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530_343lecture13 - ME 530.343 Design and Analysis of...

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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
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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.
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