3.5.48 · HinglishGuidance, Navigation & Control (GNC)
Reaction wheels — momentum management, zero-crossing
3.5.48· Physics › Guidance, Navigation & Control (GNC)
Reaction wheels KYUN exist karte hain?
Dikkat yeh hai: internal trick ek external torque ko hamesha ke liye cancel nahi kar sakti. External torques poore system mein real angular momentum inject karte hain. Wheel use absorb karta hai, tez aur tez ghoomke — jab tak redline nahi hit karta. Isliye momentum management exist karta hai.
KYA conserved hota hai — core law
KAISE: body–wheel coupling scratch se derive karein
Socho ek single wheel jiska spin axis body mein fixed hai, spin inertia hai, wheel speed hai. Maano body ki us axis ke baare mein inertia hai aur rate hai. Spin axis ke along:
Yeh form kyun? Angular momentum hota hai (inertia)×(angular rate), aur ek common axis ke along yeh scalars ki tarah add hote hain.
Differentiate karo. Koi external torque nahi → :
\quad\Rightarrow\quad \boxed{\;\dot\omega_b = -\frac{I_w}{I_b}\,\dot\Omega_w\;}$$ > [!intuition] Minus sign padho > Wheel spin up karo ($\dot\Omega_w>0$) aur body *ulti* direction mein accelerate karti hai ($\dot\omega_b<0$). Isi tarah turn karte hain: wheel ka motor torque $\tau_m = I_w\dot\Omega_w$ body ke against react karta hai — isliye "**reaction** wheel." Ab ek external torque $\tau_{\text{ext}}$ axis ke along add karo: $$\frac{d}{dt}\big(I_b\omega_b + I_w\Omega_w\big) = \tau_{\text{ext}}$$ Agar controller body ko still rakhta hai ($\omega_b\approx 0$, $\dot\omega_b\approx0$), tab: $$I_w\,\dot\Omega_w = \tau_{\text{ext}} \quad\Rightarrow\quad \Omega_w(t) = \Omega_w(0) + \frac{1}{I_w}\int_0^t \tau_{\text{ext}}\,dt'$$ > [!formula] Wheel momentum accumulation > Wheel ka stored momentum equal hota hai **external torque ke time-integral** ke: > $$H_w(t) = H_w(0) + \int_0^t \tau_{\text{ext}}\,dt'$$ > Ek chota lekin *constant-sign* torque (jaise fixed solar-pressure offset) $H_w$ ko bina kisi bound ke badhata hai → **saturation**. --- ## Momentum management (desaturation / "momentum dumping") > [!definition] Momentum management > Ek **external** actuator (thrusters ya magnetic torquers) use karke accumulated wheel momentum hatane ka process, taaki wheel ek safe speed pe wapas aaye aur body pointed rahe. > [!intuition] Kuch external KYUN chahiye > Wheels sirf momentum ko body aur wheel ke beech *shuffle* karte hain — sum fixed rehta hai. **Total** change karne ke liye, aapko ek real external torque apply karna hoga. Magnetorquers Earth ke field $\vec B$ ke against push karte hain; thrusters mass bahar fekte hain. Dono $\vec H_{\text{tot}}$ change karte hain, controller ko wheel dheera karne dete hain. Magnetorquer dump law (LEO mein sabse zyada use hone wala): $$\vec\tau_{\text{mag}} = \vec m \times \vec B, \qquad \vec m = -\,k\,(\vec H_w - \vec H_{\text{target}})\times \vec B$$ Yeh ek aisa torque produce karta hai jo $\vec H_w$ ko target ki taraf drive karta hai (ek sirf cross-product control — yeh $\vec B$ ke along component ko touch nahi kar sakta, isliye $\vec B$ ki orbital variation pe rely karte hain). --- ## Zero-crossing — subtle dushman > [!definition] Zero-crossing > Woh instant jab wheel ki speed $\Omega_w = 0$ se guzarti hai (sign change karti hai) jab uska momentum zero ke paas regulate hota hai. > [!intuition] Zero ek bura jagah kyun hai > $\Omega_w=0$ ke paas do buri cheezein hoti hain: > 1. **Bearing friction sign reverse karti hai** aur highly nonlinear hoti hai (stiction). Zero ke aas-paas friction torque direction flip karta hai, toh motor ka needed torque discontinuous ho jaata hai → wheel "hang" karta hai, phir jerk maarta hai. Yeh spacecraft mein ek **jitter** disturbance inject karta hai — telescope ke liye deadly. > 2. **Motor torque ripple / cogging** zero rate ke paas sabse bura hota hai jahan rotation pe averaging nahi hoti. > > Result: zero speed pe dawdle kar raha wheel micro-vibrations produce karta hai bilkul tab jab aapko finest pointing chahiye. > [!intuition] Kyun hum deliberately wheels spinning rakhte hain (bias momentum) > Zero-crossing *avoid* karne ke liye hum har wheel ko ek **non-zero bias speed** $\Omega_{\text{bias}}$ pe run karte hain. Controller $H_w$ ko zero ki jagah ek bias value ki taraf regulate karta hai, toh friction kabhi reverse nahi hoti. 4-wheel pyramid mein aap biases set kar sakte ho taaki koi bhi wheel kisi maneuver ke dauran zero cross na kare. > [!mistake] "Bas wheel ko zero ke through command karo, yeh sirf friction hai." > **Kyun sahi lagta hai:** friction motor torque ke comparison mein chota lagta hai, toh kya fark padta hai. **Kyun galat hai:** zero speed pe, static (stick) friction us tiny torque command se *zyada* ho sakti hai jo fine pointing ke liye chahiye; wheel stick karta hai aur pointing error badhti hai jab tak motor stiction overcome nahi karta, phir snap maarta hai — ek impulsive disturbance. **Fix:** wheels ko zero se door bias karo, ya redundant array mein doosre wheels ke through required momentum route karo taaki koi single wheel zero cross na kare. > [!mistake] "Reaction wheels momentum store karte hain, toh secular torques ko hamesha ke liye solve karte hain." > **Kyun sahi lagta hai:** aap kisi bhi torque ko thodi der ke liye absorb *kar* sakte ho. **Kyun galat hai:** ek constant-sign external torque bina bound ke integrate hota hai; wheel finite time mein saturate ho jaata hai. **Fix:** ek external actuator ke saath periodic momentum dumping mandatory hai — wheels ek *buffer* hain, *sink* nahi. ![[3.5.48-Reaction-wheels-—-momentum-management,-zero-crossing.png]] --- ## Worked examples > [!example] 1 — Ek wheel se satellite slew karo > $I_b = 500\ \text{kg·m}^2$, wheel $I_w = 0.05\ \text{kg·m}^2$. Hum wheel ko $0$ se $\Omega_w = 300\ \text{rad/s}$ tak $10\ \text{s}$ mein spin karte hain. Kya body rate milegi (koi external torque nahi, rest se shuru)? > > **Step — conservation:** $I_b\omega_b + I_w\Omega_w = 0$. > *Kyun?* Koi external torque nahi → total $H$ apni initial value $0$ pe rehta hai. > $$\omega_b = -\frac{I_w\Omega_w}{I_b} = -\frac{0.05\times 300}{500} = -0.03\ \text{rad/s}$$ > Body dheere-dheere wheel ke ulti direction mein rotate karti hai. **Minus sign kyun matter karta hai:** ek target ki *taraf* point karne ke liye wheel ko us se *door* spin karo. > > **Motor torque used:** $\tau_m = I_w\dot\Omega_w = 0.05 \times (300/10) = 1.5\ \text{N·m}$. > [!example] 2 — Solar pressure se saturation ka time > Ek constant disturbance $\tau_{\text{ext}} = 2\times10^{-5}\ \text{N·m}$. Wheel max momentum $H_{\max} = 0.05\ \text{kg·m}^2 \times 600\ \text{rad/s} = 30\ \text{N·m·s}$. Bias $H_w(0)=0$ se shuru. > > **Step — integrate:** $H_w(t) = \tau_{\text{ext}}\,t$. > *Kyun?* Constant torque → linear accumulation ($\int \text{const}\,dt = \text{const}\cdot t$). > $$t_{\text{sat}} = \frac{H_{\max}}{\tau_{\text{ext}}} = \frac{30}{2\times10^{-5}} = 1.5\times10^{6}\ \text{s} \approx 17.4\ \text{days}$$ > **Kyun matter karta hai:** ~17 days se pehle ek dump schedule karna chahiye, warna wheel max out ho jaata hai aur *attitude control kho deta hai*. > [!example] 3 — Magnetorquer dump feasibility > Earth field $B = 3\times10^{-5}\ \text{T}$, magnetorquer dipole $m = 30\ \text{A·m}^2$ $B$ ke perpendicular. > > **Step — max torque:** $\tau = mB\sin\theta$, $\theta=90^\circ$ pe max. > *Kyun?* $\vec\tau=\vec m\times\vec B$; cross product tab peak karta hai jab perpendicular ho. > $$\tau_{\max} = 30\times 3\times10^{-5} = 9\times10^{-4}\ \text{N·m}$$ > $30\ \text{N·m·s}$ dump karne ke liye: $t = 30 / 9\times10^{-4} \approx 3.3\times10^{4}\ \text{s} \approx 9.3\ \text{h}$. > **Kyun matter karta hai:** dumping slow hai aur sirf $\vec B$ ke perpendicular act kar sakta hai, isliye ise orbit ke across spread karte hain jaise $\vec B$ rotate karta hai. --- > [!recall]- Feynman: 12-saal ke bachche ko samjhao > Socho tum ek ghoomne wali office chair pe baitha ho aur ek bicycle wheel pakad rakha hai. Agar wheel ko ek taraf ghoomao, TUM doosri taraf ghoomte ho — isi tarah satellite bina kuch bahar push kiye khud ko turn karta hai. Lekin wheel sirf itni tez ghoom sakta hai. Agar ek tiny steady wind (space mein sunlight aur gravity ke gentle "winds" hote hain) usi tarah nudge karte rahe, wheel tez aur tez ghoomta hai use fight karne ke liye, jab tak woh aur fast nahi ja sakta — woh "full" ho jaata hai. Tab tumhe kuch *bahar* se gently push karna padta hai (Earth ka magnet ya gas ka ek puff) wheel ko waapas empty karne ke liye. Aur agar wheel kabhi ruk ke reverse spin karne ki koshish kare, ushe thodi si "stuck-then-jerk" feeling aati hai (jaise ek squeaky door hinge). Woh jerk tumhara telescope hilata hai. Isliye hum wheels ko hamesha thoda ghoomte rakhte hain, unhe zero pe freeze nahi hone dete. > [!mnemonic] BUFFER + BIAS > **B**uffer, sink nahi (wheels fill up hote hain). **U**se karo external torque dump ke liye. **F**riction zero pe flip hoti hai. **F**inite time mein saturate hota hai. **E**mpty karo redline se pehle. **R**un karo **BIAS** speed pe zero-crossing se bachne ke liye. --- ## #flashcards/physics Reaction wheel ko ek taraf ghoomane se spacecraft doosri taraf kyun rotate karta hai? ::: Total angular momentum ka conservation: $I_b\omega_b + I_w\Omega_w = \text{const}$, isliye $\dot\omega_b = -(I_w/I_b)\dot\Omega_w$. Kya reaction wheels ek external torque ko indefinitely cancel kar sakte hain? ::: Nahi — wheel momentum external torque ka time-integral hota hai; ek constant-sign torque finite time mein wheel ko saturate kar deta hai. Momentum management (desaturation) kya hota hai? ::: Ek external actuator (thrusters/magnetorquers) use karke accumulated wheel momentum hatana taaki wheel ek safe speed pe wapas aaye. Reaction wheels khud total angular momentum kyun nahi change kar sakte? ::: Yeh sirf momentum ko internally body aur wheel ke beech redistribute karte hain; total change karne ke liye external torque chahiye. Zero-crossing kya hai aur yeh haanikaarak kyun hai? ::: Wheel ki speed ka zero se guzarna; bearing stiction nonlinearly sign reverse karti hai, ek stick–slip jitter cause karti hai jo fine pointing kharaab karta hai. Zero-crossing se kaise bachein? ::: Wheels ko non-zero bias momentum pe chalao (ya momentum ko redundant wheel array ke through route karo) taaki kisi wheel ki speed reverse na ho. Magnetorquer torque law? ::: $\vec\tau = \vec m \times \vec B$; sirf $\vec B$ ke perpendicular act karta hai, isliye dumping orbit ke saath field vary hone pe rely karta hai. Constant torque $\tau$ ke liye saturation ka time? ::: $t_{\text{sat}} = H_{\max}/\tau$ jahan $H_{\max}=I_w\Omega_{\max}$. Wheel accelerate karne ke liye motor torque kitna chahiye? ::: $\tau_m = I_w\dot\Omega_w$ (body ke against react karta hai). --- ## Connections - [[Attitude Determination and Control System (ADCS)]] - [[Conservation of Angular Momentum]] - [[Control Moment Gyroscopes (CMG)]] — related, gimbaled spinning wheels use karta hai - [[Magnetorquers and Earth's Magnetic Field]] - [[Gravity Gradient Torque]] · [[Solar Radiation Pressure]] · [[Aerodynamic Drag Torque]] - [[PID and Feedback Control]] — woh loop jo wheel torque command karta hai - [[Bearing Friction and Stiction]] ## 🖼️ Concept Map ```mermaid flowchart TD RW[Reaction wheel flywheel] AM[Total angular momentum conserved] POINT[Precise pointing no fuel] EXT[External torques drag solar gravity] ACC[Wheel momentum accumulation] SAT[Wheel saturation redline] MM[Momentum management] DUMP[Dump via thrusters or torquers] MINUS[Body spins opposite to wheel] RW -->|obeys| AM AM -->|enables| POINT RW -->|spin up gives| MINUS EXT -->|inject real| AM EXT -->|integrated over time| ACC ACC -->|grows toward| SAT SAT -->|requires| MM MM -->|performs| DUMP DUMP -->|removes| ACC POINT -->|limited by| SAT ``` ## 🔬 Deep Dive > [!intuition] Aur gehraai mein jao — visual, zero se > Is topic ke step-by-step 3Blue1Brown-style breakdowns. - [[3.5.48 D1 Foundations|D1 · Foundations — har symbol zero se]] - [[3.5.48 D2 Visual Walkthrough|D2 · Visual walkthrough — derivation pictures mein]] - [[3.5.48 D3 Worked Examples|D3 · Worked examples — har scenario]] - [[3.5.48 D4 Exercises|D4 · Exercises — graded, full solutions]] - [[3.5.48 D5 Question Bank|D5 · Question bank — concept traps]]