Spring-Mass Systems

Differential Equations · Axiom Academy

Spring-Mass Systems: Damped Harmonic Oscillator A mass on a spring, slowed by friction, obeys one second-order ODE — and the sign of a single number decides whether it bounces or just settles. Pull a mass down on a spring and let go. Depending on how much resistance (damping) it feels, it can oscillate and decay , snap back with no bounce , or creep back slowly — same spring, same mass, three completely different motions. Three things fight it out: the spring force pulling the mass back, the damping force resisting its velocity, and its own inertia . Newton's second law bundles all three into one differential equation. Drag the sliders and watch the mass move. Assume a solution . Substituting into the ODE turns a differential equation into an ordinary quadratic — — and its roots decide everything. Drag the damping and watch the roots move on the complex plane. Slide damping from zero to heavy and watch all three response curves at once. Underdamped overshoots and rings; overdamped creeps; critically damped is the fastest path home with no overshoot at all . The roots of the characteristic equation are the physics: a complex pair means the imaginary part drives oscillation while the real part drives decay; a repeated real root is the fastest non-oscillatory return; two distinct real roots mean pure exponential decay. From a shock absorber to an RLC circuit, this one root-sign argument predicts the motion before you ever build the system.

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