Biomedical Engineering Reference
In-Depth Information
Z
L
F
es
.y.x
0
//.x
x
0
/dx
0
M.x/
D
.1=L/
x
0
Dx
Z
L
.x
x
0
/=Œy.x
0
/
C
.t="/
2
dx
0
:
D
"
0
W
(2.41)
x
0
Dx
Introducing (
2.40
)into(
2.39
), the cantilever bending equation becomes
Z
L
d
2
y=dx
2
D
V
2
.6"
0
=
Et
3
/
.x
x
0
/=Œy.x
0
/
C
.t="/
2
dx
0
:
(2.42)
x
0
D
x
Considering further that y.x
0
/
h, the solution of (
2.42
) can be approximated by
y.x/
D
h
V
2
.6"
0
=
Et
3
/ŒL
2
x
2
=4
Lx
3
=6
C
x
4
=24=Œh
C
.t="/
2
;
(2.43)
and the maximum deflection, which corresponds to x
D
L,isgivenby
y
max
D
y.L/
D
h
V
2
.3"
0
=4
Et
3
/L
4
=Œh
C
.t="/
2
:
(2.44)
The deflection due to a force concentrated in a single point x (
Petersen 1978
)has
the expression
y
conc
D
x
2
.3L
x/
Wf
.x/dx=6
EI
;
(2.45)
whereas if the electrostatic force is distributed uniformly along the cantilever, the
corresponding deflection is
Z
L
.3L
x/x
2
f
es
.x/.6
EI
/
1
dx;
y
unif
D
W
(2.46)
0
where f
es
.x/
D
."
0
=2/V
2
=Œt
y.x/
2
is the electrostatic force per unit area.
For a square-law cantilever bending y.x/
D
.x=L/
2
y
max
, the deflection in 2.46
is found by solving numerically the equation
"
0
WL
4
V
2
=2
EIh
3
D
.4
N
y
max
/Œ.2=3/.1
y
max
/
tanh
1
.
N
y
max
/
1=2
=.
N
y
max
/
1=2
ln.1
y
max
/=3.
N
y
max
/
1
;
(2.47)
where
N
y
max
D
y
max
=h is the normalized deflection. Figure
2.24
represents the
voltage dependence of normalized deflection calculated from (
2.47
).
The cantilever is in dynamical equilibrium when the elastic F
el
and electrostatic
forces are balanced, i.e., when
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