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Damped & Driven Oscillations and Resonance

You’ll be able to

Adding a velocity-dependent force

Real oscillators lose energy. Modeling the loss as a drag force F = −bv proportional to velocity gives the equation m d²x/dt² = −kx − b dx/dt, whose solution for light damping is x(t) = A₀e^(−bt/2m) cos(ω′t + φ). The oscillation continues at a slightly reduced frequency inside an exponentially decaying envelope. The amplitude falls by the same factor in each successive equal time interval, which is the signature of exponential decay and is what a damped oscilloscope trace shows.

Damped oscillation
x(t) = A₀e^(−bt/2m) cos(ω′t + φ) ω′ = √(ω₀² − (b/2m)²) E ∝ A² so E decays as e^(−bt/m)
Because energy goes as amplitude squared, it decays at twice the rate of the amplitude. The damped frequency ω′ is always below the natural frequency ω₀ and falls to zero at critical damping.

Three damping regimes

Underdamped (b small): the system oscillates with decaying amplitude — a plucked guitar string, a child's swing left alone. Critically damped (b at the threshold where ω′ = 0): the system returns to equilibrium in the shortest possible time without overshooting, which is why door closers, analogue meter needles and vehicle suspensions are designed near this point. Overdamped (b large): the system returns without oscillating but sluggishly, taking longer than critical damping. The common misconception is that more damping always means faster settling; beyond critical it means slower.

Driving and resonance

Apply a periodic external force at frequency ω_d and the system settles into a steady oscillation at the driving frequency, not its own. The steady-state amplitude depends sharply on how close ω_d is to the natural frequency ω₀: it peaks at resonance, ω_d ≈ ω₀, where each push arrives in phase with the motion and adds energy on every cycle. How large the peak becomes is set by the damping — light damping gives a tall, narrow resonance peak, heavy damping a low, broad one. With no damping at all the mathematical amplitude grows without bound, which is why every real structure is designed with damping and why soldiers break step crossing a bridge.

Worked example

A damped oscillator's amplitude falls to half its initial value after 10 complete oscillations. Describe qualitatively what happens to its total mechanical energy over the same interval, and to the amplitude after 20 oscillations.

  1. 1.Amplitude decays exponentially, so equal numbers of cycles produce equal fractional reductions.
  2. 2.After another 10 oscillations the amplitude halves again, reaching one quarter of the original.
  3. 3.Energy is proportional to A², so when A has fallen to ½A₀ the energy is (½)² = ¼ of its initial value.
  4. 4.After 20 oscillations, A = ¼A₀ and the energy is (¼)² = 1/16 of the original.
Answer: Amplitude is ½A₀ then ¼A₀; energy is ¼E₀ then E₀/16 — energy decays at twice the exponential rate of amplitude because E ∝ A²
Checkpoint

A car suspension is designed to be approximately critically damped so that it —

On the exam

Resonance is defined by the driving frequency matching the natural frequency, not by any property of the driving force's strength. A weak force applied at resonance can produce a larger response than a strong force applied far from it — which is the whole reason resonance is worth a name.

Checkpoint

Increasing the damping of a driven oscillator affects the resonance peak by making it —

Answer the 2 checkpoints as you read.

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