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assn2 - 1.685J/2.0345/18.377J Nonlinear Dynamics and Waves...

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1.685J/2.0345/18.377J Nonlinear Dynamics and Waves Spring 2007 Problem Set No. 2 Out: Thursday, March 1, 2007 Due: Thursday, March 15, 2007 in class Problem 1 The cylinder rolls back and forth without slip as shown in the figure below. (a) Show that the equation of motion can be written in the form where w2 = 2L/3M and 1 is the free length of the spring. All lengths were made dimen- sionless with respect to the radius r. (b) Sketch the potential energy as a function of x for (i) 1 5 1 (ii) 1 > 1 Show the equilibrium positions and indicate whether thcy are stable or unstable. (c) For 1 = 4, obtain a two-term frequency-amplitude relationship for small oscil- lations around the equilibrium position.
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Problem 2 Consider a simple pendulum with a dashpot as shown below. (a) Show that the equation of motions is ml10 = fiml10cos2(/3 - 0). Then show that (1) can also be written as (b) Expanding and retaining through the cubic terms, show that (2) becomes 1 + w2(1 - + 2p820 = 0
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