The Nature of Mathematical Modelling
Consider the motion of a damped, driven harmonic oscillator (such as a mass on a spring, a ball in a well, or a pendulum making small motions)
(a) Under what conditions will the governing equations for small displacements of a particle around an arbitrary 1D potential minimum be simple undamped harmonic motion?
(b) Find the solution to the homogeneous equation, and comment on the possible cases. How does the amplitude depend on the frequency?
(c) Find a particular solution to the inhomogeneous problem by assuming a response at the driving frequency, and plot its magnitude and phase as a function of the driving frequency for m = k = 1, γ = 0.1.
(e) Now find the solution to equation (3.58) by using Laplace transforms. Take the initial condition as x(0) = x˙(0) = 0.