Lab Assignment 1: Damped Harmonic Oscillation with a Driving Force

Assignment from Wednesday's class

Consider the differential equation for damped harmonic oscillation with a sinusoidal driving force:  

(You can choose to use another version of this equation, if you have the urge to call your constants something else...)

Use Mathematica to generate a series of graphs (or animations, if you want to get fancy) that illustrate the following:

Beating

Show an example of a solution that yields "beats".  Give an example of both a damped and an undamped case and briefly describe how they differ.  (For the damped case, you will probably want to stick with a relatively small damping coefficient.)

Resonance

Show an example of a solution that demonstrates "resonance".  Give an example of both a damped and an undamped case and briefly describe how they differ.  (If you run into a lot of problems with complex solutions, you could always use NDSolve instead of DSolve.)

Damping

Show a series of examples (set of graphs, all on one graph, or animation - your choice) that demonstrate the basic damping states (underdamped, critically damped, overdamped).  Show one set of examples for the undriven case (i.e., make the equation homogeneous) and one set for the driven case (as given) and briefly describe how they differ.  (You might want to use NDSolve to approximate the solutions for the damped case.  Solving for the general form with a parameter of in it takes an awful long time on my pathetic home computer...)