Question bank — Thrust vector control — single-gimbal, dual-gimbal; TVC angles

False. Engine wahi magnitude $T$ banata hai; TVC sirf us vector ko rotate karta hai. Tilting se $T$ do parts mein split hota hai — forward part $T\cos \delta$ aur sideways part $T\sin \delta$ — kuch bhi add nahi hota.

True. $M=\ell T\sin 0=0$. Ek thrust jo seedha CoM line se guzar raha ho uska koi sideways lever arm nahi hota, isliye woh vehicle ko twist nahi kar sakta.

True. $T\cos 0=T$, jo $\cos$ ki sabse badi value hai. Koi bhi tilt forward push ko sirf ghata sakti hai, isliye straight-back direction sabse zyada fuel-efficient hai.

True. $M=\ell T\sin \delta$ mein $\ell$ linear hai. CoM se pivot tak ka lamba lever matlab hai ki wahi sideways force zyada twist karta hai — yeh Torque and Moment Arm ka kaam hai.

False. Yeh sirf tab hold karta hai jab $\sin \delta \approx \delta$ ho (roughly ~$8^{\circ }$ se neeche ~1% accuracy ke saath). $45^{\circ }$ par, $\sin \delta =0.707$ lekin $\delta =0.785$ rad hai — yeh 11% error hai. Exact $\sin$ use karo.

False. Yeh pitch aur yaw handle karta hai, lekin thrust line roll axis ke paas se guzarti hai, isliye roll ke baare mein moment arm ~zero hai → koi roll torque nahi. Roll ke liye canted engines, differential gimbaling, ya Reaction Control System (RCS) chahiye.

False. ${\delta }_p$ aur ${\delta }_y$ ek physical tilt ke perpendicular components hain, isliye geometric tilt ${\delta }_{\text{tot}}=\sqrt{{\delta }_p^2+{\delta }_y^2}$ hai. Woh magnitude — sum nahi — mechanical stop se takrati hai.

False. Loss $=1-\cos \delta \approx {\delta }^2/2$ — yeh quadratic hai, $\delta$ mein second order. Isliye hi small-angle steering almost free hai jabki torque linearly badhta hai.

True. Chhote $\delta$ par, aap first-order twist ($\propto \delta$) khareedte ho jabki sirf second-order fuel cost ($\propto {\delta }^2$) dete ho. Ratio (benefit/cost) $\delta \to 0$ jaane par blow up ho jaata hai.

False. Tab $\ell =0$, isliye $M=\ell T\sin \delta =0$ chahe $\delta$ kuch bhi ho. Koi lever arm nahi hone par, sideways force push karta hai lekin twist nahi kar sakta — dekhein Center of Mass Migration kyun $\ell$ flight mein badalta hai.

Dono $T$ aur $\delta$ torque badhate hain ($M=\ell T\sin \delta$), lekin $T$ Rocket Thrust Equation aur mission profile se set hota hai, steering ke liye freely dial-able nahi hota. Controllable knob $\delta$ hai; wahi Attitude Control Autopilot move karta hai.

Minus sirf cross product $\vec{r}\times \vec{F}$ se aaya ek sign convention hai; yeh rotation ki direction encode karta hai, koi weakening nahi. Magnitude $\ell T\sin \delta$ hi real strength hai.

Combined tilt ${\delta }_{\text{tot}}=\sqrt{5^2+5^2}=7.07^{\circ }>6^{\circ }$ hai, isliye yeh physical stop violate karta hai aur clip karna padega (dono ko $6/7.07$ se scale karo). Individual angles ka legal hona kaafi nahi hai.

Torque angular acceleration deta hai $M=I\dot{\omega }$ ke through (rotational Newton's law), angle directly nahi. Aapko Rigid Body Rotational Dynamics chahiye: $\dot{\omega }$ ko do baar integrate karo, aur $I$ (moment of inertia) bhi matter karta hai.

$\sin \delta$ sideways (steering) fraction hai. Forward thrust ki loss $1-\cos \delta$ hai, jo chhote angles par bahut kam hoti hai (${\delta }^2/2$ vs $\delta$).

RCS sirf control ke liye extra propellant jalaata hai aur mass add karta hai; TVC main engine ka thrust reuse karta hai, max angle par sirf ~$1\%$ lose karta hai. TVC usually kaafi sasta hai — RCS tab shine karta hai jab main engine off ho ya fine/roll authority chahiye.

Kyunki steering ke liye ek sideways force chahiye torque banane ke liye, aur ek fixed-length vector ko rotate karne se exactly woh transverse component $T\sin \delta$ milta hai jabki magnitude fixed rehti hai.

Transverse component thrust components ke right triangle mein tilt angle ke opposite hai, aur opposite/hypotenuse exactly $\sin \delta$ hai. $\cos \delta$ adjacent (forward) part deta hai.

Bahut chhote $\delta$ ke liye sine curve almost apni khud ki tangent line origin se hoti hai, isliye $\sin \delta \approx \delta$ (Small-Angle Approximation). Ek linear law $M\approx \ell T\delta$ controller ke liye invert aur stabilize karna kaafi aasaan hai.

Forward thrust $T\cos \delta$ mein $\ell$ bilkul nahi hai. Torque $=\text{force}\times \text{lever arm}$ hai, isliye sirf twisting effect $\ell$ ke saath scale karta hai; push nahi karta.

Physical gimbal sirf itna hi center se kisi bhi direction mein swing kar sakta hai; woh maximum reach ek radius hai, aur actual tilt vector sum $\sqrt{{\delta }_p^2+{\delta }_y^2}$ hai, jo woh radius hit karta hai.

Ek dual-gimbal engine ek hi actuator set se dono planes mein torque deta hai — do alag single-axis engines mount karne aur coordinate karne se kaafi lighter aur simpler hai.

$M=\ell T\sin 90^{\circ }=\ell T$ — maximum possible torque, lekin forward thrust $T\cos 90^{\circ }=0$, isliye sab twist milega aur zero push. Real gimbals kabhi yahan nahi pahunchte; yeh ek limiting sanity check hai.

Yeh diverge karta hai: torque $\sim \ell T\delta$ over loss $\sim T{\delta }^2/2$ $2\ell /\delta \to \infty$ ki tarah scale karta hai. Chhote angles maximally efficient hain per unit torque — yeh "steering cheap hai" ka mathematical heart hai.

Woh khatam ho jaati hai: $M=\ell T\sin \delta =0$, isliye engine vehicle ko bilkul twist nahi kar sakta. Propellant drain hone se Center of Mass Migration ek real reason hai kyun $\ell$ (aur isliye authority) ek burn ke dauran badlata hai.

Nahi. $M=\ell \cdot 0\cdot \sin \delta =0$ — koi thrust nahi toh redirect karne ke liye koi force nahi, isliye koi torque nahi. Coasting phases Reaction Control System (RCS) par rely karti hain.

Pehla exactly edge par hai (${\delta }_{\text{tot}}=6^{\circ }$, feasible). Doosra ${\delta }_{\text{tot}}=\sqrt{6^2+0.1^2}\approx 6.0008^{\circ }>6^{\circ }$ deta hai, isliye clip karna padega — doosre axis ka ek whisper bhi stop se aage push karta hai.

$\delta =0$ ke saath nahi — us se zero torque milta hai, isliye existing spin continue karta hai (Newton: koi torque nahi, $\omega$ mein koi change nahi). Rotation rokne ke liye autopilot ko ek counter-deflection command karni padegi.