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.

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.