Page 80 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_79_08/30/05

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

Basic Members: Lumped- and Distributed-Parameter Modeling and Design 79
quite acceptable. As a consequence, one gets the simplified resonant
frequency corresponding to axial vibrations by combining Eqs. (2.107)
and (2.112):

1
2
Ȧ * = 3.46 E(t — t ) (2.115)
a,e l
ȡ(3t + t )ln(t
t )
1 2 2/ 1
In torsion, the stiffness corresponding to the free end of the trapezoid
microcantilever of Fig. 2.26 is found by means of the axial stiffness
[Eq. (2.107)] and the connection Eq. (2.85).
Again, the lumped inertia will be calculated by using the two variants
utilized in determining the effective axial inertia, as performed above.
The effective moment of inertia that results by using the exact
distribution function of Eq. (2.91) is

2
2
2
2
2
2
2
ȡlw{(t í t )(t + t +8w ) í 4t ln(t t ) t +4w 2
2 1 1 2 1 2/ 1 1
t ) }
2
2
+2(t +2w )ln(t 2/ 1 (2.116)
1
J =
t,e 2
384(t í t )ln (t t )
2 1 2/ 1
Similarly, the effective mechanical moment of inertia calculated with
the constant-cross-section distribution function of Eq. (2.48) is
3
2
2
3
ȡlw 10t +6t t +3t t + t +5(3t + t )w 2
* 1 1 2 1 2 2 1 2 (2.117)
J t,e = 720
The errors that are generated by using the approximate Eq. (2.117)
instead of the exact Eq. (2.116) can be assessed by following the path
utilized in the previous comparison corresponding to the effective
mass in axial vibrations, and the conclusions are very similar. Note
that, again, both Eqs. (2.116) and (2.117) reduce to Eq. (2.55) when
t 2 ĺ t 1 , so they are both valid. As a consequence, the torsion-related
resonant frequency of the microcantilever of Fig. 2.26 can be calculated
as
2
3
/
G {ȡ(t + t ) 10t +6t t
2
1 2
1
1
21.91t t
1 2 +3t t + t +5(3t + t )w } (2.118)
2
3
2
Ȧ * = 1 2 2 1 2
t,e l
The lumped-parameter stiffness corresponding to the out-of-plane
bending of the trapezoid microcantilever of Fig. 2.26 is
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