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Applied Calculus of Variations for Engineers, 2nd Edition
book

Applied Calculus of Variations for Engineers, 2nd Edition

by Louis Komzsik
June 2014
Intermediate to advanced content levelIntermediate to advanced
233 pages
5h 42m
English
CRC Press
Content preview from Applied Calculus of Variations for Engineers, 2nd Edition
182 Applied calculus of variations for engineers
where
˙q
=
˙q
1
...
˙q
n
,
we obtain
E
k
=
V
1
2
ρ ˙u
T
˙udV =
1
2
˙q
t
V
N
T
ρNdV ˙q.
Introducing the mass matrix
M =
V
N
T
ρNdV,
the final form of the kinetic energy becomes
E
k
=
1
2
˙q
T
M ˙q.
Now let’s focus on the strain energy. Note that the strain is now also ex-
pressed in terms of the basis functions. Hence
(N)=
n
i=1
q
t
i
∂N
∂x
n
i=1
q
t
i
∂N
∂y
n
i=1
q
t
i
∂N
∂z
n
i=1
q
t
i
(
∂N
∂z
+
∂N
∂y
)
n
i=1
q
t
i
(
∂N
∂z
+
∂N
∂x
)
n
i=1
q
t
i
(
∂N
∂y
+
∂N
∂x
)
,
or in matrix form
(N)=Bq,
where the columns of B are
B
i
=
∂N
i
∂x
∂N
i
∂y
∂N
i
∂z
∂N
i
∂z
+
∂N
i
∂y
∂N
i
∂z
+
∂N
i
∂x
∂N
i
∂y
+
∂N
i
∂x
.
With this, the integral becomes
V
T
(N)D(N)dV =
V
q
T
B
T
DBqdV.
Computational mechanics ...
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Publisher Resources

ISBN: 9781482253597