Pitch program — open-loop pitch-over — Hinglish notes

3.4.14 · Physics › Rocket Flight Mechanics

WHAT is a pitch program?

WHY do it this way? (first principles)

WHY pitch over at all? Orbit ke liye bahut badi horizontal speed chahiye. Agar rocket vertical rehta, toh saara thrust gravity se ladhta aur woh bas upar jaata aur wapas girta. Usse thrust ko sideways redirect karna padta hai.

WHY start vertical? Do reasons:

Pad ke paas hawa sabse dense hoti hai — aap chahte ho ki aap iske through fast aur body axis ke saath punch karo taaki aerodynamic loads kam rahen.
Gravity loss sirf thodi der ke liye minimize hoti hai; aap time waste nahi kar sakte — lekin aap bina dangerous angle of attack generate kiye jaldi turn bhi nahi kar sakte.

WHY open-loop in the atmosphere? Sabse bada dushman hai angle of attack $α$ (velocity vector aur body axis ke beech ka angle). Aerodynamic side force $\propto α \cdot q$ , jahan $q$ dynamic pressure hai. High $q$ par bada $α$ vehicle ko tod sakta hai. Toh goal yeh hai ki poore dense atmosphere mein $α \approx 0$ rakha jaaye. Isko guarantee karne ka elegant trick hai gravity turn.

HOW: deriving the flight-path equations

Rocket ko vertical plane mein ek point mass maano. Maano:

$v$ = speed, $γ$ = flight-path angle (horizontal ke upar velocity ka angle),
$T$ = thrust (body axis ke saath), $m$ = mass, $g$ = gravity, $D$ = drag.

Pure gravity turn mein body axis velocity ke saath aligned hoti hai ( $α = 0$ ), toh thrust $v$ ke saath act karta hai.

Velocity ke saath (tangential): $m \overset{v}{˙} = T - D - m g sin γ$

Kyun? Thrust aur drag flight path ke saath hain; gravity ka component path ke saath $m g sin γ$ hai (climbing ko oppose karta hai).

Velocity ke perpendicular (normal): $v$ ko turn karne wala sirf gravity ka normal component $m g cos γ$ hai (thrust aur drag on-axis hain, koi side force nahi): $m v \overset{γ}{˙} = - m g cos γ$

Minus kyun? Gravity neeche point karti hai, jo $γ$ ko reduce karti hai (path ko horizontal ki taraf mod deti hai). $m$ cancel karo: $\overset{γ}{˙} = - \frac{g cos γ}{v}$

HOW initial kick poori trajectory set karta hai. Time eliminate karne ke liye tangential ko normal se divide karo:

= \frac{(T-D)/m - g\sin\gamma}{-g\cos\gamma/v}. $$ Drag neglect karo aur $T/m = a$ lo (roughly constant, $n = a/g$): $$ \frac{dv}{d\gamma} = \frac{v\,(n - \sin\gamma)}{-\cos\gamma}. $$ Yeh dikhata hai: **kick angle** (initial $\gamma_0$ just below $90^\circ$) aur thrust-to-weight $n$ choose karo, aur **poora** $v(\gamma)$ curve determined ho jaata hai. Yahi open-loop program ka essence hai: **$\gamma_0$ (aur $n$) choose karo, aur ascent shape physics se follow karta hai.** > [!intuition] Why the kick is "just a nudge" > Near vertical, $\cos\gamma\approx 0$, toh $\dot\gamma$ tiny hai — rocket barely turn karta hai. > Lekin woh tiny turn ek growing $\cos\gamma$ seed karta hai, jo turning ko speed up karta hai, jo > $\cos\gamma$ grow karta hai... ek slow self-amplifying bend. Bahut chhota kick → burnout par bhi upar ja raha hai > (horizontal kabhi nahi pahunchta). Bahut bada → bahut jaldi turn ho jaata hai, dives, high drag/heating. ![[3.4.14-Pitch-program-—-open-loop-pitch-over.png]] --- ## Worked examples > [!example] 1) Instantaneous pitch rate at kick-over > Ek rocket $v = 120\ \text{m/s}$, $\gamma = 89^\circ$, $g = 9.8\ \text{m/s}^2$ par hai. $\dot\gamma$ find karo. > > **Step:** $\dot\gamma = -\dfrac{g\cos\gamma}{v} = -\dfrac{9.8\cos 89^\circ}{120}$. > *Yeh step kyun?* Gravity-turn law directly state se turn rate deta hai. > $\cos 89^\circ = 0.01745$, toh $\dot\gamma = -\dfrac{9.8(0.01745)}{120} \approx -0.00143\ \text{rad/s}$ > $\approx -0.082^\circ/\text{s}$. > **Padho:** high speed aur near-vertical par, turn barely $0.08°$ per second hai — koi bhi turn hone ke liye tumhe kick shuru karni HI PADEGI. > [!example] 2) Why a slow rocket needs a smaller kick > Same $\gamma=89°$ lekin $v = 60\ \text{m/s}$ (heavier, low thrust). > $\dot\gamma = -\dfrac{9.8\cos89°}{60} = -0.00285\ \text{rad/s} \approx -0.16°/\text{s}$. > *Kyun?* $v$ ko half karne se turn rate **double** ho jaata hai ($\dot\gamma \propto 1/v$). Toh ek sluggish vehicle > twice as fast gravity-turn karta hai → isse over-rotating se bachne ke liye **smaller** initial kick chahiye. > [!example] 3) Kick angle from desired burnout > Hum chahte hain $\gamma \approx 0°$ (horizontal) burnout tak. Qualitatively, > $\dot\gamma = -g\cos\gamma/v$ integrate karo. Kyunki $\dot\gamma$ shrink hota hai jab $v$ badhta hai, zyaadatar turning > **early aur low** hoti hai jab $v$ chhota hota hai. *Yeh kyun matter karta hai:* program designer turn ko front-load karta hai — ~10–20 s par pitch-over angle poori trajectory shape dominate karta hai. $\gamma_0$ > (e.g. $88.5°$ vs $89.5°$) tune karo jab tak simulated burnout $\gamma$ target hit na kare. Yahi tuning **hai** open-loop pitch program design karna. --- ## Common mistakes > [!mistake] "Engine rocket ko gravity turn ke dauran turn karne ke liye gimbal karta hai." > **Kyun sahi lagta hai:** turn karne ka matlab usually steering hota hai. **Fix:** ek *ideal* gravity > turn mein engine **seedha body axis ke saath** point karta hai — gravity turning karti hai. Gimbal sirf brief initial **kick** ke liye aur stabilization ke liye use hota hai, pitch-over force karne ke liye nahi. > Thrust se turn karne se angle of attack aur side loads create hote. > [!mistake] "Open-loop matlab bilkul koi control system nahi." > **Kyun sahi lagta hai:** "open-loop" sunne mein "no controller" lagta hai. **Fix:** ek attitude > control loop HAI jo vehicle ko *commanded angle par* rakhta hai; jo open-loop hai woh yeh hai ki > **command itself** ($\theta_c(t)$) ek fixed schedule hai, navigation state se computed nahi. > [!mistake] "Horizontal speed gain karne ke liye jitna jaldi aur jitna hard ho sake pitch over karo." > **Kyun sahi lagta hai:** orbit ko horizontal speed chahiye, toh jaldi turn karo! **Fix:** high $q$ par jaldi turn karne se huge $\alpha$ aur aerodynamic loads build hote hain → structural failure ya gravity/steering > losses. Aap **gently** pitch karte ho, low, taaki max-Q ke through $\alpha\approx 0$ rahe. > [!mistake] "Bada $v$ rocket ko faster turn karta hai." > **Kyun sahi lagta hai:** faster = zyaada kuch ho raha hai. **Fix:** $\dot\gamma \propto 1/v$ — higher > speed gravity ko (ab zyaada inertial) velocity vector ko **kam** bend karne deta hai. --- ## #flashcards/physics What is a pitch program? ::: Commanded pitch angle $\theta_c(t)$ ka time ke saath ek pre-computed schedule, jo ascent ke dauran open-loop fly hota hai. What does "open-loop" refer to specifically? ::: Pitch *command* time ka ek fixed function hai, measured navigation state se computed nahi (though ek attitude loop command track karta rehta hai). What is the pitch-over / pitch kick? ::: Vertical se woh chhoti si initial rotation, tower clear ke just baad, jo horizontal ki taraf turn shuru karta hai. State the gravity-turn pitch-rate equation. ::: $\dot\gamma = -\,g\cos\gamma / v$. Derive $\dot\gamma$: which force provides normal acceleration? ::: Gravity ka component velocity ke normal, $mg\cos\gamma$; $mv\dot\gamma=-mg\cos\gamma$ set karne se $\dot\gamma=-g\cos\gamma/v$ milta hai. Why keep angle of attack $\alpha\approx0$ in the atmosphere? ::: Aerodynamic side force $\propto \alpha\,q$; high dynamic pressure par bada $\alpha$ structural failure cause karta hai. In an ideal gravity turn, where does thrust point? ::: Velocity vector ke saath (body axis $\vec v$ ke saath aligned); gravity turning karti hai. How does $\dot\gamma$ depend on speed $v$? ::: Inversely: $\dot\gamma\propto 1/v$, toh faster rockets zyaada slowly turn karte hain. Effect of too-small a pitch kick? ::: Rocket burnout par bhi climb kar raha hota hai (bahut steep) — horizontal / orbit kabhi nahi pahunchta. Effect of too-large a pitch kick? ::: Over-rotate ho jaata hai, dives, high $\alpha$/drag/heating generate hoti hai. Why start the ascent vertical? ::: Dense low atmosphere mein body axis ke saath fast punch karo taaki aero loads minimize hon. Open-loop vs closed-loop guidance in ascent — which phase uses which? ::: Atmospheric ascent: open-loop program (safe aero). Upper stage/exo-atmosphere: closed-loop (e.g. PEG/IGM). --- > [!recall]- Feynman: explain to a 12-year-old > Socho ek ball seedha upar throw kar rahe ho. Isse sirf upar-neeche ki jagah door sideways jaane ke liye, > apna throw thoda tilt karo. Space mein jaane wale rocket ko *sideways* bahut fast fly karna hota hai end mein. > Toh launch ke right baad woh apna nose thoda sa tilt karta hai — jaise ek gentle bow. Us > nudge ke baad, **gravity slowly nose ko neeche kheenchti rehti hai**, toh woh apne aap curve ho jaata hai, > smoothly, jaise ek ball field mein arc karte hue. Rocket turn karne ke liye fight nahi karta (thick low air mein woh isse tod deta); > woh bas ek baar lean karta hai aur Earth ko arc mein pull karne deta hai. Aur > woh ek **pre-written timetable** follow karta hai — "10 seconds par, itna lean karo" — bina dekhe ki > woh actually kahan hai, jaise ghadi dekhte hue recipe follow karna. > [!mnemonic] > **"KICK once, let Gravity STEER."** Aur formula ke sign+shape ke liye: > **$\dot\gamma = -g\cos\gamma/v$** → *"Gravity Cuts the angle, Speed Slows the cut."* > (minus = angle drops; $\cos\gamma$ = vertical par zero toh barely turn; $/v$ = faster ⇒ slower turn.) --- ## Connections - [[Gravity turn trajectory]] — woh physics jisko open-loop program approximate karta hai. - [[Angle of attack and dynamic pressure (max-Q)]] — woh constraint jo $\alpha\approx0$ force karta hai. - [[Thrust-to-weight ratio]] — $dv/d\gamma$ relation mein $n=a/g$ set karta hai. - [[Gravity loss and steering loss]] — trade-offs jo pitch schedule optimize karta hai. - [[Closed-loop ascent guidance (PEG / IGM)]] — atmosphere ke baad takeover karta hai. - [[Attitude control and thrust vectoring (gimbal)]] — kick physically kaise execute hoti hai. - [[Tsiolkovsky rocket equation]] — woh $\Delta v$ budget jisko losses khaate hain. ## 🖼️ Concept Map ```mermaid flowchart TD ORBIT[Orbit needs horizontal speed 7.8 km/s] -->|requires| PITCH[Pitch program] PITCH -->|schedules| THETA[Commanded angle theta_c of time t] PITCH -->|flown| OPEN[Open-loop no feedback] OPEN -->|contrast| CLOSED[Closed-loop from measured state] PITCH -->|starts with| KICK[Pitch-over kick] START[Start vertical] -->|reason| DENSEAIR[Dense air near pad] KICK -->|initiates| GTURN[Gravity turn] GTURN -->|keeps| ALPHA[Angle of attack near zero] ALPHA -->|limits| SIDEFORCE[Side force prop to alpha times q] SIDEFORCE -->|threatens| STRUCT[Structural failure at high q] DENSEAIR -->|forces| OPEN GTURN -->|gravity turns| GAMMA[Flight-path angle gamma decreases] ```