Page 111 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH03_110_08/30/05

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

110 Chapter Three

12l ln(w 2/ 1
w )
C = (3.6)
r
3
Et (w í w )
2 1
By applying now Eqs. (3.2), the compliances of the microcantilever shown in
Fig. 3.4b are obtained (when the fixed and free ends interchange and the
compliances are determined with respect to the new free end) as
3 2
w )
6l (w í w )(w í 3w ) +2w ln(w 2/ 1
1
1
2
2
2
Ҡ
C = (3.7)
l 3 3
Et (w í w )
2
1
2
12l w í w + w ln(w w )
Ҡ 1 2 2 2/ 1
C = (3.8)
c
3
Et (w í w ) 2
2 1
Equations (3.7) and (3.8) are also obtained when one is directly calculating
ƍ
ƍ
the compliances C l and C c by applying the definition equations of linear
direct and cross compliances of Chap. 2 and by starting the required
integrations from the thicker end (assumed free).
3.2.2 Compliances in arbitrarily translated
reference frames
At times, integrals that are of the form pertaining to the axial, torsion,
and bending compliances need to be calculated with respect to reference
frames that are translated at a specific distance from one end of the
member. These compliances, as shown in the following, can be ex-
pressed in terms of the normally defined compliances. Figure 3.5 shows
a generic microcantilever whose compliances, evaluated with respect to
a translated frame x 1 O 1 y 1 , need to be expressed in terms of the compli-
ances calculated with respect to the reference frame xOy.
The relationships between the two coordinates as well as between
their corresponding differentials are
x = a + x 1 dx = dx 1 (3.9)
It can be shown that the bending-related compliances calculated with re-
spect to the translated frame x 1 O 1 y 1 can be expressed in terms of the nor-
mal ones that are formulated in the endpoint reference frame xOy as
ҡ
ҡ
ҡ
2
C = C +2aC + a C C = C C = C + aC (3.10)
l l c r r r c c r

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