Graphics Programs Reference
In-Depth Information
The simply supportedbeam of length
L
is resting onanelasticfoundation of
stiffness
k
N/m
2
. The displacement
v
of the beamdue to the uniformlydistributed
load of intensity
w
0
N/mis givenbythe solution of the boundary value problem
E I
d
4
v
dx
4
dx
2
x
=
0
=
dx
2
x
=
L
=
d
2
y
d
2
v
+
kv
=
w
0
,
v
|
x
=
0
=
v
|
x
=
L
=
0
The nondimensionalform of the problemis
dx
2
ξ
=
0
=
dx
2
ξ
=
1
=
d
4
y
d
d
2
y
d
2
y
+
γ
y
=
1,
y
|
ξ
=
0
=
y
|
ξ
=
1
=
0
4
ξ
where
kL
4
E I
x
L
E I
w
0
L
4
v
ξ
=
y
=
γ
=
10
5
and plot
y
vs.
.
16.
SolveProb. 15 if the ends of the beam arefree and the load isconfined to the
middle half of the beam. Consider only the left half of the beam, in which case
the nondimensionalform of the problemis
Solvethis problembythe finite difference methodwith
γ
=
ξ
0 in 0
d
4
y
d
<ξ<
/
1
4
+
γ
y
=
4
ξ
1in1
/
4
<ξ<
1
/
2
ξ
=
0
=
ξ
=
0
=
ξ
=
1
/
2
=
ξ
=
1
/
2
=
d
2
y
d
d
3
y
d
d
3
y
d
dy
d
0
2
3
ξ
3
ξ
ξ
ξ
17.
The generalform of a linear, second-orderboundary value problemis
y
=
t
(
x
)
y
r
(
x
)
+
s
(
x
)
y
+
or
y
(
a
)
y
(
a
)
=
α
=
α
or
y
(
b
)
y
(
b
)
=
β
=
β
Write aprogram that solves this problemwith the finite difference method for
any user-specified
r
(
x
),
s
(
x
) and
t
(
x
). Test the program by solving Prob. 8.
MATLAB Functions
MATLAB hasonly the following function for solution of boundary value problems:
sol = bvp4c(@dEqs,@residual,solinit)
uses a high-order finite difference
methodwith an adaptive mesh to solve boundary value problems. The out-
put
sol
is a
structure
(a MATLAB data type) createdby
bvp4c
. The first two
input arguments arehandles to the following user-supplied functions:
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