Damping and Resonance

Before we can understand damping and resonance, we need to distinguish between two fundamentally different ways an oscillator can move.

Free Oscillations

A free oscillation occurs when a system oscillates at its natural frequency f₀ with no external driving force after an initial displacement. The frequency depends only on the physical properties of the system — for a mass–spring system, f₀ = (1/2π)√(k/m); for a simple pendulum, f₀ = (1/2π)√(g/L).

In practice, all free oscillations are eventually damped by friction or air resistance, but we describe them as "free" when no external periodic force is being applied.

Forced Oscillations

A forced oscillation occurs when a periodic external driving force is applied to a system. The system oscillates at the driving frequency f_d, which may be different from the natural frequency.

The natural frequency is sometimes called the resonant frequency — the frequency at which a system "wants" to vibrate when left alone.

Damping is the process by which the amplitude of an oscillation decreases over time. Energy is transferred from the oscillating system to the surroundings (usually as thermal energy due to friction or drag). In all real systems, some damping is present.

Levels of Damping

The OCR specification requires you to understand three categories of damping. They differ in how quickly the system returns to equilibrium and whether it overshoots.

Displacement–Time Graphs

The graph below shows how displacement varies with time for each type of damping, starting from the same initial displacement.

Resonance occurs when the driving frequency equals the natural frequency of the oscillator: f_d = f₀. At this frequency, energy is transferred from the driver to the oscillator most efficiently, producing the maximum amplitude of oscillation.

How Resonance Works

When you push a child on a swing, you push at the right moment each cycle. If you push at the natural frequency of the swing, each push adds energy at the point of maximum displacement where the driving force is in phase with the velocity. The amplitude grows with each cycle.

If you push at a different frequency, sometimes you push when the swing is already moving away — your push might even slow it down. The amplitude stays small.

Amplitude–Driving Frequency Graphs

The resonance curve (also called a response curve) shows how the amplitude of a forced oscillator depends on the driving frequency. The shape of the curve depends critically on the amount of damping.

• Light damping → very sharp, tall peak at f₀. Small changes in frequency cause large changes in amplitude.
• Heavy damping → broad, low peak. The amplitude varies less dramatically with frequency.
• No damping → theoretically infinite amplitude at resonance (the system would eventually break).

The graph below shows resonance curves for different levels of damping.

So far we have treated damping and resonance separately. In reality, they are deeply connected. When a system is damped, both its behaviour at resonance and its effective time period change.

How Damping Changes the Time Period

For an undamped oscillator, the time period is T₀ = 2π/ω₀. When damping is present, the damped angular frequency ω' is:

where γ is the damping coefficient. Since ω' < ω₀, the damped period T' is longer than the undamped period T₀:

For light damping, γ is small compared to ω₀, so T' ≈ T₀. This is the standard assumption at A Level and the one you should use in calculations unless the question specifically asks about heavy damping. For heavier damping, the period does increase.

How Damping Shifts the Resonant Frequency

In a forced, damped system, the frequency at which maximum amplitude occurs is not exactly f₀. For light damping, the peak is very close to f₀, but as damping increases:

• The peak amplitude shifts to a slightly lower frequency than f₀
• The peak becomes broader and lower
• For heavy damping, the concept of a distinct resonant peak becomes meaningless

This is because the damping effectively reduces the restoring force's dominance over the motion, so the system responds most strongly at a slightly lower frequency.

Summary of Effects

Barton's Pendulums

This is the classic demonstration of resonance for A Level. Several pendulums of different lengths hang from a taut string. One "driver" pendulum is pulled back and released — it drives all the others through the connecting string.

• The pendulum whose length matches the driver oscillates with the largest amplitude (resonance)
• Pendulums with different lengths oscillate with smaller amplitudes
• The resonant pendulum oscillates in phase with the driver
• Pendulums shorter than the driver oscillate roughly in antiphase
• Pendulums longer than the driver oscillate with no consistent phase relationship

Other Examples

• Radio tuning — an LC circuit is tuned to resonate at the frequency of the desired radio station
• Musical instruments — the body of a guitar resonates at certain frequencies, amplifying specific notes
• MRI scanners — use resonance of protons in a magnetic field
• Tacoma Narrows Bridge (1940) — aeroelastic flutter (a form of resonance) caused the bridge to oscillate with increasing amplitude until it collapsed
• Vibration in vehicles — engine vibrations at certain RPMs can cause resonance in panels and mirrors
• Glass shattering with sound — a singer matching the natural frequency of a wine glass can cause it to resonate and break

A mass is released from displacement and allowed to oscillate freely. Adjust the damping to see how it affects the decay. All three curves are plotted simultaneously so you can compare light, critical and heavy damping.

A rotating cam drives a mass–spring system. Adjust the cam frequency and watch how the amplitude of oscillation changes. When the driving frequency matches the natural frequency, resonance occurs and the amplitude is maximum. The graph on the right records each measurement — sweep the frequency to build the resonance curve yourself.