Environmental testing — thermal vacuum (TVAC), vibration, acoustic, EMC - EMI: Visual walkthrough

This is the "movie" version of the parent topic's vibration section. We go slower and draw everything.

Step 1 — A satellite part is really just a weight on a springy stalk

WHAT. Look at any box bolted onto a spacecraft panel: a circuit board on standoffs, a camera on a bracket. Zoom out and it is a lump of stuff (the box) held up by something bendy (the bracket, the bolts, the panel). We replace this with the simplest possible cartoon: a block that can slide, tied to a wall by a spring.

WHY. We cannot solve a full satellite by hand — millions of tiny pieces. But a single block-on-spring already contains the entire secret of resonance. If we understand this cartoon completely, the real thing is just "many cartoons stacked". This is the single-degree-of-freedom (SDOF) model — "one degree of freedom" means the block can only move one way (left–right), so one number $x$ fully describes where it is.

PICTURE. Below: the block of mass $m$ , the spring of stiffness $k$ , and a little dashpot (shock-absorber) of strength $c$ . The floor underneath is what the shaker table grabs and wiggles.

Figure — Environmental testing — thermal vacuum (TVAC), vibration, acoustic, EMC - EMI

Step 2 — Write down Newton's law for the block

WHAT. We now let the table push the block with a wiggling force and ask: how does the block move? Newton says (mass)(acceleration) = (sum of forces). Three forces act on the block: the spring, the damper, and the driving push.

WHY. We need an equation before we can find resonance. Every force gets translated into "how it depends on the block's motion", and that turns a sentence into algebra we can actually solve.

PICTURE. Each arrow in the figure is one force; watch which way it points as the block moves right.

Let $x$ be how far the block has slid from rest (metres). Then:

$\overset{x}{˙}$ (read "x-dot") is the block's velocity — how fast $x$ changes each second. The dot means "rate of change".
$\overset{x}{¨}$ ("x-double-dot") is the acceleration — how fast the velocity changes.

Now translate each force:

$mass \times accel m \overset{x}{¨} = spring pulls back - k x damper drags - c \overset{x}{˙} + table’s push F_{0} sin (ω t)$

Reading it term by term:

$- k x$ : spring force. Minus sign because the spring always pulls the block back toward home. Slid right → pulls left.
$- c \overset{x}{˙}$ : damper force. Minus sign because it always opposes the direction of motion.
$F_{0} sin (ω t)$ : the shaker's push. $F_{0}$ is how hard (peak newtons), $ω$ is how fast it wiggles (angular frequency, radians per second), $t$ is time. $sin$ makes it swing smoothly back and forth.

Move everything to the left to get the standard form quoted in the parent note:

$m \overset{x}{¨} + c \overset{x}{˙} + k x = F_{0} sin (ω t)$

Step 3 — Turn off the table: the part has its OWN favourite rhythm

WHAT. Kill the push ( $F_{0} = 0$ ) and ignore damping for a moment ( $c = 0$ ). Pluck the block once and let go. It oscillates forever at one special frequency.

WHY. We need a reference frequency to compare the table's shaking against. This natural rhythm is the star of the show — resonance happens when the table's rhythm matches this one. So we must find it first.

PICTURE. A stiff spring snaps back fast (high pitch); a heavy block is sluggish (low pitch). The figure shows both extremes.

With $c = 0, F_{0} = 0$ the equation is $m \overset{x}{¨} + k x = 0$ . The motion that solves this is a pure sine wave, and plugging it in forces the frequency to be:

$ω_{n} = \frac{k}{m} (radians per second)$

$ω_{n}$ : the natural angular frequency. The subscript $n$ = "natural".
$k$ on top: stiffer spring → snaps back harder → faster wiggle. Makes sense.
$m$ on bottom: heavier block → more sluggish → slower wiggle. Also makes sense.

Engineers prefer cycles-per-second (hertz), so divide by one full circle's worth of radians, $2 π$ :

$f_{n} = \frac{ω _{n}}{2 π} = \frac{1}{2 π} \frac{k}{m} (Hz)$

Step 4 — First name the friction number $ζ$ , then push at frequency $ω$

WHAT. Before any amplification formula, we bottle the damping $c$ into one clean, unitless number $ζ$ . Then we switch the table back on, pushing at frequency $ω$ , and watch the steady wiggle that results.

WHY. In Step 5 the friction $c$ will appear inside a square root as $2 ζ r$ . If we do not first define $ζ$ , that symbol would ambush the reader. So we define it here, on its own, before it is ever used.

PICTURE. Input arrow (table) small and steady; output arrow (block) growing as $ω$ creeps toward $ω_{n}$ .

The friction number. Take the raw damping $c$ and compare it to the "critical" amount $2 k m$ (the exact amount that would kill all oscillation). Their ratio is the damping ratio:

$ζ = \frac{c}{2 k m}$

$ζ$ (Greek "zeta") measures how much friction the part has, as a clean unitless fraction.
$ζ = 0$ = no friction at all; $ζ = 1$ = so much friction the block just crawls home without overshooting.
Spacecraft structure is lightly damped: $ζ \approx 0.02$ – $0.05$ .

The two dials. Now define the other clean dial, the frequency ratio:

$r = \frac{ω}{ω _{n}}$

$r$ tells us "how does the push frequency compare to the part's favourite frequency?"
$r < 1$ : pushing slower than natural. $r = 1$ : pushing exactly at natural. $r > 1$ : pushing faster.

What we are hunting. After a moment the block settles into a steady sine wiggle at the same frequency $ω$ , but with amplitude $X$ . The ratio of that output wiggle to the input wiggle is the transmissibility $T$ . If $T = 1$ , the part shakes exactly as hard as the table. If $T = 20$ , a gentle $1$ g input becomes a brutal $20$ g at the part. Both $ζ$ and $r$ are now on the table — we are ready to build $T$ .

Step 5 — Build the transmissibility with a rotating-arrow (phasor) picture

WHAT. We derive $T (r)$ without heavy algebra by drawing each force as a rotating arrow (a "phasor") and adding the arrows tip-to-tail.

WHY. The formula's three terms — $k x$ , $c \overset{x}{˙}$ , $m \overset{x}{¨}$ — do not just add like plain numbers; they point in different directions because velocity and acceleration are "a quarter-turn ahead" of position. A picture of three arrows makes the square root obvious instead of magical.

PICTURE. Below: the driving push (yellow) must equal the tip-to-tail sum of the three response arrows. Read the right triangle it forms.

Here is the whole idea in three sentences:

When the block wiggles as $x = X sin (ω t)$ , its spring force $k x$ points along the motion, size $k X$ .
Its velocity $\overset{x}{˙}$ is a quarter-turn ( $9 0^{\circ}$ ) ahead, so the damper force $c \overset{x}{˙}$ points sideways, size $c ω X$ .
Its acceleration $\overset{x}{¨}$ is a half-turn ( $18 0^{\circ}$ ) round, so the inertia force $m \overset{x}{¨}$ points backward along the motion, size $m ω^{2} X$ .

The table's push $F_{0}$ must balance the sum of these. The two forces along the line of motion combine to $(k - m ω^{2}) X$ ; the sideways one is $c ω X$ . They are at right angles, so by Pythagoras the push magnitude is the hypotenuse:

$F_{0} = X (k - m ω^{2})^{2} + (c ω)^{2}$

$(k - m ω^{2})$ : spring pull minus inertia — the "along-motion" leg of the triangle. It vanishes exactly when $ω^{2} = k / m$ , i.e. at resonance.
$(c ω)$ : the friction leg, always sideways, never zero while the block moves.

Solve for the output amplitude $X$ , and compare it to the amplitude the table would give a rigid part, $X_{0} = F_{0} / k$ (push divided by stiffness). Their ratio is $T$ :

$T = \frac{X}{X _{0}} = \frac{k}{( k - m ω ^{2} ) ^{2} + ( c ω ) ^{2}}$

Now divide top and bottom by $k$ and use our two dials. Since $m ω^{2} / k = (ω / ω_{n})^{2} = r^{2}$ and $c ω / k = 2 ζ r$ (using $ζ = c / (2 k m)$ and $ω_{n} = k / m$ ), everything collapses to:

$T = \frac{1}{( 1 - r ^{2} ) ^{2} + ( 2 ζ r ) ^{2}}$

Term-by-term inside the square root:

$(1 - r^{2})$ : the along-motion leg, the "distance from resonance". When $r = 1$ this is zero — the danger point.
$(2 ζ r)$ : the friction leg. The only thing stopping $T$ from blowing up when $(1 - r^{2}) = 0$ .

Step 6 — Walk through every case of $r$

WHAT. Feed three regions of $r$ into the formula and read off what happens.

WHY. The contract: the reader must never meet a case we did not show. Below resonance, at resonance, above resonance — each behaves completely differently, and each matters in real testing.

PICTURE. The famous transmissibility curve, with all three zones coloured.

Case A — slow push, $r < 1$ (far below resonance). Then $r^{2}$ is small, $(1 - r^{2}) \approx 1$ , friction term tiny, so $T \approx \frac{1}{1} = 1.$ The block just rides along with the table. No amplification, no danger. The part feels exactly what it's given.

Case B — matched push, $r = 1$ (AT resonance). Set $r = 1$ : the $(1 - r^{2})^{2}$ term vanishes completely. Only friction survives: $T = \frac{1}{0 + ( 2 ζ \cdot 1 ) ^{2}} = \frac{1}{2 ζ} .$ This peak value has its own name, the quality factor: $Q = \frac{1}{2 ζ} .$ With $ζ = 0.02$ , $Q = 25$ . A 1 g input becomes a 25 g monster. This is where hardware dies. Notice: without friction ( $ζ = 0$ ) the formula gives $T = 1/0 = \infty$ — the block would shake itself apart. Friction is the only thing saving it.

Case C — fast push, $r > 2$ (above resonance). Now $r^{2} > 2$ , so $(1 - r^{2})^{2}$ grows large, the denominator is big, and $T < 1.$ The part shakes less than the table — the mass is too sluggish to keep up. This is the principle of vibration isolation: mount a delicate part on a soft spring so its $ω_{n}$ sits well below the shaking, pushing you into this safe zone.

Step 7 — The degenerate & heavy-damping cases

WHAT. Push the formula to its limits: no friction, rigid support, huge mass — and the important opposite extreme, so much friction that the peak disappears entirely.

WHY. Edge cases are where intuition breaks and bugs hide. Spacecraft brackets are lightly damped, but rubber isolators, foams and fluid dampers used elsewhere can be heavily damped — the reader must know what those look like too.

PICTURE. Left: three quick thought-experiments. Right: the peak flattening out as $ζ$ grows past $1/ 2$ .

Zero damping, $ζ \to 0$ : at $r = 1$ , $T = 1/ (2 ζ) \to \infty$ . Infinite amplitude — the block accelerates without bound. Real parts never hit true zero damping, but low damping is exactly why light structures are so dangerous.
Very stiff support, $k \to \infty$ : $ω_{n} = k / m \to \infty$ , so its natural frequency is way above any launch shaking. Then $r ≪ 1$ always → $T \approx 1$ , no amplification. A rock-solid rigid mount never resonates in the test band. (This is the design goal: push $f_{n}$ above 50 Hz so the random-vibration band cannot ring it.)
Very heavy block, $m \to \infty$ : $ω_{n} \to 0$ , so the part is always in the $r > 1$ isolation zone — but it also barely responds to anything. Massive, sluggish, safe but useless as a structure.
Heavy damping, $ζ \geq 1/ 2 \approx 0.707$ : the peak disappears. To have a peak, $T$ must rise above $1$ somewhere; the maximum of $T$ actually sits at $r_{p e ak} = 1 - 2 ζ^{2}$ , which only exists (is a real, non-zero number) while $2 ζ^{2} < 1$ , i.e. $ζ < 1/ 2$ . Once $ζ \geq 1/ 2$ , $T$ just slides downward from $1$ with no bump at all — the system is too gooey to resonate. This is why a well-designed rubber isolator can be pushed through its own natural frequency with barely a wobble.

Recall Why can't the "at resonance" amplitude be infinite in real life?

Because real parts always have some friction ( $ζ > 0$ ), the denominator $2 ζ r$ never truly reaches zero, so $T$ stays finite at $Q = 1/ (2 ζ)$ . ::: Friction (damping) caps the peak.

Recall At what damping does the resonant peak vanish entirely?

When $ζ \geq 1/ 2 \approx 0.707$ , because the peak location $r_{p e ak} = 1 - 2 ζ^{2}$ stops being a real number. ::: $ζ \geq 1/ 2$ .

Step 8 — Putting a number on it: the cubesat from the parent note

WHAT. Use $Q = 1/ (2 ζ)$ with the parent's spacecraft damping.

WHY. So the abstract " $T = 20$ " becomes a real force you can feel.

The one-picture summary

Now walk your eye across the final board left-to-right; here is exactly what to look for at each spot.

Intuition How to read the summary figure

Far left — the cartoon. The block+spring is the whole model. Everything else is derived from it. Point to $m$ , $k$ , $c$ and remember: mass, stiffness, friction.
The arrow to $ω_{n} = k / m$ . This is Step 3 — the part's own rhythm, born from just $k$ and $m$ . It fixes where on the horizontal axis "resonance" lives.
The curve $T (r)$ . This is Steps 5–6. Trace it from left ( $T \approx 1$ , safe), up to the spike (danger), down through $r = 2$ (the dashed crossover), into the isolation tail ( $T < 1$ ).
The label $Q = 1/ (2 ζ)$ on the peak. This is Step 6B — how tall the spike is, set entirely by friction. Low $ζ$ → tall, deadly spike; high $ζ$ → no spike at all (Step 7).
The takeaway to say aloud: "Match the rhythm and the force multiplies by $Q$ ; friction is the only thing that caps it."

Recall Feynman retelling — say it in plain words

Picture a kid on a swing. If you shove at random moments (wrong frequency, $r \neq = 1$ ) the swing barely moves — that's $T \approx 1$ . But if you shove exactly in time with the swing's natural back-and-forth ( $r = 1$ ), each little push adds up and the kid flies dangerously high — that's resonance, $T = Q$ . The only thing keeping the swing from going over the top is air drag and the rusty chain — that's damping $ζ$ , and it sets the maximum height $Q = 1/ (2 ζ)$ . If the chain were coated in thick mud ( $ζ \geq 1/ 2$ ) the swing would barely move at all — no resonance peak. A satellite part is a swing bolted to a rocket. The rocket shoves it at every frequency during launch. If any part's natural swing-rhythm lives inside the shaking band, that part gets pumped up 10–25× and snaps. So on the ground we deliberately sweep the shaker across all frequencies to find each part's rhythm, then either damp it, stiffen it (raise $f_{n}$ out of the band), or isolate it on a soft mount (drop into the $r > 2$ zone). That is the whole game.

Connections

Parent: the full environmental-testing note
Launch Vehicle Dynamics — where the shaking comes from
Structural Mechanics — where $k$ and $ω_{n}$ come from
Reliability Engineering & Quality Assurance in Aerospace — why we test to find failures on the ground
Related heat-side story: Spacecraft Thermal Control Systems