In this assignment we will solve differential equations and use
Maple to explore their solutions. This problem was taken from
this project description (Simiode is an acronym for
"a Systemic Initiative for Modeling Investigations and Opportunities
with Differential Equations"). They have a huge set of interesting projects
that we can explore. The one we will look at here is one about resonance
and how it can lead to catastrophic behavior in a system.
Equations and mathematics can be used to model physical systems, and although
the questions we will just be asking the computer to solve differential equations,
we can then ask if the equations represent what is happening in our system that
we are trying to model.
The question we are going to explore with equations is: "When can resonance in
a system lead to solutions with "catastrophic" behavior?"
An example of this is using the human voice to shatter a wine glass
by matching the frequency of the sound with the resonant frequency of the glass.
Yes, it is possible to do this. An experiment was done on MythBusters and you
can watch a
YouTube video to see how it they
made it happen.
We will look at differential equations that are used to model the behavior of
these systems. For most values of the parameters in these equations, the graph
the solution looks bounded, but if you change those parameters the
graphs can look very different and those are exactly the solutions
that correspond to what we observe as "extreme" behavior in our physical system.
If position of our equations will be represented by an equation $y(t)$,
then the speed or velocity is $\frac{d}{dt}y(t)$ and the acceleration is
$\frac{d^2}{dt^2} y(t)$.
For example try the following commands and ensure that you understand what they do:
> S1:=dsolve(diff(y(t),t)=-y(t)+t);
> S2:=dsolve(diff(diff(y(t),t),t)=-y(t)+t);
> subs(_C1 = 4, subs(S1, y(t)));
> subs([_C1 = 4, _C2 = -1], subs(S2, y(t)));
Question #1:
Explore the equation
$$y''(t) + 16 y(t) = 3 sin(\omega t)$$
where $\omega$ is a parameter and with the initial conditions
$y(0) = y'(0) = 0$. The interpretation of this equation
in terms of forces on a bridge is described at
this project description and it corresponds to a system of equations where
there is no damping.
Solve and plot the solutions to this equation for
$\omega \in \{1,2,3,4,5,6,7,8\}$ in the range of $0 \leq t \leq 20$.
(a) Looking at the solution when $\omega = 8$, give a description of the graph.
(b) Describe the solution when $\omega = 5$ and how it differs from the solution when $\omega = 8$.
(c) Do this again for the case when $\omega = 4$ and compare it to the previous two solutions.
Question #2:
The damping in the system is taken into account in the equation by
adding a term which is proportional to $\frac{d}{dt}y(t)$.
Consider the equation
$$y''(t) + 2 y'(t) + 16 y(t) = 3 sin(\omega t)$$
where again $\omega$ is a parameter.
Solve and plot the solution for this equation in the following cases.
(a) $\omega = 8$, $y(0) = y'(0) = 0$
(b) $\omega = 4$, $y(0) = y'(0) = 0$
(c) $\omega = 4$, $y(0) = 0$, $y'(0) = 5$
(d) $\omega = 4$, $y(0) = 1$, $y'(0) = 0$
Give a description of the graphs of the solutions and describe
what you would expect to see if this equation were modelling the
behavior of a bridge in each of these cases.
You should open up a new worksheet and start from scratch. You will have to save
your work in a file and upload that file on to the course
moodle. Your
solution should be a sequence of commands where it is easy to change the input
string and after you execute the sequence of commands you should have the
correct output string. Please add documentation to your worksheet to explain how it
works. Just a few sentences is sufficient, but imagine that you were opening up the
worksheet for the first time and wanted to know what it did. You will be marked down
if what you write is not clear and coherent.
You should finish your assignment before class Thursday, November 15 by 11:59pm.
Assignments submitted after this date will be assessed a penalty of 10% per day.