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bonus_1

# bonus_1 - vector ﬁeld corresponding to the above equation...

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MEE 201 Advanced Dynamics. 10/16/2007. Bonus set No. 1 For extra credit, present 10/23/2007 in class. 1. Consider the following equation of motion for a mass-spring system: m ¨ x = kx - δx 3 . where m is the mass, k is the spring constant and δ is a positive constant representing the strength of the qubic nonlinearity. Write this as a dy- namical 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
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Unformatted text preview: 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. 2. A pendulum is acted upon by a torque T = a sin 2 ( θ ) where θ is the angle that the pendulum of mass m makes with the vertical axis and a is a constant. Write the second-order equations of motion using Newton’s law. Represent the equations in phase-space form. Find the constant of motion. Plot the vector ﬁeld on the phase space on top of energy contours. 1...
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