Foundations — Reaction control system — thruster selection, plume impingement limits

3.5.46 · D1 · Physics › Guidance, Navigation & Control (GNC) › Reaction control system — thruster selection, plume impingem

Yeh ek foundations page hai. Parent note the RCS topic arrows, cross products, matrices, aur cosines ko aise use karta hai jaise aap unhe pehle se jaante ho. Yahan hum unhe ek-ek karke, ek aisi order mein banate hain jahan har brick apne pehle wali brick par tikti hai. Agar aapne kabhi hat wala arrow nahi dekha, to line one se shuru karo — kuch miss nahi hoga.

0. Woh picture jisme poora topic rehta hai

Figure dekho. Ek boxy spacecraft black mein float kar raha hai. Beech mein ek dot iske balance point ko mark karta hai. Chote nozzles corners se bahar nikal rahe hain, aur ek fire ho raha hai — ek colored spray nozzle se nikalti hai, aur ship ko ulti direction mein nudge milta hai. Us ek image mein lagbhag woh saare symbols hain jo hum abhi name karne wale hain. Baar baar isme nazar daalte rehna.

1. Space mein ek point — position, aur woh arrow jo uski taraf point karta hai

Picture: upar wali figure mein, center dot se ek nozzle tak ek arrow draw karo. Woh arrow hi ${\mathbf{r}}_i$ hai — thruster number $i$ kahan bolt hai, balance point se measure karke.

Topic ko yeh kyun chahiye: balance point se door apply ki gayi force ship ko twist karti hai; seedha uspe apply ki gayi force sirf shoves karti hai. Isliye topic kabhi turning ki baat nahi kar sakta jab tak yeh na jaane ki har nozzle kahan hai. Woh "kahan" hi ${\mathbf{r}}_i$ hai.

2. Sirf direction — hat wala unit vector $\hat{\mathbf{u}}$

Picture: kisi bhi arrow ko shrink ya stretch karo jab tak woh exactly ek grid-square lamba na ho, uski heading wahi rakhte hue. Woh chota hua arrow hat wala version hai. Figure mein, har nozzle par woh chota colored arrow jo dikhata hai exhaust kis taraf jaata hai woh ${\hat{\mathbf{u}}}_i$ hai.

Topic ko yeh kyun chahiye: har thruster ek fixed direction mein fire karta hai. Hum us direction ko ek baar store karna chahte hain (${\hat{\mathbf{u}}}_i$) aur alag se strength ($F_i$) upar-neeche dial karna chahte hain. "Direction" ko "amount" se alag karna hi woh poora trick hai jo baad mein hume force ko $F_i{\hat{\mathbf{u}}}_i$ likhne deta hai.

3. Force $\mathbf{F}$ aur number $F$ — kitna hard, kis taraf

Picture: figure mein ship par recoil arrow. Lamba arrow = hard push; chota arrow = gentle nudge.

Ab hum wo clean split likh sakte hain jo upar promise ki thi: ${\mathbf{F}}_i=F_i{\hat{\mathbf{u}}}_i.$ Ise zor se padho: "thruster $i$ ki force uski strength $F_i$ times uski direction ${\hat{\mathbf{u}}}_i$ ke barabar hai." Strength unit arrow ko scale karti hai.

$F\ge 0$ kyun (one-sidedness): ek nozzle sirf gas bahar phek sakta hai — woh push kar sakta hai, pull nahi. Isliye $F$ $0$ tak aa sakta hai (off) lekin kabhi negative nahi ja sakta. Yahi ek fact ($F\ge 0$) ship ko opposite directions mein nozzles carry karne par majboor karta hai, kyunki aap "ek jet reverse" nahi kar sakte; uski jagah uske neighbour ko fire karo.

4. Torque $\mathbf{\tau }$ — turning, aur cross product kyun chahiye

Ab iska dil: side mein lagi force ship ko spin karati hai. "Spinning effort" ka measure torque hai, likha jaata hai $\mathbf{\tau }$ (Greek letter tau, bold kyunki yeh bhi ek arrow hai).

Figure dekho. Center se, arrow $\mathbf{r}$ nozzle tak pahunchta hai. Nozzle par, force arrow $\mathbf{F}$ push karta hai. Turning $\mathbf{r}$ ke relative $\mathbf{F}$ ke sideways part par depend karta hai. Jab $\mathbf{F}$, $\mathbf{r}$ ke saath line up hoti hai (center se seedha bahar point karti hai), twist zero hai — pure shove. Jab $\mathbf{F}$, $\mathbf{r}$ ke perpendicular hai, twist sabse bada hai.

Cross product kyun, aur ordinary multiplication kyun nahi? Do lengths ki ordinary multiplication unke beech ke angle ke baare mein nahi jaan sakti, aur ek nayi direction (spin axis) produce nahi kar sakti. Humein ek aisi machine chahiye jo do arrows khaaye aur output kare:

• ek size jo arrows ke perpendicular hone par sabse badi ho aur parallel hone par zero ho, aur
• ek direction = woh axis jis par ship spin kare.

Yahi exact machine cross product hai, likha jaata hai $\mathbf{r}\times \mathbf{F}$.

Cross product ka number padhna: uski length hai $|\mathbf{r}\times \mathbf{F}|=|\mathbf{r}||\mathbf{F}|\sin \theta ,$ jahan $\theta$ do arrows ke beech ka angle hai. $\sin \theta$ exactly "kitna sideways" hai — yeh $0$ hai jab parallel ($\theta =0$) aur $1$ jab perpendicular ($\theta =90^{\circ }$), jo wrench intuition se match karta hai. (Isliye topic baad mein sines aur angles ki parwah karta hai.)

Direction padhna — right-hand rule: apni right fingers pehle arrow $\mathbf{r}$ ke saath point karo, unhe doosre arrow $\mathbf{F}$ ki taraf curl karo; aapka thumb $\mathbf{\tau }$, spin axis, ke saath point karega. Isliye opposite-side thrusters ship ko same taraf twist kar sakte hain — dono ka thumb same direction mein point karta hai. (Parent ka worked "pure roll couple" dekho.)

Turning bodies ki deeper machinery ke liye, Cross Product & Rigid-Body Torque aur Attitude Dynamics — Euler's Equations dekho.

5. Force aur torque stack karna — wrench $\mathbf{w}$ aur vector notation

Ab hamare paas ek thruster describe karne wale do arrows hain: shove $\mathbf{F}$ (3 numbers) aur twist $\mathbf{\tau }$ (3 numbers). Topic unhe 6 numbers ki ek tall list mein stack karta hai:

Picture: 6 sliders wala ek control panel imagine karo — 3 "kitna shove karna hai (3 directions mein se har ek mein)" aur 3 "kitna twist karna hai (3 axes mein se har ek ke around)" ke liye. Pilot jo bhi command chahta ho, woh un 6 sliders ki ek setting hai. Woh setting ${\mathbf{w}}_{\mathrm{cmd}}$ hai (commanded wrench).

Unhe bundle kyun karein: kyunki har control command — aage nudge, roll, still rehna — poori tarah se in 6 numbers se describe hota hai. Bundle karne se hum saara selection problem ek clean line mein likh sakte hain chhe bikre hue lines ki jagah.

6. Matrix $B$ — ek table jo "kaun se nozzles" ko "kya wrench" mein badal deti hai

Har thruster, strength $1$ par fire kiya gaya, apna chhota wrench contribute karta hai: $[{\hat{\mathbf{u}}}_i;{\mathbf{r}}_i\times {\hat{\mathbf{u}}}_i]$ — 6 numbers ka ek column. Saare $N$ thrusters ke columns side by side lagao aur aapko numbers ki ek table milti hai, matrix $B$:

$B=\left[\begin{array}{ccc}{\hat{\mathbf{u}}}_1& \cdots & {\hat{\mathbf{u}}}_N\\ {\mathbf{r}}_1\times {\hat{\mathbf{u}}}_1& \cdots & {\mathbf{r}}_N\times {\hat{\mathbf{u}}}_N\end{array}\right](6\times N).$

Picture: $B$ ke columns ko $N$ ingredient arrows ki tarah socho aur $\mathbf{f}$ ko "har ingredient ka kitna amount daalna hai" ki tarah. $B\mathbf{f}$ final mixed wrench hai. Yeh exactly parent ka "saare thrusters par sum" hai, compactly likha gaya.

Yeh Control Allocation & Pseudo-inverse ki machinery hai: usually nozzles control karne wali 6 cheezons se zyada hote hain ($N>6$), isliye kai mixes kaam karte hain aur hum sabse sasta choose karte hain — Rocket Equation & Specific Impulse dekho ki "sabse sasta" ka matlab "least propellant" kyun hai.

7. Angles aur plume — $\theta$, $\cos$, aur $1/d^2$ falloff

Exhaust nozzle par gayab nahi hota — woh ek spray cone mein fan out hota hai. Do naye symbols describe karte hain ki woh spray kahan jaati hai.

• $d$ = nozzle mouth se jo bhi spray hit karta hai (ek panel, ek target ship) tak ki distance. Figure mein, cone ke center ke saath length.
• $\theta$ = centerline se angle — ek surface "seedhe aage" se kitna side mein hai. $\theta =0$ matlab spray ke dead-center mein; bada $\theta$ matlab cone ke rim ki taraf bahar.

Dono ko milakar parent ka pressure law milta hai: $P_{\mathrm{imp}}(d,\theta )\approx P_0{\left(\frac{d_0}{d}\right)}^2{\cos }^k\theta .$

Topic ko yeh sab kyun chahiye: yeh akela formula safety gate hai. Agar kisi fragile surface par $P_{\mathrm{imp}}$ uski limit $P_{\max }$ se zyada ho, to woh nozzle forbidden hai — uski strength selection run hone se pehle $0$ par lock ho jaati hai. Yahi "keep-out cone" hai. Deeper modeling Rarefied Gas Dynamics / Plume Modeling mein rehta hai, aur yeh close approach ke dauran sabse zyada matter karta hai — Rendezvous and Docking dekho.

8. Foundations topic ko kaise feed karti hain

Us map mein har arrow ka matlab hai "yeh samajhne se pehle, woh samjho." Do streams — allocation stream (left) aur plume-safety stream (right) — final selection par milti hain.

Equipment checklist

Right side cover karo aur reveal karne se pehle zor se jawaab do.