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Exploring the Stability of Inverted Multilinked Pendulums
Peder Thusgaard Ruhoff
Odense University
Download talk material:
Recently, Mullin and Acheson (Nature 366, 215-216, 1993) demonstrated that it
is possible in practice to stabilize an inverted triple pendulum by rapid vertical
oscillations of its suspension point. More recently, they even showed that one can
stabilize a "stiff" rope using the same technique (New Scientist
157, 1998).
In a remarkable theorem, Acheson (Proc. Roy. Soc. Lond. A 443, 239-245,
1993) gave a theoretical explanation for the above observations. He showed that in the
domain of linear theory the stability is related to the largest and smallest of the normal
mode frequencies of the multilinked pendulum in the noninverted equilibrium state. In this
talk, I demonstrate how Mathematica can be used to explore these stability
properties. First, I present the stability theorem by Acheson. Then I introduce a package
for normal mode analysis which is used in determining the regions of linear stability. I
then proceed to demonstrate how the stability regions can be computed by numerically
integrating the equations of motion. Finally, I discuss how to visualize these
phenomena using animations, phase space plots, sounds, polar symmetry plots, and other
techniques.
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