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Unformatted text preview: ME201 Advanced Dynamics (Fall 2007) Bonus-1 Solutions ( 100 pts ) 1. ( 30 pts )Consider the following equation of motion for a mass-spring system; m x = kx- x 3 where m is the mass, k is the spring constant and is a positive constant representing the strength of the quadratic nonlinearity. Write this as a dynamical system in the phase space. Find the equilibrium points. Find the kinetic and potential energy. Show by an explicit calculation that the sum of the kinetic and potential energy is conserved during the motion. Can you solve the resulting system explicitly? Using MATLAB, plot the vector field corresponding to the above equation and, on the same plot, the contours of total energy of the system. Note how the vector field is tangential to the contours of the energy. To get the system in phase space form, let z 1 = x , z 2 = x . Differentiate these equations as needed and plug into the equation given results in: z 1 = z 2 z 2 = k m z 1- m z 3 1 The fixed points are found by setting the right hand sides of these equations equal to zero. This results in two fixed points z 2 = (0 , 0) and z 1 = (0 , q k ). Note that the mass does not impact the location of the fixed points. The kinetic energy for this system is: T = 1 2 mz 2 2 The potential energy is obtained as: V =- Z z 1 z 1 (0) f ( ) , f ( ) = k-...
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- Fall '08