Page 133 - Mechanics of Microelectromechanical Systems

120 Chapter 2
The bending stiffness of a constant rectangular cross-section microbridge
is given in Eq. (2.185), and consequently, the ratio of the two microbridges,
can be expressed just in terms of the parameters b and w, as
plotted in Fig. 2.38. The bending stiffness of the right elliptic microbridge
can be 10 times larger than the stiffness of the constant cross-section design,
as illustrated in Fig. 2.38.

5.2.2 Torsion

A torque is applied at point 2 of the microbridge sketched in Fig
2.37 in order to determine the torsion-related stiffness at that point. The two
end-point torque reactions are equal to half the torque that is applied at
midpoint 2. The torsional stiffness is defined as:

The angular displacement at point 2 is found by means of Castigliano’s
displacement theorem as:

By applying considerations similar to the ones presented for bending, the
torsional stiffness can be expressed as:

where the compliance of Eq. (2.188) corresponds to the 2-3 interval of the
microbridge.
For a thin, constant cross-section configuration, the torsional stiffness of
Eq. (2.170) simplifies to:

which is indeed the known relationship.

5.3 Compound Designs

Microbridges can be formed of different compliant segments that are
connected serially and are fixed at the extremities of the chain, as pictured in
Fig. 2.39 for instance. Figure 2.39 gives the geometric dimensions of a