Random vibration — PSD, RMS acceleration: Foundations

This page assumes you know nothing. We will build every letter, squiggle, and picture the parent note (Random Vibration) throws at you — in an order where each idea rests on the one before it.

0. The picture we keep coming back to

Before any symbol: imagine a sensor glued to a spacecraft panel. It measures acceleration — how hard the panel is being yanked — many thousands of times per second. Plot those readings against time and you get a jagged, never-repeating scribble.

Look at figure s01: the teal curve is $x (t)$ . Notice it crosses zero constantly and never repeats — the orange arrow marks a spot where you simply cannot guess the next value from the past. That un-guessability is exactly what "random" means, and it is why we give up predicting the curve and instead measure its statistics. Everything below is a tool for taming that scribble.

1. Time $t$ and the signal $x (t)$

Plain words: "the acceleration at moment $t$ ".
Picture: the height of the jagged curve in figure s01 above the time-axis.
Why the topic needs it: it is the raw thing the sensor records. Everything else is derived from it.

2. Acceleration and the unit " $g$ "

Picture: stack $8.9$ people-weights of yank on the panel.
Why the topic needs it: engineers describe vibration strength in $g$ because it instantly says "how brutal, compared to gravity."

3. Frequency $f$ — "how fast it wiggles"

Picture: a slow ocean swell is low $f$ ; a buzzing mosquito is high $f$ .
Why the topic needs it: launch noise contains all frequencies mixed together. To make sense of it we must sort the mess by frequency.

Look at figure s02: the teal wave fits only $3$ cycles into one second (low $f$ ); the orange wave crams $25$ cycles into the same second (high $f$ ). Same axis, same duration — frequency is simply how densely the wiggles are packed along time.

4. The infinitesimal step $d t$ — "a sliver of time"

Before any integral we must earn the tiny symbol $d t$ , which appears at the end of the Fourier formula.

Picture: stand a row of ultra-thin vertical strips side by side under a curve; the width of any single strip is $d t$ .
Why the topic needs it: whenever we "add up a quantity spread continuously" (energy over time, energy over frequency), we cannot count in whole chunks — the world is smooth. $d t$ and $df$ are the widths of those vanishing chunks, so that "height $\times$ width, summed" becomes exact. This is the same idea as $df$ in §8's integral — we simply meet it here first because the Fourier formula uses it.

5. The imaginary unit $i$ , $e^{i θ}$ , and Euler's formula

The parent note writes $e^{- i 2 π f t}$ . Two brand-new characters hide in there: the letter $i$ and the number $e$ . Let's earn both before using them.

Picture: a dot glued to the rim of a spinning wheel of radius $1$ . Its shadow on the floor traces $cos θ$ ; its shadow on the wall traces $sin θ$ . That is why $e^{i θ}$ is a pure wiggle — its shadows are perfect sine and cosine waves.
Why the topic needs it: a single symbol $e^{i θ}$ carries both the cosine and sine wiggle at once, which is far tidier than juggling two separate waves. This compactness is why every frequency formula is written with $e$ instead of $cos$ and $sin$ .

Look at figure s02b: the plum dot sits on the unit circle at angle $θ$ ; its drop onto the horizontal axis (teal) is $cos θ$ , onto the vertical axis (orange) is $sin θ$ . As the dot spins, those two shadows sweep out the two waves — that is Euler's formula made visible.

6. The pure tone $e^{- i 2 π f t}$ — the building block

Now the whole kernel makes sense piece by piece.

Breaking the squiggle down:

$2 π$ — one full circle is $2 π$ "radians." It converts cycles into angle swept.
$2 π f$ — angle swept per second. This "angle swept per second" gets its own name and symbol, angular frequency $ω$ (Greek "omega"): $ω = 2 π f$ , measured in radians per second. It is the same information as $f$ , just counted in radians instead of whole cycles — so we can always swap $2 π f$ for $ω$ and back. (You will meet $ω$ again in the response equations of §12, where $ω_{n} = 2 π f_{n}$ .)
$2 π f t = ω t$ — total angle swept after $t$ seconds — this is the $θ$ inside Euler's formula.
$e^{i (angle)}$ — the spinning-wheel dot from §5.
the minus sign ( $- i$ ): it makes the dot spin clockwise. This is not arbitrary. The forward transform (signal $\to$ frequencies) uses $- i$ ; the inverse transform (frequencies $\to$ signal) uses $+ i$ . Choosing opposite signs is what lets the two operations undo each other: spin one way to measure a frequency, spin the other way to rebuild the signal. Pick $- i$ for the forward direction and you are locked into $+ i$ for the inverse.

Why the topic needs it: to sort vibration by frequency, we first need the frequency-bricks themselves. That is what a tone is.

7. The Fourier transform $X (f)$ — the sorting machine

Read the forward transform as a question: "How much of the pure tone at frequency $f$ is hidden inside my jagged signal $x (t)$ ?"

$\int \dots d t$ — the integral, "add up over all time," where $d t$ (§4) is the width of each vanishing time-sliver. (See §9 for the full picture of what an integral is.)
$x (t) e^{- i 2 π f t}$ — line up the signal against the tone of frequency $f$ ; where they march in step the product is big, where they clash it cancels.
Result $X (f)$ : a number (generally complex, a point $a + bi$ ) saying how strongly frequency $f$ is present.
Picture: a prism splitting white light into a rainbow — $x (t)$ is the white light, $X (f)$ is the brightness of each color.
Why the topic needs it: it converts the useless time-scribble into a frequency recipe.

Definition Why

$X (- f)$ mirrors $X (f)$ for a real signal The integral runs from $- \infty$ to $+ \infty$ , so it produces values at negative $f$ too. Here is why the negatives mirror the positives. Because $e^{- i 2 π (- f) t} = e^{+ i 2 π f t}$ is the exact complex-conjugate (the " $i \to - i$ flip") of $e^{- i 2 π f t}$ , and because the signal $x (t)$ is a real number (an acceleration reading, never imaginary), replacing $f$ by $- f$ inside the integral simply conjugates the whole result: $X (- f) = \overline{X (f)}$ (a bar denotes complex conjugate — flip the sign of the imaginary part). Conjugating a complex number leaves its distance from the origin unchanged, so $∣ X (- f) ∣ = ∣ X (f) ∣$ : the two sizes are equal. This size-symmetry is what will later force the PSD (built in §10 from $∣ X ∣^{2}$ ) to obey $G_{xx} (- f) = G_{xx} (f)$ — but we defer that statement until §10, once $G_{xx}$ actually exists. Practical upshot: negative frequencies carry no new information, so engineers fold the two halves together and quote a one-sided PSD over $f \geq 0$ only, at twice the height so total energy is preserved. That is why every PSD table (like the parent's launch spectrum) starts at a positive $f$ such as $20 Hz$ .

8. Magnitude, the square $∣ X (f) ∣^{2}$ , and $E [\cdot]$

Picture: flip the same rattling launch a hundred times, measure each, and take the mean — that mean is $E$ .
Why the topic needs it: randomness means a single measurement is noisy. $E$ washes the noise out and leaves the statistical character — the only thing that's actually predictable.

9. What an integral $\int$ really is (the workhorse)

Look at figure s03: the plum curve is a PSD. The single orange strip is one rectangle — height $G (f)$ times width $df$ . The integral is nothing more than the total teal-shaded area you get by laying infinitely many such strips side by side; that area is the grand-total energy.

Why this exact tool? We want a grand total of energy, but energy is spread continuously across frequency (not in neat chunks). Ordinary addition can't sum infinitely many infinitely-thin slices — the integral is the machine invented precisely for that job.
Where the topic uses it: turning a per-frequency PSD into total mean-square acceleration.

10. PSD $G_{xx} (f)$ — energy sorted by frequency

Now every ingredient assembles into the star symbol, and we can finally write it as a formula, not just words.

Definition The assumptions that make the definition legal: stationary + ergodic

Two properties of the shaking must hold before the formula below even makes sense:

Stationary — the signal's statistics (its average, its spread, its frequency mix) do not drift over time. The launch buffeting looks statistically the same in the first second as in the tenth. Without this, "the" PSD would keep changing and the $T \to \infty$ limit could never settle.
Ergodic — a single long recording eventually explores all the behaviour the random process can produce, so a time-average over one record equals the ensemble-average $E$ over many records. This is what lets engineers estimate a PSD from one test run instead of relaunching a hundred rockets.

The parent note's phrase "stationary random process" is precisely this promise. We assume it throughout.

Reading the formula piece by piece (each piece was built above):

$∣ X_{T} (f) ∣^{2}$ — the energy the frequency- $f$ tone carries in one $T$ -second recording (§7, §8).
$\frac{1}{T}$ — divide by the record length to turn total energy into energy per second (a rate, "power"). Record twice as long and $∣ X_{T} ∣^{2}$ roughly doubles; dividing by $T$ cancels that growth so the number settles down.
$E [\cdot]$ — average over many recordings (§8) so the randomness washes out.
$lim_{T \to \infty}$ — let the record grow forever so the rate reaches its steady value.

Picture: the height of the curve in figure s03 — tall where lots of shaking energy concentrates, short where the shaking is gentle.
The units decoded: $g^{2}$ because it's energy-like (squared), per $Hz$ because it's density — spread over frequency. Multiply by a bandwidth in $Hz$ and the $Hz$ cancels, leaving $g^{2}$ .
The promised symmetry (now that $G_{xx}$ exists): because §7 showed $∣ X (- f) ∣ = ∣ X (f) ∣$ , squaring gives $G_{xx} (- f) = G_{xx} (f)$ — the PSD is a mirror image about $f = 0$ , which is exactly why the one-sided (positive- $f$ -only) version is used in practice.
Why the topic needs it: it is the single object that fully describes random vibration for engineering — the "fingerprint" you test hardware against. It feeds directly into Fatigue Analysis and Acoustic Loading predictions.

11. Mean-square $\overline{a^{2}}$ and RMS $a_{RMS}$ — and the link to PSD

Why the square root at the end? Mean-square is in $g^{2}$ — un-physical. The square root drags it back to plain $g$ , giving one honest number: "the shaking is as if a steady $a_{RMS}$ acceleration."
Why not just average $x (t)$ ? The signal wiggles equally positive and negative, so its plain average is $\approx 0$ — useless. Squaring first makes everything positive, so nothing cancels. That is the whole reason RMS exists.
Picture: the total area under the PSD (figure s03), square-rooted, collapsed into one bar height.

12. Symbols for the structure's response (preview)

The parent note's later half introduces the oscillator that reacts to the shaking. Meet the letters here — the full mechanics live in the sibling deep dives and in Modal Analysis.

Symbol	Plain words	Picture
$f_{n}$	natural frequency — the frequency the part "wants" to vibrate at if you flick it	a plucked guitar string's own note
$ω_{n} = 2 π f_{n}$	natural angular frequency — the same $f_{n}$ counted in radians/sec (the $ω$ from §6)	same note, angle units
$ζ$ (zeta)	damping ratio — how quickly wiggles die out ( $0$ = rings forever, $1$ = no bounce)	see Structural Damping
$Q = 1/ (2 ζ)$	quality factor — how sharp/tall the resonance peak is	height of the spike in figure s04
$\overset{x}{¨}, \overset{y}{¨}$	two dots = acceleration (base input $x$ , mass output $y$ )	how hard base vs. mass is yanked
$z = y - x$	relative displacement — how far the mass moved past its base; this bends metal and causes fatigue	the stretch in the spring
$T (f)$	transmissibility — output shake ÷ input shake at each $f$	see Transmissibility, Frequency Response Function (FRF)

Look at figure s04: the orange curve is the transmissibility $∣ T (f) ∣^{2}$ — output shake divided by input shake at each frequency. Far below the natural frequency it hugs the teal dashed "unity" line (structure passes the shaking straight through, no amplification). At $f_{n} = 200 Hz$ (plum dotted line) it erupts into a sharp spike reaching $\approx 101$ — this is resonance, and it is the reason a flat input PSD comes out peaked. That spike is what drives stress, ties into Fatigue Analysis, and is hunted for in Sine Vibration Testing and Shock Response Spectrum (SRS).

How the foundations feed the topic

Equipment checklist

Cover the right side and answer aloud before revealing.

What does

x (t)

mean, and why is it NOT a multiplication?

"The acceleration value at time

t

" —

x

is a function fed the input

t

, not a product.

What is

1 g

in everyday terms?

The acceleration of Earth's gravity,

9.81 m/s^{2}

; a strength yardstick.

What does frequency

f

(in

Hz

) count?

Full back-and-forth cycles per second.

How does angular frequency

ω

relate to

f

ω = 2 π f

— the same information counted in radians per second instead of whole cycles.

What does the differential

d t

(or

df

) stand for?

The width of an infinitely thin sliver of time (or frequency) used when summing a continuous quantity.

What is the rule defining the imaginary unit

i

i^{2} = - 1

; it lives on an axis perpendicular to the real number line.

State Euler's formula and its picture.

e^{i θ} = cos θ + i sin θ

; a dot on the unit circle at angle

θ

whose shadows are cosine and sine.

Why does the forward Fourier transform use

- i

and not

+ i

The forward transform spins one way (

- i

) so the inverse can spin the opposite way (

+ i

) and undo it; the signs must be opposite to be inverses.

What is a "pure tone" and why do we care?

A single-frequency wiggle

e^{- i 2 π f t}

; tones are the building bricks any signal can be summed from.

In one sentence, what question does the Fourier transform

X (f)

answer?

"How much of frequency

f

is inside my signal?"

Why does

∣ X (- f) ∣ = ∣ X (f) ∣

for a real signal?

Flipping

f \to - f

conjugates the transform (

X (- f) = \overline{X (f)}

), and conjugation leaves the distance from the origin unchanged.

Write the definition of the PSD as a formula.

G_{xx} (f) = lim_{T \to \infty} \frac{1}{T} E [∣ X_{T} (f) ∣^{2}]

Which two assumptions on

x (t)

make that PSD limit well-defined?

Stationary (statistics don't drift in time) and ergodic (one long record's time-average equals the ensemble-average).

In that PSD formula, why divide by

T

and why take

E

Dividing by

T

turns growing total energy into a steady rate (power); $\m