Homework 8
1.
Problem 7.2 in the textbook.
Derive equations of motion first by using
Newton’s second law, and then by using Lagrange’s equations.
(They
should agree!)
Then do problem 7.21 in the textbook.
You can use the
“eig” command in MATLAB, but check that
0
)
det(
=

M
K
i
λ
for each of
the eigenvalues found and that
u
u
M
K
λ
=
for each eigenpair
)
,
(
u
λ
, and
finally that
KU
U
T
and
MU
U
T
yield diagonal matrices.
Plot modes as in
Fig. 7.5.
Finally, “massnormalize” them so that they satisfy Eqs. (7.92).
2.
Problem 7.6 in the textbook.
The second sentence of the problem should
say, “Assume that mass
3
m
undergoes small angular displacements…”
Ignore weight in this problem.
You will need to express rotation
θ
and the
mass center displacement
C
x
of
3
m
in terms of
1
x
and
2
x
.
You should get
EOMs in the form
0
x
x
x
=
+
+
K
C
M
.
Are the matrices symmetric?
Next
derive EOMs by using Lagrange’s equations.
Derive
nc
W
δ
carefully by
imagining the virtual work done by the dashpot forces through all four
virtual displacements.
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 Spring '10
 Staff
 Energy, Kinetic Energy, Mass, Velocity, Lagrange’s equations

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