lec16 (1)

# lec16 (1) - Example: Falling Stick (Continued) 1...

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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 Example: Falling Stick (Continued) 2.003J/1.053J Dynamics and Control I, Spring 2007 Professor Thomas Peacock 4/11/2007 Lecture 16 Lagrangian Dynamics: Examples Example: Falling Stick (Continued) Figure 1: Falling stick. The surface on which the stick rests is frictionless, so the stick slips. Figure by MIT OCW. A stick slides with out friction as it falls. Length: L Mass: M C: Center of Mass Assume uniform mass distribution.

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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: Falling Stick (Continued) x c , φ : Generalized Coordinates Figure 2: The falling stick is subject to a normal force, N at the point of contact. Figure by MIT OCW. Holonomic (constraint forces do no work). No tangential forces at P. Normal force is a constraint force. N does no work. Kinematics L r c = x c ˆ ı + y c j ˆ= x c ˆ ı + sin φj ˆ 2 L v c = x ˙ c ˆ ı + φ ˙ cos φj ˆ 2 Energy 1 1 T = m | v c | 2 + I | ω | 2 (1) 2 2 = 1 m ( ˙ x 2 c + L 2 φ ˙ 2 cos 2 φ ) + 1 I c φ ˙ 2 (2) 2 4 2 (3) L V = mg sin φ 2 Generalized Forces Ξ x = 0, Ξ φ = 0. (Professor Sarma uses Q for generalized forces.) Professor Williams uses Ξ. c
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]. ± ± ± ± 3 Example: Falling Stick (Continued) Forces: Conservative [gravity] + Nonconservative [normal]. The constraint force (normal force) does no work. As we change x c and φ , no virtual work (no dis- placement in direction of force). Lagrangian

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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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lec16 (1) - Example: Falling Stick (Continued) 1...

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