Page 325 - Mechanical design of microresonators _ modeling and applications

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Microcantilever and Microbridge Systems for Mass Detection

324 Chapter Six
96(EI )
y 1
k b,0 = (6.65)
l 3
The stiffness after mass deposition becomes

96(EI ) (EI ) [ l (EI ) + (l – l )(EI )
p
y
p
y
y e
y e
k = 1 1 (6.66)
b 4 2 2
(EI ) 2 l +2l (l – l )(2l – ll + l )(EI ) (EI ) + (l – l ) 4 (EI ) 2
y 1 p p p p p y 1 y e p y e
The effective mass of the original half-length microbridge was given in
Eqs. (4.54) as
13
m = m (6.67)
b,0 70 1
where m 1 is the total mass of the substrate. The modified mass is
expressed based on Eq. (4.52) as
13
m = (m + f ǻm) (6.68)
b 70 1 p
with f p being defined in Eq. (4.53).
The bending resonant frequency ratio can now be formulated by using
Eqs. (6.65) through (6.68) in the form:

2 4
(EI ) l +2l (l í l )(2l 2
y 1 p p p
2
íll + l (EI ) (EI )
)
p p y 1 y e (6.69)
4
Ȧ m + f ǻ m +(l í l )(EI ) 2
p
y e
b,0 =0.277 1 p ×
Ȧ m 3
b 1 l (EI ) l (EI ) + (l í l )(EI )
y e p y 1 p y e
By using the nondimensional parameters
ǻm t p l p E p ȡ p
f = c = c = c = c = (6.70)
m m 1 t t 1 l l E E 1 ȡ ȡ 1
Eq. (6.69) becomes
3
Ȧ c (1+ c c )
b,0 = 0.784 1+ f f (1 í c ) + l E t +A
3
Ȧ b p m l 1+ c c {4+ c 6
E t t (6.71)
+c (4+ c c ) }
t E t

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