Page 259 - Mechanical design of microresonators _ modeling and applications

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Resonant Micromechanical Systems

258 Chapter Five

TWO-SPRING RESONATORS

PLANAR OUT-OF-THE-PLANE

FRONTAL LATERAL ROTARY BENDING TORSION

Figure 5.31 Two-spring microresonators.

J
2x 0
M = (5.86)
0 J 1y

and stiffness matrix is

2k t2 0
K = (5.87)
0 2k t1

This system is fully decoupled as both the mass matrix and the stiffness ma-
trix are in diagonal form. The eigenvalues of the dynamic matrix which
corresponds to the mass and stiffness matrices of Eqs. (5.86) and (5.87) yield
the following resonant frequencies:

k t1 k t2
Ȧ =2 Ȧ =2 (5.88)
1 J 2 J
1y 2x
5.3 Spring-Type Microresonators

Micro- and nanoresonators can be built with several complex elastic
suspensions, including those based on beam-type springs, as previously
discussed. The simplest multispring microresonators utilize two
springs to achieve the resonant motion, as indicated in Fig. 5.31.
The planar designs are capable of motion in a plane parallel to the
substrate, and Fig. 5.32 shows the sketches of frontal and lateral motion
solutions. The x direction vibration corresponds to the frontal solution
whereas the y direction solution is the lateral motion design.
In out-of-the-plane microresonators, the motion is no longer parallel
to the substrate and can be produced by either bending or torsion; the
corresponding lumped-parameter models are sketched in Fig. 5.33a
and b.

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