Forced vibrations and resonance (A-level only)

What you'll learn:

• The difference between a free vibration and a forced vibration.
• The condition for resonance to occur.
• How damping affects the amplitude and sharpness of a resonance curve.
• Real-world examples of resonance in mechanical systems and stationary waves.

Free vs Forced Vibrations

When an oscillating system is displaced and released, it will oscillate at its natural frequency.

If you pluck a guitar string or tap a pendulum, you are giving the system some initial energy and then leaving it alone. Because there are no external continuous forces acting on it (apart from a little bit of air resistance to eventually stop it), this is called a free vibration.

However, in many real-world situations, a system doesn't just vibrate on its own; it is constantly pushed or pulled by an external source. Think of pushing a child on a swing. You apply a periodic push. This causes the swing to undergo forced vibrations.

During a forced vibration, the system is forced to vibrate at the driving frequency $f$, regardless of what its natural frequency $f_0$ is.

Resonance

What happens to the amplitude of the forced vibrations as you change the driving frequency?

• If you push the system at a frequency very different from its natural frequency, the amplitude stays small.
• But, as the driving frequency $f$ gets closer to the natural frequency $f_0$, the amplitude of the system's oscillations increases dramatically.

Barton's Pendulums

A classic laboratory demonstration of resonance is Barton's Pendulums. It consists of a thick, slightly flexible horizontal string from which several light pendulums (usually paper cones) of different lengths are hung. Alongside them, one heavy pendulum (the "driver") is also hung.

When the heavy driver pendulum is set swinging, it sends small periodic forces along the horizontal string. This forces all the lighter pendulums to vibrate at the driver's frequency.

Because the time period (and therefore frequency) of a simple pendulum depends only on its length, the light pendulum that is exactly the same length as the driver has the same natural frequency. This specific pendulum will resonate, exhibiting an enormous amplitude compared to the others.

The Effect of Damping on Resonance

In the real world, systems experience frictional forces that remove energy over time. This is called damping. Damping doesn't just slowly stop a free vibration; it dramatically changes how a system behaves during forced vibrations.

We can plot a resonance curve to show how the amplitude of the driven system changes as we vary the driving frequency.

Look carefully at the graph above. As the degree of damping increases, three distinct things happen to the resonance curve:

1. The maximum amplitude decreases. More energy is dissipated as heat due to friction, so less energy is stored in the oscillation.
2. The peak becomes wider and flatter. We say the sharpness of resonance decreases. A lightly damped system will only resonate strongly if the driving frequency is exactly equal to $f_0$. A heavily damped system will respond moderately well to a much wider range of driving frequencies.
3. The resonant frequency shifts slightly to the left. For a lightly damped system, the peak occurs exactly at $f_0$. But as heavy damping is introduced, the peak amplitude occurs at a driving frequency slightly lower than the natural frequency.

Useful and Destructive Resonance

Resonance isn't just an abstract concept; it surrounds us in everyday physics. Sometimes we want to maximise it (useful resonance), and sometimes we want to avoid it at all costs (destructive resonance).

Examples of Useful Resonance:

• Stationary waves in instruments: The bodies of string instruments (like acoustic guitars) or the air columns in wind instruments are designed to resonate at specific frequencies, amplifying the sound.
• Microwave ovens: Microwaves are emitted into a cavity designed to set up stationary waves, resonating and transferring maximum energy to the water molecules in food.
• Tuning a radio: The electrical circuit inside a radio is tuned so that its natural electrical frequency perfectly matches the broadcasting frequency of the radio station you want to listen to.

Examples of Destructive Resonance:

• Bridges: Wind, or even the synchronized footsteps of a crowd, can act as a driving force. If the driving frequency matches the natural frequency of the bridge, the bridge can resonate with a dangerously high amplitude (e.g., the famous Millennium Bridge wobble). This is mitigated by installing heavy damping mechanisms (like shock absorbers) to flatten the resonance curve.
• Car suspensions: If a car drives over evenly spaced bumps at a speed where the bump frequency matches the natural frequency of the suspension springs, the car will bounce violently. Car shock absorbers provide heavy damping to prevent this.