Page 41 - Mechanics of Microelectromechanical Systems
P. 41
28 Chapter 1
The following boundary conditions apply for the fixed-free beam:
The differential equation for the equilibrium position of the analyzed
cantilever can be written by combining Eqs. (1.52) and (1.53) as:
The solution to Eq. (1.90) is a third degree polynomial in x whose 4
unknown coefficients can be found by applying the boundary conditions of
Eq. (1.89). This solution can be put into the following form:
where the distribution functions and (b stands for bending, d for
deflection and s for slope, respectively) are dependent on the geometry of the
analyzed microcantilever. For the specific case being analyzed here, the
distribution functions are:
The slope function is the derivative of the deflection function in terms of the
space variable x, and therefore its equation derives from Eq. (1.91):
The tip force can be expressed by using Eqs. (1.86), (1.88), (1.91) and
(1.93) as:
