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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 Fall '10
 NA
 Chaos Theory, Equations, Nonlinear system, Parametric equation, load parameter values

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