Vibration refers to any motion that repeats itself periodically, such as a pendulum swinging back and forth or a plucked string oscillating. There are several types of vibration including free vibration where a system vibrates on its own after an initial disturbance, forced vibration where an external repeating force causes the vibration, and damped vibration where energy is lost during oscillations. Vibrations can also be classified as longitudinal, transverse, or torsional depending on the direction of motion of the vibrating particles. Proper vibration analysis is important for machine maintenance to identify faults and prevent damage.

1. Vibration
Any motion that repeats itself after an interval of time is called vibration or
oscillation.
Or
Periodic back-and-forth motion of the particles of an elastic body or
medium, commonly resulting when almost any physical system is displaced
from its equilibrium condition and allowed to respond to the forces that tend
to restore equilibrium.
The swinging of a pendulum and the motion of a plucked string are typical
examples of vibration. The theory of vibration deals with the study of
oscillatory motions of bodies and the forces associated with them.

2. Important of vibration analysis
The primary goal of vibration analysis is to identify faults within a machine and
then alert personnel that some type of action needs to occur. Problems start to
occur when the needed frequency of the data collection is not aligned with the
maintenance strategy.
Causes of machine vibration
1- Alignment problems
When two or more rotating machines are connected, the correct alignment
is crucial.
2- Unbalance
When the center of gravity of a rotating object is not exactly in the center
line, it causes machine unbalance resulting in vibration.
3- Loose parts
Loose bearings, loose bolts and corrosion can cause the machine to vibrate
excessively. Due to the mechanical forces in the machine, loose parts can rapidly
cause damage.

3. Periodic motion
Motion repeated in equal intervals of time. Periodic motion is performed,
for example, the Earth in its orbit around the Sun.
Harmonic Motion
The simplest type of periodic motion is harmonic motion.
Simple harmonic motion, in physics, repetitive movement back and forth
through an equilibrium, or central, position, so that the maximum
displacement on one side of this position is equal to the maximum
displacement on the other side. The time interval of each complete vibration
is the same.
A specific example of a simple harmonic oscillator is the vibration of a mass
attached to a vertical spring.

4. Cycle
The movement of a vibrating body from its undisturbed or equilibrium
position to its extreme position in one direction, then to the equilibrium
position, then to its extreme position in other direction and back to
equilibrium position.
Amplitude
The maximum displacement of a vibrating body from its equilibrium
position.
Period of Oscillation
The time taken to complete one cycle of motion, denoted by τ.
Frequency of oscillation
The number of cycles per unit time, denoted by f.


 2
1


f
Here is called the circular frequency to distinguish it from the linear
frequency
f=
The variable denotes the angular velocity of the cyclic motion;
f is measured in cycles per second (hertz),
while is measured in radians per second.
Natural frequency
If a system, after an initial disturbance, is left to vibrate on its own, the
frequency with which it oscillates without external forces is known as its
natural frequency.

5. Phase angle , Ф.
Consider two vibratory motion:
)
sin(
sin
2
2
1
1






t
A
x
t
A
x
These two harmonic motion are called synchronous because they have the
same frequency or angular velocity, ω.

7. Elementary parts of vibrating systems
Vibratory system, in general includes:
a- A mean for storing kinetic energy (mass or inertia).
b- A mean for storing potential energy (springs or elastic members).
c- A mean by which energy is gradually lost. (dampers).
Potential energy is the energy held by an object because of its position relative to
other objects,
Kinetic energy of an object is the energy that it possesses due to its motion
The vibration of a system involves the transfer of its potential energy to kinetic energy
and kinetic energy to potential energy, atternately. If the system is damped, some
energy is dissipated in each cycle of vibration and must be replaced by an external
source if a state of steady vibration is to be maintained.

8. CLASSIFICATION OF VIBRATION
1. Free Vibration:
A system is left to vibrate on its own after an initial disturbance and no
external force acts on the system. E.g. simple pendulum
2. Forced Vibration:
A system that is subjected to a repeating external force. e.g. oscillation
arises from diesel engines
Resonance occurs when the frequency of the external force
coincides with one of the natural frequencies of the system
3. Undamped Vibration:
When no energy is lost or dissipated in friction or other resistance during
oscillations
4. Damped Vibration:
When any energy is lost or dissipated in friction or other resistance during
oscillations
5. Deterministic Vibration:
If the value or magnitude of the excitation (force or motion) acting on a
vibratory system is known at any given time
6. Nondeterministic or random Vibration:
When the value of the excitation at a given time cannot be predicted.

9. In addition, can be classified according to motion in to:
1. Longitudinal vibrations : When the particles of a bar or disc move
parallel to the axis of the shaft, then the vibrations are known as
longitudinal vibrations as shown in fig. (a). The bar is elongated and
shortened alternately and thus the tensile and compressive stresses are
inducted in the bar. The motion of spring mass system is longitudinal
vibrations.
2. Transverse Vibrations : When the particles of the bar or disc move
approximately perpendicular to the axis of the bar, then the vibrations
are known as transverse vibrations as shown in fig.(b). In this case, bar
is straight and bent alternately. Bending stresses are induced in the bar.
3. Torsional Vibrations : When the particles of the bar or disc get
alternately twisted and untwisted on account of vibratory motion of
suspended body, it is said to be undergoing torsional vibrations as
shown in fig. (c). In this case, torsional shear stresses are induced in
the bar.