Forced Spring Systems

Part 3: Resonance

1. Continue changing w to get even closer to k. What happens to the amplitude of the solution? What happens to the frequency of the beats? What sort of solution function do you seem to be approaching? You are exploring the onset of resonance, which means driving the system at its natural frequency.
2. Redefine the initial value problem so that w = k, that is, so the system is being driven at its natural frequency:
   y'' + k² y = F₀ cos kt, y(0) = 0, y'(0) = 0.

   Find the symbolic solution of this initial value problem. You may use your helper application.

3. Plot the solution and its trajectory in the phase plane for k = 5 and F₀ = 1. What feature of the symbolic solution explains the growing amplitudes? Will the growth in amplitude ever stop (as it did in the beats setting)? Explain.