Differential Equations · step 11 of 14

Oscillations, damping & resonance

The mass–spring equation my″ + cy′ + ky = F(t) covers mechanical and electrical oscillation alike. The discriminant c² − 4mk classifies the unforced behaviour: positive gives overdamping (no oscillation), zero gives critical damping (fastest return without overshoot), negative gives underdamping (decaying oscillation). Forcing at the natural frequency with no damping produces resonance and unbounded growth.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Compute the natural frequencyω₀ = √(k/m) for the undamped system.
Classify with the discriminantc² > 4mk overdamped; c² = 4mk critically damped; c² < 4mk underdamped.
Split the forced solutionTransient (the decaying complementary function) plus steady state (the particular solution, which persists).
Check for resonanceIf the forcing frequency equals ω₀ and c = 0, the particular solution needs a factor of t and the amplitude grows without bound.

Worked example

For y″ + 2y′ + 5y = 0, classify the damping and describe the motion.

Here m = 1, c = 2, k = 5. Discriminant: c² − 4mk = 4 − 20 = −16 < 0.
So the system is underdamped.
Roots: r = (−2 ± 4i)/2 = −1 ± 2i.
Solution: y = e^(−t)(A cos 2t + B sin 2t).

Answer. Underdamped: oscillation at angular frequency 2 with amplitude decaying like e^(−t), so it crosses equilibrium infinitely often while dying away.

Where marks get dropped

These are the specific errors that cost credit on oscillations, damping & resonance questions — QED's rubric penalises each of them separately.

Confusing critical damping with "no oscillation happens fastest". Critical damping returns to equilibrium fastest WITHOUT overshoot; overdamping is slower.
Expecting unbounded resonance in a damped system. Any c > 0 keeps the amplitude finite, though it peaks near ω₀.
Forgetting the transient. The full solution is transient plus steady state, and only the transient dies out.

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Oscillations, damping & resonance — frequently asked questions

What exactly is resonance?

Forcing at the natural frequency, so energy is added in phase every cycle. Undamped, the amplitude grows linearly with t; damped, it peaks at a large but finite value.

Why is critical damping used in engineering?

Because it returns to equilibrium fastest with no overshoot — exactly what you want in car suspensions, door closers and measuring instruments.

How does this apply to circuits?

An RLC circuit obeys Lq″ + Rq′ + q/C = V(t), identical in form. Resistance plays the role of damping and inductance the role of mass.

The rest of Differential Equations

Solving and modelling with ODEs, from separable to systems. Each subtopic below has its own method, worked example and mark-losing traps.

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