Page 115 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH03_114_08/30/05

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

114 Chapter Three
Equation (3.21) reduces to Eq. (2.55) when l 1 = l 2 = l/2, w 2 = w 1 (con-
stant), and t 1 = t 2 , when the serial microcantilever of Fig. 3.7 becomes
a constant-cross-section member for the above-mentioned conditions.
In bending, it can be shown that the stiffness associated to the free
end 1 is calculated as

1
k b,e = (1) (2) 2 (2) (2) (3.22)
C + C + l C +2l C
l l 1 r 1 c
Equation (3.22) is verified, too, because when l 1 = l 2 = l/2, w 2 = w 1
(constant), and t = t , it reduces to Eq. (2.61), which characterizes a
1
2
constant rectangular cross-section microcantilever of length l. The
lumped effective mass is
{ l 1 1 2 l +l 2 (2) 2 }
1
m = ȡ t 1ฒ w (x) f (x) dx + t 2 ฒ w (x) f (x) dx (3.23)
b,e 0 1 b l 1 2 b

(1)
(2)
where f b and f b are the bending-related distribution functions, which
are potentially different. Equation (2.66), which gives the effective iner-
tia of a constant-cross-section microcantilever of length l, is retrieved
from Eq. (3.23) when l 1 = l 2 = l/2, w 2 = w 1 (constant), and t 1 = t 2 .
In the particular class of micromembers formed of two compliant
segments, the two components are identical and placed in mirror, which
means the resulting member has a symmetry axis, as sketched in
Fig. 3.7. Purposely, the member of this structure has no axial symmetry
about the x axis because this is not a prerequisite, although in the great
majority of practical situations, microhinges and microcantilevers do
also have axial symmetry. In such cases, the expression of bending
stiffness, which is expressed generically in Eq. (3.22), can be simplified.
The bending stiffness, which is calculated with respect to point 1 in
Fig. 3.7, can be formulated by first expressing the linear direct bending
compliance of the first (right-side) segment in terms of reversed
frames—the first of Eqs. (3.2), namely,
2 (1)
l C
(1) Ҡ (1) (1) r
C = C í lC + (3.24)
l l c 4
and then applying the series-connection rule of Eq. (3.22). The bending
stiffness of the flexible member of Fig. 3.7 becomes
1
k b,e = (2) (3.25)
2 (1)
2(C + l C r / 4)
l

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com)
Copyright © 2004 The McGraw-Hill Companies. All rights reserved.
Any use is subject to the Terms of Use as given at the website.