Coupled Oscillators: Normal Modes and the Physics of Beating

Two springs, two masses, one shared story

Take two identical masses on springs, each anchored to a wall, then connect the two masses with a third, weaker spring. Each mass alone would swing at its own natural frequency, the square root of k over m. Couple them, and something new appears: the system no longer has one natural frequency - it has exactly two, called normal modes, and every possible motion of the two masses is some combination of just those two patterns.

The equations of motion

With spring constants k1, k2 for the outer springs, coupling spring k_c between the masses, and equal masses m, Newton's second law for each mass gives a pair of coupled linear ODEs:

m·x1'' = -k1·x1 - k_c·(x1 - x2)
m·x2'' = -k2·x2 - k_c·(x2 - x1)

symmetric mode:      x1 = x2      → ω_sym  = √(k/m)              (coupling spring never stretches)
antisymmetric mode:  x1 = -x2     → ω_anti = √((k + 2·k_c)/m)     (coupling spring fights both masses)

In the symmetric mode both masses move in lockstep, in the same direction with the same amplitude, so the coupling spring's length never changes and it contributes nothing to the restoring force - the pair oscillates at exactly the single-mass frequency. In the antisymmetric mode the masses move oppositely, so the coupling spring stretches and compresses twice as hard as it would for one mass alone, raising the frequency. These two frequencies, and only these two, are the system's normal modes: initial conditions that are pure combinations of x1=x2 or x1=-x2 stay purely sinusoidal forever, at a single frequency, with no beating.

Beating: what happens for any other start

Almost any real starting condition - say, pulling only one mass aside and releasing both from rest - is not a pure normal mode. It is a superposition of both, and the general solution combines both mode frequencies using the product-to-sum trigonometric identity:

x1(t) = A·cos(ω_sym·t) + A·cos(ω_anti·t)
x2(t) = A·cos(ω_sym·t) - A·cos(ω_anti·t)

using the product-to-sum identity:
x1(t) = 2A·cos((ω_anti-ω_sym)/2 · t) · cos((ω_anti+ω_sym)/2 · t)

That last line is the signature of beating: a fast oscillation at the average frequency, with its amplitude slowly modulated by a much slower envelope at the difference frequency. Because the coupling spring k_c is typically weak, the two mode frequencies are close together, so the envelope period is long compared to the oscillation period - you hear (or see) rapid wiggling whose overall size swells and shrinks slowly. Energy is being traded back and forth between the two masses at the beat frequency: when mass 1's amplitude is near maximum, mass 2's is near zero, and a beat period later the roles swap.

Driving off-resonance

Add a periodic driving force to one mass at frequency omega_d, and the steady-state response has resonant peaks near both normal-mode frequencies - a coupled two-mass system has two resonances where a single oscillator has one. Drive exactly between them and the response is a mixture of both normal modes excited simultaneously, again producing beating, but now sustained indefinitely by the external drive rather than decaying with any single mode's phase relationship. Drive close to one normal mode and that mode dominates the response, with the other barely excited - a direct way to isolate and measure a single mode experimentally.

Why this generalises to almost everything

Two masses is the simplest nontrivial case of a much bigger idea: any linear system of N coupled oscillators has exactly N normal modes, each an independent, non-beating solution, and any motion of the system is some superposition of them. Add a third mass and you get three modes; take the limit of infinitely many, infinitesimally coupled masses and the normal modes become the standing waves of a continuous string or the resonances of a room. Molecular vibrations, coupled pendulum clocks (famously synchronising via a shared support, as Huygens noticed in 1665), and the vibrational spectra used in infrared spectroscopy are all instances of the same normal-mode mathematics that starts with two masses and two springs.

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