Mass properties — CG location, inertia tensor changing with propellant depletion: Foundations

3.4.6 · D1 · Physics › Rocket Flight Mechanics › Mass properties — CG location, inertia tensor changing with

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0. Position, aur yahan "vector" ka matlab kya hai

Mass se pehle, twisting se pehle, hume ye batana hoga ki cheezein kahan hain.

Picture: nose par khade ho, rocket mein kisi bolt ki taraf point karo — woh pointing arrow hi $r$ hai.

Topic ko ye kyun chahiye: rocket mein har mass element kisi jagah par hai. Positions ka average nikalne ke liye ya axis-se-distance measure karne ke liye, pehle hume "kahan" ki ek language chahiye.

Kyunki real rocket lamba aur patla hota hai, aksar hum sirf uski length ke saath ek coordinate ki parwah karte hain. Usse hum $x$ kehte hain — ek plain number (ek "station") jo nose se metres mein measure hota hai.

Figure — Mass properties — CG location, inertia tensor changing with propellant depletion

0b. Teen axes: hamaara reference frame

"Ek coordinate $x$ " teen mein badhne se pehle, hume yeh tay karna hoga ki hum kaunsi teen directions mein measure karte hain.

Picture: ek kamre ka kona — teen edges right angles par milte hue. Rocket us kone mein baitha hai; har tukda "along, sideways, up" se locate hota hai.

Topic ko ye kyun chahiye: parent likhta hai $I_{xx} = \int (y^{2} + z^{2}) d m$ . Woh $(y^{2} + z^{2})$ tabhi kuch maayane rakhta hai jab tum jaante ho ki $y$ aur $z$ woh do directions hain jo $x$ -axis se perpendicular hain — do tarike jis par ek crumb roll axis se door baith sakta hai. Fixed frame ke bina, tensor ke symbols bekar hain.

1. Mass aur mass element $d m$

Picture: rocket ko lakhon tiny crumbs mein slice karo. Har crumb ka apna chhota weight $d m$ hai aur apni position $r$ .

Topic ko ye kyun chahiye: CG aur inertia dono har crumb ka sum hain. Jab crumbs count kiye ja sakte hain hum $m_{i}$ likhte hain ( $i$ -va crumb); jab woh ek continuum mein blur ho jaate hain hum $d m$ likhte hain aur integral use karte hain (§4).

Recall Subscript

$i$ ka matlab kya hai? Yeh ek label hai: $m_{1}, m_{2}, m_{3}, \dots$ — "crumb number 1, crumb number 2, …". $\sum_{i}$ ka matlab hai "saare crumbs par add karo."

2. Summation sign $\sum$

Picture: boxes ki ek queue; $\sum$ ho tum, queue mein walk karte hue har ek ko ek bucket mein daal rahe ho.

Topic ko ye kyun chahiye: CG formula $\sum_{i} m_{i} r_{i}$ literally hai "har crumb ke liye, uski mass ko uski position se multiply karo, phir saare wo products add karo."

3. Ek number ko ek arrow se multiply karna, aur "average position" ka idea

Agar ek crumb ki mass $m_{i}$ hai aur position $r_{i}$ hai, toh $m_{i} r_{i}$ woh arrow hai jo number $m_{i}$ se stretch hua hai. Bhaari crumbs ko lambe arrows milte hain; halke crumbs ko chhote.

Picture: imagine karo har position arrow, lekin jitna bhaari crumb utna mota/lamba drawn. "Balance point" mote arrows ki taraf jhukta hai.

Ab parent ka central formula padhne layak hai: $r_{cg} = \frac{\sum _{i} m _{i} r _{i}}{\sum _{i} m _{i}}$ Upar = "har crumb ke liye (mass × position) add karo." Neeche = "saari masses add karo" $= M$ . Divide karne se weighted sum ek weighted average position ban jaata hai — balance point.

4. Integral $\int \dots d m$ — infinitely many crumbs wala sum

Ye tool kyun, sirf $\sum$ nahi? Ek solid rocket mein koi natural "crumbs" nahi hote — mass continuously pheli hoti hai. Integral exact tool hai "continuously pheli cheez par koi quantity add karne" ke liye. Yeh sawal ka jawaab deta hai "infinitely many infinitely small pieces hone par position-weighted mass ka total kya hai?" — jo ek finite sum nahi kar sakta.

Picture: §1 ke crumbs ko dust tak shrink karo; §2 ki queue ek smooth flow ban jaati hai; $\int$ woh smooth flow ek bucket mein hai.

Toh continuous CG, $r_{cg} = \frac{1}{M} \int r d m,$ bilkul wahi weighted-average idea hai, ab smooth bodies ke liye.

Recall

$M$ se divide kyun karte hain, $\sum m_{i}$ se kyun nahi? Dono ek hi cheez hain — $M = \int d m = \sum_{i} m_{i}$ total mass hai. Usse divide karna hi average banata hai, sirf sum nahi.

5. Arrows ke do special multiplications

Inertia tensor do operations se bana hai. Dono pehle earn karne honge.

Ye tool kyun? Inertia ko parwah hai ki mass spin axis se kitni door hai — aur "door" matlab length, toh dot product distance-squared nikalne ka natural tarika hai.

Agle operation se pehle hume ek chhota sa symbol chahiye: transpose.

Picture: wahi teen-number list, ek baar khadi (column $r$ ), ek baar leti hui (row $r^{⊤}$ ).

Picture: $r \cdot r$ ek number hai (length-squared); $r r^{⊤}$ pairwise coordinate products ki poori $3 \times 3$ table hai.

Dono kyun aate hain: twisting resistance distance aur offset ki direction par depend karti hai. Dot product distance-squared part deta hai; outer product woh piece isolate karta hai jo axis ke saath hai (jo us axis ke baare mein twist ko resist nahi karta). Hum in dono ko subtract karne wale hain — wahan tensor, aur uske minus signs, janm lete hain.

6. Inertia tensor $I$ banana

Picture: hammer ko head se pakdo vs. handle ke end se. Same hammer, lekin door ka mass use "swing karne mein bhaari" feel karaata hai. Woh $d^{2}$ hi reason hai ki parent ke $I = \int (y^{2} + z^{2}) d m$ mein squares hain.

Ek number ki jagah poora tensor kyun chahiye: ek rocket ko pitch, yaw, ya roll — teen alag axes — ke baare mein twist kiya ja sakta hai — aur ek lopsided (asymmetric) body pitch-twist ko roll response mein spill bhi kar sakti hai. Ek number yeh capture nahi kar sakta; $3 \times 3$ grid kar sakta hai. Woh grid mass ka rotational version hai, aur hum ise $I$ likhte hain.

Grid ko term by term assemble karna

Ab hum §5 ke dono products combine karte hain. Pehle $1$ chahiye:

Ab, ek crumb $d m$ ke liye jo position $r = (x, y, z)$ par hai, yeh quantity banao: $[(r \cdot r) 1 - r r^{⊤}] d m .$

Yeh KYA hai: "har diagonal slot par distance-squared, minus §5 ka coordinate-products ka grid." Likha jaaye toh pehla piece $(x^{2} + y^{2} + z^{2})$ hai har diagonal par; outer-product grid subtract karne par milta hai:

$(r \cdot r) 1 - r r^{⊤} = y^{2} + z^{2} - x y - x z - x y x^{2} + z^{2} - y z - x z - y z x^{2} + y^{2} .$

Diagonal ek square kyun khota hai (top-left $y^{2} + z^{2}$ hai, $x^{2} + y^{2} + z^{2}$ nahi): $x$ -axis diagonal par, dot product ne teeno squares diye, lekin outer product ki apni diagonal entry $x^{2}$ subtract ho jaati hai. Yeh deliberate hai — mass jo $x$ -axis ke saath baitha hai woh us axis se zero distance par hai, isliye $x$ -twisting-laziness mein count nahi hona chahiye. Subtraction exactly wahi hai jo along-axis part hataati hai, sirf do perpendicular distances $y^{2} + z^{2}$ bachti hain. Dono products introduce karne ki poori wajah yehi hai.

Off-diagonals mein minus sign kahan se aata hai: diagonal (dot-product) part off-diagonal par kuch contribute nahi karta — $1$ diagonal ke bahar zero hai. Toh har off-diagonal entry sirf subtracted outer product se aati hai, aur outer product ki $(x, y)$ entry $+ x y$ hai. Use subtract karne par $- x y$ milta hai. Minus isliye convention nahi jo hum choose karte hain — yeh " $(\cdot) - (\cdot)$ " structure se forced hai. Saare crumbs par sum karne par:

Recall

$- \int x y d m$ ka sign sirf ek bookkeeping choice kyun nahi hai? Kyunki tensor defined hai (dot-product diagonal) minus (outer product) ke roop mein. Diagonal ke bahar sirf outer product bachta hai, aur woh $+ x y$ hai; built-in minus use $- x y$ kar deta hai. Sign ko haath se badlo aur $I$ correct twisting response nahi dega.

7. CG-se-distance aur offset $d$

Parallel-axis theorem ko ek aur symbol chahiye.

Picture: CG ek dot hai; jis axis ke baare mein tum actually spin karte ho woh ek aur dot hai; $d$ chhota arrow hai jo unhe jodhta hai, $D$ uski length.

Topic ko ye kyun chahiye: ek flying rocket mein har part (dry structure, propellant) apna inertia apne centre ke baare mein jaanta hai, lekin sab kuch vehicle ke CG ke baare mein add karna hota hai. Offset $d$ (aur $M D^{2}$ correction) bridge hai — aur kyunki CG fuel jalne par move karta hai, $d$ aur $D$ har instant badlte hain. Yahi poore topic ki dhadkan hai.

8. Time-dependence: chhota sa $(t)$

Topic ko ye kyun chahiye: notation ka yeh ek tukda hi reason hai ki note exist karta hai. Ek textbook static body mein koi $(t)$ nahi hota; ek rocket mein almost sab par $(t)$ hota hai, isliye autopilot gains aur thrust vectoring ko mid-flight re-tune karna padta hai.

Prerequisite map

Har box ek tool hai jo ab tumhare paas hai; arrows dikhate hain ki ye parent topic ki live-in-flight mass properties mein kaise stack hote hain. Related downstream ideas — Tsiolkovsky Rocket Equation, Propellant Slosh Dynamics — yahi foundation reuse karte hain.

Equipment checklist

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Bold symbol

r

ka matlab kya hai, aur

x

se kaise alag hai?

r

ek poora arrow hai (direction + length) origin se ek point tak;

x

sirf ek number hai, uska length-wise coordinate.

Teen body axes

x, y, z

ke naam batao aur unke roles.

x

length ke saath (roll),

y

side bahar (pitch),

z

upar/neeche (yaw) — mutually perpendicular, ek orthonormal frame.

d m

kya hai?

Ek infinitely small mass ka crumb — "mass ka ek chhota sa hissa" — integrals ke andar use hota hai.

\sum_{i} m_{i} r_{i}

ko plain words mein padhno.

Har crumb ke liye, uski mass ko uski position se multiply karo, phir saare woh products add karo.

CG ek mass-weighted average kyun hai, plain average kyun nahi?

Kyunki bhaari crumbs balance point ko zyada kheenchna chahiye; position ko mass se multiply karna bhaari parts ko zyada vote deta hai.

\sum_{i}

kab

\int d m

ban jaata hai, aur kyun?

Jab mass continuously smeared ho aur koi countable crumbs na hon; integral infinitely many infinitesimal pieces par sum karne ka exact tool hai.

Transpose

r^{⊤}

kya karta hai?

Teen numbers ki ek khadi column ko wahi numbers ki ek leti hui row mein tipa deta hai, taaki column times row ek grid banaye.

r \cdot r

kya compute karta hai, aur inertia ko yeh kyun chahiye?

Arrow ki length ka square; inertia axis-se-distance par depend karta hai, toh distance-squared exactly wahi hai jo chahiye.

Inertia tensor

I

ka poora formula kya hai?

I = \int [(r \cdot r) 1 - r r^{⊤}] d m

— dot-product diagonal minus outer-product grid.

Diagonal term

I_{xx} = \int (y^{2} + z^{2}) d m

kyun ban jaata hai,

\int (x^{2} + y^{2} + z^{2}) d m

kyun nahi?

Outer product apna

x^{2}

subtract kar deta hai, along-axis part hata deta hai; sirf do perpendicular distances bachti hain.

I_{x y} = - \int x y d m

mein minus sign kahan se aata hai?

Identity part off-diagonal par zero hota hai, toh wahan sirf subtracted outer product bachta hai (

+ x y

become

- x y

ho jaata hai); yeh forced hai, chosen nahi.

Offset

d

(length

D

) kya represent karta hai?

Woh arrow jo ek body ke apne CG se common point (vehicle CG) tak jaata hai jiske baare mein saari inertias add karni hain.

Trailing

(t)

kya signify karta hai, aur yahan central kyun hai?

Quantity time ke saath badlti hai; ek rocket mein

m_{p}

x_{cg}

aur

I

sab par

(t)

lagta hai kyunki propellant har second leak ho raha hai.

Door ka mass twisting ko zyada kyun resist karta hai near mass se?

Moment of inertia distance squared ke saath badhta hai, toh axis se door mass disproportionately zyada contribute karta hai.