530_343lecture13

530_343lecture13 - ME 530.343 Design and Analysis of Dynamic Systems Spring 2009 Lecture 13 Lagranges Equations Monday Todays Objectives Dene

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
ME 530.343: Design and Analysis of Dynamic Systems Spring 2009 Lecture 13 - Lagrange’s Equations Monday, March 30, 2009 Today’s Objectives Define generalized coordinates Lagrange’s equation Lagrange example Generalized Coordinates and Forces - Equations of motion can be formalized in a number of different coordinate systems. - n independent coordinates are necessary to describe the motion of a system having n degrees of freedom - Any set of n independent coordinates is called generalized coordinates: q 1 , q 2 , . . . q n - These coordinates may be lengths, angles, etc. Consider the triple pendulum: We could use ( x j , y j ), where j = 1 , 2 , 3, to specify configuration of the system. However, ( x j , y j ) are not independent, but constrained by: x 2 1 + y 2 1 = l 2 1 ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 = l 2 2 ( x 3 - x 2 ) 2 + ( y 3 - y 2 ) 2 = l 2 3 1
Background image of page 1

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

View Full DocumentRight Arrow Icon
Thus, ( x j , y j ), j = 1 , 2 , 3 cannot be called generalized coordinates. The constraints eliminate 3 dof. A good choice for generalized coordinates is θ j , where j = 1 , 2 , 3 q j = θ j , where j = 1 , 2 , 3 When external forces act on the system, the configuration changes: Generalized coordinates q j change by δq j , j = 1 , 2 , . . . , n If U j is the work done in changing q j by δq j , the corresponding generalized force is: Q j = U j δq j , where j = 1 , 2 , . . . , n Q j is a force/moment and q j is a linear/angular displacement. An Energy Method for Creating Coupled Equations of Motion:
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 03/02/2010 for the course MECH 530.343 taught by Professor Sun during the Spring '08 term at Johns Hopkins.

Page1 / 5

530_343lecture13 - ME 530.343 Design and Analysis of Dynamic Systems Spring 2009 Lecture 13 Lagranges Equations Monday Todays Objectives Dene

This preview shows document pages 1 - 3. Sign up to view the full document.

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