Damping and Resonance

A $\textbf{free oscillation}$ occurs when a system is displaced and released with no further external force. It vibrates at its $\textbf{natural frequency}$ $f_{0}$, determined by its properties — such as $m$ and $k$ for a spring.

A $\textbf{forced oscillation}$ occurs when a periodic external force drives the system. After an initial settling period, it oscillates at the $\textbf{driving frequency}$, not its natural frequency.

The amplitude of a forced oscillation depends on how close the driving frequency is to the natural frequency, and on the damping.

$\textbf{Damping}$ is the loss of energy from an oscillating system caused by resistive forces such as air resistance or friction. The energy is usually transferred to thermal energy in the surroundings. It reduces the amplitude over time.

$\textbf{Light damping}$: the amplitude decreases gradually over many oscillations, with the period almost unchanged. A pendulum in air is lightly damped.

$\textbf{Critical damping}$: the system returns to equilibrium in the shortest possible time without oscillating. Car suspension is designed to be close to critically damped.

$\textbf{Heavy damping}$ (overdamping): the system returns to equilibrium without oscillating, but more slowly than with critical damping — like a pendulum moving through thick oil.

$\textbf{Resonance}$ occurs when the driving frequency equals the natural frequency of the system. Energy is then transferred from the driver to the oscillator most efficiently, and the amplitude reaches a maximum.

At resonance, the driver and oscillator are in step so that every push adds energy — the driving force is in phase with the velocity.

A graph of amplitude against driving frequency peaks at the natural frequency. With $\textbf{less damping}$ the peak is taller and narrower. With $\textbf{more damping}$ it is lower and broader, and the peak shifts slightly below $f_{0}$.

With no damping at all, the amplitude would grow without limit; in practice something breaks or damping always intervenes.

$\textbf{Useful}$: musical instruments use resonance in air columns and strings to amplify particular notes; radios are tuned by adjusting a circuit’s natural frequency to match a station; MRI scanners use resonance of hydrogen nuclei in a strong magnetic field.

$\textbf{Harmful}$: bridges and buildings can resonate with wind or footsteps. London’s Millennium Bridge swayed when pedestrians fell into step with its sideways motion and had to be fitted with dampers. Tall buildings in earthquake zones use tuned mass dampers to absorb energy at their natural frequency.

Engineers avoid harmful resonance by changing a structure’s natural frequency away from likely driving frequencies, or by adding damping to flatten the response.