J. K. NARWAL B.Tech (Gold-Medalist), M.Tech(ME) Assistant Professor Depatment of Mechanical Engineering Geeta Engineering College, Panipat

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2 MECHANICALVIBRATIONS J. K. NARWAL B.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 India 2/25, Ansari Road, Darya Ganj, New Delhi

3 Mechanical Vibrations Copyright J.K. NARWAL ISBN: 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 INDIA 2/25, Ansari Road, Darya Ganj, New Delhi Ph.: , Fax: vei@veiindia.com,vayueducation1@gmail.com Web:

4 Er. Ankush Bansal FOREWORD 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

5 Preface 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.

6 viii Mechanical Vibrations 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.

7 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

8 Contributors 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

9 Preface 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

10 Contents 1. FUNDAMENTAL OF VIBRATIONS Introduction Causes of Vibrations Effects of Vibrations Benefits of Vibrations Elimination of Undesirable Vibrations Elements of a Vibrating System Types of Vibrations Various Definitions Method of Vibration Analysis Simple Harmonic Motion and Periodic Motion Analytical methods of Addition of Harmonic... motions Graphical Method of Addition of two Harmonic Motions Sinusoidal method Representation of Harmonic Motion in Complex Number Vectorial Representation of Harmonic Motions Average Amplitude Root Mean Square Amplitude Beats Phenomenon Work Done by a Harmonic Force on a Harmonic Motion Harmonic Series...53 Exercise for Practice UNDAMPED FREE VIBRATIONS Introduction Newton s Method Response to an Initial Disturbance D Alembert s Principle Energy Method Rayleigh s Method Displacement, Velocity and Acceleration

11 xiv Mechanical Vibrations 2.9 Inertia Effect of the Mass of the Spring on the Natural Frequency of the System Free Torsional Vibrations (Single Rotor) Inertia Effect of Mass of Shaft Equivalent Stiffness of Spring Combinations Simple Pendulum Compound Pendulum Floating and Immersed Bodies Buoyancy Archimede s Principle Multifilar System Bifilar Supension Trifilar Suspension Exercise for Practice DAMPED VIBRATIONS Introduction Single Degree of Freedom System with Viscous Damper or Free Vibration with Viscous Damping Logarithmic Decrement Types of Damping Viscous Damping Coulomb Damping Structural Damping Slip or Interfacial Damping Comparison Between Viscous Damping and Coulomb Damping Exercise for Practice FORCED VIBRATION Introduction Equation of Motion with Harmonic Force Total Response Characteristics Curves a. Frequency Response Curve b. Phase Frequency Response Curve Variation of Frequency Ratio Sharpness of Resonance (Quality Factor and Bandwidth) Forced Vibration Without Damping Excitation due to Unbalance Rotating Unbalance Reciprocating Unbalance Harmonic Motion of Support Absolute Motion Relative Motion...297

12 Contents xv 4.7 Energy Dissipated by Damping Forced Vibrations with Coulomb Damping Forced Vibration with Structural Damping Forced Vibrations with Coulomb and Viscous Damping Vibration Isolation Transmissibility Force Transmissibility Motion Transmissibility Vibration Measuring Instruments Vibrometer Accelerometer Frequency Measuring Devices Exercise for Practice TWO-DEGREES OF FREEDOM SYSTEM Introduction Undamped Free Vibrations and Principal Modes of Vibrations Free Vibration in Terms of Initial Conditions Semi-Definite System Two Masses Connected by a Spring Forced Harmonic Vibration Damped Free Vibrations in Two Degrees of Freedom System Forced Virbations of Undamped System Forced Vibrations with Damping Coordinate Coupling and Principal Coordinates Static Coupling Dynamic Coupling Static and Dynamic Coupling Vibration Absorber Centrifugal Pendulum Vibration Absorber Torsional Vibration Absorber Untuned Vibration Dampers Lanchester Damper Houdaille Damper Lagrange Equations and Generalized Coordinates Orthogonality Principle Exercise for Practice MULTI DEGREES OF FREEDOM SYSTEM Introduction Free Vibration : Equations of Motion for Multidegree of Freedom System Influence Coefficients Influence Coefficient for Rectilinear System...441

13 xvi Mechanical Vibrations 6.4 Dunkerley's Method Rayleigh's Method Beam Carrying Several Point Loads Beam Carrying Distributed Loads Rayleigh-Ritz Method Holzer Method Matrix Method Orthogonality Principle Matrix Interation Method Stodola s Method Generalized Coordinates Exercise for Practice TORSIONAL VIBRATIONS Introduction Analogy Between Rectilinear and Torsional Vibrations Undamped Free Vibrations Torsionally EquIvalent Shaft Fixed Shaft with a Rotor Torsional Vibrations of Two-rotor System Torsional Vibrations of Three rotors System Vibrations of Geared Systems Torsional Vibrations of Branched Geared System Exercise for Practice WHIRLING OF SHAFTS Introduction Critical Speed of A Light-Flexible Shaft Carrying Single Rotor with Damping Critical Speeds of A Shaft Carrying Multiple Discs... (Without Damping) Secondary Critical Speed Exercise for Practice CONTINUOUS SYSTEMS Introduction Vibration of Strings Solution of Wave Equation Longitudinal Vibrations of Bars Torsional Vibration of Circular Shafts Transverse Vibration of Beams Exercise for Practice

14 Contents xvii 10. TRANSIENT VIBRATIONS Introduction Laplace Transformation Transforms of Particular Functions Impulse Excitation Arbitary Excitation Base Excitation Response Spectrum Runge-Kutta Method Phase-Plane Method Exercise for Practice Appendices References Index

15 1 Fundamental of Vibrations 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

16 Mechanical Vibrations By J.K. Narwal Publisher : Vayu Education ISBN : Author : J.K. Narwal Type the URL : Get this ebook