# lec18 - Example: Cart with Pendulum and Spring (Continued)...

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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 4/23/2007 Lecture 18 Lagrangian Dynamics: Equilibrium Analysis - Cart with Pendulum and Spring and Spinning Hoop with Sliding Mass Examples Example: Cart with Pendulum and Spring (Continued) Recall: Figure 1: Cart with pendulum and spring. Figure by MIT OCW. Equations of Motion ( M + m x + m s sin θ + 2 ˙ ˙ cos θ + ¨ cos θ ˙ 2 sin θ ) = 0 (1) ¨ + 2 ˙ ˙ + ¨ x cos θ + g sin θ = 0 (2) ms ¨+ mx ¨ sin θ msθ ˙ 2 mg cos θ + k ( s l ) = 0 (3)

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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) 2 Equilibria s ¨ = s ˙ = θ ¨ = θ ˙ = x ¨ = x ˙ = 0 Set all variables to 0 except for position variables. Conﬁguration: mg θ 0 = 0, s 0 = l + k (Stable) = = l mg (Stable or unstable? Expect it to be unstable based on θ 0 π , s 0 k physical intuition.) x for both can be any value. Linearize Equations (1), (2), and (3) about θ 0 = π and s o = l mg/k θ = θ 0 + φ , s = s 0 + θ ˙ = φ ˙ , φ ¨ = θ ¨ , s ˙ = ˙, s ¨ = ¨ Taylor Series d cos( θ 0 + φ ) = cos θ 0 + cos θ φ + . . . ≈ − 1 + 0 for θ = π θ 0 d sin( θ 0 + φ ) = sin θ 0 + sin
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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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lec18 - Example: Cart with Pendulum and Spring (Continued)...

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