Biomedical Engineering Reference
In-Depth Information
Equation (7.7) can be rearranged algebraically to
32
/
2
2
∂
∂
M
EI
∂
∂
v
x
(7.8)
()
v
=−
1
+
2
x
Solving equation (7.7) for a constant moment,
M
, will produce a function,
v
(
x
),
that is the beam deflection of each point on the beam as a function of the distance
from the wall. Determining the solution for equation (7.7) subject to a constant
moment,
M
, can be accomplished in five steps:
1.
Change the independent variable (temporarily) from
x
to
t
:
32
/
2
∂
∂
2
M
EI
∂
∂
v
t
(7.9)
()
v
=−
1
+
t
2
This can be rewritten as
v
″
= -(
M
/
EI
) [1 + (
v
′
)
2
]
3/2
(7.10)
2.
Change the second-order differential equation (7.10) into a set of first-
order differential equations:
set
x
1
=
v
(7.11)
then
x
1
′
=
v
′
(7.12)
set
x
2
=
x
1
′
=
v
′
(7.13)
then
x
2
′
=
x
1
″
=
v
″
= -(
M
/
EI
) [1 + (
v
′
)
2
]
3/2
(7.14)
or
v
″
= -(
M
/
EI
) [1 + (
x
2
)
2
]
3/2
(7.15)
3.
Model the set of first-order differential equations using Matlab's Simulink
tool, as shown in figure 7.20.
4.
Integrate the Simulink model. Simulink defaults to a fourth-order
Runge-Kutta integration method, which was used for this effort. The time
step used during integration was constrained to be constant, since this
would correspond (after the transformation described in step 5) to a fixed-
length fraction of the beam.
5.
Change the independent variable back to
x
from
t
:
v
(
x
) =
v
(
t
)
(7.16)
