Page 23 - Mechanical design of microresonators _ modeling and applications

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Design at Resonance of Mechanical Microsystems

22 Chapter One

paddle bridge

rotary motion

anchor

translatory motion

anchor
Figure 1.22 Microbridge as a 2-DOF system.

1 =෸ 1
Q overall i Q i (1.61)

where the Q i are all identifiable and quantifiable individual quality
factors.

1.3 Multiple-Degree-of-Freedom Systems
Mechanical microresonators that are capable of more than one motion
are modeled as multiple-DOF members. A paddle microbridge, such as
the one shown in Fig. 1.22 and which can be used at detecting extra-
neous substances through shifts in the resonant frequencies (more
details on mass addition detection are offered in Chap. 6), may vibrate
in bending and in torsion, and therefore it needs to be modeled as a 2-
DOF system.
This section first discusses the approximation methods of Rayleigh
and Dunkerley that permit evaluation of the upper and lower bounds
on the resonant frequencies of a multiple-DOF system. Presented next
are the notions of eigenvalues, eigenvectors, and eigenmodes (or mode
shapes) as well as the static and/dynamic coupling. Lagrange’s equa-
tions are studied subsequently as a tool of formulating the equations of
motion of a vibrating system. Mechanical-electrical analogies are
presented in the end with the main notions of the Laplace transform,
the transfer function, and a mechanical resonator filter application.

1.3.1 Approximate methods for resonant
frequencies calculation
In many instances only the extreme resonant frequencies are relevant
in the design of a microresonator which is modeled as a multiple-DOF

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