Coupled spring-mass systems
A chain of masses connected by springs. Each mass pulls its neighbors, energy sloshes back and forth, and hidden in the chaos are the normal modes — collective oscillation patterns where every mass moves at the same frequency.
m·ẍi = k(xi+1 − 2xi + xi−1) − b·ẋi
Normal modes
A system of N coupled masses has exactly N normal modes — patterns of collective vibration where every mass oscillates at the same frequency (though with different amplitudes). In a normal mode, the entire system moves as one. Any initial condition can be decomposed as a superposition of normal modes, and the time evolution is just the independent evolution of each mode.
The eigenvalue problem
Finding the normal modes is equivalent to finding the eigenvalues and eigenvectors of the stiffness matrix. For N masses on springs with fixed walls, the eigenfrequencies are ωn = 2√(k/m) sin(nπ / 2(N+1)) for n = 1, 2, ..., N. The lowest mode is the fundamental: all masses move together. The highest mode has adjacent masses moving in opposite directions.
Energy transfer and beats
When you displace a single mass, you excite a superposition of all normal modes. The energy appears to “slosh” back and forth between masses — a phenomenon called beating. The beat frequency equals the difference between adjacent normal mode frequencies. With more masses and more modes, the energy redistribution becomes increasingly complex.
From discrete to continuous
Take the limit: infinite masses, infinitesimal spacing, and the coupled spring-mass chain becomes a continuous vibrating string governed by the wave equation. The normal modes become the sinusoidal standing waves familiar from musical instruments. Every vibrating string is, at the atomic level, a coupled spring-mass system.