Page 157 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH03_156_08/30/05

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

156 Chapter Three

20
r ω 3 × 10 -6
14
5. × 10 -7 t2 [m]
t1 [m] -6
1.5 × 10
1.5 × 10 -6
Figure 3.33 Torsional-to-bending resonant frequency ratio in terms of layer thicknesses.

Example: Study the influence of geometry on the bending and torsional res-
onant frequency of a bimorph when the following parameters are known:
E 1 = 150 GPa, G 1 = 60 GPa, E 2 = 180 GPa, G 2 = 72 GPa. The densities of the
two materials are equal. Consider that the geometric parameters range as
follows: t 1 ĺ [0.5 “m, 1.5 “m], t 2 ĺ [1.5 “m, 3 “m], l ĺ [800 “m, 1500 “m],
w ĺ [50 “m, 150 “m].
The torsion-to-bending resonant frequency ratio is formed by using
Eqs. (3.146) and (3.153). By first considering that l = 1200 “m and w = 100
“m, the thickness parameters are allowed to vary within their specified
ranges, and the plot of the resonant frequency ratio r Ȧ in Fig. 3.33 is obtained.
For the thickness ranges and all other constant material and geometry
parameter values, the torsional resonant frequency is 14 to 20 times higher
than the bending resonant frequency, as Fig. 3.33 indicates. The plot also
shows that the torsion-to-stiffness resonant frequency ratio decreases with
the increase of the thickness of the deposited (thinner) layer 1 and increases
with the increase of the thickness of the substrate (thicker) layer.
A similar numerical simulation is presented in Fig. 3.34, where the fol-
lowing values have been selected for the layer thicknesses: t 1 = 1 “m and
t 2 = 2 “m, whereas the parameters l and w have been allowed to vary within
their specified ranges.
Varying the length and width of the sandwich makes the torsion-to-
bending resonant frequency ratio vary from 10 to 30, as shown in Fig. 3.34.
Increasing the length and decreasing the width of the bimorph contribute to
increasing the resonant frequency ratio.

For relatively short microcantilevers, Timoshenko’s model needs to
be employed to determine the first bending-related resonant frequency.
The stiffness of a multimorph can be put in a form similar to that given
in Eq. (2.79) for a homogeneous microcantilever, namely,
3(EI ) (GA) e
y e
k = (3.156)
b,e 2
l (GA) l +3ț(EI )
e
y e

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