In the beam, so the two forces

In this
report the design of our system is chosen to be as following: cantilever beam
“a beam with fixed end” and two piezoelectric layers to be pounded on the beam.
One of the piezoelectric layers is to be installed on the top surface of the
cantilever beam and the other layer to be fixed on the bottom surface of the
cantilever beam, the piezoelectric layers are to be used as Sensor/Actuator. If
we consider the beam is divided into three equal parts then the piezoelectric
layers are to be pounded on the last third of the cantilever beam near the
fixed end where, it is proven it is the optimal location for the piezoelectric
layers to be installed for the best results of controlling the vibration of the
cantilever beam. The cantilever beam is considered to be a uniform beam where,
the piezoelectric layers are too thin, so their thickness and their stiffness
are to be neglected.

            In our design we consider active
vibration control where the vibration of the beam is caused due to external
force applied on the beam and to be controlled by applying a counter force on
the cantilever beam using the piezoelectric layers where the counter force is
equal to the original force applied on the beam, so the two forces cancel each
other thus the beam stops vibration. So as explained before the piezoelectric
material has the ability to transform mechanical energy to electrical energy
and vice-versa so that how the piezoelectric layers will be working as sensor
and an actuator, as sensor: when the vibration of the beam occurs the
piezoelectric layers will transform that mechanical energy into electric energy
in the form of “voltage” which can be measure and as an actuator: by giving the
piezoelectric layers some electric energy again the form of voltage which
transform to mechanical energy in the form of vibration or “force”, so a
controller is needed to control this process of sensing, actuating process.

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            The first step of building a
controller is creating a mathematical model of the vibration. Finite element
analysis is to be used to create the mathematical model but before that we have
to obtain the equation of motion for beam and to start with that we to make
some assumptions.