homework_4

homework_4 - x,y = x . (b) Consider the energy of the...

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MEE 201 Advanced Dynamics. 11/8/2007. Homework set No. 4 Due Thursday, 11/15/2007 in morning class. 1. Consider a mathematical pendulum of length l and mass m , moving in the field of gravity. The pendulum is moving in a viscous liquid which exerts a force of magnitude c ˙ θ in the direction opposite to the pendulum motion. (a) Plot the vector field of the dissipative ( c = 1 say) system on top of level sets of h . (b) Prove that the equilibrium position straight down is stable. (c) Derive action-angle coordinates of the pendulum in the part of the phase-space around the stable equilibrium. 2. Consider the following system with dissipation and forcing arising in non- linear circuit theory (van der Pol oscillator): ¨ x = - x + ± (1 - x 2 ) ˙ x, (1) where ± > 0 is assumed to be small. You can think of this as a linear spring perturbed by a small dissipation ± ˙ x and small forcing ±x 2 ˙ x . (a) Write this as a dynamical system in terms of variables
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Unformatted text preview: x,y = x . (b) Consider the energy of the unperturbed ( = 0) system, E = ( x 2 + y 2 ) / 2. Find the region in phase space where E increases in time and the region where E decreases in time. (c) Introduce action-angle coordinates I = ( x 2 + y 2 ) / 2 , = arctan( y/x ) of the unperturbed harmonic oscillator. (d) Obtain the equations for I (which you should already have from 2b) and . (e) Average the equation for I with respect to (i.e. integrate the right-hand side with respect to , from = 0 to = 2 and divide by 2 ). Find out for which E the resulting averaged expression vanishes. (f) Numerically simulate the system obtained in 2a, using small . Plot trajectories in the phase space, starting from various initial condi-tions. Do you nd any limit cycles in the phase space? Where? 1...
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