Page 324 - Mechanics of Microelectromechanical Systems

5. Static response of MEMS 311
The length of the beam can be expressed by integrating the differential length
ds of Eq. (5.133), namely:

The right-hand side of Equation (5.135) can be expressed in terms of an
integral of the form:

which is known as an elliptic integral of the first kind where c is a constant.
Equation (5.135) enables to find the force F that corresponds to a tip slope
The maximum tip deflection can be found by solving the first Eq. (5.131),
namely:

Finding implies numerically solving an elliptic integral of the second kind,
which is defined as:

Similarly, the tip position is determined by integrating the second Eq.
(5.131) as:

An example will be studied now to highlight the differences between the
deformations of a beam when calculated by the large-displacement
hypotheses versus the small-displacement theory.

Example 5.19
Consider a microcantilever of length and having a cross-
section defined by and being acted upon by a force F as