Resonance: Natural Frequency and Forced Vibration

Pushing a swing

To get a little cousin swinging high at the park, you don't shove hard. You give small pushes at just the right moment, once every swing. Each push adds a bit of energy, and the swing goes higher and higher. Push at the wrong moments and you fight the swing instead. This is resonance, one of the most useful and most dangerous ideas in physics.

Natural frequency

Pull a swing back and let go, pluck a ruler, or tap a wine glass. Each one vibrates at its own natural frequency, f₀. The natural frequency depends on the object's properties:

• a pendulum's length (longer → lower frequency),
• a string's length, tension and mass,
• an object's mass and stiffness (heavier or less stiff → lower frequency).

Many objects have several natural frequencies: a string or an air column has a whole series of harmonics, which you met in the standing-wave lessons.

Forced vibration and resonance

A forced vibration happens when an outside force pushes an object back and forth at the driving frequency. Usually the response is small. But when the driving frequency equals the natural frequency, each push arrives in step with the motion, energy builds up cycle after cycle, and the amplitude becomes very large. That is resonance.

Three conditions for resonance:

1. an object that can vibrate with a natural frequency,
2. a repeated driving force,
3. the driving frequency matching the natural frequency (or a harmonic of it).

Damping (friction, air resistance) removes energy. Light damping gives a tall, narrow resonance peak; heavy damping gives a low, broad one.

Sympathetic vibration

Sound waves can drive resonance too. Put two identical tuning forks, each on a wooden resonance box, facing each other. Strike one, then silence it with your hand: the other fork is now humming. The sound waves pushed it at exactly its natural frequency. A fork of a different frequency would barely respond. This is sympathetic vibration. A singer who produces a loud, sustained note exactly at a crystal glass's natural frequency can make the glass vibrate so strongly that it shatters.

Resonance at work (E2.7)

• Musical instruments: a guitar's body and a violin's wooden box resonate with the strings and make the sound much louder. Air columns in wind instruments resonate at chosen frequencies.
• Tuning-fork resonance boxes are cut to the length that resonates with the fork (a closed air column with L ≈ λ/4).
• Clocks and watches: a quartz watch keeps time with a tiny quartz crystal that vibrates at its natural frequency (usually 32 768 Hz).
• Playing with friends: pushing a swing, rocking a car out of a snowbank by pushing in rhythm.

When resonance is dangerous

• Bridges: soldiers have traditionally "broken step" when crossing bridges so their marching cannot drive the bridge at a natural frequency. When London's Millennium Bridge opened in 2000, it swayed sideways as crowds of walkers fell into step with its motion; it was closed and fitted with dampers.
• The Tacoma Narrows Bridge (Washington State, U.S.) twisted wildly and collapsed in November 1940 in a steady wind of about 64 km/h. It is often called resonance, but engineers explain it as aeroelastic flutter: the bridge's own twisting changed how the wind pushed on it, so the wind kept feeding energy into the bridge's natural twisting motion. Since then, bridges are tested in wind tunnels and designed to be stiffer and more aerodynamic.
• Buildings and earthquakes: an earthquake shakes the ground over a range of frequencies. A building whose natural frequency matches can sway dangerously. Engineers change stiffness and mass, add dampers, or use base isolation (flexible bearings) so the building's natural frequency avoids the shaking. Some skyscrapers, such as Taipei 101 in Taiwan, use a huge tuned mass damper.
• Machines and vehicles: a car part that rattles only at one speed is resonating; rubber mounts add damping.