Laplace transforms find broad applications in the modelling of oscillators for energy harvesting (EH). Consider the displacement of a mass-spring oscillator resting on a frictionless surface, governed by the ODE Ay + By = f(t) .... (1) (a) Suppose A = B = 1, and the force f(t) represents a push to the left of the mass over a time period from t = 2 to 1 = 2x + a for some a> 2. Let f(t) be f(t) = -[u(t-2n) - u(t (2 + a))] such that a small a corresponds to a push of short duration, while a large a is a push of long duration. The function u(t) is the unit step function. Use Laplace Transform to solve the initial value problem y(0) = 1, y(0) = 0. (b) Use MATLAB to plot the solution in part (a). 16 32' c) Suppose A = B = 2 and the force f(t) represents an impulse force. Write down the governing equations for the displacement of the mass, if the mass is initially released from rest at 3 displacement units from the equilibrium position, and then struck by 4 force units at t = 2 time units later. [Solve the governing equation for practice!] Consider now the more realistic scenario of a frictional surface, so that the displacement of the mass-spring oscillator is damped.