DblPendulum Project UofM

# DblPendulum Project UofM - The Chaotic Motion of a Double...

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University of Michigan Department of Physics 1 The Chaotic Motion of a Double Pendulum Carl W. Akerlof September 26, 2012 The following notes describe the kinematics of the double pendulum. The starting point is a pendulum consisting of two point masses, m , and m 2 , suspended by massless wires of length l 1 and l 2 . The treatment of this case can be found at: For a real system, the equations of motion depend in a more complicated way on the distribution of mass that is essential for modeling the physical pendulum used in this experiment. Figure 1. Point mass double pendulum. Figure 2. Extended mass double pendulum.

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University of Michigan Department of Physics 2 Double pendulum with point masses: 1 1 1 1 1 1 2 1 2 2 2 1 1 2 2 sin cos sin sin cos cos x l y l x l l y l l     1 1 1 1 1 1 1 1 2 1 1 1 2 2 2 2 1 1 1 2 2 2 cos sin cos cos sin sin x l y l x l l y l l         2 2 2 1 1 1 2 2 2 2 2 2 1 1 1 2 1 2 1 2 2 2 2 cos r l r l l l l   Double pendulum with distributed masses: 1 1 1 1 1 1 1 1 1 1 2 1 1 2 2 2 2 2 1 1 2 2 2 2 sin cos cos sin sin sin cos cos cos sin x u v y u v x l u v y l u v     1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 2 2 2 2 2 2 1 1 1 2 2 2 2 2 2 cos sin sin cos cos cos sin sin sin cos x u v y u v x l u v y l u v             2 2 2 2 2 2 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 1 1 1 2 1 2 2 1 2 1 2 2 2 2 2 2 ; 2 cos sin ; r r r u v r l l u v r r u v   Assume mirror symmetry: 1 2 0 v v 2 2 2 2 2 2 1 1 1 2 1 1 2 2 1 2 1 2 2 2 1 2 2 cos KE m r m l m u l m r 1 1 1 2 1 1 2 2 2 cos cos cos PE m u g m l g m u g  
University of Michigan Department of Physics 3 Thus, the Lagrangian for the system is: 2 2 2 2 2 2 1 1 1 2 1 1 2 2 1 2 1 1 2 2 2 2 1 1 1 2 1 1 2 2 2 1 2 2 cos cos cos cos L T V m r m l m u l m r m u g m l g m u g     This leads directly to the equations of motion which we shall investigate shortly.

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