homework_2

homework_2 - described by a function y = h ( x ). If the...

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MEE 201 Advanced Dynamics. 10/23/2007. Homework set No. 2 Due Tuesday, 10/30/2007 in class. 1. Consider the system of two oscillating equal masses given by m ¨ x 1 = - kx 1 + c 1 ( x 1 - x 2 ) + c 2 ( x 1 - x 2 ) 3 , (1) m ¨ x 2 = - kx 2 - c 1 ( x 1 - x 2 ) - c 2 ( x 1 - x 2 ) 3 . (2) a Is this a conservative system? b Find explicit dependence of y 1 = x 1 - x 2 ,y 2 = x 1 + x 2 on time. Rewrite the equations as a system of first-order ordinary differential equations (or a dynamical system ) for y 1 , ˙ y 1 = v 1 ,y 2 , ˙ y 2 = v 2 . Find the fixed points of this system. Describe the qualitative nature of solutions with various initial conditions in the 4-dimensional phase space. 2. Consider a spring-mass system moving above a surface of unequal height
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Unformatted text preview: described by a function y = h ( x ). If the spring is linear (with equilibrium position for the mass y eq ) and surface acts on the mass with an attractive force proportional to the inverse of the distance of the mass from the surface, derive the equation of motion for the mass (neglect gravity). a Draw the phase portrait. b How do the equations of motion change if the surface starts moving in the horizontal direction with small constant speed e ? c How would you use this system to measure the surface height h ( x )? 1...
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This note was uploaded on 08/06/2010 for the course ME 201 taught by Professor Mezic,i during the Fall '08 term at UCSB.

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