homework_3

# homework_3 - sin θ T af t(a Let a = 0 Find the energy of...

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MEE 201 Advanced Dynamics. 10/30/2007. Homework set No. 3 Due Tuesday, 11/6/2007 in class. 1. Consider the following linear dynamical system on R 2 : ˙ x = ax + by, (1) ˙ y = cx + dy. (2) In matrix form, this system can be written as ˙ x = A x for a matrix A , where x = ( x,y ). Find the matrix A . Find the eigenvalues and gen- eralized eigenvectors (if they exist) for any value of a,b,c,d . Draw the corresponding phase portraits. Show that the nature of the phase portrait is determined by Det(A) and Tr(A), the determinant and trace of the ma- trix A . In Det(A)-Tr(A) plane, draw the various phase portraits. Do not forget the lines where you have zero eigenvalues. If needed, use help of numerical computation of trajectories. 2. Consider the system (pendulum in ﬁeld of gravity with possibly time- dependent torque T + af ( t ) where a and T are constant and f is a function of time: ¨ θ = - g/l
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Unformatted text preview: sin( θ ) + T + af ( t ) . (a) Let a = 0. Find the energy of the system. Plot the energy level sets. Find the ﬁxed points. Linearize the system around the ﬁxed points. Find eigenvectors and eigenvalues (possibly numerically) of the ﬁxed points. Draw the ”skeleton” of the phase space, localized around the ﬁxed points. (b) Let T = 0 ,g/l = 1 ,f ( t ) = cos ( ωt ) ,a = { , . 5 , 1 . } , where ω = 2 π . Set up a grid of initial conditions on the square [-π,π ] × [-1 , 1]. Set θ = ωt . Plot the trajectories of the Poincar´ e map for θ = 0 (don’t forget to take the angle mod 2 π . Describe the results. (c) Is it possible to execute an arbitrary motion θ ( t ) between time 0 and T by choosing appropriately f ( t )? If so, describe how. 1...
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