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Unformatted text preview: ME201 Advanced Dynamics (Fall 2007) HW-4 Solution ( 100 pts ) 1. 50pts 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. The equations of motion are: ˙ θ = ω ˙ ω =- c mL 2 ω- g L sin θ The level sets are defined by the Hamiltonian of the conservative system ( c = 0): H = 1 2 mL 2 ω 2 + mgL (1- cos θ )-4-3-2-1 1 2 3 4-5-4-3-2-1 1 2 3 4 5 θ ϖ Damped Pendulum, c = 0-4-3-2-1 1 2 3 4-5-4-3-2-1 1 2 3 4 5 θ ϖ Damped Pendulum, c = 5 Figure 1: Problem 1: Phase space of pendulum with and without damping The problem statement asks for c = 1 while here we show c = 5 because it highlights the impact of damping better. Here you can see that the vector field crosses the level sets with nonzero damping. (b) Prove that the equilibrium position straight down is stable. At the straight down position, θ = 0. Choosing the Lyapunov function V ( θ,ω ) = 1 2 ω 2- g l cos θ + g l The derivative of the Lyapunov function is: ˙ V ( θ,ω ) = ω ˙ ω + g l sin θ ˙ θ =- c ml ω 2 < , 2200 ( θ,ω ) 6 = ( θ eq ,ω eq ) So this equilibrium is stable. This condition refers to nonlinear stability and linear stability can be shown by investigating the Jacobian evaluated at the same equilibrium. ME201 Advanced Dynamics 1 HW4 Solution (c) Derive action-angle coordinates of the pendulum in the part of the phase-space around the stable equilibrium. The action angle coordinate transform is beneficial because it is symmplectic and preserves the structure/symmetries of an original Hamiltonian system. To generate the action- angle coordinate transform rules using the nonlinear equations is difficult. So what is usually done is develop the transforms with linear equations, then plug these into the original nonlinear system. In this way, the nonlinearity plays a perturbation role to the action and angle variables.system....
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