Page 203 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH04_202_08/30/05

Microbridges: Lumped-Parameter Modeling and Design

202 Chapter Four
y

x

l1
l2/2
Figure 4.20 Half-model of paddle microbridge.

the torsional and bending resonant frequencies, by directly using
Castigliano’s displacement theorem for the lumped-parameter stiffness
calculation and Rayleigh’s principle for effective inertia fraction deter-
mination. Figure 4.20 pictures the half-length paddle microbridge and
also indicates the guided boundary condition at the midspan which has
to be utilized in bending calculations.
In torsion, the stiffness of the half-microbridge is

3
2Gt w w
1 2
k t,e = 3(w l +2w l (4.118)
2 1
1 2
Equation (4.118) simplifies to Eq. (4.24) – giving the torsional stiffness
of a homogeneous, constant-cross-section half-microcantilever – when
w 2 = w 1 , l 1 = l/4 and l 2 = l/2 (such that l 1 + l 2 /2 = l/2).
The equivalent mechanical moment of inertia which is dynamically
equivalent to the distributed-parameter half-microbridge undergoing
free torsional vibrations is calculated by applying Rayleigh’s principle,
which has been detailed thus far. It is worth noting that the distribution
function connecting the rotation/deformation angle at an abscissa x
measured from the guided end in Fig. 4.20 to the maximum rotation/
deformation angle is

x
f (x) =1 Ì (4.119)
t
2/
l + l 2
1
and therefore of the form corresponding to a fixed-free microbar (fixed-
free beam). The effective moment of inertia is calculated as

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.