3.5.44
Ek rocket flight mein kam speed par almost koi aerodynamic control nahi hota (fins ko push karne ke liye hawa nahi hoti). Toh aap use steer kaise karte ho? Aap engine ko tilt karte ho torque create karta hai jo vehicle ko rotate karta hai. Push ko hi steer karna Thrust Vector Control (TVC) kehlata hai.
KYUN kaam karta hai: torque = × KYA hai yeh "vector": thrust ek vector hai; TVC iska direction change karta hai, (zyaadatar) magnitude nahi.KAISE quantify karte hain: TVC deflection angles se (single-gimbal = 1 angle; dual-gimbal = full pitch+yaw authority ke liye 2 angles).
Axes ko rocket se attach karo: x b roll axis (nose direction), y b z b − x b gimbal point ke baare mein pivot karta hai jo center of mass (CoM) se ℓ peeche hota hai.
Gimbal point ek mechanical pivot hai. Moment arm ℓ x b
Hum woh moment derive karte hain jo thrust ko tilt karne se produce hota hai. Pehle 2D mein karo (single-gimbal), phir generalize karo.
Intuition Formula ko samjhoM = ℓ T δ engine thrust T moment arm ℓ deflection δ ℓ T chota sa piece, T sin δ
Yahi key trade hai: torque δ δ — chote angles par ek zabardast deal.
Engine ek axis ke baare mein pivot karta hai → ek deflection angle δ Ek plane mein torque produce karta hai (e.g. pitch ya yaw). Sasta, halka, lekin full 3-axis control ke liye ek partner chahiye (doosra engine, vernier, ya roll control).
Definition Dual-gimbal (do axes mein gimbaled)Engine do orthogonal axes ke baare mein pivot karta hai → do angles: pitch deflection δ p δ y dono planes mein torque deta hai.
Note: ek single centered engine roll torque nahi bana sakta (thrust line roll axis ke paas se guzarti hai). Roll ke liye canted thrusters, multiple engines ka differential gimbaling, fins, ya RCS chahiye.
Worked example Example 1 — commanded angular acceleration ke liye required gimbal angleEk rocket mein T = 800 kN ℓ = 6 m I = 1.2 × 1 0 5 kg⋅m 2 ω ˙ = 0.5 rad/s 2 δ
Step 1: required torque M = I ω ˙ = 1.2 × 1 0 5 × 0.5 = 6.0 × 1 0 4 Kyun? Newton ka rotational law M = I ω ˙ Step 2: M = ℓ T δ δ = ℓ T M = 6 × 8 × 1 0 5 6.0 × 1 0 4 = 0.0125 Kyun? Small-angle law hume linearly solve karne deta hai.Step 3: convert karo: δ = 0.0125 × π 180 ≈ 0.7 2 ∘ Yeh kyun matter karta hai: confirm karta hai ki sirf chote gimbal angles chahiye → halke actuators, chota thrust loss.
Worked example Example 2 — max deflection par thrust lossMax gimbal δ max = 8 ∘ T = 800
Step 1: loss = 1 − cos 8 ∘ = 1 − 0.990 = 0.0097 = 0.97% Kyun? Axial thrust T cos δ Step 2: side force = T sin 8 ∘ = 800 × 0.139 = 111 Kyun? Transverse component steer karta hai.Insight: ~1% loss par 111 kN steering force milti hai — ek zabardast trade.
Worked example Example 3 — dual-gimbal combined commandAutopilot δ p = 3 ∘ δ y = 4 ∘ 6 ∘
Step 1: δ tot = 3 2 + 4 2 = 5 ∘ Kyun? Orthogonal deflections ka vector sum.Step 2: 5 ∘ < 6 ∘ Agar bajaaye δ p = 5 ∘ , δ y = 5 ∘ δ tot = 7.0 7 ∘ > 6 ∘ clip karna hoga (dono ko 6/7.07
Common mistake "Zyaada forward thrust ke liyeδ Kyun sahi lagta hai: zyaada gimbal = zyaada "engine action," toh surely zyaada push. Reality: gimbaling axial thrust kam karta hai (T cos δ δ = 0 Fix: yaad rakho thrust ek fixed-magnitude vector hai jise aap rotate kar rahe ho, amplify nahi.
Common mistake "Total limit ke liye bas donon gimbal angles add karo:δ p + δ y Kyun sahi lagta hai: limits usually add hoti hain. Reality: yeh ek tilt ke perpendicular components hain → yeh δ p 2 + δ y 2 Fix: deflection ko ek 2D vector samjho; magnitude mechanical stop hit karta hai.
Common mistake "Ek single centered gimbaled engine roll bhi control kar sakta hai."Kyun sahi lagta hai: yeh pitch aur yaw control karta hai, toh roll kyun nahi? Reality: thrust line roll axis se hokar/paas se guzarti hai, isliye roll ke baare mein moment arm ~zero hai → koi roll torque nahi. Fix: roll ke liye canted engines, differential gimbaling, ya RCS chahiye.
Common mistake "Kisi bhi angle ke liyeM = ℓ T δ Kyun sahi lagta hai: yahi boxed formula hai. Reality: yeh small-angle form hai; exact law M = ℓ T sin δ Fix: ~10° se zyaada par sin δ
Recall Quick self-test (answers chhupa lo)
TVC kaunsi physical quantity change karta hai? → thrust vector ki direction .
Exact torque formula? → M = ℓ T sin δ
Steering "sasti" kyun hai? → torque O ( δ ) O ( δ 2 )
Single vs dual gimbal mein kya difference hai? → 1 angle/1 plane vs 2 angles (pitch+yaw)/2 planes.
Ek centered engine roll control kyun nahi kar sakta? → roll axis ke baare mein zero moment arm.
Combined gimbal limit rule? → δ p 2 + δ y 2 ≤ δ max
Recall Feynman: 12-saal ke bache ko samjhaoSocho tum peeche se ek stick se shopping cart push kar rahe ho. Agar aap seedha beech se push karo, toh yeh seedha jaati hai. Agar aap thoda side se push karo, toh cart mud jaati hai. Ek rocket bhi yehi karta hai: uski flame uski "push" hai. Engine ko thoda tilt karke, push thodi sideways point karti hai aur rocket ko wahan point karne ke liye spin karta hai jahan hum chahte hain. Sirf kuch degrees tilting kaafi hai kyunki engine bahut powerful hai aur rocket ke balance point se kaafi peeche baitha hai — isliye ek chota sa tilt bada turn banata hai jabki rocket almost slow nahi hota.
"TILT karo TURN ke liye, seedha raho STAY ke liye."
Aur trade ke liye: "Torque is Tiny-angle-Linear, Loss is Little-and-squared." (M ∝ δ ∝ δ 2
Thrust Vector Control thrust ke baare mein physically kya change karta hai? Iska direction (thrust vector tilt hota hai), primarily iska magnitude nahi.
Exact single-gimbal control torque formula? M = ℓ T sin δ ℓ T δ
Control torque ka small-angle form? M ≈ ℓ T δ sin δ ≈ δ
TVC steering thrust mein sasti kyun hai? Torque linearly badhta hai (∝ δ 1 − cos δ ≈ δ 2 /2
δ T cos δ
Single-gimbal vs dual-gimbal? Single = 1 pivot axis, 1 angle, ek plane mein torque; dual = 2 orthogonal axes, angles δ p , δ y
Dual-gimbal deflections hard limit ke against kaise combine hote hain? Vector magnitude ke roop mein
δ p 2 + δ y 2 ≤ δ max .
Commanded angular acceleration ω ˙ δ = ℓ T I ω ˙ M = I ω ˙ = ℓ T δ
Ek single centered gimbaled engine roll control kyun nahi kar sakta? Iska thrust line roll axis ke along/paas hoti hai → moment arm ≈ 0 → koi roll torque nahi; roll ke liye canted engines/differential gimbal/RCS chahiye.
Forward thrust kis deflection par maximum hota hai? δ = 0 T cos δ
No aero control at low speed
Thrust Vector Control TVC
Engine about gimbal point
Vector, changes direction
Axial T cos delta + transverse T sin delta
Control torque M = ell T sin delta
