Page 224 - Mechanics of Microelectromechanical Systems

4. Microtransduction: actuation and sensing 211

with:

where F is a dummy force applied to the microcantilever at the free end 1. By
applying the assumptions that the deflection varies according to a quadratic
distribution over the overlapping length (see Kovacs [3] for instance),
namely:

it is possible to simplify Eq. (4.54) – which contains and as
unknowns – to an equation which only contains as unknown. Although
simpler, this equation is still an integral-differential one, which can be solved
only numerically. The final solution is complex and is not given here, but an
example will be studied next to better illustrate this problem.

Example 4.7
Determine the free tip deflection of a microcantilever defined by
and when a voltage U = 50 V acts electrostatically
on the overlap length The initial gap between the microcantilever and its
corresponding fixed actuation plate is The microcantilever’s
material has a Young’s modulus of E = 130 GPa, and the permittivity of the
free space is Assume that the overlap length can range
in the interval.

Solution:
The solution to Eq. (4.54) was obtained by using the calculation
procedure that has previously been outlined, based on the numerical values
of this problem. When the overlap length was given values in the specified
range, the tip deflection values that are plotted in Fig. 4.31 have been
obtained. As Fig. 4.31 indicates, the tip deflection of the microcantilever
increases quasi-linearly with the overlap length increasing.