# Formula and examples (Word file)

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15 Νοε 2013 (πριν από 5 χρόνια και 1 μήνα)

142 εμφανίσεις

1
. Electrostatic Force:

Capacitance due to air medium:

D
A
C
0

0
=8.85 e
-
6 pF/

m

For the beam to be simulated in Tutorial #1,

C = 8.85e
-
6 pF/

m⁸
㠰8

m

m ㈠

m‽
㌮㔠3
-
㌠3F.

El
ectrostatic potential energy corresponding to a voltage V:

2
2
1
CV
U

U
F


2
2
0
2
D
AV
F

With a moving plate, replacing D by D
-

where

is the displacement of top plate:

2
2
0
)
(
2

D
AV
F

2
. At the pull
-
down point, Electrostatic force = Mechanical restoring force:

nk
F

where n is the number of springs

So,
A
nk
D
V
0
2
1
)
(

3.
The critical d
isplacement

corresponding to the pull
-
down point:

k

k

D

S

A

C
1

C
2

3
-
D view

Parallel plates

LCP

Bonding
film

Dielectric

Copper

Duroid

0

d
dV

yields pull
-
down displacement

3
D

For the beam to be simulated,

㴠㈠

m⼳‽‰⸷

m

4
. Pull
-
down voltage:

A
nkD
V
0
3
27
8

5.
Spring constant:

EI
Fl
y
3
3

3
12
1
bt
I

b: beam width and t: beam thickness

3
3
4
l
Ebt
y
F
k

For the beam to be simulated,

k = 7.7e4 MPa x 10

m砠⠰⸵

3
/(4 x (
40

3
) =
0.38

m

〮㌸0

m

3
)/(27 x 8.85e
-
6 pF/

m⁸

m

m)]

†
㴠ㄱ⸳⁖潬=

F‽‸⸸㕥
-
㘠6F/

m⁸
㠰8

m

m⁸
⠱ㄮ(嘩
2
/(2 x (2
-
0.7

2
) =
0.27

䥦we⁮eed⁴漠潮癥r琠F楮瑯⁵tif潲m⁰re獳sre⁬潡o楮本⁴桥⁰牥獳sre⁷ou汤⁢e

F⽁‽/
〮㈷0

N

㠰8

m

x‱〠

m⤠㴠
〮〰〴0Ma
℠To

⡴he⁩pu琠灲t獳sre⁩猠

-

〮〰㔠0p愩

F

6.
Spring constant for another possible configuration:

w
x
l
= F

EI
wl
y
384
4

and
EI
Fl
y
384
3

3
12
1
bt
I

b: beam width and t: beam thickness

3
3
4
)
3
/
384
(
l
Ebt
y
F
k

increased by 128 times!

For the beam to be simulated,

k = 7.7e4 MPa x 10

m砠⠰⸵

3
/(4 x
(80

3
) x 128

=
6

m

N
/

m⁸⠲(

3
)/(27 x 8.85e
-
6 pF/

m⁸‸〠

m⁸‱〠

m)]

†
㴠㌮㘠噯汴⁸=
S兒Q ㄲ)

㐵⁖潬4

F‽‸⸸㕥
-
㘠6F/

m⁸‸〠

m⁸‱〠

m⁸
⠴(嘩
2
/(2 x (2
-
0.7

2
) = 0.027

ㄲ1
㴠㐮㈠

N

䥦we⁮eed⁴漠潮癥r琠F楮瑯⁵tif潲m⁰re獳sre⁬潡o楮本⁴桥⁰牥獳sre⁷ou汤⁢e

F⽁/
㴠㐮㈠

N
⼨㠰/

m⁸㄰1

m⤠砠‽)
〮〰㔠Ma

Bu琬⁴he⁤楳i污捥men琠t潲re獰潮d楮朠瑯‰⸰〵⁍
愠楳⁯n汹‰⸳㌠

mⰠ,h楣栠i猠such

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