lec14 (1) - Cite as: Thomas Peacock and Nicolas...

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Unformatted text preview: 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 Admissible Variations/Virtual Displacements 2.003J/1.053J Dynamics and Control I, Spring 2007 Paula Echeverri, Professor Thomas Peacock 4/4/2007 Lecture 14 Lagrangian Dynamics: Virtual Work and Generalized Forces Reading: Williams, Chapter 5 L = T V d L L dt q i q i = Q i All q i are scalars. q i : Generalized Coordinates L : Lagrangian Q i : Generalized Forces Admissible Variations/Virtual Displacements Virtual Displacement: Admissible variations: hypothetical (not real) small change from one geometri- cally admissible state to a nearby geometrically admissible state. Bead on Wire Figure 1: Bead on a wire. Figure by MIT OCW. 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 Admissible Variations/Virtual Displacements Both x and y are admissible variations. Hypothetical geometric configuration displacement. = d x = dx dx implies t involved. y = f ( x ) df dy = dx dx df ( x ) y = x dx Generalized Coordinates Minimal, complete, and independent set of coordinates s is referred to as complete : capable of describing all geometric configurations at all times. s is referred to as independent : If all but one coordinate is fixed , there is a continuous range of values that the free one can take. That corresponds to the admissible system configurations. Example: 2-Dimensional Rod Figure 2: 2D rod with fixed translation in x and y but free to rotate about . Figure by MIT OCW. 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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lec14 (1) - Cite as: Thomas Peacock and Nicolas...

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