Page 159 - Mechanics of Microelectromechanical Systems

146 Chapter 3
Equation (3.42) indicates that the model including bending effects generates a
compliance about the main direction of action which is very slightly larger
than the compliance yielded by the model that adds axial effects to bending.
Figure 3.13 is the plot of the compliance ratio of Eq. (3.42) as a function of
and when and Similarly, Fig. 3.14 is the plot of the
same compliance ratio in terms of w and when As both
figures indicate, the compliance ratio is in the very close vicinity of 1 when
the design variables of Eq. (3.42) span relatively wide ranges, which indicates
that neglecting the axial effects has little influence on the main compliance.

Example 3.6
Find the definition stiffness of a U-spring about the y-direction in the case
where the middle leg has a small length which implies considering the
additional shearing effects and associated deformations. Compare the
resulting stiffness with the regular one determined by means of the
compliance of Eq. (3.38) in the case where and

Solution:
When the length is only about 3-5 times greater than the largest cross-
sectional dimension, the deformation produced by the shearing force
has to be accounted for in addition to bending. The displacement at point 1
about the y-direction in Fig. 3.12 is calculated by means of Castigliano’s
displacement theorem as:

where the subscripts in bending moments M and shearing force S indicate the
specific segment out of the three ones making up together the U-spring. The
linear stiffness about the y-direction can be expressed according to its
definition as:

whereas the same stiffness which only considers bending is:

By constructing the ratio of the y-axis stiffness in Eq. (3.44) to the stiffness of
Eq. (3.45), the plot of Fig. 3.15 can be drawn in terms of the lengths and