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In the previous lessons we built the theory of an ideal simple harmonic oscillator with no energy loss. In the real world, every oscillating system eventually stops — the pendulum swings with decreasing amplitude until it hangs still, the guitar string fades away after being plucked, the building sways a little then settles after an earthquake. Energy is dissipated to the surroundings, usually as heat, sound or air resistance. This energy loss is called damping.
But damping is only half the story. If we drive an oscillator with a periodic external force, something remarkable happens when the driving frequency matches the oscillator's natural frequency: the amplitude grows dramatically. This is resonance. Understanding damping and resonance is essential for everything from the design of car suspension to the engineering of skyscrapers, and it is OCR A-Level Physics A Module 5.3.
A free oscillation is what happens when you pluck a guitar string, displace a mass on a spring and release it, or pull a pendulum aside and let it go. The system oscillates at its own natural frequency f₀ — a property of the system, determined by its mass, stiffness and geometry.
In the ideal (no-friction) case, a free oscillation would continue forever with constant amplitude. In practice, energy leaks away and the oscillation dies down.
A damped oscillation is a free oscillation in which a resistive force — typically air resistance, friction, or viscous drag — opposes the motion and removes energy from the system with each cycle.
The resistive force almost always acts opposite to velocity. It is zero when the oscillator is momentarily stationary at the extremes and maximum when the oscillator is moving fastest at equilibrium. This is why each cycle of a damped oscillator is slightly smaller in amplitude than the previous one.
OCR categorises damping into three important regimes:
| Type | Description | Appearance |
|---|---|---|
| Light damping | Small resistive force. Amplitude decays slowly (exponentially) over many cycles. | Familiar sine wave inside a decaying envelope. |
| Critical damping | Resistive force just large enough to return the system to equilibrium in the shortest time without oscillating. | Exponential decay, no overshoot. |
| Heavy (over) damping | Resistive force so large the system returns to equilibrium slowly and without oscillating. | Very slow exponential approach to equilibrium. |
graph TD
A[Displacement − time graphs]
A --> L[Light damping:<br/>oscillates with slowly<br/>decreasing amplitude]
A --> C[Critical damping:<br/>returns to zero in<br/>shortest time, no overshoot]
A --> O[Over/heavy damping:<br/>returns slowly, no overshoot]
Critical damping is the Goldilocks case — just right for engineering applications where you want fast return to equilibrium without any wobble. Examples: car suspension, door closers, analogue voltmeters.
For lightly damped SHM, the amplitude of successive cycles decays exponentially:
A(t) = A₀ e^(−kt)
for some positive decay constant k (not the spring constant; the notation is unfortunate). The number of cycles over which the amplitude decays to 1/e of its initial value is a measure of how heavily the system is damped. The frequency of the lightly damped oscillation is almost the same as the natural frequency — a tiny bit lower — but for A-Level purposes you may treat them as equal.
Exam Tip: OCR often asks candidates to "sketch the displacement-time graph of a lightly damped oscillator". The mark scheme expects: (i) an oscillation with the correct period, (ii) successive peaks that decrease in amplitude, (iii) a smooth exponential envelope enclosing the peaks and troughs symmetrically.
A forced oscillation occurs when an external periodic driver pushes the system. Examples:
The key feature is that the driver has its own frequency — the driving frequency f_d — which is generally different from the natural frequency f₀ of the oscillator.
After an initial transient, the driven oscillator settles into steady-state motion at the driver's frequency f_d, with some amplitude determined by how closely f_d matches f₀ and how heavily the system is damped.
Resonance is what happens when the driving frequency is equal to the natural frequency of the oscillator: f_d = f₀. At resonance the amplitude of the steady-state motion reaches a maximum — often dramatically larger than the driving amplitude itself.
At resonance, every push from the driver arrives in phase with the natural motion of the oscillator, so every push adds energy. In any other regime, the pushes sometimes add energy and sometimes remove it, and the amplitude settles to a smaller value. At f_d = f₀, the pushes always arrive in phase, so the energy builds up cycle by cycle until losses to damping exactly balance the energy input.
The steady-state amplitude in a lightly damped system can easily be 10×, 100× or even 1000× the driving amplitude. In a system with no damping at all, the amplitude would grow without bound — but real systems always have some damping.
Plotting amplitude A against driving frequency f_d produces the characteristic resonance curve. Its shape depends on the damping:
f₀; amplitude falls off rapidly either side.f₀ (approximately).f₀.graph TD
A[Amplitude−frequency curves<br/>for a driven oscillator]
A --> L[Light damping:<br/>tall, narrow peak at f₀]
A --> M[Moderate damping:<br/>shorter, broader peak]
A --> H[Heavy damping:<br/>low, very broad peak,<br/>slightly below f₀]
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