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# exam1 - 1.685512.0345118.377J Nonlinear Dynamics and Waves...

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1.685 512.034 5118.377 J Nonlinear Dynamics and Waves Take-Home Exam Thursday, April 5,2007 This is a closed-book exum. You may use your own class notes, problem sets and solutions posted on the 1.685/2.034/18.377 website only. You are not allowed to discuss this exum with anyone else. If you need clarification regarding the problems, please ask me. The exam is due on Tuesday,l loth, 2007 before 5:00pm, Problem 1 (10 points) A particle of mass m is constrained to move around a smooth circular wire of radius a fixed in a vertical plane. The particle is acted on by gravity and is attached to one end of a linear spring of natural length I ( < 2a) and stiffness the other end being fixed to the highest point of the circle. (a) Derive the governing equations of motion. (b) Find the number of equilibrium points and discuss their stability as k varies. (c) Sketch the bifurcation diagram describing these equilibrium states as a function of k. (d) Sketch the phase plane for representative values of k. Problem 2 (10 points) The motion of a particle restrained by a linear spring and under the combined influence of

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• Spring '07
• TriantaphyllosAkylas
• Equilibrium point, Stability theory, Bifurcation theory, linear stability analysis, phase plane, smooth circular wire

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exam1 - 1.685512.0345118.377J Nonlinear Dynamics and Waves...

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