05 / CHAPTER 06
Coupled oscillators
Find collective modes in a small system of interacting oscillators.
ANCHOR EQUATION
Normal modes
In symbols: x(t) = A cos(ωt + φ)
A coupled system can be decomposed into simple collective motions.
Coupled oscillators share collective modes
When oscillators interact, they do not generally move independently; they can organize into normal modes in which every coordinate oscillates at one frequency with fixed amplitude ratios.
Diagonalize the two-mass equations
For two identical masses m joined by springs k, the equations are m x₁¨ = −2kx₁ + kx₂ and m x₂¨ = kx₁ −2kx₂. Try x_i = A_i cosωt. The in-phase choice A₁ = A₂ gives ω₁² = k/m; the out-of-phase choice A₁ = −A₂ gives ω₂² = 3k/m. These vectors are eigenmodes of the coupling matrix.
With m = 1 kg and k = 4 N/m, ω₁ = 2 rad/s and ω₂ = √12 = 3.46 rad/s. An arbitrary initial displacement is a sum of these two patterns, so the motion can beat.
Modes depend on boundaries and symmetry
Unequal masses or spring constants change both frequencies and amplitude ratios. The power of the method remains: find collective eigenvectors rather than forcing each coordinate into one sinusoid.
Work it: For two equal masses with k = 9 N/m, calculate both mode frequencies and sketch the displacement pattern of each. Which mode has the larger spring extension?