lec17 (1) - Example: Cart with Pendulum and Spring...

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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]. ± Example: Cart with Pendulum and Spring (continued) 1 2.003J/1.053J Dynamics and Control I, Spring 2007 Professor Peacock 4/18/2007 Lecture 17 Lagrangian Dynamics: Examples and Equilibrium Analysis Example: Cart with Pendulum and Spring (continued) Figure 1: Cart with pendulum and spring. Figure by MIT OCW. x , θ , s : Generalized Coordinates Lagrangian L = T V = 1 ( M + m ) ˙ x 2 + 1 m s 2 + s 2 θ ˙ 2 + 2 ˙ x s sin θ + ˙ cos θ )] + mgs cos θ 1 k ( s l ) 2 2 2 2 Equations of Motion x: d ∂L ∂L = Ξ x dt ∂x ˙ ∂x
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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 Example: Cart with Pendulum and Spring (continued) ( M + m x + ms ¨sin θ + 2 ms ˙ θ ˙ cos θ + ms θ ¨ cos θ msθ ˙ 2 sin θ = 0 (1) θ : d ∂L ∂L = Ξ θ dt ∂θ ˙ ∂θ ¨ + 2 ˙ ˙ + ¨ x cos θ + g sin θ = 0 (2) s: d ∂L ∂L = Ξ s dt ∂s ˙ ∂s ∂L = ms ˙ + mx ˙ sin θ ∂s ˙ ∂L = msθ ˙ 2 + mx ˙ θ ˙ cos θ + mg cos θ k ( s l ) ∂s d ∂L = ms ¨+ mx ¨sin θ + mx ˙ θ ˙ cos θ msθ ˙ 2 mx ˙ θ ˙ cos θ mg cos θ k ( s l ) = 0 dt ∂s ˙ ms ¨+ mx ¨ sin θ msθ ˙ 2 mg cos θ + k ( s l ) = 0 (3) These equations are
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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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lec17 (1) - Example: Cart with Pendulum and Spring...

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