Formula and examples (Word file)

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15 Νοε 2013 (πριν από 3 χρόνια και 10 μήνες)

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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

砠㄰x

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

嘠㴠卑剔嬨㠠砠
〮㌸0



m

砠⠲x


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

m⁸


m

砠x〠

m)]

†
㴠ㄱ⸳⁖潬=


F‽‸⸸㕥
-
㘠6F/

m⁸
㠰8

m

砠㄰x

m⁸
⠱ㄮ(嘩
2
/(2 x (2
-
0.7


2
) =
0.27




䥦weeed⁴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
漠汯o
⡴he⁩pu琠灲t獳sre⁩猠
慲潵nd‰⸰〱
-

〮〰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


嘠㴠卑剔嬨㠠砠


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

丠砠
ㄲ1
㴠㐮㈠

N


䥦weeed⁴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
獭慬aer⁴桡⁴桥‰⸷.

m⁥硰ec瑥d⁦潲‰⸰〵⁍偡潡楮朮⁗h礿