Page 147 - Mechanical design of microresonators _ modeling and applications

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Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

146 Chapter Three

w2 l2
w1

11
Figure 3.26 Top view of hollow rectangular microcantilever.

3.4.1 Rectangular microcantilevers
Figure 3.26 shows the top view of a hollow microcantilever of rectan-
gular envelope. It consists of two arms that are flexible and a connecting
arm which can be considered rigid.
Under the assumptions that only the thinner side bars are compliant,
the resulting stiffness of the structure, be it axial, torsional, or bending-
related, will be twice the corresponding stiffness of a single side bar,
because the two flexible parts behave as two springs in parallel.
The axial stiffness of the hollow rectangular microcantilever is thus

2Ew t
1
k a,e = (3.111)
l
1
The lumped-parameter mass which corresponds to the distributed
parameter inertia of the axial free vibrations consists of three fractions:
two identical terms that result from the flexible parts and the mass of
the rigid transverse member. Such situations, where effective mass
factions coming from flexible parts are added to actual masses of rigid
members, is studied more thoroughly in subsequent chapters of this
book. The total mass undergoing free axial vibrations is

2 2)
2 (
2 2l w
1 1
m a,e = 3 m 1,a + m = ȡt 3 + l w (3.112)
The axial-related resonant frequency is
Ew 1
Ȧ =2.45 (3.113)
a,e ȡl (2l w + l w )
1 1 1 2 2
The torsional stiffness of this hollow configuration (its compliance is
5
given by Lobontiu and Garcia ) is

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