Page 64 - Mechanics of Microelectromechanical Systems

1. Stiffness basics 51
Example 1.13
Two microcantilevers are fabricated as shown in Fig. 1.28 (a) and (b)
from the same material and they also have the same thickness t. Which of the
two designs is the most compliant (in terms of the direct linear compliance
about the z-axis, which is perpendicular to the planar microcantilevers at
their free ends) when

Figure 1.28 Two microcantilevers formed by serial connection of rectangular segments

Solution:
The linear direct compliance of the microcantilever sketched in Fig. 1.28
(a) can be obtained by applying Castigliano’s displacement theorem in the
presence of a force acting at the free end about a direction perpendicular to
the microcantilever’s plane. Its equation is:

Similarly, the compliance of the microcantilever of Fig. 1.28 (b) is:

By taking into account that:

the ratio of the two compliances of Eqs. (1.201) and (1.202) is:

As a conclusion, the configuration of Fig. 1.28 (a) is approximately 1.66
times more compliant than the design of Fig. 1.28 (b).