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This report deals with vibration control of a cantilever beam using
piezoelectric material, where two piezoelectric layers to be installed at fixed
end of the cantilever beam where one layer will be installed on top of the beam
and the other to be fixed at the bottom. One of the piezoelectric layers will
be used a sensor and the other will be used as an actuator.

The forces could be in the form of longitudinal or distributed
forces and the vibration could be in the form of (free or forced vibration),
where the report will mainly focus on the forced vibration.

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The study will be demonstrated through simulations via MATLAB where
Proportional Integral Derivative controller “PID” controller will be used. The
entire structure will be modelled using Euler-Bernoulli beam theory, The Lagrange’s Equations, analytical analysis, finite element
method (FEM), concept of piezoelectric theory, and state space techniques.











The piezoelectric effect is a molecular phenomenon that can be
observed at the macroscopic level as a change in electric potential that is
created when a piezoelectric substance is deformed. This was first observed by
Carl Linaeus and Franz Aepinus in the mid-1800s, but it was not truly
understood until it was demonstrated by two French physicists, Jacques and
Pierre Curie in 1880. Certain materials such as quartz, salt, and even sugar
would generate a voltage when placed under stress. These materials all had
characteristic crystal structures formed from a lattice of molecules with asymmetric
dipole moments that would respond to mechanical pressure. Thus, they were named
piezoelectric crystals after the Greek word piezo meaning press. Later it was
found that alternatively, the crystals would vibrate when a current was passed
through them. Depending on the wave form of the voltage, sound could be










Modern applications


After the piezo electric was discovered and been known a wide range
of researches has been done to make this discovery more useful in a verity of
application. Moreover, in the early 1900s, the converse of this was used in the
first submarine sonars. There are common examples of using piezoelectric as we
are going to mention some of them such as Pressure Sensors, Sonar Equipment, Ultrasonic Cleaning etc. here are some clarification of the useful of each one of them:

1-Pressure Sensors — In nearly any application requiring the measurement
of dynamic pressure changes, using piezoelectric pressure sensors yields more
reliable results than using conventional electromechanical pressure sensors.

2-Sonar Equipment — Depth sounders and sonar equipment rely extensively
on piezoelectric sensors to transmit and receive ultrasonic “pings” in the
50-200kHz range. Besides having an ideal frequency response for such
applications, piezoelectric transducers have a high-power density that enables
large amounts of acoustic power to be transmitted from a small package.

3-Fast Response Solenoids — piezoelectric actuators are often able to provide
fast response and low power consumption in small packages, compared to
electromagnetic solenoids.

4-Ultrasonic Cleaning — Piezoelectric actuators are also used for
ultrasonic cleaning applications. To perform ultrasonic cleaning, objects are
immersed in a solvent (water, alcohol, acetone, etc.). A piezoelectric
transducer then agitates the solvent. Many objects with inaccessible surfaces
can be cleaned using this methodology.

The application of piezoelectric is
expanded to include the industries of structures and airplane, where
controlling the vibration of a cantilever beams, wings of an aircraft and
buildings are getting more as a research intrest. More detail explanation is
discussed below:

Piezoelectric materials are very much
suitable for producing electricity from a very high frequency vibration. In
aircraft a very high frequency vibration is produced in the surface of the
aircraft. This vibration is a loss of energy that is produced by inside
machinery of the aircraft and also due to the effect of ramming air. If we can
use this vibration to produce electricity, then it would be reusing or
recycling this energy loss. This could be a cost-free source of electricity.

 Another application is used for
structures such as A cement-based piezoelectric sensor for civil engineering structure

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