Page 311 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH06_310_08/30/05

Microcantilever and Microbridge Systems for Mass Detection

310 Chapter Six

l
l
1 y1(
s
1
ฒ
1
u = E I ฒ M ˜M 1 dx + M ˜M 2 dx
1z 1 ˜F 2 ˜F
0 1z l 1z
s
l l +l p
1
˜M 4 1 ˜M 3
dx +
+ ฒ M 4 ˜F 1z ) (EI ) ฒ M 3 ˜F dx
l +l y e l 1z
1 p l 1
1 y1( s ˜M 1 l 1 ˜M 2
1
ฒ
ș = E I ฒ M dx + M dx
1y 1 ˜M 2 ˜M
0 1y l 1y
s
(6.27)
l +l
l ˜M 1 p ˜M
1
4
dx +
+ ฒ M 4 ˜M 1y ) (EI ) ฒ M 3 ˜M 3 dx
l +l y e l 1y
1 p 1
1 y1( l 1 ˜M 2 l ˜M 2z )
4
1
u 2z = E I ฒ M 2 ˜F dx + ฒ M 4 ˜F dx
l 2z l +l
s 1 p
l +l
1 p
1 ˜M 3
+ ฒ M dx
(EI ) 3 ˜F
y e 2z
l
1
The equivalent rigidity is calculated over the length l p where the
attached mass and microcantilever do superimpose and, as shown in a
previous chapter, is equal to
(EI ) = E I + z A (z – z ) + E I + z A (z – z )
y e 1 y1 1 1 1 N p yp p p p N (6.28)
The bending moments M through M , which enter Eqs. (6.27), are
4
1
M =– M – F x
1 1y 1z
M =– M – F x – F (x – l )
2 1y 1z 2z s
q (x – l ) 2
M =– M – F x – F (x – l ) – z 1 (6.29)
3 1y 1z 2z s 2
z p (
2 )
1
M =– M 1y – F x – F (x – l ) – q l x – l + l p
4
1z
2z
s
and the corresponding partial derivatives of the same Eqs. (6.27) can
simply be calculated from Eqs. (6.29).
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