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Physics · University

Driven oscillations, damping and resonance

A linear mass-spring oscillator obeys mx″+kx=0. Its natural angular frequency is sqrt(k/m), not k/m.

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Prerequisites

Second-order differential equations and complex numbers.

Study notes

Free motion

A linear mass-spring oscillator obeys mx″+kx=0. Its natural angular frequency is sqrt(k/m), not k/m.

Damping

A viscous damping force −bx′ converts the equation to mx″+bx′+kx=0. In underdamping, the oscillation envelope decays exponentially.

Driving

A periodic external force can transfer energy to the oscillator. The steady response depends on drive frequency, damping and phase.

Resonance

A finite drive does not imply infinite amplitude in a damped physical oscillator. Transient motion and nonlinear limits matter.

Worked examples

For m=2 kg and k=18 N/m, find the undamped angular frequency.

ω0=sqrt(k/m)=sqrt(9).

3 rad/s

Which force represents linear viscous damping?

The force opposes velocity and removes mechanical energy.

−b times velocity with b positive

Check your understanding

  1. An underdamped amplitude follows A(t)=8 exp(−t/2) cm. Find the exponential decay time constant.

  2. What distinguishes a transient from a steady driven response?

    1. A transient must last forever
    2. A steady response has no frequency
    3. A transient depends on initial conditions and may decay
    4. A steady response cannot lose energy

  3. Why does energy not grow without limit in an ordinary damped oscillator driven near resonance?

  4. If k is multiplied by 4 at fixed mass, by what factor does ω0 change?

Answers and marking guidance

Answers and marking notes
  1. 2 s

    Write the envelope as A0 exp(−t/τ); τ=2 s.

  2. A transient depends on initial conditions and may decay

    Damping can remove the initial-condition contribution while driving maintains a steady oscillation.

  3. In steady state, average driving power balances dissipated power. Damping limits the linear response, and large-amplitude nonlinearities can further change it.

    Credit energy balance and avoidance of an undamped infinite-amplitude generalization.

  4. 2

    ω0 scales as sqrt(k).

Further reading and review references

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