Normal modes — coupled oscillators, normal coordinates

2.1.20 · Physics › Analytical Mechanics

1. Setup (KYA solve kar rahe hain)

Classic case lo: do equal masses $m$ jo walls se aur ek doosre se springs se jude hain.

Figure — Normal modes — coupled oscillators, normal coordinates

Bahari springs ka constant $k$ ho aur coupling spring $k_{c}$ . Equilibrium se displacements $x_{1}, x_{2}$ .

2. Equations of motion derive karna (KAISE, scratch se)

Energy kyun use karein? Forces mein signs bahut jaldi confusing ho jaati hain. Lagrangian mechanics mein sign error kabhi nahi hoti.

Kinetic energy: $T = \frac{1}{2} m \overset{x}{˙}_{1}^{2} + \frac{1}{2} m \overset{x}{˙}_{2}^{2}$

Potential energy — ye step kyun? har spring $\frac{1}{2} k (stretch)^{2}$ store karta hai. Beech wale spring ki stretch $(x_{2} - x_{1})$ hai: $V = \frac{1}{2} k x_{1}^{2} + \frac{1}{2} k x_{2}^{2} + \frac{1}{2} k_{c} (x_{2} - x_{1})^{2}$

Lagrangian $L = T - V$ . Euler–Lagrange apply karo $\frac{d}{d t} \frac{\partial L}{\partial x ˙ _{i}} = \frac{\partial L}{\partial x _{i}}$ :

$m \overset{x}{¨}_{1} = - k x_{1} + k_{c} (x_{2} - x_{1})$ $m \overset{x}{¨}_{2} = - k x_{2} - k_{c} (x_{2} - x_{1})$

Ye step kyun? $\partial V / \partial x_{1}$ mein wall spring ( $k x_{1}$ ) aur coupling term dono hain, $(x_{2} - x_{1})^{2}$ se chain-rule wala sign lekar. $k_{c}$ term dono equations mein opposite sign ke saath appear hota hai — yahi coupling hai.

3. Trick: single-frequency motion try karo (Forecast-then-Verify)

Forecast: agar normal modes exist karte hain, to kuch aisi special motions hoti hain jahan dono masses ek hi frequency $ω$ par fixed proportion mein oscillate karte hain. Guess karo: $x_{j} (t) = A_{j} cos (ω t + ϕ)$

Substitute karo. Har $\overset{x}{¨}_{j} = - ω^{2} x_{j}$ . Cosine divide karne par algebraic equations milti hain:

$- m ω^{2} A_{1} = - (k + k_{c}) A_{1} + k_{c} A_{2}$ $- m ω^{2} A_{2} = + k_{c} A_{1} - (k + k_{c}) A_{2}$

Matrix form mein $(K - ω^{2} M) A = 0$ :

$(k + k_{c} - m ω^{2} - k_{c} - k_{c} k + k_{c} - m ω^{2}) (A_{1} A_{2}) = 0$

Expand karo: $(k + k_{c} - m ω^{2})^{2} - k_{c}^{2} = 0 ⟹ k + k_{c} - m ω^{2} = \pm k_{c}$

To: $ω_{1}^{2} = \frac{k}{m} ω_{2}^{2} = \frac{k + 2 k _{c}}{m}$

4. Mode shapes dhundhna (eigenvectors)

Har $ω$ ko wapas plug karo aur ratio $A_{2} / A_{1}$ nikalo.

Mode 1 ( $ω_{1}^{2} = k / m$ ): bracket $k + k_{c} - m ω_{1}^{2} = k_{c}$ hai, to $k_{c} A_{1} - k_{c} A_{2} = 0 \Rightarrow A_{2} = A_{1}$ . → In-phase / symmetric: masses saath chalte hain, coupling spring kabhi stretch nahi hoti, isliye $k_{c}$ feel nahi hoti → frequency sirf $k / m$ hai.

Mode 2 ( $ω_{2}^{2} = (k + 2 k_{c}) / m$ ): bracket $= - k_{c}$ hai, to $A_{2} = - A_{1}$ . → Anti-phase / antisymmetric: masses opposite direction mein chalte hain, coupling spring maximum work hoti hai → zyada stiff → higher frequency.

5. Normal coordinates (punchline)

Naye variables define karo jo equations ko decouple karte hain: $η_{1} = \frac{1}{2} (x_{1} + x_{2}) η_{2} = \frac{1}{2} (x_{1} - x_{2})$

Ye kyun? Ye symmetric aur antisymmetric combinations hain — exactly eigenvectors. Dono EOMs ko add aur subtract karo:

Add karo: $m (\overset{x}{¨}_{1} + \overset{x}{¨}_{2}) = - k (x_{1} + x_{2}) \Rightarrow \overset{η}{¨}_{1} = - ω_{1}^{2} η_{1}$

Subtract karo: $m (\overset{x}{¨}_{1} - \overset{x}{¨}_{2}) = - (k + 2 k_{c}) (x_{1} - x_{2}) \Rightarrow \overset{η}{¨}_{2} = - ω_{2}^{2} η_{2}$

6. Worked examples

7. Common mistakes

Recall Feynman: ek 12 saal ke bachche ko explain karo

Do swings imagine karo jo rubber band se bande hain. Agar tum unhe ek hi direction mein ek saath push karo, rubber band saath chal leta hai — wo relaxed hai, aur wo apni normal slow speed se swing karte hain. Agar tum unhe opposite direction mein push karo, rubber band bar bar stretch aur snap back hota rehta hai, unse fight karta hai, to wo zyada tez swing karte hain. Ye do special tarike "normal modes" hain. Koi bhi aur crazy swinging jo tum shuru karte ho wo actually in do special taridon ka mix hai jo ek saath ho raha hai. In special taridon ko pakad ke, confusing motion do simple motions ban jaati hai.

Flashcards

Normal mode kya define karta hai?

Ek collective motion jisme har coordinate ek single common frequency par fixed amplitude ratio mein oscillate karta hai (

K - ω^{2} M

ka eigenvector).

Normal-mode frequencies konsi equation deti hai?

Secular equation

det (K - ω^{2} M) = 0

Do equal masses ke liye bahari

k

, coupling

k_{c}

ke saath do frequencies kya hain?

ω_{1}^{2} = k / m

(symmetric) aur

ω_{2}^{2} = (k + 2 k_{c}) / m

(antisymmetric).

Symmetric mode ki frequency kam kyun hoti hai?

Dono masses saath chalte hain isliye coupling spring kabhi stretch nahi hoti; sirf

k

restoring force deta hai.

Normal coordinate kya hota hai?

Original coordinates ka ek linear combination jo uncoupled SHM equation

\overset{η}{¨}_{a} = - ω_{a}^{2} η_{a}

follow karta hai.

Do equal masses ke liye normal coordinates?

η_{1, 2} = \frac{1}{2} (x_{1} \pm x_{2})

N-DOF system mein kitne normal modes hote hain?

Exactly N.

Coupled oscillators mein 'beats' kab aate hain?

Do nearly equal frequencies wale modes ke superposition se; energy masses ke beech slow envelope frequency

(ω_{2} - ω_{1}) /2

se transfer hoti hai.

Ek single pure mode excite kaise karte hain?

Initial conditions us mode ke eigenvector ke equal rakh do (jaise symmetric ke liye

x_{1} = x_{2}

Eigenvectors konsi orthogonality follow karte hain?

Mass-weighted:

A_{a}^{T} M A_{b} = 0

jab

a \neq = b

ho.

Connections

Lagrangian Mechanics — jisse humne EOMs derive kiye bina force-sign errors ke
Simple Harmonic Motion — har normal coordinate EK SHM hai
Eigenvalues and Eigenvectors — secular equation = generalized eigenvalue problem
Beats and Superposition — do close modes ka observable consequence
Small Oscillations — $V$ ko quadratic order tak Taylor-expand karne par $K$ milta hai
Phonons and Lattice Vibrations — solids mein coupled oscillators ka N→∞ limit