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- 1. de Silva, Clarence W. “Frontmatter”Vibration: Fundamentals and PracticeClarence W. de SilvaBoca Raton: CRC Press LLC, 2000
- 2. VIBRATIONFundamentals and Practice Clarence W. de Silva CRC Press Boca Raton London New York Washington, D.C.
- 3. Library of Congress Cataloging-in-Publication DataDe Silva, Clarence W. Vibration : fundamentals and practice / Clarence W. de Silva p. cm. Includes bibliographical references and index. ISBN 0-8493-1808-4 (alk. paper) 1. Vibration. I. Title. TA355.D384 1999 620.3—dc21 99-16238 CIP This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted withpermission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publishreliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materialsor for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical,including photocopying, microﬁlming, and recording, or by any information storage or retrieval system, without priorpermission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating newworks, or for resale. Speciﬁc permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 Corporate Blvd., N.W., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are only used foridentiﬁcation and explanation, without intent to infringe.Cover art is the U.S. Space Shuttle and the International Space Station. (Courtesy of NASA Langley ResearchCenter, Hampton, VA. With permission.)© 2000 by CRC Press LLCNo claim to original U.S. Government worksInternational Standard Book Number 0-8493-1808-4Library of Congress Card Number 99-16238Printed in the United States of America 1 2 3 4 5 6 7 8 9 0Printed on acid-free paper©2000 CRC Press
- 4. PrefaceThis book provides the background and techniques that will allow successful modeling, analysis,monitoring, testing, design, modiﬁcation, and control of vibration in engineering systems. It issuitable as both a course textbook for students and instructors, and a practical reference tool forengineers and other professionals. As a textbook, it can be used in a single-semester course forthird-year (junior) and fourth-year (senior) undergraduate students, or for Master’s level graduatestudents in any branch of engineering such as aeronautical and aerospace, civil, mechanical, andmanufacturing engineering. But, in view of the practical considerations, design issues, experimentaltechniques, and instrumentation that are presented throughout the book, and in view of the simpliﬁedand snapshot-style presentation of fundamentals and advanced theory, the book will also serve asa valuable reference tool for engineers, technicians, and other professionals in industry and inresearch laboratories. The book is an outgrowth of the author’s experience in teaching undergraduate and graduatecourses in Dynamics, Mechanical Vibration, Dynamic System Modeling, Instrumentation andDesign, Feedback Control, Modern Control Engineering, and Modal Analysis and Testing in theU.S. and Canada (Carnegie Mellon University and the University of British Columbia) for morethan 20 years. The industrial experience and training that he received in product testing andqualiﬁcation, analysis, design, and vibration instrumentation at places like Westinghouse ElectricCorporation in Pittsburgh, IBM Corporation in Boca Raton, NASA’s Langley and Lewis ResearchCenters, and Bruel and Kjaer in Denmark enabled the author to provide a realistic and practicaltreatment of the subject. Design for vibration and control of vibration are crucial in maintaining a high performancelevel and production efﬁciency, and prolonging the useful life of machinery, structures, and indus-trial processes. Before designing or controlling an engineering system for good vibratory perfor-mance, it is important to understand, represent (i.e., model), and analyze the vibratory characteristicsof the system. Suppression or elimination of undesirable vibrations and generation of requiredforms and levels of desired vibrations are general goals of vibration engineering. In recent years,researchers and practitioners have devoted considerable effort to studying and controlling vibrationin a range of applications in various branches of engineering. With this book, designers, engineers,and students can reap the beneﬁts of that study and experience, and learn the observation, instru-mentation, modeling, analysis, design, modiﬁcation, and control techniques that produce mechan-ical and aeronautical systems, civil engineering structures, and manufacturing processes that areoptimized against the effects of vibration. The book provides the background and techniques that will allow successful modeling, analysis,design, modiﬁcation, testing, and control of vibration in engineering systems. This knowledge willbe useful in the practice of vibration, regardless of the application area or the branch of engineering.A uniform and coherent treatment of the subject is presented, by introducing practical applicationsof vibration, through examples, in the very beginning of the book, along with experimental tech-niques and instrumentation, and then integrating these applications, design, and control consider-ations into fundamentals and analytical methods throughout the text. To maintain clarity and focusand to maximize the usefulness of the book, an attempt has been made to describe and illustrateindustry-standard and state-of-the-art instrumentation, hardware, and computational techniquesrelated to the practice of vibration. As its main features, the book:©2000 CRC Press
- 5. • Introduces practical applications, design, and experimental techniques in the very begin- ning, and then uniformly integrates them throughout the book • Provides 36 “Summary Boxes” that present key material covered in the book, in point form, within each chapter, for easy reference and recollection (these items are particularly suitable for use by instructors in their presentations) • Outlines mathematics, dynamics, modeling, fast Fourier transform (FFT) techniques, and reliability analysis in appendices • Provides over 60 worked examples and case studies, and over 300 problems • Will be accompanied by an Instructor’s Manual, for instructors, that contains complete solutions to all the end-of-chapter problems • Describes sensors, transducers, ﬁlters, ampliﬁers, analyzers, and other instrumentation that is useful in the practice of vibration • Describes industry-standard computer techniques, hardware, and tools for analysis, design, and control of vibratory systems, with examples • Provides a comprehensive coverage of vibration testing and qualiﬁcation of products • Offers analogies of mechanical and structural vibration, to other oscillatory behavior such as in electrical and ﬂuid systems, and contrasts these with thermal systems.A NOTE TO INSTRUCTORSThe book is suitable as the text for a standard undergraduate course in Mechanical Vibration or fora specialized course for ﬁnal-year undergraduate students and Master’s level graduate students.Three typical course syllabuses are outlined below.A. A Standard Undergraduate CourseAs the textbook for an undergraduate (3rd year or 4th year) course in Mechanical Vibration, it maybe incorporated into the following syllabus for a 12 week course consisting of 36 hours of lecturesand 12 hours of laboratory experiments:LecturesChapter 1 (1 hour)Sections 8.1, 8.2, 8.4, 9.1, 9.2, 9.8 (3 hours)Chapter 2 (6 hours)Chapter 3 (6 hours)Section 11.4 (2 hours)Chapter 5 (6 hours)Chapter 6 (6 hours)Sections 12.1, 12.2, 12.3, 12.4, 12.5 (6 hours)Laboratory Experiments The following four laboratory experiments, each of 3-hour duration, may be incorporated. 1. Experiment on modal testing (hammer test and other transient tests) and damping mea- surement in the time domain (see Section 11.4) 2. Experiment on shaker testing and damping measurement in the frequency domain (see Section 11.4) 3. Experiment on single-plane and two-plane balancing (see Section 12.3) 4. Experiment on modal testing of a distributed-parameter system (see Section 11.4)©2000 CRC Press
- 6. B. A Course in Industrial VibrationChapter 1 (1 hour)Chapter 4 (3 hours)Chapter 7 (5 hours)Chapter 8 (5 hours)Chapter 9 (4 hours)Chapter 10 (6 hours)Chapter 11 (6 hours)Chapter 12 (6 hours)A project may be included in place of a ﬁnal examination.C. A Course in Modal Analysis and TestingChapter 1 (1 hour)Chapter 4 (3 hours)Chapter 5 (6 hours)Chapter 6 (6 hours)Chapter 7 (5 hours)Chapter 10 (6 hours)Chapter 11 (6 hours)Section 12.6 (hours)A project may be included in place of a ﬁnal examination. Clarence W. de Silva Vancouver, Canada©2000 CRC Press
- 7. The AuthorClarence W. de Silva, Fellow ASME and Fellow IEEE, is Professor of Mechanical Engineeringat the University of British Columbia, Vancouver, Canada, and has occupied the NSERC Chair inIndustrial Automation since 1988. He obtained his ﬁrst Ph.D. from the Massachusetts Institute ofTechnology in 1978 and, 20 years later, another Ph.D. from the University of Cambridge, England.De Silva has served as a consultant to several companies, including IBM and Westinghouse in theU.S., and has led the development of many industrial machines. He is recipient of the EducationAward of the Dynamic Systems and Control Division of the American Society of MechanicalEngineers, the Meritorious Achievement Award of the Association of Professional Engineers ofBritish Columbia, the Outstanding Contribution Award of the IEEE Systems, Man, and CyberneticsSociety, the Outstanding Large Chapter Award of the IEEE Industry Applications Society, and theOutstanding Chapter Award from the IEEE Control Systems Society. He has authored 14 technicalbooks, 10 edited volumes, over 120 journal papers, and a similar number of conference papers andbook chapters. He has served on the editorial boards of 12 international journals, and is the Editor-in-Chief of the International Journal of Knowledge-Based Intelligent Engineering Systems, SeniorTechnical Editor of Measurements and Control, and Regional Editor, North America, of the Inter-national Journal of Intelligent Real-Time Automation. He has been a Lilly Fellow, Senior FulbrightFellow to Cambridge University, ASI Fellow, and a Killam Fellow.©2000 CRC Press
- 8. AcknowledgmentPreparation of this book would not have been possible if not for the support of many individualsand organizations, but it is not possible to list all of them here. I wish to recognize the followingspeciﬁc contributions:Financial assistance for my research and professional activities has been provided primarily by: • Ministry of Advanced Education, Training and Technology, Province of British Columbia, for the Network of Centres of Excellence Program • Natural Sciences and Engineering Research Council of Canada (NSERC) • Network of Centres of Excellence (Institute of Robotics and Intelligent Systems) • Advanced Systems Institute of British Columbia • Science Council of British Columbia • Ministry of Environment of British Columbia • British Columbia Hydro and Power Authority • National Research Council • Killam Memorial Faculty Fellowship Program • B.C. Packers, Ltd. • Neptune Dynamics, Ltd. • Garﬁeld Weston Foundation Special acknowledgment should be made here of the Infrastructure Grant from the Ministry ofAdvanced Education, Training and Technology, Province of British Columbia, which made part ofthe secretarial support for my work possible. The Department of Mechanical Engineering at theUniversity of British Columbia provided me with an excellent environment to carry out myeducational activities, including the preparation of this book. My graduate students, researchassociates, teaching assistants, and ofﬁce staff have contributed directly and indirectly to the successof the book. Particular mention should be made of the following people: • Ricky Min-Fan Lee for systems assistance • Hassan Bayoumi and Jay Choi for graphics assistance • Yuan Chen, Scott Gu, Iwan Kurnianto, Farag Omar, and Roya Rahbari for teaching assistance • Marje Lewis and Laura Gawronski for secretarial assistance. I wish to thank the staff of CRC Press, particularly the Associate Editor, Felicia Shapiro and theProject Editor, Sara Seltzer, for their ﬁne effort in the production of the book. Encouragement ofvarious authorities in the ﬁeld of engineering — particularly, Professor Devendra Garg of DukeUniversity, Professor Mo Jamshidi of the University of New Mexico, and Professor Arthur Murphy(DuPont Fellow Emeritus) — is gratefully acknowledged. Finally, my family deserves an apologyfor the unintentional “neglect” that they may have suffered during the latter stages of production ofthe book.©2000 CRC Press
- 9. Source CreditsThe sources of several photos, ﬁgures, and tables are recognized and given credit, as follows: • Figure 1.1: Courtesy of Ms. Kimberly Land, NASA Langley Research Center, Hampton, Virginia. • Figure 1.3: Courtesy of Professor Carlos E. Ventura, Department of Civil Engineering, the University of British Columbia, Vancouver, Canada. • Figure 1.4: Courtesy of Ms. Heather Conn of BC Transit, Vancouver, Canada. Photo by Mark Van Manen. • Figure 1.5: Courtesy of Ms. Jeana Dugger, Key Technologies, Inc., Walla Walla, Wash- ington. • Figure 1.8: Courtesy of Mechanical Engineering magazine, from article “Semiactive Cone Suspension Smooths the Ride” by Bill Siuru, Vol. 116, No. 3, page 106. Copyright, American Society of Mechanical Engineers International, New York. • Table 7.5: Reprinted from ASME BPVC, Section III-Division 1, Appendices, by permis- sion of The American Society of Mechanical Engineers, New York. All rights reserved. • Figure 8.8: Courtesy of Bruel & Kjaer, Naerum, Denmark. • Figure 9.36 and Figure 9.38: Courtesy of Ms. Beth Daniels. Copyright 1999 Tektronix, Inc. All rights reserved. Reproduced by permission. • Figure 11.6, Figure 11.8, and Figure 12.15: Experimental setups used by the author for teaching a fourth-year course in the Undergraduate Vibrations Laboratory, Department of Mechanical Engineering, the University of British Columbia, Vancouver, Canada. • Figure 12.34, Figure 12.35, and Table 12.2: Courtesy of Dr. George Wang. Extracted from the report “Active Control of Vibration in Wood Machining for Wood Recovery” by G. Wang, J. Xi, Q. Zhong, S. Abayakoon, K. Krishnappa, and F. Lam, National Research Council, Integrated Manufacturing Technologies Institute, Vancouver, Canada, pp. 5, 8, 25-28, May 1998.©2000 CRC Press
- 10. Dedication Professor David N. Wormley. “For the things we have to learn before we can do them, we learn by doing them.” — Aristotle (Author of Mechanics and Acoustics, 384–322 B.C.)©2000 CRC Press
- 11. Table of ContentsChapter 1 Vibration Engineering1.1 Study of Vibration1.2 Application Areas1.3 History of Vibration1.4 Organization of the BookProblemsReferences and Further Reading Author’s Work Other Useful PublicationsChapter 2 Time Response2.1 Undamped Oscillator 2.1.1 Energy Storage Elements Inertia Spring Gravitational Potential Energy 2.1.2 Conservation of Energy System 1 (Translatory) System 2 (Rotatory) System 3 (Flexural) System 4 (Swinging) System 5 (Liquid Slosh) System 6 (Electrical) Capacitor Inductor 2.1.3 Free Response Example 2.1 Solution2.2 Heavy Springs 2.2.1 Kinetic Energy Equivalence Example 2.2 Solution2.3 Oscillations in Fluid Systems Example 2.3 Solution2.4 Damped Simple Oscillator 2.4.1 Case 1: Underdamped Motion Initial Conditions 2.4.2 Logarithmic Decrement Method 2.4.3 Case 2: Overdamped Motion 2.4.4 Case 3: Critically Damped Motion 2.4.5 Justiﬁcation for the Trial Solution©2000 CRC Press
- 12. First-Order System Second-Order System Repeated Roots 2.4.6 Stability and Speed of Response Example 2.4 Solution2.5 Forced Response 2.5.1 Impulse Response Function 2.5.2 Forced Response 2.5.3 Response to a Support Motion Impulse Response The Riddle of Zero Initial Conditions Step Response Liebnitz’s RuleProblemsChapter 3 Frequency Response3.1 Response to Harmonic Excitations 3.1.1 Response Characteristics Case 1 Case 2 Case 3 Particular Solution (Method 1) Particular Solution (Method 2): Complex Function Method Resonance 3.1.2 Measurement of Damping Ratio (Q-Factor Method) Example 3.1 Solution3.2 Transform Techniques 3.2.1 Transfer Function 3.2.2 Frequency-Response Function (Frequency-Transfer Function) Impulse Response Case 1 (ζ < 1) Case 2 (ζ > 1) Case 3 (ζ = 1) Step Response 3.2.3 Transfer Function Matrix Example 3.2 Example 3.3 Example 3.4 Solution3.3 Mechanical Impedance Approach Mass Element Spring Element Damper Element 3.3.1 Interconnection Laws Example 3.5 Example 3.63.4 Transmissibility Functions 3.4.1 Force Transmissibility©2000 CRC Press
- 13. 3.4.2 Motion Transmissibility System Suspended on a Rigid Base (Force Transmissibility) System with Support Motion (Motion Transmissibility) 3.4.3 General Case Example 3.7 3.4.4 Peak Values of Frequency-Response Functions3.5 Receptance Method 3.5.1 Application of Receptance Undamped Simple Oscillator Dynamic AbsorberProblemsChapter 4 Vibration Signal Analysis4.1 Frequency Spectrum 4.1.1 Frequency 4.1.2 Amplitude Spectrum 4.1.3 Phase Angle 4.1.4 Phasor Representation of Harmonic Signals 4.1.5 RMS Amplitude Spectrum 4.1.6 One-Sided and Two-Sided Spectra 4.1.7 Complex Spectrum4.2 Signal Types4.3 Fourier Analysis 4.3.1 Fourier Integral Transform (FIT) 4.3.2 Fourier Series Expansion (FSE) 4.3.3 Discrete Fourier Transform (DFT) 4.3.4 Aliasing Distortion Sampling Theorem Aliasing Distortion in the Time Domain Anti-Aliasing Filter Example 4.1 4.3.5 Another Illustration of Aliasing Example 4.24.4 Analysis of Random Signals 4.4.1 Ergodic Random Signals 4.4.2 Correlation and Spectral Density 4.4.3 Frequency Response Using Digital Fourier Transform 4.4.4 Leakage (Truncation Error) 4.4.5 Coherence 4.4.6 Parseval’s Theorem 4.4.7 Window Functions 4.4.8 Spectral Approach to Process Monitoring 4.4.9 Cepstrum4.5 Other Topics of Signal Analysis 4.5.1 Bandwidth 4.5.2 Transmission Level of a Bandpass Filter 4.5.3 Effective Noise Bandwidth 4.5.4 Half-Power (or 3 dB) Bandwidth 4.5.5 Fourier Analysis Bandwidth©2000 CRC Press
- 14. 4.6 Resolution in Digital Fourier Results4.7 Overlapped Processing Example 4.3 4.7.1 Order Analysis Speed Spectral Map Time Spectral Map Order TrackingProblemsChapter 5 Modal Analysis5.1 Degrees of Freedom and Independent Coordinates 5.1.1 Nonholonomic Constraints Example 5.1 Example 5.25.2 System Representation 5.2.1 Stiffness and Flexibility Matrices 5.2.2 Inertia Matrix 5.2.3 Direct Approach for Equations of Motion5.3 Modal Vibrations Example 5.35.4 Orthogonality of Natural Modes 5.4.1 Modal Mass and Normalized Modal Vectors5.5 Static Modes and Rigid Body Modes 5.5.1 Static Modes 5.5.2 Linear Independence of Modal Vectors 5.5.3 Modal Stiffness and Normalized Modal Vectors 5.5.4 Rigid Body Modes Example 5.4 Equation of Heave Motion Equation of Pitch Motion Example 5.5 First Mode (Rigid Body Mode) Second Mode 5.5.5 Modal Matrix 5.5.6 Conﬁguration Space and State Space State Vector5.6 Other Modal Formulations 5.6.1 Non-Symmetric Modal Formulation 5.6.2 Transformed Symmetric Modal Formulation Example 5.6 Approach 2 Approach 35.7 Forced Vibration Example 5.7 First Mode (Rigid Body Mode) Second Mode (Oscillatory Mode)5.8 Damped Systems 5.8.1 Proportional Damping Example 5.8©2000 CRC Press
- 15. 5.9 State-Space Approach 5.9.1 Modal Analysis 5.9.2 Mode Shapes of Nonoscillatory Systems 5.9.3 Mode Shapes of Oscillatory Systems Example 5.9ProblemsChapter 6 Distributed-Parameter Systems6.1 Transverse Vibration of Cables 6.1.1 Wave Equation 6.1.2 General (Modal) Solution 6.1.3 Cable with Fixed Ends 6.1.4 Orthogonality of Natural Modes Example 6.1 Solution 6.1.5 Application of Initial Conditions Example 6.2 Solution6.2 Longitudinal Vibration of Rods 6.2.1 Equation of Motion 6.2.2 Boundary Conditions Example 6.3 Solution6.3 Torsional Vibration of Shafts 6.3.1 Shaft with Circular Cross Section 6.3.2 Torsional Vibration of Noncircular Shafts Example 6.4 Solution Example 6.5 Solution6.4 Flexural Vibration of Beams 6.4.1 Governing Equation for Thin Beams Moment-Deﬂection Relation Rotatory Dynamics (Equilibrium) Transverse Dynamics 6.4.2 Modal Analysis 6.4.3 Boundary Conditions 6.4.4 Free Vibration of a Simply Supported Beam Normalization of Mode Shape Functions Initial Conditions 6.4.5 Orthogonality of Mode Shapes Case of Variable Cross Section 6.4.6 Forced Bending Vibration Example 6.6 Solution Example 6.7 Solution 6.4.7 Bending Vibration of Beams with Axial Loads 6.4.8 Bending Vibration of Thick Beams©2000 CRC Press
- 16. 6.4.9 Use of the Energy Approach 6.4.10 Orthogonality with Inertial Boundary Conditions Rotatory Inertia6.5 Damped Continuous Systems 6.5.1 Modal Analysis of Damped Beams Example 6.8 Solution6.6 Vibration of Membranes and Plates 6.6.1 Transverse Vibration of Membranes 6.6.2 Rectangular Membrane with Fixed Edges 6.6.3 Transverse Vibration of Thin Plates 6.6.4 Rectangular Plate with Simply Supported EdgesProblemsChapter 7 Damping7.1 Types of Damping 7.1.1 Material (Internal) Damping Viscoelastic Damping Hysteretic Damping Example 7.1 Solution 7.1.2 Structural Damping 7.1.3 Fluid Damping Example 7.2 Solution7.2 Representation of Damping in Vibration Analysis 7.2.1 Equivalent Viscous Damping 7.2.2 Complex Stiffness Example 7.3 Solution 7.2.3 Loss Factor7.3 Measurement of Damping 7.3.1 Logarithmic Decrement Method 7.3.2 Step-Response Method 7.3.3 Hysteresis Loop Method Example 7.4 Solution 7.3.4 Magniﬁcation-Factor Method 7.3.5 Bandwidth Method 7.3.6 General Remarks7.4 Interface Damping Example 7.5 Solution 7.4.1 Friction In Rotational Interfaces 7.4.2 InstabilityProblems©2000 CRC Press
- 17. Chapter 8 Vibration Instrumentation8.1 Vibration Exciters 8.1.1 Shaker Selection Force Rating Power Rating Stroke Rating Example 8.1 Solution Hydraulic Shakers Inertial Shakers Electromagnetic Shakers 8.1.2 Dynamics of Electromagnetic Shakers Transient Exciters8.2 Control System 8.2.1 Components of a Shaker Controller Compressor Equalizer (Spectrum Shaper) Tracking Filter Excitation Controller (Amplitude Servo-Monitor) 8.2.2 Signal-Generating Equipment Oscillators Random Signal Generators Tape Players Data Processing8.3 Performance Speciﬁcation 8.3.1 Parameters for Performance Speciﬁcation Time-Domain Speciﬁcations Frequency-Domain Speciﬁcations 8.3.2 Linearity 8.3.3 Instrument Ratings Rating Parameters 8.3.4 Accuracy and Precision8.4 Motion Sensors and Transducers 8.4.1 Potentiometer Potentiometer Resolution Optical Potentiometer 8.4.2 Variable-Inductance Transducers Mutual-Induction Transducers Linear-Variable Differential Transformer (LVDT) Signal Conditioning Example 8.2 Solution 8.4.3 Mutual-Induction Proximity Sensor 8.4.4 Self-Induction Transducers 8.4.5 Permanent-Magnet Transducers 8.4.6 AC Permanent-Magnet Tachometer 8.4.7 AC Induction Tachometer 8.4.8 Eddy Current Transducers 8.4.9 Variable-Capacitance Transducers Capacitive Displacement Sensors©2000 CRC Press
- 18. Capacitive Angular Velocity Sensor Capacitance Bridge Circuit 8.4.10 Piezoelectric Transducers Sensitivity Example 8.3 Solution Piezoelectric Accelerometer Charge Ampliﬁer8.5 Torque, Force, and Other Sensors 8.5.1 Strain-Gage Sensors Equations for Strain-Gage Measurements Bridge Sensitivity The Bridge Constant Example 8.4 Solution The Calibration Constant Example 8.5 Solution Data Acquisition Accuracy Considerations Semiconductor Strain Gages Force and Torque Sensors Strain-Gage Torque Sensors Deﬂection Torque Sensors Variable-Reluctance Torque Sensor Reaction Torque Sensors 8.5.2 Miscellaneous Sensors Stroboscope Fiber-Optic Sensors and Lasers Fiber-Optic Gyroscope Laser Doppler Interferometer Ultrasonic Sensors Gyroscopic Sensors8.6 Component Interconnection 8.6.1 Impedance Characteristics Cascade Connection of Devices Impedance-Matching Ampliﬁers Operational Ampliﬁers Voltage Followers Charge Ampliﬁers 8.6.2 Instrumentation Ampliﬁer Ground Loop NoiseProblemsChapter 9 Signal Conditioning and Modiﬁcation9.1 Ampliﬁers 9.1.1 Operational Ampliﬁer Example 9.1 Solution©2000 CRC Press
- 19. 9.1.2 Use of Feedback in Op-amps 9.1.3 Voltage, Current, and Power Ampliﬁers 9.1.4 Instrumentation Ampliﬁers Differential Ampliﬁer Common Mode Ampliﬁer Performance Ratings Example 9.2 Solution Common-Mode Rejection Ratio (CMRR) AC-Coupled Ampliﬁers9.2 Analog Filters 9.2.1 Passive Filters and Active Filters Number of Poles 9.2.2 Low-Pass Filters Example 9.3 Solution Low-Pass Butterworth Filter Example 9.4 Solution 9.2.3 High-Pass Filters 9.2.4 Bandpass Filters Resonance-Type Bandpass Filters Example 9.5 Solution 9.2.5 Band-Reject Filters9.3 Modulators and Demodulators 9.3.1 Amplitude Modulation Modulation Theorem Side Frequencies and Side Bands 9.3.2 Application of Amplitude Modulation Fault Detection and Diagnosis 9.3.3 Demodulation9.4 Analog/Digital Conversion 9.4.1 Digital-to-Analog Conversion (DAC) Weighted-Resistor DAC Ladder DAC DAC Error Sources 9.4.2 Analog-to-Digital Conversion (ADC) Successive-Approximation ADC Dual-Slope ADC Counter ADC 9.4.3 ADC Performance Characteristics Resolution and Quantization Error Monotonicity, Nonlinearity, and Offset Error ADC Conversion Rate 9.4.4 Sample-and-Hold (S/H) Circuitry 9.4.5 Multiplexers (MUX) Analog Multiplexers Digital Multiplexers 9.4.6 Digital Filters©2000 CRC Press
- 20. Software Implementation and Hardware Implementation9.5 BridgeCircuits 9.5.1 Wheatstone Bridge 9.5.2 Constant-Current Bridge 9.5.3 Bridge Ampliﬁers Half-Bridge Circuits 9.5.4 Impedance Bridges Owen Bridge Wien-Bridge Oscillator9.6 Linearizing Devices 9.6.1 Linearization by Software 9.6.2 Linearization by Hardware Logic 9.6.3 Analog Linearizing Circuitry 9.6.4 Offsetting Circuitry 9.6.5 Proportional-Output Circuitry Curve-Shaping Circuitry9.7 Miscellaneous Signal-Modiﬁcation Circuitry 9.7.1 Phase Shifter 9.7.2 Voltage-to-Frequency Converter (VFC) 9.7.3 Frequency-to-Voltage Converter (FVC) 9.7.4 Voltage-to-Current Converter (VCC) 9.7.5 Peak-Hold Circuit9.8 Signal Analyzers and Display Devices 9.8.1 Signal Analyzers 9.8.2 Oscilloscopes Triggering Lissajous Patterns Digital OscilloscopesProblemsChapter 10 Vibration Testing10.1 Representation of a Vibration Environment 10.1.1 Test Signals Stochastic versus Deterministic Signals 10.1.2 Deterministic Signal Representation Single-Frequency Signals Sine Sweep Sine Dwell Decaying Sine Sine Beat Sine Beat with Pauses Multifrequency Signals Actual Excitation Records Simulated Excitation Signals 10.1.3 Stochastic Signal Representation Ergodic Random Signals Stationary Random Signals Independent and Uncorrelated Signals Transmission of Random Excitations©2000 CRC Press
- 21. 10.1.4 Frequency-Domain Representations Fourier Spectrum Method Power Spectral Density Method 10.1.5 Response Spectrum Displacement, Velocity, and Acceleration Spectra Response-Spectra Plotting Paper Zero-Period Acceleration Uses of Response Spectra 10.1.6 Comparison of Various Representations10.2 Pretest Procedures 10.2.1 Purpose of Testing 10.2.2 Service Functions Active Equipment Passive Equipment Functional Testing 10.2.3 Information Acquisition Interface Details Effect of Neglecting Interface Dynamics Effects of Damping Effects of Inertia Effect of Natural Frequency Effect of Excitation Frequency Other Effects of Interface 10.2.4 Test-Program Planning Testing of Cabinet-Mounted Equipment 10.2.5 Pretest Inspection10.3 Testing Procedures 10.3.1 Resonance Search 10.3.2 Methods of Determining Frequency-Response Functions Fourier Transform Method Spectral Density Method Harmonic Excitation Method 10.3.3 Resonance-Search Test Methods Hammer (Bump) Test and Drop Test Pluck Test Shaker Tests 10.3.4 Mechanical Aging Equivalence for Mechanical Aging Excitation-Intensity Equivalence Dynamic-Excitation Equivalence Cumulative Damage Theory 10.3.5 TRS Generation 10.3.6 Instrument Calibration 10.3.7 Test-Object Mounting 10.3.8 Test-Input Considerations Test Nomenclature Testing with Uncorrelated Excitations Symmetrical Rectilinear Testing Geometry versus Dynamics Some Limitations Testing of Black Boxes©2000 CRC Press
- 22. Phasing of Excitations Testing a Gray or White Box Overtesting in Multitest Sequences10.4 Product Qualiﬁcation Testing 10.4.1 Distribution Qualiﬁcation Drive-Signal Generation Distribution Spectra Test Procedures 10.4.2 Seismic Qualiﬁcation Stages of Seismic Qualiﬁcation 10.4.3 Test Preliminaries Single-Frequency Testing Multifrequency Testing 10.4.4 Generation of RRS SpeciﬁcationsProblemsChapter 11 Experimental Modal Analysis11.1 Frequency-Domain Formulation 11.1.1 Transfer Function Matrix 11.1.2 Principle of Reciprocity Example 11.111.2 Experimental Model Development 11.2.1 Extraction of the Time-Domain Model11.3 Curve-Fitting of Transfer Functions 11.3.1 Problem Identiﬁcation 11.3.2 Single-Degree-of-Freedom and Multi-Degree-of-Freedom Techniques 11.3.3 Single-Degree-of-Freedom Parameter Extraction in the Frequency Domain Circle-Fit Method Peak Picking Method 11.3.4 Multi-Degree-of-Freedom Curve Fitting Formulation of the Method 11.3.5 A Comment on Static Modes and Rigid Body Modes 11.3.6 Residue Extraction11.4 Laboratory Experiments 11.4.1 Lumped-Parameter System Frequency-Domain Test Time-Domain Test 11.4.2 Distributed-Parameter System11.5 Commercial EMA Systems 11.5.1 System Conﬁguration FFT Analysis Options Modal Analysis ComponentsProblemsChapter 12 Vibration Design and Control Shock and Vibration12.1 Speciﬁcation of Vibration Limits 12.1.1 Peak Level Speciﬁcation 12.1.2 RMS Value Speciﬁcation©2000 CRC Press
- 23. 12.1.3 Frequency-Domain Speciﬁcation12.2 Vibration Isolation Example 12.1 Solution 12.2.1 Design Considerations Example 12.2 Solution 12.2.2 Vibration Isolation of Flexible Systems12.3 Balancing of Rotating Machinery 12.3.1 Static Balancing Balancing Approach 12.3.2 Complex Number/Vector Approach Example 12.3 Solution 12.3.3 Dynamic (Two-Plane) Balancing Example 12.4 Solution 12.3.4 Experimental Procedure of Balancing12.4 Balancing of Reciprocating Machines 12.4.1 Single-Cylinder Engine 12.4.2 Balancing the Inertia Load of the Piston 12.4.3 Multicylinder Engines Two-Cylinder Engine Six-Cylinder Engine Example 12.5 Solution 12.4.4 Combustion/Pressure Load12.5 Whirling of Shafts 12.5.1 Equations of Motion 12.5.2 Steady-State Whirling Example 12.6 Solution 12.5.3 Self-Excited Vibrations12.6 Design Through Modal Testing 12.6.1 Component Modiﬁcation Example 12.7 Solution 12.6.2 Substructuring12.7 Passive Control of Vibration 12.7.1 Undamped Vibration Absorber Example 12.8 Solution 12.7.2 Damped Vibration Absorber Optimal Absorber Design Example 12.9 Solution 12.7.3 Vibration Dampers12.8 Active Control of Vibration 12.8.1 Active Control System 12.8.2 Control Techniques State-Space Models©2000 CRC Press
- 24. Example 12.10 Solution Position and Velocity Feedback Linear Quadratic Regulator (LQR) Control Modal Control 12.8.3 Active Control of Saw Blade Vibration12.9 Control of Beam Vibrations 12.9.1 State-Space Model of Beam Dynamics 12.9.2 Control Problem 12.9.3 Use of Linear Dampers Design ExampleProblemsAppendix A Dynamic Models and AnalogiesA.1 Model DevelopmentA.2 AnalogiesA.3 Mechanical Elements A.3.1 Mass (Inertia) Element A.3.2 Spring (Stiffness) ElementA.4 Electrical Elements A.4.1 Capacitor Element A.4.2 Inductor ElementA.5 Thermal Elements A.5.1 Thermal Capacitor A.5.2 Thermal ResistanceA.6 Fluid Elements A.6.1 Fluid Capacitor A.6.2 Fluid Inertor A.6.3 Fluid Resistance A.6.4 Natural OscillationsA.7 State-Space Models A.7.1 Linearization A.7.2 Time Response A.7.3 Some Formal Deﬁnitions A.7.4 Illustrative Example A.7.5 Causality and Physical RealizabilityAppendix B Newtonian and Lagrangian MechanicsB.1 Vector Kinematics B.1.1 Euler’s Theorem Important Corollary Proof B.1.2 Angular Velocity and Velocity at a Point of a Rigid Body Theorem Proof B.1.3 Rates of Unit Vectors Along Axes of Rotating Frames General Result Cartesian Coordinates Polar Coordinates (2-D)©2000 CRC Press
- 25. Spherical Polar Coordinates Tangential-Normal (Intrinsive) Coordinates (2-D) B.1.4 Acceleration Expressed in Rotating Frames Spherical Polar Coordinates Tangential-Normal Coordinates (2-D)B.2 Newtonian (Vector) Mechanics B.2.1 Frames of Reference Rotating at Angular Velocity ω B.2.2 Newton’s Second Law for a Particle of Mass m B.2.3 Second Law for a System of Particles — Rigidly or Flexibly Connected B.2.4 Rigid Body Dynamics — Inertia Matrix and Angular Momentum B.2.5 Manipulation of Inertia Matrix Parallel Axis Theorem — Translational Transformation of [I] Rotational Transformation of [I] Principal Directions (Eigenvalue Problem) Mohr’s Circle B.2.6 Euler’s Equations (for a Rigid Body Rotating at ω ) B.2.7 Euler’s AnglesB.3 Lagrangian Mechanics B.3.1 Kinetic Energy and Kinetic Coenergy B.3.2 Work and Potential Energy Examples B.3.3 Holonomic Systems, Generalized Coordinates, and Degrees of Freedom B.3.4 Hamilton’s Principle B.3.5 Lagrange’s Equations Example Generalized Coordinates Generalized Nonconservative Forces Lagrangian Lagrange’s EquationsAppendix C Review of Linear AlgebraC.1 Vectors and MatricesC.2 Vector-Matrix Algebra C.2.1 Matrix Addition and Subtraction C.2.2 Null Matrix C.2.3 Matrix Multiplication C.2.4 Identity MatrixC.3 Matrix Inverse C.3.1 Matrix Transpose C.3.2 Trace of a Matrix C.3.3 Determinant of a Matrix C.3.4 Adjoint of a Matrix C.3.5 Inverse of a MatrixC.4 Vector Spaces C.4.1 Field (Ᏺ) C.4.2 Vector Space (ᏸ) Properties Special Case C.4.3 Subspace of ᏸ©2000 CRC Press
- 26. C.4.4Linear Dependence C.4.5Basis and Dimension of a Vector Space C.4.6Inner Product C.4.7Norm Properties C.4.8 Gram-Schmidt Orthogonalization C.4.9 Modiﬁed Gram-Schmidt ProcedureC.5 Determinants C.5.1 Properties of Determinant of a Matrix C.5.2 Rank of a MatrixC.6 System of Linear EquationsReferencesAppendix D Digital Fourier Analysis and FFTD.1 Uniﬁcation of the Three Fourier Transform Types D.1.1 Relationship Between DFT and FIT D.1.2 Relationship Between DFT and FSED.2 Fast Fourier Transform (FFT) D.2.1 Development of the Radix-Two FFT Algorithm D.2.2 The Radix-Two FFT Procedure D.2.3 Illustrative ExampleD.3 Discrete Correlation and Convolution D.3.1 Discrete Correlation Discrete Correlation Theorem Discrete Convolution TheoremD.4 Digital Fourier Analysis Procedures D.4.1 Fourier Transform Using DFT D.4.2 Inverse DFT Using DFT D.4.3 Simultaneous DFT of Two Real Data Records D.4.4 Reduction of Computation Time for a Real Data Record D.4.5 Convolution of Finite Duration Signals Using DFT Wraparound Error Data-Record Sectioning in ConvolutionAppendix E Reliability Considerations for Multicomponent UnitsE.1 Failure Analysis E.1.1 Reliability E.1.2 Unreliability E.1.3 Inclusion–Exclusion Formula ExampleE.2 Bayes’ Theorem E.2.1 Product Rule for Independent Events E.2.2 Failure Rate E.2.3 Product Rule for ReliabilityAnswers to Numerical Problems©2000 CRC Press
- 27. de Silva, Clarence W. “Vibration Engineering”Vibration: Fundamentals and PracticeClarence W. de SilvaBoca Raton: CRC Press LLC, 2000
- 28. 1 Vibration EngineeringVibration is a repetitive, periodic, or oscillatory response of a mechanical system. The rate of thevibration cycles is termed “frequency.” Repetitive motions that are somewhat clean and regular,and that occur at relatively low frequencies, are commonly called oscillations, while any repetitivemotion, even at high frequencies, with low amplitudes, and having irregular and random behaviorfalls into the general class of vibration. Nevertheless, the terms “vibration” and “oscillation” areoften used interchangeably, as is done in this book. Vibrations can naturally occur in an engineering system and may be representative of its freeand natural dynamic behavior. Also, vibrations may be forced onto a system through some formof excitation. The excitation forces may be either generated internally within the dynamic system,or transmitted to the system through an external source. When the frequency of the forcing excitationcoincides with that of the natural motion, the system will respond more vigorously with increasedamplitude. This condition is known as resonance, and the associated frequency is called the resonantfrequency. There are “good vibrations,” which serve a useful purpose. Also, there are “bad vibra-tions,” which can be unpleasant or harmful. For many engineering systems, operation at resonancewould be undesirable and could be destructive. Suppression or elimination of bad vibrations andgeneration of desired forms and levels of good vibration are general goals of vibration engineering. This book deals with 1. Analysis 2. Observation 3. Modiﬁcationof vibration in engineering systems. Applications of vibration are found in many branches ofengineering such as aeronautical and aerospace, civil, manufacturing, mechanical, and even elec-trical. Usually, an analytical or computer model is needed to analyze the vibration in an engineeringsystem. Models are also useful in the process of design and development of an engineering systemfor good performance with respect to vibrations. Vibration monitoring, testing, and experimentationare important as well in the design, implementation, maintenance, and repair of engineering systems.All these are important topics of study in the ﬁeld of vibration engineering, and the book will coverpertinent 1. Theory and modeling 2. Analysis 3. Design 4. Experimentation 5. Control In particular, practical applications and design considerations related to modifying the vibrationalbehavior of mechanical devices and structures will be studied. This knowledge will be useful in thepractice of vibration regardless of the application area or the branch of engineering; for example, inthe analysis, design, construction, operation, and maintenance of complex structures such as theSpace Shuttle and the International Space Station. Note in Figure 1.1 that long and ﬂexible compo-nents, which would be prone to complex “modes” of vibration, are present. The structural designshould take this into consideration. Also, functional and servicing devices such as robotic manipu-©2000 CRC Press
- 29. FIGURE 1.1 The U.S. Space Shuttle and the International Space Station with the Canadarm. (Courtesy ofNASA Langley Research Center, Hampton, VA. With permission.)lators (e.g., Canadarm) can give rise to vibration interactions that need to be controlled for accurateperformance. The approach used in the book is to introduce practical applications of vibration inthe very beginning, along with experimental techniques, and then integrate these applications anddesign considerations into fundamentals and analytical methods throughout the text.1.1 STUDY OF VIBRATIONNatural, free vibration is a manifestation of the oscillatory behavior in mechanical systems, as aresult of repetitive interchange of kinetic and potential energies among components in the system.Such natural oscillatory response is not limited, however, to purely mechanical systems, and is foundin electrical and ﬂuid systems as well, again due to a repetitive exchange of two types of energyamong system components. But, purely thermal systems do not undergo free, natural oscillations,primarily because of the absence of two forms of reversible energy. Even a system that can holdtwo reversible forms of energy may not necessarily display free, natural oscillations. The reason forthis would be the strong presence of an energy dissipation mechanism that could use up the initialenergy of the system before completing a single oscillation cycle (energy interchange). Such dissi-pation is provided by damping or friction in mechanical systems, and resistance in electrical systems.Any engineering system (even a purely thermal one) is able to undergo forced oscillations, regardlessof the degree of energy dissipation. In this case, the energy necessary to sustain the oscillations willcome from the excitation source, and will be continuously replenished. Proper design and control are crucial in maintaining a high performance level and productionefﬁciency, and prolonging the useful life of machinery, structures, and industrial processes. Beforedesigning or controlling an engineering system for good vibratory performance, it is important tounderstand, represent (model), and analyze the vibratory characteristics of the system. This can be©2000 CRC Press
- 30. FIGURE 1.2(a) An elevated guideway transit system.accomplished through purely analytical means, computer analysis of analytical models, testing andanalysis of test data, or a combination of these approaches. As an example, a schematic diagramof an innovative elevated guideway transit system is shown in Figure 1.2(a). This is an automatedtransit system that is operated without drivers. The ride quality, which depends on the vibratorymotion of the vehicle, can be analyzed using an appropriate model. Usually, the dynamics (inertia,ﬂexibility, and energy dissipation) of the guideway, as well as the vehicle, must be incorporatedinto such a model. A simpliﬁed model is shown in Figure 1.2(b). It follows that modeling, analysis,testing, design, and control are all important aspects of study in mechanical vibration. The analysis of a vibrating system can be done either in the time domain or in the frequencydomain. In the time domain, the independent variable of a vibration signal is time. In this case,the system itself can be modeled as a set of differential equations with respect to time. A modelof a vibrating system can be formulated by applying either force-momentum rate relations (New-ton’s second law) or the concepts of kinetic and potential energies. Both Newtonian (force-motion)and Lagrangian (energy) approaches will be utilized in this book. In the frequency domain, the independent variable of a vibration signal is frequency. In thiscase, the system can be modeled by input-output transfer functions which are algebraic, rather thandifferential, models. Transfer function representations such as mechanical impedance, mobility,receptance, and transmissibility can be conveniently analyzed in the frequency domain, and effec-tively used in vibration design and evaluation. Modeling and vibration-signal analysis in both timeand frequency domains will be studied in this book. The two domains are connected by the Fouriertransformation, which can be treated as a special case of the Laplace transformation. Thesetransform techniques will be studied, ﬁrst in the purely analytical and analog measurement situationof continuous time. In practice, however, digital electronics and computers are commonly used insignal analysis, sensing, and control. In this situation, one needs to employ concepts of discretetime, sampled data, and digital signal analysis in the time domain. Correspondingly, then, conceptsof discrete or digital Fourier transformation and techniques of fast Fourier transform (FFT) will beapplicable in the frequency domain. These concepts and techniques are also studied in this book. An engineering system, when given an initial disturbance and allowed to execute free vibrationswithout a subsequent forcing excitation, will tend to do so at a particular “preferred” frequencyand maintaining a particular “preferred” geometric shape. This frequency is termed a “naturalfrequency” of the system, and the corresponding shape (or motion ratio) of the moving parts ofthe system is termed a “mode shape.” Any arbitrary motion of a vibrating system can be representedin terms of its natural frequencies and mode shapes. The subject of modal analysis primarilyconcerns determination of natural frequencies and mode shapes of a dynamic system. Once the©2000 CRC Press
- 31. FIGURE 1.2(b) A model for determining the ride quality of the elevated guideway transit system.modes are determined, they can be used in understanding the dynamic nature of the systems, andalso in design and control. Modal analysis is extremely important in vibration engineering, andwill be studied in this book. Natural frequencies and mode shapes of a vibrating system can bedetermined experimentally through procedures of modal testing. In fact, a dynamic model(an experimental model) of the system can be determined in this manner. The subject of modaltesting, experimental modeling (or model identiﬁcation), and associated analysis and design isknown as experimental modal analysis. This subject will also be treated in this book. Energy dissipation (or damping) is present in any mechanical system. It alters the dynamicresponse of the system, and has desirable effects such as stability, vibration suppression, powertransmission (e.g., in friction drives), and control. Also, it has obvious undesirable effects such asenergy wastage, reduction of the process efﬁciency, wear and tear, noise, and heat generation. For©2000 CRC Press
- 32. these reasons, damping is an important topic of study in the area of vibration, and will be coveredin this book. In general, energy dissipation is a nonlinear phenomenon. But, in view of well-knowndifﬁculties of analyzing nonlinear behavior, and because an equivalent representation of the overallenergy dissipation is often adequate in vibration analysis, linear models are primarily used torepresent damping in the analyses herein. However, nonlinear representations are discussed as well;and how equivalent linear models can be determined for nonlinear damping are described. Properties such as mass (inertia), ﬂexibility (spring-like effect), and damping (energy dissipa-tion) are continuously distributed throughout practical mechanical devices and structures to a largeextent. This is the case with distributed components such as cables, shafts, beams, membranes,plates, shells, and various solids, as well as structures made of such components. Representation(i.e., modeling) of these distributed-parameter (or continuous) vibrating systems will require inde-pendent variables in space (spatial coordinates) in addition to time; these models are partialdifferential equations in time and space. The analysis of distributed-parameter models will requirecomplex procedures and special tools. This book studies vibration analysis, particularly modalanalysis, of several types of continuous components, as well as how approximate lumped-parametermodels can be developed for continuous systems, using procedures such as modal analysis andenergy equivalence. Vibration testing is useful in a variety of stages in the development and utilization of a product.In the design and development stage, vibration testing can be used to design, develop, and verifythe performance of individual components of a complex system before the overall system is built(assembled) and evaluated. In the production stage, vibration testing can be used for screening ofselected batches of products for quality control. Another use of vibration testing is in productqualiﬁcation. Here, a product of good quality is tested to see whether it can withstand variousdynamic environments that it may encounter in a specialized application. An example of a large-scale shaker used for vibration testing of civil engineering structures is shown in Figure 1.3. Thesubject of vibration testing is addressed in some detail in this book. Design is a subject of paramount signiﬁcance in the practice of vibration. In particular, mechan-ical and structural design for acceptable vibration characteristics will be important. Modiﬁcationof existing components and integration of new components and devices, such as vibration dampers,isolators, inertia blocks, and dynamic absorbers, can be incorporated into these practices. Further-more, elimination of sources of vibration — for example, through component alignment andbalancing of rotating devices — is a common practice. Both passive and active techniques are usedin vibration control. In passive control, actuators that require external power sources are notemployed. In active control, vibration is controlled by means of actuators (which need power) tocounteract vibration forces. Monitoring, testing, and control of vibration will require devices suchas sensors and transducers, signal conditioning and modiﬁcation hardware (e.g., ﬁlters, ampliﬁers,modulators, demodulators, analog-digital conversion means), and actuators (e.g., vibration excitersor shakers). The underlying subject of vibration instrumentation will be covered in this book.Particularly, within the topic of signal conditioning, both hardware and software (numerical)techniques will be presented.1.2 APPLICATION AREASThe science and engineering of vibration involve two broad categories of applications: 1. Elimination or suppression of undesirable vibrations 2. Generation of the necessary forms and quantities of useful vibrationsUndesirable and harmful types of vibration include structural motions generated due to earthquakes,dynamic interactions between vehicles and bridges or guideways, noise generated by constructionequipment, vibration transmitted from machinery to its supporting structures or environment, and©2000 CRC Press
- 33. FIGURE 1.3 A multi-degree-of-freedom hydraulic shaker used in testing civil engineering structures.(Courtesy of Prof. C.E. Ventura, University of British Columbia. With permission.)damage, malfunction, and failure due to dynamic loading, unacceptable motions, and fatigue causedby vibration. As an example, dynamic interactions between an automated transit vehicle and abridge (see Figure 1.4) can cause structural problems as well as degradation in ride quality. Rigorousanalysis and design are needed, particularly with regard to vibration, in the development of theseground transit systems. Lowering the levels of vibration will result in reduced noise and improvedwork environment, maintenance of a high performance level and production efﬁciency, reductionin user/operator discomfort, and prolonging the useful life of industrial machinery. Desirable typesof vibration include those generated by musical instruments, devices used in physical therapy andmedical applications, vibrators used in industrial mixers, part feeders and sorters, and vibratorymaterial removers such as drills and polishers (ﬁnishers). For example, product alignment for©2000 CRC Press
- 34. FIGURE 1.4 The SkyTrain in Vancouver, Canada, a modern automated transit system. (Photo by Mark VanManen, courtesy of BC Transit. With permission.)FIGURE 1.5 An alignment shaker. (Key Technology, Inc., of Walla Walla, WA. With permission.)©2000 CRC Press
- 35. industrial processing or grading can be carried out by means of vibratory conveyors or shakers, asshown in Figure 1.5. Concepts of vibration have been used for many centuries in practical applications. Recentadvances of vibration are quite signiﬁcant, and the corresponding applications are numerous. Manyof the recent developments in the ﬁeld of vibration were motivated perhaps for two primary reasons: 1. The speeds of operation of machinery have doubled over the past 50 years and, conse- quently, the vibration loads generated due to rotational excitations and unbalances would have quadrupled if proper actions of design and control were not taken. 2. Mass, energy, and efﬁciency considerations have resulted in lightweight, optimal designs of machinery and structures consisting of thin members with high strength. Associated structural ﬂexibility has made the rigid-structure assumption unsatisfactory, and given rise to the need for sophisticated procedures of analysis and design that govern distrib- uted-parameter ﬂexible structures.One can then visualize several practical applications where modeling, analysis, design, control,monitoring, and testing, related to vibration are important. A range of applications of vibration can be found in various branches of engineering: partic-ularly civil, mechanical, aeronautical and aerospace, and production and manufacturing. Modalanalysis and design of ﬂexible civil engineering structures such as bridges, guideways, tall buildings,and chimneys directly incorporate theory and practice of vibration. A ﬁne example of an elongatedbuilding where vibration analysis and design are crucial is the Jefferson Memorial Arch, shown inFigure 1.6. In the area of ground transportation, vehicles are designed by incorporating vibration engineer-ing, not only to ensure structural integrity and functional operability, but also to achieve requiredlevels of ride quality and comfort. Speciﬁcations such as the one shown in Figure 1.7, where limitson root-mean-square (rms) levels of vibration (expressed in units of acceleration due to gravity, g)for different frequencies of excitation (expressed in cycles per second, or hertz, or Hz) and differenttrip durations, are used to specify ride quality requirements in the design of transit systems. Inparticular, the design of suspension systems, both active and passive, falls within the ﬁeld ofvibration engineering. Figure 1.8 shows a test setup used in the development of an automotivesuspension system. In the area of air transportation, mechanical and structural components ofaircraft are designed for good vibration performance. For example, proper design and balancingcan reduce helicopter vibrations caused by imbalance in their rotors. Vibrations in ships can besuppressed through structural design, propeller and rudder design, and control. Balancing of internalcombustion engines is carried out using principles of design for vibration suppression. Oscillation of transmission lines of electric power and communication signals (e.g., overheadtelephone lines) can result in faults, service interruptions, and sometimes major structural damage.Stabilization of transmission lines involves direct application of the principles of vibration in cablesand the design of vibration dampers and absorbers. In the area of production and manufacturing engineering, mechanical vibration has directimplications of product quality and process efﬁciency. Machine tool vibrations are known to notonly degrade the dimensional accuracy and the ﬁnish of a product, but also will cause fast wearand tear and breakage of tools. Milling machines, lathes, drills, forging machines, and extruders,for example, should be designed for achieving low vibration levels. In addition to reducing the toollife, vibration will result in other mechanical problems in production machinery, and will requiremore frequent maintenance. Associated downtime (production loss) and cost can be quite signiﬁcant.Also, as noted before, vibrations in production machinery will generate noise problems and alsowill be transmitted to other operations through support structures, thereby interfering with theirperformance as well. In general, vibration can degrade performance and production efﬁciency of©2000 CRC Press
- 36. FIGURE 1.6 Jefferson Memorial Arch in St. Louis, MO.FIGURE 1.7 A typical specification of vehicle ride quality for a specified trip duration.©2000 CRC Press
- 37. FIGURE 1.8 Cone suspension system installed on a Volvo 480ES automobile for testing. (Copyright Mechan-ical Engineering magazine; the American Society of Mechanical Engineers International. With permission.)manufacturing processes. Proper vibration isolation (e.g., mountings) will be needed to reduce thesetransmissibility problems. Heavy machinery in the construction industry (e.g., cranes, excavators, pile drivers, impactingand compacting machinery, and bulldozers) rely on structural integrity, reliability, and safety. Theirdesign must be based on sound principles of engineering. Although the dynamic loading in thesemachines is generally random, it is also quite repetitive from the point of view of both the excitationgenerated by the engine and the functional operation of the tasks performed. Design based onvibration and fatigue is an important requirement for these machines: for maintaining satisfactoryperformance, prolonging the useful life, and reducing the cost and frequency of maintenance.1.3 HISTORY OF VIBRATIONThe origins of the theory of vibration can be traced back to the design and development of musicalinstruments (good vibration). It is known that drums, ﬂutes, and stringed instruments existed inChina and India for several millennia B.C. Also, ancient Egyptians and Greeks explored sound andvibration from both practical and analytical points of view. For example, while Egyptians had knownof a harp since at least 3000 B.C., the Greek philosopher, mathematician, and musician Pythagoras(of the Pythagoras theorem fame) who lived during 582 to 502 B.C., experimented on soundsgenerated by blacksmiths and related them to music and physics. The Chinese developed a mechan-ical seismograph (an instrument to detect and record earthquake vibrations) in the 2nd century A.D. The foundation of the modern-day theory of vibration was probably laid by scientists andmathematicians such as Robert Hooke (1635–1703) of the Hooke’s law fame, who experimentedon the vibration of strings; Sir Isaac Newton (1642–1727), who gave us calculus and the laws ofmotion for analyzing vibrations; Daniel Bernoulli (1700–1782) and Leonard Euler (1707–1783),who studied beam vibrations (Bernoulli-Euler beam) and also explored dynamics and ﬂuid mechan-ics; Joseph Lagrange (1736–1813), who studied vibration of strings and also explored the energyapproach to formulating equations of dynamics; Charles Coulomb (1736–1806), who studied©2000 CRC Press
- 38. torsional vibrations and friction; Joseph Fourier (1768–1830), who developed the theory of fre-quency analysis of signals; and Simeon-Dennis Poisson (1781–1840), who analyzed vibration ofmembranes and also analyzed elasticity (Poisson’s ratio). As a result of the industrial revolutionand associated developments of steam turbines and other rotating machinery, an urgent need wasfelt for developments in the analysis, design, measurement, and control of vibration. Motivationfor many aspects of the existing techniques of vibration can be traced back to related activitiessince the industrial revolution. Much credit should go to scientists and engineers of more recent history, as well. Among thenotable contributors are Rankine (1820–1872), who studied critical speeds of shafts; Kirchhoff(1824–1887), who analyzed vibration of plates; Rayleigh (1842–1919), who made contributions tothe theory of sound and vibration and developed computational techniques for determining naturalvibrations; de Laval (1845–1913), who studied the balancing problem of rotating disks; Poincaré(1854–1912), who analyzed nonlinear vibrations; and Stodola (1859–1943), who studied vibrationsof rotors, bearings, and continuous systems. Distinguished engineers who made signiﬁcant contri-butions to the published literature and also to the practice of vibration include Timoshenko, DenHartog, Clough, and Crandall.1.4 ORGANIZATION OF THE BOOKThis book provides the background and techniques for modeling, analysis, design, instrumentationand monitoring, modiﬁcation, and control of vibration in engineering systems. This knowledge willbe useful in the practice of vibration, regardless of the application area or the branch of engineering.A uniform and coherent treatment of the subject is given by introducing practical applications ofvibration in the very beginning of the book, along with experimental techniques and instrumentation,and then integrating these applications, design and experimental techniques, and control consider-ations into fundamentals and analytical methods throughout the text. The book consists of 12 chapters and 5 appendices. The chapters have summary boxes for easyreference and recollection. Many worked examples and problems (over 300) are included. Somebackground material is presented in the appendices, rather than in the main text, in order to avoidinterference with the continuity of the subject matter. The present introductory chapter provides some background material on the subject of vibrationengineering, and sets the course for the study. It gives the objectives and motivation of the studyand indicates key application areas. A brief history of the ﬁeld of vibration is given as well. Chapter 2 provides the basics of time response analysis of vibrating systems. Both undampedand damped systems are studied. Also, analysis of both free (unforced) and forced response isgiven. The concept of a state variable is introduced. Some analogies of purely mechanical andstructural vibrating systems — speciﬁcally, translatory, ﬂexural, and torsional; to electrical andﬂuid oscillatory systems — are introduced. An energy-based approximation of a distributed-parameter system (a heavy spring) to a lumped-parameter system is developed in detail. Thelogarithmic decrement method of damping measurement is developed. Although the chapter pri-marily considers single-degree-of-freedom systems, the underlying concepts can be easily extendedto multi-degree-of-freedom systems. Chapter 3 concerns frequency response analysis of vibrating systems. First, the response of avibrating system to harmonic (sinusoidal) excitation forces (inputs) is analyzed, primarily using thetime-domain concepts developed in Chapter 2. Then, its interpretation in the frequency domain is given.The link between the time domain and the frequency domain, through Fourier transform, is highlighted.In particular, Fourier transform is interpreted as a special case of Laplace transform. The responseanalysis using transform techniques is presented, along with the associated basic ideas of convolutionintegral, and the impulse response function whose Laplace transform is the transfer function, andFourier transform is the frequency response function. The half-power bandwidth approach of measuringdamping is given. Special types of frequency transfer functions — speciﬁcally, force transmissibility,©2000 CRC Press
- 39. motion transmissibility, and receptance — are studied and their complementary relationships arehighlighted. Their use in the practice of vibration, particularly in vibration isolation, is discussed. Chapter 4 presents the fundamentals of analyzing vibration signals. First, the idea of frequencyspectrum of a time signal is given. Various types and classiﬁcations of signals encountered invibration engineering are discussed. The technique of Fourier analysis is formally introduced andlinked to the concepts presented in Chapter 3. The idea of random signals is introduced, and usefulanalytical techniques for these signals are presented. Practical issues pertaining to vibration signalanalysis are raised. Computational techniques of signal analysis are given and various sources oferror, such as aliasing and truncation, are indicated; and ways of improving the accuracy of digitalsignal analysis are given. Chapter 5 deals with the modal analysis of lumped-parameter vibrating systems. The basicassumption made is that distributed effects of inertia and ﬂexibility in a vibrating system can berepresented by an interconnected set of lumped inertia and spring elements. The total number ofpossible independent, incremental motions of these inertia elements is the number of degrees offreedom of the system. For holonomic systems, this is also equal to the total number of independentcoordinates needed to represent an arbitrary conﬁguration of the system; but for non-holonomicsystems, the required number of coordinates will be larger. For this reason, the concepts ofholonomic and non-holonomic systems and the corresponding types of constraints are discussed.The representation of a general lumped-parameter vibrating system by a differential equation modelis given, and methods of obtaining such a model are discussed. Apart from the Newtonian andLagrangian approaches, the inﬂuence coefﬁcient approach is given for determining the mass andstiffness matrices. The concepts of natural frequencies and mode shapes are discussed, and theprocedure for determining these characteristic quantities, through modal analysis, is developed.The orthogonality property of natural modes is derived. The ideas of static modes and rigid bodymodes are explored, and the causes of these conditions will be indicated. In addition to the standardformulation of the modal analysis problem, two other modal formulations are developed. Theanalysis of the problem of forced vibration, using modal analysis, is given. Damped lumped-parameter vibrating systems are studied from the point of view of modal analysis. The conditionsof existence of real modes for damped systems are explored, with speciﬁc reference to proportionaldamping. The state-space approach of representing and analyzing a vibrating system is presented.Practical problems of modal analysis are presented. Chapter 6 studies distributed-parameter vibrating systems such as cables, rods, shafts, beams,membranes, and plates. Practical examples of associated vibration problems are indicated. Vibrationof continuous systems is treated as a generalization of lumped-parameter systems, discussed inChapter 5. In particular, the modal analysis of continuous systems is addressed in detail. The issueof orthogonality of modes is studied. The inﬂuence of system boundary conditions on the modalproblem in general and the orthogonality in particular is discussed, with special emphasis on“inertial” boundary conditions (e.g., continuous systems with lumped masses at the boundaries).The inﬂuence of damping on the modal analysis problem is discussed. The analysis of response toa forcing excitation is performed. Chapter 7 exclusively deals with the problem of energy dissipation or damping in vibratingsystems. Various types of damping present in mechanical and structural systems are discussed, withpractical examples, and particular emphasis on interface damping. Methods of representation ormodeling of damping in the analysis of vibrating systems are indicated. Techniques and principlesof measurement of damping are given, with examples. Chapter 8 studies instrumentation issues in the practice of vibration. Applications range frommonitoring and fault diagnosis of industrial processes, to product testing for quality assessmentand qualiﬁcation, experimental modal analysis for developing experimental models and for design-ing of vibrating systems, and control of vibration. Instrumentation types, basics of operation,industrial practices pertaining to vibration exciters, control systems, motion sensors and transducers,torque and force sensors, and other types of transducers are addressed. Performance speciﬁcation©2000 CRC Press
- 40. of an instrumented system is discussed. Issues and implications of component interconnection inthe practical use of instrumentation are addressed. Chapter 9 addresses signal conditioning and modiﬁcation for practical vibration systems. Theseconsiderations are closely related to the subject of instrumentation discussed in Chapter 8 andsignal analysis discussed in Chapter 4. Particular emphasis is given to commercial instruments andhardware that are useful in monitoring, analyzing, and control of vibration. Speciﬁc devicesconsidered include ampliﬁers, analog ﬁlters, modulators and demodulators, analog-to-digital con-verters, digital-to-analog converters, bridge circuits, linearizing devices, and other types of signalmodiﬁcation circuitry. Commercial spectrum analyzers and digital oscilloscopes commonlyemployed in the practice of vibration are discussed as well. Chapter 10 deals with vibration testing. This is a practical topic that is directly applicable toproduct design and development, experimental modeling, quality assessment and control, andproduct qualiﬁcation. Various methods of representing a vibration environment in a test programare discussed. Procedures that need to be followed prior to testing an object (i.e., pre-test procedures)are given. Available testing procedures are presented, with a discussion of appropriateness, advan-tages, and disadvantages of various test procedures. The topic of product qualiﬁcation testing isaddressed in some length. Chapter 11 studies experimental modal analysis, which is directly related to vibration testing(Chapter 10), experimental modeling, and design. It draws from the analytical procedures presentedin previous chapters, particularly Chapters 5 and 6. Frequency domain formulation of the problemis given. The procedure of developing a complete experimental model of a vibrating system ispresented. Procedures of curve ﬁtting of frequency transfer functions, which are essential in modelparameter extraction, are discussed. Several laboratory experiments in the area of vibration testing(modal testing) are described, giving details of the applicable instrumentation. Features and capa-bilities of several commercially available experimental modal analysis systems are described, anda comparative evaluation is given. Chapter 12 addresses practical and analytical issues of vibration design and control. Theemphasis here is in the ways of designing, modifying, or controlling a system for good performancewith regard to vibration. Ways of speciﬁcation of vibration limits for proper performance of anengineering system are discussed. Techniques and practical considerations of vibration isolationare described, with an emphasis on the use of transmissibility concepts developed in Chapter 3.Static and dynamic balancing of rotating machinery is studied by presenting both analytical andpractical procedures. The related topic of balancing multi-cylinder reciprocating machines isaddressed in some detail. The topic of whirling of rotating components and shafts is studied. Thesubject of design through modal testing, which is directly related to the material in chapters 10and 11, is discussed. Both passive control and active control of vibration are studied, givingprocedures and practical examples. The background material that is not given in the main body of the text, but is useful incomprehending the underlying procedures, is given in the appendices. Reference is made in themain text to these appendices, for further reading. Appendix A deals with dynamic models andanalogies. Main steps of developing analytical models for dynamic systems are indicated. Analogiesbetween mechanical, electrical, ﬂuid, and thermal systems are presented, with particular emphasison the cause of free natural oscillations. Development procedure of state-space models for thesesystems is indicated. Appendix B summarizes Newtonian and Lagrangian approaches to writingequations of motion for dynamic systems. Appendix C reviews the basics of linear algebra. Vector-matrix techniques that are useful in vibration analysis and practice are summarized. Appendix Dfurther explores the topic of digital Fourier analysis, with a special emphasis on the computationalprocedure of fast Fourier transform (FFT). As the background theory, the concepts of Fourier series,Fourier integral transform, and discrete Fourier transform are discussed and integrated, which leadsthe digital computation of these quantities using FFT. Practical procedures and applications ofdigital Fourier analysis are given. Appendix E addresses reliability considerations for multicom-©2000 CRC Press
- 41. ponent devices. These considerations have a direct relationship to vibration monitoring and testing,failure diagnosis, product qualiﬁcation, and design optimization.PROBLEMS1.1 Explain why mechanical vibration is an important area of study for engineers. Mechanical vibrations are known to have harmful effects as well as useful ones. Brieﬂy describe ﬁve practical examples of good vibrations and also ﬁve practical examples of bad vibrations.1.2 Under some conditions it may be necessary to modify or redesign a machine with respect to its performance under vibrations. What are possible reasons for this? What are some of the modiﬁcations that can be carried out on a machine in order to suppress its vibrations?1.3 On the one hand, modern machines are designed with sophisticated procedures and computer tools, and should perform better than the older designs, with respect to mechan- ical vibration. On the other hand, modern machines have to operate under more stringent speciﬁcations and requirements in a somewhat optimal fashion. In general, design for satisfactory performance under vibration takes an increased importance for modern machinery. Indicate some reasons for this.1.4 Dynamic modeling — both analytical and experimental (e.g., experimental modal anal- ysis) — is quite important in the design and development of a product, for good perfor- mance with regard to vibration. Indicate how a dynamic model can be utilized in the vibration design of a device.1.5 Outline one practical application of mechanical vibration in each of the following branches of engineering: 1. Civil engineering 2. Aeronautical and aerospace engineering 3. Mechanical engineering 4. Manufacturing engineering 5. Electrical engineeringREFERENCES AND FURTHER READINGThe book has relied on many publications, directly and indirectly, in its evolution and development.The author’s own work as well as other excellent books have provided a wealth of knowledge.Although it is not possible or useful to list all such material, some selected publications are listed below.AUTHOR’S WORK 1. De Silva, C.W., Dynamic Testing and Seismic Qualiﬁcation Practice, D.C. Heath and Co., Lexington, MA, 1983. 2. De Silva C.W. and Wormley, D.N., Automated Transit Guideways: Analysis and Design, D.C. Heath and Co., Lexington, MA, 1983. 3. De Silva, C.W., Control Sensors and Actuators, Prentice-Hall, Englewood Cliffs, NJ, 1989. 4. De Silva, C.W., Control System Modeling, Measurements and Data Corp., Pittsburgh, PA, 1989. 5. De Silva, C.W., A technique to model the simply supported timoshenko beam in the design of mechanical vibrating systems, International Journal of Mechanical Sciences, 17, 389-393, 1975. 6. Van de Vegte, J. and de Silva, C.W., Design of passive vibration controls for internally damped beams by modal control techniques, Journal of Sound and Vibration, 45(3), 417-425, 1976. 7. De Silva, C.W., Optimal estimation of the response of internally damped beams to random loads in the presence of measurement noise, Journal of Sound and Vibration, 47(4), 485-493, 1976.©2000 CRC Press
- 42. 8. De Silva, C.W., Dynamic beam model with internal damping, rotatory inertia and shear deformation, AIAA Journal, 14(5), 676-680, 1976. 9. De Silva, C.W. and Wormley, D.N., Material optimization in a torsional guideway transit system, Journal of Advanced Transportation, 13(3), 41-60, 1979. 10. De Silva, C.W., Buyukozturk, O., and Wormley, D.N., Postcracking compliance of RC beams, Journal of the Structural Division, Trans. ASCE, 105(ST1), 35-51, 1979. 11. De Silva, C.W., Seismic qualiﬁcation of electrical equipment using a uniaxial test, Earthquake Engineering and Structural Dynamics, 8, 337-348, 1980. 12. De Silva, C.W., Loceff, F., and Vashi, K.M., Consideration of an optimal procedure for testing the operability of equipment under seismic disturbances, Shock and Vibration Bulletin, 50(5), 149-158, 1980. 13. De Silva, C.W. and Wormley, D.N., Torsional analysis of cutout beams, Journal of the Structural Division, Trans. ASCE, 106(ST9), 1933-1946, 1980. 14. De Silva, C.W., An algorithm for the optimal design of passive vibration controllers for ﬂexible systems, Journal of Sound and Vibration, 74(4), 495-502, 1982. 15. De Silva, C.W., Matrix eigenvalue problem of multiple-shaker testing, Journal of the Engineering Mechanics Division, Trans. ASCE, 108(EM2), 457-461, 1982. 16. De Silva, C.W., Selection of shaker speciﬁcations in seismic qualiﬁcation tests, Journal of Sound and Vibration, 91(2), 21-26, 1983. 17. De Silva, C.W., Shaker test-ﬁxture design, Measurements and Control, 17(6), 152-155, 1983. 18. De Silva, C.W., On the modal analysis of discrete vibratory systems, International Journal of Mechan- ical Engineering Education, 12(1), 35-44, 1984. 19. De Silva, C.W. and Palusamy, S.S., Experimental modal analysis — A modeling and design tool, Mechanical Engineering, ASME, 106(6), 56-65, 1984. 20. De Silva, C.W., A dynamic test procedure for improving seismic qualiﬁcation guidelines, Journal of Dynamic Systems, Measurement, and Control, Trans. ASME, 106(2), 143-148, 1984. 21. De Silva, C.W., Hardware and software selection for experimental modal analysis, The Shock and Vibration Digest, 16(8), 3-10, 1984. 22. De Silva, C.W., Computer-automated failure prediction in mechanical systems under dynamic loading, The Shock and Vibration Digest, 17(8), 3-12, 1985. 23. De Silva, C.W., Henning, S.J., and Brown, J.D., Random testing with digital control — Application in the distribution qualiﬁcation of microcomputers, The Shock and Vibration Digest, 18(9), 3-13, 1986. 24. De Silva, C.W., The digital processing of acceleration measurements for modal analysis, The Shock and Vibration Digest, 18(10), 3-10, 1986. 25. De Silva, C.W., Price, T.E., and Kanade, T., A torque sensor for direct-drive manipulators, Journal of Engineering for Industry, Trans. ASME, 109(2), 122-127, 1987. 26. De Silva, C.W., Optimal input design for the dynamic testing of mechanical systems, Journal of Dynamic Systems, Measurement, and Control, Trans. ASME, 109(2), 111-119, 1987. 27. De Silva, C.W., Singh, M., and Zaldonis, J., Improvement of response spectrum speciﬁcations in dynamic testing, Journal of Engineering for Industry, Trans. ASME, 112(4), 384-387, 1990. 28. De Silva, C.W., Schultz, M., and Dolejsi, E., Kinematic analysis and design of a continuously-variable transmission, Mechanism and Machine Theory, 29(1), 149-167, 1994. 29. Bussani, F. and de Silva, C.W., Use of ﬁnite element method to model machine processing of ﬁsh, Finite Element News, 5, 36-42, 1994. 30. Caron, M., Modi, V.J., Pradhan, S., de Silva, C.W., and Misra, A.K., Planar dynamics of ﬂexible manipulators with slewing deployable links, Journal of Guidance, Control, and Dynamics, 21(4), 572-580, 1998.OTHER USEFUL PUBLICATIONS 1. Beards, C.F., Engineering Vibration Analysis with Application to Control Systems, Halsted Press, New York, 1996. 2. Bendat, J.S. and Piersol, A.G., Random Data: Analysis and Measurement Procedures, Wiley-Inter- science, New York, 1971. 3. Blevins, R.D., Flow-Induced Vibration, Van Nostrand Reinhold, New York, 1977. 4. Brigham, E.O., The Fast Fourier Transform, Prentice-Hall, Englewood Cliffs, NJ, 1974.©2000 CRC Press
- 43. 5. Broch, J.T., Mechanical Vibration and Shock Measurements, Bruel and Kjaer, Naerum, Denmark, 1980. 6. Buzdugan, G., Mihaiescu, E., and Rades, M., Vibration Measurement, Martinus Nijhoff Publishers, Dordrecht, The Netherlands, 1986. 7. Crandall, S.H., Karnopp, D.C., Kurtz, E.F., and Prodmore-Brown, D.C., Dynamics of Mechanical and Electromechanical Systems, McGraw-Hill, New York, 1968. 8. Den Hartog, J.P., Mechanical Vibrations, McGraw-Hill, New York, 1956. 9. Dimarogonas, A., Vibration for Engineers, 2nd edition, Prentice-Hall, Upper Saddle River, NJ, 1996. 10. Ewins, D.J., Modal Testing: Theory and Practice, Research Studies Press Ltd., Letchworth, England, 1984. 11. Inman, D.J., Engineering Vibration, Prentice-Hall, Englewood Cliffs, NJ, 1996. 12. Irwin, J.D. and Graf, E.R., Industrial Noise and Vibration Control, Prentice-Hall, Englewood Cliffs, NJ, 1979. 13. McConnell, K.G., Vibration Testing, John Wiley & Sons, New York, 1995. 14. Meirovitch, L., Computational Methods in Structural Dynamics, Sijthoff & Noordhoff, Rockville, MD, 1980. 15. Meirovitch, L., Elements of Vibration Analysis, 2nd edition, McGraw-Hill, New York, 1986. 16. Randall, R.B., Application of B&K Equipment to Frequency Analysis, Bruel and Kjaer, Naerum, Denmark, 1977. 17. Rao, S.S., Mechanical Vibrations, 3rd edition, Addison-Wesley, Reading, MA, 1995. 18. Shearer, J.L. and Kulakowski, B.T., Dynamic Modeling and Control of Engineering Systems, Mac- Millan Publishing, New York, 1990. 19. Shearer, J.L., Murphy, A.T., and Richardson, H.H., Introduction to System Dynamics, Addison-Wesley, Reading, MA, 1971. 20. Steidel, R.F., An Introduction to Mechanical Vibrations, 2nd edition, John Wiley & Sons, New York, 1979. 21. Volterra, E. and Zachmanoglou, E.C., Dynamics of Vibrations, Charles E. Merrill Books, Columbus, OH, 1965.©2000 CRC Press
- 44. de Silva, Clarence W. “Time Response”Vibration: Fundamentals and PracticeClarence W. de SilvaBoca Raton: CRC Press LLC, 2000
- 45. 2 Time ResponseVibrations are oscillatory responses of dynamic systems. Natural vibrations occur in these systemsdue to the presence of two modes of energy storage. Speciﬁcally, when the stored energy is convertedfrom one form to the other, repeatedly back and forth, the resulting time response of the systemis oscillatory in nature. In a mechanical system, natural vibrations can occur because kinetic energy,which is manifested as velocities of mass (inertia) elements, can be converted into potential energy(which has two basic types: elastic potential energy due to the deformation in spring-like elements,and gravitational potential energy due to the elevation of mass elements against the Earth’s grav-itational pull) and back to kinetic energy, repetitively, during motion. Similarly, natural oscillationsof electrical signals occur in circuits due to the presence of electrostatic energy (of the electriccharge storage in capacitor-like elements) and electromagnetic energy (due to the magnetic ﬁeldsin inductor-like elements). Fluid systems can also exhibit natural oscillatory responses as theypossess two forms of energy. But purely thermal systems do not produce natural oscillations becausethey, as far as anyone knows, have only one type of energy. These ideas are summarized in AppendixA. Note, however, that an oscillatory forcing function is able to make a dynamic system respondwith an oscillatory motion (usually at the same frequency as the forcing excitation) even in theabsence of two forms of energy storage. Such motions are forced responses rather than natural orfree responses. This book concerns vibrations in mechanical systems. Nevertheless, clear analogiesexist with electrical and ﬂuid systems as well as mixed systems such as electromechanical systems. Mechanical vibrations can occur as both free (natural) responses and forced responses innumerous practical situations. Some of these vibrations are desirable and useful, and others areundesirable and should be avoided or suppressed. The sound that is generated after a string of aguitar is plucked is a free vibration, while the sound of a violin is a mixture of both free and forcedvibrations. These sounds are generally pleasant and desirable. The response of an automobile afterit hits a road bump is an undesirable free vibration. The vibrations felt while operating a concretedrill are desirable for the drilling process itself, but are undesirable forced vibrations for the humanwho operates the drill. In the design and development of a mechanical system, regardless of whetherit is intended for generating desirable vibrations or for operating without vibrations, an analyticalmodel of the system can serve a very useful function. The model will represent the dynamic system,and can be analyzed and modiﬁed more quickly and cost effectively than one could build and testa physical prototype. Similarly, in the control or suppression of vibrations, it is possible to design,develop, and evaluate vibration isolators and control schemes through analytical means before theyare physically implemented. It follows that analytical models (see Appendix A) are useful in theanalysis, control, and evaluation of vibrations in dynamic systems, and also in the design anddevelopment of dynamic systems for desired performance in vibration environments. An analytical model of a mechanical system is a set of equations, and can be developed eitherby the Newtonian approach where Newton’s second law is explicitly applied to each inertia element,or by the Lagrangian or Hamiltonian approach, which is based on the concepts of energy (kineticand potential energies). These approaches are summarized in Appendix B. A time-domain analyticalmodel is a set of differential equations, with respect to the independent variable time (t). Afrequency-domain model is a set of input-output transfer functions with respect to the independentvariable frequency (ω). The time response will describe how the system moves (responds) as afunction of time. Both free and forced responses are useful. The frequency response will describethe way the system moves when excited by a harmonic (sinusoidal) forcing input, and is a functionof the frequency of excitation. This chapter introduces some basic concepts of vibration analysis©2000 CRC Press
- 46. FIGURE 2.1 A mechanical dynamic system.using time-domain methods. The frequency-domain analysis will be studied in subsequent chapters(Chapters 3 and 4, in particular).2.1 UNDAMPED OSCILLATORConsider the mechanical system that is schematically shown in Figure 2.1. The inputs (or excitation)applied to the system are represented by the force f(t). The outputs (or response) of the system arerepresented by the displacement y. The system boundary demarcates the region of interest in thisanalysis. This boundary could be an imaginary one. What is outside the system boundary is theenvironment in which the system operates. An analytical model of the system can be given by oneor more equations relating the outputs to the inputs. If the rates of changes of the response (outputs)are not negligible, the system is a dynamic system. In this case, the analytical model in the timedomain becomes one or more differential equations rather than algebraic equations. System param-eters (e.g., mass, stiffness, damping constant) are represented in the model, and their values shouldbe known in order to determine the response of the system to a particular excitation. State variablesare a minimum set of variables that completely represent the dynamic state of a system at anygiven time t. These variables are not unique (more than one choice of a valid set of state variablesis possible). The concepts of state variables and state models are introduced in Appendix A andalso in this chapter. For a simple oscillator (a single-degree-of-freedom mass-spring-damper systemas in Figure 2.1), an appropriate set of state variables would be the displacement y and the velocity ˙ ˙ y . An alternative set would be y and the spring force. This chapter provides an introduction to the response analysis of mechanical vibrating systems inthe time domain. In this introductory chapter, single-degree-of-freedom systems that require only onecoordinate (or one independent displacement variable) in their model, are considered almost exclusively.Higher-degree-of-freedom systems will be analyzed elsewhere in the book (e.g., Chapter 5). Mass(inertia) and spring are the two basic energy storage elements in a mechanical vibrating system. Amass can store gravitational potential energy as well when located against a gravitational force. Theseelements are analyzed ﬁrst. In a practical system, mass and stiffness properties can be distributed(continuous) throughout the system. But in this present analysis, lumped-parameter models areemployed where inertia, ﬂexibility, and damping effects are separately lumped into single parameters,with a single geometric coordinate used to represent the location of each lumped inertia. This chapter section ﬁrst shows that many types of oscillatory systems can be represented bythe equation of an undamped simple oscillator. In particular, mechanical, electrical, and ﬂuid systemsare considered. Please refer to Appendix A for some foundation material on this topic. The conser-vation of energy is a straightforward approach for deriving the equations of motion for undampedoscillatory systems (or conservative systems). The equations of motion for mechanical systems can©2000 CRC Press

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