the spring-mass-damper system
Picture a weight hanging from a spring, with a small shock absorber (like the piston on a screen door) attached so the bouncing dies down. Pull the weight, let go, and it wobbles up and down, but a little less each swing until it settles. Nearly every vibration in engineering — a car suspension, a building swaying in wind, a guitar string, a microphone diaphragm — is, at heart, this same picture.
Newton's second law (force equals mass times acceleration) turns the picture into one equation: m x'' + c x' + k x = F(t). Here x(t) is the displacement from rest, m is the mass, k is the spring stiffness (the spring pulls back with force k x, proportional to how far you stretch it), and c is the damping (the absorber pushes back with force c x', proportional to how fast you move). F(t) is any outside push. The three terms are simply inertia, friction, and restoring force, added up to equal the applied force.
This single second-order linear equation with constant coefficients is the workhorse of the whole subject. Set F = 0 and you study how a plucked system rings down (free vibration); set c = 0 and you get an idealized frictionless bounce (simple harmonic motion); switch F on and you study how the system answers being shaken (forced oscillation). The same equation, with the names of the letters swapped, also describes an electrical RLC circuit — which is why one theory serves two worlds.
A 2 kg mass on a spring with k = 50 N/m and a damper c = 4 N·s/m, released from rest 0.1 m below equilibrium, obeys 2 x'' + 4 x' + 50 x = 0; it oscillates while its amplitude shrinks toward zero.
Inertia (2 x''), damping (4 x'), and restoring force (50 x) balance to zero because no outside force is applied.
The model assumes the spring force is exactly proportional to stretch (Hooke's law) and damping is proportional to velocity. Real springs stiffen at large stretch and real friction is messier; the linear model is excellent for small motions and only an approximation for large ones.