j. k. narwal - · pdf fileforeword my heartiest congratulations to er. j k narwal on bringing...

J. K. NARWALB.Tech (Gold-Medalist), M.Tech(ME)

Assistant Professor

Depatment of Mechanical Engineering

Geeta Engineering College, Panipat

AN ISO 9001:2008 CERTIFIED COMPANY

Vayu Education of India2/25, Ansari Road, Darya Ganj, New Delhi-110 002

MECHANICALVIBRATIONS

Mechanical Vibrations

Copyright © J.K. NARWAL

ISBN: 978-93-83758-18-0

First Edition: 2012

Second Edition: 2014

Price: 200/-

All rights reserved. No part of this publication may be reproduced, stored in a retrieval

system, or transmitted, in any form or by any means, electronic, mechanical,

photocopying, recording or otherwise, without the prior permission of the Author.

Published by:

VAYU EDUCATION OF INDIA2/25, Ansari Road, Darya Ganj, New Delhi-110 002

Ph.: 91-11-43526600, 41564445

Fax: 91-11-41564440

E-mail: [email protected],[email protected]

Web: www.veiindia.com

FOREWORDFOREWORDFOREWORDFOREWORD

My heartiest congratulations to Er. J K Narwal on bringing out this excellent book

on Mechanical Vibrations. The book written by Er. Narwal appears to be a good addition

to the existing list of books on the similar topic. The author is well known in the field

of engineering. He has put forth the best efforts to bringing out a book which is

comprehensive and will be useful to the students. The best thing about the author is

that he has explained each and every topic in depth and details. The book will be useful

for the students who intend to have hands on practical experience of the topics. The

book is indeed an example of acknowledgement in the field of engineering and deserves

the professional's attention. His passion for teaching, his diligence in organizing the

best of events for his students and his sheer enthusiasm towards his work are compelling

qualities which no doubt would take him very far in his career.

I am sure that Er. Narwal will go on to write many books in order to pass on his

knowledge and skills to budding technologists.

With Best Wishes

Ankush Bansal

Vice Chairman,

Geeta Group of Institutions, Panipat

Er. Ankush Bansal

Preface to the Second EditionPreface to the Second EditionPreface to the Second EditionPreface to the Second Edition

I am pleased to present the second edition of the book "Mechanical Vibrations".

The entire book has been thoroughly revised and some additions have been made, to

make the book still comprehensive and complete unit in all respects. I am thankful to

the readers, who has adopted this as text or reference book. Numerous solved and

unsolved examples have been added for the benefit of the student community and

teaching faculty.

The book contain 10 chapters. These chapters have been written to serve the

present need and also the possible future enrichment of the syllabi of the various

technologist institutes and universities in the country. Each chapter in this book consists

of fundamentals theory, followed by a number of selected illustrative worked examples.

Chapter 1 gives a review of the fundamentals of vibrations, elements of vibrating

systems, classification of vibrations along with periodic motion analysis.

Chapter 2 deals with the analysis of single degree of freedom, undamped,

translational and torsional systems, floating bodies.

In chapter 3 the concept of free damped vibrations of single degree of system is

explained. Different types of damping are also explained in this chapter.

Chapter 4 described the forced vibrations along with the basic formulations for

rotating unbalance, harmonic motion of support, magnification factor, transmissibility

and vibrations measuring instruments.

In chapter 5 coordinates coupling, principal coordinates and orthogonality of

modes of two degrees of freedom system is explained. This chapter prepares the basis

for the next chapter.

Chapter 6 explained the vibration analysis of several degrees of freedom system

with the help of Dunkerley's Method, Rayleigh's method, Rayleigh Ritz Method, Holzer's

Method, Matrix Method, Matrix Iteration Method, Stodola Method.

Chapter 7 deals with torsional vibrations of single and two rotors along with

geared system.

Mechanical Vibrationsviii

In chapter 8 whirling of shaft is explained in details.

Chapter 9 provides an introduction to the theory of vibrations of continuous

system. The vibration analysis of continuous systems including strings, bars, shafts,

beams are examined.

In chapter 10 response analysis of system subjected to transient vibration is

studied. Laplace Transformation method, Runge-Kutta method are also explained.

Appendices are also given in the end of the book.

The book has been written in a simple and easy to follow language, so that even an

average student can grasp the subject by self study.

I wish to thank to the Hon'ble Management members, specially Sh. Ankush Bansal,

Vice Chairman, Geeta Group of Institutions for writing foreword and providing me the

environment for my work.

It would have been very difficult to write without the enthusiastic support, advise and

inspiration given to me by Dr.Aman Aggarwal (Director, Geeta Engineering College,

Panipat), Dr. J.S. Dilawari (VP, Continental group of Institutions, Rajpura),

Dr. K.L. Miglani (Chief Administrative Officer), Prof. Mukesh Verma

(Head, Department of Mechanical Engineering, SSIET, Dera Bassi), and Prof. Deepak Juneja

(Head, Department of Mechanical Engineering, GEC). I am grateful for their kind support

and blessings.

I wish to thank Dr. G.N. Verma (Director, SSIET Dera Bassi), Dr. Rohit Garg

(Director- Principal, IIET Jind), Dr.Sanjay Bhardwaj ( Director, IITM Murthal),

Dr. Rajesh Bansal (Principal, PPIMT Hisar), Prof. Jaswant Jangra, Sh. Ajay Jangra

(CPA Global, Gr. Noida), Dr. Gurucharan Das ( Principal, JCDMCOE, Sirsa),

Dr. Vikas Gupta (CDCOE, Panniwala Mota), Prof. Harish Mahendru

(Dean , HEC Jagadhari) for their blessings and guidance.

I am also thankful to my friends Er. Abhishek Gupta, Prof. Ravi Kumar Malik

(HOD, ECE). Er. Anish Shandilya, Er. Garima Anand, Er. Jitender Chatra

(AP, MED), Er. Vinit Saluja (AP, MED) for checking the proof of this text and

valuable suggestions during the preparation of this book.

I acknowledge the best support from my father Sh. Satbir Singh Narwal, My

mother Smt. Birmati Narwal, my uncle Sh. Shyam Kumar Narwal, my aunt

Smt. Shakuntala Narwal, Smt. Nirmala Devi, my younger brothers

Narender, Anil, Sunil,Vinay and my sisters Anju and Shilky Devi.

Special thanks to Ms. Kanchan Bakshi (Head Librarian), Mr. Rohit

(Asst. Librarian) for providing me number of books from the Institute Library. I am

also thankful to my students

Mr. Sarvesh Dutt, Mr. Simranjeet Singh, Mr. Vivek Pundir, Mr. Harmeet,

Mr. Kartik for their support during the preparation of this text.

Contents ix

I wish to thank Mr. Hitesh Kumar (Scientific Solutions, Ambala) for providing

me some instruments for my practical study.

Thank to Mr. Ayaz Khan, Editorial team member of VAYU EDUCATION OF

INDIA for his technical support. At last but not the least, Dr. R. K. Jain, the Publisher

and Managing Director, deserve special thanks for his untiring effort in bringing out

this book.

Thanks are due to those Author's, whose books and articles have been referred.

— J K NARWAL

My sense of gratitude are due to all contributors, their valuable suggestions help

me to prepare this edition in a simple manner. I am highly grateful to all specially,

Prof. Atul Aggarwal- SSIET , Dera Bassi

Prof. Shanti Prakash- HEC , Jagadhari

Prof. Arpan Manchanda-Rawal Institute of Engineering and Technology, Faridabad

Prof. Rajesh Dudi- Om Group of Colleges, Hisar

Prof. Tarun Anand- JCDMCOE, Sirsa

Prof. Rajesh Kaushik-Doon Valley Group of Institutions, Karnal

Prof. I.C. Ramola- ACE, Mithapur Ambala

Er. Kuldeep Goyat- GITM, Gurgaon

Er. Dharmender- Northern India Engineering College, New Delhi

Er. Ankur Sharma- HMR Institute of Technology and Management, New Delhi

Er. Pardeep Raman- HMR Institute of Technology and Management, New Delhi

Er. Rohtash Sharma- Narayani Group of Institutions, Karnal

Er. Neeraj Kumar– M.M. Group of Institutions, Karnal

Er. Suneev Bansal- RIMT, Chandigarh

Er. Amit Bhatia- PPIMT, Hisar

Er. Virender Chahal-NCCE, Israna

Er. Nirmal Garg- NCIT, Israna

Er. Mohit - SGI, Samalkha

Er. Baljeet Singh-E-Max College, Ambala

Er. Hemant Rajput- JECRC, Jaipur

Er. Devi Ram- DRCET, Panipat

Er. Rahul Kumar- ACE, Mithapur Ambala

Er. Ashwani Kaushik- NIT Kurukshetra

Er. Saurabh Mittal- NCCE, Israna

Er. Ajay Kumar( Neetu)- Maneta Automotives Ltd., Sonepat

Er. Hardeep Singh- Baddi University, Baddi

Er. Sandeep Sheoran- Universal Institute of Technology, Hansi

Er. Amit Kaushik- Geeta Engineering College, Panipat

Er. Akash Sharma- M.M. UNIVERSITY, Mullana

— J K NARWAL

ContributorsContributorsContributorsContributors

Preface to the First EditionPreface to the First EditionPreface to the First EditionPreface to the First Edition

This book is written primarily for the undergraduate students of Mechanical

Engineering. Vibration is the study of oscillatory motions. The ultimate goals of this

study are to determine the effects of vibration on the performance and safety of

systems and to control its effects.

This study has be come a part of most engineering curricula. The course contents

of the book are according to the prescribed syllabus of Kurukshetra University Kurukshetra,

Maharishi Dayanand University, Rohtak, Guru Jambheshwar University, Hisar. It is

observed that the students face difficulties in understanding clearly the mathematical

treatment of this subject. This book meets all the basic requirements of the students.

The present book is an effort in the direction of presenting the concepts of Mechanical

Vibrations and its applications in clean, concise and systematic manner at one

place. Presentation is made in very simple and easily understandable language and

well supported with wide range of illustrations, solved numerical problems and unsolved

numerical problems.

I hope that the students and teachers referring this book will find it useful.

— J K NARWAL

ContentsContentsContentsContents

1. FUNDAMENTAL OF VIBRATIONS ........................................... 1-71

1.1 Introduction ..............................................................................................1

1.2 Causes of Vibrations .................................................................................1

1.3 Effects of Vibrations ................................................................................2

1.4 Benefits of Vibrations ...............................................................................2

1.5 Elimination of Undesirable Vibrations .......................................................2

1.6 Elements of a Vibrating System ...............................................................3

1.7 Types of Vibrations ..................................................................................4

1.8 Various Definitions ...................................................................................7

1.9 Method of Vibration Analysis ................................................................. 11

1.10 Simple Harmonic Motion and Periodic Motion ......................................13

1.11 Analytical methods of Addition of Harmonic .............................................

motions ...................................................................................................15

1.12 Graphical Method of Addition of two Harmonic Motions .....................16

1.13 Sinusoidal method ..................................................................................28

1.14 Representation of Harmonic Motion in Complex Number .....................29

1.15 Vectorial Representation of Harmonic Motions ......................................34

1.15.1 Average Amplitude .................................................................36

1.15.2 Root Mean Square Amplitude ................................................ 37

1.16 Beats Phenomenon ................................................................................43

1.17 Work Done by a Harmonic Force on a Harmonic Motion ...................46

1.18 Harmonic Series .....................................................................................53

Exercise for Practice ............................................................................. 69

2. UNDAMPED FREE VIBRATIONS ........................................ 72-194

2.1 Introduction ............................................................................................72

2.2 Newton’s Method...................................................................................72

2.3 Response to an Initial Disturbance .........................................................81

2.4 D’Alembert’s Principle ...........................................................................86

2.5 Energy Method ..................................................................................... 101

2.6 Rayleigh’s Method ................................................................................ 118

2.7 Displacement, Velocity and Acceleration .............................................. 119

Mechanical Vibrationsxiv

2.9 Inertia Effect of the Mass of the Spring on the Natural Frequency of the

System..................................................................................................130

2.10 Free Torsional Vibrations (Single Rotor) ..............................................131

2.11 Inertia Effect of Mass of Shaft ............................................................132

2.12 Equivalent Stiffness of Spring Combinations ......................................143

2.13 Simple Pendulum ..................................................................................164

2.14 Compound Pendulum ........................................................................... 165

2.15 Floating and Immersed Bodies .............................................................168

2.15.1 Buoyancy .............................................................................168

2.15.2 Archimede’s Principle ..........................................................169

2.16 Multifilar System ..................................................................................178

2.16.1 Bifilar Supension .................................................................. 178

2.16.2 Trifilar Suspension ...............................................................180

Exercise for Practice ........................................................................... 184

3. DAMPED VIBRATIONS ................................................. 195-253

3.1 Introduction ..........................................................................................195

3.2 Single Degree of Freedom System with Viscous Damper or Free Vibration

with Viscous Damping .........................................................................195

3.3 Logarithmic Decrement .......................................................................212

3.4 Types of Damping ................................................................................236

3.4.1 Viscous Damping .................................................................236

3.4.2 Coulomb Damping ...............................................................239

3.4.3 Structural Damping .............................................................. 248

3.4.4 Slip or Interfacial Damping ..................................................249

3.5 Comparison Between Viscous Damping and Coulomb Damping ........ 250


4. FORCED VIBRATION .................................................. 254-346

4.1 Introduction ..........................................................................................254

4.2 Equation of Motion with Harmonic Force ...........................................255

4.2.1 Total Response .....................................................................260

4.2.2 Characteristics Curves ......................................................... 261

4.2.2 a. Frequency Response Curve .............................................261

4.2.2. b. Phase Frequency Response Curve ..................................262

4.2.3 Variation of Frequency Ratio ................................................263

4.3 Sharpness of Resonance (Quality Factor and Bandwidth) ..................265

4.4 Forced Vibration Without Damping .....................................................278

4.5 Excitation due to Unbalance .................................................................283

4.5.1 Rotating Unbalance .............................................................. 284

4.5.2 Reciprocating Unbalance ......................................................289

4.6 Harmonic Motion of Support ...............................................................293

4.6.1 Absolute Motion ...................................................................294

4.6.2 Relative Motion .................................................................... 297

Contents xv

4.7 Energy Dissipated by Damping ............................................................305

4.8 Forced Vibrations with Coulomb Damping .........................................307

4.9 Forced Vibration with Structural Damping..........................................309

4.10 Forced Vibrations with Coulomb and Viscous Damping .....................310

4.11 Vibration Isolation ................................................................................312

4.12 Transmissibility ....................................................................................313

4.12.1 Force Transmissibility..........................................................313

4.12.2 Motion Transmissibility........................................................317

4.13 Vibration Measuring Instruments .........................................................334

4.13.1 Vibrometer ............................................................................334

4.13.2 Accelerometer ...................................................................... 335

4.13.3 Frequency Measuring Devices ............................................336


5. TWO-DEGREES OF FREEDOM SYSTEM ............................. 347-426

5.1 Introduction ..........................................................................................347

5.2 Undamped Free Vibrations and Principal Modes of Vibrations ............347

5.3 Free Vibration in Terms of Initial Conditions ...................................... 375

5.4 Semi-Definite System........................................................................... 377

5.4.1 Two Masses Connected by a Spring ................................... 377

5.5 Forced Harmonic Vibration ..................................................................381

5.6 Damped Free Vibrations in Two Degrees of Freedom System ...........382

5.7 Forced Virbations of Undamped System ............................................. 385

5.8 Forced Vibrations with Damping .........................................................387

5.9 Coordinate Coupling and Principal Coordinates ...................................391

5.9.1 Static Coupling .....................................................................393

5.9.2 Dynamic Coupling ...............................................................394

5.9.3 Static and Dynamic Coupling .............................................. 395

5.10 Vibration Absorber ................................................................................401

5.11 Centrifugal Pendulum Vibration Absorber ............................................ 406

5.12 Torsional Vibration Absorber ................................................................408

5.13 Untuned Vibration Dampers .................................................................408

5.13.1 Lanchester Damper ..............................................................408

5.13.2 Houdaille Damper ................................................................. 410

5.14 Lagrange’ Equations and Generalized Coordinates ..............................417

5.15 Orthogonality Principle .........................................................................422


6. MULTI DEGREES OF FREEDOM SYSTEM .......................... 427-570

6.1 Introduction ..........................................................................................427

6.2 Free Vibration : Equations of Motion for Multidegree of Freedom

System..................................................................................................427

6.3 Influence Coefficients ..........................................................................439

6.3.1 Influence Coefficient for Rectilinear System....................... 441

Mechanical Vibrationsxvi

6.4 Dunkerley's Method ............................................................................448

6.5 Rayleigh's Method ................................................................................465

6.5.1 Beam Carrying Several Point Loads .....................................465

6.5.2 Beam Carrying Distributed Loads ........................................466

6.6 Rayleigh-Ritz Method ...........................................................................482

6.7 Holzer Method ......................................................................................486

6.8 Matrix Method ...................................................................................... 507

6.9 Orthogonality Principle .........................................................................517

6.10 Matrix Interation Method .....................................................................518

6.11 Stodola’s Method .................................................................................546

6.12 Generalized Coordinates .......................................................................559


7. TORSIONAL VIBRATIONS ............................................. 571-593

7.1 Introduction ..........................................................................................571

7.2 Analogy Between Rectilinear and Torsional Vibrations ........................571

7.3 Undamped Free Vibrations ...................................................................572

7.4 Torsionally EquIvalent Shaft ................................................................573

7.5 Fixed Shaft with a Rotor ......................................................................577

7.6 Torsional Vibrations of Two-rotor System ..........................................577

7.7 Torsional Vibrations of Three rotors System .......................................581

7.8 Vibrations of Geared Systems ..............................................................584

7.9 Torsional Vibrations of Branched Geared System .............................. 586


8. WHIRLING OF SHAFTS ............................................... 594-631

8.1 Introduction ..........................................................................................594

8.2 Critical Speed of A Light-Flexible Shaft Carrying Single Rotor with

Damping ...............................................................................................605

8.3 Critical Speeds of A Shaft Carrying Multiple Discs ...................................

(Without Damping) ...............................................................................621

8.4 Secondary Critical Speed .....................................................................627


9. CONTINUOUS SYSTEMS ............................................... 632-683

9.1 Introduction ..........................................................................................632

9.2 Vibration of Strings ..............................................................................632

9.2.1 Solution of Wave Equation ...................................................634

9.3 Longitudinal Vibrations of Bars ............................................................641

9.4 Torsional Vibration of Circular Shafts ..................................................660

9.5 Transverse Vibration of Beams ............................................................668


Contents xvii

10. TRANSIENT VIBRATIONS.............................................. 684-703

10.1 Introduction .......................................................................................... 684

10.2 Laplace Transformation .......................................................................684

10.3 Transforms of Particular Functions ....................................................686

10.4 Impulse Excitation ................................................................................ 688

10.5 Arbitary Excitation ...............................................................................690

10.6 Base Excitation ..................................................................................... 691

10.7 Response Spectrum .............................................................................692

10.8 Runge-Kutta Method ............................................................................ 693

10.9 Phase-Plane Method .............................................................................695

Exercise for Practice ...........................................................................701

Appendices ................................................................................... 705-725

References ............................................................................................727

Index ............................................................................................ 729-731

1.1 INTRODUCTION

A body is said to vibrate if it has to and fro motion. Any motion that repeats itself after

an equal interval of time is called vibration or oscillation. Mechanical vibration is the

oscillation of structural or machine components rapidly and continuously under the

excitation of forces which may be inherent in the component or applied externally.

Usually vibrations are due to elastic forces. The vibration system involves the transfer

of potential energy to kinetic energy and kinetic energy to the potential energy, 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.

“Vibration”, a term that tends to induce the same in once’s mind when someone

hears it. Invariably everything vibrates in the world, some vibrations are good and

useful, some are really negligible, some are tolerable, some annoying and the rest fall

under the dangerous category.

A vibrating guitar tends to produce soothing music for our ears, a vibrating sieve

separates particles of fine sand and coarse sand, the vibrating mode in our mobile

phone to alert us without disturbing others.

Well some examples of unwanted things vibration does to us, an old running engine

vibrating a lot creating a lot of noise unbearable to us, the most dreaded vibration of all

the “Earthquake” a random vibration of the earth.

These small vital examples tend to give a glance of the importance that needs to be

given to this field.

1.2 CAUSES OF VIBRATIONS

Following are the causes of Vibrations:

• Unbalanced forces in the machine because of faulty design and poor manufacture.

• External excitations

Fundamental of

Vibrations

1

Mechanical Vibrations By J.K. Narwal

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