1.685 512.034 5118.377
J
Nonlinear Dynamics and Waves
TakeHome Exam
Thursday, April 5,2007
This is a closedbook exum. You may use your own class notes, problem sets and solutions
posted on the
1.685/2.034/18.377
website only. You are not allowed to discuss this exum
with anyone else.
If you need clarification regarding the problems, please ask me. The
exam is due on Tuesday,l loth, 2007
before 5:00pm,
Problem
1
(10 points)
A particle of mass
m
is constrained to move around a smooth circular wire of radius
a
fixed
in a vertical plane. The particle is acted on by gravity and is attached to one end of a linear
spring of natural length
I ( <
2a)
and stiffness
the other end being fixed to the highest
point of the circle.
(a) Derive the governing equations of motion.
(b) Find the number of equilibrium points and discuss their stability as
k
varies.
(c) Sketch the bifurcation diagram describing these equilibrium states as a function of
k.
(d) Sketch the phase plane for representative values of
k.
Problem
2
(10 points)
The motion of a particle restrained by a linear spring and under the combined influence of
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 Spring '07
 TriantaphyllosAkylas
 Equilibrium point, Stability theory, Bifurcation theory, linear stability analysis, phase plane, smooth circular wire

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