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MEM 423
–
Mechanics of Vibrations
–
Patel, Shwarz, Esposito, Acklin
–
FINAL EXAM REPORT
MEM
–
423
MECHANICS OF VIBRATIONS
TEAM
–
C
Lakir Patel
Chris Esposito
Erica Shwarz
Sara Acklin
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–
Mechanics of Vibrations
–
Patel, Shwarz, Esposito, Acklin
–
FINAL EXAM REPORT
Introduction:
The study of Vibration analysis is crucial to science, due to the fact that most human
activities involve some sort of vibrations.
Such examples are the vibration of lungs when we
breathe, periodic oscillations of the feet when we walk as well as vibration is the ear that
allows us to hear. Especially when dealing with Mechanical system or any system for that
matter, each system has its own natural frequency. These systems can be impacted via an
external frequency causing it to vibrate at a certain artificial frequency, when the two
frequencies coincide, a phenomenon known as resonance occurs. At this point in time, excessive
gains in amplitude can result, causing system failure due to material wear and fatigue. In our
specific case, we are dealing with simple beam that is supported on both ends, which has a
natural frequency of 2 rad/sec and has force acting on it of 1000 N, which is denoted as
F = F
o
sin(ω
o
t).
Theory:
The vibration of a system involves the transfer of it potential energy to kinetic energy
and the transfer of its kinetic energy to its potential energy. For such a simple definition,
various states of vibration exist. If a system, after an initial disturbance, is left to vibrate on
its own, the ensuing vibration is known as Free Vibration and no external forces act on the
system. If a system is subjected to an external force, subsequently it is referred to as Forced
Vibration. If any energy is lost during vibration, it is called damped vibration.
MEM 423
–
Mechanics of Vibrations
–
Patel, Shwarz, Esposito, Acklin
–
FINAL EXAM REPORT
Any system can be simplified for the sake of understanding the system dynamics, thus
the Degree of Freedom (DOF) of system. A DOF is a minimum number of independent coordinates
required to determine completely the positions of all parts of a system at any instant of time
defines the degree of freedom of a system (Mechanical Engineering, Rao). Any system can then
be characterized with the help of Equations of Motion, which describes the behavior of the
system as a function of time.
For our system of the beam, The basic equation of motion will be:
M
°
+ Kq = Eu, where q represents the Degree of Freedom, DOF.
In the equation, M represents the MassMatrix and K represents the Stiffness Matrix.
The Mass Matrix could be generated with values in diagonal. Stiffness Matrix, K can be obtained
from the Flexibility Matrix. For approximation of masses placed on the beam in such a way that
masses are evenly distributed throughout the beam, thus the weight distribution stays even
completely across. The beam is not considered a rigid body, thus making it a continuous system
with infinite DOF. In order to practically analyze the system, as requested both 2 and 4 Mass
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This note was uploaded on 01/23/2009 for the course MEM 423 taught by Professor Yousuff during the Summer '07 term at Drexel.
 Summer '07
 Yousuff

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