Structural loads — axial (thrust), bending (wind shear), dynamic (vibration, acoustics, shock)

3.6.1 · Physics › Spacecraft Structures & Systems Engineering

1. Axial load — thrust

Axial force ko ek station par derive kaise karein. Socho rocket ko height $x$ par cut kar rahe ho. Cut ke upar ka sab kuch mass $m (x)$ ka hai aur use upar $a$ acceleration se accelerate kiya ja raha hai. Us upper chunk ka free-body:

$F_{axial} (x) - m (x) g = m (x) a$

Internal force ke liye solve karo jo structure ko $x$ par transmit karni padegi:

$F_{axial} (x) = m (x) (g + a)$

2. Bending load — wind shear ("max-Q" / gust)

Bending stress kaise aati hai. Vehicle par ek transverse distributed load $w$ (N/m) ek bending moment $M (x)$ deta hai. Flexure formula (bending ki geometry se derive, neeche) hai:

$σ_{bend} = \frac{M c}{I}$

$σ = M c / I$ ko first principles se derive karna. Jab ek beam bend karti hai, toh neutral axis se $y$ distance par ek fiber strain $ε = y / R$ (radius of curvature $R$ ) se stretch hoti hai. Hooke: $σ = E ε = E y / R$ . Internal moment area par stress×lever-arm ka sum hai:

$M = \int y σ d A = \frac{E}{R} \int y^{2} d A = \frac{E}{R} I, I \equiv \int y^{2} d A .$

Isliye $E / R = M / I$ , aur $σ = E y / R$ mein substitute karne par $σ = M y / I$ milta hai; maximum outer fiber $y = c$ par hai:

$σ_{bend} = \frac{M c}{I}$

Figure — Structural loads — axial (thrust), bending (wind shear), dynamic (vibration, acoustics, shock)

3. Dynamic loads — vibration, acoustics, shock

Hum ek structure ko spring–mass ki tarah model kaise karte hain. Single degree of freedom:

$m \overset{x}{¨} + c \overset{x}{˙} + k x = F (t), ω_{n} = k / m (rad/s), f_{n} = \frac{1}{2 π} \frac{k}{m}$

$ω_{n} = k / m$ kyun? $F = 0$ , $c = 0$ set karo: $m \overset{x}{¨} = - k x$ SHM hai jiska angular frequency $k / m$ hai $\overset{x}{¨} = - ω^{2} x$ match karne se.

Common mistakes (Steel-man → Fix)

Feynman

Recall Ek 12-saal ke bachche ko explain karo

Ek soda can imagine karo. Upar se apna angutha dabao = thrust use squeeze kar raha hai (axial). Use sideways bend karo = hawa lambe rocket par push kar rahi hai (bending). Ab use tez flick karo taaki woh hum kare — agar aap bilkul sahi speed par flick karo toh woh pagalon ki tarah rattle karta hai: yahi vibration/resonance hai. Ek chamach se achanak whack = shock. Rocket engineers can ko thumb + bend ke liye ek saath itna strong banate hain, aur woh ensure karte hain ki woh ek speed par hum kare jo rocket kabhi reach nahi karta taaki woh rattle karke alag na ho jaaye.

Active recall

Axial load factor

n

a

aur

g

ko kaise relate karta hai?

g + a = n g

, isliye

n = 1 + a / g

(dimensionless "g's").

Station

x

par axial force ka formula?

F_{axial} (x) = m (x) (g + a) = m (x) n g

Axial compression base ki taraf kyun badhti hai?

Lower stations ke upar zyada mass hota hai, isliye zyada inertial force transmit karni padti hai.

Flexure formula aur uske terms?

σ = M c / I

;

M

=bending moment,

c

=outer fiber tak distance,

I = \int y^{2} d A

σ = M c / I

derive karna shuru karo?

ε = y / R \Rightarrow σ = E y / R

;

M = \int y σ d A = E I / R

;

R

eliminate karo:

σ = M y / I

Thin cylindrical shell ke liye

I

I = π R^{3} t

Max-Q par worst-case combined stress?

σ_{total} = F / A + M c / I

(woh compressive fiber par add hote hain).

SDOF ki natural frequency?

f_{n} = \frac{1}{2 π} k / m

Resonant magnification factor

Q

Q = 1/ (2 ζ)

; typical spacecraft

Q \approx 10

–

25

Teen dynamic load types?

Vibration, Acoustics, Shock.

Large lightweight panels par kaun sa dynamic load dominate karta hai?

Acoustics (pressure waves).

Kaun sa load type mainly electronics ko damage karta hai na ki structure ko?

Shock (pyro separation transients).

Miles' equation RMS g response ke liye?

g_{RMS} = \frac{π}{2} f_{n} Q W

;

3 σ

tak design karo.

"Bas use stiffer banao" vibration ke liye hamesha safer kyun nahi hai?

Stiffness

f_{n}

ko upar uthati hai; tabhi helpful hai jab yeh aapko excitation band se door le jaaye, kisi PSD peak mein nahi.

Connections

Rocket Equation & Thrust — axial load ka source.
Max-Q and Dynamic Pressure — jab bending peak karti hai.
Beam Bending & Second Moment of Area — $σ = M c / I$ ki origin.
Modal Analysis & Natural Frequencies — $f_{n}$ compute karna.
Random Vibration & PSD — jahan Miles' equation rehti hai.
Factor of Safety & Margins of Safety — stress ko pass/fail mein badalna.
Pyrotechnic Separation Systems — shock ka source.