Page 191 - Mechanics of Microelectromechanical Systems
P. 191
178 Chapter 3
are substituted into Eq. (3.137)‚ which defines the torsion compliance. The
corresponding definition stiffness is the inverse of the compliance. The plot of
Fig. 3.45 is drawn in terms of and It can be seen that the influence of
is not marked in comparison to The latter parameter should be small in
order to reduce to stiffness (and implicitly to increase the compliance).
3.2.2 Spiral Spring with Large Number of Turns
The case of a spiral with a large number of thin turns‚ as the one pictured
in Fig. 3.46‚ is studied now.
Figure 3.46 Spiral with a relatively large number of turns
The outer end is clamped whereas the inner end is fixed to a shaft that can
rotate under the action of a torque The torsion stiffness (relating to
the rotation angle is of interest here. The bending moment at a generic
point of coordinates x and y is:
The following approximate equilibrium equation applies:
By combining Eqs. (3.141) and (3.142)‚ the bending moment becomes:
By applying Castigliano’s displacement theorem‚ the forces and can be
expressed in terms of and‚ in the end‚ the rotational stiffness of the spiral
spring is found to be (as also shown in Chironis [2] and Wahl [3]):
