Visual walkthrough — Random vibration — PSD, RMS acceleration

3.6.11 · D2 · Physics › Spacecraft Structures & Systems Engineering › Random vibration — PSD, RMS acceleration

Koi bhi symbol aane se pehle, ek vaada: har letter pehle ek picture hai. Chalte hain.

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Step 1 — Hilaav: ek random signal aakhir hota kya hai

KYA HAI. Hum samay ke saath acceleration record karte hain. Time $t$ par record ki gayi value ko letter $x(t)$ kehte hain — padhte hain "$x$ of $t$", matlab "woh number $x$ jo moment $t$ par depend karta hai". Yahan $x$ acceleration hai g mein measured (Earth ki gravity ke multiples, $1\text{g}\approx 9.81{\text{m/s}}^2$), aur $t$ time hai seconds mein.

KYUN. Ek musical note ki tarah (ek smooth repeating wave) nahi, launch signal bina kisi pattern ke kaampti hai — ek second aage predict nahi kar sakte. Lekin agar aap uska spread dekho — woh kitni typically zero se door jaati hai — toh woh spread stable hai. Wahi stability ek cheez hai jisko hum pakad sakte hain.

PICTURE. Figure mein, blue line hai $x(t)$: ek jagged trace jo gray zero line ke upar aur neeche bounce karta hai. Orange band uske typical swing ko mark karta hai. Notice karo ki line kabhi repeat nahi hoti, phir bhi orange band ki width poore record mein same rehti hai. Wahi constant width ka matlab hai "stationary random process".

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Step 2 — Hilaav ko pure tones mein todna

KYA HAI. Hum claim karte hain ki woh jagged blue line secretly bahut saari frequencies ki smooth waves ka sum hai jo ek saath add hoti hain. Frequency $f$ (hertz mein, $\text{Hz}$ = cycles per second) ki ek pure tone ek aisi wave hai jo har second $f$ baar repeat hoti hai.

KYUN yeh tool — Fourier transform. Hume ek tarika chahiye yeh poochhne ka ki "is mess mein frequency $f$ kitni hai?" Fourier transform exactly wahi measuring device hai. Woh ek specific sawaal ka jawab deta hai: har frequency $f$ ke liye, $x(t)$ ke andar chhipi us frequency ki wave ki size aur timing kya hai? Hum ise choose karte hain, simple averaging ko nahi, kyunki averaging frequency information ko destroy kar deti hai jabki Fourier transform use preserve karta hai.

$X(f)={\int }_{-\infty }^{\infty }\underset{\text{the wiggle}}{\underbrace{x(t)}}\underset{\text{a probe tone at freq }f}{\underbrace{e^{-i2\pi ft}}}dt$

• $X(f)$ — frequency $f$ ki maatra jo present hai (ek complex number: uski size strength hai, uska angle timing hai).
• $e^{-i2\pi ft}$ — "probe": ek spinning unit arrow jo $f$ times per second ghoomta hai. Isse multiply karke add up karna (woh $\int dt$) poochhta hai "kya hilaav is tone ke saath step mein spin karti hai?"
• ${\int }_{-\infty }^{\infty }\dots dt$ — "saare time par add up karo".

PICTURE. Figure mein wahi jagged blue line left par dikhti hai, aur right par teen smooth colored sine waves (low, middle, high frequency) jo usse add up karke banati hain. "Fourier transform" label wala arrow mess se uske ingredients ki taraf point karta hai.

Prerequisite: yeh decomposition wahi idea hai jo Frequency Response Function (FRF) aur Modal Analysis mein use hoti hai — kisi bhi motion ko frequency ingredients mein tod do.

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Step 3 — "Amount" se "power" tak: square karna

KYA HAI. $X(f)$ ki size lo aur square karo: $|X(f){|}^2$. Bars $|\cdot |$ ka matlab hai "arrow ki length" (uski timing/angle ignore karo); square karna us length ko ek power-jaisi quantity mein convert karta hai.

KYUN square karte hain? Physics mein power amplitude squared ke saath jaati hai (jaise kinetic energy $\propto$ velocity²). Hume parwah nahi ki wave upar push karti hai ya neeche — sirf itna ki woh kitni strongly shake karti hai. Square karna sign throw away kar deta hai aur strength rakhta hai. Yeh "spread" measure karne ke tarike se bhi match karta hai (variance squares ka mean hota hai).

PICTURE. Figure $|X(f){|}^2$ ko $f$ ke against plot karta hai: ek bumpy curve, jahan strong frequencies rehti hain wahan high, baaki jagah near zero. Har vertical bar ek frequency ki squared strength hai.

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Step 4 — Length fix karna: $T$ se divide karo, phir average karo

KYA HAI. Hamari recording sirf finite time $T$ (seconds) tak chali. $|X(f,T){|}^2$ likhte hain record length yaad rakhne ke liye. Hum banate hain $\frac{|X(f,T){|}^2}{T}.$

KYUN $T$ se divide karte hain? Zyada lambi record karo toh aap zyada cycles accumulate karte ho, toh $|X(f,T){|}^2$ badhta hai sirf isliye kyunki aapne zyada der dekha — physics stronger nahi hua. $T$ se divide karna ek badhte hue total ko ek stable rate mein convert karta hai: power per second of record.

KYUN average karte hain, aur $T\to \infty$ lete hain? Ek recording random dice ka ek throw hai. Alag alag launches thodi alag curves dete hain. Hum bahut saari (expected value $\mathbb{E}[\cdot ]$, padhte hain "saari possible recordings ka mean") par average karte hain aur $T\to \infty$ lete hain taaki rate apni sahi steady value par settle ho jaye.

$G_{xx}(f)={\lim }_{T\to \infty }\frac{1}{T}\mathbb{E}[|X(f,T){|}^2]$

• $G_{xx}(f)$ — PSD: settled power per hertz, units ${\text{g}}^2/\text{Hz}$.
• $\frac{1}{T}$ — total ko rate mein convert karta hai (per second of record).
• $\mathbb{E}[\cdot ]$ — bahut saari recordings par average (random luck ko khatam karta hai).
• ${\lim }_{T\to \infty }$ — record ko forever badhne do taaki rate wobbling band kare.

PICTURE. Figure teen patli jittery curves (teen recordings) aur ek moti smooth curve dikhata hai — unka average — jo $G_{xx}(f)$ hai. Individual runs ka jitter ek clean shape mein wash out ho jaata hai.

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Step 5 — PSD ke neeche area = total mean-square shaking

KYA HAI. Saari shaking power pane ke liye, har thin frequency slice ko add karo. Frequency $f$ par width $df$ ki har slice mein mean-square power ka $G_{xx}(f)df$ hota hai. Slices ko sum karna integration hai: $\overline{a^2}={\int }_{f_1}^{f_2}{G}_{xx}(f)df.$

KYUN integration — aur kyun yahi tool? Hamare paas ek quantity hai jo frequency par continuously spread hai (power density), aur hum uska total chahte hain. Tiny contributions ke continuum ko add karna exactly wahi kaam karta hai jo integral karta hai; ek plain sum tabhi kaam karta jab hamare paas frequencies ki finite list hoti. $a^2$ ke upar bar, $\overline{a^2}$, ka matlab hai "mean square" — acceleration squared ka average.

• $G_{xx}(f)df$ — ek thin slice mein power (height $\times$ width).
• ${\int }_{f_1}^{f_2}$ — $f_1$ se $f_2$ tak saari slices ko glue karo.
• $\overline{a^2}$ — total mean-square acceleration, units ${\text{g}}^2$.

PICTURE. Figure $f_1$ aur $f_2$ ke beech PSD curve ke neeche poore area ko shade karta hai. Ek thin slice orange mein highlight hai jisme uski height $G_{xx}(f)$ aur width $df$ labelled hai, dikhata hai ki yeh ek tiny rectangle hai. Shaded total $\overline{a^2}$ hai.

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Step 6 — Square root: honest units mein wapas

KYA HAI. Area ${\text{g}}^2$ mein hai. $\text{g}$ mein wapas aane ke liye square root lo: $a_{\text{RMS}}=\sqrt{\overline{a^2}}=\sqrt{{\int }_{f_1}^{f_2}{G}_{xx}(f)df}.$

KYUN. "RMS" = Squares ka Mean ka Root. Square karna (Step 3) aur average karna (Step 5) hume ${\text{g}}^2$ mein chhod gaya; root squaring ko undo karta hai taaki answer ek real, feel kiya ja sakne wala acceleration $\text{g}$ mein ho — ek "typical" shake magnitude.

PICTURE. Figure ek simple bar hai: ek tall ${\text{g}}^2$ box (area) jisme ek arrow " ​" hai jo use ek shorter $\text{g}$ bar (RMS) mein collapse karta hai. Words "${\text{g}}^2\to \text{g}$" arrow par hain.

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Step 7 — Edge aur degenerate cases

KYA HAI & KYUN. Ek formula jo sirf friendly inputs par kaam kare woh ek trap hai. Corners check karte hain:

• Har jagah zero PSD ($G_{xx}=0$): area $=0$, toh $a_{\text{RMS}}=0$. Shaking in nahi, shaking out nahi. Sahi hai.
• Ek spike (single tone): agar saari energy exactly ek frequency par baithe, toh wahan density $G_{xx}$ infinite hogi jabki uski width zero hai — PSD ek pure tone describe nahi kar sakta. Isliye single tones Sine Vibration Testing se belong karte hain, random analysis se nahi.
• Bahut narrow band: agar $f_2\to f_1$, area $\to 0$, toh $a_{\text{RMS}}\to 0$. Ek whisker-thin band almost koi power nahi carry karta.
• Piecewise input: area ko pieces mein split karo aur add karo — kuch naya nahi, sirf Step 5 har segment par apply hota hai. Region areas ${\text{g}}^2$ mein add hote hain, aur sirf final total ko root milta hai.

PICTURE. Figure parent Example 2 ka launch spectrum teen colored rectangles ke roop mein dikhata hai ($0.8$, $16.0$, $15.0{\text{g}}^2$). Unki heights PSD levels hain, unki widths bands hain, aur ek bracket dikhata hai ki areas $31.8{\text{g}}^2$ mein sum hote hain ek final root se pehle jo $5.64\text{g}$ deta hai.

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Step 8 — Resonance picture ke saath kya karta hai

KYA HAI. Real structures shaking ko bina badlaav ke pass-through nahi karte. Natural frequency $f_n$ (uski favourite ringing frequency) aur damping ratio $\zeta$ (kitni jaldi settle hota hai) wala ek part $f_n$ ke paas input ko amplify karta hai. Response PSD hai $G_{\ddot{y}\ddot{y}}(f)=|T(f){|}^2{G}_{\ddot{x}\ddot{x}}(f),$ jahan $|T(f){|}^2$ hai transmissibility — frequency-by-frequency gain.

KYUN RMS badal jaata hai. Yahan tak ki ek flat input ek peaked output ban jaata hai: response curve ke neeche area $f_n$ par ek tall spike se dominate hota hai. Toh response ka $a_{\text{RMS}}$ input se bada hota hai, resonance ke paas concentrated.

PICTURE. Figure ek flat blue input PSD aur orange response PSD ko overlay karta hai jisme $f_n=200\text{Hz}$ par ek sharp peak hai. Resonance par $|T{|}^2=1+1/(4{\zeta }^2)=101$ for $\zeta =0.05$, toh peak response density $0.02\times 101=2.02{\text{g}}^2/\text{Hz}$ hai — ek red dot ise mark karta hai.

Dekho Transmissibility, Structural Damping, aur Fatigue Analysis ke liye jahan yeh spike apna damage karta hai.

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Ek picture mein summary

Sab kuch ek canvas par: wiggle → tones → square → average (PSD) → area → root → RMS, resonance ke saath jo curve ko reshape karta hai.

Recall Feynman retelling — seedha bolo

Main ek shaky line record karta hoon. Main use predict nahi kar sakta, lekin uska spread steady hai. Main poochhta hoon "usme har pitch kitni hai?" — woh Fourier transform hai. Main jawab ko square karta hoon strength pane ke liye (mujhe parwah nahi upar ya neeche), divide karta hoon kitna lamba dekha uspe taaki yeh ek fair rate ho, aur bahut saari tries par average karta hoon taaki luck cancel ho jaye. Isse mujhe PSD milta hai: har frequency par shaking-power ki ek curve. Ek number pane ke liye, main us curve ke neeche area dhoondhta hoon — ${\text{g}}^2$ mein total mean-square shaking — aur plain $\text{g}$ wapas paane ke liye square root leta hoon. Woh number, $a_{\text{RMS}}$, mera "typical shake" hai. Agar part kisi frequency par ring karna pasand karta hai, toh woh wahan curve ko swell kar deta hai, toh uska apna RMS jo aaya usse bada hota hai. Yeh poori kahani hai, wiggle se ek honest number tak.

Recall Quick self-test

$|X(f,T){|}^2$ ko $T$ se kyun divide karte hain? ::: Lambi records total ko sirf zyada der dekhne se inflate karti hain; divide karne se ek stable rate milta hai (power per second). Integrate karne se pehle square kyun karte hain? ::: Sign ki parwah kiye bina strength measure karne ke liye, aur is liye bhi ki power/variance kaise define hoti hai usse match kare. Ek final square root kyun, per segment nahi? ::: Powers (${\text{g}}^2$) add hoti hain; root honest $\text{g}$ units restore karta hai aur sirf total par apply hona chahiye. Flat input, resonant part — output RMS bada hoga ya chhota? ::: Bada; $|T{|}^2$ $f_n$ par spike karta hai aur response PSD ke neeche area ko swell karta hai.