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RE: Forced damped pendulum



Oren,
I see your point and coming up with the diff. eq. is the easy
part. It is the analysis that is harder. My point is that you can start
simple, with simple approximations, and then eliminate them as you go
along. The phase plots offer a great deal of intuitive analysis. In
fact, for a high school student it is probably better than some awkward
mathematical expression. Plus, they are easy to learn and make the plots.
The differential equation to start with should be theta double dot +
damping coeff. * theta dot + theta = driving term (usually sinusoidal).
However, the real case is a simple nonlinear problem. In fact, the
pendulum is the example that my teacher of a nonlinear dynamics used all
semester long and built on until we reached a driven, damped pendulum. I
am not saying that the student has to learn all of it, but as an advanced
high school student with decent math abilities, he should be able to
tackle this with some help. You are mostly learning differential equation
techniques but then go on to make the solution graphical for an easier
analysis especially of the physics of the problem. You should get
different behaviors based on the amplitude of the driver because the
damping sucks energy out at a constant rate but the driver puts it back in
at varying rates.
I think it is a nice project for an advnaced high school student
and wish that person luck and "happy hunting" no matter what their goals
are, because there are so many things one can do with it.


Sam Held


sheld@utk.edu





On Mon, 8 Jun 1998, Quist, Oren Phys wrote:

I agree.

But, it is my opinion (humble though it may be) that it is the normal
solution to the standard differential equation that best leads to an
intuitive understanding of the significance of the various terms --
damping, driving force, driving frequency, or whatever. Numerical
techniques can yield interesting answers, but much of the intuitive
connections get lost in the details.
Oren Quist

----------
From: Sam Held
Sent: Tuesday, June 09, 1998 12:45 PM
To: QuistO; RAUBERJ; phys-l
Subject: RE: Forced damped pendulum


Oren,
After giving this problem thought, I noticed no one corrected me
that we cannot use Lagrangians here due to the damping term. I guess I
wsn't the only one without the morning coffee before answering e-mail.
However, learning simple nonlinear diff eq. methods like phase plots
should not be hard. One can even lessen the learning curve by first
teaching the regular pendulum, then damped (or dirven), and finally
driven
(or damped). Then most numerical method books will tell you how to
numerically solve ODEs (Ordinary Differential Equations) and some do
PDEs (Partial Differential Equations). Learning some about a computing
language (like C++ for example) would be good to put on the old college
application.


Sam Held

sheld@utk.edu



On Mon, 8 Jun 1998, Quist, Oren Phys wrote:

I suspect that worrying about types of damping, non-linear terms, and


using Lagrangians is way off base for this high school student. I
think
referencing a source on solving the normal DSHM equations would be
sufficient.

I would recommend:

MECHANICS, by Symon, Addison Wesley, 1960 (also 1971), p 47 +

or a book by Becker -- I can't find mine -- who borrowed it??

or I suspect most classical mechanics books (Marion?) etc. will handle
the solution particulars of the resulting DSHM differential equation.

I hope this helps.
Oren Quist, SDSU


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