Coupled Oscillators

A coupled oscillator system is one of the most fundamental models in classical mechanics. Take two masses connected by springs to each other and to fixed walls, and you get a surprisingly rich set of behaviours: energy sloshes back and forth between the masses, new collective frequencies emerge, and the system can oscillate in coordinated patterns called normal modes.

The Setup

Consider two masses $m_0$ and $m_1$ connected by three springs with spring constants $k_0$ , $k_1$ , and $k_2$ :

\text{wall} \;—\!\!\overset{k_0}{=\!=\!=}\!\!—\; m_0 \;—\!\!\overset{k_1}{=\!=\!=}\!\!—\; m_1 \;—\!\!\overset{k_2}{=\!=\!=}\!\!—\; \text{wall}

Let $x_0$ and $x_1$ denote the displacement of each mass from its equilibrium position. The spring $k_1$ couples the two masses together --- without it, they would oscillate independently.

Equations of Motion

Applying Newton's second law to each mass:

m_0 \, \ddot{x}_0 = -k_0 \, x_0 - k_1 (x_0 - x_1)

m_1 \, \ddot{x}_1 = -k_2 \, x_1 + k_1 (x_0 - x_1)

The first mass feels a restoring force from its wall spring ( $k_0$ ) and from the coupling spring ( $k_1$ ) which depends on the relative displacement between the two masses. The second mass similarly feels its wall spring ( $k_2$ ) and the coupling spring, but with the opposite sign.

Normal Modes

A normal mode is a pattern of motion in which all parts of the system oscillate at the same frequency. For two coupled oscillators there are exactly two normal modes:

Symmetric (in-phase) mode --- both masses move in the same direction at the same time. The coupling spring is neither stretched nor compressed, so only the wall springs matter.
Antisymmetric (out-of-phase) mode --- the masses move in opposite directions. The coupling spring is maximally stretched or compressed, raising the effective stiffness and hence the frequency.

For the special case of equal masses ( $m_0 = m_1 = m$ ) and equal wall springs ( $k_0 = k_2 = k$ ), the normal-mode frequencies are:

\omega_{\text{sym}} = \sqrt{\frac{k}{m}}, \qquad \omega_{\text{anti}} = \sqrt{\frac{k + 2k_1}{m}}

Beats and Energy Transfer

When you displace only one mass and release the system, neither normal mode is excited in isolation --- instead, both are superimposed. Because the two modes have slightly different frequencies, they go in and out of phase over time, producing beats. You will see the first mass gradually come to rest while the second mass picks up all the energy, and then the process reverses. The beat frequency is:

\omega_{\text{beat}} = |\omega_{\text{anti}} - \omega_{\text{sym}}|

A weak coupling spring ( $k_1 \ll k$ ) produces slow beats with a long energy-transfer period, while a strong coupling spring leads to rapid exchange.

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Mass 1 (m₀)1

Mass 2 (m₁)1.50

Spring k₀1

Spring k₁ (coupling)0.50

Spring k₂1

Initial x₀1

Initial x₁0

Things to Try

Classic beats --- Use the defaults ( $m_0 = 1$ , $m_1 = 1.5$ , $k_1 = 0.5$ ). Only Mass 1 is displaced. Watch energy flow back and forth between the two masses.
Equal masses --- Set both masses to 1.0 and both wall springs to 1.0. The beat pattern becomes perfectly periodic, and the energy transfer is complete (one mass stops entirely at the beat minimum).
Strong coupling --- Increase $k_1$ to 3 or higher. The beat frequency rises and the energy exchange happens much faster.
Weak coupling --- Lower $k_1$ toward 0.1. The beats slow down dramatically, and the masses oscillate almost independently.
In-phase mode --- Set both initial displacements to 1.0 (equal sign and magnitude). If the masses and wall springs are equal, the coupling spring is never stretched and both masses oscillate at the same frequency forever.
Out-of-phase mode --- Set the initial displacements to 1.0 and $-1.0$ . The coupling spring is maximally active, and the system oscillates at the higher antisymmetric frequency.
Asymmetric springs --- Try $k_0 = 1$ , $k_2 = 3$ . The two normal modes are no longer simple symmetric/antisymmetric patterns, and the energy transfer is incomplete.

Why It Matters

Coupled oscillators appear throughout physics. Atoms in a crystal lattice behave as coupled oscillators, giving rise to phonons --- the quantum of lattice vibration. Molecular vibrations in chemistry, the response of coupled electrical circuits, and even the synchronisation of pendulum clocks on a shared beam are all instances of the same underlying mathematics. Understanding two coupled oscillators is the first step toward understanding waves, normal modes of continuous systems, and ultimately quantum field theory.

A note on the numerics --- This simulation uses Euler integration with a fixed step of $dt = 0.01$ over 5 000 steps (50 seconds of simulated time). Euler's method is not energy-conserving, so you may notice a slow drift in amplitude over long runs. For more accurate results, a symplectic integrator (such as Verlet or leapfrog) would be preferable.