]> COSC1229/1479 Computational Science

Computational Science
COSC1229/1479
Assignment 2
Title: Linear and Non-Linear (Chaotic) Oscillator Systems
Due Date : 8/10/2004, 5pm
Assessment : 30%

Overview

The assignment is to investigate linear and non-linear oscillators - or more simply spring-mass systems.

You have seen in the lectures the example of a one-spring and one mass system produce simple harmonic motion. This means that only a single mode frequency of ω 2 = k m exists, and you can see that in the simulations on the My Physics Lab website shown in lectures. You also saw that for two masses and three springs, there exist two modes, namely ω 1 2 = k m and ω 2 2 = 3 k m so that if we were to look at the frequency spectrum of the oscillation, we should see two peaks appear, one at frequency ω , and at 3 ω

In this assignment, we will be looking at the behaviour of spring systems, including linear springs, coupled linear springs, and non-linear springs in which chaotic motion occurs. There are five parts, each worth 6 marks and each of which requires some programming and a report.

Details

  1. Extend the linear spring-mass code, or write your own, to allow position curves, velocity curves and phase plots. Investigate a single spring-mass system and compare with results in Acheson chapter 4.
  2. As discussed in lectures, coupled linear oscillators have normal modes. Extend the program to allow a system of three springs and two masses and investigate it as discussed in Acheson Chapter 5. In particular look for the slow normal mode and the fast normal modes with frequencies ω and 3 ω . For the slow mode ω , move both masses in one direction and release them. For the fast mode 3 ω , move the masses in opposite directions and release them. Compare your results with the theory and results discussed in Acheson section 5.4.
  3. Using supplied or other code for Fourier Transform apply the Fourier transform to x and x ¨ and see if the expected normal modes are present. Extend the oscillator program to include a (dynamically updated) frequency graph.
  4. Generalise your spring-mass/oscillator program to handle n masses and n+1 springs and explore how many modes are present and whether they are expected.
  5. Now we move to non-linear springs. Acheson discusses a non-linear spring - the forced cubic oscillator - in Chapter 11 on page 152. Implement the forced cubic oscillator and investigate position curves and phase plots for a variety of initial conditions, including initial conditions which lead to chaotic behaviour. Are normal modes observed?

You may use supplied program code as a basis for your assignment. For reports you may need to use different tools to your program for generating graphs.Your reports should use MathML for mathematics.

Assessment

Assessment will be based on reports and program code. As a guide, the code and reports will be equally weighted - both 15 marks.

Submission

Your assignment submission must include: a tar file containing your report(s), source code and any necessary data which must be submitted using turnin in a similar way to assignment 1. Instructions on how to do this here.

Valid XHTML 1.1!
Last modifed: $Date: 2004/09/14 10:30:31$