Forced vibrations, resonance and dampingAQA A-Level Physics: Revision notes

A free vibration happens when a system is displaced and then left to oscillate with no external force. It oscillates at its natural frequency $f_0$, which depends on the system itself, for example $f_0 = \frac{1}{2\pi}\sqrt{k/m}$ for a mass on a spring.

A forced vibration happens when a periodic driving force is applied. In the steady state the system oscillates at the frequency of the driver, not at its own natural frequency. The amplitude depends on how close the driving frequency is to the natural frequency.

A swing pushed once is free. Pushed repeatedly at a steady rate, it is forced.

Resonance occurs when the driving frequency equals the natural frequency of the system. The amplitude is then a maximum, because the driver transfers energy to the system as efficiently as possible: the driving force is always in the direction of the velocity, so it does positive work through the whole cycle.

On a graph of amplitude against driving frequency (for constant driving force), the amplitude is small at low frequency, rises steeply to a peak at resonance, then falls to small values above the natural frequency, as the driver is changing direction too quickly for the system to respond.

Barton's pendulums show this: several pendulums hang from a taut string with one heavy driver pendulum. The pendulum with the same length, and so the same natural frequency, as the driver swings with the largest amplitude.

Damping dissipates energy from the oscillator, usually as thermal energy, because resistive forces do work against the motion.

As damping increases the resonance curve changes:

• the peak amplitude falls
• the peak becomes broader and less sharp
• the peak moves slightly to a lower frequency for heavy damping

Light damping gives a tall, sharp peak. Heavy damping gives a low, flat peak. At resonance the energy lost per cycle equals the energy gained from the driver, which is why the amplitude levels off rather than growing for ever.

Resonance can be a problem that damping is added to reduce:

• Bridges and buildings: wind, marching feet or earthquakes can drive a structure at its natural frequency, so engineers use dampers and tuned mass dampers.
• Car suspension and shock absorbers are designed with close to critical damping so bumps die away quickly.
• Machine parts can vibrate loudly at certain speeds, so they are balanced or isolated with rubber mountings.

Resonance can also be useful: a swing pushed at its natural frequency, the sound box of a guitar amplifying the strings, and the air in a flute or organ pipe resonating to give a loud note.

Critical damping returns the system to equilibrium in the shortest time without oscillating, so a door closer or a car suspension is designed close to this.

A stationary wave forms when waves reflected at the end of a medium superpose with incident waves. The medium resonates when the driving frequency equals one of its natural frequencies, giving a large-amplitude stationary wave.

For a tube closed at one end with a tuning fork at the open end, a node forms at the closed end and an antinode at the open end. The first resonance is when $L = \lambda/4$, the next at $L = 3\lambda/4$, and so on.

Other examples: standing waves on guitar strings driven by plucking, and resonance of the air in organ pipes and wind instruments. Standing waves in a microwave oven are why the food has hot and cold spots.

Worked example: for a 512 Hz fork and $v = 340$ m s⁻¹, $\lambda = 340/512 = 0.664$ m, so the first resonant length is $0.166$ m.