3.5.48 · D1Reaction wheels — momentum management, zero-crossing
Ek spinning wheel aur spacecraft body ek fixed "turning ki matra" share karte hain — ek ko push karo, doosra peeche ghoomta hai, lekin total apne aap kabhi nahi badalta. Is total ko actually change karne ke liye tumhe bahar jaana padega (Earth ka magnetic field ya thruster), aur jab tak nahi jaate, koi bhi steady bahari nudge silently wheel ko tab tak bharta rehta hai jab tak woh aur fast spin nahi kar sakta.
Yeh page kuch bhi assume nahi karta . Parent note padhne se pehle, har woh symbol jo wahan use hota hai, yahan ek picture se build kiya gaya hai. Upar se neeche padho; har idea sirf uske upar wale ideas use karta hai.
Is topic ki har baat spinning cheezoon ke baare mein hai. Toh pehle: hum spin kaise measure karte hain?
θ angle hai. Hum ise "hours" ya degrees mein nahi balki radians mein measure karte hain: ek radian woh angle hota hai jab tumhara trace kiya hua arc radius ke exactly utna hi lamba ho.
Poora circle = 2 π ≈ 6.28
Symbol: θ
ω t θ turning ki rate yeh hai ki ek second mein kitna angle:
ω = change in t change in θ [ radians per second ] ω ω = 0
Yeh topic isko kyun zaroorat hai: spacecraft body rate ω b body ) aur wheel rate Ω w wheel ). Same idea, do objects. Wheel ke liye capital Ω
ω Ω ω Ω
x ˙ woh symbol per second kitni tezi se change ho raha hai. " Yeh usi "change ÷ time" idea ka shorthand hai jo humne abhi use kiya.
Ω ˙ w speed kitni tezi se change ho rahi hai = wheel speed up ya slow down ho rahi hai.ω ˙ b
Intuition Har baar fraction kyun nahi, dot kyund t d Ω w
Jo tool introduce ho raha hai: yeh dot hi derivative hai — woh mathematical machine jo is sawaal ka jawaab deta hai ki "is pal, meri quantity kitni tezi se chal rahi hai?" Hum ise (plain subtraction ke bajaye) isliye choose karte hain kyunki yahan torques aur speeds continuously change hote hain, pal pal, jumps mein nahi.
Derivative (dot) ek total ko rate mein todta hai. Integral bilkul ulta karta hai: woh rate ko wapas total mein jodta hai.
∫ 0 t f d t ′ f 0 t ∫ d t ′
Intuition Yahan integral ki zaroorat kyun haiEk steady tiny bahari push (ek torque) momentum ki ek rate add karta hai. Yeh jaanne ke liye ki ghanton ya dinon baad kitna momentum ikatha ho gaya , humen woh rate us saare time par sum karni padegi — exactly yahi ∫ τ ext d t ′
Definition Moment of inertiaI push karna kitna mushkil hai. Moment of inertia I twist karna (ya spin se bahar nikalna) kitna mushkil hai. Mota, bhaari, phela hua object bada I I
I b body (poori spacecraft) ki inertia — badi, kyunki spacecraft badi hai.I w wheel ki inertia — tiny, kyunki wheel chhoti aur halki hai.
Ek broomstick ko uski length ke baare mein spin karo (aasaan, tiny I I I I b ≫ I w 500 0.05 dheere ghoomata hai.
Ab rate aur inertia ko us ek quantity mein combine karte hain jise poora topic conserve karta hai.
Definition Angular momentumH inertia × rate :
H = I ω [ kg⋅m 2 / s = N⋅m⋅s ] H H
Intuition Ek axis par plain numbers ki tarah kyun add hota haiJab body aur wheel same line (spin axis) ke baare mein spin karte hain, toh unke stored spins simply add hote hain:
H tot = H body I b ω b + H wheel I w Ω w + −
Gehra fact — Conservation of Angular Momentum mein proven — yeh hai ki bahar se kuch push nahi kar raha toh H tot Woh ek sentence poore topic ka engine hai.
Mnemonic Momentum ka see-sawH tot doosri taraf spin karna. Yahi ek image mein reaction wheel hai.
τ torque ek twisting effort hai — ek force leverage ke saath lagaaya jaata hai jo kisi cheez ki spin change karne ki koshish karta hai. Yeh angular momentum ka wahi hai jo push ordinary motion ka hai:
τ = d t d H = H ˙ H H
Intuition Is topic mein torque ke do flavours
Internal / motor torque τ m H move karta hai; yeh kabhi total change nahi kar sakta.External torque τ ext sunlight , gravity gradient , thin-air drag . Yeh H tot zaroor change karte hain, aur yahi slowly wheel ko bharta hai.
Topic inhe itne carefully kyun alag karta hai: poora "momentum management" problem precisely yahi hai ki internal torque woh nahi undo kar sakta jo external torque karta hai — tumhe ek doosra external torque (ek magnetorquer ya thruster) laana hoga use wapas nikalne ke liye.
Dump law τ = m × B × cross product hai, aur humen isko isliye chahiye kyunki torque yahan ek direction rakhta hai jo do aur directions par depend karti hai.
Definition Vector (chhota arrow)v size aur direction dono rakhta hai (sirf number nahi). B m
a × b teesra arrow, dono ke perpendicular , jiska length hai
∣ a × b ∣ = ∣ a ∣ ∣ b ∣ sin θ θ a b
sin θ sin θ = 1 perpendicular hoon (θ = 9 0 ∘ zero jab woh parallel hoon (θ = 0 koi twist nahi. Woh geometric fact poora reason hai ki magnetorquer "component along B sin θ Magnetorquers and Earth's Magnetic Field .
Yeh naya math nahi hai, sirf Ω w
Definition Teen wheel states
Saturation : Ω w Ω m a x H H w = H m a x Bias speed Ω bias Zero-crossing : woh moment jab Ω w 0 bearing friction reverse hoti hai aur wheel jerk karta hai — ek jitter source jiske around hum design karte hain.
Integral means sum over time
Angular momentum H = I omega
Torque = rate of change of H
Reaction wheel body-wheel coupling
Magnetorquer momentum dump
Saturation bias and zero-crossing
Reaction Wheels topic 3.5.48
Har jawab cover karo aur khud ko test karo. Agar koi atka de, us section ko dobara padho.
Angle θ Koi cheez kitna ghoom gayi; radians mein measure hota hai (poora circle = 2 π
ω Angular rate — ek second mein kitna angle sweep hota hai.
Wheel ki rate Ω w ω b Sirf tez wheel ko slow body se alag karne ke liye; same physical quantity hai.
Kisi symbol ke upar dot, jaise Ω ˙ w Us quantity ki per second rate of change (ek derivative).
Words mein, ∫ 0 t τ d t ′ Woh total momentum jo torque ka contribution har sliver of time par 0 t
Constant torque c ∫ 0 t c d t ′ c ⋅ t c t
Moment of inertia I Kisi object ki spin change karna kitna mushkil hai (rotational "heaviness").
Is topic par I b ≫ I w Poori spacecraft body badi aur bhaari hai; wheel chhota aur halka hai.
Angular momentum I H = I ω
Body aur wheel ke momenta plain numbers ki tarah kyun add hote hain? Yeh same axis ke baare mein spin karte hain, isliye direction sirf ek + −
Kaun sa ek law kehta hai ki H tot Conservation of angular momentum — koi external torque nahi matlab frozen total.
Torque τ Angular momentum ki rate of change, τ = H ˙
Internal motor torque aur external torque mein farq? Internal sirf H H tot
Cross product m × B Dono ke perpendicular ek vector, jiska length m B sin θ
Magnetorquer ka torque maximum kab hota hai, aur zero kab? Maximum jab
m ⊥ B (
sin 9 0 ∘ = 1 ); zero jab parallel (
sin 0 = 0 ).
Saturation kya hai? Wheel apni max speed tak pahunch gayi hai aur aur momentum absorb nahi kar sakta.
Wheel ko zero ki jagah bias speed par kyun rakho? Zero-crossing se bachne ke liye, jahan friction reverse hoti hai aur pointing jitter hoti hai.
