Shock response spectrum (SRS): Visual walkthrough — Hinglish deep dive

3.6.13 · D2 · Physics › Spacecraft Structures & Systems Engineering › Shock response spectrum (SRS)

Hum assume karte hain ki tum sirf push, spring, aur wobble jaante ho. Baaki sab hum khud construct karenge.

Step 1 — Ek chota wobbler ek hilte hue floor par

KYA. Hum wobbler ki teen properties name karte hain:

$m$ = block ka mass (kitna material hai, kg mein). Bada $m$ = dhakka dena mushkil.
$k$ = spring ki stiffness (woh kitni zor se pushback karta hai per metre stretch par, N/m). Bada $k$ = snappy spring.
$c$ = piston ki damping (woh kitni zor se speed ko resist karta hai, N per m/s). Bada $c$ = wobble jaldi khatam ho jaata hai.

YEH TEEN KYUN. Yahi woh ingredients hain jo ek bell ko bajaane aur phir chup hone ke liye chahiye: ek mass jo momentum le jaaye, ek spring jo use wapas kheenche (yahi ringing hai), ek damper jo energy bleed kare (yahi fading hai). Har real bracket, board, ya tank mount apni main resonance ke paas isi mein reduce hota hai.

PICTURE. Figure mein, floor ek jagged shock carry karta hai. Block floor se peeche reh jaata hai — woh peeche rehna hi woh cheez hai jise hum track karna chahte hain.

Relative kyun? Kyunki spring aur damper sirf block aur floor ke beech ka stretch feel karte hain — unhe poore room ki parwah nahi, sirf yeh ki unke do ends kitne door hain.

Step 2 — Newton hilte hue floor par: equation of motion milti hai

KYA. Block par har force ko jodo. Newton kehta hai mass × acceleration = forces ka sum.

Spring force $= - k z$ — wapas kheenchta hai, stretch ke ulti direction mein (isliye minus).
Damper force $= - c \overset{z}{˙}$ — motion resist karta hai, speed ke ulti direction mein.
Inertial (fictitious) force $= - m a (t)$ — accelerating floor par sawari karne ka peeche ki taraf dhakka. Yahan $a (t)$ hai floor ki acceleration (woh shock jo humein di gayi hai).

$ma (t)$ par minus kyun. Jab floor ek taraf accelerate karta hai, block dusri taraf phenka gaya feel karta hai — bilkul car wali example. Figure mein pink arrow dekho jo floor ki motion ke ulte point karta hai.

Ab har term ko $m$ se divide karo taaki mass nikl jaaye aur sirf ratios bachein:

Recall

$ω_{n} = k / m$ kyun aur $k / m$ kyun nahi? $k / m$ kahan se aata hai ::: Damping aur forcing ko zero karo; $\overset{z}{¨} = - (k / m) z$ ka solution ek sine hai jis ki frequency $k / m$ hai — classic "acceleration proportional to minus position" swing.

Dekho Quality factor Q and damping: engineers aksar $Q = \frac{1}{2 ζ}$ quote karte hain, toh $ζ = 0.05$ matlab $Q = 10$ .

Step 3 — Ek single kick kya karta hai (building block)

KYA. Wobbler ko time zero par ek instantaneous kick (impulse) do. Woh decaying sine se respond karta hai: woh swing karta hai, aur har swing pehle se choti hoti hai.

DECAYING SINE KYUN. Spring use swing karata hai (sine); damper har swing ko chota karata hai (ek shrinking multiplier). Do effects, do factors — unhe multiply karo.

"Exponential × sine" tool kyun aur, say, polynomial kyun nahi? Kyunki yeh wahi shape hai jis ki acceleration khud ka aur apni speed ka ek fixed combination equal hoti hai — bilkul wahi jo EOM demand karta hai. Koi aur fit nahi hota.

Step 4 — Saare kicks ko jodo: Duhamel's integral

KYA. Ek kick ke response ko slide karo taaki woh time $τ$ par start ho (uski "age" hai $t - τ$ ), use us kick ki strength $- a (τ) d τ$ se scale karo, aur start ( $0$ ) se ab ( $t$ ) tak integrate karo.

PICTURE. Figure mein, teen sample kicks mein se har ek apna shrinking sine launch karta hai; black curve unka running sum hai — woh sum hi $z (t)$ hai.

Step 5 — Relative wobble se us acceleration tak jo cheezein actually torti hain

KYA. True (absolute) acceleration = floor ne kya kiya + block ne floor ke relative kya kiya: $a_{abs} (t) = a (t) + \overset{z}{¨} (t)$

Ab Step 2 se EOM use karo taaki ugly $\overset{z}{¨}$ mit jaaye. $\overset{z}{¨} + 2 ζ ω_{n} \overset{z}{˙} + ω_{n}^{2} z = - a (t)$ ko rearrange karne par $\overset{z}{¨} = - a (t) - 2 ζ ω_{n} \overset{z}{˙} - ω_{n}^{2} z$ milta hai. Substitute karo:

YEH CANCEL KYUN HUA. Floor ka shove block mein sirf spring aur damper ke through enter karta hai — koi aur raasta nahi hai. Toh jab ek baar hum stretch aur speed jaante hain, toh floor ne block ke saath jo bhi kiya woh sab hum jaante hain.

Step 6 — Sabse bura moment lo: woh single number hi SRS point hai

Step 7 — Har frequency sweep karo: spectrum janam leta hai

KYA — aur woh saare cases jo tumhe kabhi surprise nahi karne chahiye:

Bahut low $f_{n}$ (slow wobbler). Uska period pulse se bahut lamba hai; pulse khatam ho jaata hai block ke zyada move karne se pehle. Block barely respond karta hai → SRS choti hai, $f_{n}^{2}$ ki tarah rise karti hai (yahi $+ 40$ dB/decade hai). Left rising branch dekho.
Knee, $f_{n} \approx \frac{1}{π τ}$ ke paas ( $τ$ = pulse duration ke saath). Yahan wobbler pulse ke dauran lagbhag ek swing complete karta hai — "ring up" ke liye perfect timing. SRS peak karta hai. Chota pulse yeh knee zyada frequency par push karta hai.
Bahut high $f_{n}$ (fast wobbler). Woh itni tezi se swing karta hai ki woh floor peak ko simply track karta hai; aur koi amplification gain karne ke liye nahi bachi. SRS flat ho jaata hai raw peak acceleration ki taraf ( $0$ dB/octave).

Ek-picture summary

Poori chain: shock $a (t)$ → kicks mein kaato → Duhamel sum → sachi acceleration ke liye subtract karo → peak pakdo → $f_{n}$ sweep karo → spectrum.

Recall Feynman retelling — plain words mein khud se bol ke dekho

Ek bolt blast hota hai aur spacecraft ka floor ek jagged, milliseconds-long jhatke mein lurches karta hai. Main jaanna chahta hoon: ek chote gadget ke liye jo kisi ek speed par wobble karna pasand karta hai, use kitni zor se slammed kiya jaata hai? Toh main us gadget ko ek block on one spring with a shock-absorber ke roop mein imagine karta hoon. Main hilte hue floor par chad jaata hoon use dekhne ke liye, jo ek nakli peeche ka dhakka ijaad karta hai — wahi dhakka shock feed in ho raha hai. Main figure out karta hoon ki block ek single instantaneous flick ka jawab kaise deta hai: woh swing karta hai aur swings choti hoti jaati hain. Kyunki flicks interfere nahi karte, main poore messy shock ko flicks ki barish mein kaatta hoon aur har jawab ko add karta hoon — yahi Duhamel's integral hai, jo mujhe floor ke relative block ki motion deta hai. Lekin damage sachi acceleration ki parwah karta hai room mein, toh main floor ki motion wapas add karta hoon; magically spring stretch aur damper speed akele hi mujhe woh sachi acceleration bata dete hain. Main poori ring-down scan karta hoon aur sirf sabse bada slam rakhta hoon — yahi ek dot hai. Phir main yeh sab ek slow wobbler ke liye, ek medium ke liye, ek fast ke liye — hundreds of them — dobara karta hoon aur dots connect karta hoon. Slow ones barely feel karte hain (rising branch), jo ek pulse per swing ke saath tuned hai woh sabse zyada ring up hota hai (knee peak), aur fast ones sirf floor ka peak ride karte hain (flat top). Woh connected curve shock response spectrum hai — ek picture jo har component ko bataati hai ki woh jeega ya marega.

Related: Pyroshock environments · Shock testing methods · MIL-STD-810 Method 516 · Modal analysis · Mechanical impedance · Force limiting in shock testing · Vibration power spectral density (PSD)