Page 94 - Mechanical design of microresonators _ modeling and applications

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Basic Members: Lumped- and Distributed-Parameter Modeling and Design

Basic Members: Lumped- and Distributed-Parameter Modeling and Design 93
By combining the definition of ȕ [Eq. (2.161)] and the value of ȕl corre-
sponding to the first mode [Eq. (2.164)], the natural (first resonant)
frequency is

EI y
Ȧ =3.52 (2.165)
ml 3

A similar reasoning is applied to a microbridge (fixed-fixed member),
and by enforcing the zero slope and zero deflection conditions at the two
ends, the following equation is obtained:
1 í cosh (ȕl) cos (ȕl) =0 (2.166)

The numerical values of ȕl corresponding to the first three modes are

{ 4.73004
ȕl = 7.8532 (2.167)
10.9956

Again, the natural frequency corresponding to the first mode is

EI y
Ȧ = 22.373 (2.168)
ml 3

Short microcantilevers and bridges. The differential Eqs. (2.72) describ-
ing the behavior of short beams by means of Timoshenko’s theory can
be written into the alternative form:

3
d ș (x)
y
2
EI y íȡAȦ u (x) =0
z
dx 3
(2.169)
2
du (x) EI y d ș (x)
z
y
ș (x) í =
y dx țAG 2
dx
By combining Eqs. (2.169) the following differential equation is obtained:
4
2
d ș (x) d ș (x)
y + Ȗ 2 y íȕ ș (x) =0 (2.170)
4
dx 4 dx 2 y
with ȕ given in Eq. (2.61) and Ȗ defined as
2
Ȗ = ȡȦ 2 (2.171)
țG

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