Random vibration — PSD, RMS acceleration: Foundations

3.6.11 · D1 · Physics › Spacecraft Structures & Systems Engineering › Random vibration — PSD, RMS acceleration

Is page mein assume kiya gaya hai ki tumhe kuch nahi pata. Hum har letter, squiggle, aur picture build karenge jo parent note (Random Vibration) mein use hoti hai — ek aisi order mein jahan har idea pichle idea ke upar tikaa ho.

0. Woh picture jise hum baar baar dekhte hain

Koi bhi symbol aane se pehle: socho ek sensor spacecraft panel pe chipka hua hai. Yeh acceleration measure karta hai — panel ko kitni zyada jhatke se khicha ja raha hai — hazaaron baar per second. Un readings ko time ke against plot karo to ek ulajha hua, kabhi na repeating hone wala scribble milta hai.

Figure s01 dekho: teal curve $x (t)$ hai. Notice karo yeh baar baar zero cross karta hai aur kabhi repeat nahi hota — orange arrow ek aisi jagah mark karta hai jahan tum past se agle value ka andaza simply nahi laga sakte. Yahi un-guessability "random" ka matlab hai, aur isliye hum curve predict karna chhod dete hain aur uski statistics measure karte hain. Neeche sab kuch us scribble ko kabu karne ke tools hain.

1. Time $t$ aur signal $x (t)$

Seedhi baat: " $t$ moment pe acceleration".
Picture: figure s01 mein time-axis ke upar jagged curve ki height.
Topic ko iske liye kyun chahiye: yeh raw cheez hai jo sensor record karta hai. Baaki sab kuch isi se derived hota hai.

2. Acceleration aur unit " $g$ "

Picture: panel pe $8.9$ log-ke-weight ke barabar ka jhatka daalo.
Topic ko iske liye kyun chahiye: engineers vibration strength ko $g$ mein describe karte hain kyunki yeh turant bata deta hai "kitna brutal, gravity se compare karke."

3. Frequency $f$ — "kitni tezi se wiggle karta hai"

Picture: slow ocean swell low $f$ hai; buzzing mosquito high $f$ hai.
Topic ko iske liye kyun chahiye: launch noise mein sabhi frequencies mixed hoti hain. Isse samajhne ke liye hume gande mix ko frequency ke hisaab se sort karna hoga.

Figure s02 dekho: teal wave ek second mein sirf $3$ cycles fit karti hai (low $f$ ); orange wave usi second mein $25$ cycles thoos deti hai (high $f$ ). Same axis, same duration — frequency simply yeh hai ki wiggles time ke saath kitne densely packed hain.

4. Infinitesimal step $d t$ — "time ka ek sliver"

Kisi bhi integral se pehle hume chhota sa symbol $d t$ earn karna hoga, jo Fourier formula ke end mein aata hai.

Picture: ek curve ke neeche ultra-thin vertical strips ki row khadi karo; kisi bhi single strip ki width $d t$ hai.
Topic ko iske liye kyun chahiye: jab bhi hum "kisi continuously pheli quantity ka total add up karte hain" (time pe energy, frequency pe energy), hum whole chunks mein count nahi kar sakte — duniya smooth hai. $d t$ aur $df$ un vanishing chunks ki widths hain, taaki "height $\times$ width, summed" exact ho jaaye. Yeh same idea hai jaise §8 ke integral mein $df$ — hum ise yahan pehle milte hain kyunki Fourier formula ise use karta hai.

5. Imaginary unit $i$ , $e^{i θ}$ , aur Euler's formula

Parent note likhta hai $e^{- i 2 π f t}$ . Usme do brand-new characters chhupe hain: letter $i$ aur number $e$ . Dono ko use karne se pehle earn karte hain.

Picture: ek dot radius $1$ ke spinning wheel ke rim pe chipka hua. Uska floor pe shadow $cos θ$ trace karta hai; wall pe shadow $sin θ$ trace karta hai. Isliye $e^{i θ}$ ek pure wiggle hai — uske shadows perfect sine aur cosine waves hain.
Topic ko iske liye kyun chahiye: ek single symbol $e^{i θ}$ dono — cosine aur sine wiggle — ek saath carry karta hai, jo do alag waves juggle karne se kaafi cleaner hai. Yahi compactness ki wajah se har frequency formula $cos$ aur $sin$ ki jagah $e$ se likha jaata hai.

Figure s02b dekho: plum dot unit circle pe angle $θ$ pe baitha hai; iska horizontal axis (teal) pe girna $cos θ$ hai, vertical axis (orange) pe $sin θ$ hai. Jaise jaise dot spin karta hai, woh do shadows do waves sweep out karte hain — yahi Euler's formula visible form mein hai.

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

Ab poora kernel piece by piece samajh aata hai.

Squiggle ko tod ke dekhte hain:

$2 π$ — ek full circle $2 π$ "radians" hota hai. Yeh cycles ko angle swept mein convert karta hai.
$2 π f$ — angle swept per second. Is "angle swept per second" ka apna naam aur symbol hai, angular frequency $ω$ (Greek "omega"): $ω = 2 π f$ , radians per second mein measured. Yeh $f$ jitni hi information hai, bas whole cycles ki jagah radians mein count ki gayi hai — isliye hum hamesha $2 π f$ ko $ω$ se replace kar sakte hain aur wapas. (Tum $ω$ phir miloge §12 ke response equations mein, jahan $ω_{n} = 2 π f_{n}$ .)
$2 π f t = ω t$ — $t$ seconds ke baad total angle swept — yeh Euler's formula ke andar $θ$ hai.
$e^{i (angle)}$ — §5 ka spinning-wheel dot.
minus sign ( $- i$ ): yeh dot ko clockwise spin karata hai. Yeh arbitrary nahi hai. Forward transform (signal $\to$ frequencies) $- i$ use karta hai; inverse transform (frequencies $\to$ signal) $+ i$ use karta hai. Opposite signs choose karna hi woh cheez hai jo dono operations ko ek doosre ko undo karne deti hai: ek direction mein spin karo frequency measure karne ke liye, doosri direction mein spin karo signal rebuild karne ke liye. Forward direction ke liye $- i$ choose karo aur tum inverse ke liye $+ i$ pe lock ho jaate ho.

Topic ko iske liye kyun chahiye: vibration ko frequency ke hisaab se sort karne ke liye, pehle frequency-bricks chahiye. Yahi tone hai.

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

Forward transform ko ek sawaal ki tarah padho: "Mere jagged signal $x (t)$ ke andar frequency $f$ ka pure tone kitna chhupa hua hai?"

$\int \dots d t$ — integral, "saare time pe add karo," jahan $d t$ (§4) har vanishing time-sliver ki width hai. (Ek integral kya hota hai iska poora picture §9 mein dekho.)
$x (t) e^{- i 2 π f t}$ — signal ko frequency $f$ ke tone ke against align karo; jahan woh ek saath march karte hain product bada hota hai, jahan clash hote hain woh cancel ho jaata hai.
Result $X (f)$ : ek number (generally complex, ek point $a + bi$ ) jo bataata hai ki frequency $f$ kitni strongly present hai.
Picture: ek prism white light ko rainbow mein split karta hai — $x (t)$ white light hai, $X (f)$ har color ki brightness hai.
Topic ko iske liye kyun chahiye: yeh useless time-scribble ko ek frequency recipe mein convert karta hai.

Definition Real signal ke liye

$X (- f)$ kyun $X (f)$ mirror karta hai Integral $- \infty$ se $+ \infty$ tak run karta hai, isliye yeh negative $f$ pe bhi values produce karta hai. Yahan kyun negatives positives mirror karte hain. Kyunki $e^{- i 2 π (- f) t} = e^{+ i 2 π f t}$ exactly complex-conjugate (yani " $i \to - i$ flip") hai $e^{- i 2 π f t}$ ka, aur kyunki signal $x (t)$ ek real number hai (ek acceleration reading, kabhi imaginary nahi), integral ke andar $f$ ko $- f$ se replace karna simply poore result ko conjugate kar deta hai: $X (- f) = \overline{X (f)}$ (bar complex conjugate denote karta hai — imaginary part ka sign flip karo). Ek complex number ko conjugate karne se uski origin se doori unchanged rehti hai, isliye $∣ X (- f) ∣ = ∣ X (f) ∣$ : dono sizes barabar hain. Yahi size-symmetry baad mein PSD (§10 mein $∣ X ∣^{2}$ se banti hai) ko $G_{xx} (- f) = G_{xx} (f)$ follow karne par majboor karti hai — lekin yeh statement tab tak defer karte hain jab tak §10 mein $G_{xx}$ actually exist na kare. Practical upshot: negative frequencies koi naya information nahi dete, isliye engineers dono halves ko fold karke sirf $f \geq 0$ ke upar ek one-sided PSD quote karte hain, double height par taaki total energy preserve ho. Isliye har PSD table (jaise parent ka launch spectrum) ek positive $f$ jaise $20 Hz$ se start hoti hai.

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

Picture: usi rattling launch ko sau baar flip karo, har baar measure karo, aur mean lo — woh mean $E$ hai.
Topic ko iske liye kyun chahiye: randomness matlab ek single measurement noisy hoti hai. $E$ noise wash kar deta hai aur statistical character chhodta hai — woh cheez jo actually predictable hai.

9. Integral $\int$ asal mein kya hota hai (workhorse)

Figure s03 dekho: plum curve ek PSD hai. Single orange strip ek rectangle hai — height $G (f)$ times width $df$ . Integral kuch nahi hai bas woh total teal-shaded area jo infinitely many aisi strips side by side rakhne se milti hai; woh area grand-total energy hai.

Yahi exact tool kyun? Hume energy ka grand total chahiye, lekin energy frequency ke across continuously pheli hai (neat chunks mein nahi). Ordinary addition infinitely many infinitely-thin slices sum nahi kar sakta — integral woh machine hai jo precisely usi kaam ke liye invent ki gayi.
Topic ise kahan use karta hai: per-frequency PSD ko total mean-square acceleration mein badalne ke liye.

10. PSD $G_{xx} (f)$ — frequency ke hisaab se sort ki gayi energy

Ab har ingredient star symbol mein assemble hota hai, aur hum ise finally ek formula ke roop mein likh sakte hain, sirf words mein nahi.

Definition Woh assumptions jo definition ko legal banate hain: stationary + ergodic

Shaking ki do properties honi chahiye pehle se taaki neeche diya formula sense bane:

Stationary — signal ki statistics (uska average, spread, frequency mix) time ke saath drift nahi karte. Launch buffeting pehle second mein statistically waise hi lagti hai jaisi daswe second mein. Iske bina, "the" PSD badalta rehta aur $T \to \infty$ limit kabhi settle nahi ho sakti.
Ergodic — ek single long recording eventually woh sab behaviour explore kar leti hai jo random process produce kar sakti hai, isliye ek record pe time-average ensemble-average $E$ (kaafi records ke upar) ke barabar hota hai. Yahi engineers ko ek test run se PSD estimate karne deta hai, sau rockets relaunch kiye bina.

Parent note ka phrase "stationary random process" precisely yahi promise hai. Hum yeh throughout assume karte hain.

Formula ko piece by piece padhna (har piece upar banayi gayi hai):

$∣ X_{T} (f) ∣^{2}$ — ek $T$ -second recording mein frequency- $f$ tone kitni energy carry karta hai (§7, §8).
$\frac{1}{T}$ — record length se divide karo taaki total energy ko energy per second (ek rate, "power") mein convert karo. Twice as long record karo aur $∣ X_{T} ∣^{2}$ roughly double ho jaata hai; $T$ se divide karna us growth ko cancel karta hai taaki number settle ho jaaye.
$E [\cdot]$ — kaafi recordings par average karo (§8) taaki randomness wash ho jaaye.
$lim_{T \to \infty}$ — record ko forever badhne do taaki rate apni steady value tak pahunche.

Picture: figure s03 mein curve ki height — jahan bahut saari shaking energy concentrate ho wahan tall, jahan shaking gentle ho wahan short.
Units decoded: $g^{2}$ kyunki yeh energy-like hai (squared), per $Hz$ kyunki yeh density hai — frequency par spread hai. $Hz$ mein bandwidth se multiply karo aur $Hz$ cancel ho jaata hai, $g^{2}$ bachta hai.
Promised symmetry (ab ki $G_{xx}$ exist karta hai): kyunki §7 ne dikhaya $∣ X (- f) ∣ = ∣ X (f) ∣$ , square karne se $G_{xx} (- f) = G_{xx} (f)$ milta hai — PSD $f = 0$ ke baare mein mirror image hai, yahi wajah hai ki practice mein one-sided (sirf positive- $f$ ) version use hota hai.
Topic ko iske liye kyun chahiye: yeh woh single object hai jo engineering ke liye random vibration fully describe karta hai — "fingerprint" jiske against hardware test kiya jaata hai. Yeh directly Fatigue Analysis aur Acoustic Loading predictions mein feed hota hai.

11. Mean-square $\overline{a^{2}}$ aur RMS $a_{RMS}$ — aur PSD se link

End mein square root kyun? Mean-square $g^{2}$ mein hai — un-physical. Square root ise wapas plain $g$ pe le aata hai, ek honest number deta hai: "shaking aise hai jaise ek steady $a_{RMS}$ acceleration."
Why not just $x (t)$ average karo? Signal equally positive aur negative wiggle karta hai, isliye uska plain average $\approx 0$ hota hai — useless. Pehle square karna sab kuch positive bana deta hai, taaki kuch cancel na ho. Isliye RMS exist karta hai.
Picture: PSD ke neeche total area (figure s03), square-rooted, ek bar height mein collapse hua.

12. Structure ke response ke symbols (preview)

Parent note ki baad ki half woh oscillator introduce karta hai jo shaking par react karta hai. Yahan letters se milo — poori mechanics sibling deep dives mein aur Modal Analysis mein hai.

Symbol	Plain words	Picture
$f_{n}$	natural frequency — woh frequency jis par part "khud se" vibrate karna chahta hai agar tum use flick karo	ek plucked guitar string ki apni note
$ω_{n} = 2 π f_{n}$	natural angular frequency — wahi $f_{n}$ radians/sec mein count ki gayi (§6 ka $ω$ )	same note, angle units
$ζ$ (zeta)	damping ratio — wiggles kitni jaldi die out hoti hain ( $0$ = hamesha ring karta hai, $1$ = koi bounce nahi)	dekho Structural Damping
$Q = 1/ (2 ζ)$	quality factor — resonance peak kitni sharp/tall hai	figure s04 mein spike ki height
$\overset{x}{¨}, \overset{y}{¨}$	do dots = acceleration (base input $x$ , mass output $y$ )	base vs. mass kitna yanked ho raha hai
$z = y - x$	relative displacement — mass apne base se aage kitna move hua; yeh metal bend karta hai aur fatigue cause karta hai	spring mein stretch
$T (f)$	transmissibility — output shake ÷ input shake har $f$ pe	dekho Transmissibility, Frequency Response Function (FRF)

Intuition

$Q = 1/ (2 ζ)$ kyun — quality damping ka inverse hai Ek swing socho. Damping $ζ$ friction hai: high friction matlab har push jaldi die out hoti hai, to swing kabhi zyada amplitude build nahi kar paata. Quality $Q$ ulta measure karta hai — resonance kitni tall build hoti hai aur fade hone se pehle kitna lamba ring karti hai. Low friction (small $ζ$ ) energy ko kaafi cycles par ek huge, sharp peak mein pile up karne deta hai (big $Q$ ); heavy friction (large $ζ$ ) energy bleed kar deti hai to peak chhoti rehti hai (small $Q$ ). Woh opposites hain, isliye ek doosre ka reciprocal hota hai — aur factor of $2$ isliye aata hai kyunki $ζ$ resonance denominator $(2 ζ f / f_{n})^{2}$ ke andar baitha hai. Exact resonance pe $∣ T ∣$ ki peak height $\approx Q$ hoti hai, isliye ek lightly damped box ( $ζ = 0.05 \Rightarrow Q = 10$ ) apna input tenfold amplify karta hai.

Figure s04 dekho: orange curve transmissibility $∣ T (f) ∣^{2}$ hai — output shake divided by input shake har frequency pe. Natural frequency se bahut neeche yeh teal dashed "unity" line se chipka rehta hai (structure shaking seedha pass karta hai, koi amplification nahi). $f_{n} = 200 Hz$ (plum dotted line) pe yeh ek sharp spike mein phoot ta hai jo $\approx 101$ tak pahunchta hai — yeh resonance hai, aur yahi wajah hai ki ek flat input PSD peaked output deta hai. Woh spike stress drive karta hai, Fatigue Analysis se connect hota hai, aur Sine Vibration Testing aur Shock Response Spectrum (SRS) mein dhundha jaata hai.

Foundations topic ko kaise feed karte hain

Equipment checklist

Cover the right side and answer aloud before revealing.

x (t)

ka kya matlab hai, aur yeh multiplication kyun NAHI hai?

t

time pe acceleration value" —

x

ek function hai jise input

t

diya jaata hai, product nahi.

1 g

everyday terms mein kya hai?

Earth's gravity ka acceleration,

9.81 m/s^{2}

; strength ka yardstick.

Frequency

f

(

Hz

mein) kya count karta hai?

Per second poore aage-peechhe cycles.

Angular frequency

ω

f

se kya relation hai?

ω = 2 π f

— wahi information whole cycles ki jagah radians per second mein count ki gayi.

Differential

d t

(ya

df

) kya represent karta hai?

Time (ya frequency) ka infinitely thin sliver ki width jo continuous quantity sum karte waqt use hoti hai.

Imaginary unit

i

define karne wala rule kya hai?

i^{2} = - 1

; yeh real number line ke perpendicular ek axis pe rehta hai.

Euler's formula batao aur uski picture.

e^{i θ} = cos θ + i sin θ

; angle

θ

pe unit circle pe ek dot jiske shadows cosine aur sine hain.

Forward Fourier transform

- i

kyun use karta hai,

+ i

nahi?

Forward transform ek taraf spin karta hai (

- i

) taaki inverse doosri taraf (

+ i

) spin karke undo kar sake; ek doosre ke inverse hone ke liye signs opposite hone chahiye.

"Pure tone" kya hai aur hum care kyun karte hain?

Ek single-frequency wiggle

e^{- i 2 π f t}

; tones woh building bricks hain jinhe jodhke koi bhi signal banaaya ja sakta hai.

Ek sentence mein, Fourier transform

X (f)

kaunsa sawaal answer karta hai?

"Mere signal mein frequency

f

kitni hai?"

Real signal ke liye

∣ X (- f) ∣ = ∣ X (f) ∣

kyun?

f \to - f

flip karne se transform conjugate ho jaata hai (

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

), aur conjugation origin se doori unchanged chhodta hai.

PSD ko formula ke roop mein define karo.

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

x (t)

pe kaunsi do assumptions woh PSD limit well-defined banati hain?

Stationary (statistics time mein drift nahi karte) aur ergodic (ek long record ka time-average ensemble-average ke barabar hai).

Us PSD formula mein

T

se divide kyun karte hain aur

E

kyun lete hain?

T

se divide karna badhti total energy ko ek steady rate (power) mein badalta hai;

E

randomness wash karta hai.