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

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Unformatted text preview: 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 2 + a 2 2 ) + ( mga cos ) 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. Example: Spinning Hoop with Sliding Mass (Continued) 2 Lagranges Equation for 2 g sin cos + sin = (1) a Equilibrium Points = 0 , , arccos g a 2 The third point only exists if g a 2 1. Stability Analysis Stability around e = arccos( g/a 2 ) = arccos g e 2 a stable. = e + + 2 sin 2 e = 0 Oscillatory Behavior. Stability around e = 0 e = 0 consider small changes = e + , = , = g 2 g + = 0 + ( 2 ) = (2) a a 2 : Controlled parameter. If 2 is small, behavior is stable. If 2 > g a , behav- ior is unstable. Stable: 2 < g a Unstable: 2 > g a If we look for a solution to Equation 1 of the form = Ae t , we have: 2 2 A t g e + ( ) Ae t = 0 a g = ( 2 ) a If 2 < g a , is imaginary oscillation. 2 g If > a , 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....
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This note was uploaded on 11/29/2011 for the course CIVIL 1.018j taught by Professor Markusbuehler during the Fall '08 term at MIT.

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

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