Page 114 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH03_113_08/30/05

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design 113
1
k t,e = (1) (2) (3.17)
C + C
t t
and, again, its equation reduces to Eq. (2.51) which defines a constant-
cross-section microcantilever in the case where l = l = l/2, w = w 1
2
2
1
(constant), and t 1 = t 2 .
It is interesting to check whether a relationship exists between the
axial and torsional stiffnesses of a two-segment microcantilever in the
case where the two segments have identical thicknesses t 1 = t 2 = t.
Equation (3.15) can be rewritten as
Et
k =
a,e l 1 l 2 (3.18)
1 ฒ
ฒ dx w (x) + dx w (x)
/
/
2
0 0
Similarly, Eq. (3.17) can be written in the form:
Gt 3
k =
t,e l 1 l 2 (3.19)
1 ฒ
/
/
3 ฒ dx w (x) + dx w (x)
2
0 0
Comparison of Eqs. (3.18) and (3.19) results in the following relation-
ship:
Gt 2
k = k (3.20)
t,e 3E a,e
Equation (3.20) is actually identical to Eq. (2.85), which applied for sin-
gle-curve micocantilevers, as shown in Chap. 2. Equation (3.20) is not
valid when the two segments have identical widths w = w = w and
2
1
different thicknesses; and as a consequence, the torsional stiffness will
be explicitly calculated for the designs having this particular feature.
The lumped-parameter mechanical moment of inertia which cor-
responds to the free torsional vibrations of the serially compounded
microcantilever of Fig. 3.6 is

{ 2 2 2
ȡ l 1 (1)
J = t 1ฒ w (x) ƒ (x) w (x) + t dx
t,e 12 0 1 a 1 1
(3.21)
l +l 2
1
2
(2)
+ t 2 ฒ w (x) ƒ (x) 2 w (x) + t 2 2 dx }
a
2
2
l
1
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