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Exploring the Stability of Inverted Multilinked Pendulums

Peder Thusgaard Ruhoff
Odense University

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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.