistake Sign convention forQ in Q out Q in Q out Q out whichever way the temperature difference points . If the "hot" wall is actually colder than the battery, what you labelled Q rad,in out — its value goes negative and it silently becomes an outflow. The safe habit: write every term as a signed function of temperature difference (e.g. Q = G ( T other − T b ) Thermal Math Modeling the moment a node can be either warmer or cooler than its neighbours.
This is the seed of Thermal Math Modeling .
Definition Steady-periodic stateA steady-periodic state is when temperature stops drifting over the long run and just repeats the same wobble every orbit. Picture a swing pushed once per lap — after a while each lap looks identical.
Definition RMS, mean, and sigmaσ Heads-up: this σ not the stress from Level 3 — same letter, different meaning. Here σ standard deviation .
RMS ("root-mean-square") is a fair "average size" of a jiggling signal — square it, average, square-root. It tells you the typical strength of random shaking.Mean = the plain average, the centre of a spread.σ width of the bell-shaped spread.
Common mistake Adding sigmas the wrong way0.1 + 0.1 = 0.2 wrong for independent spreads. The correct 0. 1 2 + 0. 1 2 ≈ 0.14 smaller , because independent errors don't all peak at once.
Bending moment M=F times L
Integral adds tiny slices
Test level and 1.25 factor
Four Verification Methods
Downstream, every proof feeds a Traceability Matrix , models must pass Model Validation , and every change is tracked by Configuration Management .
Reveal each and check you can answer it cold.
What does F = ma The force on each part, F = m ( n g )
What is a quasi-static load? A steady acceleration that feels like extra gravity — a constant pretend-weight, not a vibration.
What are the three parts of a requirement? A quantity, a limit, and a condition (when it must hold).
What is the neutral axis? The line in a bending beam that neither stretches nor squashes; y
Why does stress vary linearly with y Fibres stretch in proportion to y σ ( y ) = k y
Why does the second moment of area use y 2 Far material is both stretched more and has a longer lever, two distance effects multiplying to distance-squared.
What is I I = b h 3 /12
Give the final launch stress formula. σ = b h 2 6 m n g L
For a downward load, which fibres are in tension? The top fibres (positive σ
When is a design safe by Margin of Safety? When MoS = σ y / σ applied − 1 > 0
What does d t d T = 0 Energy in equals energy out; temperature is steady.
What sign convention keeps heat terms honest? Write each as Q = G ( T other − T b )
How do two independent ± 10% 0. 1 2 + 0. 1 2 ≈ 0.14 , assuming independence and roughly Gaussian spreads.
Why accept 1.25 × 1.42 × Inputs are already conservative, acceptance tests and margins catch the tail, and over-test/cost argue against 1.42
Which method proves a mass requirement? Inspection (you weigh it) — not test.
