MODULE-I --- INTRODUCTION VIBRATION ENGINEERING 2014

VTU-NPTEL-NMEICT

Project Progress Report

The Project on Development of Remaining Three Quadrants to NPTEL Phase-I under grant in aid NMEICT, MHRD, New Delhi

DEPARTMENT OF MECHANICAL ENGINEERING,

GHOUSIA COLLEGE OF ENGINEERING,

RAMANARAM -562159

Subject Matter Expert Details

SME Name : Dr.MOHAMED HANEEF

PRINCIPAL, VTU SENATE MEMBER

Course Name: Vibration engineering

Type of the Course

web

Module I

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MODULE-I --- INTRODUCTION VIBRATION ENGINEERING 2014

CONTENTS

Sl. No. DISCRETION

1. Lecture Notes (Introduction to Vibration Engineering).

2. Quadrant -2

a. Animations.

b. Videos.

c. Illustrations.

3. Quadrant -3

a. Wikis.

b. Open Contents

c. Case Studies.

4. Quadrant -4

a. Problems.

b. Assignments

c. Self Assigned Q & A.

d. Test your Skills.

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Module-I

INTRODUCTION TO VIBRATION ENGINEERING

1. LECTURE NOTES

Overview of the course, practical applications and research trends:

The study of vibrations is concerned with the oscillatory motion of bodies and the forces associated

with them. All bodies possessing mass and elasticity are capable of vibrating. Thus most engineering

machines and structures experiences vibration to some degree and their design generally requires

consideration of their oscillatory behavior. The oscillatory motion of the system may be objectionable or

necessary for performing a task.

The objective of the designer is to control the vibration when it is objectionable and to enhance the

vibration when it is useful. Objectionable or undesirable vibration in machine may cause the loosening of

parts, its malfunctioning or its failure. The useful vibration helps in the design of shaker in foundries,

vibrators in testing machines etc. Sometimes vibrations are bad and other times they are good.

1.1. Causes of vibration: -

The main causes of vibration are:-

1) Unbalanced forces in the machine. These forces are produced from within the machine itself

because of non-uniform material distribution in a rotating machine element.

2) Dry friction between the two mating surfaces: This is what known as self-exited vibration.

3) External excitations. These excitations may be periodic, random or of the nature of an impact

produced external to the vibrating system.

4) Elastic nature of the system

5) Earth quakes. These are responsible for the failure of many buildings, dams etc.

6) Winds. These may cause the vibrations of transmission and telephone lines under certain

conditions.

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The effect of vibrations is excessive stresses, undesirable noise, looseness of parts and partial or complete

failure of parts. Inspite of these harmful effects the vibration phenomenon does have, some uses also e.g.

in musical instruments, vibrating screens, shakers, stress relieving.etc.

1.2. Methods to reduce vibrations

Elimination or reduction of the undesirable vibrations can be obtained by one or more of the following

methods

1) Removing the cause of vibrations

2) Putting in screens if noise is the only objection

3) Resting the machinery in proper type of isolators

4) Shock absorbers.

5) Dynamic vibration absorbers

1.3 Research trends in vibration Engineering and Practical Applications

1. Earthquake-resistant structures are structures designed to withstand earthquakes. While no

structure can be entirely immune to damage from earthquakes, the goal of earthquake-resistant

construction is to erect structures that fare better during seismic activity than their conventional

counterparts.

According to building codes, earthquake-resistant structures are intended to withstand the largest

earthquake of a certain probability that is likely to occur at their location. This means the loss of

life should be minimized by preventing collapse of the buildings for rare earthquakes while the

loss of functionality should be limited for more frequent ones.

To combat earthquake destruction, the only method available to ancient architects was to build

their landmark structures to last, often by making them excessivelystiff and strong, like the El

Castillo pyramid at Chichen Itza.

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Fig 1: Snapshot from shake-table video of testing , base-isolated (right) and regular (left)

building model 2. SCADA (supervisory control and data acquisition) is a type of industrial control

system (ICS). Industrial control systems are computer-controlled systems that monitor and control industrial processes that exist in the physical world. SCADA systems historically distinguish themselves from other ICS systems by being large-scale processes that can include multiple sites, and large distances.[1] These processes include industrial, infrastructure, and facility-based processes, as described below:

• Industrial processes include those of manufacturing, production, power generation, fabrication, and refining, and may run in continuous, batch, repetitive, or discrete modes.

• Infrastructure processes may be public or private, and include water treatment and distribution, wastewater collection and treatment, oil and gas pipelines, electrical power transmission and distribution,wind farms, civil defense siren systems, and large communication systems.

• Facility processes occur both in public facilities and private ones, including buildings, airports, ships, and space stations. They monitor and control heating, ventilation, and air conditioning systems (HVAC), access, and energy consumption.

Fig 2: SCADA overview

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Seismology is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other planet-like bodies. The field also includes studies of earthquake effects, such as tsunamis as well as diverse seismic sources such as volcanic, tectonic, oceanic, atmospheric, and artificial processes (such as explosions). A related field that uses geology to infer information regarding past earthquakes is paleoseismology. A recording of earth motion as a function of time is called aseismogram. A seismologist is a scientist who does research in seismology. Fig 3: Seismogram records showing the three components of ground motion. The red line marks

the first arrival of P-waves; the green line, the later arrival of S-waves. 3. Condition monitoring (or, colloquially, CM) is the process of monitoring a parameter of

condition in machinery (vibration, temperature etc), in order to identify a significant change which is indicative of a developing fault. It is a major component of predictive maintenance. The use of conditional monitoring allows maintenance to be scheduled, or other actions to be taken to prevent failure and avoid its consequences. Condition monitoring has a unique benefit in that conditions that would shorten normal lifespan can be addressed before they develop into a major failure. Condition monitoring techniques are normally used on rotating equipment and other machinery (pumps, electric motors, internal combustion engines, presses), while periodic inspection using non-destructive testing techniques and fit for service (FFS)[1] evaluation are used for stationary plant equipment such as steam boilers, piping and heat exchangers.

4. The process of implementing a damage detection and characterization strategy for

engineering structures is referred to as Structural Health Monitoring (SHM). Here damage

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is defined as changes to the material and/or geometric properties of a structural system, including changes to the boundary conditions and system connectivity, which adversely affect the system’s performance. The SHM process involves the observation of a system over time using periodically sampled dynamic response measurements from an array of sensors, the extraction of damage-sensitive features from these measurements, and the statistical analysis of these features to determine the current state of system health. For long term SHM, the output of this process is periodically updated information regarding the ability of the structure to perform its intended function in light of the inevitable aging and degradation resulting from operational environments. After extreme events, such as earthquakes or blast loading, SHM is used for rapid condition screening and aims to provide, in near real time, reliable information regarding the integrity of the structure.

5. Foldback is the use of rear-facing loudspeakers known as monitor speakers or stage

monitors on stage during live music performances. The sound is amplified with power amplifiers or a public address system and the speakers are aimed at the on-stage performers rather than the audience. This sound signal may be produced on the same mixing console as the main mix for the audience (called the "front of house" mix), or there may be a separate sound engineer and mixing console on or beside the stage creating a separate mix for the monitor system.

Fig: 5.1: This small venue's stage shows an example of a typical monitor speaker set-up:

2. Harmonic and periodic motion, vibration terminology

a) Harmonic Motion

Oscillatory motion may repeat itself regularly as in the balance wheel of a watch, or display

considerable irregularity, as in earthquakes. When the motion is repeated in equal intervals of

time T, it is called periodic motion. The repetition time T is called the period of the oscillation,

and its reciprocal f = 1/T is called the frequency. If the motion is designated by the time function

X (t), then any periodic motion must satisfy the relationship X (t) = X (t+T).

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The simplest form of periodic motion is harmonic motion. It can be demonstrated by a mass

suspended from a light spring, as shown in fig (1). If the mass is displaced from its rest position

and released, it will oscillate up and down, by placing a light source on the oscillating mass; its

motion can be recorded on a light – sensitive film strip which is made to move past it at constant

speed.

The motion recorded on the film strip can be expressed by the equation,

X = A Sin (2π t/T)

Where A is the amplitude of oscillation, measured from the equilibrium position of the mass and T is the

period. The motion is repeated when t = T.

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 (2) with the angular speed of the line designated by ω, the

displacement X can be written as

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X = A Sin ω t ---------- (1)

The quantity ω is generally measured in rad/sec, and is referred to as the circular frequency. Since the

motion repeat itself in 2 π radians.

We have ω = 2π / T = 2π f

Where T is period in sec and f is frequency in cycles/sec

The velocity and acceleration of harmonic motion can be determined by differentiation of equation (1).

i.e. x = A.Cos ω t. ω = A ω Sin (ω t + π / 2) ------- (2)

x = - A ω. Sin ω t. ω = A ω 2 Sin (ω t + π ) ------- (3)

Thus the velocity and acceleration are also harmonic with the same frequency of oscillation, but lead

the displacement by π/2 and π radians respectively.

Fig (3) shows both the time variation and vector phase relationship between displacement, velocity

and acceleration in harmonic motion.

From equations (1) and (3) x = - �̈� ω2

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b) Addition of harmonic motion

When we add two harmonic motions of the same frequency, we get the resultant motion as harmonic.

Let us have the harmonic motions of amplitudes A1 and A2 the same frequency ω and phase difference φ

as

X1 = A1 Sin ω t

X2 = A2 Sin (ω t + φ)

The resultant motion is given by adding the above equations

X = X1 + X2 = A1 Sin ω t + A2 Sin (ω t + φ)

= A1 Sin ω t + A2 [Sin ω t Cos φ + Sin φ Cos ω t]

= A1 Sin ω t + A2 Sin ω t Cos φ + A2 Sin φ Cos ω t

X = (A1 + A2 Cos φ) Sin ω t + A2 Sin φ Cos ω t

Let A1 + A2 Cos φ = A Cos θ

A2 Sin φ = A Sin θ

Then X = A Cosθ Sin ω t + A Sin θ Cos ω t

X = A Sin (ω t + θ)

The above equation shows that the resultant displacement is also simple harmonic motion of

amplitude A and phase θ. To find out the value of A consider A Cosθ = A1 + A2 Cos φ

A Sinθ = A22 Sin φ � − − −−(1)

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Squaring and adding

A2 Cos2θ + A2 Sin2θ = (A1 + A2 Cos φ)2 + (A2 Sin φ)2

A2 = A12 + A2

2 Cos2 φ + 2 A1 A2 Cos φ + A22 Sin2 φ

A2 = A12 + A2

2 + 2 A1 A2 Cos φ

A = � 𝐴12 + 𝐴22 + 2𝐴1𝐴2Cos φ

The resultant phase difference can be determine from the above equation (I)

tan θ = A2 Sin φ / (A1+A2 Cos φ)

θ = tan-1 [A2 Sin φ / (A1 + A2 Cos φ)]

The graphical method of addition of two simple harmonic motion is shown in figure.

3. Vibration terminology:

1) Periodic motion: A motion which repeats itself after equal intervals of time is known as periodic

motion. Any periodic motion can be represented by function x (t) in the period T. the function x(t) is

called periodic function.

2) Time period: - Time taken to complete one cycle is called time period.

3) Frequency: - The number of cycles per unit time is known as frequency.

4) Natural frequency: - When no external force acts on the system after giving it an initial

displacement, the body vibrates. These vibrations are called free vibrations and their frequency as

natural frequency. It is expressed in c/s or hertz

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5) Amplitude: - The max displacement of a vibrating body from its equilibrium position is called

amplitude.

6) Fundamental mode of vibration: - The fundamental mode of vibration of a system is the mode

having the lowest natural frequency.

7) Resonance: - When the frequency of external exitation is equal to the natural frequency of a vibrating

body, the amplitude of vibration becomes excessively large. This concept is known as resonance.

8) Mechanical systems: - The systems consisting of mass shiftness and damping are known as

mechanical systems.

9) Continuous and discrete systems: - Most of mechanical systems include elastic members which

have infinite number of degree of freedom. Such systems are called continuous systems. Continuous

systems are also known as distributed systems. Ex. Cantilever, Simply supported beam etc. Systems

with finite number of degrees of freedom are called discrete or lumped systems.

10) Degree of freedom: - The minimum no of independent co-ordinates required to specify the motion of

a system at any instant is known as degree of freedom of the system. Thus a free particle undergoing

general motion in space will have three degree of freedom, while a rigid body will have six degree of

freedom i.e. three components of position and three angles defining its orientation. Furthermore a

continuous body will require an infinite number of co-ordinates to describe its motion; hence its

degree of freedom must be infinite.

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11) Simple harmonic motion (SHM) A periodic motion of a particle whose acceleration is always

directed towards the mean position and is proportional to its distance from the mean position is

known as SHM. It may also be defined as the motion of a projection of a particle moving round a

circle with uniform angular velocity, on a diameter.

12) Phase difference It is the angle between two rotating vectors representing simple harmonic motion of

the same frequency.

2.2.2 Classification of vibrations

Mechanical vibrations may broadly be classified the following types 1) Free and forced vibration

2) Linear and nonlinear vibration

3) Damped and un damped vibration

4) Deterministic and random vibration

5) Longitudinal, transverse and torsional vibration

6) Transient vibration

1) Free and Forced vibration

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Free vibration takes place when system oscillates under the action of forces inherent in the system

itself and when external impressed forces are absent. The system under free vibration will vibrate at

one or more of its natural frequencies.

Vibration that takes place under the exitation of external forces is called forced vibration. When the

exitation is oscillating the system is forced to vibrate at the exitation frequency. If the frequency of

excitation coincides with one of the natural frequency of the system, a condition of resonance is

encountered, and dangerously large oscillations may result.

2) Linear and Non-linear vibration

If in a vibrating system mass, spring and damper behave in a linear manner, the vibrations caused

are known as linear in nature. Linear vibrations are governed by linear differential equations. They

follow law of superposition.

On the other hand, if any of the basic components of a vibrating system behaves non-linearly, the

vibration is called non-linear. Linear vibration becomes non-linear for very large amplitude of

vibration. It does not follow the law of super- positions.

3) Damped and Undamped vibration

If the vibrating system has a damper, the motion of the system will be opposed by it and the energy of the system will be dissipated in friction. This type of vibration is called damped vibration. The system having no damper is known as undamped vibration

4) Deterministic and Random vibration

If in the vibrating system the amount of external excitation is known in magnitude, it causes

deterministic vibration. Contrary to it the non-deterministic vibrations are known as random

vibrations.

5) Longitudinal, Transverse and Torsional vibration

Fig represents a body of mass ‘m’ carried on one end of a weightless spindle, the other end being

fixed. If the mass moves up and down parallel to the spindle and it is said to execute longitudinal

vibrations as shown in fig (1).

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When the particles of the body or spindle move approximately perpendicular to the axis of the spindle as

shown in fig (2) the vibrations so caused are known as transverse vibrations if the spindle gets alternately

twisted and untwisted on account of vibrating motion of the suspended disc, it is said to be undergoing

torsional vibrations as shown in fig(3).

6) Transient Vibration

In ideal system the free vibrations continue indefinitely as there is no damping. The amplitude of

vibration decays continuously because of damping (in a real system) and vanishes ultimately. Such

vibration in a real system is called transient vibration.

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QUADRANT-2 Animations (Animation links related ,Introduction to Vibration Engineering)

1. Illustration of vibrations of a drum. http://en.wikipedia.org/wiki/File:Drum_vibration_mode21.gif

2. Illustration of a Simple harmonic oscillator http://en.wikipedia.org/wiki/File:Simple_harmonic_oscillator.gif

3. Illustration of The mode shapes of a cantilevered I-beam: 1st lateral bending http://en.wikipedia.org/wiki/File:Beam_mode_1.gif

Videos (Animation links related ,Introduction to Vibration Engineering)

• youtube.com/playlist?list=PLAC668A0566953FB5

• www.youtube.com/watch?v=OyqHoJ2Wd8Q

• www.youtube.com/watch?v=VnGkoMoUkgI

• www.youtube.com/watch?v=9Qm2qRxnJwY

• www.youtube.com/watch?v=9HHRMCBQ2NU • www.youtube.com/watch?v=SZ541Luq4nE • www.youtube.com/watch?v=cw9M57Z8hCM

• www.youtube.com/watch?v=r8ffD9ZvYaY

• www.youtube.com/watch?v=zz-027WAUVw

• ww.youtube.com/watch?v=U1cSvRGZYcM

• www.youtube.com/watch?v=51aivZiP9vA

• www.youtube.com/watch?v=ULaCiRUo8Wk • www.tcyonline.com/video-lectures-mechanical-vibrations/777968/2

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ILLUSTRATIONS 1. What do you by vibration? Explain the causes of Vibration.

Ans) Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium

point. The oscillations may be periodic such as the motion of a pendulum or random such as the

movement of a tire on a gravel road.

Causes of vibration: - The main causes of vibration are:-

Unbalanced forces in the machine. These forces are produced from within the machine itself

because of non-uniform material distribution in a rotating machine element.

Dry friction between the two mating surfaces: This is what known as self-exited vibration.

External excitations. These excitations may be periodic, random or of the nature of an impact

produced external to the vibrating system.

Earth quakes. These are responsible for the failure of many buildings, dams etc.

Winds. These may cause the vibrations of transmission and telephone lines under certain

conditions.

The effect of vibrations is excessive stresses, undesirable noise, looseness of parts and partial or

complete failure of parts. Inspite of these harmful effects the vibration phenomenon does have,

some uses also e.g. in musical instruments, vibrating screens, shakers, stress relieving.etc.

2. Define the following:

a) Periodic Motion b) Time Period c) Frequency d) Amplitude

Ans) Periodic motion: A motion which repeats itself after equal intervals of time is

known as periodic motion. Any periodic motion can be represented by function x

(t) in the period T. the function x(t) is called periodic function.

Time period: - Time taken to complete one cycle is called time period.

Frequency: - The number of cycles per unit time is known as frequency.

Amplitude: - The max displacement of a vibrating body from its equilibrium

position is called amplitude.

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3. Define SHM and explain how do you represent harmonic motions by vector method.

Ans) Any Motion which repeats itself in equal intervals of time is known as periodic motion. A

SHM is the simplest form of periodic motion. Oscillatory motion may repeat itself regularly as in

the balance wheel of a watch, or display considerable irregularity, as in earthquakes. When the

motion is repeated in equal intervals of time T, it is called periodic motion. The repetition time T

is called the period of the oscillation, and its reciprocal f = 1/T is called the frequency. If the

motion is designated by the time function X (t), then any periodic motion must satisfy the

relationship X (t) = X (t+T).

The simplest form of periodic motion is harmonic motion. It can be demonstrated by a mass

suspended from a light spring, as shown in fig (1). If the mass is displaced from its rest position

and released, it will oscillate up and down, by placing a light source on the oscillating mass; its

motion can be recorded on a light – sensitive film strip which is made to move past it at constant

speed.

The motion recorded on the film strip can be expressed by the equation X = A Sin (2π

t/T).Where A is the amplitude of oscillation, measured from the equilibrium position of the mass

and T is the period. The motion is repeated when t = T.

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 (2) with the angular speed of the line

designated by ω, the displacement X can be written as

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X = A Sin ω t ---------- (1)

Fig (2) Harmonic motion as projection of a point moving as a circle

The quantity ω is generally measured in rad/sec, and is referred to as the circular frequency.

Since the motion repeat itself in 2 π radians.

We have ω = 2π / T = 2π f

Where T is period in sec and f is frequency in cycles/sec

The velocity and acceleration of harmonic motion can be determined by differentiation of

equation (1).

i.e. x = A.Cos ω t. ω = A ω Sin (ω t + π / 2) ------- (2)

x = - A ω. Sin ω t. ω = A ω 2 Sin (ω t + π ) ------- (3)

Thus the velocity and acceleration are also harmonic with the same frequency of oscillation, but

lead the displacement by π/2 and π radians respectively.

Fig (3) shows both the time variation and vector phase relationship between displacement,

velocity and acceleration in harmonic motion. ..

From equations (1) and (3) x = - X ω2

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4. Explain any five types of vibrations, with examples.

Ans) Free and Forced vibration

Free vibration takes place when system oscillates under the action of forces inherent in the

system itself and when external impressed forces are absent. The system under free vibration

will vibrate at one or more of its natural frequencies.

Vibration that takes place under the exitation of external forces is called forced vibration.

When the exitation is oscillating the system is forced to vibrate at the exitation frequency. If

the frequency of excitation coincides with one of the natural frequency of the system, a

condition of resonance is encountered, and dangerously large oscillations may result.

Linear and Non-linear vibration

If in a vibrating system mass, spring and damper behave in a linear manner, the vibrations

caused are known as linear in nature. Linear vibrations are governed by linear differential

equations. They follow law of superposition.

On the other hand, if any of the basic components of a vibrating system behaves non-linearly,

the vibration is called non-linear. Linear vibration becomes non-linear for very large

amplitude of vibration. It does not follow the law of super- positions.

Damped and Undamped vibration

If the vibrating system has a damper, the motion of the system will be opposed by it and the energy of the system will be dissipated in friction. This type of vibration is called damped vibration.The system having no damper is known as undamped vibration

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Deterministic and Random vibration

If in the vibrating system the amount of external excitation is known in magnitude, it causes

deterministic vibration. Contrary to it the non-deterministic vibrations are known as random

vibrations.

Longitudinal, Transverse and Torsional vibration

Fig represents a body of mass ‘m’ carried on one end of a weightless spindle, the other end

being fixed. If the mass moves up and down parallel to the spindle and it is said to execute

longitudinal vibrations as shown in fig (1).

When the particles of the body or spindle move approximately perpendicular to the axis of the

spindle as shown in fig (2) the vibrations so caused are known as transverse vibrations .if the

spindle gets alternately twisted and untwisted on account of vibrating motion of the suspended

disc, it is said to be undergoing torsional vibrations as shown in fig(3).

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QUADRANT-3

Wikis: (This includes wikis related to introduction to Vibration engineering contains practical application and research trends ,Harmonic & Periodic Motions)

1. http://en.wikipedia.org/wiki/Vibration 2. http://nengvib.sydneyinstitute.wikispaces.net/file/view/BA767412.pdf/33225085/BA767

412.pdf 3. http://physics.doane.edu/hpp/Resources/Fuller3/pdf/F3Chapter_15.pdf 4. http://nptel.ac.in/courses/Webcourse-contents/IIT-%20Guwahati/ve/index.htm 5. http://nptel.ac.in/courses/112103112/1 6. http://nptel.ac.in/courses/112103112/2

Open Contents: (This includes wikis related to introduction to Vibration engineering contains practical application and research trends ,Harmonic & Periodic Motions)

• Mechanical Vibrations, S. S. Rao, Pearson Education Inc, 4th edition, 2003. • Mechanical Vibrations, V. P. Singh, Dhanpat Rai & Company, 3rd edition, 2006. • Mechanical Vibrations, G. K.Grover, Nem Chand and Bros, 6th edition, 1996 • Theory of vibration with applications ,W.T.Thomson,M.D.Dahleh and C

Padmanabhan,Pearson Education inc,5th Edition ,2008 • Theory and practice of Mechanical Vibration : J.S.Rao&K,Gupta,New Age International

Publications ,New Delhi,2001 Case Studies

• Earthquake-resistant structures • Siesmology • Condition Monitoing • Structural Health Monitoring

1. Earthquake-resistant structures are structures designed to withstand earthquakes.

While no structure can be entirely immune to damage from earthquakes, the goal

of earthquake-resistant construction is to erect structures that fare better during seismic

activity than their conventional counterparts.

According to building codes, earthquake-resistant structures are intended to withstand the largest earthquake of a certain probability that is likely to occur at their location. This means the loss of

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life should be minimized by preventing collapse of the buildings for rare earthquakes while the loss of functionality should be limited for more frequent ones.

To combat earthquake destruction, the only method available to ancient architects was to build their landmark structures to last, often by making them excessivelystiff and strong, like the El Castillo pyramid at Chichen Itza.

Fig 1: Snapshot from shake-table video of testing , base-isolated (right) and regular (left) building model

2. SCADA (supervisory control and data acquisition) is a type of industrial control system (ICS). Industrial control systems are computer-controlled systems that monitor and control industrial processes that exist in the physical world. SCADA systems historically distinguish themselves from other ICS systems by being large-scale processes that can include multiple sites, and large distances.[1] These processes include industrial, infrastructure, and facility-based processes, as described below:

• Industrial processes include those of manufacturing, production, power generation, fabrication, and refining, and may run in continuous, batch, repetitive, or discrete modes.

• Infrastructure processes may be public or private, and include water treatment and distribution, wastewater collection and treatment, oil and gas pipelines, electrical power transmission and distribution,wind farms, civil defense siren systems, and large communication systems.

• Facility processes occur both in public facilities and private ones, including buildings, airports, ships, and space stations. They monitor and control heating, ventilation, and air conditioning systems (HVAC), access, and energy consumption.

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Fig 2: SCADA overview 3. Seismology is the scientific study of earthquakes and the propagation of elastic

waves through the Earth or through other planet-like bodies. The field also includes studies of earthquake effects, such as tsunamis as well as diverse seismic sources such as volcanic, tectonic, oceanic, atmospheric, and artificial processes (such as explosions). A related field that uses geology to infer information regarding past earthquakes is paleoseismology. A recording of earth motion as a function of time is called aseismogram. A seismologist is a scientist who does research in seismology.

Fig 3: Seismogram records showing the three components of ground motion. The red line marks

the first arrival of P-waves; the green line, the later arrival of S-waves.

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4. Condition monitoring (or, colloquially, CM) is the process of monitoring a parameter of condition in machinery (vibration, temperature etc), in order to identify a significant change which is indicative of a developing fault. It is a major component of predictive maintenance. The use of conditional monitoring allows maintenance to be scheduled, or other actions to be taken to prevent failure and avoid its consequences. Condition monitoring has a unique benefit in that conditions that would shorten normal lifespan can be addressed before they develop into a major failure. Condition monitoring techniques are normally used on rotating equipment and other machinery (pumps, electric motors, internal combustion engines, presses), while periodic inspection using non-destructive testing techniques and fit for service (FFS)[1] evaluation are used for stationary plant equipment such as steam boilers, piping and heat exchangers.

5. The process of implementing a damage detection and characterization strategy for

engineering structures is referred to as Structural Health Monitoring (SHM). Here damage is defined as changes to the material and/or geometric properties of a structural system, including changes to the boundary conditions and system connectivity, which adversely affect the system’s performance. The SHM process involves the observation of a system over time using periodically sampled dynamic response measurements from an array of sensors, the extraction of damage-sensitive features from these measurements, and the statistical analysis of these features to determine the current state of system health. For long term SHM, the output of this process is periodically updated information regarding the ability of the structure to perform its intended function in light of the inevitable aging and degradation resulting from operational environments. After extreme events, such as earthquakes or blast loading, SHM is used for rapid condition screening and aims to provide, in near real time, reliable information regarding the integrity of the structure.

6. Foldback is the use of rear-facing loudspeakers known as monitor speakers or stage

monitors on stage during live music performances. The sound is amplified with power amplifiers or a public address system and the speakers are aimed at the on-stage performers rather than the audience. This sound signal may be produced on the same mixing console as the main mix for the audience (called the "front of house" mix), or there may be a separate sound engineer and mixing console on or beside the stage creating a separate mix for the monitor system.

Fig: 5.1: This small venue's stage shows an example of a typical monitor speaker set-up

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QUADRANT-4 Problems

1. A simple harmonic motion has an amplitude of 3 cm and a period of 2 secs. Determine the maximum velocity and acceleration.

Solution: We have x = 𝐴 sinω𝑡

𝑥 = 𝐴𝜔 sın �𝜔𝑡 + 𝜋2�̇

𝑥 = 𝐴 𝜔2 ̈ (sin𝜔𝑡 + 𝜋)

Now, A= 3 cm; T = 2 sec = 2𝜋𝜔

; ∴ 𝜔 = 2𝜋𝑇

= 2𝜋2

= 𝜋 𝑟𝑎𝑑/𝑠𝑒𝑐. Maximum velocity �̇� Rmax = A𝜔 = 3 × 𝜋 = 9.425 𝑐𝑚/𝑠𝑒𝑐

Maximum Acceleration �̈� max = A𝜔2 = 3 × 𝜋2 = 29.609 𝑐𝑚/𝑠𝑒𝑐2 2. A harmonic motion has a frequency of 10 Hz and its maximum velocity is 2.5 m/sec.

Determine its amplitude, period and maximum acceleration.

Solution:

Frequency 𝑓 = 𝜔2𝜋𝐻𝑧

i.e, 10 = 𝜔2𝜋

; ∴ 𝜔 = 62.832 𝑟𝑎𝑑/𝑠𝑒𝑐.

Time period 𝑇 = 1𝑓

= 110

= 0.1 𝑠𝑒𝑐.

Maximum Velocity �̇� Rmax = 𝜔.𝐴 i.e, 2.5 = (62.832) A

∴ Amplitude A = 0.03979m = 3.979 cm.

Maximum Acceleraton �̈� max = 𝜔2𝐴 = (62.832)2 (0.03979)

= 157.08 m/sec2

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3. Add the following harmonics analytically and check the solution graphically. x1 = 3 sin (𝝎𝒕 + 30o) x2 = 4 cos (𝝎𝒕 + 10o)

Solution: Analytical Method The resultant motion is given by

x=x1 + x2 = 3 sin (𝜔𝑡 + 30o) + 4 cos (𝜔𝑡 + 10o) Since the frequency is same for both x1 and x2, the resultant motion can also be written as x= A sin (ωt + θ)

∴ A sin (ωt + θ) = 3 sin (𝜔𝑡 + 30o) + 4 cos (𝜔𝑡 + 10o)

i.e, A sin 𝜔𝑡 𝑐𝑜𝑠𝜃 + 𝐴 cos𝜔𝑡 𝑠𝑖𝑛𝜃 = 3 sin𝜔𝑡 𝑐𝑜𝑠30𝑜 + 3 cos𝜔𝑡 𝑠𝑖𝑛30𝑜

+ 4 cos𝜔𝑡 𝑐𝑜𝑠10𝑜 – 4 sin𝜔𝑡 𝑠𝑖𝑛10𝑜

i.e, sin 𝜔𝑡 (𝐴 cos 𝜃) + cos𝜔𝑡 (𝐴 sin𝜃) = 1.9035 sin𝜔𝑡 + 5.4392 cos𝜔𝑡

By equating the corresponding coefficients of cos𝜔𝑡 𝑎𝑛𝑑 𝑠𝑖𝑛 𝜔𝑡 𝑜𝑛 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒𝑠,

𝐴 cos 𝜃 = 1.9035

𝐴 sin𝜃 = 5.4392 Now squaring and adding A2 cos2 θ + A2 sin2 θ = 1.90352 + 5.43922

i.e, A2 (cos2 θ + sin2 θ) = 33.20815

∴ A = 5.76265

tan 𝜃 = 5.43921.9035

∴ 𝜃 = 70.712o

∴ x = 5.76265 sin (𝝎𝒕 + 𝟕𝟎.𝟕𝟏𝟐P

o) Graphical Method x1 = 3 sin (𝜔𝑡 + 30o) x2 = 4 cos (𝜔𝑡 + 10o) = 4 sin (𝜔𝑡 + 10o + 90o) = 4 sin (𝜔𝑡 + 100o)

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The vector diagram can be drawn as shown in Figure. On measurement, the sum of the two vectors is 5.8 at an angle of 71o. ∴ x = 5.8 sin (𝜔𝑡 + 71o) which agrees closely with the analytical results.

4. A harmonics motion is given by the equation 𝒙(𝒕) = 𝟓 𝐬𝐢𝐧 �𝟏𝟓𝒕 − 𝝅𝟒� cm where

phase angle is in radians and in seconds. Find i) Period of motion ii) Frequency iii) Maximum displacement, velocity and acceleration.

Solution: 𝑥(𝑡) = 5 sin �15𝑡 − 𝜋

4�

Let the harmonics motion be in the from x = A sin (𝜔𝑡 - φ) where A = Amplitude and 𝜔 = frequency in radians. Comparing the two equations, A = 5 and 𝜔 = 15 rad/sec. i) Period of motion T = 2𝜋

𝜔 = 2𝜋

15 = 0.4189

ii) Frequency 𝑓 = 𝜔2𝜋

= 152𝜋

= 2.387 Hz.

iii) Maximum displacement xmax = A = 5 cm.

iv) Maximum velocity �̇� Rmax = A ω = 5 x 15 = 75 cm/sec2

v) Maximum Acceleration �̈� max = A ω2 = 5 x 152 = 1125 cm/sec.

5. Find the Foruier series expansion for the impact force generated by the forging

hammer shown in Figure

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Soluton: Let the Fourier series of the above periodic motion be represented by, 𝑥(𝑡) = 𝑎𝑜 + 𝑎1 cos𝜔𝑡 + 𝑎2 cos 2 𝜔𝑡 + ---- + b1 sin𝜔𝑡 + 𝑏2 𝑠𝑖𝑛2 𝜔𝑡 +----- = 𝑎𝑜 + ∑ (𝑎𝑛𝛼

𝑛=1 cos𝑛𝜔𝑡 + 𝑏𝑛 sin𝑛𝜔𝑡) The equation of the curve for the cycle is

x(t) = A sin ωt 0 ≤ t ≤ 𝑇2

= 0 𝑇2 ≤ t ≤ T

ω = 2𝜋𝑇

∴ T = 2𝜋𝜔

𝑎0 = 𝜔2𝜋

� 𝐴 sin𝜔𝑡 𝑑𝑡 = 𝜔2𝜋

�–𝐴 cos𝜔𝑡

𝜔�0

𝜋𝜔

𝑇2�

0

= − 𝐴2𝜋

�cos𝜔 �𝜋𝜔� − cos 0� = − 𝐴

2𝜋 (−2)

∴𝑎𝑜 = 𝐴𝜋

𝑎𝑛 = 𝜔𝜋� 𝑥(𝑡) cos(𝑛𝜔𝑡)𝑑𝑡 =

𝜔𝜋

𝑇2�

0

� 𝐴 sin𝜔𝑡 𝑐𝑜𝑠 𝜔𝑡 𝑑𝑡

𝑇2�

0

= 𝜔𝐴𝜋

∫{sin(1+𝑛)𝜔𝑡+sin(1−𝑛)𝜔𝑡}𝑑𝑡

2

𝜋 𝜔⁄0

= 𝜔𝐴2𝜋

�– cos (1+𝑛)𝜔𝑡(1+𝑛)𝜔

− cos(1−𝑛)𝜔𝑡(1−𝑛)𝜔

�0

𝜋 𝜔⁄

When n=1

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𝑎1 = 𝜔𝐴2𝜋�− cos2 𝜔𝑡

2𝜔�0

𝜋 𝜔⁄ = 0

When n = 2, 3, 4, -----

𝑎𝑛 = 𝜔𝐴2𝜋

�–�cos(1 + 𝑛)𝜔. 𝜋𝜔

(1 + 𝑛)𝜔+

cos(1 − 𝑛)𝜔 𝜋𝜔

(1 − 𝑛)𝜔−

1(1 + 𝑛)𝜔

− 1

(1 − 𝑛)𝜔��

= 𝜔𝐴2𝜋�1−cos(1+𝑛)𝜋

𝜔(1+𝑛)+ 1−cos(1−𝑛)𝜋

𝜔(1−𝑛)�

= 𝐴2𝜋�1−cos(1+𝑛)𝜋

(1+𝑛)+ 1−cos(1−𝑛)𝜋

(1−𝑛)�

= 𝐴2𝜋�− cos(1+𝑛)𝜋−1

𝑛+1+ cos(1−𝑛)𝜋−1

𝑛−1�

= − 2𝐴(𝑛+1)(𝑛−1)𝜋

if n is even. = 0 if n is odd.

Similarly 𝑏𝑛 = 𝜔𝜋

∫ 𝑥(𝑡) sin(𝑛𝜔𝑡)𝑑𝑡 = 𝜔𝜋

∫ 𝐴 sin 𝜔𝑡 𝑠𝑖𝑛 𝑛𝜔𝑡𝑑𝑡 𝜋 𝜔⁄0

𝑇2�

0

= 𝐴𝜔𝜋

∫{cos(1−𝑛)𝜔𝑡−cos (1+𝑛)𝜔𝑡} 𝑑𝑡

2

𝜋 𝜔⁄0

When n = 1

𝑏1 = 𝐴𝜔2𝜋

∫ {1 − 𝑐𝑜𝑠2𝜔𝑡} 𝑑𝑡 = 𝜋 𝜔⁄0

𝐴𝜔2𝜋�𝑡 − 𝑠𝑖𝑛2𝜔𝑡

2𝜔�0

𝜋 𝜔⁄

= 𝐴𝜔2𝜋�𝜋𝜔− sin2𝜋

2𝜔− 0 + 0� = 𝐴

2

When n = 2, 3, 4, -----

𝑏𝑛 = 𝜔𝐴2𝜋�sin(1−𝑛)𝜔𝑡

(1−𝑛)𝜔− sin(1+𝑛)𝜔𝑡

(1+𝑛)𝜔�0

𝜋 𝜔⁄ = 0

Hence, the harmonic series of the given motion is,

𝑥(𝑡) = 𝐴𝜋

+ ∑ 2𝐴(𝑛+)(𝑛−1)𝜋

cos𝑛𝜔𝑡𝛼𝑛=2,4,6,−− + 𝐴

2sin𝜔𝑡

= 𝐴𝜋 - 2𝐴

𝜋∑ cos𝑛𝜔𝑡

(𝑛2−1)𝛼𝑛=2,4,6,−− + 𝐴

2sin𝜔𝑡

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Frequently asked Questions.

1. What is vibration? Explain different types of vibrations with examples.

2. Explain causes of vibrations and Methods to reduce vibrations

3. Define the following:

a. Periodic motion b. Amplitude c. SHM

d. Time period e. Resonance f. Degree of freedom

4. Define SHM and explain how do you represent harmonic motions by vector method.

5. How do you analyse harmonic motions using Fourier series? Explain.

6. Explain the terminology of vibration?

7. Explain any five types of vibrations, with examples

8. Distinguish between following;

a. Free and forced vibrations b. Damped and un-damped vibrations c. Linear and Non-linear vibrations d. Longitudinal, Transverse and Torsional vibrations

9. Find the Foruier series expansion for the impact force generated by the forging hammer shown in Figure

10.Determine the natural frequency of a compound pendulum.

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Self Answered Question & Answer 1. Explain any five types of vibrations, with examples.

Ans) Free and Forced vibration

Free vibration takes place when system oscillates under the action of forces inherent in the

system itself and when external impressed forces are absent. The system under free vibration

will vibrate at one or more of its natural frequencies.

Vibration that takes place under the exitation of external forces is called forced vibration.

When the exitation is oscillating the system is forced to vibrate at the exitation frequency. If

the frequency of excitation coincides with one of the natural frequency of the system, a

condition of resonance is encountered, and dangerously large oscillations may result.

Linear and Non-linear vibration

If in a vibrating system mass, spring and damper behave in a linear manner, the

vibrations caused are known as linear in nature. Linear vibrations are governed by linear

differential equations. They follow law of superposition.

On the other hand, if any of the basic components of a vibrating system behaves non-

linearly, the vibration is called non-linear. Linear vibration becomes non-linear for very large

amplitude of vibration. It does not follow the law of super- positions.

Damped and Undamped vibration

If the vibrating system has a damper, the motion of the system will be opposed by it and the energy of the system will be dissipated in friction. This type of vibration is called damped vibration.The system having no damper is known as undamped vibration Deterministic and Random vibration

If in the vibrating system the amount of external excitation is known in magnitude, it causes

deterministic vibration. Contrary to it the non-deterministic vibrations are known as random

vibrations.

Longitudinal, Transverse and Torsional vibration

Fig represents a body of mass ‘m’ carried on one end of a weightless spindle, the other end

being fixed. If the mass moves up and down parallel to the spindle and it is said to execute

longitudinal vibrations as shown in fig (1).

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2. A harmonics motion is given by the equation 𝒙(𝒕) = 𝟓 𝐬𝐢𝐧 �𝟏𝟓𝒕 − 𝝅

𝟒� cm where

phase angle is in radians and in seconds. Find i) Period of motion ii) Frequency iii) Maximum displacement, velocity and acceleration.

Solution: 𝑥(𝑡) = 5 sin �15𝑡 − 𝜋

4�

Let the harmonics motion be in the from x = A sin (𝜔𝑡 - φ) where A = Amplitude and 𝜔 = frequency in radians. Comparing the two equations, A = 5 and 𝜔 = 15 rad/sec. vi) Period of motion T = 2𝜋

𝜔 = 2𝜋

15 = 0.4189

vii) Frequency 𝑓 = 𝜔2𝜋

= 152𝜋

= 2.387 Hz.

viii) Maximum displacement xmax = A = 5 cm.

ix) Maximum velocity �̇� Rmax = A ω = 5 x 15 = 75 cm/sec2

x) Maximum Acceleration �̈� max = A ω2 = 5 x 152 = 1125 cm/sec.

3. A harmonic motion has a frequency of 10 Hz and its maximum velocity is 2.5 m/sec. Determine its amplitude, period and maximum acceleration.

Solution:

Frequency 𝑓 = 𝜔2𝜋𝐻𝑧

i.e, 10 = 𝜔2𝜋

; ∴ 𝜔 = 62.832 𝑟𝑎𝑑/𝑠𝑒𝑐.

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Time period 𝑇 = 1𝑓

= 110

= 0.1 𝑠𝑒𝑐.

Maximum Velocity �̇� Rmax = 𝜔.𝐴 i.e, 2.5 = (62.832) A

∴ Amplitude A = 0.03979m = 3.979 cm.

Maximum Acceleraton �̈� max = 𝜔2𝐴 = (62.832)2 (0.03979)

= 157.08 m/sec2

4. What do you by vibration? Explain the causes of Vibration.

Ans) Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium

point. The oscillations may be periodic such as the motion of a pendulum or random such as the

movement of a tire on a gravel road.

Causes of vibration: - The main causes of vibration are:-

Unbalanced forces in the machine. These forces are produced from within the machine itself

because of non-uniform material distribution in a rotating machine element.

Dry friction between the two mating surfaces: This is what known as self-exited vibration.

External excitations. These excitations may be periodic, random or of the nature of an impact

produced external to the vibrating system.

Earth quakes. These are responsible for the failure of many buildings, dams etc.

Winds. These may cause the vibrations of transmission and telephone lines under certain

conditions.

The effect of vibrations is excessive stresses, undesirable noise, looseness of parts and partial or

complete failure of parts. Inspite of these harmful effects the vibration phenomenon does have,

some uses also e.g. in musical instruments, vibrating screens, shakers, stress relieving.etc.

5. Define the following:

b) Periodic Motion b) Time Period c) Frequency d) Amplitude

Ans) Periodic motion: A motion which repeats itself after equal intervals of time is

known as periodic motion. Any periodic motion can be represented by function x

(t) in the period T. the function x(t) is called periodic function.

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Time period: - Time taken to complete one cycle is called time period.

Frequency: - The number of cycles per unit time is known as frequency.

Amplitude: - The max displacement of a vibrating body from its equilibrium

position is called amplitude.

Test Your Skills: Fill in the Blanks 1. A motion which repeats itself after a certain interval of time may be called a __________

2. A motion which repeats itself after equal interval of Time is known as ___________

3. The number of cucle per unit time is called________

4. The resistance offered to the motion of the vibrating body_____________

5. _________ is the reciprocating motion.

6. The number of independent co-ordinates required to describe the motion of a system is called

___________

7. An external force that acts on the system causes __________

8. __________ is the angle between two rotating vectors representing simple harmonic motions

of the same frequency.

8. When the body vibrates under the influence of external force ,then the body is said to be under

__________

9. If the magnitude of the excitation force acting on a vibrating system is known as __________

10. When the frequency of external excitation is equal to the natural frequency of System

___________

Answers

1. Vibration 2. Periodic motion 3. Frequency 4. Damping

5. Simple Harmonic Motion 6. Degree of Freedom 7. Forced Vibration 8.Phase difference 9. Random Vibration 10. Resonance

Dr.MOHAMED HANEEF ,PRINCIPAL,GHOUSIA COLLEGE OF ENGINEERING ,RAMANAGARAM

VTU-NPTEL-N

MEICT Proj

ect

Page 35 of 37

MODULE-I --- INTRODUCTION VIBRATION ENGINEERING 2014

True /False:

1. The root cause of vibration is balanced forces in the machines. 2. A motion which repeats itself after equal interval of time is Known as periodic Motion. 3. If the vibratory system has no damper then the vibration is called damped vibration. 4. The simple harmonic motion can be expressed by the equation x=Asinwt. 5. When the particles of the shaft moves parallel to the axis of shaft, then the vibrations are

known as longitudinal vibration. 6. When the particles of the shaft moves approximately perpendicular to the axis of the

shaft, then the vibrations are known as transverse vibration. 7. Vibration occurse when a system is displaced from a position of unstable equilinrium. 8. The time taken to complete one cycle is called frequency. 9. The free vibrations continue indefinateley in an ideal system as there is no damping. 10. Self excited vibrations are periodic and deterministic.

Answers

1.(x) 2.() 3. (x) 4 () 5 () 6. () 7. (x) 8.(x) 9. () 10. ()

MATCH THE FOLLOWING 1. Dry friction between

2. Unbalanced

3. Mass shiftness and damping are known

4. External impressed forces are absent

5. Damped vibration

6. Velocity of harmonic motion

7. Acceleration of harmonic motion

8. Harmonic force

9. Musical instruments

10. Very large amplitude of vibration

1. Forces in the machine

2. Free vibration

3. - A ω. Sin ω t. ω

4. Energy of the system will be dissipated in

friction

5. Mechanical systems

6. Fo Sin ωt

7. Two mating surfaces

8. A ω Sin (ω t + π / 2)

9. Linear vibration becomes non-linear vibration

10. Good vibration

Answers

1-7 , 2-1 ,3-5 ,4-2 ,5-4 ,6-8 ,7-3 ,8-6 ,9-10 ,10-9

Dr.MOHAMED HANEEF ,PRINCIPAL,GHOUSIA COLLEGE OF ENGINEERING ,RAMANAGARAM

VTU-NPTEL-N

MEICT Proj

ect

Page 36 of 37

MODULE-I --- INTRODUCTION VIBRATION ENGINEERING 2014

Dr.MOHAMED HANEEF ,PRINCIPAL,GHOUSIA COLLEGE OF ENGINEERING ,RAMANAGARAM

VTU-NPTEL-N

MEICT Proj

ect

Page 37 of 37