Page 72 - Mechanics of Microelectromechanical Systems

1. Stiffness basics 59
uniform pressure p, the maximum displacement is registered at its center, and
it can be calculated by solving the following third degree equation – see den
Hartog [7] for instance:

where is the center deflection and D is the rigidity which is defined as:

It can be seen that Eq. (1.227) incorporates both the small-deformation
bending effects through the linear term in and the membrane (stretching)
effects through the non-linear term in the same
For bending-dominated cases, where the membrane effects can be
ignored, the differential equation of deflection is:

The maximum deflection for the circumferentially-clamped circular plate
under uniform pressure is:

The maximum deflection of the same plate under a concentrated load acting
normally at the disc’s center is:

For rectangular plates, the solution to Eq. (1.229) is found by using the
Fourier series expansion, and thus the solution is only approximate. For a
rectangular plate of dimensions and which is fixed on its edges and is
acted upon by a pressure p, the maximum center deflection is:

where the summation only retains the odd counters i and j. The maximum
deflection of this plate when acted upon by a force, perpendicularly to the
plate’s plane, is similarly calculated as: