Shock response spectrum (SRS)

3.6.13 · Physics › Spacecraft Structures & Systems Engineering

Shock Response Spectrum Kya Hai?

SDOF oscillators kyun? Har real component (circuit board, fuel tank mount, antenna bracket) ko uski fundamental resonance ke paas ek mass-spring-damper ki tarah approximate kiya ja sakta hai. SRS sabhi possible resonances ke liye worst-case response ko pehle se compute kar leta hai.

SRS Ko First Principles Se Derive Karna

Step 1: Single Oscillator Ke Liye Equation of Motion

Socho ek component jisme mass $m$ , stiffness $k$ , damping $c$ hai, jo ek base se attached hai jo acceleration $a (t)$ experience kar raha hai. $z (t)$ ko relative displacement maano (component position minus base position).

Free-body diagram se milta hai: $m \overset{z}{¨} + c \overset{z}{˙} + k z = - ma (t)$

Minus sign kyun? Base acceleration $a (t)$ ek inertial force hai jo base motion direction ke opposite act karta hai. Socho tum ek accelerating car mein ho—tumhe peeche dhakka lagta hai.

$m$ se divide karo: $\overset{z}{¨} + 2 ζ ω_{n} \overset{z}{˙} + ω_{n}^{2} z = - a (t)$

jahan:

$ω_{n} = k / m$ natural frequency hai (rad/s)
$ζ = c / (2 k m)$ damping ratio hai
$f_{n} = ω_{n} / (2 π)$ natural frequency hai Hz mein

Step 2: Duhamel's Integral (Convolution Solution)

Underdamped system ( $ζ < 1$ ) ke liye, relative displacement response hai:

$z (t) = - \frac{1}{ω _{d}} \int_{0}^{t} a (τ) e^{- ζ ω_{n} (t - τ)} sin [ω_{d} (t - τ)] d τ$

jahan $ω_{d} = ω_{n} 1 - ζ^{2}$ damped natural frequency hai.

Yeh form kyun? Time $τ$ par har impulse $a (τ) d τ$ ek decaying oscillation create karta hai; hum $τ = 0$ se $t$ tak ke saare contributions ko sum (integrate) karte hain.

Step 3: Absolute Acceleration

Component ka absolute acceleration (jo actually cheezein todta hai) hai: $a_{abs} (t) = a (t) + \overset{z}{¨} (t)$

$a (t)$ add kyun karte hain? Relative acceleration $\overset{z}{¨}$ moving base se measure hoti hai; humein inertial frame mein total acceleration chahiye.

EOM se: $\overset{z}{¨} = - a (t) - 2 ζ ω_{n} \overset{z}{˙} - ω_{n}^{2} z$ , toh: $a_{abs} (t) = - 2 ζ ω_{n} \overset{z}{˙} (t) - ω_{n}^{2} z (t)$

Step 4: Maximum Absolute Acceleration

Natural frequency $f_{n}$ aur damping $ζ$ par SRS value hai: $SRS (f_{n}, ζ) = max_{t \geq 0} ∣ a_{abs} (t) ∣$

Computational procedure:

Ek natural frequency $f_{n}$ chuno (jaise 100 Hz)
$z (t)$ aur $\overset{z}{˙} (t)$ paane ke liye Duhamel convolution ko numerically integrate karo
$a_{abs} (t) = - 2 ζ ω_{n} \overset{z}{˙} - ω_{n}^{2} z$ compute karo
Peak dhundo: $max ∣ a_{abs} (t) ∣$
~10 Hz se 10,000 Hz tak 100–1000 frequencies ke liye repeat karo
Envelope plot karo

Physical Interpretation

SRS curve padhna:

Low frequencies (10–100 Hz): Bade structural modes; yahan oscillator period pulse se kaafi lambi hoti hai, isliye SRS frequency ke saath steeply rise karta hai (approximately $+ 40$ dB/decade, yaani $+ 12$ dB/octave, $f_{n}^{2}$ ke proportional)
Mid frequencies (100–1000 Hz): Secondary structure (panels, brackets); pyroshock ke liye knee aur peak amplification aksar yahan hoti hai
High frequencies (1000–10,000 Hz): Component-level resonances (PCBs, relays); knee ke baad ek simple pulse SRS peak acceleration ki taraf flatten hota hai (≈ $0$ dB/octave), kyunki yeh fast oscillators simply base peak ko track karte hain
Knee frequency ≈ $1/ (π τ)$ duration $τ$ ke half-sine ke liye; chota pulse knee ko higher frequency par push karta hai

Worked example Worked Example 1: Stage Separation Shock

Given: Explosive bolt firing se 500g peak shock pulse create hoti hai jo 0.5 ms tak chalti hai. Ek circuit board ka $f_{n} = 200$ Hz, $ζ = 0.03$ hai. Find: Kya board 1000g se zyada dekhega?

Solution:

Pulse duration ko oscillator period se compare karo. $f_{n} = 200$ Hz ke liye, $T_{n} = 1/200 = 5$ ms. Toh $0.5$ ms $\approx T_{n} /10$ — pulse ek oscillation period se kaafi chhoti hai. Yeh step kyun? Jab pulse $T_{n}$ ke relative choti hoti hai, oscillator sirf ek impulse "dekhta" hai; lekin $f_{n} τ = 200 \times 0.0005 = 0.1$ par hum abhi bhi SRS ke rising (low-frequency) part par hain, toh amplification modest hai.
Response estimate karo. Rising branch par SRS roughly $f_{n}$ times velocity change ke saath scale hota hai. Low damping ke liye yahan ek typical amplification factor ≈ 1.5–2× hai (knee ke paas hone par hi poora resonant 5–10× milta hai).
Estimate: $SRS (200 Hz) \approx 2 \times 500 g = 1000 g$ — bilkul limit par; marginal.
Yeh matter kyun karta hai? Board SRS ke rising part par baitha hai. Agar instead $f_{n}$ knee ke paas hota ( $\approx 1/ (π τ) \approx 640$ Hz), toh yeh kaafi zyada amplification dekhta. Engineering action: $f_{n}$ low rakho (knee se door) ya damping add karo (potting compound) peak response kam karne ke liye.

Worked example Worked Example 2: Component Qualification Ke Liye SRS Ka Use

Scenario: Supplier datasheet kehta hai ek relay 2000 Hz par 1500g SRS survive karta hai. Tumhara spacecraft pyroshock spec (flight environment) 2000 Hz par 1800g hai, dono $ζ = 0.05$ par quoted hain.

Question: Kya relay qualified hai?

Solution:

Apples-to-apples compare karo: Dono SRS values same $ζ$ use karein (yahan 5%). ✓
Relay capability: 1500g < Requirement: 1800g → Qualified nahi jaise-hai.
Yeh step kyun? Relay flight mein 1800g experience karega lekin sirf 1500g survive karta hai → failure risk.
Options:
- Relay ko shock isolate karo (soft mount add karo taaki uska $f_{n}$ 2000 Hz se door shift ho)
- Zyada robust relay select karo (jaise 3000g rating)
- Relay ko actual environment mein test karo (hidden margin discover ho sakta hai)

Worked example Worked Example 3: Pulse Shape Se SRS Estimate Karna

Idealized half-sine pulse: Duration $τ = 0.2$ ms $= 0.0002$ s, amplitude $A_{0} = 300 g$ .

$f_{n} = 2000$ Hz par Approximate SRS:

Key dimensionless number compute karo: $f_{n} τ = 2000 \times 0.0002 = 0.4$ . Yeh step kyun? $f_{n} τ$ decide karta hai hum SRS ke kis branch par hain—1 se kaafi neeche matlab low-frequency (rising) branch, 1 ke paas/upar matlab hum knee ke paas ya usse aage hain.
Low-frequency asymptote (valid jab $f_{n} τ ≪ 1$ ho) hai $SRS \approx A_{0} \cdot \frac{π ^{2} f _{n}^{2} τ ^{2}}{2}$ , jo $f_{n}^{2}$ ke saath rise karta hai (yaani $+ 40$ dB/decade). Yeh low-frequency behavior hai, high-frequency wala nahi.
Yahan $f_{n} τ = 0.4$ knee ki taraf badh raha hai ( $f_{n} τ \approx 1/ π \approx 0.32$ knee ke liye), toh simple $f_{n}^{2}$ formula break hona shuru ho raha hai aur curve apne flat, peak-amplification region ki taraf transition kar raha hai.
Knee ke paas practical estimate: amplification ≈ 1.6–1.8× → $SRS (2000 Hz) \approx 1.8 \times 300 g \approx 540 g$ .
Yeh step kyun? 2000 Hz par oscillator pulse ke dauran sirf ek cycle se thoda kam complete karta hai, toh yeh partially "ring up" hota hai — rising branch aur flat high-frequency branch ke beech ka classic knee behavior.

Common Mistakes

Engineers SRS Kyun Use Karte Hain Raw Time Histories Ki Jagah

Teen reasons:

Universality: Ek SRS curve sabhi possible component resonances ke liye shock ko characterize karta hai. SRS envelope tak testing karna har part qualify karta hai, uske specific $f_{n}$ se regardless.
Comparability: Tum alag-alag events (launch, stage sep, docking) ke SRS overlay kar sakte ho aur design spec ke roop mein worst-case envelope le sakte ho.
Test reproducibility: Pyroshock time histories chaotic aur unrepeatable hote hain. Lekin tum ek synthetic shock pulse generate kar sakte ho (shaker ya pyro simulator se) jo SRS match kare—alag waveforms ke saath same damage potential achieve hota hai.

Analogy: SRS shock ke liye wahi hai jo aerodynamics mein "design load factor" hota hai—ek single number (ya curve) jo reality ki complexity ko bound karta hai.

Recall Ek 12-Saal Ke Bachche Ko Explain Karo

Socho tumhare paas 100 alag-alag ghante hain, har ek alag pitch (frequency) par bajta hai. Ab tum us table ko jis par sab rakhe hain hammer se maaro (yeh tumhara shock hai). Kuch ghante SUPER LOUD bajenge kyunki hammer ki maar unki special pitch se match karti hai—ise resonance kehte hain. Doosre ghante muskil se awaaz karte hain.

Shock Response Spectrum ek report card ki tarah hai jo batata hai: "50 Hz par bajne wale ghante ko 200g tak hilaya gaya, 100 Hz wale ghante ko 500g mila, 1000 Hz wale ghante ko 800g mila..." Har possible ghante ki pitch ke liye tumhe ek number milta hai.

Yeh useful kyun hai? Kyunki tumhari spacecraft mein hazaaron parts hain, har ek alag ghante ki tarah. Har part ko alag-alag hammer se test karne ki jagah, tum bas SRS curve dekho aur kaho, "Theek hai, mera circuit board 150 Hz par bajta hai, toh curve mujhe bata raha hai ki use 300g milega. Kya yeh 300g survive kar sakta hai? Haan? Badiya, ho gaya!"

SRS ek cheat sheet hai jo sirf ek hammer hit se har single part ke liye worst shaking predict karta hai.

Connections

Pyroshock environments — Spacecraft mein severe SRS specs ki origin
Single-degree-of-freedom systems — Foundation: har SRS point ek SDOF response hai
Duhamel's integral — Transient response compute karne ka mathematical engine
Quality factor Q and damping — $Q = 1/ (2 ζ)$ ; SRS peak sharpness se relate karta hai
Mechanical impedance — SRS is se tie karta hai ki components base structure ko kaise "load" karte hain
Modal analysis — Real structure = SDOF modes ka superposition; SRS har ek ko test karta hai
Shock testing methods — Lab mein SRS kaise reproduce karein (drop tables, resonant plates)
Vibration power spectral density (PSD) — Shocks ke liye SRS, random vibration ke liye PSD
MIL-STD-810 Method 516 — Military shock test standard; SRS test profiles specify karta hai
Acceleration response in structures — SRS iska worst-case envelope hai
Force limiting in shock testing — High frequencies par over-testing se bachata hai

#flashcards/physics

SRS curve par har point kya represent karta hai? :: Ek single-degree-of-freedom (SDOF) oscillator us natural frequency aur specified damping par jo maximum absolute acceleration experience karta hai jab shock input diya jaaye.

Spacecraft shock specs ke liye raw time histories ki jagah SRS kyun prefer kiya jaata hai?

SRS ek universal envelope provide karta hai jo sabhi possible component resonances ko cover kare, alag-alag shock sources ke beech comparison enable karta hai, aur synthetic shocks ke saath reproducible testing allow karta hai jo same damage potential match kare.

Base excitation

a (t)

ke under SDOF system ke liye equation of motion likho.

\overset{z}{¨} + 2 ζ ω_{n} \overset{z}{˙} + ω_{n}^{2} z = - a (t)

, jahan

z

relative displacement hai aur minus sign inertial force direction account karta hai.

SRS analysis mein absolute acceleration aur relative motion mein kya relation hai?

a_{abs} (t) = a (t) + \overset{z}{¨} (t) = - 2 ζ ω_{n} \overset{z}{˙} (t) - ω_{n}^{2} z (t)

, base acceleration aur base ke relative component ke acceleration ko combine karta hai.

Spacecraft SRS specifications ke liye standard damping ratio kya hai?

ζ = 0.05

(5%), jise Q=10 bhi kehte hain, typical lightly-damped aerospace structures represent karta hai.

Duration

τ

ke half-sine shock pulse ke liye SRS knee roughly kahan hoti hai?

Near

f_{n} \approx 1/ (π τ)

; knee ke neeche SRS

f_{n}^{2}

ke saath rise karta hai (+40 dB/decade), knee ke upar peak acceleration ki taraf flatten hota hai.

Ek short pulse ke liye SRS ka low-frequency asymptotic behavior kya hai?

Yeh

f_{n}^{2}

ke saath rise karta hai (approximately

+ 40

dB/decade, ya

+ 12

dB/octave),

SRS \approx A_{0} π^{2} f_{n}^{2} τ^{2} /2

se describe hota hai jab

f_{n} τ ≪ 1

Knee ke baad simple-pulse SRS ka high-frequency behavior kya hota hai?

Yeh flatten hota hai (≈ 0 dB/octave) peak base acceleration ki taraf, kyunki bahut fast oscillators simply base peak ko track karte hain.

Shock pulse ke SRS aur FFT mein kya difference hai?

FFT input signal ki frequency content (energy distribution) dikhata hai; SRS har frequency par resonant systems ke liye maximum response amplification dikhata hai.

SRS specifications mein damping ratio hamesha kyun include karna chahiye?

SRS magnitude damping par strongly depend karta hai—5% damping 10% damping se 2-3× higher peaks deta hai—isliye comparisons ke liye matched damping assumptions zaroori hain.

SRS actual component natural frequencies se kaise relate karta hai?

Har component ek SDOF system ki tarah resonate karta hai; component ki natural frequency par SRS value uska peak acceleration predict karta hai, design aur qualification guide karta hai.