(1) Consider the Un-damped Oscillator with Forcing defined by the initial value problem
u'' + 4u = F(t), u(0) = 0 and u'(0) = 0
with forcing terms F(t) described below.
(a) The forcing term is periodic with frequency equal to the natural frequency. (Resonance)
(b) The forcing term is periodic with frequency "close to" the natural frequency. (Beats)
For each case:
(i) Define a forcing term matching the description.
(ii) Find the solution.
(iii) Plot the solution over a long enough time period that a pattern is apparent.
(iv) Describe the behavior of the solution in terms of oscillations and amplitudes.
(2) Consider the Damped Oscillator with Forcing defined by the initial value problem
u'' + 2u' + 5u = F(t), u(0) = 0 and u'(0) = 0
with forcing terms F(t) described below.
(a) Find a nonzero forcing term which results in a solution that tends to zero as t goes to infinity.
(b) Find a forcing term which results in a solution that becomes unbounded as t goes to infinity.
(c) Find a forcing term which results in an oscillatory solution that does not tend to zero or become unbounded at t goes to infinity.
For each case:
(i) Define a forcing term which results in a solution with the requested properties.
(ii) Plot the solution over a long enough period that the behavior is apparent.
Notes on Format:
- You may work in groups of no more than 4 people.
- Five of the points will be based on presentation.
- You should hand in three pages:
one page for problem 1(a)
one page for problem 1(b)
one page for problem 2 (you should be able to fit all three parts on one page).
- You may write up the lab in any text editor you want but make sure you can cut and paste Maple graphics into the document and describe mathematical equations.
- I suggest you hand in a printed Maple document.
- Only hand in the requested information. I do not want to see the Maple commands.
- To enter the text editor in Maple, click on the "T" button from the tool bar.
- To get a Maple prompt, click on the "[>" button from the tool bar.