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lec19 (2) - Example Spinning Hoop with Sliding...

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Example: Spinning Hoop with Sliding Mass (Continued) 1 2.003J/1.053J Dynamics and Control I, Spring 2007 Professor Thomas Peacock 4/25/2007 Lecture 19 Lagrangian Dynamics: Spinning Hoop with Sliding Mass, Linearization of Equations of Motion, and Bifurcations Example: Spinning Hoop with Sliding Mass (Continued) Lagrangian 1 L = m ( a 2 sin 2 θ Ω 2 + a 2 ˙ θ 2 ) + ( mga cos θ ) 2 Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J Dynamics and Control I, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Figure 1: Spinning hoop with sliding mass. Figure by MIT OCW.
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Example: Spinning Hoop with Sliding Mass (Continued) 2 Lagrange’s Equation for θ 2 g ¨ θ sin θ cos θ Ω + sin θ = 0 (1) a Equilibrium Points θ = 0 ,π, arccos g a Ω 2 The third point only exists if g a Ω 2 1. Stability Analysis Stability around θ e = arccos( g/a Ω 2 ) θ = arccos g e 2 a Ω stable. θ = θ e + ¨+ Ω 2 sin 2 θ e = 0 Oscillatory Behavior. Stability around θ e = 0 ˙ ¨ θ e = 0 consider small changes θ = θ e + , θ = ˙ , θ = ¨ g ¨ Ω 2 g + = 0 ¨ + ( Ω 2 ) = 0 (2) a a Ω 2 : Controlled parameter. If Ω 2 is small, behavior is stable. If Ω 2 > g a , behav- ior is unstable. Stable: Ω 2 < g a Unstable: Ω 2 > g a If we look for a solution to Equation 1 of the form = Ae λt , we have: λ 2 2 A λt g e + ( Ω ) Ae λt = 0 a g λ = ± 2 ) a If Ω 2 < g a , λ is imaginary oscillation. 2 g If Ω > a , λ is real exponential growth. Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J Dynamics and Control I, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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3 Example: Spinning Hoop with Sliding Mass (Continued) Stability around θ e = π θ e = π θ = θ e + = π + From Equation (1) ¨ sin( π + )cos( π + 2 + g sin( π + ) = 0 a sin( π + ) = sin π cos + cos π sin ≈ − cos( π + ) =
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  • Fall '08
  • MarkusBuehler
  • MIT OpenCourseWare, Massachusetts Institute of Technology, Equilibrium point, Stability theory, DD Month YYYY, Nicolas Hadjiconstantinou

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