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Transcript of Theory of vibrations
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Theory of vibration with
application
Assist. Lecture. ALI F. F.
2009-2010
Textbook: W.T. Thomson, Theory of vibration with application, 5th
ED., PRENTICE HALL, 2001
Catalog data: mathematical analysis of physical problem in the
vibration of mechanical system. Topics include linear- free vibration,
forced vibration, and damping in single degree of freedom system,
transient vibration, critical speed and whirling of rotating shaft,
dynamic balancing, and multidegree of freedom system with lumped
parameters.
Lecture-1-
Any motion that repeated itself an interval of time is calledvibration or oscillation motion.
Oscillatory motion
The theory of vibration deals with the study of oscillatory motionof bodies and the forces associated with them.
All bodies possessing mass and elasticity are capable vibration.
Classification of vibration
Free vibration: without external force acts on the system. Theoscillation of a simple pendulum is an example of free vibration
fig.1.
lecture-1
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Theory of vibration with
application
Assist. Lecture. ALI F. F.
2009-2010
Textbook: W.T. Thomson, Theory of vibration with application, 5th
ED., PRENTICE HALL, 2001
Forced vibration: if a system is subjected to an external force, theresulting vibration is known as forced vibration such as diesel
engines. Fig.2
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Theory of vibration with
application
Assist. Lecture. ALI F. F.
2009-2010
Textbook: W.T. Thomson, Theory of vibration with application, 5th
ED., PRENTICE HALL, 2001
Degree of freedom : the minimum number of independent coordinaterequired to determine complete the position of all parts of a system at any
instant of time defines degree of freedom of the system fig.3, 4, and 5.
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Theory of vibration with
application
Assist. Lecture. ALI F. F.
2009-2010
Textbook: W.T. Thomson, Theory of vibration with application, 5th
ED., PRENTICE HALL, 2001
Resonance state :
if the frequency of the external force coincide
with one of the natural frequencies of the system, a condition
known as resonance accrues. And dangerously large oscillations
may result.
Undammed and damped vibration : if no energy is lost or
dissipated in friction or other resistance during oscillation, the
vibration is known as undammed vibration fig.6. If any energy is
lost in this way, on the other hand, it is called damping vibration
fig.7.
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Theory of vibration with
application
Assist. Lecture. ALI F. F.
2009-2010
Textbook: W.T. Thomson, Theory of vibration with application, 5th
ED., PRENTICE HALL, 2001
1.1 harmonic motion:
It can be demo started by a mass suspended from a light spring, as
shown in fig.(8). If the mass displaced from its rest position and
released, it will oscillate up and down.
if the motion is repeated after equal intervals
of time() period, it is called periodic motion. The simplest type ofperiodic motion is harmonic motion, and the reciprocal = iscalled the frequency measured in cycles per second (hertz-HZ.)
Harmonic motion is often represented as the projection on a straight
line of a point that is moving on a circle at constant speed, as shown
in fig.(9)
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Theory of vibration with
application
Assist. Lecture. ALI F. F.
2009-2010
Textbook: W.T. Thomson, Theory of vibration with application, 5th
ED., PRENTICE HALL, 2001
From fig.9:
= (1) = = () = (2)Where:
= . = (, ,, , . )
The motion repeats itself in
We have the relationship:
= = = = (3)Where are the period and frequency of the harmonic motionin secand and cycles per second (Hz), respectively.
The velocity and acceleration of harmonic motion are fig. 10:
() = () = = = +
(4)
() = = ( + ) (5)
Hint: examination of Esq. (2) and (5):
() = ()
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Theory of vibration with
application
Assist. Lecture. ALI F. F.
2009-2010
Textbook: W.T. Thomson, Theory of vibration with application, 5th
ED., PRENTICE HALL, 2001
Exponential form and complex algebra.
Any vector in the xy plane can be represented as a complex number[Eulers eq.]:
= ( + ) = = , fig.11
cture-
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Theory of vibration with
application
Assist. Lecture. ALI F. F.
2009-2010
Textbook: W.T. Thomson, Theory of vibration with application, 5th
ED., PRENTICE HALL, 2001
= + (6)
= + = () = + (8) = = . = , . =Results :
= (+) ; = ; / =// ; =
()
1.2If () is a periodic function with period , its Fourier seriesrepresentation is given by:
() = + ( + )= (9)
= = = ; =
= ;.
: The fundamental frequency(
,,, . . ,,, ) Are constant coefficients.
To determine the we multiply Eq.(9) by , and integrate over the one period . weobtain
periodic motion (Fourier series):
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Theory of vibration with
application
Assist. Lecture. ALI F. F.
2009-2010
Textbook: W.T. Thomson, Theory of vibration with application, 5th
ED., PRENTICE HALL, 2001
0=
2
(
)
2
2
= 2 () cos1 2
2
= 2 () sin1 2
2
Properties of oscillatory motion:
The simplest of these are the peak value, average value and the mean
square value:
peak value:
indicates the maximum stress thus the vibration part
is undergoing
Average value: indicates a steady or static value, somewhat like
the D.C. level of an electrical current. It can be found by the time
integral.
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Theory of vibration with
application
Assist. Lecture. ALI F. F.
2009-2010
Textbook: W.T. Thomson, Theory of vibration with application, 5th
ED., PRENTICE HALL, 2001
= lim
()
0
For example, see fig.12 the average value, for a complete cycle of sin
wave,, is zero.
= lim1
sin = 0
0
Whereas, its average value for a half cycle is:
= lim
1
sin /2
0
A
2A/PI
Fig.12
X(t)
wt
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Theory of vibration with
application
Assist. Lecture. ALI F. F.
2009-2010
Textbook: W.T. Thomson, Theory of vibration with application, 5th
ED., PRENTICE HALL, 2001
=
=
[ ] =
[+ ]
=
The mean square value:
The M.S.V. of a time function x(t) is found from the average of the
squared values integrated over some time interval:
2 = lim 2()
0
If() = sin, its mean square value is:
=
() =
Hint: the root mean square (rms) value is the square root of the mean
square value. From the previous example the rms of the sin wave of
amplitude A is (/2) = 0.707.
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Theory of vibration with
application
Assist. Lecture. ALI F. F.
2009-2010
Textbook: W.T. Thomson, Theory of vibration with application, 5th
ED., PRENTICE HALL, 2001
Decibel : (dB)
The decibel is a unit of measurement that is frequently used in
vibration measurements FIG.13.
It is defined in terms of a power ratio:
= ()
[]
= =
dB
Fig.13
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Theory of vibration with
application
Assist. Lecture. ALI F. F.
2009-2010
Textbook: W.T. Thomson, Theory of vibration with application, 5th
ED., PRENTICE HALL, 2001
Frequency
Hz.
Octave:
When the upper limit of a frequency range is twice its lower limit the
frequency span is said to be a octave fig.15. For example
Band
wide
10-20 10
20-40 20
40-80 40
40
20
10
5
10
20
Fig. 15
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lecture-3
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