Foundations — Modal analysis — natural frequencies, mode shapes

3.6.10 · D1 · Physics › Spacecraft Structures & Systems Engineering › Modal analysis — natural frequencies, mode shapes

Pehle aap parent note Modal Analysis padh sakein, uske liye uski equations mein har ek squiggle ka matlab aapko pata hona chahiye. Yeh page har ek ko zero se build karta hai — pehle saadhe alfaaz, phir ek picture, phir kyun yeh topic us cheez ke bina jee nahi sakta. Upar se neeche padho; har block sirf usse upar defined cheezein use karta hai.

1. Ek "degree of freedom" (DOF)

Picture. Neeche do manke dekho. Kisi bhi pal poori picture jaanene ke liye aapko do number batane padte hain: manka 1 kitna hila aur manka 2 kitna hila. Yeh "kitne number" ki count DOF ki sankhya hai, yahan $2$.

Topic ko kyun chahiye. Ek asli spacecraft ek smooth continuous body hai jisme infinite points hain — infinitely many DOF. Hum infinity ke saath compute nahi kar sakte, isliye Finite element analysis (FEA) ise $n$ DOF ki ek finite sankhya mein kaatta hai. Sab kuch aage ($n$ frequencies, $n$ shapes) DOF mein gina jaata hai.

2. Displacement $x$ aur displacement vector $\{x\}$

Jab kai DOF hote hain toh hum sabke displacements ko ek lambe numbers ke column mein stack karte hain — ek vector — aur ise curly braces ke saath likhte hain:

Topic ko kyun chahiye. Newton ka law poore structure ke liye $\{x\}$ use karke ek baar likha jaayega, na ki $n$ alag alag scribbles mein.

3. Velocity $\dot{x}$ aur acceleration $\ddot{x}$ — dots

Picture. $x$ ka time ke against graph par, $\dot{x}$ curve ki steepness hai aur $\ddot{x}$ yeh hai ki woh steepness khud kitni tezi se badlati hai — curviness.

Topic ko kyun chahiye. Newton ka law hai $\text{force}=\text{mass}\times \text{acceleration}$, isliye acceleration $\ddot{x}$ zaroori hai. Sab accelerations stack karne par $\{\ddot{x}\}$ milta hai.

4. Stiffness $k$ aur spring force

Minus sign maayane rakhta hai. Force hamesha rest ki taraf point karta hai: daayein kheencho ($x>0$) toh baayein kheenchta hai (negative force). Yahi restoring push hai jo cheezein hilne par majboor karta hai drift away hone ki jagah.

Topic ko kyun chahiye. Spacecraft ke metal panels aur struts stiff springs ki tarah behave karte hain. Stiffness decide karti hai woh kitni tezi se hilna chahta hai — high stiffness natural frequency ko upar dhakelta hai, Launch vehicle load environments mein bataaye gaye dangerous launch frequencies se door.

5. Mass matrix $[M]$ aur stiffness matrix $[K]$

Jab springs kai masses ko connect karte hain, toh har mass ko apne springs aur padosiyon se forces milti hain. Har mass ke liye ek Newton equation likhne aur unhe stack karne par, numbers ke do grids saamne aate hain. Square brackets $[]$ flag karte hain ki matrix hai — ek grid, rows × columns.

Do-bead example ke liye jo parent note use karta hai:

Do words jinpar parent depend karta hai:

Topic ko kyun chahiye. Motion ka poora equation $[M]\{\ddot{x}\}+[K]\{x\}=\{0\}$ theek in do grids se bana hai. Symmetry baad mein hume Orthogonality of mode shapes ka khoobsoorat result muft mein de deta hai.

6. Frequency: $\omega$, $f$, aur wobble clock

Subscript $n$ ${\omega }_n$ mein bas natural ka matlab hai — "woh frequency jo structure akele choose karta hai jab chod diya jaaye," koi baahri push nahi.

Topic ko kyun chahiye. Modal analysis jis ek cheez ko compute karne ke liye exist karti hai woh yahi natural frequencies ki list hai, taaki hum launch ki forcing frequencies ko unse door rakh sakein.

7. Harmonic (sine) motion $\sin (\omega t+\theta )$

Topic ko kyun chahiye. Yahi exact time-shape assume karna woh trick hai jo differential equation ko ek saadha algebra problem (eigenvalue problem) mein badal deti hai, kyunki do baar differentiate kiya hua sine bas $-{\omega }^2$ times itself hai.

8. Mode shape $\{\varphi \}$

Picture. Do-bead system ke liye parent do shapes dhundhta hai:

• $\{{\varphi }_1\}=\left[\begin{array}{c}1\\ 1.618\end{array}\right]$ — dono beads ek hi taraf jhoolte hain (in-phase), bead 2 zyada door.
• $\{{\varphi }_2\}=\left[\begin{array}{c}1\\ -0.618\end{array}\right]$ — beads opposite taraf jhoolte hain (out-of-phase).

Topic ko kyun chahiye. Pair $({\omega }_i,\{{\varphi }_i\})$ hi mode $i$ hai. Shape jaanne se engineers ko pata chalta hai ki spacecraft ka kaunsa hissa sabse zyada hilta hai, taaki stable rehne ki zaroorat wale instruments ko calm points (nodes) par rakha ja sake.

9. Transpose $\{{\}}^T$ aur determinant $\det$

Derivation mein do operations aate hain; yahan batate hain har ek kya karta hai.

Topic ko kyun chahiye. Determinant ko zero set karna woh machine hai jo natural frequencies bahar nikalti hai; transpose woh tool hai jo baad mein equations ko simple one-DOF oscillators mein alag kar deta hai.

Prerequisite map

Equipment checklist

Daayein taraf chhupaao aur zyubaan par jawaab do; check karne ke liye reveal karo.