Coupled Spring-Mass

Differential Equations · Axiom Academy

Two masses, three springs, one wall on each side — displace them and something surprising happens: the whole system only ever moves in two clean patterns. Two equal masses m sit between two walls, linked by three identical springs (stiffness k ) — outer-left, middle (the coupling spring), outer-right. Pull them apart and let go: eigenvalues of the system tell you the two frequencies it's allowed to ring at. However you start it, this system settles into exactly two possible motion patterns — a normal mode. Pick one, hit Play, and watch the masses ring at that mode's own frequency. Any motion is a mix of the two modes Displace the masses any way you like, and the resulting motion is always just some blend of Mode 1 and Mode 2 running at once. Drag the mix and watch the pattern shift from clean to complex. Where the frequencies come from Assume a solution of the form x = v·cos( t) and the equations of motion collapse into an eigenvalue problem det(K − I) = 0. Sweep and watch it cross zero exactly at the two frequencies you already watched ring.

This is the written version of the interactive lesson above. See the full Differential Equations course.