Page 297 - Mechanics of Microelectromechanical Systems

284 Chapter 5

Equations (5.49) and (5.52) reduce to the equations of Sattler et al. [2], who
treated the particular case where and
As mentioned previously, the active moment of either Eq. (5.49) or Eq.
(5.52) is opposed by the elastic spring moment produced through torsion of
the two supporting hinges. The spring moment is:

where is the torsional stiffness of a hinge. At equilibrium, and
should balance each other, and the point of stable equilibrium, as discussed
for a single-spring microsystem, can be found by solving the equilibrium
equations:

where:

The solution to Eqs. (5.54) is the set for the voltage-control problem,
or the set for the charge-control problem (the p subscript denotes
pull-in, as mentioned previously).

Example 5.10
The torsion microdevice of Fig. 5.18 is used to determine the magnitude
of an electromagnetic field B which acts in the plane of the middle sensing
plate and of the two identical circular corner-filleted microhinges. The
rotation angle of the plate is determined experimentally to be 3° when a
current I = 20 mA passes through the circular loop of radius The
shear modulus of the hinges is G = 56.7 GPa and the hinges are defined by
r = 1/8, and Find the external magnetic
disturbance B.

Solution:
The torque M that is produced by the interaction between the current I
and the external magnetic field B is given in Eq. (4.65) of Chapter 4. This
torque rotates the microdevice of Fig. 5.18 by an angle according to Eq.