Page 116 - The Mechatronics Handbook

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FIGURE 8.3 Surface micromachined beams: (a) Two-layer composite beam with electrostatic actuation; (b) two-
layer composite bridge with electrostatic actuation.
where w is the width of the beam, t 1 the thickness of the top beam, t 2 the thickness of insulating layer
(silicon oxide, silicon nitride), l the length of the beam, l e the length of fixed electrode, E 1 the Young’s
modulus of the top layer, E 2 the Young’s modulus of insulating layer.

Torsional Springs
Torsion of beams is used primarily in rotating structures such as micromirrors for optical scanning, or
projection displays. The micromirror array developed by Texas Instruments for example uses polycrys-
talline silicon beams as hinges of the micromirror plate.
The torsion problems can be solved in a closed form for beams with elliptical or triangular cross sections
[Mendleson 1968]. In the case of an elliptical cross section, the moment required to produce an angular
twist (angle or rotation per unit length of the beam) α [rad/m] is equal to
3 3
pa b
M = ----------------Ga (8.25)
a + b 2
2
where G is the elastic shear modulus, and a and b are the lengths of the two semi-axes of the ellipse. The
maximum shear stress in this case is
2
-------------------,
t max = 2Gaa b a > b (8.26)
a + b 2
2
The torsional stiffness of rectangular cross-section beams can be obtained in terms of infinite power
series [Hopkins 1987]. If the cross-section has dimension a × b, b < a, the first three term of this series
result in an equation similar to (8.25)
4
b


3 1
b
M = 2KGα, where K = ab -- – 0.21-- 1 – ---------- . (8.27)
4
3 a  12a
Thin Plates
Pressure sensors are one of the most popular electromechanical transducers. The basic structure used to
convert mechanical pressure into electrical signal is a thin plate subjected to a pressure differential.
Piezoresistive gauges are used to convert the strain in the membrane into change of resistance, which is
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