  The dynamic properties of membranes have been object of many researches since
they can be used as sensor heads in different devices. Some methods have been
proposed to solve the problem of determining the resonance frequencies and
their dependence on the stress caused by forces applied on the membrane
surface. The problem of the vibrating rectangular membrane under a stress
caused by a uniform in-plane force is well known. However, the resonance
frequency behaviour when the force is out-of-plane instead of in-plane, is not
so well understood and documented. A gradiometer which uses a silicon square
membrane with a magnet fixed on it as a sensor head has been developed in a
previous work. This device reports a quadratic dependence of the frequency on
the out-of-plane magnetic force. In this work, simulations to obtain the
dependence of the frequency of the fundamental flexural mode on the stress have
been performed. It has been studied the influence of in-plane and out-of-plane
forces applied to the membrane. As expected, a square root dependence has been
found for in-plane forces. Nevertheless, the problem is more complex when
out-of plane forces are considered. Out-of-plane forces gives rise to an
initial quadratic dependence which turns into a square root dependence from a
certain stress value. The quadratic range increases and the rate of change of
the frequency decreases as the surface of the magnet fixed on the membrane
increases. The study has addressed these problems and both, experimental and
simulated results have been compared and a good agreement between experimental
and simulated results has been found
