Page 125 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH03_124_08/30/05

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

124 Chapter Three

6
2
rωb
0
0.1
c1
cw
0.2
0.9
Figure 3.15 Bending resonant frequency comparison between the microcantilever of
Fig. 3.11 and the reversed configuration of Fig. 3.12.

constant-cross-section segment is larger than that of the trapezoid (for values
of c l that are larger than 1).
It can be shown by using Eqs. (3.49) and (3.51) that the bending resonant
frequency ratio can be expressed as

rev
Ȧ b,e
rȦ = = rk rk m (3.52)
b
b
Ȧ
b,e
and Fig. 3.15 is the three-dimensional plot of this ratio.
The trend displayed by both the stiffness and the mass ratios is also fol-
lowed by the resonant frequency ratio, which is plotted in Fig. 3.15, as this
ratio is the square root of the product of the stiffness and mass ratios defined
in Eqs. (3.49) and (3.51).
The paddle microcantilever of Fig. 3.16 is similar to the one previ-
ously studied, and it combines a rectangular portion at the root with a
trapezoid at its free end. The difference consists in the trapezoid
unit having its thickness variable. The two basic units have the same
width w.
The axial stiffness is

Et (t – t )w
2 1
2
k a,e = (3.53)
l (t – t ) + l t ln(t 1/ 2
t )
1 2
2
2 1
The effective axial mass is

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.