brian@hpcvlx.cv.hp.com (Brian Phillips) (06/01/91)
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
to a new semi-stable angle? What constitutes stability in such a
system? While I am interested in examining system behavior with
regard to its transition from stability to (chaos?) instability, I
am most curious about the parameter domains for values creating
these transitions. ie. If I vary the length of the pendulum slightly
and a previously stable pendulum simulation becomes otherwise, what
will I see if I refine such changes? What does a region about a stable
choice that includes unstable selections look like? Are the boundaries
well-behaved or are they more interesting? What if I look at the
frequency of the forcing function much the same way?
I'm not particularly interested in a purely mathematical exercise
where the results are merely artifacts derived from the simulation
method and its accuracy. Any guidance would be appreciated.
Brian Phillips
brian@hpcvxbjp.cv.hp.com
steve@Pkg.Mcc.COM (Steve Madere) (06/04/91)
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