Coupled Oscillators

The setup

Two identical masses m on identical springs of stiffness k, joined by a softer coupling spring of stiffness k_c. Let x₁ and x₂ be the displacements from each mass's own equilibrium. Newton's law for each mass:

mẍ₁ = −k x₁ − k_c (x₁ − x₂)
mẍ₂ = −k x₂ − k_c (x₂ − x₁)

The system is linear, so the matrix form M ẍ + K x = 0 has solutions of the form x(t) = v · cos(ωt + φ) where (K − ω²M) v = 0. The eigenvalues give the squared mode frequencies; the eigenvectors give the mode shapes.

The two normal modes

Adding and subtracting the equations of motion isolates two new coordinates that decouple:

q₊ = (x₁ + x₂)/2  →  q̈₊ = −(k/m) q₊                       ω₊ = √(k/m)
q₋ = (x₁ − x₂)/2  →  q̈₋ = −((k + 2k_c)/m) q₋               ω₋ = √((k + 2k_c)/m)

The symmetric mode q₊ moves both bobs together at the natural frequency ω₀ — the coupling spring is never stretched, so it does not enter the frequency. The antisymmetric mode q₋ moves them opposite, stretching the coupling spring by 2·q₋ and raising the effective stiffness.

Beats — energy ping-pongs between the bobs

If the left bob is released from amplitude A while the right is held at rest, the initial state decomposes as q₊(0) = q₋(0) = A/2. The subsequent motion is:

x₁(t) = (A/2)[cos(ω₊ t) + cos(ω₋ t)] = A cos((ω₋−ω₊)/2 · t) · cos((ω₋+ω₊)/2 · t)
x₂(t) = (A/2)[cos(ω₊ t) − cos(ω₋ t)] = A sin((ω₋−ω₊)/2 · t) · sin((ω₋+ω₊)/2 · t)

The fast oscillation is at the average frequency (ω₋+ω₊)/2 ≈ ω₀; the slow envelope is at the half-difference (ω₋−ω₊)/2. Energy thus transfers entirely from bob 1 to bob 2 in time T_beat / 2 = π / (ω₋ − ω₊).

Worked example — Δm = 5 g pendulums, k_c sliding the coupling thread

Two 5 g pendulums of length 0.30 m give ω₀ = √(g/L) = 5.72 rad/s, period T₀ = 1.099 s. Connect them with a horizontal thread giving coupling stiffness k_c = 0.05 · k. Then:

Coupling fraction k_c/k	ω₋/ω₊ − 1	T_beat	Beats per minute
0.01 (very weak)	1.005 % (0.01)	1.83 min	0.55
0.05 (weak)	2.47 %	22.2 s	2.7
0.20 (medium)	9.54 %	5.76 s	10.4
1.00 (strong)	41.4 %	1.33 s	45.1

Lab demonstrations use the medium range — a beat every few seconds is fast enough to see and slow enough to count.

Variants

• Three or more pendulums. Add masses and you add modes. Three identical pendulums coupled identically have modes (+, +, +), (+, 0, −), and (+, −2, +) — wavelike shapes. N pendulums give N modes at frequencies ω_n = √(ω₀² + 4ω_c² sin²(nπ/(2(N+1)))).
• Continuous limit — a string. N → ∞ with finite total length gives the wave equation; normal modes become sinusoidal standing waves with wavelengths 2L/n. The discrete-to-continuum transition is a tidy textbook proof that strings are just lots of coupled pendulums.
• Wilberforce pendulum. A mass on a vertical spring that also twists. The translational mode and the torsional mode have slightly different frequencies; the spring's helical geometry couples them. Energy moves between bouncing and spinning in a few seconds — a classroom favorite.
• LC ladder filter. Repeated LC sections couple electrically just like coupled pendulums couple mechanically — same eigenvalue problem, same dispersion ω(k). The cutoff frequency above which no mode exists is the basis of analog lowpass filters and acoustic waveguides.
• Diatomic molecules. Two atoms joined by a chemical bond modeled as a spring. The center-of-mass mode is uncoupled translation (a free particle); the internal stretch mode has ω = √(k/μ) with the reduced mass μ = m₁m₂/(m₁+m₂). Infrared absorption frequencies in CO and HCl come straight from this formula.
• Huygens synchronization. Two pendulum clocks on the same wall couple through the wall's microvibrations. Huygens observed in 1665 that they slowly phase-lock to antisymmetric motion. Nonlinear damping plus weak coupling generally selects one mode preferentially — a precursor to Kuramoto's synchronization model for laser arrays, fireflies, and neurons.

JavaScript — coupled simulation

// Analytic solution for two identical coupled masses
function coupledAnalytic(A, k, kc, m, t) {
  const w_plus  = Math.sqrt(k / m);
  const w_minus = Math.sqrt((k + 2 * kc) / m);
  const w_fast  = (w_minus + w_plus) / 2;
  const w_beat  = (w_minus - w_plus) / 2;
  return {
    x1: A * Math.cos(w_beat * t) * Math.cos(w_fast * t),
    x2: A * Math.sin(w_beat * t) * Math.sin(w_fast * t),
    w_plus, w_minus,
    T_beat: 2 * Math.PI / (w_minus - w_plus),
  };
}

// Beat period for a textbook demo
const demo = coupledAnalytic(0.05, 10, 0.5, 0.005, 0);
console.log(`T_beat = ${demo.T_beat.toFixed(2)} s`);  // ~13.2 s
console.log(`Modes: ${demo.w_plus.toFixed(2)} and ${demo.w_minus.toFixed(2)} rad/s`);

// Velocity Verlet integrator for arbitrary coupled chains
function stepChain(x, v, k, kc, m, dt) {
  const N = x.length;
  const a = new Array(N);
  for (let i = 0; i < N; i++) {
    a[i] = -k * x[i] / m;
    if (i > 0)     a[i] -= kc * (x[i] - x[i-1]) / m;
    if (i < N - 1) a[i] -= kc * (x[i] - x[i+1]) / m;
  }
  for (let i = 0; i < N; i++) {
    v[i] += a[i] * dt;
    x[i] += v[i] * dt;
  }
}

// Eigenmodes of an N-mass chain — analytic formula
function chainModes(N, k, kc, m) {
  const modes = [];
  for (let n = 1; n <= N; n++) {
    const w_n = Math.sqrt((k + 2 * kc * (1 - Math.cos(n * Math.PI / (N + 1)))) / m);
    modes.push({ n, omega: w_n, shape: Array.from({length: N}, (_, i) =>
      Math.sin((i + 1) * n * Math.PI / (N + 1))) });
  }
  return modes;
}

Where coupled oscillators show up

• Molecular vibrations. IR spectroscopy frequencies are normal-mode frequencies of polyatomic molecules — CO₂ at 667 and 2349 cm⁻¹, water at 1595 / 3657 / 3756 cm⁻¹.
• Lattice phonons. Atoms in a crystal are a 3D coupled-oscillator network. Their normal modes (phonons) carry sound, conduct heat, and dominate low-temperature heat capacity (Debye model).
• Coupled pendulum clocks. Huygens 1665, and a 350-year tradition since: synchronization phenomena now span neurons, fireflies, lasers, and power grids.
• Tuned mass dampers. Taipei 101's 660-tonne ball is a deliberate coupled oscillator with the building itself, tuned to drain energy from the tower's lowest sway mode.
• Coupled-cavity lasers. Two optical cavities sharing a mirror form coupled modes; tuning the coupling stabilizes mode-locked operation in modern frequency combs.
• Power-grid synchronization. Hundreds of generators across a continent behave as coupled oscillators on a network. Stable operation requires all phases to track within a narrow range — failures cascade as modes go unstable.
• Quantum bits. Two superconducting qubits coupled through a resonator are described by exactly this Hamiltonian. Gate fidelity depends on tuning the coupling and timing the swap of a single energy quantum.

Common mistakes

• Calling the individual-bob motion a "mode." Single-bob oscillation is a superposition of two modes. Modes are the orthogonal motions that diagonalize the equations.
• Assuming weak coupling means no effect. Weak coupling does not change frequencies much, but it sets the beat period — full energy still transfers, just slowly.
• Forgetting the factor of 2 in the antisymmetric frequency. ω₋² = (k + 2k_c)/m, not (k + k_c)/m. The coupling spring stretches by 2·q₋ in the antisymmetric motion.
• Mixing up beat frequency and beat period. Beat angular frequency is Δω = ω₋ − ω₊. The envelope half-cycle is π/Δω; one full envelope cycle (left → right → left) is 2π/Δω.
• Using unequal masses with the same mode equations. For different masses, you diagonalize K − ω²M with a non-identity M — eigenvectors are no longer pure (+, +) and (+, −).
• Confusing parametric resonance with coupling. The Wilberforce demo is true coupling; a child pumping a swing is parametric driving — different physics with similar visual effect.

Performance notes — modal vs direct integration

For N coupled oscillators, modal analysis is O(N³) to find eigenmodes once, then O(N) per timestep per mode — so O(N²) per timestep total once decomposed. Direct integration of the coupled system is also O(N²) per timestep if the coupling matrix is dense, but O(N) for nearest-neighbor coupling. Engineers typically use modal analysis when (a) only the lowest few modes are excited (truncate at, e.g., 20 modes for a thousand-DOF building), or (b) repeated solves with different forcings are needed. For impulsive or nonlinear problems, direct integration wins because high modes matter and superposition fails.