Page 172 - Mechanical design of microresonators _ modeling and applications

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Microbridges: Lumped-Parameter Modeling and Design

Microbridges: Lumped-Parameter Modeling and Design 171

l 2
/
13
2
m b,e = ȡA ฒ f (x) dx = 70m (4.4)
b
0
with m being the total mass of the microbridge. The corresponding res-
onant frequency becomes
EI y
Ȧ = 22.736 (4.5)
b,e 3
ml
Compared to the exact value of
EI y
Ȧ = 22.373 3 (4.6)
b
ml
which is the solution to a partial differential equation, the approximate
value of Eq. (4.5), which is higher, introduces a relative error of only
1.62 percent.
For a full-length microbridge model, the bending stiffness that is
associated with the midspan is

192EI y
k b,e = (4.7)
l 3
To find the effective mass, which needs to be placed at the midspan
of the full-length microbridge, a method with initially unknown coeffi-
cients can be applied as shown in the following. The distribution function
which relates the deflection of a generic point on the microbridge (at an
abscissa x measured from one of the fixed ends) to the maximum
deflection (at the midspan) can be found by assuming the following form
of the deflection:

2 3 4
u (x) = a + bx + cx + dx + ex (4.8)
z
The reason for choosing the particular polynomial of Eq. (4.8) with five
unknown coefficients is that five separate boundary condition equa-
tions are available, namely, zero deflections at both ends and zero
slopes at the ends and at the midpoint. The slope is the x-dependent
derivative of the deflection, and therefore

du (x)
z
2
ș (x) = dx = b +2cx +3dx +4ex 3 (4.9)
y
By using the five boundary condition equations

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