Graphics Programs Reference
In-Depth Information
or, using vectornotation
θ
(
u
)
=
β
(8.7)
These aresimultaneous(generallynonlinear)equationsthatcan be solvedbythe
Newton-RaphsonmethoddiscussedinArt. 4.6. It must be pointed out again that
intelligent estimates of
u
1
and
u
2
are neededif the differentialequationis not linear.
EXAMPLE 8.4
w
0
x
L
v
The displacement
v
of the simply supportedbeam can beobtainedbysolving the
boundary value problem
d
4
v
dx
4
d
2
v
dx
2
w
0
E I
x
L
=
v
=
=
0 at
x
=
0 and
x
=
L
where
E I
is the bending rigidity. Determine by numerical integration the slopes at
the twoends and the displacement at mid-span.
Solution
Introducing the dimensionless variables
x
L
E I
w
0
L
4
v
ξ
=
y
=
the problemistransformed to
d
4
y
d
d
2
y
d
=
ξ
y
=
=
0 at
ξ
=
0 and
ξ
=
1
4
2
ξ
ξ
The equivalent first-order equations and the boundary conditions are (the prime
denotes
d
/
d
ξ
)
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
y
1
y
2
y
3
y
4
y
2
y
3
y
4
ξ
y
=
=
=
=
=
y
1
(0)
y
3
(0)
y
1
(1)
y
3
(1)
0
The program listedbelowissimilar to the one in Example 8.1. With appropriate
changes in functions
dEqs(x,y)
,
inCond(u)
and
residual(u)
the program can
solve boundary value problemsofany ordergreater than two.For the problemat
hand we chose the Bulirsch-Stoeralgorithm to do the integrationbecause it gives us
control over the printout(we need
y
preciselyat mid-span). The nonadaptive Runge-
Kuttamethod couldalso be used here, but we would havetoguess a suitable step
size
h
.
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