# The Nature of Mathematical Modelling

## 3.1

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.