Past exam of the physics 1 paper of the University of Oxford / 2015 / 10 / c

As $\omega$ approaches ${\omega }_0=\sqrt{g/l}$, the driving motion couples resonantly to the pendulum. The particular-solution amplitude has denominator ${\omega }_0^2-{\omega }^2$, so a steady harmonic-response formula alone is singular. For a specified finite initial state, the near-resonant motion instead develops beats, and its exact resonant limit has a growing envelope.

For example, with the vertical zero-angular-velocity initial condition above, taking the limit givesThe ideal undamped linear theory predicts amplitude growing linearly with time, not an infinite displacement at the first instant. In reality the motion leaves the small-angle regime, and damping and nonlinear changes of the pendulum's oscillation period alter the response. Below and above resonance, the sign of the harmonic response coefficient changes, corresponding to a phase reversal.