Page 174 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH04_173_08/30/05

Microbridges: Lumped-Parameter Modeling and Design

Microbridges: Lumped-Parameter Modeling and Design 173
2

1.5
1
e kb [%]
0.5
0
0 0.01 0.02 0.03 0.04 0.05
Figure 4.5 Relative errors between long and short microbridges in terms of stiffness.

/
Fl ț (GA) + l 2 / (48EI )
y
u = u (0) = 2 (4.16)
z
z
The stiffness of the half-model is
sh F 96EGI A
y
k b,e = = 2 (4.17)
u
1 l(48țEI + GAl )
y
Example: Compare the stiffness of a long configuration to that of a similar
short configuration by analyzing one-half the length of a microbridge of
constant cross section (defined by w and t) and length l.
If the error function is formulated as

sh
k b,e – k b,e
e = 100 (4.18)
kb k
b,e
the following equation is obtained by using Eqs. (4.13) and (4.17):
1 100Į
e = 100 =
kb 2 2 2 (4.19)
1+ Gl /(4țEt ) 0.12 + Į
where it has been considered that

t
Į = (4.20)
l
and that ț = ൣ (for a rectangular cross section) and Í = 0.25 (Poisson’s ratio
for polysilicon) and that E and G are related according to Eq. (2.70). The
relative error function of Eq. (4.19) is plotted in Fig. 4.5, which shows that
the maximum deviation between the long- and short-configuration model
predictions is approximately 2 percent.

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