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Every real oscillating system loses energy. Air resistance slows a swinging pendulum, internal friction warms a vibrating spring, viscous losses in suspension fluid drain the motion of a car after it crosses a speed bump. This loss is damping, and its severity is engineered — too little damping and a mechanical system rings dangerously; too much and it cannot respond quickly enough. Driving a damped oscillator at a particular frequency produces resonance, a phenomenon used constructively in radio tuning and MRI scanners, and avoided destructively in bridges, buildings and aircraft. This lesson develops both topics and connects them through the amplitude-frequency response curve.
Spec mapping: This lesson sits under AQA 7408 section 3.6.1.2. It covers the qualitative description of free and forced oscillations, the three damping regimes (light, critical, heavy), the displacement-time graphs that distinguish them, the natural frequency f₀ of a free oscillator, the driving frequency f_d of a forced system, the condition f_d = f₀ for resonance, the qualitative effect of damping on the amplitude-frequency response, and engineering applications of both phenomena. The Q-factor is mentioned synoptically. (Refer to the official AQA specification document for exact wording.)
Synoptic links:
- Section 3.6.1.2 (SHM): the undamped, free SHM oscillator of the previous lessons is the limiting case of light damping, and the resonance phenomenon depends on the system's natural angular frequency ω₀ = √(k/m) or √(g/L).
- Section 3.5 (electricity — LC and LCR circuits): the LC oscillator is the electromagnetic analogue of the undamped SHM oscillator; adding resistance R turns it into an LCR circuit, the direct electrical analogue of a damped mechanical oscillator. Radio tuning works by varying C (or L) to bring the LCR's natural frequency into resonance with the broadcast frequency.
- Section 3.11 (medical physics — MRI): the resonance of hydrogen nuclei in a magnetic field at the Larmor frequency is exactly the resonance phenomenon discussed here, applied at quantum-mechanical scale.
A free oscillation occurs when an oscillating system is displaced from equilibrium and then left to oscillate without any external periodic driving force. The system oscillates at its natural frequency f₀ (or natural angular frequency ω₀), set entirely by the system's intrinsic properties (mass, spring constant, length, gravity).
Examples:
In an ideal (frictionless) free oscillation, the amplitude is constant and the oscillation continues forever. In reality, energy is dissipated and the amplitude decays.
A forced oscillation occurs when an external periodic force is applied to an oscillating system. The driving force has its own frequency, the driving frequency f_d. After any transient behaviour dies away, the system oscillates at the driving frequency, not at its natural frequency.
Examples:
The amplitude of the forced oscillation depends on how close the driving frequency is to the natural frequency. The closer they are, the larger the amplitude — and the closest case, f_d = f₀, is resonance.
In any real oscillating system, energy is lost to the surroundings — to heat through friction, to air resistance, to sound, to material strain. This loss is damping, and it reduces the amplitude of the oscillation over time.
Damping is the dissipation of mechanical energy from an oscillating system to its surroundings, causing the amplitude of oscillation to decrease over time.
There are three qualitative regimes of damping, each with a characteristic displacement-time signature.
In light damping, the dissipative force is small. The system continues to oscillate at approximately its natural frequency, but with an amplitude that decays gradually over many cycles. The displacement-time graph shows a sinusoidal oscillation with an exponentially decreasing amplitude envelope.
Mathematically, the displacement of a lightly damped oscillator can be written
x(t) = A₀ × exp(−γt) × cos(ω_d t + φ)
where γ is a damping coefficient (positive) and ω_d is the damped angular frequency (slightly less than ω₀). The envelope A₀ × exp(−γt) is an exponential decay; the cosine oscillates inside it.
Examples:
In critical damping, the dissipative force is precisely large enough to return the system to equilibrium as quickly as possible without overshoot. The system does not oscillate; it relaxes monotonically to equilibrium in a time of order 1/ω₀.
The displacement-time graph shows a smooth, exponential-like decay from the initial displacement to zero, without crossing the equilibrium line.
Examples:
Critical damping is the fastest return to equilibrium that does not produce overshoot. Lighter damping would produce a quicker first crossing of zero but the system would then continue to oscillate; heavier damping would prevent oscillation but would take longer to settle.
In heavy damping, the dissipative force is large enough to prevent oscillation but excessive — the system returns to equilibrium very slowly, taking much longer than the critical-damping case.
The displacement-time graph shows a very slow exponential-like decay from the initial displacement, again without crossing the equilibrium line.
Examples:
In most engineering applications, heavy damping is undesirable — the system is "sluggish" and slow to respond.
flowchart TD
A["Oscillating system<br/>displaced and released"] --> B{"Damping level?"}
B -->|"Very low (light)"| C["Light damping<br/>Oscillates many cycles<br/>Exponential amplitude decay"]
B -->|"Just enough (critical)"| D["Critical damping<br/>Returns to equilibrium quickest<br/>No overshoot, no oscillation"]
B -->|"Excessive (heavy)"| E["Heavy damping<br/>Slow return, no oscillation<br/>Sluggish"]
C --> F["Examples: pendulum,<br/>tuning fork, guitar string"]
D --> G["Examples: car suspension,<br/>galvanometer, door closer"]
E --> H["Examples: door with friction,<br/>pendulum in oil"]
style C fill:#3498db,color:#fff
style D fill:#27ae60,color:#fff
style E fill:#e67e22,color:#fff
For a pulled-and-released system displaced to +A₀ at t = 0:
When a damped oscillating system is driven by an external periodic force, the steady-state amplitude depends on the relationship between the driving frequency f_d and the natural frequency f₀ of the system.
Resonance occurs when the driving frequency equals the natural frequency: f_d = f₀ (or equivalently ω_d = ω₀).
At resonance, the driving force is "in step" with the natural oscillation of the system. Every push delivers energy in the direction the system is already moving, so the amplitude builds up. The amplitude is limited only by the rate at which damping dissipates the input energy.
A plot of steady-state amplitude against driving frequency, for fixed amplitude of driving force, has a characteristic peaked shape:
The height and width of the resonance peak depend on the damping:
| Damping | Peak height | Peak width | Peak frequency |
|---|---|---|---|
| None (theoretical) | Infinite | Infinitesimal | Exactly f₀ |
| Light | Very tall | Narrow | Just below f₀ |
| Moderate | Lower | Broader | Noticeably below f₀ |
| Heavy | Very low | Very broad | Significantly below f₀ |
The lighter the damping, the sharper and taller the peak (high "Q" — see below). The heavier the damping, the lower and broader the peak.
Key observation. Damping reduces the peak resonance amplitude, broadens the resonance curve, and shifts the peak frequency slightly lower than f₀.
A useful measure of the "sharpness" of a resonance peak is the Q-factor (quality factor), defined as
Q = ω₀/Δω = f₀/Δf
where Δω is the full width of the resonance peak at half-maximum amplitude (or, more rigorously, at half-power, which is amplitude/√2 of the peak). Higher Q means sharper resonance, lower damping, more cycles before the free oscillation decays significantly.
Approximate Q-factors of typical systems:
| System | Q |
|---|---|
| Tuning fork | 1,000 – 10,000 |
| Quartz crystal oscillator | 10,000 – 10⁵ |
| Atomic clock (caesium) | 10⁹ |
| Suspension bridge | ~50 |
| Car shock absorber (critical) | ~0.5 |
| MRI nuclear resonance | 10³ – 10⁴ |
A high-Q system rings for many cycles; a low-Q system damps out quickly.
A radio receiver uses an LC tuning circuit. The inductor has L = 10 μH and the variable capacitor can be adjusted between 50 pF and 500 pF. Calculate (a) the range of resonant frequencies the circuit can tune, and (b) the bandwidth of an FM broadcast at 100 MHz given a circuit Q of 50.
(a) ω₀ = 1/√(LC), so f₀ = 1/(2π√(LC)).
At C = 500 pF: f₀ = 1/(2π√(10⁻⁵ × 5 × 10⁻¹⁰)) = 1/(2π × √(5 × 10⁻¹⁵)) = 1/(2π × 7.07 × 10⁻⁸) = 2.25 MHz.
At C = 50 pF: f₀ = 1/(2π√(10⁻⁵ × 5 × 10⁻¹¹)) = 1/(2π × √(5 × 10⁻¹⁶)) = 1/(2π × 2.24 × 10⁻⁸) = 7.12 MHz.
So the circuit tunes from about 2.25 MHz to 7.12 MHz — the AM/short-wave range.
(b) For a tuner at f₀ = 100 MHz and Q = 50: bandwidth Δf = f₀/Q = 100/50 = 2 MHz. The tuner accepts a band 2 MHz wide around 100 MHz — exactly the bandwidth of a single FM broadcast channel.
A pedestrian bridge has a natural lateral oscillation frequency of 0.95 Hz. The average stride frequency of a walking person is about 1 step per second. Comment on the likelihood of pedestrian-induced resonance.
The walker's stride frequency at 1 Hz is very close to the bridge's natural frequency of 0.95 Hz. The frequency mismatch is only about 5%, which is well within the typical bandwidth of an under-damped bridge's resonance peak. A crowd of walkers would tend to synchronise their strides with the bridge's sway (a known feedback effect), driving the bridge ever closer to resonance. The bridge would oscillate strongly and require additional damping — exactly the issue that occurred at the Millennium Bridge.
Specimen question modelled on the AQA paper format. Nine marks.
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