This document is a course outline for a mechanical vibrations class. It includes an introduction to mechanical vibrations that discusses oscillatory motion in dynamic systems and the causes and effects of vibrations. It also summarizes different types of vibration systems including free and forced vibrations. Key concepts from the course like harmonic motion, Fourier analysis, and beats are defined. The document provides an overview of the topics to be covered in the class through multiple unit sections.

1. Mechanical Department
COURSE NAME: MECHANICAL VIBRATIONS
Prepared By:
MD ATEEQUE KHAN
(Assistant Professor)
Mechanical Engineering Department
JIT,Barabanki,U.P. INDIA
12/31/2016 1NME-013 MA KHAN

2. Table of Contents
Unit-1.1
1. Introduction
2. Importance of Mechanical Vibrations
3. Causes
4. Classification of Vibration Systems
5. Harmonic motion
6. Vector representation of harmonic motion
7. Natural frequency & response
8. Effects of vibration
9. superposition of simple harmonic motions
10. beats
11. Fourier analysis-analytical and numerical methods
12. Objective Questions
12/31/2016 2NME-013 MA KHAN

3. Unit-1: Mechanical Vibrations
Introduction
• Mechanical vibrations are basically a oscillatory motion of dynamic systems and forces associated with
them.
• Any body having mass and elasticity is capable of vibrations.
• For an ideal system , continuous conversion of kinetic energy in to potential energy and vice versa ,is
responsible for continuous oscillatory motion.
• In case of machines and structures vibrations are undesirable, as they are responsible for producing :
Excessive stresses, wear, unwanted noise, and premature failure of components.
• Due to the mechanical vibration it is very difficult to attain accuracy in size, shape and surface finish etc.
12/31/2016 3NME-013 MA KHAN

4. Unit-1: Mechanical Vibrations
Importance/Applications
• Our heart beats, lungs oscillate,
• We speak due to our larynges can vibrate
• We hear due to ear drum oscillate
• In washing machine, mechanical shakers.
• In Vibratory conveyors and comparators
• In musical instruments
• Stress relieving techniques etc.
12/31/2016 4NME-013 MA KHAN

5. Unit-1: Mechanical Vibrations
Causes
• Unbalance
• Misalignment
• Looseness
• Dry Friction between two rubbing surfaces
• Wind –induced vibration
• Oil whirl
• External Excitation
12/31/2016 5NME-013 MA KHAN

6. Unit-1: Mechanical Vibrations
Classification
• Free Vibration: After initial disturbance system left to vibrate without influence of external force.
• Forced Vibration: Vibratory systems are continuously excited by external forces.
• Undamped Vibration: In case of undamped vibration no energy being dissipated or dissipated energy is
very small and can be neglected. Amplitude remains constant.
• Damped Vibration: Energy being dissipated in damped vibration. Amplitude decays.
• Linear Vibration: Vibratory elements( Mass, Spring, Damper) show linear behavior. Superposition holds
for double excitation level or double response level, Mathematical relation also well defined.
• Non linear Vibration: At least one elements behaves non linear trend and superposition does not hold,
mathematical relation also not defined.
• Deterministic Vibration: It can be described by implicit mathematical function as a function of time.
• Random Vibration: It can not be predicted. Process can be explained by statistical means.
12/31/2016 6NME-013 MA KHAN

7. Unit-1: Mechanical Vibrations
Harmonic Motion
• Simple harmonic motion is a type of periodic motion or oscillatory motion where the restoring force is directly
proportional to the displacement and acts in the direction opposite to that of displacement.
tX
t
v
a
tX
t
x
v
tXx



sin
cos
sin
2









2
tXx sin

12/31/2016 7NME-013 MA KHAN

8. Unit-1: Mechanical Vibrations
superposition of simple harmonic motion:
Resultant Motion:
By expanding the second term:
)sin(
sin
22
11




tXx
tXx
21 xxx 
)(sinsin 21   tXtX
)sin(
cos)sin(sin)cos( 221




tXx
tXtXXx
12/31/2016 8NME-013 MA KHAN

9. Unit-1: Mechanical Vibrations
superposition of simple harmonic motion:
Where




cos
sin
tan
cos2
21
2
212
2
1
2
XX
X
XXXXX



t


x
y
O
1 X
2X
X
12/31/2016 9NME-013 MA KHAN

10. Unit-1: Mechanical Vibrations
Beats
Beat is a phenomena which arises when two almost similar frequencies
imposed together, i.e. heart beat.
Time Period of Beat:
21
2





12/31/2016 10NME-013 MA KHAN

11. Unit-1: Mechanical Vibrations
Fourier analysis
• Many oscilatory motion are periodic but not harmonic.
• Any periodic function can be represented by Fourier series in terms of sinusoidal and co-
sinusoidal series.
To find out an and bn series is multiplied by cos(nωt) and sin(nωt) respectively and
integrated over one period.
......2sinsin...................2coscos
2
2121
0
)(  tbtbtata
a
x t 
 



n
n
nn tnbtna
a
1
0
)sin()cos(
2

12/31/2016 11NME-013 MA KHAN

12. Unit-1: Mechanical Vibrations
Objective Questions
12/31/2016 12NME-013 MA KHAN