The simplest oscillator
Hang a weight on a spring, pull it down and let go: it bounces up and down at a steady rhythm. This spring-mass system is the basic model for vibration in physics and engineering. Car suspensions, washing machine mounts, building isolators, seismometers and even atoms in a crystal are analyzed the same way.
The formulas
For a mass m on a spring of stiffness k, without friction:
- Angular frequency: ω = √(k ÷ m)
- Period: T = 2π√(m ÷ k)
- Natural frequency: f = 1 ÷ T
For a motion of amplitude A:
- Maximum speed = A × ω, as the mass passes the rest position
- Maximum acceleration = A × ω², at the ends of the motion
- Energy = ½ × k × A², constantly exchanged between the spring and the motion
Under gravity, a vertical spring also stretches by the static deflection δ = m × g ÷ k, but the period is unchanged.
Example
A 10 lb mass on a spring of 10 lbf/in, pulled down 1 inch:
- Static deflection: 1 in
- Natural frequency: 3.13 Hz, a period of 0.32 s
- Maximum speed: 1.64 ft/s
- Maximum acceleration: 1 g
- Energy: 0.56 J
A useful shortcut
Because ω² = k ÷ m = g ÷ δ, the natural frequency depends only on how far the spring sags under the load: f ≈ 3.13 ÷ √δ with δ in inches. A car whose suspension compresses about 4 inches under its weight bounces at about 1.5 Hz, which riders find comfortable. A stiff sports car with 1 inch of sag bounces at about 3 Hz.
Resonance and isolation
When something shakes a spring-mass system at a frequency close to its natural frequency, the motion builds up, sometimes dangerously. To isolate a machine from the floor, engineers choose mounts that give a natural frequency well below the frequency of the vibration. Isolation only starts when the vibration is more than √2, about 1.4, times the natural frequency, and in practice designers aim for a ratio of 3 or more.
Limits
Real springs have mass of their own, which lowers the frequency slightly: add about a third of the spring’s mass to m for a better estimate. Friction and damping make the oscillation die out, and very large stretches can take a spring beyond its linear range, where Hooke’s law no longer applies.