Introduction
Coupled oscillators exchange energy. Normal modes are collective motions where all parts oscillate at same frequency with fixed phase relation. Analyzing normal modes simplifies complex coupled systems and is fundamental to solid-state physics and molecular vibrations.
Two Coupled Masses on String
Two masses m connected by springs k to walls and k_c between them. Equations: máº1 = -kx1 - k_c(x1-x2), máº2 = -kx2 - k_c(x2-x1). Coupling creates energy exchange between masses.
Normal Modes
Symmetric mode (both masses move together): x1 = x2 = A cos(ω1t), ω1 = √(k/m). Antisymmetric mode (opposite motion): x1 = -x2 = A cos(ω2t), ω2 = √((k+2k_c)/m).
General Solution
Arbitrary motion is superposition: x1(t) = A1cos(ω1t + φ1) + A2cos(ω2t + φ2), x2(t) = A1cos(ω1t + φ1) - A2cos(ω2t + φ2). Constants determined by initial conditions. Energy transfers between masses at beat frequency (ω2-ω1)/2.
\nIntroduction
Coupled oscillators exchange energy. Normal modes are collective motions where all parts oscillate at same frequency with fixed phase relation. Analyzing normal modes simplifies complex coupled systems and is fundamental to solid-state physics and molecular vibrations.
Two Coupled Masses on String
Two masses m connected by springs k to walls and k_c between them. Equations: máº1 = -kx1 - k_c(x1-x2), máº2 = -kx2 - k_c(x2-x1). Coupling creates energy exchange between masses.
Normal Modes
Symmetric mode (both masses move together): x1 = x2 = A cos(ω1t), ω1 = √(k/m). Antisymmetric mode (opposite motion): x1 = -x2 = A cos(ω2t), ω2 = √((k+2k_c)/m).
General Solution
Arbitrary motion is superposition: x1(t) = A1cos(ω1t + φ1) + A2cos(ω2t + φ2), x2(t) = A1cos(ω1t + φ1) - A2cos(ω2t + φ2). Constants determined by initial conditions. Energy transfers between masses at beat frequency (ω2-ω1)/2.
\nGeneral Theory
N coupled oscillators have N normal modes. Each mode has characteristic frequency ωᵢ and pattern of displacements (eigenvector). Total motion is sum of modes. Matrix formulation: MẠ+ Kx = 0 becomes eigenvalue problem. Important for structural engineering, acoustics, and quantum mechanics.
Applications
Applications: molecular vibrations (normal modes are vibrational spectra), solid-state physics (phonons are lattice normal modes), bridge and building dynamics, coupled pendulums, musical instruments (coupled strings, air columns), electrical circuits (coupled LC oscillators).
\nSolved Example: Coupled Oscillator Frequencies
Two 1 kg masses connected: each to wall by springs k = 10 N/m, and to each other by coupling spring k_c = 5 N/m. Find normal mode frequencies and beat period. Solution: ω1 = √(k/m) = √(10/1) = √10 = 3.16 rad/s. ω2 = √((k+2k_c)/m) = √((10+10)/1) = √20 = 4.47 rad/s. Frequencies: f1 = 3.16/(2π) = 0.503 Hz, f2 = 4.47/(2π) = 0.712 Hz. Beat frequency: f_beat = (ω2-ω1)/(2π) = (4.47-3.16)/(2π) = 0.209 Hz. Beat period T_beat = 1/f_beat = 4.78 s. This means if mass 1 is initially displaced and released (mass 2 at rest), energy will transfer completely to mass 2 after half beat period (2.39 s), then back to mass 1 after full beat period (4.78 s), continuing to exchange. The system exhibits this energy sloshing between masses due to coupling.
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