Simple harmonic motion occurs whenever a restoring force is proportional to displacement. A mass on a spring is the classic example. Pull it away from equilibrium, and the spring pulls it back with a force proportional to how far you stretched it. The result is a smooth back-and-forth oscillation that, in an ideal system with no damping, continues forever.
The key parameters are amplitude (maximum displacement), period (time for one complete cycle), and frequency (cycles per second). For a mass-spring system, the period is T equals 2 pi times the square root of m over k. Heavier masses oscillate more slowly. Stiffer springs oscillate faster. A 1 kg mass on a spring with k equals 100 N/m has a period of about 0.63 seconds, or a frequency of about 1.6 Hz.
A pendulum is another example of simple harmonic motion, approximately. The period of a simple pendulum is T equals 2 pi times the square root of L over g, where L is the pendulum length and g is gravitational acceleration. Notice that the period depends only on length and gravity, not on mass. A heavy pendulum and a light one, both on 1-meter strings, swing with the same period. This is why pendulum clocks work reliably regardless of the mass of the bob.
Galileo discovered this isochronism around 1602, reportedly by watching a chandelier swing in the Pisa cathedral. Christiaan Huygens built the first pendulum clock in 1656, achieving accuracy of about 15 seconds per day. This was revolutionary for navigation, as accurate timekeeping was essential for determining longitude at sea.
Damping causes real oscillations to decay over time. In a car suspension, shock absorbers provide damping to prevent the car from bouncing endlessly after hitting a bump. Without damping, a car would oscillate several times after each bump, making the ride uncomfortable and unsafe. The damping ratio determines whether the system is underdamped (oscillates before stopping), critically damped (returns to equilibrium fastest), or overdamped (returns slowly without oscillating).
Resonance occurs when a driving frequency matches the natural frequency of a system. Push a swing at the right rhythm, and the amplitude grows with each push. This is useful in some contexts. Radio tuners use resonance to select a specific frequency. Musical instruments use resonance to amplify sound. But resonance can also be destructive. The Tacoma Narrows Bridge collapsed in 1940 when wind-driven oscillations matched the bridge’s natural frequency.