Page 170 - Mechanics of Microelectromechanical Systems

3. Microsuspensions 157
The definition stiffness about the z-direction is determined by applying a
force (not shown in Fig. 3.24) and by calculating the corresponding
displacement (when bending and torsion are taken into consideration), as
done with previous spring designs. The equation of this stiffness is:

There are situations where the rigid body which attaches frontally to two
spiral springs undergoes a rotation about its longitudinal (x) axis. In this case,
each spiral spring will be subject to torsion, and therefore the torsional
stiffness is of interest. By applying a moment at the free end 1 – this
moment is not drawn in Fig. 3.24 – the torsional stiffness of a serpentine
spring unit is:

Equation (3.82) took into consideration that the long legs are subject to
bending about the in-plane direction y, whereas the short legs are loaded in
torsion about their longitudinal axes.

Example 3.9
Design a basic serpentine spring in such a manner that the following
stiffness relationship apply: (both stiffnesses are calculated
in the definition sense). The rectangular cross-section is constant is defined
by and Poisson’s ratio is

Solution:
By using the following equations for the cross-sectional properties:

the stiffness condition, which is:

results in an equation that can be solved for the length of the short leg in
terms of the length of the other leg as plotted in Fig. 3.28. It can be seen
that in order to satisfy the requirements of the example, the length varies
almost linearly with