Page 185 - Mechanics of Microelectromechanical Systems

172 Chapter 3

where is the out-of-the-plane stiffness of one curved spring. The six
out-of-the-plane compliances of a thin curved member have explicitly been
formulated in Eqs. (1.139) to (1.143) and Eq. (1.163), respectively. The
linear stiffness can be found by inverting the compliance matrix of Eq.
(1.137), which results in:

As a consequence, the individual stiffness is:

Example 3.15
Find the tip angle of a curved microbeam which is part of a rotary
(torsional) spring suspension which connects an inner shaft of diameter d to
an outer hub of interior diameter D (D = 2d), in a way that would maximize
the spring’s compliance with respect to an external torque for a given
rectangular cross-section and material properties. Consider that
and E =125 GPa.

Solution:
The tip angle of the curved spring can be expressed as:

and therefore this condition has to be used in Eq. (3.119), which gives the
torsional stiffness of such a spring. The stiffness of interest is plotted in Fig.
3.42 in terms of the shaft diameter d and the radius of the curved spring R by
utilizing the given numerical values. As the figure shows, the stiffness is
larger for larger values of d and smaller values of R. As a consequence, one
has to select these parameters accordingly, namely small values for d and
large values for R.