Damping and Resonance
Specification: OCR A H556 | Section: 5.3.3 | Teaching time: ~4 hours
- Distinguish between free oscillations and forced oscillations
- Describe the effects of damping on an oscillatory system, including light, heavy and critical damping
- Explain resonance and identify the natural frequency of an oscillator
- Sketch and interpret amplitude–driving frequency graphs for forced oscillators with different amounts of damping
- Describe practical examples of forced oscillations and resonance, including Barton's pendulums
- Explain how damping affects the resonant frequency and the time period of an oscillating system
Free and Forced Oscillations
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 fd, which may be different from the natural frequency.
Free oscillation: system vibrates at its own natural frequency after being displaced and released.
Forced oscillation: system is made to vibrate by an external periodic force at the driving frequency.
The natural frequency is sometimes called the resonant frequency — the frequency at which a system "wants" to vibrate when left alone.
Damping
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.
| Type | Behaviour | Example |
|---|---|---|
| Light damping | Oscillates with gradually decreasing amplitude. Takes many cycles to die out. | Car suspension, pendulum in air |
| Critical damping | Returns to equilibrium in the shortest possible time without oscillating. | Car shock absorbers, galvanometer |
| Heavy (over-) damping | Returns to equilibrium very slowly without oscillating. Slower than critical damping. | Door with very stiff closer, overdamped scales |
Displacement–Time Graphs
The graph below shows how displacement varies with time for each type of damping, starting from the same initial displacement.
Heavy damping does not mean the oscillation stops sooner. Heavy damping returns to equilibrium more slowly than critical damping. Critical damping is the optimum — fastest return without overshooting.
Figure 1 — Displacement–time graphs for a damped oscillator. Light damping (blue) shows decaying oscillations. Critical damping (orange) returns to equilibrium fastest without overshooting. Heavy damping (red) returns to equilibrium more slowly than critical.
Knowledge Check
- Free oscillation: system oscillates at its natural frequency without a periodic external force ✓
- Forced oscillation: system is driven by an external periodic force at the driving frequency ✓
- Critical damping returns the system to equilibrium in the shortest possible time ✓
- Without oscillating / overshooting — the car doesn't bounce up and down repeatedly after a bump ✓
- Total energy decreases over time as energy is transferred to the surroundings (e.g. as thermal energy) ✓
Resonance
Resonance occurs when the driving frequency equals the natural frequency of the oscillator: fd = 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.
Damping reduces the amplitude at every frequency, not just near resonance. But the effect is most dramatic near the peak — light damping produces a much taller peak than heavy damping.
Figure 2 — Resonance curves for a forced oscillator. Light damping (blue) gives a sharp peak with large maximum amplitude. Heavy damping (red) gives a broad, low peak. The natural frequency f₀ is where the peak occurs.
Knowledge Check
- The maximum amplitude at resonance decreases ✓
- The resonance curve becomes broader / less sharp ✓
- At resonance, the driving force is in phase with the velocity of the oscillator ✓
- Power transferred to the oscillator = driving force × velocity, and this is maximised when they are in phase ✓
- Maximum power transfer means maximum rate of energy input, so the amplitude reaches its largest value ✓
Damping, Resonance and Time Period
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:
Light damping does not change the time period. At A Level, when a system is described as lightly damped, you should assume T' = T0. The change in period is negligible for light damping and examiners expect you to state this. Only for heavy damping does the period increase noticeably (T' > 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.
If asked "how does damping affect the time period?", the correct A Level answer is: light damping has no significant effect on the period (T' ≈ T0). Only mention the period increase for heavy damping if the question specifically asks about it. Many students lose marks by saying "damping increases the period" without qualifying that this only applies to significant damping.
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
| Quantity | Effect of Increasing Damping |
|---|---|
| Amplitude (at resonance) | Decreases |
| Time period T' | Increases |
| Resonant frequency (peak) | Decreases slightly |
| Sharpness of resonance peak | Broadens (Q-factor decreases) |
| Total energy | Decreases faster |
Practical Examples of Resonance
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
Resonance can be dangerous. The amplitude at resonance can become so large that the system is damaged or destroyed. This is why engineers design structures to avoid resonant frequencies matching common driving forces (wind, traffic, earthquakes). Damping is deliberately added to reduce the resonant amplitude.
Damping Explorer
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.
Resonance Explorer
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.
Exam-Style Questions
(a) Which of the other pendulums will oscillate with the largest amplitude? Explain your answer.
(b) Describe the phase relationship between the driver pendulum and the pendulum that resonates with it.
- (a) The pendulum with length 40.0 cm (same length as the driver) ✓
- Because it has the same natural frequency as the driver, so it is driven at its resonant frequency ✓
- Maximum energy transfer occurs at resonance, producing the largest amplitude ✓
- (b) The resonant pendulum oscillates in phase with the driver pendulum ✓
(a) Sketch a graph of amplitude against driving frequency for this system with (i) light damping and (ii) heavy damping. Label the natural frequency on your graph.
(b) Explain why the maximum amplitude occurs at a frequency slightly below f0 for a damped system.
- (a) Both curves peak near f0, with amplitude on the y-axis and driving frequency on the x-axis ✓
- Light damping: tall, narrow peak ✓
- Heavy damping: short, broad peak ✓
- f0 labelled at the position of the peak(s) ✓
- (b) Damping effectively reduces the restoring force's dominance, so the system responds most strongly at a slightly lower frequency ✓
- The damped period T' > T0, so the frequency at which maximum amplitude occurs is slightly less than f0 ✓
(a) Explain why the designer adds damping to the cone mechanism.
(b) The damping is increased. Describe two effects this has on the response of the cone across the frequency range.
- (a) Without damping, the cone would resonate strongly at 120 Hz ✓
- This would cause a large, unwanted peak in the frequency response, producing distortion and potentially damaging the cone ✓
- (b) The maximum amplitude at resonance decreases ✓
- The resonance peak becomes broader / the frequency response is more uniform across the range ✓
Extended Response (6 marks)
Discuss the physics of this situation. In your answer, you should:
• explain why the bridge vibrates dangerously
• describe what is meant by resonance in this context
• explain how adding dampers would reduce the problem
• discuss any disadvantages of adding significant damping
[6 marks, QWC]
- The marching troops apply a periodic driving force to the bridge at their stepping frequency ✓
- When this driving frequency matches the natural frequency of the bridge, resonance occurs ✓
- At resonance, maximum energy is transferred to the bridge, causing large amplitude vibrations ✓
- Damping dissipates energy from the bridge's oscillations, reducing the amplitude of vibration at all frequencies ✓
- The resonance peak becomes lower and broader, so the bridge is less sensitive to any particular driving frequency ✓
- A disadvantage is that heavy damping makes the bridge feel "dead" — it may not respond well to normal loads and can add cost and maintenance requirements ✓
Topic Summary
Free vs Forced
Free oscillations happen at the natural frequency. Forced oscillations occur at the driving frequency of an external periodic force.
Damping Types
Light: oscillates with decaying amplitude. Critical: fastest return to equilibrium, no oscillation. Heavy: slow return, no oscillation.
Resonance
Maximum amplitude when fd = f0. Energy transfer is most efficient. Damping reduces and broadens the resonance peak.
Damping Effects
Damping increases the time period (T' > T0) and shifts the resonant frequency slightly below f0.