Page 99 - Mechanical design of microresonators _ modeling and applications

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Basic Members: Lumped- and Distributed-Parameter Modeling and Design

98 Chapter Two
EI y EI y
Ȧ =4.468 Ȧ = 22.93 (2.193)
1 3 2 3
ml ml
It can be seen that while the second frequency is closer to the exact one,
the natural frequency is a less precise approximation, compared to the
one given by the first set of shape functions, Eqs. (2.189). A better ap-
proximation to both the first and second resonant frequencies can be
produced by using three shape functions: the two in Eqs. (2.189) and
the second Eqs. (2.192).

2.3.2 Circular rings
The circular ring can be used as a resonant microgyroscope, for in-
9
stance, as discussed and implemented by Ayazi and Najafi. This reso-
nant microdevice is discussed in Chap. 5. A circular ring can vibrate
radially, in bending, or in torsion or may have a combined torsional/
bending mode, as discussed next.
The ring vibrates radially as shown in Fig. 2.36a. It can be assumed
that the ring is only subject to axial deformations along its circum-
ference during the radial vibrations.
In this case, it can be shown that the axial force that produces a radius
change of u r is
EAu r
N = (2.194)
R

The strain energy is therefore

R

(a) (b)
Figure 2.36 Circular ring undergoing (a) radial vibrations and (b) bending vibrations.

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