Free and forced oscillations and resonanceEdexcel International A Level Physics: Revision notes

In a free oscillation a system is displaced and then left to oscillate with no further external periodic force. It oscillates at its natural frequency $f_0$, which depends only on the system. For a mass on a spring, $f_0 = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$; for a simple pendulum, $f_0 = \frac{1}{2\pi}\sqrt{\frac{g}{l}}$.

In a forced oscillation a periodic driving force is applied continuously, for example by a vibration generator or an engine. After initial transients the system oscillates at the driving frequency, not its natural frequency, and the amplitude depends on how close the driving frequency is to the natural frequency.

Resonance occurs when the driving frequency equals the natural frequency of the system. The amplitude of the forced oscillation is then a maximum, because the driver transfers energy to the oscillator most efficiently, and the energy supplied per cycle balances the energy dissipated by damping.

Examples: a loose dashboard panel rattling at a certain engine speed, a child's swing pushed at its natural frequency, and a bridge or building excited by wind, marching feet or an earthquake. Resonance can be harmful (structural failure) or useful (tuning a radio, musical instruments).

Plotting amplitude of a forced oscillation against driving frequency (driving amplitude constant) gives a resonance curve. The amplitude is small when the driving frequency is much lower or much higher than $f_0$, and rises to a peak close to $f_0$.

The amount of damping controls the shape of the peak. Light damping: a tall, sharp (narrow) peak. Heavier damping: a lower, flatter, broader peak, with the maximum at a slightly lower frequency. The more energy is dissipated each cycle, the less the amplitude can build up. In very heavily damped systems there is no distinct peak.

To reduce the danger of resonance, engineers increase damping (for example tuned dampers in tall buildings, shock absorbers, rubber mounts) or change the natural frequency of a structure so it differs from likely driving frequencies. Increasing damping reduces the maximum amplitude but it also dissipates more energy, which is why a car shock absorber stops a wheel bouncing at resonance.

Method. Hang a known mass from a spring (or clamp a mass on a flexible strip) driven by a vibration generator connected to a signal generator. Keep the driving amplitude constant. Vary the frequency in small steps, especially near the expected natural frequency, and judge the amplitude using a ruler or fiducial marker at the equilibrium position. The resonant frequency $f$ is where the amplitude is a maximum. Repeat for several known masses.

Analysis. Since $f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$, we have $\frac{1}{f^2} = \frac{4\pi^2}{k}m$. A graph of $1/f^2$ against $m$ is a straight line through the origin with gradient $4\pi^2/k$. Find the unknown mass by measuring its resonant frequency and reading from the graph.

Worked example. Gradient = 1.58 s² kg⁻¹ gives $k = 4\pi^2/1.58 = 25$ N m⁻¹. An unknown mass resonating at 2.40 Hz has $1/f^2 = 0.174$ s², so $m = 0.174/1.58 = 0.110$ kg.