Me 553 Advanced Vibrations

ME 553 :
Advanced Vibrations
Dr. A. Aziz Bazoune ME 553 ADVANCED VIBRATIONS Ch-01-LEC 02, Slide 1

Chapter 1
INTRODUCTION
Dr. Abdelaziz Bazoune

Mechanical Engineering Department
King Fahd University of Petroleum & Minerals

After you have finished this lecture you will be able to:
1. Know what is meant by vibration
2. Know the importance of vibration
3. Know the desirable effects of vibration
4. Know the undesirable effects of vibration
5. Classify the different types of vibration

Preliminary Remarks
Brief History of Vibration
Importance of the Study of Vibration
Basic Concepts of Vibration
Classification of Vibration
Vibration Analysis Procedure
Spring Elements
Mass or Inertia Elements
Damping Elements

Vibration is the back and forth (up and down) motion of a machine or
machine part from its equilibrium position.
Examples:
Remark: All machines can be

represented as ( spring-mass
system) since they have
weight, and spring-like
properties.

Vibration:
Oscillatory motion of bodies, such as acceleration, velocity and
displacement of bodies, and the forces associated with them.
All bodies possessing mass and elasticity are capable of
vibration.
Vibration could be regular like the pendulum, or could be

irregular like the earthquake. The simplest vibration type is
the Simple Harmonic Motion (SHM).

Vibration can lead to excessive deflections and failure on the machines
and structures.
To reduce vibration through proper design of machines and their
mountings.
To improve the efficiency of certain machining, casting, forging &
welding processes.
To stimulate earthquakes for geological research and conduct studies
in design of nuclear reactors.

Most of Engineering Branches are confronted with vibration
Widespread in Energy daily life

Home appliances
Trains, Cars, etc
Rotating Machines : Turbines, Pumps, Compressors
Reciprocating Machines: Internal Combustion Engines (ICE), Reciprocating

Compressors
Can be used in Fault Diagnosis Analysis (FDA)

In Recreation
In Machinery

In Defense In Transportation

In Aerospace In Automobiles

NVH (Noise, Vibration, & Harshness)
Sound/Noise in Automobiles is a Top Priority for the BIG 3 Auto
Industry.
Several hundred million dollars have

been invested in infrastructure &
human resource development in
this area over the past 30 years.

• Origins of vibration:
582-507 B.C. –
Pythagoras, the Greek philosopher and
mathematician, is the first to investigate
musical sounds on a scientific basis. He
conducted experiments on a vibrating
string by using a simple apparatus called
a monochord. He further developed the
concept of pitch.

Around 350 B.C. –
Aristotle wrote treatises on music and sound
In 320 B.C. –
Aristoxenus wrote a three-volume work entitled Elements of Harmony
In 300 B.C. –
Euclid wrote a treatises Introduction to
Harmonics
A.D. 132 –
Zhang Heng invented the world’s first
seismograph to measure earthquakes

Galileo to Rayleigh:
Galileo Galilei (1564 – 1642)
- founder of modern experimental science
- started experimenting on simple pendulum
- published a book, Discourses Concerning Two New Sciences, in

1638, describing resonance, frequency, length, tension and density
of a vibrating stretched string
Robert Hooke (1635 – 1703)
- found relation between pitch and frequency of vibration of a

string

Joseph Sauveur (1653 – 1716)
- coined the word “acoustics” for the science of sound
- founded nodes, loops, harmonics and the fundamental frequency
- calculated the frequency of a stretched string from the measured

sag of its middle point
Sir Isaac Newton (1642-1727)
- published his monumental work, Philosophiae

Naturalis Principia Mathematica, in 1686,
discovering three laws of motion

Joseph Lagrange (1736 – 1813)
- found the analytical solution of the vibrating
string and the wave equation
Simeon Poisson (1781 – 1840)

- solved the problem of vibration of a
rectangular flexible membrane
R.F.A. Clebsch (1833 – 1872)

- studied the vibration of a circular membrane
Lord Baron Rayleigh (1842-1919)

- founded Rayleigh-Ritz method, used to find
frequency of vibration of a conservative
system and multiple natural frequencies

Recent contributions
1902 – Frahm investigated the importance of torsional vibration

study in the design of propeller shafts of steamships
Aurel Stodola (1859 – 1943)

- contributed to the study of vibration of beams, plates, and
membranes.
- developed a method for analyzing vibrating beams which is
applicable to turbine blades
C.G.P. De Laval (1845 – 1913)

- presented a practical solution to the problem of vibration of
an unbalanced rotating disk

1892 – Lyapunov laid the foundations of modern stability
theory which is applicable to all types of dynamical
systems
1920 – Duffling and Van der Pol brought the first definite
solutions into the theory of nonlinear vibrations and
drew attention to its importance in engineering
– Introduction of the correlation function by Taylor
1950 – advent of high-speed digital computers

– generate approximate solutions

1950s – developed finite element method enabled engineers to conduct
numerically detailed vibration analysis of complex mechanical,
vehicular, and structural systems displaying thousands of degrees of
freedom with the aid of computers
– Turner, Clough, Martin and Topp presented the finite element method as
known today

Fohi (3000 B.C.) Consonances, book on music
Thales of Miletos (640-546 B.C.) Scientific method
Pythagoras of Samos Natural frequency, experimental

(ca. 570-497 B.C.) physics; theory of numbers, Vibration
of the taut string
Aristophanes (450-388 B.C.) Oscillator
Platon ( ca. 429-347 B.C.) Sympathetic Vibration
Aristoteles (384-322 B.C.) Laws of motion, Book on acoustics
Euclides (330-275 B.C.) Pendulum use as a vibration meter

Alexander of Aphrodisias Kinetic and potential energy
(Early Third Century B.C.)
Archimedes ( ca. 287-212 B.C.) Laws of Statics, hydrostatics
Vitruvius (First Century B.C.) Chinese seismograph
Leonardo da Vinci (1452-1519)
Galileo Galilei (1564-1642) Measurement of pendulum frequency
Christian Huygens ( 1629-1695) Pendulum clock, nonlinearity
Issac Newton (1642-1727) Laws of motion
Joseph Sauveur (1653-1716) Vibration Harmonics
Daniel Bernoulli (1700-1782) Wave equation

Leonhard Euler( 1707-1783) Principle of superposition
Jean le Rond D’Alembert D’Alembert principle
(1717-1783)
Joseph Louis Lagrange Lagrange’s equation
(1736-1783)
C. A. coulomb (1736- 1806) Torsional vibration
Sophie Germain(1776-1831) Vibration of plates
Claude Louis Marie Henri Navier Vibration of solids
(1785-1836)
W. J. M. Rankine (1820-1872) Critical Speeds of Shafts

John William Strutt, Lord Rayleigh Energy methods; first vibration
( 1842-1919) treatise
Carl G. P. deLaval (1845-1913) Rotor dynamics
Henri Poincare’ (1854-1912) Nonlinear Vibrations
Heinrich Hertz (1857-1894) Rotor-bearing dynamics
A. Stodola (1859-1942) Vibration of plates
Stephen P. Timoshenko Timoshenko beam equation
(1878-1972)

Our heart beats, our lungs oscillate, we hear because our ear
drum vibrates … Vibration even makes us snore!!
http://www.freehearingtest.com/about_animated.shtml
The light waves which permit us to see & sound waves

through which we hear entail vibration
We cannot even say “Vibration” without vibration of larynges,

vocal cord
We move by oscillating our legs

== IMPORTANT
We limit out discussion to
“MECHANICAL VIBRATION”… i.e. Vibration of

Dynamic Systems.
What is a Dynamic System ???

Any System that contain Mass and Elasticity.

Desirable Effects
Washing Machines
Material Handling, Conveyors,
Hoppers, Compactors, Pneumatic
Drills
Seismic Instruments
Can be used in Fault Diagnosis
It is one of the best indicators of
M/c health conditions
Medical fields, massages,
ultrasonic processes
Musical Instruments
Clocks, watches
Mixing of dispersions and
aggregates- concrete mixing
for example

Bad or Harmful Effects
Undesirable noise
For the Previous Harmful
Tiring to people
Resonance effects Effects
Excessive vibration leads to loosening on
parts, noise & eventual failure Vibration must be avoided if
Structural damage possible. (Its occurrence
Equipment malfunction
indicates some kind of faults or
Effects of vibration on human body:
Discomfort, fatigue, performance, loss of defects).
efficiency, and the health of people
Minimize its harmful Effects.
subjected to it, as in sickness due to ship
(or high rise building) oscillation.

Basic Concepts of vibration
Vibration = the back and forth (up and down) motion of a machine or
machine part from its equilibrium position. The swinging of a pendulum
and the motion of a plucked string are typical examples of vibration.
The study of vibration deals with the study of oscillatory motions of
bodies and the forces associated with them.
Vibratory System consists of:

1) spring or elasticity
2) mass or inertia
3) damper
Involves transfer of potential energy to kinetic energy and vice versa

Degrees of Freedom and Generalized Coordinates
The minimum number of independent coordinates needed to describe a
system completely is the number of degrees of freedom (DOF) of the
system.
Any such set of kinematically independent coordinates is called a set of

generalized coordinates.
The number of degrees of freedom is unique, whereas the choice of a set

of generalized coordinates is not unique.
Kinematics quantities such as displacements, velocities and accelerations

are written as functions of the generalized coordinates and their time
derivatives.




Discrete Systems and Continuous Systems
Systems with a finite number of degrees of freedom are termed
discrete or lumped parameter systems
Infinite number of degrees of freedom system are termed continuous or
distributed systems
Example of Infinite-number-of-degrees-of-freedom system:
More accurate results obtained by increasing number of degrees of

freedom

Free Vibration:
A system is left to vibrate on its own
 after an initial disturbance (shock received, or initial displacement)
from equilibrium
 NO external force acts on the system.

Forced Vibration:
Forced vibrations occur if a system is continuously driven by an external
excitation. A simple example is a child’s swing that is pushed on each
downswing. Of special interest are systems undergoing SHM and driven by
sinusoidal forcing.
This leads to the important phenomenon of resonance. Resonance occurs

when the frequency of the external force coincides with one of the
natural frequencies of the system. The result is ….. ???? =====

The Tacoma Narrows Bridge in Washington state, was with
1.9 km length one of the largest suspended bridges built at
the time. The bridge connecting the Tacoma Narrows
channel collapsed in a dramatic way on Thursday November
7, 1940. Winds of 35-46 mi/hours =65-75 km/hr) produced
an oscillation which eventually broke the construction.
Below you see some movies taken during that event. The
bridge began first to vibrate the bridge torsionally, giving it a
twisting motion. Later the vibrations entered a natural
resonance frequency of the bridge which started to increase
their amplitude.
http://www.brown.edu/Departments/Engineering/Courses/En4/java/forced.html
http://abel.math.harvard.edu/archive/21b_fall_03/tacoma/
http://rotorlab.tamu.edu/me617/some%20videos/050606_GroundResonance.wmv

Undamped Vibration:
When no energy is lost or dissipated in friction or other resistance during
oscillations.
Damped Vibration:
When any energy is lost or dissipated in
friction or other resistance during oscillations.
http://www.lon-capa.org/~mmp/applist/damped/d.htm

Linear Vibration:
When all basic components of a vibratory system, i.e. the spring, the
mass and the damper behave linearly. The superposition principle
applies for linear systems.
Nonlinear Vibration:
If any of the components of a vibratory system behave nonlinearly. The
superposition principle does not apply for this type of systems.

Stationary Signals can be divided into Deterministic Signals
and Random Signals.
Stationary deterministic signals are made up entirely of

sinusoidal components at discrete frequencies.
Deterministic Vibration:
If the value or magnitude of the excitation (force or
motion) acting on a vibratory system is known at any
given time.
Nondeterministic or Random Vibration:

When the value of the excitation at a given time cannot
be predicted.

Non-stationary Signals can be divided
into Continuous and Transient signals.
Continuous Non-stationary Vibration:

has some similarities with both transient
and stationary vibrations. During analysis
continuous non-stationary signals should
normally be treated as random signals or
separated into the individual transient and
treated as transients.
Transient Vibration: signals which

commence and finish at a constant level,
normally zero, within the analysis time.

Longitudinal Vibration:
Transverse Vibration:
Torsional Vibration:

Step 1: Mathematical Modeling
Use mechanical elements such as spring, mass, dashpot
Step 2: Derivation of Governing Equations
Using Newton’s second law of motion for translation and/or rotation
Using Energy Method for conservative systems
Using Lagrange Equations
Step 3: Solution of the Governing Equations
Direct solution
Laplace Transform Solution
Numerical Solutions
Step 4: Interpretation of the Results

All mechanical systems contain the
three basic elements: spring,
damper, and mass (inertia element).
When each of these in turn is

exposed to a constant force they
react with a constant displacement,
a constant velocity and a constant
acceleration respectively.
Similarly we can obtain the same

relationships for rotational
(torsional) elements.

http://www.brown.edu/Departments/Engineering/Courses/En4/java/shm.html

Example of the modeling of a forging hammer:
Anvil: heavy block of iron

or steel with a smooth,
flat top on which metals
are shaped by hammering

Example of the
modeling of an
automobile
(a) Model representation of an automobile.
(b) A simplified representation.

A simplified, multiple model
for a human body standing on
a vibrating platform.

Aristoteles (384-322 B.C.) Archimedes ( ca. 287-212 B.C.)
Euclides (330-275 B.C.) Galileo Galilei (1564-1642)

Christian Huygens ( 1629-1695) Issac Newton (1642-1727)
Pythagoras (582-507BC) Bernoulli (1700-1782)

Leonhard Euler (1707-1783) Joseph Louis Lagrange (1736-1813)
Lord Rayleigh (1842-1919) Stephen P. Timoshenko (1878-1972)

QUESTIONS … ...?

Thank U … ...!

Me 553 Advanced Vibrations

Uploaded by

Copyright:

Available Formats

Me 553 Advanced Vibrations

Uploaded by

Document Information

Original Description:

Original Title

Copyright

Available Formats

Share this document

Share or Embed Document

Sharing Options

Did you find this document useful?

Is this content inappropriate?

Copyright:

Available Formats

Me 553 Advanced Vibrations

Uploaded by

Copyright:

Available Formats

ME 553 :

Dr. A. Aziz Bazoune ME 553 ADVANCED VIBRATIONS Ch-01-LEC 02, Slide 1

Dr. Abdelaziz Bazoune

Dr. A. Aziz Bazoune ME 553 ADVANCED VIBRATIONS Ch-01-LEC 02, Slide 2

Dr. A. Aziz Bazoune ME 553 ADVANCED VIBRATIONS Ch-01-LEC 02, Slide 3

Dr. A. Aziz Bazoune ME 553 ADVANCED VIBRATIONS Ch-01-LEC 02, Slide 4

Remark: All machines can be

Dr. A. Aziz Bazoune ME 553 ADVANCED VIBRATIONS Ch-01-LEC 02, Slide 5

Vibration could be regular like the pendulum, or could be

Dr. A. Aziz Bazoune ME 553 ADVANCED VIBRATIONS Ch-01-LEC 02, Slide 6

Dr. A. Aziz Bazoune ME 553 ADVANCED VIBRATIONS Ch-01-LEC 02, Slide 7

Widespread in Energy daily life

Trains, Cars, etc

Rotating Machines : Turbines, Pumps, Compressors

Reciprocating Machines: Internal Combustion Engines (ICE), Reciprocating

Can be used in Fault Diagnosis Analysis (FDA)

Dr. A. Aziz Bazoune ME 553 ADVANCED VIBRATIONS Ch-01-LEC 02, Slide 8

Dr. A. Aziz Bazoune ME 553 ADVANCED VIBRATIONS Ch-01-LEC 02, Slide 11

Dr. A. Aziz Bazoune ME 553 ADVANCED VIBRATIONS Ch-01-LEC 02, Slide 12

Dr. A. Aziz Bazoune ME 553 ADVANCED VIBRATIONS Ch-01-LEC 02, Slide 13

Several hundred million dollars have

Dr. A. Aziz Bazoune ME 553 ADVANCED VIBRATIONS Ch-01-LEC 02, Slide 14

Dr. A. Aziz Bazoune ME 553 ADVANCED VIBRATIONS Ch-01-LEC 02, Slide 15

Dr. A. Aziz Bazoune ME 553 ADVANCED VIBRATIONS Ch-01-LEC 02, Slide 16

- founder of modern experimental science

- started experimenting on simple pendulum

- published a book, Discourses Concerning Two New Sciences, in

Robert Hooke (1635 – 1703)

- found relation between pitch and frequency of vibration of a

Dr. A. Aziz Bazoune ME 553 ADVANCED VIBRATIONS Ch-01-LEC 02, Slide 17

- coined the word “acoustics” for the science of sound

- founded nodes, loops, harmonics and the fundamental frequency

- calculated the frequency of a stretched string from the measured

Sir Isaac Newton (1642-1727)

- published his monumental work, Philosophiae

Dr. A. Aziz Bazoune ME 553 ADVANCED VIBRATIONS Ch-01-LEC 02, Slide 18

Simeon Poisson (1781 – 1840)

R.F.A. Clebsch (1833 – 1872)

Lord Baron Rayleigh (1842-1919)

Dr. A. Aziz Bazoune ME 553 ADVANCED VIBRATIONS Ch-01-LEC 02, Slide 19

1902 – Frahm investigated the importance of torsional vibration

Aurel Stodola (1859 – 1943)

C.G.P. De Laval (1845 – 1913)

Dr. A. Aziz Bazoune ME 553 ADVANCED VIBRATIONS Ch-01-LEC 02, Slide 20

1950 – advent of high-speed digital computers

Dr. A. Aziz Bazoune ME 553 ADVANCED VIBRATIONS Ch-01-LEC 02, Slide 21

Dr. A. Aziz Bazoune ME 553 ADVANCED VIBRATIONS Ch-01-LEC 02, Slide 22

Thales of Miletos (640-546 B.C.) Scientific method

Pythagoras of Samos Natural frequency, experimental

Aristophanes (450-388 B.C.) Oscillator

Platon ( ca. 429-347 B.C.) Sympathetic Vibration

Aristoteles (384-322 B.C.) Laws of motion, Book on acoustics

Euclides (330-275 B.C.) Pendulum use as a vibration meter

Dr. A. Aziz Bazoune ME 553 ADVANCED VIBRATIONS Ch-01-LEC 02, Slide 23

Archimedes ( ca. 287-212 B.C.) Laws of Statics, hydrostatics

Vitruvius (First Century B.C.) Chinese seismograph

Leonardo da Vinci (1452-1519)

Galileo Galilei (1564-1642) Measurement of pendulum frequency