Computational Methods Junior Honours : Checkpoint 4

Example code:

There is no example code for this checkpoint as the coding is relatively straightforwards.
Your solution for Checkpoint 3, Test1 should be a good starting point.

The main purpose of the checkpoint is to explore the system and demonstrate that the numerical simulation approach works for a system we know how to solve analytically before we apply it to systems that are not solvable.

Your task: Driven Damped Simple harmonic motion

Review your notes from Physics 1Ah, vibrations and waves, to find the equations and long time solution for the driven, damped oscillator.

o Note that, unlike Scientific Programming checkpoint 4, your simulation will not implement this equation directly and so that checkpoint does not provide a good starting point for this exercise.

· Review the summary of simple harmonic motion in the background notes.

· Write a code to numerically integrate the equation of motion for a particle mass m subject to a damping force, a spring force and a sinusoidal driving force.

$\begin{displaymath} m\ddot{x} = -kx + F_D \cos (\omega_D t) -b \dot{x}\end{displaymath}$

You should once again use the Particle class with all y and z positions, velocities, forces set to zero and use the leap-frog algorithm.

o You should produce graphs of position and velocity against time.

o Note that the particle position, velocity and acceleration can be expressed by a single number rather than a vector.

§ Use the x-component of your vectors to represent this

o As usual you should write your code to read its initial state from an input file
Think what this file should contain. E.g. b, F_D, m, k, ω_D, ω₀ , timestep and particle parameters

Study the spring system:

Convince yourself you solve SHM correctly
Try starting from rest and also starting with non-zero initial position and velocity - explain the transients to your demonstrator.
Set m=1 , k=1 and F_D =2 and study how quickly the system reaches its steady state behaviour for various values of ω _Dand b .
for b=1 and b=3 which value of ω _D gives the largest response - i.e. maximum amplitude at long times (answer to two sig.figs)?

Save a plot of the trajectory for each of these situations

With b=0, Add a second particle which starts at rest. Introduce a force on the particles F_12 = - F_21 = C (x1 – x2)
Take C=0.05 and plot the trajectory of both

Advanced:

Compute, via simulation, the amplitude of the steady state solution as a function of ω _D for b=0.1 and 2
Compare this to the formula for the amplitude response of damped SHM in the background notes
Modify your code to produce a plot of the amplitude frequency-response curves for this data