To study the effects of the parameters 
and 
on the differential equation y''(t) + 2
y'(t) + 2y(t)
= F cos(
t).
This lab begins with students solving the forced free harmonic oscillator
| y''(t) + 2y(t)
= F cos(t), |
(1) |
| y(0) = 0, y'(0) = 0. | (2) |
Using trigonometric identities they rewrite the equation in the form
y(t) = A(,
t) sin( t)
sin(t), |
(3) |
where = (
- )/2
is assumed to be 'small'. They plot the envelope A(,
t)sin(t)
and the approximate solution (3) for
= 
and F = 5
on the same graph for various values of .
This part of the lab emphasizes the concept of beats.
The second part of the lab has students looking at the forced damped harmonic oscillator
| y''(t) + y'(t)
+ 2y(t)
= F cos(t), |
(4) |
| y(0) = 1, y'(0) = 0. | (5) |
The solution of (4), (5) is plotted for various values of .
All the other variables, ,
, and
F, are fixed.
Students are required to write a report on the findings. In particular,
they need to describe the behavior (1), (2) as
tends to 0. They also need to explain the meanings of steady state solution
and transient solution for (4), (5) in their report, supporting
these definitions with the graphs they generated.
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