Foundations — 6DOF equations — translational (Newton), rotational (Euler's equations)

3.4.5 · D1 · Physics › Rocket Flight Mechanics › 6DOF equations — translational (Newton), rotational (Euler's

Parent note padhne se pehle, tumhe har us symbol ki poori pakad chahiye jo woh tumhare saamne phenk ta hai. Yeh page har ek ko zero se build karta hai — simple words, ek picture, aur woh reason ki topic uske bina exist nahi kar sakti. Upar se neeche padho; har block uske upar wale par lean karta hai.

0. "Rigid body" kya hota hai?

Topic ko yeh kyun chahiye: "point moves" + "body twists" ka saaf split ek theorem hai jo sirf rigid bodies ke liye hold karta hai. Ek noodle ko infinitely many numbers ki zaroorat hogi.

1. Vectors aur arrow picture

3D mein ek vector actually teen numbers hote hain stacked — ek har axis ke liye. Figure dekho: arrow $\vec{A}$ har axis par ek shadow daalti hai, aur woh teen shadow-lengths $A_x,A_y,A_z$ uske components hain. $\vec{A}=A_x{\hat{e}}_x+A_y{\hat{e}}_y+A_z{\hat{e}}_z$ likhne ka matlab bas yahi hai ki "$A_x$ steps $x$ along chalo, phir $A_y$ steps $y$ along, phir $A_z$ steps $z$ along, aur tum arrow tip par land karoge."

Topic ko yeh kyun chahiye: 6DOF mein har quantity (force, velocity, spin) ek vector hai, aur transport theorem derive hota hai yeh poochh kar ki jab rocket turn karta hai toh basis vectors ${\hat{e}}_i$ khud kaise move karte hain.

2. Rate of change: dot aur $\frac{d}{dt}$

Topic ko yeh kyun chahiye: Newton's law aur Euler's law dono rate of change ke baare mein statements hain — momentum ki rate push ke barabar hai. Dot ke bina koi law nahi.

3. Velocity $\vec{v}=(u,v,w)$ aur angular velocity $\vec{\omega }=(p,q,r)$

Topic ko yeh kyun chahiye: $\vec{v}$ Newton's equation ka star hai, $\vec{\omega }$ Euler's ka. Dono equations mein ek "$\vec{\omega }\times$" term hai, isliye pehle yeh jaanna zaroori hai ki $\vec{\omega }$ kya hai.

4. Cross product $\vec{\omega }\times \vec{A}$

Figure dekho: $\vec{\omega }$ par spinning body mein position $\vec{A}$ par ek point green arrow $\vec{\omega }\times \vec{A}$ ke along swing karta hai — apne circle ka tangent, axis aur radius dono ke right angles par.

Topic ko yeh kyun chahiye: body-frame equations ground-frame equations se alag kyun hain — iska poora reason yahi $\vec{\omega }\times$ term hai. Yeh hai transport theorem ka punchline.

5. Do frames: inertial $I$ aur body $B$

Topic ko yeh kyun chahiye: yahi central tension hai jo parent note resolve karta hai. $B$ mein measure karo, lekin law $I$ mein rehta hai — transport theorem bridge hai. Deeper jaane ke liye Reference frames and rotation matrices aur Transport theorem (rotating frames) dekho.

6. Mass $m$, momentum $\vec{p}$, aur force $\vec{F}$

Topic ko yeh kyun chahiye: ye teeno translational equation ke poore cast hain.

7. Inertia tensor $\mathbf{I}$ aur angular momentum $\vec{H}=\mathbf{I}\vec{\omega }$

Topic ko yeh kyun chahiye: $\vec{M}=\dot{\vec{H}}{|}_I=\mathbf{I}\dot{\vec{\omega }}{|}_B+\vec{\omega }\times (\mathbf{I}\vec{\omega })$ hi Euler's equation hai. Isme har symbol — $\mathbf{I}$, $\vec{\omega }$, $\vec{H}$, $\vec{M}$, cross product, do frames — ab tumhari pakad mein hain.

8. Yeh sab topic ko kaise feed karta hai

Sab kuch transport theorem mein funnel hota hai, jo phir parent note the 6DOF topic ki Newton aur Euler equations dono produce karta hai.

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