Page 306 - Mechanical design of microresonators _ modeling and applications

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Microcantilever and Microbridge Systems for Mass Detection

Microcantilever and Microbridge Systems for Mass Detection 305
By using the notation of Fig. 6.11, the following static equations can
be written:

F (l – l ) 2 (2l + l )
x
x
z
u 1z =
6EI
y
F (l – l ) 2
ș = z x (6.14)
1y 2EI
y
F l (l – l )
z y
x
ș 1x = GI
t
By solving the equation system above, the following solution is obtained:
3
2EI ș
y 1y
F =
z
9(lș – u ) 2
1y 1z
3u 1z
l = ș –2l (6.15)
x
1y
3GI ș (lș 1y – u 1z )
t 1x
l =
y
2EI ș 2
y 1y
When the detection system is set up to capture gravitational effects, the
force F is
z
F = ǻmg (6.16)
z
and therefore the added mass can be determined from the first of
Eqs. (6.15).
Resonant approach. Mass addition to a microcantilever modifies its res-
onant response, and it has been mentioned that the bending resonant
frequency is the smallest one in an ascending series also containing the
torsional and axial resonant frequencies. Figure 6.12 shows schemati-
cally a pointlike mass which is attached to a microcantilever at a
distance a measured from the free end.
It was shown in Chap. 5 that the exact bending resonant frequency
of a microcantilever is
EI y
Ȧ =3.52 (6.17)
b,0 3
ml

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