Propulsive forces — thrust misalignment, gimbal angle

3.4.12 · Physics › Rocket Flight Mechanics

KYA ho raha hai

KYUN thrust direction matter karta hai — torque derive karna

Setup (scratch se derivation)

CoM ko origin par rakhte hain. Engine gimbal pivot, CoM se $L$ door peeche rocket axis par baitha hai ($x$-axis, nose = $+x$). Thrust magnitude $T$ hai, rocket axis se $\delta$ angle par jhuka hua.

Thrust force components (body frame): $\vec{T}=(\underset{\text{along axis}}{\underbrace{-T\cos \delta }},\underset{\text{lateral}}{\underbrace{-T\sin \delta }})$

Minus signs kyun? Nozzle peeche ki taraf point karta hai ($-x$ ki taraf), isliye thrust rocket ko $+x$ ki taraf push karta hai — ruko, dhyaan se: rocket par force aage ki taraf point karti hai. Agar nozzle exit $-x$ face kar raha hai, toh rocket par thrust $+x$ hai. Force ko seedha rocket par likhte hain: ${\vec{T}}_{rocket}=(T\cos \delta ,T\sin \delta ).$

CoM se pivot tak position vector (jahan force apply ho rahi hai): $\vec{r}=(-L,0).$

Kyun? Engine CoM ke peeche hai, isliye iska $x$-coordinate $-L$ hai.

Cross product (2D torque = scalar)

2D mein, $\tau =r_x{F}_y-r_y{F}_x$: $\tau =(-L)(T\sin \delta )-(0)(T\cos \delta )=-LT\sin \delta .$

Small-angle form kyun? Gimbals sirf kuch degrees move karte hain; $\sin \delta \approx \delta$ 10° par <0.5% accurate hai, jisse control math linear ban jaati hai.

Useful axial force loss

Engine ko tilt karne se kuch thrust "waste" ho jaata hai jo aage push nahi karta: $T_{axial}=T\cos \delta ,\text{loss}=T(1-\cos \delta )\approx \frac{1}{2}T{\delta }^2.$

$\frac{1}{2}T{\delta }^2$ kyun? Taylor se: $\cos \delta \approx 1-{\delta }^2/2$. Loss second order hai — chhota — isliye gimballing performance mein "sasti" hai lekin torque mein "strong" (torque $\delta$ mein first order hai).

KAISE use karein — angular acceleration & steering

Newton ka rotational law: $\tau =I\ddot{\theta }$, jahan $I$ pitch moment of inertia hai. $\ddot{\theta }=\frac{\tau }{I}=-\frac{LT\delta }{I}.$

Toh ek constant gimbal angle constant angular acceleration deta hai → rocket pitch over karta hai. Guidance computers $\delta (t)$ ko continuously command karte hain desired attitude maintain karne ke liye (yahi TVC — Thrust Vector Control — ka basis hai).

Worked examples

Common mistakes

Flashcards

Connections

• Newton's Third Law — thrust exhaust ki reaction hai.
• Torque and Moment of Inertia — $\tau =I\ddot{\theta }$ rotational dynamics.
• Center of Mass shift during burn — flight ke dauran $L$ change hota hai.
• Thrust Vector Control (TVC) — engineering system jo $\delta (t)$ command karta hai.
• Rocket Equation — mass flow aur $v_e$ se $T$ ka magnitude.
• Attitude Control & Stability — rocket ko sahi direction mein rakhne ke liye $\tau$ ka use.

Concept Map