This document provides an introduction to mechanical vibration, including definitions, applications, causes, and importance. It discusses elementary parts of vibrating systems like springs, masses, and dampers. Vibration can be classified based on the excitation, such as free or forced vibration. Simple harmonic motion is described as the simplest form of periodic motion. The steps for analyzing vibration are defined as: 1) defining the problem, 2) physical modeling, 3) formulating governing equations, 4) mathematically solving the equations, and 5) physically interpreting the results.

1. Chapter One
Introduction to Mechanical Vibration

2. Contents
 Definition
 Application
 Causes
 Importance
 Elementary parts
 DOF
 Classifications
 Simple Harmonic Motion
 Analysis steps

3. What is Vibration?
 Defined as an oscillatory motion of a particle or rigid body about an equilibrium
position in response to disturbance.
 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.

4. Where do we find Vibration?
Most human activities involve vibration in one form or other. For example,
 Our heart beats, our lungs oscillate, our ear drum vibrates.
 We cannot even say “Vibration” without vibration of larynges (and
tongues), vocal cord.
 We move by oscillating our legs and hands.
 The light waves which permit us to see and sound waves through which
we hear entail vibration.
But, we limit our discussion to “MECHANICAL VIBRATION”… i.e., Vibration

5. Where do we find Mechanical
Vibration?

6. What Causes Vibration?
 Vibration can be caused by one or more factors at any given time, the most
common being
§ Imbalance
§ Misalignment
§ Wear and looseness
§ External excitation
 Vibration could be desired or undesired depending on its occurrence.

7. Desired (useful) Vibration
Testing
Ultrasonic Vibration
cleaning
Compressor
Conveyor
Sieve

8. Undesired Vibration
Noise
Destruction
Wear Fatigue

9. Why do we care about Vibration?
Early scholars in the field of vibration concentrated their efforts on understanding the
natural phenomena and developing mathematical theories to describe the vibration
of physical systems. In recent times, many investigations have been motivated by
the engineering applications of vibration, such as the design of machines,
foundations, structures, engines, turbines, and control systems.
 The structure or machine component subjected to vibration can fail because of
material fatigue resulting from the cyclic variation of the induced stress.
 Vibration causes more rapid wear of machine parts such as bearings and gears and
also creates excessive noise.

10. Cont…
 Vibration can loosen fasteners such as nuts.
 In metal cutting processes, vibration can cause chatter, which leads to a poor
surface finish.
 Vibration of instrument panels can cause their malfunction or difficulty in reading
the meters
 The transmission of vibration to human beings results in discomfort and loss of
efficiency.

11. Cont…
In spite of its detrimental effects, vibration can be utilized profitably in several
consumer and industrial applications. For example,
 vibration is put to work in vibratory conveyors, hoppers, sieves, compactors,
washing machines, electric toothbrushes, dentist ‘s drills, clocks, and electric
massaging units.
 Vibration is also used in pile driving, vibratory testing of materials, vibratory
finishing processes, and electronic circuits to filter out the unwanted frequencies.
 It is employed to simulate earthquakes for geological research and also to conduct
studies in the design of nuclear reactors.

12. Elementary parts of vibrating systems
 A vibrating system consists of;
1. Spring or stiffness element: a means for storing potential energy
2. Mass or inertia : a means for storing kinetic energy
3. Damper : a means by which energy is gradually lost
 An undamped vibrating system involves the transfer of its potential energy to
kinetic energy and kinetic energy to potential energy, alternatively.
 In a damped vibrating system, some energy is dissipated in each cycle of vibration
and should be replaced by an external source if a steady state of vibration is to be
maintained.

13. Degrees of Freedom
 Minimum number of independent coordinates required to determine
completely the positions of all parts of a system at any instant of time
1. Finite number of degrees of freedom are termed discrete or lumped
parameter systems
2. Infinite number of degrees of freedom system are termed continuous
or distributed systems
More accurate results obtained by increasing number of degrees of
freedom

16. What are their DOFs? HW

17. Classification of Vibration
Vibration can be classified in several ways. Some of the important classifications are
as follows.
 By nature of excitation
1. Free vibration : vibration that occurs in the absence of external force. 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 : vibration that occurs by an external force that acts on the
system. In this case, the exciting force continuously supplies energy to the
system.

18.  By nature of the excitation force pattern:
1. Nondeterministic or random Vibration:
When the value of the excitation at a given time cannot be
predicted.
Process can be described by statistical means. An example -
the shock that is felt as a result of driving down the road and
hitting a pothole
 Do not repeat themselves
 Not related to a fundamental frequency.

19.  By nature of the excitation force pattern:
2. Deterministic Vibration:
If the value or magnitude of the excitation (force or motion)
acting on a vibratory system is known at any given time
Can be described by implicit mathematical function as a
function of time.

20.  By energy dissipation
1. Un-damped Vibration:
When no energy is lost or dissipated in friction or other resistance during
oscillations. Undamped vibration is created due to the conversion of KE to PE or vice
versa
2. Damped Vibration:
When any energy is lost or dissipated in friction or other resistance during
oscillations
In many cases, damping is (negligibly) small (1 – 1.5%). However small,
damping has critical importance when analyzing systems at or near
resonance.

21.  By nature of elements in the system
1. Linear Vibration:
When all basic components of a vibratory system, i.e. the spring, the
mass and the damper behave linearly
Superposition holds: double excitation level = double response level,
mathematical solutions are well defined.
2. Nonlinear Vibration: If any of the components behave nonlinearly
Superposition does not hold, and analysis technique not clearly
defined.

22.  By number of DOF
1. Discrete (lumped parameter) System: is a practical system
described using a finite number of degrees of freedom.
2. Continuous System: Some systems, especially those involving
continuous elastic members, have an infinite number of
degrees of freedom.

23.  By number of DOF
 Most of the time, continuous systems are approximated as
discrete systems, and solutions are obtained in a simpler manner.
Although treatment of a system as continuous gives exact results,
the analytical methods available for dealing with continuous
systems are limited to a narrow selection of problems, such as
uniform beams, slender rods, and thin plates. Hence most of the
practical systems are studied by treating them as finite lumped
masses, springs, and dampers.
 In general, more accurate results are obtained by increasing the
number of masses, springs, and dampers that is, by increasing
the number of degrees of freedom.

24. Simple Harmonic Motion
x Asin( ) A sin( t )
 
 
• Its velocity and acceleration are:
2
2 2
2
dx
A cos( t )
dt
and
d x
A sin( t ) x
dt
 
  

   
• Harmonic motion: simplest form of periodic
motion (deterministic).
• Pure sinusoidal (co-sinusoidal) motion
• Eg: Scotch-yoke mechanism rotating with angular
velocity  - simple harmonic motion:
• The motion of mass m is described by:
24

25. Cont…
• Sinusoidal motion emanates from cyclic motion
• Can be represented by a vector (OP) with a magnitude, angular
velocity (frequency) and phase.
• The projection of the tip of the vector on the vertical and horizontal
axis is given by and resply.
• The rotating vector generates a sinusoidal and a co-sinusoidal
components along mutually perpendicular axes.
25
� = ���⁡
(��) � = ���⁡
(��)

26. Cont…
• Often convenient to represent sinusoidal and co-sinusoidal components (mutually perpendicular)
in complex number format
• Where a and b denote the sinusoidal (x) and co-sinusoidal (y) components
• a and b = real and imaginary parted of vector X
26
i t
i t
i t i t
i t i t
e cos( t ) i sin( t )
e cos( t ) i sin( t )
e e
cos( t )
2
e e
sin( t )
2i


 
 
 
 





 
 





27. 27
Definitions and Terminology
ü Cycle: One revolution (i.e., angular displacement of radians) of the vector OP in scotch yoke
mechanism
ü Amplitude: The maximum displacement of a vibrating body from its equilibrium position
ü Period of oscillation (�): The time taken to complete one cycle (2�) of motion
ü Frequency of oscillation (f): The number of cycles per unit time in cycles per
second (hertz)
ü 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.
Where ω is called the circular frequency

28. 28
ü Phase angle: Consider two vibratory motions denoted by:
ü The motions given by Eqs. Above can be represented graphically as shown in Fig. below. In this figure,
the second vector leads the first one by an angle known as the phase angle.
Phase difference between two vectors

29. Vibration Analysis Procedures
The vibration analysis of a physical system may be summarized by the
five steps:
1. Define the Vibration Problem
2. Physical Modeling
3. Formulation of Governing Equations (Mathematical Modeling)
4. Mathematical Solution of the Governing Equations
5. Physical Interpretation of the Results

30. Define the Vibration Problem
 The system to be modelled is abstracted from its surroundings,
and the effects of the surroundings are noted.
 Known constants are specified.
 Parameters which are to remain variable are identified.

31. Modeling of a physical system
 The purpose of the modelling is to determine the existence and
nature of the system, its features and aspects, and the physical
elements or components involved in the physical system.
 Assumptions taken for modeling
a) A physical system can be treated as a continuous piece of
matter
b) Newton’s laws of motion can be applied by assuming that the
earth is an internal frame
c) Ignore or neglect the relativistic effects

32. Formulation of governing equations
 Apply the basic laws of nature and the principles of dynamics
and obtain the differential equations that govern the behavior
of the system.
 A basic law of nature is a physical law that is applicable to all
physical systems irrespective of the material from which the
system is constructed

33. Mathematical Solution of the Governing
Equations
 The governing equations of motion of a system are solved to
find the response of the system.
 Analytical or numerical methods can be used to solve the
governing equation.

34. Physical interpretation of the results
 Physical interpretation of the results is an important and final step
in the analysis procedure. This may involve;
1. Drawing general inferences from the mathematical solution,
2. Developing design curves,
3. Arriving at a simple arithmetic for conclusion (for a typical or
specific problem), and
4. Recommending as a regard to the significance of the results
and any changes (if any) required or desirable in the system
involved.

35. The END!
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