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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 Constraints and Degrees of Freedom 2.003J/1.053J Dynamics and Control I, Spring 2007 Professor Thomas Peacock 4/9/2007 Lecture 15 Lagrangian Dynamics: Derivations of Lagranges Equations Constraints and Degrees of Freedom Constraints can be prescribed motion Figure 1: Two masses, m 1 and m 2 connected by a spring and dashpot in parallel. Figure by MIT OCW. 2 degrees of freedom If we prescribe the motion of m 1 , the system will have only 1 degree of freedom, only x 2 . For example, x 1 ( t ) = x cos t x 1 = x 1 ( t ) is a constraint. The constraint implies that x 1 = 0. The admissible variation is zero because position of x 1 is determined. For this system, the equation of motion (use Linear Momentum Principle) is mx 2 = k ( x 2 x 1 ( t )) c ( x 2 x 1 ( t )) mx 2 + cx 2 + kx 2 = cx 1 ( t ) + kx 1 ( t ) 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 Lagranges Equations cx 1 ( t ) + kx 1 ( t ): known forcing term differential equation for x 2 ( t ): ODE, second order, inhomogeneous Lagranges Equations For a system of n particles with ideal constraints Linear Momentum p = f ext + f constraint (1) i i i N ( f ext + f constraint p ) = 0 (2) i i i i =1 f constraint = 0 i i =1 DAlemberts Principle N ( f ext p ) r i = (3) i i i =1 Choose p i = at equilibrium. We have the principle of virtual work. Hamiltons Principle Now p = m i r i , so we can write: i N ( m i r i f ext ) r i = (4) i i =1 N W = f ext r i , (5) i i =1 which is the virtual work of all active forces, conservative and nonconservative. N N d m i r i r i = m i ( r i r i ) r i r i (6) dt i =1 i =1 d (6) is obtained by using dt ( r r ) = rr + rr 3 Lagranges Equations r i r i can be rewritten as 1 2 ( r r ) by using ( r r )...
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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.
 Fall '08
 MarkusBuehler

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