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Homework1

# Homework1 - ME 680 INTRODUCTION TO BIFURCATION AND CHAOS...

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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 1 1 2 2 ( , , , )     ) around the vertical equilibrium position 12 ( 0, 0)   . (ii) Show that the equations in (i) can be put in the form     1 1 2 2 0 MK    where 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 M for which [K] has a zero eigenvalue and express these for the load P as a function of the parameters ( , ) kM .
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