Visual walkthrough — Propulsive forces — thrust misalignment, gimbal angle

3.4.12 · D2 · Physics › Rocket Flight Mechanics › Propulsive forces — thrust misalignment, gimbal angle

Hum kuch bhi assume nahi karte. Agar tumne kabhi nahi dekha ek arrow jiska matlab "force" ho, lever arm, sine, ya cross product — achha hai. Hum har cheez us waqt define karte hain jab woh chahiye hoti hai, usse pehle nahi.

Step 1 — "Thrust" aakhir hai kya (ek arrow)

KYA set up kar rahe hain: ek rocket apni side pe leta hua hai, nose right ki taraf point kar raha hai. KYU: humein ek fixed picture chahiye jis par baad ke saare ideas rakh sakein. Sab kuch — angles, levers, torques — is rocket ki apni body ke saath measure kiye jaayenge. PICTURE: thrust arrow engine se shuru hota hai (peeche, left) aur aage (right) ki taraf point karta hai. Uska size $T$ bas arrow ki length hai.

Hum rocket ke through do reference lines draw karte hain:

• roll axis (lambī centre line, nose-to-tail) — ise $x$-direction kehte hain,
• lateral direction (seedha cross karke) — ise $y$-direction kehte hain.

$\vec{T}=(\underset{\text{forward part}}{\underbrace{T_x}},\underset{\text{sideways part}}{\underbrace{T_y}})$

Yahan $T_x$ hai "push ka kitna hissa aage jaata hai" aur $T_y$ hai "kitna sideways leak hota hai". Jab engine bilkul seedha point karta hai, $T_y=0$ aur poora $T$ aage jaata hai.

Step 2 — Centre of mass, woh ek point jo matter karta hai

KYA: hum CoM ko place karte hain aur use apna origin mark karte hain — point $(0,0)$. KYU: ek force jo exactly CoM ke through push karti hai woh rocket ko sirf slide kara sakti hai, kabhi spin nahi. Ek force jo CoM ko miss kare woh spin kara sakti hai. Toh CoM woh pivot hai jiske around har twist measure hoti hai. Ise origin par rakhne se arithmetic clean rehti hai. PICTURE: CoM ek dot hai near front-middle mein. Engine pivot uske peeche baitha hai, axis ke neeche $L$ distance par.

$\vec{r}=(\underset{\text{engine CoM ke peeche hai}}{\underbrace{-L}},\underset{\text{axis par}}{\underbrace{0}})$

• $\vec{r}$ = arrow CoM se wahan tak jahan force apply ho rahi hai (pivot).
• $-L$ = pivot ka forward-coordinate. Yeh negative hai kyunki "peeche" $-x$ side hai.
• $0$ = pivot centre line par hai, koi sideways offset nahi.

Yeh length $L$ lever arm hai: pivot-point se push-point tak ki reach. Ise yaad rakho — yeh woh "lamba handle" hai jo chhoti pushes ko badi turns mein badalta hai.

Step 3 — Engine tilto: thrust arrow ko todna

KYA: thrust arrow ko $\delta$ se rotate karo aur use uske forward aur sideways parts mein todo. KYU: torque sirf sideways part $T_y$ ki parwaah karta hai — woh woh piece hai jo off-centre push karta hai. $T_y$ find karne ke liye hum tilted arrow ko components mein resolve karte hain. Yahi sine aur cosine ka kaam hai.

$\vec{T}=(\underset{\text{forward: bachta hai}}{\underbrace{T\cos \delta }},\underset{\text{sideways: turning karta hai}}{\underbrace{T\sin \delta }})$

• $T\cos \delta$ = forward thrust. $\delta =0$ par, $\cos 0=1$, toh yeh poore $T$ ke barabar hai — kuch nahi khota.
• $T\sin \delta$ = sideways thrust. $\delta =0$ par, $\sin 0=0$, toh koi side-push nahi — rocket seedha udta hai. Ise tilto, aur yeh term barhti hai.

Step 4 — "Torque" kya hai: twisting number

KYA: lever arm $\vec{r}$ (Step 2) ko force $\vec{T}$ (Step 3) ke saath combine karke ek twisting number banao. KYU: hum ek aisi single quantity chahte hain jo kahe "yeh rocket ko kitni zor se, aur kis direction mein spin karta hai?" Flat 2D mein, woh number cross product se aata hai, jo ek plane mein collapse ho jaata hai:

$\tau =r_x{T}_y-r_y{T}_x$

• $r_x{T}_y$ = (forward reach) × (sideways push) — yeh real twister hai.
• $r_y{T}_x$ = (sideways reach) × (forward push) — yahan zero hai, kyunki $r_y=0$.

PICTURE: sideways thrust lambe lever ke end par tail ko ek taraf swing karta hai, toh nose doosri taraf swing karta hai — rocket CoM dot ke around pivot karta hai.

$\vec{r}=(-L,0)$ aur $\vec{T}=(T\cos \delta ,T\sin \delta )$ plug karo:

$\tau =(-L)(T\sin \delta )-(0)(T\cos \delta )$

Doosra term khatam ho jaata hai. Jo bachta hai:

Forward part $T\cos \delta$ ne twist mein kuch contribute nahi kiya — yeh lever ke seedha along point karta hai, toh yeh sirf rocket ko aage dhakelta hai. Sirf sideways part use turn karta hai. Yeh ek fact poori derivation hai.

Step 5 — Turn "sasta" kyun hai: small-angle picture

KYA: tilt barhne par jo turn hum gain karte hain use forward-push jo hum lose karte hain se compare karo. KYU: yeh comparison TVC ka poora justification hai. Agar losses turn ki tarah hi fast badhte, toh gimballing ek bura deal hota. Woh nahi badhte — losses ek squared term hain, jo chhoti rehti hai jabki linear turn shoot karta hai. PICTURE: origin se do curves. Torque curve $\sin \delta$ seedhi line ki tarah uthti hai; loss curve $1-\cos \delta$ floor se chipki rehti hai pehle.

$\underset{\text{fast badhta hai (linear)}}{\underbrace{\tau \approx LT\delta }}\underset{\text{slowly badhta hai (squared)}}{\underbrace{\text{loss}=T(1-\cos \delta )\approx \frac{1}{2}T{\delta }^2}}$

$\delta =5^{\circ }$ par: tumhe $\sin 5^{\circ }=0.0872$ full sideways authority milti hai, lekin sirf $1-\cos 5^{\circ }=0.0038$ khote ho — yeh thrust ka 0.38% hai. Bada turn, near-zero cost.

Step 6 — Har case: kis direction mein spin hoga, aur degenerate ones

KYA: sign (turn direction) pakko aur boundary cases check karo taaki koi scenario surprise na kare. KYU: ek control law jo galat direction mein turn kare, ya $\delta =0$ par blow up kare, useless se bhi bura hai. Humein dikhana hai ki formula har jagah sahi behave karta hai.

• $\delta >0$ (nozzle $+y$ ki taraf swing): $\sin \delta >0$, toh $\tau =-LT\sin \delta <0$. Tail $+y$ ki taraf shove hota hai, toh nose $-y$ ki taraf swing karta hai — ek negative (clockwise) rotation. Minus sign bas yahi bookkeeping hai.
• $\delta <0$ (nozzle doosri taraf swing): $\sin \delta <0$, toh $\tau >0$. Nose doosri taraf swing karta hai. Controller dono directions mein steer karta hai bas $\delta$ ka sign choose karke.
• $\delta =0$ (bilkul seedha): $\sin 0=0\Rightarrow \tau =0$. Pure forward flight, koi spin nahi. Formula gracefully degenerate hota hai.
• $\delta =90^{\circ }$ (ek bakwaas extreme): $\sin 90^{\circ }=1$, sab thrust sideways, $\cos 90^{\circ }=0$, zero forward push. Maximum twist, koi acceleration nahi — isliye real gimbals hard-limited hote hain kuch degrees tak.
• $L=0$ (engine CoM par hi): $\tau =0$ chahe kitna bhi tilt karo. Koi lever nahi, koi twist nahi — tum bas sideways slide karte. Yeh woh Attitude Control & Stability reason hai kyun engines CoM ke kaafi peeche baithe hain.
• Pure offset, koi tilt nahi ($\delta =0$ lekin pivot sideways $d$ shift): ab $\vec{r}$ mein ek $y$-part hai, aur forward thrust ko ek lever milta hai. Torque ban jaata hai $\tau =dT$ — alag geometry, wahi principle: CoM ko miss karne wali force body ko twist karti hai.

Step 7 — Torque se motion tak: rocket actually pitch karta hai

KYA: twist $\tau$ ko actual pitch-rate change $\ddot{\theta }$ mein convert karo. KYU: torque tabhi useful hai jab woh nose ko move kare. Yeh last step "engine tilt" se "rocket kisi naye direction mein point karta hai" tak ka loop close karta hai.

$\ddot{\theta }=\frac{\tau }{I}=-\frac{LT\delta }{I}$

• $\ddot{\theta }$ = pitch-rate kitni jaldi build ho rahi hai (angular acceleration).
• $I$ = moment of inertia — bada $I$ (lamba bhaari rocket) sluggishly turn karta hai.
• ek held gimbal ⇒ constant $\ddot{\theta }$ ⇒ nose tezi se aur tezi se pitch over karta rehta hai.

Ek-picture summary

Sab kuch ek canvas par: rocket, CoM dot, lever arm $L$, tilted thrust jo ek surviving forward part $T\cos \delta$ aur ek turning sideways part $T\sin \delta$ mein split hai, aur resulting spin CoM ke around.

$\tau =-LT\sin \delta \approx -LT\delta ⟹\ddot{\theta }=\frac{\tau }{I}$

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