Damping and Resonance

Spec mapping: OCR H556 Module 5.3 — Oscillations (free and forced oscillations; natural frequency; light, critical and heavy damping; displacement-time graphs; resonance and amplitude-frequency curves; effect of damping on resonance peak; examples of useful and destructive resonance; PAG 11 anchor — pendulum or mass-spring SHM investigation). Refer to the official OCR H556 specification document for exact wording.

In the previous five 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 fluid drag. 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, often to levels far larger than the driving amplitude itself. This is resonance. Understanding damping and resonance is essential for everything from the design of car suspension to the engineering of skyscrapers, the tuning of radios, the choice of MRI frequencies, and the avoidance of bridge collapses. It is OCR A-Level Physics A Module 5.3 — the bridge between the theoretical SHM of lessons 3-5 and the real-world oscillating systems that engineers and physicists actually have to design.

This lesson is also the PAG 11 anchor: the practical investigation of pendulum or mass-spring SHM (period dependence on length / mass), and the qualitative observation of damping in air vs in water.

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Free Oscillations

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 $f_0$f0​ — a property of the system, determined by its mass, stiffness and geometry. For a mass on a spring, $\omega_0^2 = k/m$ω02​=k/m; for a small-angle pendulum, $\omega_0^2 = g/L$ω02​=g/L; for a charged-LC circuit, $\omega_0^2 = 1/(LC)$ω02​=1/(LC). The natural frequency is intrinsic — it does not depend on amplitude or initial conditions (for true SHM).

In the ideal (no-friction, no-air-resistance) case, a free oscillation would continue forever with constant amplitude. In practice, energy always leaks away to the surroundings — heat from internal friction in the spring, sound from a vibrating string, viscous drag from air on a pendulum bob — and the oscillation dies down. The only physical systems that come close to ideal-undamped behaviour are atomic and molecular vibrations, and even those eventually radiate energy as photons.

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Damped Oscillations

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 ($\vec F_\text{drag} = -b\vec v$Fdrag​=−bv for linear viscous damping, with $b$b a positive damping coefficient). It is zero when the oscillator is momentarily stationary at the extremes ($v = 0$v=0) and maximum when the oscillator is moving fastest at equilibrium ($v = v_\text{max}$v=vmax​). This is why each cycle of a damped oscillator is slightly smaller in amplitude than the previous one — energy is being continuously bled off whenever the oscillator is moving.

The energy lost per cycle in linear viscous damping is approximately $\Delta E = \pi b\omega A^2$ΔE=πbωA2 (for light damping), which gives the geometric decay law: each cycle loses a fixed fraction of the current energy, leading to exponential amplitude decay.

Three Degrees of Damping

OCR categorises damping into three important regimes:

Type	Description	Displacement-time appearance
Light damping	Small resistive force. Amplitude decays slowly (exponentially) over many cycles.	Familiar sine wave inside a decaying exponential envelope.
Critical damping	Resistive force just large enough to return the system to equilibrium in the shortest possible time without oscillating.	Monotonic exponential return to zero, no overshoot.
Heavy (over) damping	Resistive force so large the system returns to equilibrium slowly and without oscillating.	Very slow exponential creep back to equilibrium.

flowchart TD
  S[Damped oscillator: free release from x = A] --> Q{Damping level?}
  Q -- Very low --> L["Light damping: many oscillations,<br/>amplitude decays exponentially,<br/>frequency ≈ ω₀"]
  Q -- Just enough to prevent oscillation --> C["Critical damping: fastest<br/>return to equilibrium with<br/>NO overshoot, NO oscillation"]
  Q -- Above critical --> H["Heavy/over damping:<br/>NO oscillation,<br/>slow exponential creep"]
  L --> A[Real-world example: pendulum in air]
  C --> B[Real-world example: car suspension, door closer, voltmeter]
  H --> D[Real-world example: pendulum in oil, badly-designed shock absorber]

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 (the pneumatic kind that ease the door shut), galvanometer needles, analogue voltmeter pointers, seismometer pendulums. Engineers often specify damping ratios just slightly under critical (0.7-0.9 of critical), trading a tiny overshoot for a faster initial return.

Light Damping in Detail

For lightly damped SHM, the amplitude of successive cycles decays exponentially:

$A(t) = A_0\,e^{-\gamma t}$A(t)=A0​e−γt

for some positive decay constant $\gamma$γ (not the spring constant; the notation $k$k is unfortunate and overloaded). The number of cycles over which the amplitude decays to $1/e$1/e of its initial value (i.e. $\tau = 1/\gamma$τ=1/γ in seconds) is a measure of how heavily the system is damped. A high-quality pendulum clock might decay over hundreds of cycles ($\gamma$γ very small); a pendulum in syrup might decay in less than one ($\gamma$γ very large).

The frequency of the lightly damped oscillation is almost the same as the natural frequency — a tiny bit lower, by a factor $\sqrt{1 - (\gamma/\omega_0)^2}$1−(γ/ω0​)2​ — but for A-Level purposes you may treat them as equal. Heavy damping reduces the oscillation frequency until it stops oscillating altogether at critical damping.

The energy decays at twice the rate of the amplitude (because $E \propto A^2$E∝A2 from lesson 5):

$E(t) = E_0\,e^{-2\gamma t}$E(t)=E0​e−2γt

So if light damping makes the amplitude drop by $5\%$5% per cycle, the energy drops by about $10\%$10% per cycle — exactly the relationship explored in Worked Example 4 of lesson 5.

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 approximately the correct period, (ii) successive peaks that decrease in amplitude (each smaller than the last), (iii) a smooth exponential envelope enclosing the peaks and troughs symmetrically (mirror image above and below the time axis). Many candidates draw a linear decline instead of exponential — losing marks.

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Forced Oscillations

A forced oscillation occurs when an external periodic driver pushes the system. Examples:

• Pushing a child on a swing at regular intervals.
• Wind blowing across a suspension bridge or chimney (vortex shedding).
• An AC voltage driving a tuned radio receiver (LC circuit).
• A vibrating loudspeaker attached to a sounding box.
• Seismic ground vibration driving a building during an earthquake.
• A vibrating wing-strut driving aircraft fuselage panels.

The key feature is that the driver has its own frequency — the driving frequency $f_d$fd​ — which is generally different from the natural frequency $f_0$f0​ of the oscillator.

After an initial transient in which the oscillator's free response (at $f_0$f0​) interacts with the driven response (at $f_d$fd​), the system settles into steady-state motion at the driver's frequency $f_d$fd​, with some amplitude determined by how closely $f_d$fd​ matches $f_0$f0​ and how heavily the system is damped. The transient typically dies away in a time of order $1/\gamma$1/γ (the damping decay time).

The steady-state amplitude is set by the balance: power input from the driver equals power dissipated by damping. At low driving frequencies the driver and oscillator are in phase, and the amplitude is approximately equal to the driver's amplitude divided by stiffness (the static deflection). At very high frequencies the inertia of the mass cannot keep up with the driver, and the amplitude falls off. At resonance, the amplitude can become very large.

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Resonance

Resonance is what happens when the driving frequency is equal to the natural frequency of the oscillator: $f_d = f_0$fd​=f0​. At resonance the amplitude of the steady-state motion reaches a maximum — often dramatically larger than the driving amplitude itself.

Why Does Resonance Happen?

At resonance, every push from the driver arrives in phase with the oscillator's velocity, so every push does positive work ($\vec F \cdot \vec v > 0$F⋅v>0 throughout each cycle). In any other regime, the pushes sometimes add energy and sometimes remove it (because $F$F and $v$v go in and out of phase), and the amplitude settles to a smaller value. At $f_d = f_0$fd​=f0​, the pushes always arrive at the right moment, 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\times$10×, $100\times$100× or even $1000\times$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. The ratio of resonance amplitude to driving amplitude is called the quality factor $Q$Q of the oscillator, and it is roughly $Q \approx \omega_0/(2\gamma)$Q≈ω0​/(2γ) for light damping. High-$Q$Q oscillators (quartz crystals, atomic clocks) have very sharp resonance peaks; low-$Q$Q ones (car suspensions, building tuned-mass-dampers) have broad, flat peaks.

Amplitude-Frequency Graphs

Plotting amplitude $A$A against driving frequency $f_d$fd​ produces the characteristic resonance curve. Its shape depends critically on the damping level:

• Very light damping: tall, sharp peak at $f_0$f0​; amplitude falls off rapidly either side. Common to high-Q systems (quartz crystals, $Q \sim 10^6$Q∼106).
• Moderate damping: shorter, broader peak, peak still very close to $f_0$f0​. Common to mid-Q systems (musical instruments, $Q \sim 100$Q∼100).
• Heavy damping: very broad, low peak, shifted slightly below $f_0$f0​ (the peak frequency drops to $\omega_p = \sqrt{\omega_0^2 - 2\gamma^2}$ωp​=ω02​−2γ2​ for damped oscillators). Common to low-Q systems (car suspension, $Q \sim 1$Q∼1-$2$2).

Exam Tip: OCR commonly asks candidates to sketch three resonance curves on the same axes for different damping levels, and to identify the natural frequency. Always: (i) mark $f_0$f0​ on the frequency axis; (ii) place the peak of the lightest-damped curve at $f_0$f0​; (iii) make the other curves shorter and broader; (iv) draw the heavy-damping peak slightly to the left of $f_0$f0​ (peak frequency drops with damping). All curves should approach zero at $f_d \to \infty$fd​→∞ and a constant low value at $f_d \to 0$fd​→0.

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Everyday Resonance — Good and Bad

Resonance is sometimes useful, sometimes disastrous.

Useful resonance

• Musical instruments. Violin/guitar bodies resonate at string frequencies (Helmholtz + panel modes), amplifying sound. Organ pipes resonate at their fundamental and harmonics.
• Radios and TVs. An LC tuning circuit ($\omega_0 = 1/\sqrt{LC}$ω0​=1/LC​) is set to the desired station's frequency; the variable capacitor shifts $\omega_0$ω0​ across the band.
• Microwave ovens. $2.45$2.45 GHz microwaves resonate with water-molecule rotational modes, heating food.
• MRI scanners. RF pulses at the Larmor frequency ($\sim 64$∼64 MHz at $1.5$1.5 T) resonate with hydrogen nuclei to produce a signal.
• Quartz watches and atomic clocks. Quartz crystals resonate at $32\,768$32768 Hz ($= 2^{15}$=215); caesium-133 atomic clocks resonate at $9.192\,631\,770$9.192631770 GHz — the SI definition of the second.

Destructive resonance

• Tacoma Narrows Bridge (1940). Wind-driven torsional resonance (aeroelastic flutter); the bridge collapsed within hours.
• Millennium Bridge, London (2000). Pedestrian-step resonance at the bridge's $\sim 1$∼1 Hz lateral mode; closed within days, retrofitted with tuned mass dampers.
• Wine glass shattering to a singer. Voice matched to the glass's $400$400-$1000$1000 Hz natural frequency at $\sim 100$∼100 dB.
• Car bodywork rattle when engine vibration matches a panel's natural frequency — manufacturers tune panel stiffness to avoid this.

In engineering, damping is deliberately introduced to kill destructive resonance or broaden the resonance curve. The Taipei 101 skyscraper contains a 730-tonne tuned mass damper — a giant pendulum suspended near the top of the building, oscillating at the building's natural frequency in antiphase to absorb energy from wind and earthquake-induced oscillations. London's Millennium Bridge has similar (much smaller) dampers under the deck.

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Worked Example 1 — Resonant Frequency of a Mass on a Spring

A $0.20$0.20 kg mass on a spring of stiffness $32$32 N m$^{-1}$−1 is driven by a small motor. At what frequency does the mass exhibit maximum amplitude?

The natural angular frequency is

$\begin{aligned} \omega_0 &= \sqrt{k/m} = \sqrt{32/0.20} = \sqrt{160} = 12.65 \text{ rad s}^{-1} \\ f_0 &= \frac{\omega_0}{2\pi} = \frac{12.65}{6.283} = 2.01 \text{ Hz} \end{aligned}$ω0​f0​​=k/m​=32/0.20​=160​=12.65 rad s−1=2πω0​​=6.28312.65​=2.01 Hz​

So resonance occurs when the motor frequency is driven at about $2$2 Hz. (Recall, OCR does not require you to memorise $T = 2\pi\sqrt{m/k}$T=2πm/k​; you derive $\omega^2 = k/m$ω2=k/m from $a = -\omega^2 x$a=−ω2x and Hooke's law, as in Lesson 3.)

For light damping the resonance peak is sharp and centred almost exactly at $f_0 = 2.01$f0​=2.01 Hz. For moderate damping the peak is lower and broader; for heavy damping the peak shifts to slightly lower frequency.

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Worked Example 2 — Energy Loss at Resonance

A resonating oscillator has a steady-state amplitude of $0.040$0.040 m when driven by a force that does $0.20$0.20 J of work on it per cycle. Estimate the energy dissipated per cycle.

In steady state, energy is neither increasing nor decreasing (the amplitude is constant), so the energy dissipated per cycle must equal the energy put in per cycle:

$E_\text{dissipated per cycle} = E_\text{input per cycle} = 0.20 \text{ J}$Edissipated per cycle​=Einput per cycle​=0.20 J

This is the general principle: at steady-state resonance, the rate of energy input from the driver equals the rate of energy loss to damping. The amplitude is set by exactly this balance: if you increase the damping, the steady-state amplitude drops (so the system can dissipate $0.20$0.20 J per cycle at a smaller swing); if you decrease the damping, the amplitude rises until the balance is restored.

This is also why a heavily damped resonance peak is lower than a lightly damped one: more damping means more dissipation per cycle at a given amplitude, so for the same driving power the amplitude must be smaller.

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Worked Example 3 — Choosing a Damping Level

A car suspension is designed so that when the car hits a bump, the wheel returns to its equilibrium position as quickly as possible without bouncing. Which type of damping is required, and why is light damping unsuitable? Why is heavy damping also a problem?

Required: critical damping (or just under). This gives the shortest possible return time without oscillating.