lec15 (3) - Cite as: Thomas Peacock and Nicolas...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 )...
View Full Document

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.

Page1 / 11

lec15 (3) - Cite as: Thomas Peacock and Nicolas...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online