ME 680: INTRODUCTION TO BIFURCATION AND CHAOS SPRING 2010 HOME Work # 1 Due: February 11, 2010 Q1. Consider the double pendulum with the vertical load applied at the end. Show that for the system, with appropriate non-dimensionalization, the equations of motion are given by and Then, (i)Follow the discussion in class and derive the linear equations (for small 1122( , ,,) ) around the vertical equilibrium position 12(0,0). (ii)Show that the equations in (i) can be put in the form 11220MKwhere the mass matrix [M] is positive definite and symmetric, and the stiffness matrix [K] is symmetric with its coefficients being functions of the parameters ( , ,)k P M. (iii)Show that the matrix [K] is not positive definite. Derive the relations in parameters ( , ,)k P Mfor which [K] has a zero eigenvalue and express these for the load Pas a function of the parameters ( ,)kM.
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