Vibrations Fundamentals and Practice FM

Vibrations Fundamentals and Practice FM Maintaining the outstanding features and practical approach that led the bestselling first edition to become a standard textbook in engineering classrooms worldwide, Clarence de Silva''s Vibration: Fundamentals and Practice, Second Edition remains a solid instructional tool for modeling, analyzing, simulating, measuring, monitoring, testing, controlling, and designing for vibration in engineering systems. It condenses the author''s distinguished and extensive experience into an easy-to-use, highly practical text that prepares students for real problems in a variety of engineering fields.

de Silva, Clarence W “Frontmatter” Vibration: Fundamentals and Practice Clarence W de Silva Boca Raton: CRC Press LLC, 2000 VIBRATION Fundamentals and Practice Clarence W de Silva CRC Press Boca Raton London New York Washington, D.C Library of Congress Cataloging-in-Publication Data De 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) 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 with permission, and sources are indicated A wide variety of references are listed Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or 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 prior permission in writing from the publisher The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, 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 for identification and explanation, without intent to infringe Cover art is the U.S Space Shuttle and the International Space Station (Courtesy of NASA Langley Research Center, Hampton, VA With permission.) © 2000 by CRC Press LLC No claim to original U.S Government works International Standard Book Number 0-8493-1808-4 Library of Congress Card Number 99-16238 Printed in the United States of America Printed on acid-free paper ©2000 CRC Press Preface This book provides the background and techniques that will allow successful modeling, analysis, monitoring, testing, design, modification, and control of vibration in engineering systems It is suitable as both a course textbook for students and instructors, and a practical reference tool for engineers and other professionals As a textbook, it can be used in a single-semester course for third-year (junior) and fourth-year (senior) undergraduate students, or for Master’s level graduate students in any branch of engineering such as aeronautical and aerospace, civil, mechanical, and manufacturing engineering But, in view of the practical considerations, design issues, experimental techniques, and instrumentation that are presented throughout the book, and in view of the simplified and snapshot-style presentation of fundamentals and advanced theory, the book will also serve as a valuable reference tool for engineers, technicians, and other professionals in industry and in research laboratories The book is an outgrowth of the author’s experience in teaching undergraduate and graduate courses in Dynamics, Mechanical Vibration, Dynamic System Modeling, Instrumentation and Design, Feedback Control, Modern Control Engineering, and Modal Analysis and Testing in the U.S and Canada (Carnegie Mellon University and the University of British Columbia) for more than 20 years The industrial experience and training that he received in product testing and qualification, analysis, design, and vibration instrumentation at places like Westinghouse Electric Corporation in Pittsburgh, IBM Corporation in Boca Raton, NASA’s Langley and Lewis Research Centers, and Bruel and Kjaer in Denmark enabled the author to provide a realistic and practical treatment of the subject Design for vibration and control of vibration are crucial in maintaining a high performance level and production efficiency, and prolonging the useful life of machinery, structures, and industrial processes Before designing or controlling an engineering system for good vibratory performance, it is important to understand, represent (i.e., model), and analyze the vibratory characteristics of the system Suppression or elimination of undesirable vibrations and generation of required forms 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 vibration in 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, instrumentation, modeling, analysis, design, modification, and control techniques that produce mechanical and aeronautical systems, civil engineering structures, and manufacturing processes that are optimized 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 will be 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 applications of vibration, through examples, in the very beginning of the book, along with experimental techniques and instrumentation, and then integrating these applications, design, and control considerations into fundamentals and analytical methods throughout the text To maintain clarity and focus and to maximize the usefulness of the book, an attempt has been made to describe and illustrate industry-standard and state-of-the-art instrumentation, hardware, and computational techniques related to the practice of vibration As its main features, the book: â2000 CRC Press Introduces practical applications, design, and experimental techniques in the very beginning, 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 INSTRUCTORS The book is suitable as the text for a standard undergraduate course in Mechanical Vibration or for a specialized course for final-year undergraduate students and Master’s level graduate students Three typical course syllabuses are outlined below A A Standard Undergraduate Course As the textbook for an undergraduate (3rd year or 4th year) course in Mechanical Vibration, it may be incorporated into the following syllabus for a 12 week course consisting of 36 hours of lectures and 12 hours of laboratory experiments: Lectures Chapter (1 hour) Sections 8.1, 8.2, 8.4, 9.1, 9.2, 9.8 (3 hours) Chapter (6 hours) Chapter (6 hours) Section 11.4 (2 hours) Chapter (6 hours) Chapter (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 Experiment on modal testing (hammer test and other transient tests) and damping measurement in the time domain (see Section 11.4) Experiment on shaker testing and damping measurement in the frequency domain (see Section 11.4) Experiment on single-plane and two-plane balancing (see Section 12.3) Experiment on modal testing of a distributed-parameter system (see Section 11.4) ©2000 CRC Press B A Course in Industrial Vibration Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter (1 hour) (3 hours) (5 hours) (5 hours) (4 hours) 10 (6 hours) 11 (6 hours) 12 (6 hours) A project may be included in place of a final examination C A Course in Modal Analysis and Testing Chapter (1 hour) Chapter (3 hours) Chapter (6 hours) Chapter (6 hours) Chapter (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 The Author Clarence W de Silva, Fellow ASME and Fellow IEEE, is Professor of Mechanical Engineering at the University of British Columbia, Vancouver, Canada, and has occupied the NSERC Chair in Industrial Automation since 1988 He obtained his first Ph.D from the Massachusetts Institute of Technology 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 the U.S., and has led the development of many industrial machines He is recipient of the Education Award of the Dynamic Systems and Control Division of the American Society of Mechanical Engineers, the Meritorious Achievement Award of the Association of Professional Engineers of British Columbia, the Outstanding Contribution Award of the IEEE Systems, Man, and Cybernetics Society, the Outstanding Large Chapter Award of the IEEE Industry Applications Society, and the Outstanding Chapter Award from the IEEE Control Systems Society He has authored 14 technical books, 10 edited volumes, over 120 journal papers, and a similar number of conference papers and book chapters He has served on the editorial boards of 12 international journals, and is the Editorin-Chief of the International Journal of Knowledge-Based Intelligent Engineering Systems, Senior Technical Editor of Measurements and Control, and Regional Editor, North America, of the International Journal of Intelligent Real-Time Automation He has been a Lilly Fellow, Senior Fulbright Fellow to Cambridge University, ASI Fellow, and a Killam Fellow ©2000 CRC Press Acknowledgment Preparation of this book would not have been possible if not for the support of many individuals and organizations, but it is not possible to list all of them here I wish to recognize the following specific 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 of Advanced Education, Training and Technology, Province of British Columbia, which made part of the secretarial support for my work possible The Department of Mechanical Engineering at the University of British Columbia provided me with an excellent environment to carry out my educational activities, including the preparation of this book My graduate students, research associates, teaching assistants, and office staff have contributed directly and indirectly to the success of 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 the Project Editor, Sara Seltzer, for their fine effort in the production of the book Encouragement of various authorities in the field of engineering — particularly, Professor Devendra Garg of Duke University, Professor Mo Jamshidi of the University of New Mexico, and Professor Arthur Murphy (DuPont Fellow Emeritus) — is gratefully acknowledged Finally, my family deserves an apology for the unintentional “neglect” that they may have suffered during the latter stages of production of the book ©2000 CRC Press Source Credits The 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, Washington • 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 permission 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 Dedication Professor David N Wormley “For the things we have to learn before we can them, we learn by doing them.” — Aristotle (Author of Mechanics and Acoustics, 384–322 B.C.) ©2000 CRC Press First-Order System Second-Order System Repeated Roots 2.4.6 Stability and Speed of Response Example 2.4 Solution 2.5 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 Rule Problems Chapter Frequency Response 3.1 Response to Harmonic Excitations 3.1.1 Response Characteristics Case Case Case 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 Solution 3.2 Transform Techniques 3.2.1 Transfer Function 3.2.2 Frequency-Response Function (Frequency-Transfer Function) Impulse Response Case (ζ < 1) Case (ζ > 1) Case (ζ = 1) Step Response 3.2.3 Transfer Function Matrix Example 3.2 Example 3.3 Example 3.4 Solution 3.3 Mechanical Impedance Approach Mass Element Spring Element Damper Element 3.3.1 Interconnection Laws Example 3.5 Example 3.6 3.4 Transmissibility Functions 3.4.1 Force Transmissibility ©2000 CRC Press 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 Functions 3.5 Receptance Method 3.5.1 Application of Receptance Undamped Simple Oscillator Dynamic Absorber Problems Chapter Vibration Signal Analysis 4.1 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 Spectrum 4.2 Signal Types 4.3 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 4.2 4.4 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 Cepstrum 4.5 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 dB) Bandwidth 4.5.5 Fourier Analysis Bandwidth ©2000 CRC Press 4.6 4.7 Resolution in Digital Fourier Results Overlapped Processing Example 4.3 4.7.1 Order Analysis Speed Spectral Map Time Spectral Map Order Tracking Problems Chapter Modal Analysis 5.1 Degrees of Freedom and Independent Coordinates 5.1.1 Nonholonomic Constraints Example 5.1 Example 5.2 5.2 System Representation 5.2.1 Stiffness and Flexibility Matrices 5.2.2 Inertia Matrix 5.2.3 Direct Approach for Equations of Motion 5.3 Modal Vibrations Example 5.3 5.4 Orthogonality of Natural Modes 5.4.1 Modal Mass and Normalized Modal Vectors 5.5 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 Vector 5.6 Other Modal Formulations 5.6.1 Non-Symmetric Modal Formulation 5.6.2 Transformed Symmetric Modal Formulation Example 5.6 Approach Approach 5.7 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 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 5.9 Problems Chapter Distributed-Parameter Systems 6.1 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 Solution 6.2 Longitudinal Vibration of Rods 6.2.1 Equation of Motion 6.2.2 Boundary Conditions Example 6.3 Solution 6.3 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 Solution 6.4 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 6.4.9 Use of the Energy Approach 6.4.10 Orthogonality with Inertial Boundary Conditions Rotatory Inertia 6.5 Damped Continuous Systems 6.5.1 Modal Analysis of Damped Beams Example 6.8 Solution 6.6 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 Edges Problems Chapter Damping 7.1 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 Solution 7.2 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 Factor 7.3 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 Remarks 7.4 Interface Damping Example 7.5 Solution 7.4.1 Friction In Rotational Interfaces 7.4.2 Instability Problems ©2000 CRC Press Chapter Vibration Instrumentation 8.1 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 Exciters 8.2 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 Processing 8.3 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 Precision 8.4 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 Capacitive Angular Velocity Sensor Capacitance Bridge Circuit 8.4.10 Piezoelectric Transducers Sensitivity Example 8.3 Solution Piezoelectric Accelerometer Charge Amplifier 8.5 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 Sensors 8.6 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 Noise Problems Chapter Signal Conditioning and Modification 9.1 Amplifiers 9.1.1 Operational Amplifier Example 9.1 Solution ©2000 CRC Press 9.1.2 9.1.3 9.1.4 9.2 9.3 9.4 Use of Feedback in Op-amps Voltage, Current, and Power Amplifiers Instrumentation Amplifiers Differential Amplifier Common Mode Amplifier Performance Ratings Example 9.2 Solution Common-Mode Rejection Ratio (CMRR) AC-Coupled Amplifiers 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 Filters 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 Demodulation 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 Software Implementation and Hardware Implementation Circuits Wheatstone Bridge Constant-Current Bridge Bridge Amplifiers Half-Bridge Circuits 9.5.4 Impedance Bridges Owen Bridge Wien-Bridge Oscillator 9.6 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 Circuitry 9.7 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 Circuit 9.8 Signal Analyzers and Display Devices 9.8.1 Signal Analyzers 9.8.2 Oscilloscopes Triggering Lissajous Patterns Digital Oscilloscopes Problems 9.5 Bridge 9.5.1 9.5.2 9.5.3 Chapter 10 Vibration Testing 10.1 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 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 Representations 10.2 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 Inspection 10.3 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 Phasing of Excitations Testing a Gray or White Box Overtesting in Multitest Sequences 10.4 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 Specifications Problems Chapter 11 Experimental Modal Analysis 11.1 Frequency-Domain Formulation 11.1.1 Transfer Function Matrix 11.1.2 Principle of Reciprocity Example 11.1 11.2 Experimental Model Development 11.2.1 Extraction of the Time-Domain Model 11.3 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 Extraction 11.4 Laboratory Experiments 11.4.1 Lumped-Parameter System Frequency-Domain Test Time-Domain Test 11.4.2 Distributed-Parameter System 11.5 Commercial EMA Systems 11.5.1 System Configuration FFT Analysis Options Modal Analysis Components Problems Chapter 12 Vibration Design and Control Shock and Vibration 12.1 Specification of Vibration Limits 12.1.1 Peak Level Specification 12.1.2 RMS Value Specification ©2000 CRC Press 12.1.3 Frequency-Domain Specification 12.2 Vibration Isolation Example 12.1 Solution 12.2.1 Design Considerations Example 12.2 Solution 12.2.2 Vibration Isolation of Flexible Systems 12.3 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 Balancing 12.4 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 Load 12.5 Whirling of Shafts 12.5.1 Equations of Motion 12.5.2 Steady-State Whirling Example 12.6 Solution 12.5.3 Self-Excited Vibrations 12.6 Design Through Modal Testing 12.6.1 Component Modification Example 12.7 Solution 12.6.2 Substructuring 12.7 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 Dampers 12.8 Active Control of Vibration 12.8.1 Active Control System 12.8.2 Control Techniques State-Space Models ©2000 CRC Press Example 12.10 Solution Position and Velocity Feedback Linear Quadratic Regulator (LQR) Control Modal Control 12.8.3 Active Control of Saw Blade Vibration 12.9 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 Example Problems Appendix A Dynamic Models and Analogies A.1 Model Development A.2 Analogies A.3 Mechanical Elements A.3.1 Mass (Inertia) Element A.3.2 Spring (Stiffness) Element A.4 Electrical Elements A.4.1 Capacitor Element A.4.2 Inductor Element A.5 Thermal Elements A.5.1 Thermal Capacitor A.5.2 Thermal Resistance A.6 Fluid Elements A.6.1 Fluid Capacitor A.6.2 Fluid Inertor A.6.3 Fluid Resistance A.6.4 Natural Oscillations A.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 Realizability Appendix B Newtonian and Lagrangian Mechanics B.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 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 Angles B.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 Equations Appendix C Review of Linear Algebra C.1 Vectors and Matrices C.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 Matrix C.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 Matrix C.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 C.4.4 C.4.5 C.4.6 C.4.7 Linear Dependence Basis and Dimension of a Vector Space Inner Product Norm Properties C.4.8 Gram-Schmidt Orthogonalization C.4.9 Modified Gram-Schmidt Procedure C.5 Determinants C.5.1 Properties of Determinant of a Matrix C.5.2 Rank of a Matrix C.6 System of Linear Equations References Appendix D Digital Fourier Analysis and FFT D.1 Unification of the Three Fourier Transform Types D.1.1 Relationship Between DFT and FIT D.1.2 Relationship Between DFT and FSE D.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 Example D.3 Discrete Correlation and Convolution D.3.1 Discrete Correlation Discrete Correlation Theorem Discrete Convolution Theorem D.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 Convolution Appendix E Reliability Considerations for Multicomponent Units E.1 Failure Analysis E.1.1 Reliability E.1.2 Unreliability E.1.3 Inclusion–Exclusion Formula Example E.2 Bayes’ Theorem E.2.1 Product Rule for Independent Events E.2.2 Failure Rate E.2.3 Product Rule for Reliability Answers to Numerical Problems ©2000 CRC Press ... techniques and instrumentation, and then integrating these applications, design, and control considerations into fundamentals and analytical methods throughout the text To maintain clarity and focus and. .. Modeling, Instrumentation and Design, Feedback Control, Modern Control Engineering, and Modal Analysis and Testing in the U.S and Canada (Carnegie Mellon University and the University of British... NASA’s Langley and Lewis Research Centers, and Bruel and Kjaer in Denmark enabled the author to provide a realistic and practical treatment of the subject Design for vibration and control of