Rigid body dynamics — Euler angles, Euler's equations of motion: Foundations

2.1.21 · D1 · Physics › Analytical Mechanics › Rigid body dynamics — Euler angles, Euler's equations of mot

Parent note ki ek bhi line padhne se pehle, tumhe har wo symbol apna banana hoga jo wahan use hota hai. Yeh page har ek ko kuch nahi se banata hai — ek plain-words meaning, ek picture, aur woh reason jiske bina yeh topic exist nahi kar sakta. Upar se neeche padho; har idea usse upar wale pe lean karta hai.

1. Cheezein kahaan rehti hain: frames aur axes

Hum poore topic mein do frames use karte hain, aur is topic mein aadha confusion tab hota hai jab tum bhool jaate ho ke tum kis mein khade ho.

Figure — Rigid body dynamics — Euler angles, Euler's equations of motion

Figure mein: baayein taraf ke kale arrows fixed ground frame $(X, Y, Z)$ hain; red arrows daayein taraf body frame $(x, y, z)$ hain jo tilt hoke ek chote cube se glued hain. Dekho ke red set rotated hai — wahi rotation exactly Euler angles measure karenge.

2. Angle symbols $ϕ, θ, ψ$

Yeh lowercase Greek letters hain. Inhe bolo: phi $ϕ$ , theta $θ$ , psi $ψ$ . Har ek ek "turn ki matra" hai, radians ya degrees mein measured.

Definition Teen turns — exact z–x'–z'' order (active rotations)

Yeh active rotations hain: hum physically object ko rotate karte hain (equivalently, uska body frame) jabki ground frame fixed rehta hai. Body frame ko space frame ke bilkul upar se shuru karke, iss order mein apply karo:

$ϕ$ space axis $Z$ ke baare mein (precession). Yeh purani $x$ -axis ko ek naye direction mein swing karta hai jise line of nodes kehte hain.
$θ$ us naye line-of-nodes axis, $x^{'}$ , ke baare mein (nutation) — spin axis ko tilt karta hai.
$ψ$ final body axis $z^{''} = z$ ke baare mein (spin) — object ko apni axis pe twirl karta hai.

Har turn right-hand rule se positive hai apni axis ke baare mein: right thumb rotation axis ke saath, ungliyaan increasing angle ki direction mein curl karti hain. Yeh standard 3-1-3 (z–x'–z'') convention hai jo parent note use karta hai.

Figure mein: vertical kala arrow $Z$ hai; flat kala arc $ϕ$ sweep hai; $Z$ aur tilted red axis ke beech kala arc $θ$ hai; tip pe chota red arc $ψ$ hai body axis ke baare mein. Unhe numbered order mein follow karo.

3. Change ki rate: dot $˙$

4. Vectors aur angular velocity $ω$

Figure mein: kala disk us tarah spin karta hai jaise chote rim arrows dikhate hain; red vertical arrow $ω$ hai, spin axis ke saath spin speed ke barabar length ke saath. Red arrow wahi hai jo "spin" ban jaata hai jab hum ise ek single vector mein pack karte hain.

Derivation Turn-rates add kyun hote hain:

$ω = \dot{ϕ} \hat{Z} + \dot{θ} \overset{n}{^} + \dot{ψ} \overset{z}{^}$ KYA HAI: total spin-arrow teen alag spin-arrows ka sum hai, har Euler-turn ke liye ek — har ek apni turn ki axis ke saath point karta hai jitni us turn ki rate hai. YE ADD KYU HO SAKTE HAIN: infinitesimal rotations (ek chota sa amount ek chote time $d t$ mein) ke liye, rotations commute karte hain aur ordinary vectors ki tarah behave karte hain, isliye unki rates $\dot{ϕ}, \dot{θ}, \dot{ψ}$ arrows ki tarah tip-to-tail add hoti hain. (Finite rotations commute nahi karte — order matter karta hai — yahi exactly woh reason hai jiske liye Section 2 mein strict $z$ – $x^{'}$ – $z^{''}$ order zaroori tha.) YEH KAISA DIKHTA HAI: teen arrows $\hat{Z}, \overset{n}{^}, \overset{z}{^}$ (abhi define kiye gaye tilted, non-perpendicular axes) ke saath tip-to-tail; unka single resultant $ω$ hai. Sign convention: har rate ko apni axis ke baare mein right-hand rule se positive liya jaata hai, $ϕ, θ, ψ$ ke positive sense se match karte hue. Kyunki $\hat{Z}, \overset{n}{^}, \overset{z}{^}$ body axes ke relative tilted hain, is sum ko body frame pe project karna hi woh cheez produce karta hai jo parent ke $ω_{1}, ω_{2}, ω_{3}$ formulas mein $sin θ, cos θ$ factors deti hai — tum sirf $\dot{ϕ}, \dot{θ}, \dot{ψ}$ ko components ki tarah nahi padh sakte.

5. Cross product $\times$

Figure mein: vertical kala arrow $ω$ hai, slanted kala arrow $A$ body mein frozen hai, dashed ellipse woh circle hai jo uski tip trace karti hai, aur red arrow $ω \times A$ hai — us circle ke saath sideways point karta hua, dono ke perpendicular. Woh red sideways drag hi poora mechanism hai.

Recall

$\times$ kyun, ordinary multiplication kyun nahi? Kyunki spinning $A$ ko stretch nahi karta, balki ise rotate karta hai. Rotation tip ko sideways (perpendicular) move karta hai, jo exactly $\times$ produce karta hai. Ordinary multiplication $A$ ke saath point karta hai — bilkul wrong direction. ::: Cross product woh tool hai "perpendicular sideways drag" ke liye, exactly wahi jo rotation karta hai.

6. $d / d t$ ke do meanings — kaunsa frame dekh raha hai?

7. Mass distribution: inertia tensor $I$

8. Angular momentum $L$ aur torque $τ$

9. Sab kuch topic mein kaise feed hota hai

Baayein taraf ka har foundation apne daayein taraf ke box ko build karta hai; sab kuch Euler's equations mein funnel hota hai. Agar koi bhi left-hand box fuzzy hai, woh fuzziness baad mein confusion ke roop mein surface hogi — abhi patch karo.

Equipment checklist

Khud ko test karo. Daayein side cover karo; tumhe har ek ka ek saanson mein jawab dene aana chahiye.

Ek frame physically kya matlab hai?

Teen right-angle direction-arrows kisi cheez se glued, directions measure karne ke rulers ki tarah.

Space frame

(X, Y, Z)

aur body frame

(x, y, z)

mein kya difference hai?

Space = ground se nailed, kabhi move nahi karta; body = object ke andar glued, uske saath tumble karta hai.

Kya yeh frames left- ya right-handed hain?

Right-handed: ungliyaan

X \to Y

thumb

Z

deti hain (body frame ke liye bhi same).

Exact Euler rotation sequence batao aur kya yeh active hai.

Active:

ϕ

space

Z

ke baare mein, phir

θ

naye line of nodes

x^{'}

ke baare mein, phir

ψ

final body axis

z

ke baare mein (3-1-3, z–x'–z'').

ϕ, θ, ψ

ke domains kya hain?

0 \leq ϕ < 2 π

0 \leq θ \leq π

0 \leq ψ < 2 π

Gimbal lock kya hai aur kab attack karta hai?

θ = 0

π

par, axes

Z

aur

z

coincide karte hain, isliye

ϕ

aur

ψ

merge ho jaate hain — ek angle lost ho jaata hai aur

1/ sin θ

terms blow up karte hain.

Dot ka kya matlab hai, jaise

\dot{ϕ}

; aur

\ddot{θ}

Per second change ki rate; do dots = rate ki rate (angular acceleration).

Angular velocity

ω

kya hai aur iska sign rule kya hai?

Spin-arrow: direction = right-hand rule se spin axis, length = spin speed.

Euler rates kyun add hote hain:

ω = \dot{ϕ} \hat{Z} + \dot{θ} \overset{n}{^} + \dot{ψ} \overset{z}{^}

Infinitesimal rotations commute karte hain aur vectors ki tarah act karte hain, isliye unki rates (non-perpendicular) axes

\hat{Z}, \overset{n}{^}, \overset{z}{^}

ke saath tip-to-tail add hoti hain.

\hat{Z}, \overset{n}{^}, \overset{z}{^}

kya hain aur kya yeh perpendicular hain?

ϕ, θ, ψ

ki unit axes; mutually perpendicular NAHI — woh tilt

sin θ, cos θ

factors cause karta hai.

ω \times A

kya deta hai aur iska direction kaise fix hota hai?

Dono ke perpendicular ek arrow — tip ki sideways velocity; direction right-hand rule se, order sign flip karta hai.

d / d t

ke do meanings kya hain aur yeh kaise linked hain?

Space (ground) vs body (bug) rate;

(d A / d t)_{space} = (d A / d t)_{body} + ω \times A

se linked.

Newton ke

τ = d L / d t

ko kaunsa frame chahiye?

Inertial (space) frame.

Kya inertia tensor

I

symmetric hai, aur kitne independent numbers hain?

Haan, symmetric; 6 independent (3 moments + 3 products of inertia).

I

ω

par kaise act karta hai?

Matrix multiply:

L_{i} = \sum_{j} I_{ij} ω_{j}

;

L

ω

ke parallel hona zaroori nahi.

Principal axes kyun choose karte hain?

Wahan

I

diagonal ban jaata hai,

L_{i} = I_{i} ω_{i}

, aur saare products of inertia khatam ho jaate hain.

Master rotation law batao.

τ = (d L / d t)_{space}

— torque angular momentum ki inertial-frame rate of change ke barabar hai.

"Torque-free" ka kya matlab hai?

τ = 0

; koi body ko twist nahi kar raha.

Jab har line instant ho jaye, parent topic par wapas jao aur notation plain sentences ki tarah padhega.