Unit 3 Free vibration

Dynamics of Machines R 2017

1. Unit III
FREE
VIBRATION
DYNAMICS OF MACHINES
FREE
VIBRATION
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B.K.Parrthipan, M.E., M.B.A., (Ph.D).,
Assistant Professor / Mechanical Engineering
Kamaraj College of Engineering and Technology.

2. Introduction -Vibratory systems
When elastic bodies such as a spring, a beam and a shaft
are displaced from the equilibrium position by the application
ofexternalforces,andthenreleased,theyexecuteavibratory
motion.
This
is
due
to
the
reason
that,
when
a
body
is
displaced,
This
is
due
to
the
reason
that,
when
a
body
is
displaced,
the internal forces in the form of elastic or strain energy are
present in the body. At release, these forces bring the body to
itsoriginalposition.
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3. Introduction -Vibratory systems
When the body reaches the equilibrium position, the
whole of the elastic or strain energy is converted into kinetic
energy due to which the body continues to move in the
oppositedirection.
The
whole
of
the
kinetic
energy
is
again
converted
into
The
whole
of
the
kinetic
energy
is
again
converted
into
strain energy due to which the body again returns to the
equilibrium position. In this way, the vibratory motion is
repeatedindefinitely.
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4. Terms Used in Vibratory Motion
1. Periodofvibrationortimeperiod.
It is the time interval after which the motion is repeated
itself.Theperiodofvibrationisusuallyexpressedinseconds.
2.Cycle.
It
is
the
motion
completed
during
one
time
period
.
It
is
the
motion
completed
during
one
time
period
.
3.Frequency.
Itisthenumberofcyclesdescribedinonesecond.InS.I.
units, the frequency is expressed in hertz (briefly written as
Hz)whichisequaltoonecyclepersecond.
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5. Types of Vibratory Motion
AccordingtotheActuatingforce
1. Freeornaturalvibrations.
Whennoexternalforceactsonthebody,aftergivingitan
initial displacement, then the body is said to be under free or
natural
vibrations
.
The
frequency
of
the
free
vibrations
is
natural
vibrations
.
The
frequency
of
the
free
vibrations
is
calledfreeornaturalfrequency.
2. Forcedvibrations.
When the body vibrates under the influence of external
force, then the body is said to be under forced vibrations. The
externalforceappliedtothebodyisaperiodicdisturbingforce
createdbyunbalance.The vibrations havethe same frequency
astheappliedforce.
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6. Types of Vibratory Motion
3.Dampedvibrations.
When there is a reduction in amplitude over every cycle
ofvibration,themotionissaidtobedampedvibration.Thisis
due to the fact that a certain amount of energy possessed by
the
vibrating
system
is
always
dissipated
in
overcoming
the
vibrating
system
is
always
dissipated
in
overcoming
frictionalresistancestothemotion.
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7. Degree of freedom
The minimum number of independent coordinates
required to specify the motion of a system at any instant is
knownasDegreeoffreedom(D.O.F)ofthesystem.
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8. Single DOF
The system shown in this figure is what is known as a
Single Degree of Freedom system. We use the term degree of
freedomtorefertothenumberofcoordinatesthatarerequired
tospecifycompletelythe configurationofthesystem.Here,if
the
position
of
the
mass
of
the
system
is
specified
then
the
position
of
the
mass
of
the
system
is
specified
then
accordingly the position of the spring and damper are also
identified.Thusweneedjustonecoordinate(thatofthemass)
to specify the system completely and hence it is known as a
singledegreeoffreedomsystem.
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9. FREE VIBRATIONS
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10. Types of Free Vibrations
Accordingtothemotionofsystemwithrespecttoaxis
1. Longitudinalvibrations,
2. Transversevibrations,and
3. Torsionalvibrations.
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11. 1.Longitudinalvibrations
When the particles of the shaft or disc moves parallel to
the axis of the shaft, then the vibrations are known as
longitudinalvibrations.
2.Transversevibrations.
When
the
particles
of
the
shaft
or
disc
move
Types of Free Vibrations
When
the
particles
of
the
shaft
or
disc
move
approximately perpendicular to the axis of the shaft, then the
vibrationsareknownastransversevibrations.
3.Torsionalvibrations
When the particles of the shaft or disc move in a circle
about the axis of the shaft, then the vibrations are known as
torsionalvibrations.
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12. Natural Frequency
Itisthenumberofcyclesdescribedinone
second. In S.I. units, the frequency is
expressed
in
hertz
(briefly
written
as
Hz)
expressed
in
hertz
(briefly
written
as
Hz)
whichisequaltoonecyclepersecond.
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13. Equation of Motion –Natural
Frequency
The natural frequency of the free longitudinal vibrations
maybedeterminedbythefollowingthreemethods:
1.EquilibriumMethod
2.Energymethod
3.Rayleigh’smethod
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14. 1. Equilibrium Method
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16. 2. Energy method
•We know that the kinetic energy is due to the motion of the
body and the potential energy is with respect to a certain
datum position which is equal to the amount of work required
tomovethebodyfromthedatumposition.
•In the case of vibrations, the datum position is the mean or
equilibrium
position
at
which
the
potential
energy
of
the
body
or
the
system
is
zero
.
equilibrium
position
at
which
the
potential
energy
of
the
body
or
the
system
is
zero
.
•Inthefreevibrations,noenergyistransferredtothesystemor
from the system. Therefore the summation of kinetic energy
andpotentialenergymustbeaconstantquantitywhichissame
atallthetimes.
•Inotherwords,
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17. 2. Energy method
•We know that kinetic energy,
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18. 3. Rayleigh’s method
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19. 3. Rayleigh’s method
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20. Natural Frequency of Free Transverse
Vibrations
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21. Natural Frequency of Free Transverse
Vibrations
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24. Damped Vibration
It is the resistance to the motion of a vibrating body. The
vibrationsassociatedwiththisresistanceareknownasdamped
vibrations.
Thecoefficientofdampingisdenotedbyc.
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25. Critical damping coefficient
The critical damping coefficient (c
c)is the amount of
dampingrequiredforasystemtobecriticallydamped.
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26. Damping Factor or Damping Ratio
The ratio of the actual damping coefficient (c) to the
critical damping coefficient (c
c) is known asdamping factor
ordampingratio.
Mathematically,
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Thedampingratioisasystemparameter,denotedbyζ(zeta),
•ifζ>1,thenthe systemissaidtobeoverdamped
•ifζ=1, thenthesystemissaidtobecriticallydamped
•ifζ<1,thenthesystemissaidtobeunderdamped.

27. Logarithmic Decrement
It is defined as the natural logarithm of the amplitude
reduction factor. The amplitude reduction factor is the ratio of
any two successive amplitudes on the same side of the mean
position.
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28. Critical Speed
The speed at which the shaft runs so that the additional
deflection of the shaft from the axis of rotation becomes
infinite,isknownascriticalorwhirlingspeed.
The speed at which resonance will occur is known as
criticalspeed.Resonanceistheconditionatwhichtheinduced
frequencyisequaltotheappliedfrequency.
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29. Torsional vibrations
whentheparticlesofashaftordiscmoveinacircleabout
the axis of a shaft, then the vibrations are known as torsional
vibrations.
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30. Free Torsional Vibrations of a Single Rotor
System
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31. Free Torsional Vibrations of a Two
Rotor System
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32. Free Torsional Vibrations of a Three
Rotor System
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