Consider a rectangular thin, uniform, homogeneous plate, simply supported along two of the opposite.

Consider a
rectangular thin, uniform, homogeneous plate, simply supported along two of the
opposite sides and fixed along the remaining two opposite sides. The plate is loaded
by electrostatic forces generated by maintaining the plate at a fixed potential
with reference to the ground plane. (i) Show that the electrostatic force is
directly proportional to the first derivative of the capacitance with respect
to the transverse displacement. (ii) Also show that the loading on the plate is
directly proportional to the plate’s transverse displacement. (iii) Hence
obtain the natural frequencies and normal mode shapes of the plate.