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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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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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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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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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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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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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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= + (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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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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= 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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=

=

[ ] =

[+ ]

=

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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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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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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Theory of vibrations

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