3.4.14Rocket Flight Mechanics
Pitch program — open-loop pitch-over
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WHAT is a pitch program?
WHY do it this way? (first principles)
WHY pitch over at all? Orbit requires huge horizontal speed. If the rocket stayed vertical, all thrust fights gravity and it just goes up and falls back. It must redirect thrust sideways.
WHY start vertical? Two reasons:
- Near the pad the air is densest — you want to punch through it fast and along the body axis to keep aerodynamic loads low.
- Gravity loss is minimized only briefly; you can't waste time — but you also can't turn early without generating dangerous angle of attack.
WHY open-loop in the atmosphere? The key enemy is angle of attack (angle between the velocity vector and the body axis). Aerodynamic side force , where is dynamic pressure. Large at high can snap the vehicle. So the goal is to keep throughout the dense atmosphere. The elegant trick to guarantee this is the gravity turn.
HOW: deriving the flight-path equations
Model the rocket as a point mass in a vertical plane. Let:
- = speed, = flight-path angle (angle of velocity above horizontal),
- = thrust (along body axis), = mass, = gravity, = drag.
In a pure gravity turn the body axis is aligned with velocity (), so thrust acts along .
Along the velocity (tangential):
Why? Thrust and drag are along the flight path; gravity's component along the path is (opposes climbing).
Perpendicular to velocity (normal): the only force turning is gravity's normal component (thrust and drag are on-axis, no side force):
Why the minus? Gravity points down, which reduces (bends the path toward horizontal). Cancel :
HOW the initial kick sets the whole trajectory. Divide the tangential by normal to eliminate time:
= \frac{(T-D)/m - g\sin\gamma}{-g\cos\gamma/v}. $$ Neglecting drag and taking $T/m = a$ (roughly constant, $n = a/g$): $$ \frac{dv}{d\gamma} = \frac{v\,(n - \sin\gamma)}{-\cos\gamma}. $$ This shows: pick the **kick angle** (initial $\gamma_0$ just below $90^\circ$) and the thrust-to-weight $n$, and the **entire** $v(\gamma)$ curve is determined. That is the essence of the open-loop program: **choose $\gamma_0$ (and $n$), and the ascent shape follows from physics.** > [!intuition] Why the kick is "just a nudge" > Near vertical, $\cos\gamma\approx 0$, so $\dot\gamma$ is tiny — the rocket barely turns. > But that tiny turn seeds a growing $\cos\gamma$, which speeds up turning, which grows > $\cos\gamma$... a slow self-amplifying bend. Too small a kick → still going up at burnout > (never reaches horizontal). Too large → turns over too fast, dives, high drag/heating. ![[3.4.14-Pitch-program-—-open-loop-pitch-over.png]] --- ## Worked examples > [!example] 1) Instantaneous pitch rate at kick-over > A rocket at $v = 120\ \text{m/s}$, $\gamma = 89^\circ$, $g = 9.8\ \text{m/s}^2$. Find $\dot\gamma$. > > **Step:** $\dot\gamma = -\dfrac{g\cos\gamma}{v} = -\dfrac{9.8\cos 89^\circ}{120}$. > *Why this step?* The gravity-turn law gives turn rate directly from state. > $\cos 89^\circ = 0.01745$, so $\dot\gamma = -\dfrac{9.8(0.01745)}{120} \approx -0.00143\ \text{rad/s}$ > $\approx -0.082^\circ/\text{s}$. > **Read:** at high speed and near-vertical, the turn is barely $0.08°$ per second — you MUST > have started the kick to have any turn at all. > [!example] 2) Why a slow rocket needs a smaller kick > Same $\gamma=89°$ but $v = 60\ \text{m/s}$ (heavier, low thrust). > $\dot\gamma = -\dfrac{9.8\cos89°}{60} = -0.00285\ \text{rad/s} \approx -0.16°/\text{s}$. > *Why?* Halving $v$ **doubles** the turn rate ($\dot\gamma \propto 1/v$). So a sluggish vehicle > gravity-turns twice as fast → it needs a **smaller** initial kick to avoid over-rotating. > [!example] 3) Kick angle from desired burnout > We want $\gamma \approx 0°$ (horizontal) by burnout. Qualitatively, integrate > $\dot\gamma = -g\cos\gamma/v$. Because $\dot\gamma$ shrinks as $v$ grows, most turning happens > **early and low** when $v$ is small. *Why this matters:* the program designer front-loads the > turn — the pitch-over angle at ~10–20 s dominates the whole trajectory shape. Tune $\gamma_0$ > (e.g. $88.5°$ vs $89.5°$) until simulated burnout $\gamma$ hits the target. That tuning **is** > designing the open-loop pitch program. --- ## Common mistakes > [!mistake] "The engine gimbals to turn the rocket during the gravity turn." > **Why it feels right:** turning usually means steering. **The fix:** in an *ideal* gravity > turn the engine points **straight along the body axis** — gravity does the turning. Gimbal is > only used for the brief initial **kick** and for stabilization, not to force the pitch-over. > Using thrust to turn would create angle of attack and side loads. > [!mistake] "Open-loop means no control system at all." > **Why it feels right:** "open-loop" sounds like "no controller." **The fix:** there IS an > attitude control loop keeping the vehicle *on the commanded angle*; what's open-loop is that > the **command itself** ($\theta_c(t)$) is a fixed schedule, not computed from navigation state. > [!mistake] "Pitch over as early and hard as possible to gain horizontal speed." > **Why it feels right:** orbit needs horizontal speed, so turn fast! **The fix:** turning early > at high $q$ builds huge $\alpha$ and aerodynamic loads → structural failure or gravity/steering > losses. You pitch **gently**, low, so $\alpha\approx 0$ through max-Q. > [!mistake] "Bigger $v$ makes the rocket turn faster." > **Why it feels right:** faster = more happening. **The fix:** $\dot\gamma \propto 1/v$ — higher > speed makes gravity **less** able to bend the (now more inertial) velocity vector. --- ## #flashcards/physics What is a pitch program? ::: A pre-computed schedule of commanded pitch angle $\theta_c(t)$ vs time, flown open-loop during ascent. What does "open-loop" refer to specifically? ::: The pitch *command* is a fixed function of time, not computed from measured navigation state (though an attitude loop still tracks the command). What is the pitch-over / pitch kick? ::: The small initial rotation off vertical, just after tower clear, that begins the turn toward horizontal. State the gravity-turn pitch-rate equation. ::: $\dot\gamma = -\,g\cos\gamma / v$. Derive $\dot\gamma$: which force provides normal acceleration? ::: Gravity's component normal to velocity, $mg\cos\gamma$; setting $mv\dot\gamma=-mg\cos\gamma$ gives $\dot\gamma=-g\cos\gamma/v$. Why keep angle of attack $\alpha\approx0$ in the atmosphere? ::: Aerodynamic side force $\propto \alpha\,q$; large $\alpha$ at high dynamic pressure causes structural failure. In an ideal gravity turn, where does thrust point? ::: Along the velocity vector (body axis aligned with $\vec v$); gravity does the turning. How does $\dot\gamma$ depend on speed $v$? ::: Inversely: $\dot\gamma\propto 1/v$, so faster rockets turn more slowly. Effect of too-small a pitch kick? ::: Rocket is still climbing (too steep) at burnout — never reaches horizontal / orbit. Effect of too-large a pitch kick? ::: Over-rotates, dives, generates high $\alpha$/drag/heating. Why start the ascent vertical? ::: Punch through dense low atmosphere fast along the body axis to minimize aero loads. 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 > Imagine throwing a ball straight up. To make it go far sideways instead of just up-and-down, > you tilt your throw a little. A rocket to space needs to end up flying *sideways* really fast. > So right after launch it tips its nose over just a tiny bit — like a gentle bow. After that > nudge, **gravity keeps slowly pulling the nose down** for it, so it curves over on its own, > smoothly, like a ball arcing across a field. The rocket doesn't fight to turn (that would tear > it apart in the thick low air); it just leans once and lets the Earth pull it into the arc. And > it follows a **pre-written timetable** — "at 10 seconds, lean this much" — without looking at > where it actually is, like following a recipe by the clock. > [!mnemonic] > **"KICK once, let Gravity STEER."** And for the formula sign+shape: > **$\dot\gamma = -g\cos\gamma/v$** → *"Gravity Cuts the angle, Speed Slows the cut."* > (minus = angle drops; $\cos\gamma$ = zero at vertical so barely turns; $/v$ = faster ⇒ slower turn.) --- ## Connections - [[Gravity turn trajectory]] — the physics the open-loop program approximates. - [[Angle of attack and dynamic pressure (max-Q)]] — the constraint that forces $\alpha\approx0$. - [[Thrust-to-weight ratio]] — sets $n=a/g$ in the $dv/d\gamma$ relation. - [[Gravity loss and steering loss]] — trade-offs the pitch schedule optimizes. - [[Closed-loop ascent guidance (PEG / IGM)]] — takes over after the atmosphere. - [[Attitude control and thrust vectoring (gimbal)]] — how the kick is physically executed. - [[Tsiolkovsky rocket equation]] — the $\Delta v$ budget the losses eat into. ## 🖼️ 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] ``` ## 🔊 Hinglish (regional understanding) > [!intuition]- Hinglish mein samjho > Dekho, rocket launch pad se hamesha **seedha upar** (vertical) uthta hai, kyunki neeche > atmosphere sabse thick hoti hai aur usme fast, straight nikalna safe rehta hai. Lekin orbit me > jaane ke liye rocket ko **sideways speed** chahiye — around 7.8 km/s horizontal. Matlab kahin na > kahin nose ko upar se ghuma kar almost horizontal karna padega. Iske liye ek chhota sa initial > tilt diya jaata hai — usko **pitch kick** ya **pitch-over** bolte hain. > > Ab magic ye hai: ek chhoti si kick ke baad rocket khud se turn nahi karta forcefully; **gravity > uski velocity ko dheere-dheere neeche bend karti hai**, aur rocket bas apni velocity ke saath > align hoke chalta rehta hai. Isko **gravity turn** kehte hain. Formula: > $\dot\gamma = -g\cos\gamma/v$. Iska matlab — jab rocket almost vertical hai ($\gamma \approx 90°$), > $\cos\gamma$ almost zero, to turn bilkul slow. Aur jitni zyada speed $v$, utna slow turn (kyunki > $1/v$). Isliye asli turning **neeche, jab speed kam hoti hai** tab hoti hai — wahi initial kick sab > kuch decide kar deti hai. > > "Open-loop" ka matlab: pitch angle ka **command pehle se time ke hisaab se likha hua hota hai** > ("10 second pe itna tilt, 20 second pe itna") — rocket real-time me apni position dekh kar decide > nahi karta. Ye atmosphere me safe hota hai kyunki hume **angle of attack** ($\alpha$) ko zero ke > paas rakhna hota hai; warna high dynamic pressure ($q$) pe side-force $\propto \alpha q$ itna > zyada ho jaata hai ki rocket toot sakta hai. Isliye rule: **KICK once, phir gravity ko steer karne > do** — na zyada jaldi, na zyada zor se, warna ya to orbit miss ho jayega ya rocket damage ho jayega. ![[audio/3.4.14-Pitch-program-—-open-loop-pitch-over.mp3]]