MEE 201 Advanced Dynamics, Fall 2007:
Final Exam
12/7//2007
Exam is take home, open notes, books and homeworks, no internet please. It
is due Friday, 12/14/2007 to Melecia Valdez in the front office of Mechanical
and Environmental Engineering Department, before the office closes. You
can take as much time as you need to complete it. Include a printout of your
code for problems that include numerical simulation.
1. Consider the system shown in figure 1. The mass
M
is constrained to move vertically,
the mass on the pendulum is
m
6
=
M
and the mass of the rod and the spring are
neglected.
The spring is a nonlinear, stiffening spring with spring force given by
kx
+
k
s
x
3
,
k
and
k
s
are different constants. There is no friction. Gravity is acting
straight down.
Figure 1:
(a) Derive the equations of motion for the system using the Lagrangian approach.
The spring is assumed to have equilibrium length
l
eq
= 0.
1
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(b) Assume the angular velocity of the pendulum,
ω
, is given and constant in
time. Derive the resulting equation of motion for the mass
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 Fall '08
 Mezic,I
 phase space, Melecia Valdez, Mechanical and Environmental Engineering Department

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