Worked Problem Assignment 5#

Due 3 Nov 23

We have developed an approach to find the normal modes of a linear oscillating systems. For this problem you will choose a system that oscillates with at least 3 oscillators and explore the motion and dynamics of that system. You cannot choose coupled springs; it must be a system that you have not already explored in class.

Updated Rubric

Remember that a worked problem means the scale of a detailed (e.g., parts a-h) homework problem.

Starting your Analysis#

You will need to orient us to your problem as there are many potential electrostatic situations. You will need to develop an analytical tractable situation that can also be explored using the numerical methods we have developed. You will need to do the following:

  1. Write a paragraph or two describing the system you are investigating. Make sure to explain the parameters and variables.

  2. Explain what you are going to do and why. If you can point us to where things are being done, that’s helpful.

  3. You need to clearly set up your analytical problem. This means write down and explain the equations you are constructing.

Investigating the System#

After you set things up, please conduct an investigation using the tools we have developed in class. The order in which it proceeds is up to you, but make sure the flow is logical and clear. We want to understand how your investigations are going to help you be successful with your projects.

Assignment#

  1. Choose an oscillatory system with at least 3 oscillators that you can solve with the normal mode approach.

  2. Setup and develop the matrix representation of your problem.

  3. Find the normal modes of the system and explain them.

  4. Plot the normal modes of the system.

  5. Plot the motion of the system for a given set of initial conditions (this might involve numerical integration).

Looking ahead (earn extra credit)#

If you are looking into these systems and finding them interesting here’s two things you can do to earn extra credit.

  1. Make this work for \(N\) oscillators of your choosing

  2. (challenging) Try to find how the normal modes can describe the motion of the system for a given set of initial conditions.

Submitting your work#

You will upload your work to D2L. It should be a single PDF of your work. If you also write code, please submit the associated files as well. If you do all of this in a Jupyter notebook, please export the notebook as a PDF and submit the notebook and PDF file. If you have any questions, please ask. We want you to enjoy working on these problems.