Path: utzoo!utgpu!news-server.csri.toronto.edu!rpi!zaphod.mps.ohio-state.edu!swrinde!cs.utexas.edu!ut-emx!lad-shrike!milano!cadillac!Pkg.Mcc.COM!steve
From: steve@Pkg.Mcc.COM (Steve Madere)
Newsgroups: comp.theory.dynamic-sys
Subject: Re: Pendulum Help
Message-ID: <1991Jun3.175911@Pkg.Mcc.COM>
Date: 3 Jun 91 22:59:11 GMT
References: <110770003@hpcvlx.cv.hp.com>
Sender: news@cadillac.CAD.MCC.COM
Reply-To: steve@Pkg.Mcc.COM (Steve Madere)
Lines: 35

In article <110770003@hpcvlx.cv.hp.com>, brian@hpcvlx.cv.hp.com (Brian
Phillips) writes:
| 
| 
|        An old classical mechanics problem... 
| 
|            I've been toying with a mathematical simulation of an
|        inverted pendulum ( a rigid bar rotating in the plane about
|        a pivot subjected to an up-down forcing function ).  The
forcing
|        function is sinusoidal and the pendulum is initially ajar at
some
|        small angle from top-dead-center (unstable equilibrium).  As I
|        understand it, such a system can maintain itself without
rotation
|        if the correct conditions are established.  Is this correct?
|            Can such a pendulum remain upright?  Does it periodically
rotate

We solved this problem analytically in a graduate classical
mechanics course that I took at UCSD.  As I recall the only
condition is that the driving frequency be at least twice
the natural small angle oscillation frequency of the pendulum.
I would assume that at a frequence W slightly less than W0 one
would just see the pendulum fall to the lower half plane.
However, for W >= W0, if the pendulum starts at the top it can
remain at the top.  As I recall, the top position is actually
only meta-stable and the angle of deviation to which this
meta-stability holds is dependent on W/W0.

Send me e-mail if you need more info, I can get my class notes
and re-work it if you would like.

Steve Madere
steve@pkg.mcc.com