Damped & Forced Oscillations; Resonance

A real swing doesn’t go on forever; it dies away unless someone keeps pushing. And if the pushes come at just the right rhythm, even small ones build a huge swing.

Damping#

Friction at joints, air resistance and many other effects add a damping force that opposes the velocity and steadily removes energy. In many practical cases it is proportional to the speed, $F = -bv$, and Newton’s second law becomes

For light damping the solution is an oscillation whose amplitude shrinks exponentially:

So the system swings at almost its natural frequency while its amplitude $A = A_0e^{-bt/2m}$ decays. With enough damping it doesn’t oscillate at all: displaced, it simply returns to equilibrium without overshooting. The smallest damping that does this is critical damping, and it brings the system to rest fastest.

Forced oscillations#

Now add a periodic driving force $F_0\sin\omega t$, as when you push a swing:

After an initial messy period, the body settles into steady oscillation at the driving frequency ω (not its natural one), with the energy lost to damping supplied by the driving force. The steady amplitude is

Resonance#

Vary the driving frequency and the amplitude peaks at $\omega = \sqrt{\omega_0^2 - b^2/2m^2}$, very close to ω₀ when damping is small. This is resonance: drive a system at its own natural frequency and it responds enormously. The smaller the damping, the taller and sharper the peak; with no damping at all the formula would give infinite amplitude.