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lec23 (2)

# lec23 (2) - Wilb erforce Pendulum 1 2.003J/1.053J Dynamics...

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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]. 1 Wilberforce Pendulum 2.003J/1.053J Dynamics and Control I, Spring 2007 Professor Peacock 5/14/2007 Lecture 23 Vibrations: Two Degrees of Freedom Systems - Wilberforce Pendulum and Bode Plots Wilberforce Pendulum Figure 1: Wilberforce Pendulum. Extension of spring coupled to rotation. Fig ure by MIT OCW. - m : mass, I : moment of inertia, γ : torsional constant (rotation), k : extension, : coupling of extension and rotation

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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]. 2 Wilberforce Pendulum Focus on Spring Figure 2: Focus on spring. Figure by MIT OCW. Force: F = kx + 1 �θ (also opposing force in opposite direction) 2 Torque: τ = γθ + 2 1 �x 1 1 �θ and �x represents the coupling for small displacements 2 2 Equations of Motion Can use Lagrange, but we will use momentum principles. 1 mx ¨ = mg kx �θ (1) ¨ 1 = γθ �x (2) Equilibrium Points 1 kx 0 + �θ 0 mg = 0 1 γθ 0 + �x 0 = 0 1 θ 0 = x 0 γ mg x 0 = 2 k 4 1 γ 2 2 2 2 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].

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• Fall '08
• MarkusBuehler
• MIT OpenCourseWare, Massachusetts Institute of Technology, 2.003J/1.053J Dynamics, Nicolas Hadjiconstantinou, Wilb erforce Pendulum

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lec23 (2) - Wilb erforce Pendulum 1 2.003J/1.053J Dynamics...

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