Noise of Polyphase Electric Motors

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Noise of Polyphase Electric Motors

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Noise of Polyphase Electric Motors

Jacek F. Gieras United Technologies Corporation Hamilton Sundstrand, Applied Research Rockford, Illinois, U.S.A.

Chong Wang General Motors Corporation Milford, Michigan, U.S.A.

Joseph Cho Lai The University of New South Wales at the Australian Defence Force Academy Canberra, Australian Capital Territory, Australia

Boca Raton London New York

A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa plc.

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Published in 2006 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 0-8247-2381-3 (Hardcover) International Standard Book Number-13: 978-0-8247-2381-1 (Hardcover) Library of Congress Card Number 2005050213 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. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.

Library of Congress Cataloging-in-Publication Data Gieras, Jacek F. Noise of polyphase electric motors / Jacek F. Gieras, Chong Wang, Joseph Cho Lai. p. cm. -- (Electrical and Computer Engineering ; 129) Includes bibliographical references and index. ISBN 0-8247-2381-3 (alk. paper) 1. Electric motors, Polyphase--noise. 2. Electric Motors, Alternating current--Noise. 3. Electric motors, Alternating current--Design and construction. I. Lai, Joseph Cho. II. Wang, Chong. III. Title. IV. Series. TK2785.G535 2005 621.46--dc22

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Preface It has been estimated that more than 65% of the electrical energy produced in developed countries is consumed by electric motors. Electrical motors are the most popular machines of everyday life. They are used either as power machines providing propulsion torque or servo motors operating in a closed loop control with speed or position feedback. Electric motors are embedded in larger systems as their integral part. The noise radiated by electric motors immensely affects the overall noise of the system. Contemporary electric motors are designed with higher magnetic flux density in the air gap than motors manufactured a half century ago. Higher magnetic flux density in the air gap produces higher radial magnetic forces acting on the stator system and, consequently, higher vibration and acoustic noise. With the increased power density of electric motors and more demanding environmental requirements, the prediction of noise at the early stage of design of electrical motors has become a very important issue. Not only electromagnetic, thermal, and economic calculations, but also the level of noise and vibration must be considered, so that the overall performance can be optimized/balanced and specific requirements can be incorporated in the design to avoid large retrofit expenses. However, prediction of noise is more difficult and less accurate than, for example, torque–speed characteristics. This is because only a very small fraction of electrical energy is converted into acoustic energy and correct estimation of some mechanical and acoustic parameters is very difficult. The first book [119] on calculation of noise in electrical motors was published by Jordan in 1950. Details of harmonic field analysis including harmonic torques, noise, and vibration in induction motors are given in the book [87] by Heller and Hamata published in 1977. Analysis of noise in induction machines with emphasis on its reduction is given in monograph [248] by Yang, which was published in 1981. The most comprehensive analysis of noise and vibration in electrical machines is contained in the book [200] by Timar, Fazekas, Kiss, Miklas, and Yang published in 1989. It is also necessary to mention two books on noise and vibration in induction machines published by Russian researchers: Shubov in 1974 [187] and Astakhov, Malishev, and Ovcharenko in 1985 [10], and a book published by Polish researcher Kwasnicki in 1998 [127]. There is no book published so far on noise and vibration of permanent magnet (PM) synchronous

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motors. The demand on these motors is nowadays in second place, after the demand on induction motors. As most books on noise and vibration analysis in electrical machines were published over two decades ago, recent advances in vibro-acoustic theories and technologies are only accessible in learned journals and have not been captured in a single monograph. These advances include the development and application of numerical methods of noise computation such as the finite element method (FEM), boundary element method (BEM), and statistical energy analysis (SEA) [43, 230] to the prediction of noise in electrical machines. With the increase in the importance of noise analysis and synthesis in the modern approach to the design of electrical motors, the authors have made an attempt to prepare a modern monograph on noise calculation in induction and PM synchronous motors addressing electromagnetic, mechanical, and vibro-acoustic issues. The noise and vibration of switched reluctance motors have not been considered here. The authors have devised the book as both an electrical motor noise textbook and a handbook for electrical machine design engineers, research scientists, and graduate students. The book can also be helpful for multidisciplinary research teams working on noise prediction of systems with electrical motors, e.g., electrical vehicles, industrial electromechanical drives, HVAC (heating, ventilating, air conditioning) systems, marine propulsion systems, airborne apparatus, elevators, office equipment, health care equipment, etc. The authors hope that this book will fill the current gap in modern treatment of the analysis and reduction of noise in polyphase electric motors. Jacek F. Gieras Chong Wang Joseph C.S. Lai

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Authors Jacek F. Gieras, Ph.D., graduated with distinction in 1971 (Master’s in Engineering) from the Technical University of Lodz, Poland. He received his Ph.D. in Power Electrical Engineering (Electrical Machines) in 1975 and dr hab. degree (corresponding to D.Sc. in the United Kingdom), also in Electrical Engineering (Electromagnetic Field Theory) in 1980 from the University of Technology, Poznan, Poland. From 1971 to 1998, Dr. Gieras pursued his academic career at several universities worldwide including Poland, Canada, Jordan, and South Africa. He was also a Central Japan Railway Company visiting professor at the University of Tokyo (Endowed Chair in Transportation Systems Engineering), Japan; guest professor at Chungbuk National University, Choengju, South Korea; and visiting professor at the University of Rome La Sapienza, Italy. In 1987, he was promoted to the rank of full professor (life title given by the President of the Republic of Poland). Since 1998 Gieras has been involved in industry-oriented research, cutting-edge technologies, and innovations at United Technologies Corporation, recently in the Department of Applied Research, Hamilton Sundstrand, Rockford, Illinois. Dr. Gieras has authored and coauthored nine books, more than 220 scientific and technical papers and ten patents. His most important books of international standing include: Linear Induction Motors, Oxford University Press, 1994, United Kingdom; Permanent Magnet Motors Technology: Design and Applications, Marcel Dekker, New York, 1996, Second Edition, 2002 (coauthor M. Wing); Linear Synchronous Motors: Transportation and Automation Systems, CRC Press, Boca Raton, Florida, 1999 (coauthor Z. Piech); and Axial Flux Permanent Magnet Machines, Springer-Kluwer, Boston, 2004 (coauthors R. Wang and M. Kamper). He is a Fellow of IEEE, full member of the International Academy of Electrical Sciences, a member of numerous steering committees of international conferences, and cited by all Marquis Who’s Whos.

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Chong Wang, Ph.D., received B.S. and M.S. degrees in Acoustics from Nanjing University, China, in 1964 and 1989, respectively. He received his Ph.D. in Mechanical Engineering from the University of New South Wales, Australia, in 1999. From 1989 to 1995, he was with the Institute of Acoustical Engineering, Northwestern Polytechnical University, China, as an assistant lecturer, lecturer, and an associate professor, where he lectured on several acoustics courses and conducted research work in noise and vibration engineering. In 1999, Dr. Wang joined the Institute for Research in Construction, National Research Council, Canada, as a Canadian Government Laboratory Visiting Fellow. Since 2000, he has been employed as a senior engineer with General Motors Corporation, United States. His research experience and interests include vibro-acoustics, room acoustics, FEA/BEA, SEA, and active noise and vibration control. He has published more than 60 papers in journals and conference proceedings in these areas. Joseph Cho Lai, Ph.D., is a professor of mechanical engineering and associate dean (research) at the Australian Defense Force Academy (ADFA) Campus of the University of New South Wales (UNSW). He was appointed in January 2004. Prior to this appointment, he was the head of the School of Aerospace and Mechanical Engineering from 2001–2003. Dr. Lai obtained a B.Sc. (United Kingdom) in Mechanical Engineering with first-class honors from the University of Hong Kong in 1975. He was awarded a Master of Engineering Science in 1978 and a Ph.D. in Mechanical Engineering in 1981 from the University of Queensland, Australia. He lectured in mechanical engineering at the University of Queensland until June 1985 when he joined the Department of Mechanical Engineering at UNSW/ADFA as a lecturer. Dr. Lai’s research interests are in turbulent shear flows and acoustics and vibration control. He has published 192 papers in journals and conference proceedings in these areas. His current work focuses on the aerodynamic propulsive mechanism of flapping wings as applied to microaerial vehicles, vibro-acoustic communication in termites, and noise and vibration control of machines, including electrical machines. He is a member of four International Standard Organization Acoustics Working Groups. He is a Fellow of the Institution of Engineers, Australia, a Fellow of the Australian Acoustical Society, and an Associate Fellow of the American Institute of Aeronautics and Astronautics.

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Contents 1

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Generation and Radiation of Noise in Electrical Machines 1.1 Vibration, sound, and noise . . . . . . . . . . . . . . . . 1.2 Sound waves . . . . . . . . . . . . . . . . . . . . . . . 1.3 Sources of noise in electrical machines . . . . . . . . . . 1.3.1 Electromagnetic sources of noise . . . . . . . . . 1.3.2 Mechanical sources of noise . . . . . . . . . . . 1.3.3 Aerodynamic noise . . . . . . . . . . . . . . . . 1.4 Energy conversion process . . . . . . . . . . . . . . . . 1.5 Noise limits and measurement procedures for electrical machines . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Deterministic and statistical methods of noise prediction 1.7 Economical aspects . . . . . . . . . . . . . . . . . . . . 1.8 Accuracy of noise prediction . . . . . . . . . . . . . . .

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Magnetic Fields and Radial Forces in Polyphase Motors Fed with Sinusoidal Currents 2.1 Construction of induction motors . . . . . . . . . . . . . . . . . 2.2 Construction of permanent magnet synchronous brushless motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 A.C. stator windings . . . . . . . . . . . . . . . . . . . . . . . 2.4 Stator winding MMF . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Single-phase stator winding . . . . . . . . . . . . . . . 2.4.2 Three-phase stator winding . . . . . . . . . . . . . . . . 2.4.3 Polyphase stator winding . . . . . . . . . . . . . . . . . 2.5 Rotor magnetic field . . . . . . . . . . . . . . . . . . . . . . . 2.6 Calculation of air gap magnetic field . . . . . . . . . . . . . . . 2.6.1 Effect of slots . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Effect of eccentricity . . . . . . . . . . . . . . . . . . .

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Deformation of the stator core . . . . . . . . Frequencies and orders of magnetic pressure Radial forces in synchronous machines with slotted stator . . . . . . . . . . . . . . . . . 2.7.6 Frequencies of vibration and noise . . . . . . Other sources of electromagnetic vibration and noise 2.8.1 Unbalanced line voltage . . . . . . . . . . . 2.8.2 Magnetostriction . . . . . . . . . . . . . . . 2.8.3 Thermal stress analogy . . . . . . . . . . . . 2.8.4 FEM model . . . . . . . . . . . . . . . . . .

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3 Inverter-Fed Motors 3.1 Generation of higher time harmonics . . . . . . . . . . . . . 3.2 Analysis of radial forces for nonsinusoidal currents . . . . . 3.2.1 Stator and rotor magnetic flux density . . . . . . . . 3.2.2 Stator harmonics of the same number . . . . . . . . 3.2.3 Interaction of stator and rotor harmonics . . . . . . . 3.2.4 Rotor harmonics of the same number . . . . . . . . 3.2.5 Frequencies and orders of magnetic pressure for nonsinusoidal currents . . . . . . . . . . . . . . . . 3.2.6 Interaction of stator harmonics of different numbers . 3.2.7 Interaction of switching frequency and higher time harmonics . . . . . . . . . . . . . . . . . . . . . . . 3.2.8 Interaction of permeance and magnetomotive force (MMF) harmonics . . . . . . . . . . . . . . . . . . 3.2.9 Rectifier harmonics . . . . . . . . . . . . . . . . . . 3.3 Higher time harmonic torques in induction machines . . . . 3.3.1 Asynchronous torques . . . . . . . . . . . . . . . . 3.3.2 Pulsating torques . . . . . . . . . . . . . . . . . . . 3.4 Higher time harmonic torques in permanent magnet (PM) brushless machines . . . . . . . . . . . . . . . . . . . . . . 3.5 Influence of the switching frequency of an inverter . . . . . 3.6 Noise reduction of inverter-fed motors . . . . . . . . . . . . 4 Torque Pulsations 4.1 Analytical methods of instantaneous torque calculation . . . 4.2 Numerical methods of instantaneous torque calculation . . . 4.3 Electromagnetic torque components . . . . . . . . . . . . . 4.4 Sources of torque pulsations . . . . . . . . . . . . . . . . . 4.5 Higher harmonic torques of induction motors . . . . . . . . 4.6 Cogging torque in permanent magnet (PM) brushless motors 4.6.1 Air gap magnetic flux density . . . . . . . . . . . . 4.6.2 Calculation of cogging torque . . . . . . . . . . . . 4.6.3 Simplified cogging torque equation . . . . . . . . . 4.6.4 Influence of eccentricity . . . . . . . . . . . . . . . Copyright © 2006 Taylor & Francis Group, LLC

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xvii 4.6.5 Calculation and comparison with measurements . . . . Torque ripple due to distortion of EMF and current waveforms in permanent magnet (PM) brushless motors . . . . . . . . . . Tangential forces vs. radial forces . . . . . . . . . . . . . . . Minimization of torque ripple in PM brushless motors . . . . . 4.9.1 Slotless windings . . . . . . . . . . . . . . . . . . . . 4.9.2 Skewing stator slots . . . . . . . . . . . . . . . . . . 4.9.3 Shaping stator slots . . . . . . . . . . . . . . . . . . . 4.9.4 Selection of the number of stator slots . . . . . . . . . 4.9.5 Shaping PMs . . . . . . . . . . . . . . . . . . . . . . 4.9.6 Skewing PMs . . . . . . . . . . . . . . . . . . . . . . 4.9.7 Shifting PM segments . . . . . . . . . . . . . . . . . 4.9.8 Selection of PM width . . . . . . . . . . . . . . . . . 4.9.9 Magnetization of PMs . . . . . . . . . . . . . . . . . 4.9.10 Creating magnetic circuit asymmetry . . . . . . . . .

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Stator System Vibration Analysis 5.1 Forced vibration . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Simplified calculation of natural frequencies of the stator system . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Improved analytical method of calculation of natural frequencies . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Natural frequency of the stator core . . . . . . . . . . . 5.3.2 Natural frequency of a frame with end bells . . . . . . . 5.3.3 Natural frequency of a stator core–frame system . . . . 5.3.4 Effect of the stator winding and teeth . . . . . . . . . . 5.3.5 Analytical calculation of natural frequencies for a stator core-winding-frame system . . . . . . . . . . . . . . . 5.4 Numerical verification . . . . . . . . . . . . . . . . . . . . . . 5.4.1 FEM modeling . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Comparison of analytical calculations with the FEM . .

6 Acoustic Calculations 6.1 Sound radiation efficiency . . . . . . . . . . . . . 6.2 Plane radiator . . . . . . . . . . . . . . . . . . . . 6.2.1 Infinite plates . . . . . . . . . . . . . . . . 6.2.2 Finite plates in bending motion . . . . . . 6.3 Infinitely long cylindrical radiator . . . . . . . . . 6.4 Finite length cylindrical radiator . . . . . . . . . . 6.4.1 Acoustically thin shells . . . . . . . . . . . 6.4.2 Acoustically thick shells . . . . . . . . . . 6.4.3 Modal radiation efficiencies of acoustically thick shells . . . . . . . . . . . . . . . . . 6.4.4 Modal averaged radiation efficiency . . . . 6.4.5 Validity of using an infinite length model . Copyright © 2006 Taylor & Francis Group, LLC

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Noise and Vibration of Mechanical and Aerodynamic Origin 7.1 Mechanical noise due to shaft and rotor irregularities . . . 7.2 Bearing noise . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Rolling bearings . . . . . . . . . . . . . . . . . . 7.2.2 Sleeve bearings . . . . . . . . . . . . . . . . . . . 7.3 Noise due to toothed gear trains . . . . . . . . . . . . . . 7.4 Aerodynamic noise . . . . . . . . . . . . . . . . . . . . . 7.5 Mechanical noise generated by the load . . . . . . . . . .

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Effects of boundary conditions on the radiation efficiency . . . . . . . . . . . . . . . . . . . . Calculations of sound power level . . . . . . . . . . . 6.5.1 Sound power radiated from a stator . . . . . . 6.5.2 Total sound power of an induction motor . . . 6.5.3 Permanent magnet synchronous motors . . . .

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Acoustic and Vibration Instrumentation 8.1 Measuring system and transducers . . . . . . . . . . . . . 8.2 Measurement of sound pressure . . . . . . . . . . . . . . 8.2.1 Choice of microphones . . . . . . . . . . . . . . . 8.2.2 The sound pressure sensor–condenser microphone 8.2.3 Sound level meter . . . . . . . . . . . . . . . . . . 8.2.4 Acoustic calibrator . . . . . . . . . . . . . . . . . 8.2.5 Level recorder . . . . . . . . . . . . . . . . . . . 8.3 Acoustic measurement procedure . . . . . . . . . . . . . . 8.3.1 Effect of the operator on measurement results . . . 8.3.2 Measurement position . . . . . . . . . . . . . . . 8.3.3 Standing waves . . . . . . . . . . . . . . . . . . . 8.3.4 Measurements of ambient sound pressure levels . . 8.3.5 Corrections for background sound during source measurements . . . . . . . . . . . . . . . . . . . . 8.3.6 Polar plots . . . . . . . . . . . . . . . . . . . . . 8.4 Vibration measurements . . . . . . . . . . . . . . . . . . 8.4.1 Theory of operation of vibration-measuring transducer . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Characteristics of piezoelectric accelerometers . . 8.4.3 Other vibration transducers . . . . . . . . . . . . . 8.5 Frequency analyzers . . . . . . . . . . . . . . . . . . . . 8.6 Sound power and sound pressure . . . . . . . . . . . . . . 8.7 Indirect methods of sound power measurement . . . . . . 8.7.1 Determination of sound power in an anechoic/ semianechoic room . . . . . . . . . . . . . . . . . 8.7.2 Reverberation room . . . . . . . . . . . . . . . . . 8.7.3 Juxtaposition principle using a reference sound source . . . . . . . . . . . . . . . . . . . .

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xix Direct method of sound power measurement — sound intensity technique . . . . . . . . . . . . . . . . . . . . 8.8.1 Historical perspective . . . . . . . . . . . . . . . 8.8.2 Theoretical background . . . . . . . . . . . . . 8.8.3 Sound intensity probe . . . . . . . . . . . . . . 8.8.4 External noise suppression . . . . . . . . . . . . 8.8.5 Error considerations . . . . . . . . . . . . . . . 8.8.6 Dynamic capability and pressure-intensity index Standard for testing acoustic performance of rotating electrical machines . . . . . . . . . . . . . . . . . . . . 8.9.1 Background . . . . . . . . . . . . . . . . . . . . 8.9.2 Acoustic tests on an induction motor . . . . . . .

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9 Numerical Analysis 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 FEM model for radial magnetic pressure . . . . . . . . . . . . . 9.2.1 Induction motor . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Permanent magnet synchronous motor . . . . . . . . . . 9.3 FEM for structural modeling . . . . . . . . . . . . . . . . . . . 9.4 BEM for acoustic radiation . . . . . . . . . . . . . . . . . . . . 9.4.1 Governing equation and boundary conditions . . . . . . 9.4.2 FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3 BEM . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.4 Radiating sphere . . . . . . . . . . . . . . . . . . . . . 9.4.5 Application to the prediction of radiated acoustic power from an inverter-fed induction motor . . . . . . . . . . . 9.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Statistical Energy Analysis 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 10.2 Power flow between linearly coupled oscillators . . . . 10.2.1 Two coupled oscillators . . . . . . . . . . . . 10.2.2 Three series coupled oscillators . . . . . . . . 10.2.3 Energy exchange between groups of oscillators 10.3 Coupled multimodal systems . . . . . . . . . . . . . . 10.3.1 General SEA equations . . . . . . . . . . . . . 10.3.2 SEA model establishment . . . . . . . . . . . 10.3.3 SEA parameters . . . . . . . . . . . . . . . . 10.3.4 Limitations of SEA . . . . . . . . . . . . . . . 10.4 Experimental SEA . . . . . . . . . . . . . . . . . . . 10.4.1 General theory . . . . . . . . . . . . . . . . . 10.4.2 Recent developments . . . . . . . . . . . . . . 10.5 Application to electrical motors . . . . . . . . . . . . . 10.5.1 Subsystems of a motor structure . . . . . . . . 10.5.2 Internal and coupling loss factors . . . . . . .

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Contents 10.5.3 Input power to the stator . . . . . . . . . . . . . . . . . . 293 10.5.4 Sound power radiated from the motor structure . . . . . . 295

11 Noise Control 11.1 Mounting . . . . . . . . . . . . . . . . . . . . . 11.1.1 Foundation . . . . . . . . . . . . . . . . 11.1.2 Principles of vibration and shock isolation 11.1.3 Vibration limits . . . . . . . . . . . . . . 11.1.4 Shaft alignment . . . . . . . . . . . . . . 11.2 Standard methods of noise reduction . . . . . . . 11.3 Active noise and vibration control . . . . . . . . 11.3.1 Principles of active noise control . . . . . 11.3.2 Induction motor acoustic noise reduction 11.3.3 Active vibration isolation . . . . . . . . .

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Appendix A Basics of Acoustics A.1 Sound field variables and wave equations A.2 Sound radiation from a point source . . . A.3 Decibel levels and their calculations . . . A.4 Spectrum analysis . . . . . . . . . . . . .

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Appendix B Permeance of Nonuniform Air Gap 327 B.1 Permeance calculation . . . . . . . . . . . . . . . . . . . . . . . 327 B.2 Eccentricity effect . . . . . . . . . . . . . . . . . . . . . . . . . . 328 Appendix C Magnetic Saturation

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Appendix D Basics of Vibration 337 D.1 A mass–spring–damper oscillator . . . . . . . . . . . . . . . . . . 337 D.2 Lumped parameter systems . . . . . . . . . . . . . . . . . . . . . 339 D.3 Continuous systems . . . . . . . . . . . . . . . . . . . . . . . . . 342 Symbols and Abbreviations

347

Bibliography

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1 Generation and Radiation of Noise in Electrical Machines 1.1

Vibration, sound, and noise

Vibration is a limited reciprocating motion of a particle of an elastic body or medium in alternately opposite directions from its position of equilibrium, when that equilibrium has been disturbed. In order to vibrate, the body or system must have two characteristics: elasticity and mass. The amplitude of vibration is the maximum displacement of a vibrating particle or body from its position of rest. Sound is defined as vibrations transmitted through an elastic solid, liquid, or gas with frequencies in the approximate range of 20 to 20,000 Hz, capable of being detected by human ears. Pitch is the perceived tone of a sound which is determined by the sound wave frequency. A sound with a high frequency (short wavelength) has a high pitch, while a sound with low frequency (long wavelength) has a low pitch. Noise is disagreeable or unwanted sound. Distinction is made between airborne noise and noise traveling through solid objects. Airborne noise is the noise caused by the movement of large volumes of air and the use of high pressure. Structure-borne noise is the noise carried by means of vibrations of solid objects.

1.2

Sound waves

A sound wave is generated by a vibrating object and can be defined as a mechanical disturbance advancing with a finite speed through a medium. Sound waves are small-amplitude adiabatic oscillations characterized by wavespeed, wavelength, frequency, and amplitude (Appendix A). In air, sound waves are longitudinal waves, that is, with displacement in the direction of propagation. In other words, the motion of the individual particles of the medium is in the direction that is 1 Copyright © 2006 Taylor & Francis Group, LLC

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Noise of Polyphase Electric Motors

parallel to the direction of the energy transport. Transverse waves are those with vibrations perpendicular to the direction of travel of the wave and exist in the elastic medium. Examples of transverse waves include waves on a string and electromagnetic waves. Only transverse waves can be polarized, i.e., can have orientation. Polarized waves oscillate in only one direction perpendicular to the line of travel. For example, the polarization of an electromagnetic wave is defined as the orientation of the electric field vector. The electric field vector is perpendicular to both the direction of travel and the magnetic field vector. Polarized waves can be formed from unpolarized waves by passing them through some polarizing process, e.g., a train of unpolarized waves in a rope can be polarized by passing them through a narrow physical gap. Sound waves cannot be polarized. Unpolarized waves can oscillate in any direction in the plane perpendicular to the direction of travel and have no preferred plane of polarization. All sound waves have common behavior under a number of standard situations and exhibit: • reflection, i.e., the phenomenon of a propagating wave being thrown back from a surface between two media with different mechanical properties; • refraction, i.e., the change in direction of a propagating wave when passing from one medium to another; • diffraction, i.e., the process of spreading out of waves, e.g., when they travel through a small slit or go around an obstacle; • scattering, i.e., the change in direction of motion; • interference, i.e., mutual influence of two waves, e.g., the addition of two waves that come in to contact with each other; • absorption, i.e., the incident sound that strikes a material that is not reflected back; • dispersion, i.e., the splitting up of a wave depending on frequency. Sound amplitude can be measured as sound pressure level (SPL), sound intensity level (SIL), sound power level (SWL), and sound energy density (SED) (Appendix A). A human ear can perceive sound waves of sufficient intensity whose frequencies are approximately within the limits from 16 to 20, 000 Hz (audio-frequency range). There is a minimum sound intensity for a given frequency at which the sound can be perceived by the human ear. The minimum sound intensity is different for different frequencies and is called the threshold of audibility. Figure 1.1 shows the audibility zone for the whole audio-frequency range. The range of the sound intensity that can be perceived by the ear is from 10−12 to 1 W/m2 corresponding to 20 µPa sound pressure. The maximum sound intensity at which the ear feels a pain is called the threshold of pain. Sound amplitudes that are extremely loud (at the threshold of pain) have pressure amplitudes of only 100 Pa. Some environmental noise levels are compared in Figure 1.2. Typical sound power levels for common sounds are also given in Table 1.1

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Generation and Radiation of Noise in Electrical Machines 150

Zo ne ty

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ili

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Figure 1.1 Sound intensity and audibility zone as a function of frequency. dB 140 Threshold of Pain 130 120 Jet Airliner 110 100 Heavy City Traffic 90 Beginning of Hearing Damage 80 70 60 50 40

Average Street Traffic Normal Conversation Business Office Living Room

30 20 10 0

Threshold of Audibility

Figure 1.2 Comparison of some environmental noise levels.

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Noise of Polyphase Electric Motors Table 1.1 Typical sound power levels. Source of noise Quietest audible sound for persons under normal conditions Rustle of leaves Soft whisper, room in a quiet dwelling at midnight Voice, low Mosquito buzzing Department store, clothing department Modern elevator propulsion motor Normal conversation Bird singing Large department store Busy restaurant or canteen Voice, conversation 10-kW, 4-pole cage induction motor Normal street traffic Pneumatic tools Alarm clock ringing Buses, trucks, motorcycles Small air compressors Loud symphonic music Lawn mower Your boss complaining Heavy city traffic Air compressor Heavy diesel vehicle Permanent hearing loss (exposed full-time) Car on highway Steel plate falling Magnetic drill press Vacuum pump Hard rock music Jet passing overhead Jackhammer Jolt squeeze hammer Jet plane taking off Saturn rocket

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Sound power level, dB(A) 10 15 30 40 45 48 50 55 60 70 75 80

90

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1.3 Sources of noise in electrical machines The frequency of interest for vibration is generally within 0 to 1000 Hz, and for noise is over 1000 Hz. Vibration and noise produced by electrical machines can be divided into three categories: • electromagnetic vibration and noise associated with parasitic effects due to higher space and time harmonics, eccentricity, phase unbalance, slot openings, magnetic saturation, and magnetostrictive expansion of the core laminations; • mechanical vibration and noise associated with the mechanical assembly, in particular bearings; • aerodynamic vibration and noise associated with flow of ventilating air through or over the motor. The load induced sources of noise include: • noise due to coupling of the machine with a load, e.g., shaft misalignment, belt transmission, elevator sheave with ropes, tooth gears, coupling, reciprocating compressor; • noise due to mounting the machine on foundation or other structure. The noise from its source is transmitted through the medium (structure, air) to the recipient (human being, sensor) of the noise. The process of noise generation and transmission in electrical machines is illustrated in Figure 1.3. Basics of acoustics are explained in Appendix A.

1.3.1

Electromagnetic sources of noise

Electromagnetic vibration and noise are caused by generation of electromagnetic fields (Chapter 2). Both stator and rotor excite magnetic flux density waves in the air gap. If the stator produces Bm1 cos(ω1 t + kα + φ1 ) magnetic flux density wave and rotor produces Bm2 cos(ω2 t +lα + φ2 ) magnetic flux densisty wave, then their product is 0.5Bm1 Bm2 cos[ω1 + ω2 )t + (k + l)α + (φ1 + φ2 )] + 0.5Bm1 Bm2 cos[ω1 − ω2 )t + (k − l)α + (φ1 − φ2 )]

(1.1)

where Bm1 and Bm2 are the amplitudes of the stator and rotor magnetic flux density waves, ω1 and ω2 are the angular frequencies of the stator and rotor magnetic fields, φ1 and φ2 are phases of the stator and rotor magentic flux desnity waves, k = 1, 2, 3, . . ., and l = 1, 2, 3, . . .. The product expressed by Equation 1.1 is proportional to magnetic stress wave in the air gap with amplitude Pmr = 0.5Bm1 Bm2 , angular frequency ωr = ω1 ±ω2 , order r = k ±l and phase φ1 ±φ2 . The magnetic stress (or magnetic pressure) wave acts in radial directions on the stator and rotor active surfaces causing the deformation and, hence, the vibration and noise. The slots, distribution of windings in slots, input current waveform distortion, air gap permeance fluctuations, rotor eccentricity, and phase unbalance give

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Figure 1.3 Noise generation and transmission in electrical machines.

rise to mechanical deformations and vibration. Magnetomotive force (MMF) space harmonics, time harmonics, slot harmonics, eccentricity harmonics, and saturation harmonics, produce parasitic higher harmonic forces and torques. Especially, radial force waves in a.c. machines, which act both on the stator and rotor, produce deformation of the magnetic circuit. The stator-frame (or stator-enclosure) structure is the primary radiator of the machine noise. If the frequency of the radial force is close to or equal to any of the natural frequencies of the stator–frame system, resonance occurs, leading to the stator system deformation, vibration, and acoustic noise. Magnetostrictive noise of electrical machines in most cases can be neglected due to low frequency 2 f and high order r = 2 p of radial forces, where f is the fundamental frequency and p is the number of pole pairs. However, radial forces due to the magnetostriction can reach about 50% of radial forces produced by the air gap magnetic field. In inverter fed motors, parasitic oscillating torques are produced due to higher time harmonics in the stator winding currents. These parasitic torques are, in general, greater than oscillating torques produced by space harmonics. Moreover,

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the voltage ripple of the rectifier is transmitted through the intermediate circuit to the inverter and produces another kind of oscillating torque [200].

1.3.2

Mechanical sources of noise

Mechanical vibration and noise (Chapter 7) is mainly due to bearings, their defects, journal ovality, sliding contacts, bent shaft, rotor unbalance, shaft misalignment, couplings, U-joints, gears etc. The rotor should be precisely balanced as it can significantly reduce the vibration. The rotor unbalance causes rotor dynamic vibration and eccentricity which in turn results in noise emission from the stator, rotor, and rotor support structure. Both rolling and sleeve bearings are used in electrical machines. The noise due to rolling bearings depends on the accuracy of bearing parts, mechanical resonance frequency of the outer ring, running speed, lubrication conditions, tolerances, alignment, load, temperature, and presence of foreign materials. The noise due to sleeve bearings is generally lower than that of rolling bearings. The vibration and noise produced by sleeve bearings depends on the roughness of sliding surfaces, lubrication, stability and whirling of the oil film in the bearing, manufacture process, quality, and installation.

1.3.3

Aerodynamic noise

The basic source of noise of an aerodynamic nature (Chapter 7) is the fan. Any obstacle placed in the air stream produces noise. In nonsealed motors, the noise of the internal fan is emitted by the vent holes. In totally enclosed motors, the noise of the external fan predominates. According to the spectral distribution of the fan noise, there is broad-band noise (100 to 10, 000 Hz) and siren noise (tonal noise). Siren noise can be eliminated by increasing the distance between the impeller and the stationary obstacle.

1.4 Energy conversion process Figure 1.4 shows how the electrical energy is converted into acoustic energy in an electrical machine. The input current interacts with the magnetic field producing high-frequency forces that act on the inner stator core surface (Figure 1.5). These forces excite the stator core and frame in the corresponding frequency range and generate mechanical vibration and noise. As a result of vibration, the surface of the stator yoke and frame displaces with frequencies corresponding to the frequencies of forces. The surrounding medium (air) is excited to vibrate, too, and generates acoustic noise. The radiated acoustic power is very small, approximately 10−6 to 10−4 W for an electrical motor rated below 10 kW. It is, therefore, not easy to calculate the acoustic power with reasonable accuracy.

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Electric Power Forces Supply Electromagnetic System

Mechanical System

Displacement

Acoustic Environment

Acoustic Noise

Figure 1.4 Conversion of electric energy into acoustic energy in electrical machines. The stator and frame assembly, as a mechanical system, is characterized by a distributed mass M, damping C, and stiffness K . The electromagnetic force waves excite the mechanical system to generate vibration. The amplitude of vibration is a function of the magnitude and frequency of those forces (Appendix D). The mechanical system can be simply described by a lumped parameter model with N degrees of freedom in the following matrix form ¨ + [C]{q} [M]{q} ˙ + [K ]{q} = {F(t)}

(1.2)

where q is an (N , 1) vector expressing the displacement of N degrees of freedom, {F(t)} is the force vector applying to the degrees of freedom, [M] is the mass matrix, [C] is the damping matrix and [K ] is the stiffness matrix. Equation 1.2

Figure 1.5 Mechanism of generation of vibration and noise in electrical machines.

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can be solved using a structural finite element method (FEM) package. In practice, there are difficulties with predictions of the [C] matrix for laminated materials, physical properties of materials, and errors in calculation of magnetic forces [213].

1.5

Noise limits and measurement procedures for electrical machines

Acoustic quantities (Appendix A) can be expressed in terms of the sound pressure level (SPL) or sound power level (SWL). The sound pressure level is the most common descriptor used to characterize the loudness of an ambient sound level. In general, it is more complicated to measure the sound power level than the sound pressure level. The sound power level measurement is independent of the surface of the machine and environmental conditions. According to the National Electrotechnical Manufacture’s Association (NEMA) [162], the sound pressure level L p A can be related to the sound power level L W A in dB(A), as follows
L p A = L W A − 10 log10

2πrd2 S0

 (1.3)

where L p A is the average sound pressure level in a free-field over a reflective plane on a hemispherical surface at 1 m distance from the machine, rd = 1.0 + 0.5lm , lm is the maximum linear dimension of the tested machine in meters, and S0 = 1.0 m2 . The noise of electrical machines depends on the type of the machine, its topology, size, design, construction, enclosure, materials, manufacturing, rated power, speed, tolerances, mounting, support, foundation, coupling, bearings, supply, load, etc. Some consequences of noise as, for example, manufacturing, mounting or support are very difficult to predict. In general, the equations for sound pressure level or sound power level as functions of rotational speed n, rated output power Pout , or torque T have the following forms: L p1 = A1 + B1 log10 n

(1.4)

L p2 = A2 + B2 log10 Pout L p3 = A3 + B3 log10 T

(1.5) (1.6)

where A1 , A2 , A3 , B1 , B2 , and B3 are constants. When a motor is tested at no load under conditions specified by [162], the sound power level of the motor shall not exceed values given in Tables 1.2 and 1.3. The enclosures of motors are an open drip proof machine (ODP) type, totally enclosed fan cooled machine (TEFC) type, and weather protected type II machine (WPII) type. The WPII machine is a guarded machine with its ventilating passages at both intake and discharge so arranged that high velocity air and airborne particles blown into the machine by storms or high winds can be discharged without entering

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Noise of Polyphase Electric Motors

Table 1.2 Maximum A-weighted sound power levels L w A in dB(A) at no load for motors with rated speeds 1200 rpm and less according to NEMA [162]. Rated power kW hp 0.37 0.5 0.5 0.75 0.75 1.0 1.1 1.5 1.5 2.0 2.2 3.0 3.0 5.0 5.5 7.5 7.5 10 11 15 15 20 17 25 22 30 30 40 40 50 45 60 55 75 75 100 100 125 110 150 150 200 185 250 220 300 260 350 300 400 350 450 370 500 450 600 520 700 600 800 670 900 750 1000 930 1250 1,100 1500 1,300 1750 1,500 2000

900 rpm and less ODP TEFC WPII 67 67 67 67 69 69 69 69 70 72 70 72 73 76 73 76 76 80 76 80 79 83 79 83 81 86 81 86 84 89 84 89 87 93 87 93 93 96 92 95 97 92 95 97 92 95 97 92 98 100 96 98 100 96 98 100 96 99 102 98 99 102 98 99 102 98 99 102 98 101 105 100 101 105 100 101 105 100 101 105 100 103 107 102 103 107 102 103 107 102

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901 to 1200 rpm ODP TEFC WPII 65 65 67 67 72 72 76 76 81 81 83 83 86 86 88 88 91 91 96 99 99 99 99 102 102 102 102 102 105 105 105 105 107 107 107

64 64 67 67 71 71 75 75 80 80 83 83 86 86 90 90 94 94 98 100 100 100 100 103 103 103 103 103 106 106 106 106 109 109 109

97 97 97 97 99 99 99 99 99 101 101 101 101 103 103 103

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Table 1.3 Maximum A-weighted sound power levels L w A in dB(A) at no load for motors with rated speeds 1201 to 3600 rpm according to NEMA [162]. Rated power kW hp 0.75 1.0 1.1 1.5 1.5 2.0 2.2 3.0 4.0 5.0 5.5 7.5 7.5 10 11 15 15 20 17 25 22 30 30 40 40 50 45 60 55 75 75 100 100 125 110 150 150 200 185 250 220 300 260 350 300 400 335 450 370 500 450 600 520 700 600 800 670 900 750 1000 930 1250 1,100 1500 1,300 1750 1,500 2000 1,700 2250 1,850 2500 2,250 3000

1201 to 1800 rpm ODP TEFC WPII 70 70 70 70 70 70 72 74 73 74 76 79 76 79 80 84 80 84 80 88 80 88 84 89 84 89 86 95 86 95 89 98 89 100 93 100 93 103 103 105 99 103 105 99 103 105 99 103 105 99 106 108 102 106 108 102 106 108 102 106 108 102 108 111 104 108 111 104 108 111 104 108 111 104 109 113 105 109 113 105 109 113 105 109 113 105 110 115 106 110 115 106

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1801 to 3600 rpm ODP TEFC WPII 76 76 76 80 80 82 82 84 84 86 86 89 89 94 94 98 98 101 101 107 107 107 107 110 110 110 110 111 111 111 111 112 112 112 112 114

85 85 88 88 91 91 94 94 94 94 100 100 101 101 102 104 104 107 107 110 110 110 110 113 113 113 113 116 116 116 116 118 118 118 118 120

102 102 102 102 105 105 105 105 106 106 106 106 107 107 107 107 109

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Table 1.4 Expected incremental increase over no-load condition in A-weighted sound power levels L w A , dB(A) for rated load condition for single-speed, threephase, cage induction motors according to NEMA [162] and IEC 60034-9 Standards [93]. Rated power kW hp 1 to 11 1.0 to 15 1 to 37 15 to 50 37 to 110 50 to 150 110 to 400 150 to 500

2p = 2 2 2 2 2

Number of poles 2p = 4 2p = 6 5 7 4 6 3 5 3 4

2p = 8 8 7 6 5

the internal ventilating passages leading directly to the electric parts of the machine itself. The sound power level at rated load should be adjusted according to Table 1.4. The increase in the sound power level under load is mostly due to the change in the air gap magnetic flux density harmonic amplitudes. This effect can be expressed by the following equation [137]
 Bload 2 L W = 10 log10 (1.7) Bnoload Table 1.5 shows maximum sound pressure level at 1 m from the machine surface according to IEC 60034-9 Standards [93]. Table 1.6 shows maximum sound power level according to IEC 60034-9 Standards [93]. The sound pressure level spectrum is the distribution of effective sound pressures measured as a function of frequency in specified frequency bands. It can also be defined as the resolution of a signal into components, each of different frequency and different amplitude (Figure 1.6). If the sound pressure level spectrum is given in a form of the Fourier series p=

kmax 

Pmk sin(ωk t + φk )

(1.8)

k=1

where Pmk is the amplitude of the kth harmonic, ωk = 2π k f is the angular fequency of the kth harmonic, and φk is the phase angle for the kth harmonic, the overall sound pressure level is calculated as a sum of amplitudes squared, i.e., P=

kmax 

2 Pmk

W.

(1.9)

k=1

The sound pressure level in dB is then calculated according to Equation A.25. The broad-band noise is the noise in which the acoustic energy is distributed over a relatively wide range of frequencies. The spectrum is generally smooth and continuous. The narrow-band noise is the noise in which the acoustic energy is concentrated in a relatively narrow range of frequencies. The spectrum will generally show a localized “hump” or peak in amplitude. Narrow-band sound may be superimposed on broad-band sound.

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Table 1.5 IEC 60034-9 limits for sound pressure level at 1 m from machine surface, dB(A) [93]. Rated power kW Pout < 1.1 1.1 < Pout < 2.2 2.2 < Pout < 5.5 5.5 < Pout < 11 11 < Pout < 22 22 < Pout < 37 37 < Pout < 55 55 < Pout < 110 110 < Pout < 220 220 < Pout < 400 Rated power kW Pout < 1.1 1.1 < Pout < 2.2 2.2 < Pout < 5.5 5.5 < Pout < 11 11 < Pout < 22 22 < Pout < 37 37 < Pout < 55 55 < Pout < 110 110 < Pout < 220 220 < Pout < 400

1.6

n < 960 rpm ODP TEFC 67 69 72 72 75 75 78 77.5 79.5 78.5 80.5 82 94 85 87 86 88 1900 < n < 2360 ODP TEFC 74 78 82 81 86 83.5 87.5 85.5 89.5 88 94 90.5 93.5 93 96 94 98

960 < n < 1320 ODP TEFC 70 70 74 75 78 78 82 80.5 83.5 82.5 85.5 85 89 87 91 89 92 2360 < n < 3150 ODP TEFC 75 80 83 84 87 86.5 90.5 88.5 92.5 93 96 92.5 93.5 95 98 95 99

1320 < n < 1900 ODP TEFC 71 73 77 81 81 81.5 85.5 83 86 86 88 88.5 91.5 90.5 93.5 92.5 95.5 3150 < n < 3750 ODP TEFC 77 82 85 87 90 90 93 92 95 95.5 98.5 95 98 96 100 98 102

Deterministic and statistical methods of noise prediction

In the efforts to predict the noise emitted from an electric machine, there are two approaches: deterministic and statistical methods. In the deterministic method, shown in Figure 1.7a, the electromagnetic forces acting on a motor structure have to be calculated from the input currents and voltages using an electromagnetic analytical model [254] or the FEM model [226]. The vibration characteristics are then determined using a structural model normally based on the FEM [223, 230]. By using the vibration velocities on the motor structure predicted from the structural model, the radiated sound power level can then be calculated on the basis of an acoustic model.1 The acoustic model may be formulated using either the FEM or boundary-element method (BEM). Generally, for calculating 1 The only commercial FEM software for calculating the electromagnetic field, stator vibration mode shapes, and acoustic noise of electrical machines is JMAG-Studio developed by Japan Research Institute, Ltd., Engineering Technology Division, Tokyo-Osaka-Nagoya, http://www.iri.co.jp/proeng/jmag/e/jmg/.

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Noise of Polyphase Electric Motors Table 1.6 IEC 60034-9 limits for sound power level, dB(A) [93].

Rated power n < 960 rpm 960 < n < 1320 kW ODP TEFC ODP TEFC Pout < 1.1 76 79 1.1 < Pout < 2.2 79 80 2.2 < Pout < 5.5 82 84 5.5 < Pout < 11 82 85 85 88 11 < Pout < 22 86 89 89 93 22 < Pout < 37 89 91 92 95 37 < Pout < 55 90 92 94 97 55 < Pout < 110 94 96 97 101 110 < Pout < 220 98 100 100 104 220 < Pout < 400 100 102 103 106 Rated power 1900 < n < 2360 2360 < n < 3150 kW ODP TEFC ODP TEFC Pout < 1.1 83 84 1.1 < Pout < 2.2 87 89 2.2 < Pout < 5.5 92 93 5.5 < Pout < 11 91 96 94 97 11 < Pout < 22 94 98 97 101 22 < Pout < 37 96 100 99 103 37 < Pout < 55 99 103 101 105 55 < Pout < 110 102 105 104 107 110 < Pout < 220 105 108 107 110 220 < Pout < 400 107 111 108 112

Figure 1.6 Sound pressure level spectrum.

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1320 < n < 1900 ODP TEFC 80 83 87 88 91 92 96 94 97 97 99 99 103 103 106 106 109 3150 < n < 3750 ODP TEFC 86 91 95 97 100 100 103 102 105 104 107 106 109 108 112 110 114

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Generation and Radiation of Noise in Electrical Machines Input Data

Input Data

Electromagnetic Force Model

Electromagnetic Force Model

Force

15

Force Mobility Model Input Power

Structural Model Vibration Velocity Acoustic Model

Sound Power Level (a)

Statistical Energy Model Vibration Power Radiation Efficiency Models

Sound Power Level (b)

Figure 1.7 Flowcharts for noise prediction: (a) deterministic method; (b) statistical method. the noise radiated into a space, the BEM is preferred because only the surface of the motor needs to be discretized and the space does not have to be discretized. Although, the analytical and FEM/BEM numerical approaches seem to work well, there are quite a number of limitations for it to be applied in practice (Section 1.8). In the deterministic approach, sometimes, simplified models can be utilized and analytical calculations can be implemented by writing a Mathcad2 or Mathematica3 computer program for fast prediction of the sound power level spectrum generated by magnetic forces. The accuracy due to physical errors may not be high, but the time of computation is very short and it is very easy to introduce and manage the input data set. The main program consists of the input data file, electromagnetic module, structural module (natural frequencies of the stator system), and acoustic module. The following effects can be included: phase current unbalance, higher space 2 Industry

standard technical calculation tool for professionals, educators, and college students. integrated technical computing environment used by scientists, engineers, analysts, educators, and college students. 3 Fully

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Noise of Polyphase Electric Motors

harmonics, higher time harmonics, slot openings, slot skew, rotor static eccentricity, rotor dynamic eccentricity, armature reaction, magnetic saturation. An auxiliary program calculates the torque ripple, converts the tangential magnetic forces into equivalent radial forces, and transfers radial forces due to the torque ripple to the main program. The input data file contains the dimensions of the machine and its stator and rotor magnetic circuit, currents (including unbalanced system and higher time harmonics), winding parameters, material parameters (specific mass, Young modulus, Poisson’s ratio), speed, static and dynamic eccentricity, skew, damping factor as a function of frequency, correction factors, e.g., for the stator systems natural frequencies, maximum force order taken into consideration, minimum magnetic flux density to exclude all magnetic flux density harmonics below the selected margin. The rotor magnetic flux density waveforms are calculated on the basis of MMF waveforms and permeances of the air gap. Magnetic forces are calcualated on the basis of Maxwell stress tensor. The natural frequencies of the stator system are calculated with the aid of equations given in Chapter 5. Those values can be corrected with the aid of correction factors obtained, e.g., from the FEM structural package. Then, using the damping coeffcient as a function of frequency, amplitudes of radial velocities are calculated. The damping factor affects significantly the accuracy of computation. Detailed research has shown that the damping factor is a nonlinear function of natural frequencies. The radiation efficiency factor (Chapter 6), acoustic impedance of the air and amplitudes of radial velocities give the sound power level spectrum (narrow band noise). The overall noise can be found on the basis of Equation 1.9. The overall sound power level calculated in such a way is lower than that obtained from measurements because computations include only the noise of magnetic origin (mechanical noise caused by bearings, shaft misalignment, and fan is not taken into account) and usually, the calculation is done for low number of harmonics of magnetic flux density waves. The FEMs/BEMs, by their nature, are limited to low frequencies. This is because the number of elements required for the model increases by a factor of 8 when the upper frequency of interest is doubled and the number of vibration modes increases significantly with frequency [230]. If the FEMs/BEMs are applied to a large motor for frequencies up to 10,000 Hz, the number of elements and the computing time required will become prohibitive, as discussed in [223]. A method that is particularly suitable for calculations of noise and vibration at high frequencies is the so-called statistical energy analysis (SEA) (Chapter 10), which has been applied with success to a number of mechanical systems such as ship, car, and aircraft structures [140]. This method, however, was applied for the first time to electrical motors in 1999 [43, 223, 230]. The method basically involves dividing a structure (such as a motor) into a number of subsystems and writing the energy balance equations for each subsystem, thus allowing the statistical distribution of energies over various frequency bands to be determined. This method is normally valid for high frequencies where the modal overlap is high [140]. An outline of the calculation procedure using the statistical method is given in Figure 1.7b. The main advantage of the statistical approach is that it does not require all the

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17

details to be modeled. The accurate distribution of the electromagnetic force might not be so important; only the total force in a frequency band is required. Thus, the electromagnetic force needs not be calculated using a FEM model and an approach such as that adopted by Cho and Kim [31] might be suitable. By considering the motor as a simple cylindrical shell, the input power due to this electromagnetic force can be formulated using an analytical ”mobility” model. By invoking a statistical energy model, this input power can then be distributed as vibrational power to different subsystems which make up the motor. If the sound radiation efficiencies of these subsystems are known, then the sound power due to each subsystem can be calculated. Since a motor structure can be decomposed into simple structural elements such as cylindrical shells, plates, and beams, the radiation efficiencies of these simple structural elements can be determined analytically, as depicted in the radiation efficiencies model in Figure 1.7b.

1.7

Economical aspects

Figure 1.8 shows the distribution of the magnetic flux density in the magnetic circuit of a 4-pole permanent magnet (PM) brushless machine. The magnetic flux density in the stator return path (yoke) is proportional to the magnetic flux density in the air gap and can be a measure of both the noise of electromagnetic origin and machine 1.8795e+000 1.8012e+000 1.7229e+000 1.6446e+000 1.5663e+000 1.4879e+000 1.4096e+000 1.3313e+000 1.2530e+000 1.1747e+000 1.0964e+000 1.0181e+000 9.3975e−001 8.6144e−001 7.8313e−001 7.0481e−001 6.2650e−001 5.4819e−001 4.6988e−001 3.9156e−001 3.1325e−001 2.3494e−001 1.5663e−002 7.8313e−002 0.0000e+000

Figure 1.8 (See color insert following page 236.) Distribution of the magnetic flux density in the cross section of a 4-pole brushless machine with surface PMs, as obtained from the 2D FEM.

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Noise of Polyphase Electric Motors

Noise, dB; cost × 200 m.u.

95 90

cost, m.u.

85

noise, dB

80

1 m.u. (monetary unit) = cost of 1 kg of laminations or 4.5 kg of copper

75 70 65 60 55 50 0

0.2

1.2 1 0.8 0.6 0.4 1.4 1.6 Magnetic Flux Density in the Stator Yoke, T

1.8

2

Figure 1.9 Noise level and cost plotted against magnetic flux density (MFD) in the stator yoke for a 200 kW induction motor [2].

cost. Figure 1.9 shows the noise and total cost of an induction machine rated at 200 kW, 50 Hz, 380 V, 1480 rpm [2]. To keep the cost independent of inflation, an arbitrary monetary unit (m.u.) has been used that is equal to the price of 1 kg of steel sheets. Using this unit, the copper wire costs 4.5 m.u./kg and aluminum costs 3 m.u./kg [2]. The minimum of cost is different than the minimum of noise. Therefore, low magnetic flux density (MFD) means low level of noise and, vice versa, increased utilization of the magnetic circuit results in increased noise. The minimum of cost is for the MFD in the stator yoke in the range from 0.6 to 1.0 T. The minimum of noise is for lower MFDs; however, due to increase in dimensions and mass of active materials the cost increases sharply at low MFDs in the stator yoke.

1.8

Accuracy of noise prediction

The results of both analytical and numerical noise prediction may significantly differ from measurements. Forces that generate vibration and noise are only small fraction of the main force produced by the interaction of the fundamental current and the fundamental normal component of the magnetic flux density. The power converted into acoustic noise is only approximately 10−6 to 10−4 of the electrical input power. The accuracy of the predicted, say, sound power level spectrum depends not only on how accurate the model is, but also how accurate are the input data, e.g., level of current unbalance, angle between the stator current and q-axis (in PM brushless machines), influence of magnetic saturation on the equivalent slot opening, damping factor, elasticity modulus of the slot content (conductors, insulation, encapsulation), higher time harmonics of the input current (inverter-fed motor), etc.

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All the above input data are difficult to predict with sufficient accuracy. Listed below are the common problems encountered in the analytical and numerical noise prediction [213, 220]: 1. The most difficult task in analytical calculation of sound power level radiated by electrical machines is the accurate prediction of the natural frequencies of the stator structure. At present, the best method to calculate the natural frequencies of the stator is to use the FEM. This is the only technique that can take into account with reasonable accuracy the end bells, mountings (feets or flanges) and asymmetries due to, e.g., terminal boxes. 2. The calculation of matrices of the mass [M] and stiffness [K ] seems to be obvious in the FEM. However, the physical properties of the materials used in electrical machines design are not known. The anisotropy of laminations, internal stresses caused by manufacturing, and change in stiffness matrix [K ] due to temperature variation (differential thermal expansion of the laminations and housing) are mostly not taken into account. 3. The damping matrix [C] in the FEM is difficult to predict. There are no adequate models available for describing damping in laminated materials and structures composed of different types of materials, e.g. copper, insulation, epoxy, laminations. Practice shows that good values for the damping are absolutely essential for predicting accurate vibrational amplitudes. 4. The force vector {F(t)} has to be found in all points on the inner stator surface. Even the most accurate FEM programs introduce a lot of errors in force calculations [213]. Forces are usually calculated analytically in the preprocessor module on the basis of magnetic flux density harmonics or using a 2D FEM. 5. Because neither the analytical approach nor the FEM/BEM computations guarantee accurate results, the laboratory tests are always very important. 6. The main advantage of the SEA is that it does not require all details to be modeled. 7. The vibration and acoustic noise can be calculated on the basis of modal analysis which is free of calculating the electromagnetic forces [213]. Only flux linkages have to be calculated (Section 9.2.2). 8. The calculated noise level is rather lower than the measured noise level. The calculation is mostly done for low harmonic numbers of the air gap permeance. The measurement gives the total noise level due to all harmonics [220].

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2 Magnetic Fields and Radial Forces in Polyphase Motors Fed with Sinusoidal Currents In this chapter the magnetic field in the air gap of polyphase electrical machines will be discussed. The polyphase winding, usually three-phase, is located in the stator core slots. It has been assumed that the stator is fed with a balanced system of sinusoidal phase currents. For polyphase machines with rectangular and trapezoidal current waveforms, e.g., permanent magnet (PM) brushless machines, this assumption is equivalent to taking into account only the fundamental stator current harmonics.

2.1 Construction of induction motors An induction motor consists of two main parts: stator core and rotor core both with polyphase windings (Figure 2.1a). The stator and rotor ferromagnetic cores are stacked of thin steel laminations. Windings are located in stator and rotor slots or can be made as slotless windings. The stator polyphase winding, in most cases, the three-phase winding has independent phase windings that are star (Y) or delta () connected and fed with a three-phase a.c. current system. Each phase winding consists of multiturn coils connected in series. Series connected coils can also be groupped in two or more parallel paths. The air gap between the stator and rotor core is small, usually less than 1 mm for modern small and medium power induction motors. The rotor winding can be made of (1) three-phase Y-connected multiturn coil groups the terminals of which are connected to three slip rings or (2) in the form of a cage winding. In the first case the induction motor is called the phasewound motor or slip ring motor. Brushes are connected to a three phase variable resistance (rheostat) and slide on slip rings. The variable resistance is necessary 21 Copyright © 2006 Taylor & Francis Group, LLC

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Noise of Polyphase Electric Motors

(a)

(b)

Figure 2.1 Cross sections and magnetic flux lines of three-phase motors: (a) 4-pole induction motor; (b) 8-pole PM brushless motor. for starting to reduce the high inrush current and also for speed control. The maximum resistance is at the first instant of starting, when the stator electromotive force (EMF) E 1 = 0. As the speed increases, the stator EMF increases too and the stator and rotor currents decrease. When the speed reaches its nominal value, the slip rings can be short-circuited. The rheostat used for the speed control must be rated at larger power than that for starting. The speed decreases as the resistance increases. Speed control rheostat is designed for continuous operation while the starting resistance is smaller because it is designed only for the short time duty. In cage induction motors the rotor winding consists of axial bars short circuited by two end rings forming a “squirrel cage” winding. A cage winding made of aluminum alloy or brass is not insulated from the rotor laminated stack. The polyphase stator winding when fed with polyphase a.c. currents produces rotating magnetic field in the air gap. The stator rotating field induces currents in the rotor winding which in turn interact with the stator field to produce the electromagnetic torque. For motoring operation the rotor spins in the same direction as the stator rotating field. The speed of the stator rotating magnetic field f ns = (2.1) p where f is the stator current frequency and p is the number of pole pairs, is called the synchronous speed. The slip between the rotor and the stator magnetic field for the fundamental harmonic is defined as ns − nm s= (2.2) ns where n m = (1 − s) f / p is the rotor mechanical speed. The slip speed n s − n m expresses the speed of the rotor relative to the rotating magnetic field of the stator. The current in the rotor creates its own magnetic field rotating with a speed that is given by sf sn s = . (2.3) p

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The frequency of the current in the rotor s f is called the slip frequency. Since n s (1 −s )+sn s = n s , the stator and rotor magnetic fields rotate with the same speed. If the stator fundamental field harmonic rotates with the speed given by Equation 2.1, the stator higher space harmonic waves will rotate with the speed nsν = ±

f ns =± . ν νp

(2.4)

The “+” sign relates to the wave rotating in the same direction as the fundamental wave of the magnetomotive force (MMF) (forward-rotating wave) and the “−” sign relates to the wave rotating in the opposite direction (backward-rotating wave). The slip for the νth space harmonic is sν =

±n s ν − n m n s − (n s − n m ) = 1 ∓ ν(1 − s ) =1∓ ±n s ν n s /ν

(2.5)

where s is the slip (Equation 2.2) for the fundamental space harmonic ν = 1 and n s /ν in the denominator is the speed of the rotor νth harmonic that can have either the “+” or “−” sign. The frequency of the current induced in the rotor winding by the magnetic flux of the stator νth harmonic is f 2ν = f sν = f [1 ∓ ν(1 − s )]

(2.6)

The sign “−” in Equations 2.5 and 2.6 is for higher harmonics rotating in the same direction as the fundamental ν = 1 and the “+” sign is for harmonics rotating in the oposite direction.

2.2

Construction of permanent magnet synchronous brushless motors

The stator of a PM synchronous brushless motor is similar to that of an induction motor (Figure 2.1b). The rotor consists of laminated stack with PMs. Rotor laminations are necessary to reduce the rotor core losses due to higher harmonics. PMs can be of surface, bread loaf, interior, or interior double-layer configuration (Figure 2.2). Stators of PM brushless motors can be sinusoidally excited or trapezoidally excited. In the first case the phase windings are fed with sinusoidal waveforms shifted by 360o /m 1 one from another (m 1 is the number of stator phases) and produce rotating magnetic field. The motor works as a synchronous motor and is called sinewave motor. The trapezoidally excited motor also called square wave motor is fed from from waveforms shifted by 360o /m 1 too, but those waveforms are rectangular or trapezoidal (Figure 2.3). Waveforms according to Figure 2.3b are produced when the stator current (MMF) is precisely synchronized with the rotor instantaneous position and frequency (speed). The most direct and popular method of providing the required rotor position information is to use an absolute angular position

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Noise of Polyphase Electric Motors

q

q

d

S

N

S

S

S

S

N

q

N (b)

d

q

S

N

N

S

S

N

N

(a) S

d

S

N

d

N

S

N

S

N

N (d)

(c)

Figure 2.2 Rotors of PM brushless motors with: (a) surface PMs; (b) bread loaf PMs; (c) interior (embedded) PMs; (d) interior double-layer PMs.

i1 Phase A

i1

Phase B

Phase C ωt 0

60

12 0

18 24 0 0 (a)

30 0

360°

0

60

12 0

18 0

24 0

ωt 30 360° 0

(b)

Figure 2.3 Basic stator winding waveforms for three-phase PM brushless motors: (a) sinusoidal waveforms; (b) rectangular waveforms.

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Magnetic Fields and Radial Forces

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encoder mounted on the rotor shaft. Only two phase windings out of three conduct current simultaneously. Such a control scheme or electronic commutation is functionally equivalent to the mechanical commutation in d.c. motors. Therefore, those motors are called d.c. brushless motors.

2.3

A.C. stator windings

In a single-layer winding, only one coil side is located in each slot. The number of all coils is s1 /2 and the number of coils per phase is n c = s1 /(2m 1 ) where s1 is the number of stator slots and m 1 is the number of phases. In a double-layer winding two sides of different coils are accommodated in each slot. The number of all coils is s1 and the number of coils per phase is n c = s1 /m 1 . The number of slots per pole is s1 Q1 = (2.7) 2p where 2 p is the number of poles. The number of slots per pole per phase is q1 =

s1 . 2 pm 1

(2.8)

The number of conductors per coil can be calculated as • for a single-layer winding Nc =

a p aw N 1 a p aw N 1 a p aw N 1 = = nc s1 /(2m 1 ) pq1

(2.9)

• for a double-layer winding Nc =

a p aw N 1 a p aw N 1 a p aw N 1 = = nc s1 /m 1 2 pq1

(2.10)

where N1 is the number of turns in series per phase, a p is the number of parallel current paths, and aw is the number of parallel conductors. The number of conductors per slot is the same for both single-layer and double-layer windings, i.e., Nsl =

a p aw N 1 . pq1

(2.11)

The full coil pitch measured in terms of the number of slots is y1 = Q 1 where Q 1 is according to Equation 2.7. The short coil pitch can be expressed as y1 = β Q 1 =

wc Q1 τ

(2.12)

where w c is the coil pitch (coil span) measured in units of length and τ is the pole pitch. The coil pitch–to–pole pitch ratio is β=

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wc . τ

(2.13)

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The distribution factor of a coil group for the fundamental space harmonic ν = 1 is defined as the ratio of the phasor sum E q —to—arithmetic sum q1 E c of EMFs E c induced in each coil and expressed by the following equation: kd1 =

Eq sin(π/2m 1 ) = q1 E c q1 sin[π/(2m 1 q1 )]

(2.14)

24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

The pitch factor for the fundamental space harmonic ν = 1 is defined as the ratio of the phasor sum—to—arithmetic sum of the EMFs per coil side and expressed

U1

W2

U2

W1

V1

V2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

(a)

V2

U1

W2

V1

U2

W1 (b)

Figure 2.4 Three-phase, 4-pole windings distributed in s1 = 24 slots with full pitch coil groups (q1 = 2): (a) single-layer, (b) double-layer.

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π (2.15) k p1 = sin β 2 The winding factor for fundamental is the product of the distribution factor (Equation 2.14) times pitch factor (Equation 2.15), i.e.

as:

kw1 = kd1 k p1 .

(2.16)

The angle in electrical degrees between the neighboring slots is γ =

360o 2π p≡ p. s1 s1

(2.17)

Figure 2.4 shows simple three-phase, 4-pole windings distributed in s1 = 24 slots with full pitch coil groups. Stator winding factors for higher space harmonics ν = 2m 1 k ± 1 where k = 1, 2, 3, 4, . . . are expressed by similar equations to Equations 2.14, 2.15, and 2.16, i.e., • stator winding distribition factor for the ν-th space harmonic sin[νπ/(2m 1 )] q1 sin[νπ/(2m 1 q1 )] • stator winding pitch factor for the ν-th space harmonic  
πw  νβπ c k p1ν = sin = sin ν 2 2 τ • the resultant winding factor for the ν-th space harmonic kd1ν =

kw1ν = kd1ν k p1ν .

(2.18)

(2.19)

(2.20)

The values of the stator winding distribution factor for higher space harmonics as a function of the number of slots per pole per phase q1 (Equation 2.8) for m 1 = 3 are given in Table 2.1. Table 2.1 shows that for some space harmonics and q1 ≥ 2 the distribution factor is the same as for the fundamental harmonic, i.e., kd1ν = ±kd1 . Those harmonics s1 ν = 2m 1 q1 k ± 1 = k ± 1 (2.21) p where k = 1, 2, 3, . . . , are called stator slot harmonics. The EMF induced by slot harmonics is not reduced. For q1 = 1 all harmonics become slot harmonics. The angle between two neighboring slots (coil sides) for the ν-th space harmonic is ν times larger. For slot harmonics   2π p s1 γν = νγ = k ± 1 = 2π k ± γ (2.22) p s1 where γ for the fundamental space harmonic ν = 1 is according to Equation 2.17. Similarly, the number of rotor slot harmonics is s2 µ=k ±1 (2.23) p where s2 is the number of rotor slots.

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Noise of Polyphase Electric Motors

Table 2.1 Winding distribution factor kd1ν according to Equation 2.18 for threephase stator windings. ν 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

2.4 2.4.1

q1 2 0.966 0.707 0.259 -0.259 -0.707 -0.966 -0.966 -0.707 -0.259 0.259 0.707 0.966 0.966 0.707 0.259 -0.259 -0.707 -0.966 -0.966 -0.707 -0.259 0.259 0.707 0.966 0.966 0.707

3 0.960 0.667 0.218 -0.177 -0.333 -0.177 0.218 0.667 0.960 0.960 0.667 0.218 -0.177 -0.333 -0.177 0.218 0.667 0.960 0.960 0.667 0.218 -0.177 -0.333 -0.177 0.218 0.667

4 0.958 0.653 0.205 -0.158 -0.271 -0.126 0.126 0.271 0.158 -0.205 -0.653 -0.958 -0.958 -0.653 -0.205 0.158 0.271 0.126 -0.126 -0.271 -0.158 0.205 0.653 0.958 0.958 0.653

5 0.957 0.647 0.200 -0.149 -0.247 -0.109 0.102 0.200 0.102 -0.109 -0.247 -0.149 0.200 0.647 0.957 0.957 0.647 0.200 -0.149 -0.247 -0.109 0.102 0.200 0.102 -0.109 -0.247

6 0.956 0.644 0.197 -0.145 -0.236 -0.102 0.092 0.173 0.084 -0.084 -0.173 -0.092 0.102 0.236 0.145 -0.197 -0.644 -0.956 -0.956 -0.644 -0.197 0.145 0.236 0.102 -0.092 -0.173

∞ 0.955 0.636 0.191 -0.136 -0.212 -0.087 0.073 0.127 0.056 -0.050 -0.091 -0.041 0.038 0.071 0.033 -0.051 -0.058 -0.027 0.026 0.049 0.023 -0.022 -0.042 -0.02 0.019 0.037

Stator winding MMF Single-phase stator winding

Figure 2.5 shows a flat model of an electrical machine with smooth stator and rotor cores. The stator winding consists of one coil per double pole pitch. The relative magnetic permeability of the stator and rotor cores µr → ∞. Coils are fed with d.c. current I and connected in series to create a single-phase winding. For a single coil with Nc turns and span w c = τ the Ampere’s circuital law can be written as   · dl  = 2H g = Nc I H (2.24)

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29

t

t

t

Stator

g

Rotor Fc(a )

ν=1

Fc

ν=3 p 2

NcI

-

5 p 2

p

p

3

2

2

p

3p

a

2p

Figure 2.5 Magnetic field and MMF of a single-phase full-pitch winding. where H is the magnetic field intensity in the air gap g. The MMF of a single coil regarded as a magnetic voltage drop across a single air gap is Nc I 2 Introducing the relative permeance (per unit area) of the air gap as µ0 g = , H/m2 g Fcg = H g =

(2.25)

(2.26)

the flat-topped value of a rectangular wave of the magnetic flux density induced by coils in the air gap is B g = µ0 H =

µ0 Nc I = g Fcg . g 2

(2.27)

In Equations 2.26 and 2.27 the air gap is assumed to be smooth (no stator and rotor slots) and the magnetic circuit is assumed to be unsaturated, i.e., the relative magnetic permeability of the stator and rotor cores µr → ∞. Similar to the magnetic flux density wave, the MMF wave is also rectangular (Figure 2.5) and can be resolved into Fourier series as ∞
π   Fc (x) = (2.28) Fcmν cos ν x τ ν=1,3,5,...

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where the higher space odd harmonics ν = 1, 3, 5, . . . and the peak value of the νth harmonic 
π 
π 4 4 0.5τ Fcmν = Fcg sin ν . (2.29) Fcg cos ν x d x = τ 0 τ νπ 2 The variation of the MMF Fc (x) with the linear coordinate x can also be expressed as a variation of the MMF with the angular coordinate α. The geometrical angle α=

1π 2π x= x pτ s 1 t1

(2.30)

where 2 pτ = s1 t1 , t1 is the stator slot pitch, and νπ x/τ is replaced by νpα. Considering an a.c. machine with sinusoidal excitation, the d.c. current I should be replaced by a sinusoidal current √ i(t) = 2I cos(ωt) (2.31) where ω = 2π f is the stator current angular frequency, f is the input frequency of the sinusoidal current, and I is the stator r ms current. Putting Equations 2.25 and 2.31 into Equations 2.28 and 2.29 √
π 2 2 Fcmν = Nc I sin ν (2.32) νπ 2 and Fc (x, t) =


π  Fcmν cos(ωt) cos ν x . τ ν=1,3,5,... ∞ 

(2.33)

Equation 2.33 shows that the MMF of a phase winding consists of an infinite number of harmonic MMFs changing in time according to cosinusoidal  law cos(ωt) and changing in space also according to cosinusoidal law cos ν πτ x . A doublelayer winding with short coil span w c < τ , i.e., β < 1 can be represented as two windings with full coil spans shifted by (1−β)τ one from another. The magnitude of the ν-th harmonic of the MMF per phase is



π π Fmν = 2Fqν cos ν(1 − β) = 2q1 Fcmν kd1ν cos ν(1 − β) (2.34) 2 2 where the magnitude of the MMF of a group of coils according to Equation 2.14 is Fqν = q1 Fcmν kd1ν (2.35) and the winding distribution factor kd1ν for the νth space harmonic is given by Equation 2.18. Putting Equation 2.32 into Equation 2.34 and regarding that  
νπ 

π νβπ sin cos ν(1 − β) = sin = k p1ν 2 2 2
νπ 
νπ  = 0; sin = ±1 for ν = 1, 3, 5, . . . cos 2 2

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the magnitude of the MMF per phase (Equation 2.34) becomes √


π π 2 2 Nc I kd1ν sin ν cos ν(1 − β) Fmν = 2q1 νπ 2 2 √ 2 2 = 2q1 Nc kd1ν k p1ν I νπ

(2.36)

where the pitch factor k p1ν for the νth space harmonic is given by Equation 2.19. The number of turns Nc per coil of a double-layer stator winding is given by Equation 2.10 and the stator phase current I 1 = a p aw I

(2.37)

where I is the single conductor (aw = 1) coil current. Thus, the amplitude of the ν-th space harmonic of the MMF per phase √ 2 2 N1 kw1ν N1 kw1ν I1 ≈ 0.9 I1 . Fmν = (2.38) π νp νp The resultant winding factor kw1ν for the ν-th harmonic is determined by Equation 2.20. The MMF of a single-phase winding varying in space and time can be resolved into two waves rotating in opposite directions, i.e., F(x, t) =


π  Fmν cos(ωt) cos ν x τ ν=1,3,5,... ∞ 



1 π  1 π  Fmν cos ωt − ν x + Fmν cos ωt + ν x . (2.39) 2 ν τ 2 ν τ ∞

=

∞

Equation 2.39 for each harmonic describes two MMF waves with the same amplitudes 0.5Fmν rotating with the same angular speed ω/ p = 2π f / p in opposite directions. Harmonics 0.5Fmν cos(ωt − νπ x/τ ) rotate in the same direction as the fundamental stator magnetic field (and rotor). Those magnetic field waves are called forward-rotating waves. Harmonics 0.5Fmν cos(ωt + νπ x/τ ) rotate in the opposite direction and are called backward-rotating waves. The speed of the MMF waves can be found assuming that the waves are stationary for an observer moving with the angular speed ω, i.e.,
π  cos ωt ∓ ν x = const. τ Thus,

π π x = const; ωdt ∓ ν d x = 0 τ τ and the linear synchronous speed of the νth harmonic wave is ωt ∓ ν

v sν =

1τ 1 f D1in 1 dx =± ω = ±2π = vs dt νπ ν p 2 ν

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(2.40)

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where ω = 2π f is the angular frequency for fundamental harmonic ν = 1, n s = f / p is the rotational speed of the fundamental frequency of the magnetic field called the synchronous speed (Equation 2.1), D1in = 2 pτ/π is the inner diameter of the stator, and v s = ωD1in /(2 p) = 2π( f / p)D1in /2 is the linear synchronous speed of the fundamental harmonic.

2.4.2

Three-phase stator winding

A three-phase symmetrical stator has three the same windings shifted in space by 2π/3 electrical degrees. The stator is fed with three sinusoidal currents i 1A , i 1B , and i 1C . For a balanced input current system, the amplitude of each phase current √ 2I1 is the same and phase shift is 2π/3 electrical degrees, i.e., √ i 1A = 2I1 cos(ωt)
√ π (2.41) i 1B = 2I1 cos ωt − 2 3
√ π i 1C = 2I1 cos ωt − 4 . 3 Based on Equation 2.39 the space harmonics of the MMF produced by each phase winding are

1 π  1 π  Fmν cos ωt − ν x + Fmν cos ωt + ν x 2 τ 2 τ


 1 2π π = Fmν cos ωt − ν x + 0(ν − 1) 2 τ 3


1 2π π  + Fmν cos ωt + ν x − 0(ν + 1) 2 τ 3

F1Aν =

 2π π x− τ 3     1 2π π 2π + Fmν cos ωt − x− +ν ) 2 3 τ 3


1 2π π  = Fmν cos ωt − ν x + 1(ν − 1) 2 τ 3


1 2π π  + Fmν cos ωt + ν x − 1(ν + 1) 2 τ 3

F1Bν =

F1Cν =

1 Fmν cos 2



ωt −



2π 3





−ν

 4π π x− τ 3     1 4π π 4π + Fmν cos ωt − x− +ν ) 2 3 τ 3 1 Fmν cos 2

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ωt −

4π 3





−ν

(2.42)

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33




2π 1 π  Fmν cos ωt − ν x + 2(ν − 1) 2 τ 3


 1 2π π + Fmν cos ωt + ν x − 2(ν + 1) . 2 τ 3

According to Equation 2.42, harmonic waves of the MMF of each phase are shifted by the angle (ν ∓1)(2π )/3 one from each other. Harmonics ν = 3k = 3, 9, 15, . . . (in general, ν = m 1 k where k = 1, 3, 5, . . . and m 1 = 3) are shifted by (ν ∓ 1)(2π )/3 = (3k ∓ 1)(2π )/3 = 2π k ∓ 2π/3 or 120o . Thus, the sum of those harmonic waves is equal to zero, so that harmonics ν = 3k do not exist in three phase systems. For harmonic waves ν = 6k + 1 (in general, ν = 2m 1 k + 1) the shift angle is (ν − 1)(2π )/3 = 4kπ or 0o . The resultant wave is the arithmetic sum of waves in all three phases. For harmonic waves ν = 6k − 1 (in general, ν = 2m 1 k − 1) the shift angle is (ν − 1)(2π )/3 = 4kπ − 4π/3 or 240o . The resultant wave is 0. The resultant MMF of a three phase symmetrical winding fed with balanced three phase current system does not contain triple harmonic waves. Harmonics ν = 1, 7, 13, 19, . . . rotate forward and harmonics ν = 5, 11, 17, 23, . . . rotate backward. The linear speed of higher space harmonics is expressed by Equation 2.40. The magnitudes of harmonic MMFs according to Equations 2.38 and 2.43 are √ 3 3 2 N1 kw1ν N1 kw1ν Fmν = [Fmν ]m 1 =1 = I1 ≈ 1.35 I1 . (2.43) 2 π νp νp The total MMF of a three-phase symmetrical stator winding fed with a balanced current system ∞
 π  F1 (x, t) = (2.44) Fmν cos ωt ∓ ν x τ ν=6k±1 or F1 (α, t) =

∞ 

Fmν cos(ωt ∓ νpα) =

ν=6k±1

∞ 

Fmν cos(νpα ∓ ωt)

(2.45)

ν=6k±1

where the angle α is determined by Equation 2.30.

2.4.3

Polyphase stator winding

The time-space distribution of the MMF of the k-th phase winding of a symmetrical polyphase winding fed with a balanced current system can be expressed as


∞ 2π 1 π  F1k (x, t) = [Fmν ]m 1 =1 {cos ωt − ν x + (k − 1)(ν − 1) 2 ν=1 τ m1


 π 2π + cos ωt + ν x − (k − 1)(ν + 1) } (2.46) τ m1

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Noise of Polyphase Electric Motors

where ω = 2π f is the angular frequency for the fundamental harmonic, τ is the pole pitch and ν = 2m 1 k ± 1,

k = 0, 1, 2, 3, 4, 5, . . .

(2.47)

The “−” sign before νπ x/τ is for the forward-rotating field and the “+” sign before νπ x/τ is for the backward-rotating field. The total MMF exicted by all m 1 phases is the sum of MMFs excited by each phase. Using the trigonometric identity cos(α ± β) = cos α cos β ∓ sin α sin β, the following equation can be written on the basis of Equation 2.46 F1ν (x, t) = F11 (x, t) + F12 (x, t) + · · · + F1m 1 (x, t) ∞
1 π  = Fmν cos ωt − ν x 2 ν=1 τ 



1 2 × 1 + cos (ν − 1)2π + cos (ν − 1)2π + · · · m1 m1

 ∞
m1 − 1 1 π  cos (ν − 1)2π − Fmν sin ωt − ν x m1 2 ν=1 τ 



1 2 × 0 + sin (ν − 1)2π + sin (ν − 1)2π + · · · m1 m1

 ∞
m1 − 1 1 π  sin (ν − 1)2π + Fmν cos ωt + ν x m1 2 ν=1 τ 



1 2 × 1 + cos (ν + 1)2π + cos (ν + 1)2π + · · · m1 m1

 ∞
m1 − 1 1 π  cos (ν + 1)2π + Fmν sin ωt + ν x m1 2 ν=1 τ 



1 2 × 0 + sin (ν + 1)2π + sin (ν + 1)2π + · · · m1 m1

 m1 − 1 sin (ν + 1)2π . (2.48) m1 Series in square brackets can be written as





1 2 m1 − 1 1+cos (ν ∓ 1)2π + cos (ν ∓ 1)2π + · · · + cos (ν ∓ 1)2π m1 m1 m1

sin(ν ∓ 1)π m1 − 1 sin(ν ∓ 1)π = cos (ν ∓ 1)2π = sin[(ν ∓ 1)π/m 1 ] m1 sin[(ν ∓ 1)π/m 1 ] (2.49)

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Table 2.2 Band factor in Equation 2.51 for three-phase stator windings. ν

sin(ν−1)π sin[(ν−1)π/m 1 ]

sin(ν+1)π sin[(ν+1)π/m 1 ]

0 0 0 3 0 0 3 0 0 3 0 0 3 0

0 0 3 0 0 3 0 0 3 0 0 3 0 0

1 3 5 7 9 11 13 15 17 19 21 23 25 27







1 2 m1 − 1 0 + sin (ν ∓ 1)2π + sin (ν ∓ 1)2π + · · · + sin (ν ∓ 1)2π m1 m1 m1

sin(ν ∓ 1)π m1 − 1 = sin (ν ∓ 1)2π = 0. (2.50) sin[(ν ∓ 1)π/m 1 ] m1 Thus, the resultant MMF of a polyphase stator winding is
π  sin(ν − 1)π 1 cos ωt − ν x Fmν 2 ν=1 sin[(ν − 1)π/m 1 ] τ ∞

F1 (x, t) =


1 π  sin(ν + 1)π cos ωt + ν x . Fmν 2 ν=1 sin[(ν + 1)π/m 1 ] τ ∞

+

(2.51)

The so-called band factor in Equation 2.51 is given in Table 2.2 by calculating sin(ν − 1)π = m1 sin[(ν − 1)π/m 1 ]

for

ν = 2km 1 + 1

k = 1, 2, 3, . . .

sin(ν + 1)π = m1 sin[(ν + 1)π/m 1 ]

for

ν = 2km 1 − 1

k = 1, 2, 3, . . . .

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Noise of Polyphase Electric Motors

Otherwise, sin[(ν ∓ 1)π ]/ sin[(ν ∓ 1)π/m 1 ] = 0 including ν = 1. Finally, for higher space harmonics F1 (x, t) =

∞
1  π  [Fmν ]m 1 =1 m 1 cos ωt − ν x 2 ν=2km +1 τ 1

+

∞ 

1 2 ν=2km

1


π  [Fmν ]m 1 =1 m 1 cos ωt + ν x . τ −1

The magnitude of the νth harmonic of the polyphase stator MMF is √ m1 m 1 2 N1 kw1ν Fmν = [Fmν ]m 1 =1 = I1 . 2 π νp

(2.52)

(2.53)

The winding factor kw1ν for the νth space harmonic is given by Equation 2.20. The peak value of the armature line current density or specific electric loading is defined √ as the number of conductors in all phases 2m 1 N1 times the peak stator current 2I1 divided by the armature stack length 2 pτ , i.e., √ m 1 2N1 I1 Am1 = . (2.54) pτ

2.5

Rotor magnetic field

The amplitude of the rotor MMF of an induction machine is √ m 2 2 N2 kw2 I2 Fm2 = π p

(2.55)

where N2 is the number of the rotor turns per phase, kw2 is the rotor winding factor for the fundamental space harmonic, and I2 is the rotor r ms current. The amplitude of the rotor MMF referred to the stator system √ m 1 2 N1 kw1   Fm2 = I2 (2.56) π p where the rotor current referred to the stator system is calculated as I2 =

m 2 N2 kw2 k s I2 . m 1 N1 kw1

(2.57)

The skewing factor for the fundamental harmonic is ks =

sin[π pbs /(t1 s1 )] sin(0.5π bs /τ ) = 0.5π bs /τ π pbs /(t1 s1 )

(2.58)

where bs is the slot skew. In practice bs = 2π τ/s1 = t1 or bs = 2π τ/s2 = t2 .

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The peak value of the rotor line current density of an induction machine is expressed similar to that of the stator (Equation 2.54), i.e., √ m 2 2N2 I2 Am2 = . (2.59) pτ For a synchronous machine, as the rotor rotates at mechanical angular velocity m = , the excitation field and its space harmonics rotate all at the same mechanical angular velocity . The electrical angular frequencies of the rotor higher harmonics are all different and increase with the harmonic number, i.e., ωµ = µp

(2.60)

The magnetic flux density waveform (normal component) of the rotor of an induction machine is given by the equation b2 (α, t) =

∞ 

 B2mµ cos µpα ∓ ωµ t + φµ .

(2.61)

µ=1

For the most important space harmonics the rotor angular frequency of an induction motor is ωµ = ω[(ks2 / p)(1 − s) + 1]. The magnitude of the rotor magnetic flux density can be found as [10] √ m 1 2 N1 kw1ν kw2µ µ0 B2mµ = I1 (2.62) π µp(1 + τ2ν )kw2ν kC g where kw1ν is the stator winding factor for the ν-th space harmonic according to Equation 2.20, kw2µ is the rotor winding factor for the rotor µ-th space harmonic, kw2ν is the rotor winding factor for the stator ν-th space harmonic, τ2ν = X 2ν / X mν is the leakage coefficient of the rotor winding for the ν-th stator harmonic, X 2ν is the rotor winding leakage reactance for the ν-th harmonic, and X mν is the mutual reactance for the ν-th harmonic.

2.6 2.6.1

Calculation of air gap magnetic field Effect of slots

The relationship between the air gap magnetic flux density b(α, t) and the MMF F(α, t) for uniform air gap g is b(α, t) = F(α, t)g (α)

(2.63)

where the relative permeance g (α) of the air gap is a function of the angle α given by Equation 2.30. Assuming a trapezoidal distribution of the magnetic flux density, the amplitude of the νth space harmonic will be equal to the flat-topped value of the flux density multiplied by the following coefficient of slot opening [87] kok =

Copyright © 2006 Taylor & Francis Group, LLC

sin[kρπb14 /(2t1 )] kρπb14 /(2t1 )

(2.64)

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Noise of Polyphase Electric Motors

where t1 is the stator slot pitch, b14 is the stator slot opening and k is the number of harmonics which replace the air gap variation bounded by the active surfaces of ferromagnetic cores of the stator and rotor. The auxiliary function dependent on the relative slot opening is √ κ 2 1 + κ2 √ ρ= . (2.65) 5 + κ 1 + κ2 − 1 The relative slot opening can be estimated as follows • for induction machines

b14 g

(2.66)

b14 g + h M /µ0

(2.67)

κ= • for most PM brushless machines κ≈

where h M is the radial height of the PM per pole. The value of κ may affect the radial magnetic forces, but it is not necessarily a decisive factor in production of noise [220]. The relative permeance of the air gap in Equation 2.63 with slot opening effect being included is ∞ µ0  A0 + Ak cos(ks1 α) 2 gkC k=1,2,3   ∞  = g0 1 + Ak cos(ks1 α) = g0 λg1 (α)

g (α) =

(2.68)

k=1,2,3

where α = 2π x/(s1 t1 ) is according to Equation 2.30, s1 is the number of stator slots and t1 is the slot pitch. The permeances and Fourier coefficients in the above Equation 2.68 are • the constant component of the air gap relative permeance g0 =

A0 µ0 µ0 = =  2 kC g g

(2.69)

• the component of the relative specific permeance of the air gap varying with the angle α when only the stator core is slotted λg1 (α) = 1 +

∞ 

Ak cos(ks1 α)

(2.70)

k=1,2,3

• the relative value of harmonic permeances of the air gap dependent on the dimensions of the slot openings and air gap [40, 219], i.e.,  γ1 2 2 π Ak = g (α) cos(ks1 α)dα = −2g  kok . (2.71) π 0 t1

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It has been assumed that g (α) above the slot opening is approximated by a linear function. The second term in Equation 2.68 multiplied by g0 yields 2 g0 Ak = −2µ0 γ1 kok /t1 , i.e., as given in classical literature [40]. The angular displacement α according to Equation 2.30 can be expressed by the linear circumferential distance x. For induction machines the constant component (Equation 2.69) corresponds to the relative permeance of physical air gap g increased by Carter’s coefficient kC . For surface magnets g  = gkC + h M /µrr ec and for buried magnets totally enclosed by laminations or mild steel g  = gkC . The Carter’s coefficient kC =

t1 t1 − γ1 g

(2.72)

where

  4

0.5κ arctan(0.5κ) − ln 1 + (0.5κ)2 . (2.73) π The relative permeance due to stator slot openings can also be calculated according to the method proposed by Weber [87, 236]. The coefficient of harmonic permeances in Equation 2.70 is

14 (k)2 Ak = −β(κ) sin(1.6π k) (2.74) 0.5 + kπ 0.78 − 2(k)2 γ1 =

where or

β(κ) = 0.1κ 0.5+1/κ

(2.75)

  1 β(κ) = 0.5 1 − √ . 1 + κ2

(2.76)

The relative permeance according to Weber is g0 Ak = µ0 Ak /g  where Ak is given by Equation 2.74. The parameter κ is according to Equation 2.66 or Equation 2.67 and b14 = . (2.77) t1 If both the stator and rotor core have slots, the resultant relative air gap permeance is g (α) = g0 λg1 (α)λg2 (α) (2.78) and the resultant Carter’s coefficient is kC = kC1 kC2

(2.79)

where λg1 and λg2 are relative specific permeances of the stator and rotor derived on the assumption that only one core has slots, kC1 is Carter’s coefficient for the stator, and kC2 is Carter’s coefficient for the rotor. The center axes of the stator and rotor teeth of an induction motor are aligned at the time instant t = 0 in the origin of coordinate system, i.e., α = 0 and the rotor angular frequency ω2 = 2π(1 − s) f = 2π(1 − s)n s p = 2π n m p = pm

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Noise of Polyphase Electric Motors

where m is the mechanical angular speed of the rotor and n m is the rotational speed of the rotor in rev/s. Thus, the relative value of harmonic permance for the rotor is ∞  λg2 (α, t) = 1 + Al cos [ls2 (α − ω2 t)] . (2.80) l=1,2,3

Putting Equations 2.69, 2.70, and 2.80 into Equation 2.78 the resultant relative permeance of the air gap of slotted stator and rotor cores is g (α, t) = g0 λg (α, t) =

µ0 λg (α, t) kC g

(2.81)

where λg (α, t) = 1 + +

∞ 

Ak cos (ks1 α) +

k=1,2,3 ∞ 

∞ 

Al cos [ls2 (α − ω2 t)]

l=1,2,3

∞  1 Ak Al {cos[(ls2 + ks1 )α − ls2 ω2 t] 2 k=1,2,3 l=1,2,3

× cos[(ls2 − ks1 )α − ls2 ω2 t]}.

(2.82)

The first term in Equation 2.82 describes the permeance of the equivalent uniform air gap g  = kC g, second term describes harmonics of the stator permeance, third term describes harmonics of the rotor permeance, and the last term represents harmonics of permeance due to reciprocal effect of the stator and rotor. Harmonics of the magnetic flux density in the air gap are obtained by multiplying Equations 2.81 and 2.44 (distribution of the MMF) as given by Equation 2.63.

2.6.2

Effect of eccentricity

The variation of the air gap around the magnetic circuit periphery and with time is g(α, t) = R − r − e cos(α − ω t) = g[1 −  cos(α −  t)]

(2.83)

where g = R − r , R is the inner stator core radius and r is the outer rotor radius. The relative eccentricity is defined as =

e g

(2.84)

where e is the rotor (shaft) eccentricity and g is an ideal uniform air gap for e = 0. For the static eccentricity the angular frequency  = 0. In the case of dynamic eccentricity: • for induction motors  = (1 − s) =

Copyright © 2006 Taylor & Francis Group, LLC

f ω (1 − s) = 2π (1 − s) p p

(2.85)

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• for synchronous motors f ω = 2π . (2.86) p p Equation 2.83 has been derived in Appendix B. The constant coefficient (Equation 2.69) in Fourier series (Equation 2.68) for a machine with eccentricity becomes [10]:  =  =

µ0 1 A0 √ = H/m2 (2.87) 2 gkC 1 −  2 The effect of the static and dynamic eccentricity on the higher harmonics of the air gap performance is difficult to calculate. Most methods are limited only to the influence of eccentricity on the fundamental harmonic of the air gap permeance [87], i.e., • the air gap relative permeance for k = 1 including the static eccentricity and variation of the permeance with slot openings γ1 2 ge,k=1 (α) = g0 Ak=1 λc1 cos(α) = −2µ0 kok λc1 cos(α) (2.88) t1 in which √ 1 − 1 − 2 λc1 = 2 √ (2.89)  1 − 2 • the air gap relative permeance for k = 1 including the dynamic eccentricity and variation of the permeance with slot openings g0 =

ged,k=1 (α, t) = g0 Ak=1 λc1d cos( t − α) γ1 2 = −2µ0 kok λc1d cos( t − α) (2.90) t1 where the angular speed  equal to the rotor mechanical angular speed is according to Equation 2.85 or Equation 2.86. The angular displacement α in Equations 2.88 and 2.90 can be replaced by the linear displacement x according to Equation 2.30. To obtain λc1d , the relative static eccentricity  in Equation 2.89 is replaced by the relative dynamic eccentricity d defined as ed (2.91) d = g in which ed denotes the dynamic eccentricity (linear radial dimension). To include the eccentricity in calculation of the relative air gap permeance, the permeances (Equation 2.88) and (Equation 2.90) should be added to Equation 2.68 or Equation 2.81. The effect of static and dynamic eccentricity can be taken into account according to Appendix B using the following coefficients √ 1 1 − 1 − 2 λge (α) ≈ √ +2 √ cos(α) (2.92) 1 + 2  1 − 2 √ 1 1 − 1 − 2  λged (α) ≈  + 2 cos ( t − α) . (2.93) 1 + d2 d 1 − d2

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Noise of Polyphase Electric Motors

The resultant relative air gap permeance g (α, t) with slots and eccentricity effects taken into account is given in Appendix B by Equation B.5. Equation B.5 has been obtained by multiplying Equation 2.78 by the product λge (α)λged (α, t) which includes eccentricity. Since the relative eccentricity is small, i.e.,  ≤ 0.3, the permeance of the air gap due to eccentricity expressed in the form of Fourier series can be limited to the first two terms. For the dynamic eccentricity (neglecting the variation of the permeance with slot openings) ged (α, t) = g0 λged (α, t) 1 1 = µ0 [λoe + λc1 cos(α −  t)] ≈ µ0 [1 +  cos(α −  t)]. gkC gkC (2.94) The wave of the magnetic flux density in the air gap pulsating with the eccentricity can be obtained by multiplying the first harmonic (second term) of Equation 2.94 by the fundamental stator MMF, i.e.,  cos(α −  t)Fm1 cos( pα − ωt − φ) gkC   = µ0 Fm1 cos[α(1 + p) − (ω +  )t − φ)] + µ0 Fm1 2gkC 2gkC × cos[α(1 − p) + (ω −  )t + φ)]  

 1 = µ0 + 1 − (ω +  )t − φ) Fm1 cos pα 2gkC p  

 1 − 1 + (ω −  )t + φ) + µ0 Fm1 cos pα 2gkC p  = µ0 Fm1 cos[ν pα ∓ (ω ±  )t ∓ φ)]. (2.95) 2gkC

b1 (α, t) ≈ µ0

The stator magnetic flux density higher harmonics due to eccentricity are ν = 1 ±

1 p

(2.96)

and the number of their pole pairs is p ± 1. For p = 1 the eccentricity harmonic numbers are ν = 2 or ν = 0. The second term in Equation 2.95 for ν = 0 represents an axial homopolar flux that can appear in a two-pole machine with eccentricity (Appendix B). The angular frequency for static eccentricity is ω = 2π f . For dynamic eccentricity and induction machines

1 ω ±  = ω 1 ± (1 − s) . (2.97) p For synchronous machines s = 0 in Equation 2.97.

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The stator magnetic flux density wave due to eccentricity generate additional magnetic flux density waves in the rotor [64, 200]  b2 (α, t ) = B2m µ cos(µ p α − µ t − φµ ) (2.98) µ

where the rotor magnetic flux density higher harmonics due to eccentricity are   s2 1 µ = k + 1 ± (2.99) p p and the angular frequency (see also Equation 2.134)

s2 µ = ων + ω k (1 − s ) p

(2.100)

where ων = 2π ν f . The rotor flux density waves originate from the stator flux density waves in the air gap with pole pairs p ± 1. For a synchronous machine, the slip in s = 0, Equation 2.100, and the number of rotor slots s2 = 2 p. The eccentricity fields contribute to the vibration and noise and produce the following effects: • cause unbalance magnetic pull that bends the shaft and magnifies the magnetic pull; • induce voltages in the parallel paths of stator windings and give rise to equalizing currents [123]; • reduce the mechanical stiffness of the shaft and the first critical speed of the rotor due to increased magnetic pull [41, 200].

2.6.3

Effect of magnetic saturation

The slot leakage and residual fields of the stator and rotor MMFs are large [87]. Consequently, the tooth tips become highly saturated (reduction of the relative magnetic permeability) which is equivalent to the increase in the slot opening from b14 to b14sat (Figure 2.6) according to Equation C.10. The maximum slot opening due to the tooth tip saturation corresponds to the region (along the pole pitch) of the maximum slot current. The fictitious slot opening b14sat is a periodic function with the period π/ p, i.e., equal to half the period of the working harmonic. Thus, the first two terms of Fourier series of the fictitious slot opening can be written as b14sat = b140 − b141 cos[2( pα − ωt)]

(2.101)

where b140 is the mean value of the slot opening along the pole pitch and b141 = b140 − b14 is the amplitude of the first harmonic of the slot opening distribution. The air gap permeance with respect to the magnetic saturation can be expressed by the following simplified equation [64, 179] (α, t) ≈ 0 + sat (α, t)

Copyright © 2006 Taylor & Francis Group, LLC

(2.102)

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Noise of Polyphase Electric Motors b14 b14 sat Variation of Slot Opening

b14

t1 b14sat(a)

b14

a=

1 p x pt

t

t (a)

b140

(b)

Figure 2.6 Effect of magnetic saturation on the slot opening: (a) fictitious increase in the slot opening; (b) variation of a fictitious slot opening along the pole pitch due to saturation [87]. where sat (α, t) = −sat cos(2 pα − 2ω1 t − 2φs ).

(2.103)

Only the fundamental harmonic ν = 1 has been included.

2.6.4

Effect of rotor saliency

The effect of the PM rotor saliency can be included with the aid of the rotor magnetic flux density waveform normalized to B0 = 1 T. On the basis of Equation 2.61 the magnetic flux density waveform is b (α, t) = where

∞

b2 (α, t)  = bµ cos(µpα ± ωµ t + φµ ) B0 µ=1

(2.104)

   τ 4 1 cp sinh(α) − 6 cosh(α) c2p + (µπ/τ )2 µπ bt3    τ 2 2 +3 cosh(α) µπ τ − bp      
π π bt τ 2 1 τ 4 µπ/τ × sin µ sin µ +6 − + 2 τ τ c2p + (µπ/τ )2 µπ bt2 µπ  
π π bt × cosh(α) sin µ cos µ ; (2.105) 2 τ

4 bµ = τ



bt =

τ − bp ; 2

Copyright © 2006 Taylor & Francis Group, LLC

cp = 2

α ; bp

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b p is the width of the pole shoe, and B0 bµ = B2mµ is given by Equation 2.61. The parameter α depends on the distribution of the normal component of the magnetic flux density in the range −0.5b p ≤ x ≤ 0.5b p . For α = 0 the magnetic flux density in the range −0.5b p ≤ x ≤ 0.5b p has constant value (flat curve). For embedded magnets with orifices in pole shoes, α = 0.5 . . . 1 and the magnetic flux density in the range −0.5b p ≤ x ≤ 0.5b p is a concave curve. Thus, the resultant air gap relative permeance with the rotor saliency effect taken into account is gsal (α, t = t0 ) = |b (α, t = t0 )|λg (α, t = t0 )

(2.106)

where b (α, t = t0 ) is according to Equation 2.104, λg (α, t = t0 ) is according to Equation 2.70 or Equation 2.82, and t0 is the arbitrarily chosen time instant. The resultant relative air gap permeance g according to Equation B.5 (Appendix B) with the effect of the rotor saliency taken into account plotted against the circumferential direction x = ατ/π at t = 0 is shown in Figure 2.7.

2.7

Radial forces

2.7.1

Production of radial magnetic forces

On the basis of Equation 2.45 the space and time distribution of the MMF of a polyphase electrical machine fed with sinusoidal and balanced current system can be expressed by the following equations:

3 ⋅ 10−4 2.5 ⋅ 10−4

Λ g (x)

2 ⋅ 10−4

1 ⋅ 10−4 5 ⋅ 10−5 0

0

0.05

0.1

0.15

0.2

0.25 x, m

0.3

0.35

0.4

0.45 0.5

Figure 2.7 Distribution of the air gap relative permeance according to Equation B.5 with the rotor saliency effect taken into account on the basis of Equation 2.106 for a PM brushless motor with s1 = 36, 2 p = 10, τ = 50 mm, b14 = 3 mm, g = 1.0 mm, e = 0.1 mm, ed = 0.1 mm.

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Noise of Polyphase Electric Motors • for the stator F1 (α, t) =

∞ 

Fmν cos(νpα ∓ ωt)

(2.107)

Fmµ cos(µpα ∓ ωµ t + φ µ )

(2.108)

ν=1

• for the rotor F2 (α, t) =

∞  µ=1

where α is the angular distance from the origin of the coordinate system, p is the number of pole pairs, φµ is the angle between vectors of the stator and rotor harmonics of equal order, ν and µ are the numbers of space harmonics of the stator and rotor, respectively, ω = 2π f is the pulsation of the input current, and Fmν and Fmµ are the peak values of the νth and µth harmonics expressed by Equations 2.43 and 2.55, respectively. The product pα = π x/τ where τ is the pole pitch and x is the linear distance from a given axis results from Equation 2.30. For symmetrical polyphase stator windings and integral number of slots per pole per phase q1 , the stator space harmonics are expressed by Equation 2.47. For induction machines µ is expressed by Equation 2.23. For rotors of synchronous machines µ = 2k − 1

(2.109)

where k = 1, 2, 3, . . .. The instantaneous value of the normal component of the magnetic flux density in the air gap at a point determined by the angle α can be calculated with the aid of Equation 2.63 b(α, t) = [F1 (α, t) + F2 (α, t)]g (α, t) = b1 (α, t) + b2 (α, t)

T

(2.110)

where • for the stator b1 (α, t) = F1 (α, t)g (α, t) =

∞ 

Bmν cos(νpα ∓ ωt)

(2.111)

ν=1

• for the rotor b2 (α, t) = F2 (α, t)g (α, t) =

∞ 

Bmµ cos(µpα ∓ ωµ t + φ µ ). (2.112)

µ=1

According to the Maxwell stress tensor, the radial magnetic force per unit area or magnetic pressure waveform at any point of the air gap is pr (α, t) =

 1  2 b (α, t) − bt2 (α, t) . 2µ0

(2.113)

Since the magnetic permeability of the ferromagnetic core is much higher than that of the air gap, the magnetic flux lines are practically perpendicular to the stator and

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rotor cores. Thus, the tangential component bt (α, t ) of the magnetic flux density is much smaller than the normal component b(α, t ) and b2 (α, t ) 1 = [F1 (α, t ) + F2 (α, t )]2 2g (α, t ) 2µ0 2µ0 1 = [F 2 (α, t )2g (α, t ) + 2F1 (α, t )F2 (α, t )2g (α, t ) 2µ0 1 + F22 (α, t )2g (α, t )]

pr (α, t ) ≈

=

[b1 (α, t )]2 + 2b1 (α, t )b2 (α, t ) + [b2 (α, t )]2 . 2µ0

(2.114)

There are three groups of the infinite number of radial force waves: • the product [b1 (α, t )]2 of the stator harmonics of the same number ν, i.e., pr ν (α, t ) =

[Bm ν cos(νp α ∓ ωt )]2 B2 = m ν [1 +cos(2νp α∓2ωt )] (2.115) 2µ0 4µ0

• the mixed product 2b1 (α, t )b2 (α, t ) of the stator ν and rotor µ harmonics 2Bm ν cos(νp α ∓ ωt )Bm µ cos(µp α ∓ ωµ t + φµ ) 2µ0 1 = Bm ν Bm µ {cos[(νp α ∓ ωt ) − (µp α ∓ ωµ t + φµ )] 2µ0 + cos[(νp α ∓ ωt ) + (µp α ∓ ωµ t + φµ )]} 1 = Bm ν Bm µ {cos[ p α(ν − µ) ∓ (ω − ωµ )t − φµ ] 2µ0 + cos[ p α(ν + µ) ∓ (ω + ωµ )t + φµ ]} (2.116)

pr νµ (α, t ) =

• the product [b2 (α, t )]2 of the rotor harmonics of the same number µ, i.e., [Bm µ cos(µp α ∓ ωµ,n t + φµ,n )]2 2µ0 1 2 = B [1 + cos(2µp α ∓ 2ωµ t + 2φµ )] 4µ0 m µ

pr µ (α, t ) =

(2.117)

The following trigonometric identities cos2 α = 0.5(1 + cos 2α) and 2 cos α cos β = cos(α − β) + cos(α + β) have been used. The constant term equal to Bm2 ν /(4µ0 ) or Bm2 µ /(4µ0 ) in Equations 2.115 and 2.117 have no significance for noise generation and can be neglected because the static magnetic pressure is uniformly distributed along the air gap. In accordance with Equations 2.115, 2.116, and 2.117, the magnetic forces per unit area (Figure 2.8) can be expressed in the following general form pr (α, t) = Pmr cos(r α − ωr t)

Copyright © 2006 Taylor & Francis Group, LLC

(2.118)

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Noise of Polyphase Electric Motors 3 ⋅ 105 2.5 ⋅ 105

Pr (x)

2 ⋅ 105 1.5 ⋅ 105 1 ⋅ 105 5 ⋅ 104 0

0

0.05

0.1

0.15

0.2

0.25 x, m

0.3

0.35

0.4

0.45

0.5

Figure 2.8 Distribution of the radial magnetic pressure along the air gap periphery at no load for s1 = 36, 2 p = 10, τ = 50 mm, g = 1.0 mm and eccentricity e = 0.1 mm. where Pmr is the amplitude of the magnetic pressure, ωr is the angular frequency and and r = 0, 1, 2, 3, . . . are corresponding orders of radial magnetic forces. The radial forces circulate around the stator bore with the angular speed ωr /r and frequency fr = ωr /(2π ). For small number of the stator pole pairs the radial forces may cause the stator to vibrate and produce acoustic noise. For PM brushless motors the full load armature reaction field is normally less than 20% of the open-circuit magnetic field [254]. Thus, the effect of the first and second term in Equation 2.114 on the acoustic noise at no load is usually lower than that of the third term, i.e., in most cases harmonics in the open-circuit magnetic field have a dominant role in the noise of electromagnetic origin [254].

2.7.2

Amplitude of magnetic pressure

The amplitude Pmr of the radial magnetic pressure of the r -th order in Equation 2.118 depends on the harmonics of the magnetic flux density participating in its production, i.e., • excited by the stator harmonics of the same number ν 2 Bmν N/m2 4µ0 • excited by the stator ν and rotor µ harmonics

Pmr =

Bmν Bmµ N/m2 2µ0 • excited by the rotor harmonics of the same number µ Pmr =

Pmr =

Copyright © 2006 Taylor & Francis Group, LLC

2 Bmµ

4µ0

N/m2 .

(2.119)

(2.120)

(2.121)

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To obtain the amplitude of the radial force, the force pressure amplitude Pmr should be multiplied by π D1in L i where D1in is the stator core inner diameter and L i is the effective length of the stator core.

2.7.3

Deformation of the stator core

By only considering the pure circumferential vibration modes of the stator core, the deflection d of the stator core is an inverse function of the fourth power of the force order r [206] 1 d ∝ 4 (2.122) r Considering the product of the stator harmonics of the same number according to Equation 2.115, the lowest frequency of the magnetic force fr = 2 f varies at twice the line frequency and the order is r = 2 p. Since the force order r = 2 for a two-pole motor and r = 4 for a four-pole motor, the magnetic force in a four-pole motor will be 1/16 of that of the two-pole motor. The twice line frequency noise is generally significant only in two-pole motors because the stator deflection varies with the force order according to Equation 2.122. The largest deformation of the stator yoke ring occurs when the frequency fr is close to the natural mechanical frequency of the stator system. The most important from airborne noise point of view are low circumferential mode numbers, i.e., r = 0, 1, 2, 3 and 4. Vibration mode r = 0. For pulsating vibration mode or “breathing” mode r = 0 the radial magnetic force density p0 = Pmr =0 cos ω0 t (2.123) is distributed uniformly around the stator periphery and changes periodically with time. It causes a radial vibration of the stator core and can be compared to a cylindrical vessel with a variable internal overpressure [87]. Equation 2.123 describes an interference of two magnetic flux density waves of equal lengths (the same number of pole pairs) and different velocity (frequency). According to [154], the “breathing” mode r = 0 can cause acoustically harmful low wave number vibration, even with a high number of poles and sinusoidal stator current. This can happen when nearby low wave numbers are given a high admittance by structural discontinuities such as supports [154]. Vibration mode r = 1. For “beam bending” mode r = 1 [58] the radial pressure p1 = Pmr =1 cos(α − ω1 t)

(2.124)

produces a single-sided magnetic pull on the rotor. The angular velocity of the pull rotation is ω1 . Physically, Equation 2.124 describes an interference of two magnetic flux density waves for which the number of pole pairs differs by one.

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Noise of Polyphase Electric Motors r=0

r=1

P

r=2

P

P

r=3

P

P r=4 P

P

P P P

P P

P P

P

P P

P

P

Figure 2.9 Deformation of the core caused by the space distribution of radial forces. Vibration modes r = 2, 3, 4. For “ovaling mode” r = 2 [58] and r = 3, 4, . . . wave-shaped deflections of the stator core will occur. Figure 2.9 shows space distribution of forces producing vibrations of the order of r = 0, 1, 2, 3, 4.

2.7.4

Frequencies and orders of magnetic pressure

Angular frequencies and orders of radial magnetic pressure result from Equations 2.115, 2.116, and 2.117, i.e.: • excited by the stator harmonics of the same number ν, i.e., ωr = 2ω;

fr = 2 f ;

r = 2νp

(2.125)

• excited by the stator ν and rotor µ harmonics ωr = ω ± ωµ ;

fr = f ± f µ ;

r = (ν ± µ) p (2.126)

• excited by the rotor harmonics of the same number µ, i.e., ωr = 2ωµ ;

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fr = 2 f µ ;

r = 2µp

(2.127)

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For the fundamental space harmonic ν = 1 the frequency of the radial magnetic pressure according to Equation 2.125 is fr = 2 f and the order (circumferential vibrational mode) r = 2 p. The rotating component with 2 p pole pairs and frequency 2 f can cause significant vibration, especially, in large machines, e.g., for Bm ν=1 = 0.8 T the magnitude of the radial magnetic pressure is 127.4 kN/m2 . For higher space harmonics ν > 1 the frequency of the radial force pressure is also fr = 2 f and the order r = 2νp. For salient pole synchronous machines the numbers of higher space harmonics for the stator and rotor are ν = 2km 1 ± 1 and µ = 1 ± 2k, respectively, where k = 0, 1, 2, 3, . . . and m 1 is the number of phases. The vibration order according to Equation 2.126 is r = p[2km 1 ±1 ±(1 ±2k )] or r = 2 p[k (m 1 +1)+1] and r = 2 p[k (m 1 +1)−1], e.g., for k = 0 the vibration order is r = 2 p, for k = 1 and m 1 = 3 the order r = 10 p and r = 6 p, for k = 2 and m 1 = 3 the order r = 18 p and r = 14 p, etc. The frequency of vibration is fr = f ± µf = f [1 ± 1 ± 2k] or fr = 2 f (1 +k ) and fr = 2 f k. The frequency of the rotor space harmonics is f (1 ±2k ). The rotor of a salient pole synchronous machine without any cage winding can be considered as a rotor with the number of slots s0 = 2 p. Thus, the angular frequency of the rotor space harmonics and rotor angular speed  = 2 p f / p is   f ωµ = ω ± s0 k  = ω ± 2 pk 2π = 2π f (1 ± 2k ) (2.128) p The vibrational mode of the rotor harmonics of the same number µ according to Equation 2.127 is r = 2µp = 2(1 ± 2k ) p and frequency fr = 2 f µ . For a salient pole synchronous machine fr = 2 f (1 ± 2k ) and for induction machine fr = 2 f [1 ± k(s2 / p)(1 − s)] where µ = 1 ± 2k, k = 0, 1, 2, 3, . . . and s is the slip (Equation 2.2) for fundamental. Those harmonics of low orders produce significant radial forces, in PM synchronous motors also at zero stator current state. The frequencies of radial forces due to rotor space harmonics of the same number µ for PM synchronous motors are given in Table 2.3.

2.7.5

Radial forces in synchronous machines with slotted stator

Significant vibration and noise can be produced by interaction of the PM rotor field and slotted structure of the stator. The fictitious MMF wave excited by PMs is expressed by Equation 2.108. The PM rotor magnetic flux density waveform modulated by stator slots is a product of the MMF F2 (α, t) and relative air gap permeance λsl (α) due to slot opening varying with the angle α. According to Equations 2.68 and 2.71 sl (α) = −g0

∞ 

Ak cos(ks1 α)

(2.129)

k=1,2,3 2 where −g0 Ak = 2µ0 γ1 kok /t1 . By multiplying Equations 2.108 and 2.129 the magnetic flux density component produced as a result of the modulation of the

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Noise of Polyphase Electric Motors

Table 2.3 Frequencies fr = 2 f (1 ± 2k ) and orders r = 2(1 ± 2k ) p of forces produced by the rotor magnetic field of synchronous motors. k 0 1 2 3 4 5 6 7 8 9 10

Frequency of radial force fr 2f −2 f and 6 f −6 f and 10 f −10 f and 14 f −14 f and 18 f −18 f and 22 f −22 f and 26 f −26 f and 30 f −30 f and 34 f −34 f and 38 f −38 f and 42 f

Order of radial force r 2p −2 p and 6 p −6 p and 10 p −10 p and 14 p −14 p and 18 p −18 p and 22 p −22 p and 26 p −26 p and 30 p −30 p and 34 p −34 p and 38 p −38 p and 42 p

rotor flux density by the stator slot openings is bλ (α, t ) = −g0

∞ 

Fm µ cos(µp α ∓ ωµ t + φµ )

µ=1,3,5

=

∞ 

Ak cos(ks1 α)

k =1,2,3

∞ ∞   1 g0 Fm µ Ak cos[(µp ± ks1 )α ∓ ωµ t + φµ )]. (2.130) 2 µ=1,3,5 k =1,2,3

In the above Equation 2.130 the trigonometric identity cos α cos β = 0.5[cos(α + β) + cos(α − β)] has been used. The radial force pressure on the stator due to the rotor µth and stator permeance kth harmonic is 1 {0.5g0 Fm µ Ak cos[(µp ± ks1 )α ∓ ωµ t + φµ ]}2 2µ0 1 2 2 2 =  F A {1 + cos[2(µp ± ks1 )α ∓ 2ωµ t + 2φµ ]}. (2.131) 8µ0 g0 m µ k

pr λ,µ,k (α, t ) =

Similar to the cogging torque, radial forces according to Equation 2.131 will be present both at zero stator current state and when the motor is fed with current. The order (circumferential mode) of the force pressure pr λ (α, t) is rλ = 2(µp ± ks1 ) and frequency fr λ = 2µf . The most important are low even orders rλ = 2, 4, 6, and 8 for the fundamental stator permeance harmonic k = 1, i.e., µ=

|0.5rλ ∓ s1 | = 1, 3, 5, . . . p

and

fr λ = 2µf.

(2.132)

Table 2.4 shows values of rλ , µ, and fr λ obtained on the basis of Equation 2.132 for selected numbers of pole pairs p and stator slots s1 . The most dangerous case

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Table 2.4 Orders and frequencies of radial forces due to interaction of the rotor field and stator slotted core in three-phase synchronous and PM motors (Equation 2.132). p

rλ

2

4

2

8

3

6

4

8

5 5

2 8

s1

q1

µ

fr λ = 2µf

12 18 24 36 12 18 24 36 18 24 36 24 36 36 36

1 1.5 2 3 1 1.5 2 3 1 1.333 2 1 1.5 1.2 1.2

5 and 7 8 and 10 11 and 13 17 and 19 4 and 8 7 and 11 10 and 14 16 and 20 5 and 7 7 and 9 11 and 13 5 and 7 8 and 10 7 8

10 f and 14 f 16 f and 20 f 22 f and 26 f 34 f and 38 f 8 f and 16 f 14 f and 22 f 20 f and 28 f 32 f and 40 f 10 f and 14 f 14 f and 18 f 22 f and 26 f 10 f and 14 f 16 f and 20 f 14 f 16 f

is for p = 5 and s1 = 36 because the order (circumferential mode) is low, i.e. rλ = 2. This mode should be avoided in the design of PM brushless motors. For example, for 660 rpm, p = 5, s1 = 36 the input frequency is f = 55 Hz and fr λ = 14 × 55 = 770 Hz at rλ = 2. If the number of pole pairs is reduced to p = 4, the input frequency is f = 44 Hz and the frequency of radial forces are fr λ = 16 × 44 = 704 Hz and fr λ = 20 × 44 = 880 Hz at rλ = 8. Reduction of number of poles from 5 to 4 and keeping the same number of slots s1 = 36 increases the order of radial magnetic forces. The higher the order of radial magnetic forces, the lower the acoustic noise of magnetic origin. Similar considerations are given by Walker and Kerruish for large synchronous machines operating at no load [220]. According to [220] the frequency of predominant noise in synchronous machine is fr λ = 2µλ f where µλ is the integer nearest to s1 /(2 p). This frequency is produced by stator harmonic permeance k = 1 and rotor MMF harmonic µ = µλ . The analysis of Equation 2.131 shows that 2µp−2ks1 is least when µ−ks1 / p is least (not in agreement with [220]). For k = 1 the expression µ − s1 / p is least when µ is as close to s1 / p as possible, i.e., µ is the integer nearest to s1 / p. Considering a machine with 2 p = 10, s1 = 36, and f = 55 Hz, the ratio s1 / p = 36/5 = 7.2 and µ = µλ = 7. The frequency of the predominant noise is again fr λ = 2µλ f = 2 × 7 × 55 = 770 Hz and the order rλ = |2(µλ p − s1 )| = |2(7 × 5 − 36)| = 2.

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2.7.6

Noise of Polyphase Electric Motors

Frequencies of vibration and noise

Product of stator space harmonics of the same number According to Equation 2.125, the frequency of the radial magnetic force is equal to the double the frequency of the line fundamental frequency. The order for the stator slot harmonics is r = 2νp = 2(ks1 ± p) (2.133) where ν is according to Equation 2.21. Interaction of stator and rotor harmonics For the fundamental time harmonic the field of each stator winding harmonic at a given point with the coordinate α pulsates with the angular frequency ω = 2π f , where f is the fundamental frequency. For an induction motor the angular frequency of the rotor slot harmonics is [87]



f s2 ωµ = ω ± ks2 m = ω ± ks2 2π (1 − s) = 2π f 1 ± k (1 − s) (2.134) p p where m = 2π f (1 − s)/ p is the mechanical angular speed of the rotor and k = 1, 2, 3, . . .. Thus, the angular frequencies of radial forces due to the stator and rotor slot harmonics result from Equation 2.126

f ωr = ω ± ωµ = ω ± ω ± ks2 2π (1 − s) (2.135) p The frequencies of radial forces s2 fr = f k (1 − s) p

and

s2 fr = f k (1 − s) ± 2 p

(2.136)

The orders are also determined by Equation 2.126 r = (ν ± µ) p = ks1 ± ks2 ± 2 p

(2.137)

where ν is according to Equation 2.21 and µ is according to Equation 2.23. For ν = µ = 1 the orders are r = 0 and r = 2 p. For ν = µ the orders are r = 0 and r = 2νp = 2µp. For a PM synchronous motor with salient poles or embedded magnets and without any cage winding, the higher harmonics of the rotor µ=k

so ± 1 = 2k ± 1 p

(2.138)

are called “pole harmonics” and their “winding coefficients” are equal to “winding coefficients” of the fundamental harmonic. The number of slots for a synchronous

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motor without any cage winding so = 2 p and k = 1, 2, 3, . . .. Thus, the angular frequency of the rotor harmonics (m =  = ω/ p) is ωµ = ω ± kso  = ω ± 2 pk = ω ± 2 pk(2π

f ) = 2π f (1 ± 2k). p

(2.139)

The noise and vibration frequencies due to the stator winding and rotor harmonics are according to Equation 2.126, i.e., fr = f ± f (1 ± 2k) or

fr = 2(1 + k) f

and

fr = 2k f.

The corresponding orders of radial forces   s1 r = (ν ± µ) p = k ± 1 ± 2k ± 1 p = ks1 ± 2 p(k + 1). p

(2.140)

(2.141)

Product of rotor space harmonics of the same number According to Equation 2.127 the angular frequency and frequency of the radial magnetic forces of an induction machine are

s2 ωr = 2ωµ = 2(ω ± ks2 m ) = 4π f 1 ± k (1 − s) (2.142) p

s2 fr = 2 f 1 ± k (1 − s) (2.143) p and the order r = 2µp = 2(ks2 ± p) (2.144) where µ is according to Equation 2.23. For synchronous motors s2 = s0 = 2 p and slip s = 0, so that Equations 2.143 and 2.144 become fr = 2(1 ± 2k) f ;

r = 2 p(1 ± 2k).

(2.145)

Radial force harmonics due to eccentricity The stator and rotor magnetic flux density waves due to eccentricity are expressed by Equations 2.95 and 2.98, harmonic numbers by Equations 2.96 and 2.99, and harmonic angular frequencies by Equations 2.85, 2.86, and 2.100. The angular frequency of the magnetic flux density wave for static eccentricity is ω = 2π f and according to Equation 2.95 for dynamic eccentricity this angular frequency is ω ±  . The angular frequency of higher harmonic radial forces in an induction machine due to eccentricity for ν = 1 is calculated with the aid of Equations 2.95, 2.100, and 2.126, i.e.,

s2 r  = ω ±  ± ων=1 + ω k (1 − s) . p

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Noise of Polyphase Electric Motors

The above equation for static eccentricity ( = 0) gives the following two frequencies

s2 s2 fr  = f 2 + k (1 − s) and fr  = f k (1 − s). (2.146) p p The basic orders of eccentricity harmonics are r =1

and

r =2

(2.147)

as there is a single-sided magnetic pull due to eccentricity (bending and ovaling modes). However, on the basis of Equations 2.96, 2.99, and 2.126, there are also higher orders of eccentricity force harmonics   s2 1 1 r = (ν ± µ ) p = 1 ± ± k + 1 ± p p p p = p ± 1 ± (ks2 + p ± 1) or r = 2( p ± 1) ± ks2

and

r = ks2

(2.148)

that usually can be neglected. Vibration of zero order (r = 0) can also be produced for interference of space harmonics of the same order [10]. The angular frequency of higher harmonic radial forces in an induction machine due to dynamic eccentricity ( = 2π( f / p)(1 − s), according to Equations 2.85, 2.100, and 2.126 is

s2 1 r  = ω 1 ± (1 − s) ± 1 ± k (1 − s) . p p The above equation gives the next two frequencies of radial forces due to eccentricity

s2 s2 1 1 fr  = f 2 ± (1 − s) + k (1 − s) ; fr  = f (1 − s) + k (1 − s) p p p p (2.149) The orders for frequencies (Equation 2.149) are determined by Equations 2.147 and 2.148. For synchronous motors the frequencies of radial forces due to eccentricity are 

 1 1 fr  = 2 f (1 + k); fr  = 2 f k; fr  = f 2 (1 + k) ± ; fr  = f 2k ± . p p (2.150)

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Radial force harmonics due to stator winding parallel paths The effect of eccentricity can produce additional radial forces if the stator winding has parallel current paths. The maximum number of parallel paths is a p = 2 p. The magnitude of the magnetic flux density waveform excited by currents in parallel paths is usually not more than 10 to 15% of the magnitude of the fundamental waveform excited by the stator winding. Interaction of fundamental flux density wave of r = p and frequency f with backward rotating wave (r = p∓1, f = − f ) caused by internal short-circuit currents for  > 0 gives radial magnetic force wave of r = 1 and fr  = 2 f . The magnitude of this force increases with the load. To avoid noise, Heller and Hamata [87] recommend the following selection of the numbers of rotor slots of induction machines: 
 p   (2.151) − p(6c ± 1) =  1 if a p > 2 is odd  s2 ± a       s2 ± 2 p − p(6c ± 1) = (2.152) if a p > 2 is even   1 a |(s2 ± 2 p) − p(6c ± 1)| =  1

if

ap = 2

(2.153)

where c = 1, 2, 3, . . . is a small integer. Radial force harmonics due to magnetic saturation The saturation magnetic flux density equation is a product of the MMF for ν = 1 according to Equation 2.45 and the second term of the air gap permeance (Equation 2.102), (Equation 2.103), i.e., bsat (α, t) = F1,1 (α, t)sat (α, t) = −Fmν=1 cos( pα − ωt)sat cos(2 pα − 2ωt − 2φs ) 1 1 = − Fmν=1 sat cos(− pα + ωt + 2φs ) − Fmν=1 sat 2 2 × cos(3 pα − 3ωt − 2φs ) = −Bsat [cos( pα − ωt − 2φs ) + cos(3 pα − 3ωt − 2φs )]

(2.154)

where Bsat = 0.5Fmν=1 sat . The above Equation 2.154 describes two stator magnetic flux density saturation waves. One wave has frequency and number of poles the same as the fundamental and second wave has three times higher frequency and number of pole pairs than the fundamental. Equation 2.154 includes both the magnetic saturation of the stator teeth and core (yoke) [64, 179]. Assuming that only the harmonic ν = 3 in Equation 2.154 saturates the teeth, the flux density in the rotor due to saturation is: b2 (α, t) =

∞ 

Bsat,µ cos(µpα − ωµ − ψµ )

(2.155)

µ=1

where for induction motor µ=k

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s2 + 3. p

(2.156)

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Noise of Polyphase Electric Motors

According to Equation 2.134 the angular frequency of the rotor saturation harmonics

s2 ωµ = 3ω + ks2  = 3ω ± k 2π f (1 − s ) . p The angular frequency an order of radial magnetic forces due to saturation of the magnetic circuit can be found with the aid of Equation 2.126

s2 ωr sat = ω ± ωµ = ω ± 3ω ± k 2π (1 − s ) (2.157) p   s1 s2 r = (ν ± µ) p = k ± 1 ± k + 3 p . p p The last two equations give the frequencies and orders of radial magnetic forces due to saturation in the following form

s2 s2 fr sat = k (1 − s ) + 4 f and fr sat = k (1 − s ) + 2 f (2.158) p p r = ks1 + ks2 + 4 p

and

r = ks1 + ks2 + 2 p .

(2.159)

Frequencies of radial magnetic forces produced by higher space harmonics in cage induction and PM synchronous motors are given in Tables 2.5 and 2.6.

2.8 2.8.1

Other sources of electromagnetic vibration and noise Unbalanced line voltage

The unbalanced line voltage causes vibration the frequency of which is equal to double the line frequency, i.e., fr unb = 2 f.

(2.160)

The effect of phase unbalance can be included by assigning different values of currents for each phase in Equation 2.42 and then calculating the resultant stator MMF similar to Equation 2.43.

2.8.2

Magnetostriction

Magnetostriction is the change of physical dimensions and crystals of a material in response to changing its magnetization. When a magnetostrictive material is subjected to an alternating magnetic field, it changes its shape and dimensions. Most ferromagnetic materials show some measurable magnetostriction. The

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Table 2.5 Frequencies of radial magnetic forces produced by higher space harmonic ν = 1 (fundamental time harmonic n = 1) in induction motors. Source

Frequency Hz

Product of stator space haromics bν2 of fr = 2 f the same number ν Product of rotor space fr = 2 f [1 ± k(s2 / p)(1 − s)] harmonics bµ2 of the where s is slip for same number µ fundamental space harmonic Product of stator and rotor space harmonics bν bµ — general fr = f ± f µ equations Product of stator and rotor space harmonics bν bµ where fr = [k(s2 / p)(1 − s) ± 2] f ν = ks1 / p ± 1 and fr = [k(s2 / p)(1 − s)] f µ = ks2 / p ± 1 (the so called slot or step harmonics) Product of stator and rotor static eccentricity fr = [2 + k(s2 / p)(1 − s)] f space fr = [k(s2 / p)(1 − s)] f harmonics bν bµ Product of stator and rotor dynamic fr = [2 ± (1 − s)/ p eccentricity +k(s2 / p)(1 − s)] f space fr = [(1 − s)/ p harmonics +k(s2 / p)(1 − s)] f bν bµ Product of stator and rotor magnetic fr = [k(s2 / p)(1 − s) + 4] f saturation space fr = [k(s2 / p)(1 − s) + 2] f harmonics bν bµ

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Order (circumferential mode) r = 2νp r = 2(ks1 ± p) k = 0, 1, 2, 3, . . . r = 2µp r = 2(ks2 ± p) r = (ν ± µ) p

r = ks1 ± ks2 ± 2 p

r =1 r =2

r =1 r =2

r = ks1 + ks2 + 4 p r = ks1 + ks2 + 2 p

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Noise of Polyphase Electric Motors

Table 2.6 Frequencies of radial magnetic forces produced by higher space harmonic ν = 1 (fundamental time harmonic n = 1) in synchronous motors. Source

Frequency Hz

Order (circumferential mode) r = 2νp ν = 2km 1 ± 1 k = 0, 1, 2, 3, . . .

Product of stator space haromics bν2 of fr = 2 f the same number ν Interaction of the rotor fr = 2µλ f magnetic field and slotted where r = 2(µλ p ± s1 ) core of the stator µλ = integer [s1 / p] Product of rotor space haromics bµ2 of the fr = 2(1 ± 2k) f r = 2µp = 2 p(1 ± 2k) same number µ where µ = 1 ± 2k Product of stator and rotor space harmonics bν bµ — general fr = f ± f µ r = (ν ± µ) p equations Product of stator winding and rotor space harmonics fr = 2(1 + k) f bν bµ where fr = 2k f r = ks1 ± 2 p(1 + k) ν = ks1 / p ± 1 and µ = 2k ± 1 Product of stator and rotor static eccentricity fr = 2(1 + k) f r =1 space fr = 2k f r =2 harmonics bν bµ Product of stator and rotor dynamic eccentricity fr = [2(1 + k) ± 1/ p] f r =1 space fr = (2k ± 1/ p) f r =2 harmonics bν bµ Product of stator and rotor magnetic fr = 2(2 + k) f r = ks1 + 2 p(k + 2) saturation space fr = 2(1 + k) f r = ks1 + 2 p(k + 1) harmonics bν bµ

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magnetostrictive coefficient  is defined as the change in length l to the length l of a specimen when the magnetization increases from zero to its saturation value, i.e., l = (2.161) l The coefficient  can be positive or negative. In the first case, the increase in the magnetic flux density causes expansion of the specimen. In the second case, the specimen shrinks as the magnetic flux density increases. For pure iron the saturation magnetostriction is  = −2 × 10−5 to +2 × 10−5 , for laminated steels  = −0.1 × 10−5 to +0.5 × 10−5 , for nickel  = −5.2 × 10−5 to 0.8 × 10−5 and for polycrystal cobalt  = −6 × 10−5 to −2.5 × 10−5 . Some alloys, e.g., iron and dysprosium DyFe2 or iron and terbium TbFe2 , display “giant” magnetostriction under relatively small fields. If the magnetostriction acts to contract a specimen, then this will act against any tensile stress on the material and leads to larger value of the Young’s modulus. The core of an electrical machine or transformer in alternating magnetic fields changes its dimensions cyclically. Figure 2.10 shows a sinusoidal variation of the magnetic flux density and corresponding variation of the magnetostrictive coefficient with time. The curve (t) shown in Figure 2.10 resolved into Fourier series has a constant component and a series of harmonic functions. Those harmonics approximately visualize vibrations relative to the mean position which is determined by the constant component. The fundamental wave of those vibration has a frequency twice the frequency of the flux density f (line frequency), i.e., f ms = 2 f

(a)

(2.162)

b(t)

t

(b)

Λ(t)

t

Figure 2.10 Variation of magnetostrictive coefficient for sinusoidal magnetic flux density: (a) magnetic flux density as a function of time; (b) magnetostrictive coefficient as a function of time.

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Noise of Polyphase Electric Motors

while the order of the fundamental wave of the magnetostriction force in an a.c. rotating machine is rms = 2 p (2.163) For static electromechanical energy converters as transformers, inductors, fluorescent light ballast, etc., magnetostriction is the major cause of acoustic noise. For a.c. rotating machines magnetostriction can contribute considerably to the acoustic noise. Especially, in two pole machines ( p = 1) the magnetostriction force at the twice line frequency (Equation 2.162) can generate undesirable acoustic noise.

2.8.3

Thermal stress analogy

The effect of magnetostriction can be implemented in a similar way as thermal stresses are applied, i.e., the thermal expansion of the free body is calculated based upon the temperature distribution [45, 159]. The elastic strain in the x direction is el,x =

1 (σx − νσ y ) E

(2.164)

where σx is the external stress in the x direction, σ y is the external stress in the y direction, σz = 0, E is Young modulus and ν is Poisson ratio. Including thermal stresses when a material is heated x − αϑ ϑ =

1 (σx − νσ y ) E

(2.165)

where αϑ is the coefficient of thermal expansion, αϑ ϑ is the thermal strain and x = el,x + αϑ ϑ is the total strain (elastic and thermal). For a magnetostrictive material, the magnetostrictive strain λ(B) can also be added to the elastic strain [45], similar to the thermal expansion strain (Equation 2.165), i.e., 1 x − λ(B) = (σx − νσ y ) (2.166) E where the total strain (elastic and magnetostrictive) is x = el,x + λ(B). The magnetostrictive strain λ(B) = 10−6 B 2 (2.167)  expressed as a function of the magnetic flux density in the direction of the vector B B is sometiems called the magnetostriction curve [45].

2.8.4

FEM model

The magnetic field is affected by the effect of magnetostriction. Thus, the coupled magnetic and mechanical system must be captured in one magnetomechanical matrix. The effect of magnetostriction can be built into the coupled system using a force distribution that is added to magnetic forces [45, 159, 210, 211].

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The elastic energy of a mechanical finite-element is expressed in terms of the elastic displacement αel and mechanical stiffness matrix K U=

1 T α K αel . 2 el

(2.168)

In general cases the elastic displacement αel =  α where α is the total displacement, because the material may show magnetostrictive or thermal expansion. Including the magnetostrictive displacement αms the elastic energy is U=

1 (α − αms )T K (α − αms ). 2

(2.169)

The strains λx , λ y in the x and y directions can be converted into nodal dise (e) placements αms = (αx,i , α ey,i ) where i = 1, 2, 3 means the element nodes with coordinates xi , yi . Considering the midpoint element (xm , ym ) as fixed  

(e) αx,i (xi − xm )λx . (2.170) = (yi − ym )λ y α (e) y,i The nodal magnetostriction forces are obtained by multiplying the mechanical (e) stiffness matrix K (e) for one element by the magnetostrictive element αms of the nodes, i.e., (e) (e) Fms = K (e) αms . (2.171) The total magnetostriction force is a sum of magnetostriction forces of all elements, i.e.,  (e) Fms = Fms = K αms . (2.172) e

The magnetic forces (sum of reluctance forces and Lorentz forces) can be found using virtual works, i.e.,  A ∂M ∂W Fmag = − AT =− dA (2.173) ∂a ∂a 0 where M is the magnetic “stiffness” matrix, a is the linear displacement, and A is the magnetic vector potential component in the normal direction (z-direction). The magnetic energy 1 W = A T M A. (2.174) 2 The total energy of the magnetomechanical system is the sum of the elastic energy U according to Equation 2.168 and magnetic energy W according to Equation 2.174, i.e., E = U + W. (2.175) The magnetomechanical system is described by the matrix equations [44, 45]: • without magnetostriction



M 0 A T = (2.176) 0 K a R + Fmag

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Noise of Polyphase Electric Motors • with feedback of magnetostriction



M 0 A T = 0 K a R + Fmag + Fms

(2.177)

where R = K a represents external forces and T = M A is the magnetic source term vector. Further procedures for deformation and vibration calculations are given in [44, 45].

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3 Inverter-Fed Motors Most variable speed electric motors are fed from solid state converters (SSCs). The a.c.–a.c. SSC consists of a rectifier, intermediate circuit (filter), and inverter (Figure 3.1). The output voltage and output current of a SSC are nonsinusoidal, that is, contain higher time harmonics as a result of switching. The noise of electric machines with the stator winding nonsinusoidal current increases as compared with an equivalent machine with sinusoidal stator current. The increase in the noise of an induction motor fed from a pulse width modulation (PWM) inverter with switching frequency up to 7 kHz is from 7 to 15 dB(A) [80]. For higher switching frequencies between 7 and 16 kHz the increase in the noise is lower, usually from 2 to 7 dB(A) [80].

3.1

Generation of higher time harmonics

The stator current of an electrical machine fed from or loaded with a SSC is nonsinusoidal, that is, contains higher time harmonics n = 2m 1 k ±1 where k = 0, 1, 2, 3, . . .. The nonsinusoidal current can be expressed in terms of Fourier series as n=∞

√ n=∞ i 1 (t) = I1mn sin(ωn t − φin ) = 2 I1n sin(ωn t − φin ) (3.1) √

n=1

n=1

where I1mn = 2I1n is the peak value of the nth harmonic current. The angular frequency and frequency of higher time harmonic currents are, respectively ωn = nω = 2π n f

f n = n f.

(3.2)

Currents expressed by Equation 3.1 are shifted in each phase of the stator winding of a polyphase machine by the angle 2π/m 1 , where m 1 is the number of phases. For induction motors, the slip for the nth time harmonic is defined as sn =

nn s ∓ n s (1 − s) nn s 65

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(3.3)

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Noise of Polyphase Electric Motors Intermediate Circuit

Rectifier

Inverter VVVF

50 or

d.c.

60 Hz

IM a.c.

a.c.

Figure 3.1 Basic circuitry of a variable voltage variable frequency (VVVF) voltage source SSC for an induction motor. where nn s = n f / p is the synchronous speed for the nth time harmonic, n s (1 − s ) is the rotor speed, and s is the slip for the fundamental harmonic. The “−” sign is for the time harmonics n = 2m 1 k + 1 and the sign “+” is for the time harmonics n = 2m 1 k − 1 where k = 0, 1, 2, 3, . . .. Induction motors operate with small slip 0.02 ≤ s ≤ 0.05, so that 1 sn ≈ 1 ∓ . (3.4) n As the number of time harmonics increases, the slip (Equation 3.4) becomes close to unity. The slip sn , angular frequency ωn of rotor currents, and higher time harmonic slip ssn corresponding to the synchronous speed (Equation 2.1) are: • for forward-rotating magnetic field sn = 1 −

1 (1 − s ); n

ωn = n ωsn = ω[n − (1 − s )];

ssn = 1 − n (3.5)

• for backward-rotating magnetic field sn = 1 +

1 (1 − s ); n

ωn = n ωsn = ω[n + (1 − s )];

ssn = 1 + n (3.6)

where s is the slip for the fundamental harmonic according to Equation 2.2, ω = 2π f is the angular frequency for the fundamental harmonic, f is the input frequency of the fundamental harmonic, and n is the time harmonic number. Higher time harmonics in the stator current of permanent magnet (PM) brushless motors are discussed in Section 4.7.

3.2

Analysis of radial forces for nonsinusoidal currents

The frequencies and orders of most important radial forces produced by SSCs (inverters) are given in Table 3.1. A brief analysis of these forces follows.

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Table 3.1 Frequencies and orders of most important radial magnetic forces produced by inverter harmonics. Frequency Hz

Source Product of stator 2 harmonics b1νn of the same number n Product of rotor harmonics of the same number Product of stator and rotor space harmonics Sums and differences of inverter fundamental and higher harmonics bn bn of different numbers Interaction of the switching frequency f sw and higher time harmonics f n = n f sw ± n f Interaction of the permeance and MMF harmonics associated with time harmonics Rectifier harmonics transmitted to the motor via intermediate circuit and inverter

3.2.1

Order (circumferential mode)

fr,n = 2n f , n = 2km 1 ± 1 where k = 0, 1, 2, 3, . . . fr,n = 2n f µ fr,n = f n ± f µ see also Equation 3.13 fr,n = f ± f n n = 2km 1 ± 1

r = 2νp r = 2 p for ν = 1 r = 2µp r = 2 p for µ = 1 r = (ν ± µ) p r =0 or r = 2p

fr,n = |(± f n ) − f | n =  n

r =0

fr,n = |(± f n − f | fr,n = | f n ± µf |

r =0 r =2

fr = 2m 1 k f k = 1, 2, 3, . . .

r = 2p

Stator and rotor magnetic flux density

The space-time variation of the magnetic flux density for the n-th time harmonic can be expressed as a sum of space harmonics • for the stator b1n (α, t) =

∞


b1νn (α, t) =

ν=1

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∞
ν=1

Bmνn cos(νpα − ωn t)

(3.7)

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Noise of Polyphase Electric Motors where b1νn (α, t ) = Bm νn cos(νp α − ωn t ) • for the rotor b2n (α, t ) =

∞


b1µn (α, t ) =

µ=1

∞


Bm µn cos(µp α − ωµ,n t + φµ,n )

(3.8)

µ=1

where b1µn (α, t ) = Bm µn cos(µp α − ωµ,n + φµ,n ). In the above Equations 3.7 and 3.8, ν is the number of the stator space harmonics, µ is the number of the rotor space harmonics, n is the number of the stator time harmonics, Bm νn and Bm µn are the peak values of the magnetic flux density harmonics for the stator and rotor, respectively, α is the angular distance from the origin of the coordinate system, ωn = 2π n f is the angular frequency of the time harmonic current in stator windings, f is the fundamental frequency, ωµ,n is the angular frequency of the space harmonic µ in the rotor system for the given time harmonic n, φµ,n is the angle between vectors of the stator and rotor space harmonics for the given n, and µ0 = 0.4π × 10−6 H/m is the magnetic permeability of free space. Higher space harmonics result from the distribution of stator winding coils in slots, rotor winding slots, or geometry of rotor poles. Higher time harmonics in the stator current are generated by the power supply, that is, utility mains or SSC.

3.2.2

Stator harmonics of the same number

The radial force pressure Figure 2.8 is expressed with the aid of Maxwell stress tensor — Equation 2.114. The first term in Equation 2.114 is the product of the stator harmonics [b1νn (α, t)]2 of the same number ν or n, i.e., [b1νn (α, t)]2 [Bmνn cos(νpα ∓ ωn t]2 = 2µ0 2µ0 2 B = mνn [1 + cos(2νpα ∓ 2ωn t)]. 4µ0

pr,n (α, t) =

(3.9)

For higher time harmonics n > 1 the angular frequency of stator space harmonics is ωn = 2π f n , where f n = n f is the frequency of the n-th higher time harmonic. The constant term for each harmonic ν and n has no significance for noise generation as the magnitudes Bmνn are uniformly distributed along the air gap. For the fundamental harmonic ν = 1 and n = 1 the frequency of the radial force pressure is fr = 2 f and the order (circumferential vibrational mode) r = 2 p. For higher space harmonics ν > 1 and fundamental time harmonic n = 1 the frequency of the radial force pressure is also fr = 2 f and the mode r = 2νp. For ν = 1 and n > 1 the frequency of the radial force pressure is fr,n = 2n f and the order is r = 2 p. Of course, there will be many more harmonics of radial forces as mixed products of stator magnetic flux density harmonics of different numbers, for example, 1 and 7, also produce radial forces.

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69

Interaction of stator and rotor harmonics

The second term in Equation 2.114 is the mixed product of the stator and rotor harmonics 2b1νn (α, t)b2µn (α, t), i.e., pr,n (α, t) = = =

2b1νn (α, t)b1µn (α, t) 2µ0  ∞ 2 ν=1 Bmνn cos(νpα ∓ ωn t) ∞ µ=1 Bmµn cos(µpα ∓ ωµ,n t + φµ,n ) 2µ0 1 2µ0

∞
∞


Bmνn Bmµn {cos[(νpα ∓ ωn t) − (µpα ∓ ωµ,n t + φµ,n ]

ν=1 µ=1

+ cos[(νpα ∓ ωn t) + (µpα ∓ ωµ,n t + φµ,n ]} ∞ ∞ 1

= Bmνn Bmµn {cos[(ν − µ) pα ∓ (ωn − ωµ,n )t − φµ,n ] 2µ0 ν=1 µ=1 + cos[(ν + µ) pα ∓ (ωn + ωµ,n )t + φµ,n ]}.

(3.10)

The mixed products of the stator and rotor harmonics generate pairs of radial force pressures with pole pairs r = (ν ± µ) p and frequencies fr,n = f n ± f µ,n . For a synchronous machine the angular frequency of the rotor space harmonics for n > 1 is ωµ,n = 2π f n (1 ± k). For an induction machine with the number of rotor slots s2 , the angular frequency of the rotor winding harmonics for time harmonics n ≥ 1 and a “harmonic rotor” angular speed mn = 2π( f n / p)(1 − sn ) is   fn ωµ,n = ωn + ks2 mn = ωn ± s2 k 2π (1 − sn ) p   s2 = 2π f n 1 ± k (1 − sn ) . (3.11) p According to Equations 3.3 and 3.4, for n = 1 the slip sn=1 = s ≈ 0 and for n > 1 the slip sn ≈ 1. The frequencies of noise and vibration of induction machines due to the mixed products of b1νn (α, t)b2µn (α, t) for µ = ks2 / p ± 1 are   s2 fr,n = f n ± f µ,n = f n ± f n 1 ± k (1 − sn ) . p The last equation can also be written as   s2 fr,n = f n 1 ± 1 ± k (1 − sn ) p   s2 s2 fr,n = f n k (1 − sn ) fr,n = f n k (1 − sn ) ± 2 . p p

(3.12)

(3.13)

The orders of the mixed product harmonics are expressed by Equation 2.137.

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3.2.4

Rotor harmonics of the same number

The third term in Equation 2.114 is the product of the rotor harmonics [b2µn (α, t)2 ] of the same number [b2µn (α, t)]2 [Bmµn cos(µpα ∓ ωµ,n t + φµ,n )]2 = 2µ0 2µ0 1 = Bmµn [1 + cos(2µpα ∓ 2ωµ,n t + 2φµ,n )]. 4µ0

pr,n (α, t) =

3.2.5

(3.14)

Frequencies and orders of magnetic pressure for nonsinusoidal currents

Angular frequencies and orders of the radial magnetic pressure due to higher time and space harmonics result from Equations 3.9, 3.10, and 3.14, i.e.: • excited by the stator harmonics of the same number ν, i.e., ωr,n = 2nω;

fr,n = 2n f ;

r = 2νp

(3.15)

• excited by the stator ν and rotor µ harmonics ωr,n = (nω ±ωµ,n );

fr,n = (n f ± f µ,n );

• excited by the rotor harmonics of the same number µ, i.e., ωr,n = 2nωµ ;

fr,n = 2n f µ ;

r = (ν ±µ) p (3.16)

r = 2µp.

(3.17)

According to Equation 3.15, for the fundamental space harmonic ν = 1 and time harmonics n > 1, the frequency of the radial magnetic pressure is fr,n = 2n f and the order r = 2 p. According to Equation 3.16, for ν = 1 and time harmonics n > 1, the frequency of the radial magnetic pressure is fr,n = n( f ± f µ ) and the order r = (1 ± µ) p. For induction machines the frequencies of magnetic pressures as a function of slip sn are expressed by Equation 3.11. Putting s2 = so = 2 p and sn = 0 for a synchronous machine, the frequency becomes fr,n = f n (1±kso / p) = n f (1 ± 2k). According to Equation 3.17, for space harmonics µ > 1 and time harmonics n > 1, the frequency of the radial magnetic pressure is fr,n = 2n f µ and the order r = 2µp. For a salient pole synchronous machine fr,n = 2n f (1 ± 2k) and for induction machine fr,n = 2n f [1 ± k(s2 / p)(1 − sn )] since according to Equation 2.143 fr = 2 f [1 ± ks2 (1 − s)/ p].

3.2.6

Interaction of stator harmonics of different numbers

Inverters can supply a rich spectrum of higher time harmonics to the stator winding. Higher time harmonics of different numbers can produce significant radial forces, the frequency and order of which can be expressed by the following equations fr,n = (n ± n ) f ;

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r =0

or

r = 2p

(3.18)

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where n =  n . Most important are magnetic forces due to sums and differences of the fundamental harmonic f with major higher time harmonics of the stator current [71]. The frequencies of those forces are fr,n = (1 ± n) f where n = 2km 1 ± 1.

3.2.7

Interaction of switching frequency and higher time harmonics

Large amplitude forces are also due to interaction of the switching frequency f sw and higher time harmonics f n = n f sw ± n f [138]. If n is an odd integer, n will be an even integer and vice versa, i.e., f n = f sw ± 2 f , f sw ± 4 f , f sw ± 6 f, . . . and f n = 2 f sw ± f , 2 f sw ± 3 f , 4 f sw ± 5 f, . . .. The most important case is the interaction between the fundamental field harmonic and higher time harmonics for which the frequencies and mode of vibration are [138]: fr,n = |(± f n ) − f |;

3.2.8

r =0

(3.19)

Interaction of permeance and magnetomotive force (MMF) harmonics

Interaction of the permeance field harmonics and MMF harmonics associated with higher time harmonics of the stator current can also result in important vibration, especially at full load and when force orders are low [138]. The frequencies and orders of radial forces are [138]   s2 fr,n = |(± f n − f |; r = 0, 2. fr,n = | f n ± f 1 + k (1 − s) |; p (3.20)

3.2.9

Rectifier harmonics

Rectifier harmonics are transmitted via the intermediate circuit and inverter to the stator winding. They can produce radial forces with the following frequency and order fr,n = n f = 2m 1 k f ; r = 2p (3.21) where n = 2m 1 k, k = 1, 2, 3, . . ., and m 1 is the number of phases.

3.3 3.3.1

Higher time harmonic torques in induction machines Asynchronous torques

Asynchronous torques due to higher time harmonics are expressed by the following equations m1 (3.22) Tn = √ pN2 kw2 n I2n cos ψ2n = cT n I2n cos ψ2n 2

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or

2 (I2n ) R2n (I )2 R2n Pw2n = m 1 p 2n =p (3.23) s n n sn ωn sn ωn √ where cT = m 1 pN2 kw2 / 2, N2 is the number of rotor turns per phase, kw2 is the rotor winding factor for fundamental, n is the stator magnetic flux produced by the nth time harmonic, ψ2n is the phase shift between the electromotive force (EMF) (excited by n ) and rotor harmonic current I2n , I2n is the rotor current due to the nth time harmonic referred to the stator system, R2n is the rotor resistance referred to the stator system, Pw2n are the rotor winding losses for the nth time harmonic, and 2π n f ωn n = = (3.24) p p

Tn = m 1

is the angular speed of the stator magnetic field due to the nth time harmonic. The direction of the torque (Equation 3.22) and (Equation 3.23) depends on the number of the higher time harmonic. The torque can be in the direction of rotation (propulsion torque) or in the opposite direction (braking torque). For a three-phase machine (m 1 = 3) higher time harmonics n = 1, 7, 13, . . . generate asynchronous torques in the direction of rotation and higher harmonics n = 5, 11, 17, . . . generate braking torques. For evaluation of torques, the electric losses in the stator winding can be assumed equal to the electric losses in the rotor winding, i.e. [21], Pw2n ≈ 0.5Pwn ≈ 0.5(SC R)2

Pw Pws=1 ≈ 0.5 3 n n3

(3.25)

where SC R = 3 . . . 7 is the starting current ratio (starting–to–rated current), Pwn are stator and rotor winding losses due tho the nth time harmonic, Pw are stator and rotor winding losses at rated load, Pws=1 are winding losses at locked rotor (s = 1), and Pw Pwn = (SC R)2 3 . (3.26) n Assuming that for higher time harmonics sn ≈ 1 and n = n, the asynchronous torque is inversely proportional to n 4 , i.e., Tn = p

Pws=1 Tst Pw2n Pws=1 ≈ ≈ 4 ≈p 3 4 s n ωn 2n × 1 × nω 2n  n

(3.27)

where Tst ≈ 0.5Pws=1 /  is the starting torque due to the fundamental harmonic. Since higher harmonic asynchronous torques are inversely proportional to n 4 , they are negligible and have little effect on the vibration and noise.

3.3.2

Pulsating torques

Pulsating torques are produced as interaction of currents and fluxes of different frequencies. Their frequencies are significantly higher than the fundamental frequency and their mean value is equal to zero. The number of these torques can

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be extremely large as there is a very large number of interactions among all harmonics. The most important pulsating torques are those produced by the stator fundamental magnetic flux and rotor current higher harmonics expressed by the following equation: 2 pm 1 T p1,n = √ N2 kw2 I2n cos[(n ± 1)]ωt − ψ1,2n ] 2 = 2cT I2n cos[(n ± 1)]ωt − ψ1,2n ]

(3.28)

where  is the magnetic flux due to the fundamental harmonic n = 1 and ψ1,2n is the phase shift between the EMF for n = 1 and rotor current I2n . The “−” sign is for harmonics n = 2m 1 k + 1 and the sign “+” is for the harmonics n = 2m 1 k−1. It means that in a three-phase machine the pulsating torques due to the 5th and 7th harmonics have the same frequency 6 f and rotate in opposite directions, i.e., f − (−5 f ) = 6 f and f − 7 f = −6 f . In three-phase induction motors the major torque ripple source is the 6th harmonic component. For the 11th and 13th harmonics the frequency of pulsating torques is 12 f , i.e., f − (−11 f ) = 12 f and f − 13 f = −12 f . The magnitudes of pulsating torques at SC R = 3 . . . 5 are 7 to 12% of the rated torque for the frequency 6 f (due to the 5th and 7th harmonics) and 0.8 to 1.5% for the frequency 12 f (due to the 11th and 13th harmonics [21]. Pulsating torques may have a signifucant influence on the vibration and noise.

3.4

Higher time harmonic torques in permanent magnet (PM) brushless machines

The interaction of the stator current and magnetic flux linkage harmonics produce: • constant torques if harmonics are of the same number; • pulsating torques if harmonics are of different numbers. For trapezoidal EMF and rectangular current the electromagnetic torque is constant (no pulsating torque) [117, 177]. All odd harmonics of the magnetic flux that interact with the stator current harmonics of the same number (except of triple harmonics) produce a constant torque, for example, fundamental harmonic of flux with the fundamental harmonic of current, 5th harmonic of flux and 5th harmonic of current, 7th harmonic of flux and 7th harmonic of current, and so on. However, the contribution of the odd harmonics n > 1 to the constant torque is negligible.

3.5

Influence of the switching frequency of an inverter

There is a significant influence of the switching frequency of a pulse width modulation (PWM) inverter on the overall motor noise. The switching frequency of the inverter is established by the frequency of a triangular waveform. In a PWM

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75

Sound Pressure Level, dB(A)

70

65

60

55 Switching Frequency 1.8 kHz Switching Frequency 3.6 kHz Switching Frequency 7.2 kHz Switching Frequency 14.4 kHz

50

45

40 0

10

20

30

40 50 60 Frequency, Hz

70

80

90

100

Figure 3.2 Influence of the inverter on the sound pressure level of a 9.3 kW (12.5 hp), 60 Hz, 4-pole induction motor for four different switching frequencies [161]. inverter the sinusoidal control signal at the desired frequency is compared with a triangular waveform [156]. At speeds below the rated speed, the noise level of a small motor with a shaft driven fan decreases with the speed (Chapter 7, Equation 7.28). Below the rated speed the magnetic noise is predominant. When the speed exceeds the rated speed, the overall noise increases due to the increased noise of the fan. Figure 3.2 shows the overall noise of a 9.3 kW (12.5 hp), 4-pole induction motor fed from a PWM inverter for four different switching frequencies [161]. For 1.8-kHz switching frequency, the influence of the magnetic noise is considerable and the sound pressure level changes only ±4 dB(A) within the operation frequency range from 10 to 100 Hz. For 3.6, 7.2, and 144 kHz switching frequencies, the magnetic noise and consequently the overall noise is reduced significantly for operation frequencies below the rated frequency of 60 Hz [161]. Table 3.2 compares the sound pressure level radiated by small power induction motors fed with 60 Hz three-phase utility grid and PWM inverters [161]. It is clear that below the rated frequency 60 Hz the sound pressure level decreases as the switching frequency is increased. This is because the noise of magnetic origin

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80

Noise Level (dB)

70

4 kHz Carrier

60 50 40

16 kHz Carrier

30 20 10 0

0

10

20

30

40

50

60

Motor Frequency (Hz)

Figure 3.3 Noise level vs. induction motor operation frequency for 4 and 16 kHz carrier frequencies of X2C series programmable inverters. (Courtesy of TB Wood’s, Chambersburg, PA, U.S.A. [244].) is predominant at low operation frequencies. The overall noise at low speeds is strongly affected by the switching frequency (compare the SPL at 30 and 60 Hz). At operation frequencies above the rated frequency 60 Hz, the influence of the cooling fan on the overall noise is very strong. The increased sound power level at 100 Hz is due to the fan and is practically independent of the switching frequency. The switching frequencies can cause pure tones that are very disturbing to the human ear [147]. The vibration frequncies due to fundamental waves and switching frequencies must not match the natural frequencies of the stator system. Otherwise, the audible noise increases significantly [147]. Modern inverters operate at higher switching frequencies ( f s ≥ 12 kHz), which results in lower harmonic content, especially at low speeds. The inverter losses increase with increasing the switching frequency and motor higher harmonic losses decrease with increasing the switching frequency [147]. Thus, at low speeds the influence of the switching frequency on the efficiency of the motor–inverter system is rather small [147]. Many inverters, for example, the X2C series TB Wood’s inverters have programmable carrier frequencies from 4 to 16 kHz for quiet drive operation [244]. Low carrier frequencies can generate audible motor noise, while at 16 kHz audible noise is virtually eliminated (see Figure 3.3).

3.6

Noise reduction of inverter-fed motors

Inverter-fed induction and PM synchronous motors are noisier than those fed with purely sinusoidal current. The radial magnetic force spectrum is richer and the chance of matching the exciting frequencies with natural frequencies of the stator

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Table 3.2 Influence of inverter on sound pressure level of induction motors, dB(A) [161] Output Utility power kW

0.37 2p = 4 0.75 2p = 2 0.75 2p = 4 2.2 2p = 2 2.2 2p = 4 3.7 2p = 4 5.5 2p = 4 9.3 2p = 4 15.0 2p = 4

grid

Switching frequency of inverter 3.6 kHz

7.2 kHz

14.4 kHz

60 Hz Operation frequency Operation frequency Operation frequency Hz Hz Hz 30 60 100 30 60 100 30 60 100 46.1 52.1 50.5

58.9

42.2 49.3

58.9

37.1 46.8

58.8

61.2 61.2 68.2

73.1

45.9 60.5

72.7

44.7 60.4

72.8

44.7 60.3 52.9

58.8

47.0 57.1

59.5

34.3 46.8

58.5

67.5 60.6 68.5

80.8

54.9 68.4

80.8

53.3 68.2

81.1

53.1 68.7 60.6

63.8

56.6 65.4

64.7

46.6 52.9

63.4

54.4 56.3 58.7

66.8

53.7 65.6

67.0

44.3 54.8

66.5

56.3 61.8 59.6

68.1

52.9 63.6

68.2

43.0 56.5

68.0

61.7 65.1 71.1

73.7

57.2 68.1

73.4

46.7 61.9

73.4

69.2 68.7 73.4

80.9

54.8 69.8

80.8

52.5 69.3

80.7

system is greatly increased. The following strategies can be employed to reduce the acoustic noise of inverter-fed motors; • elimination of higher time harmonics; • operation with low commutation angle (the larger the commutation angle, the higher the magnitude of the torque pulsation); • dynamic variation of the switching frequency [147]; • high switching frequency (above 15 kHz) is a very efficient method but imposes high stress on solid switches and increased switching frequency losses; • random PWM [36, 82, 229]; • active noise control of the motor.

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4 Torque Pulsations In addition to radial forces, torque pulsations and tangential forces also contribute to vibrations and magnetic noise. Normally, tangential forces have less effect on vibrations and noise than radial forces. Tangential forces can either slightly magnify or reduce the amplitude of the first order radial forces [87]. In this chapter, calculations of torque ripple and tangential forces with special attention given to cogging torque of permanent magnet (PM) synchronous motors will be discussed.

4.1

Analytical methods of instantaneous torque calculation

The electromagnetic torque of electrical machines is calculated analytically as T =−

∂W ∂

J

(4.1)

where W is the magnetic field energy in the air gap and  is the mechanical angle measured around the air gap periphery. Equation 4.1 can be written as [87] T =−

gL i ∂ 2µ0 ∂




2π

[B(α, t)]2 dα.

(4.2)

0

Neglecting the magnetic saturation B(α, t) =

µ0 µ0 F(α, t) = [F1 (α, t) + F2 (α, t)] g g

(4.3)

where F1 (α, t) is the stator magnetomotive force (MMF) and F2 (α, t) is the rotor MMF. Thus,
2π µ0 L i ∂ T =− [F1 (α, t) + F2 (α, t)]2 dα. (4.4) 2g ∂ 0 77 Copyright © 2006 Taylor & Francis Group, LLC

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For a stationary stator and coordinate system fixed to the stator, only the rotor MMF is dependent on the rotor position [87], i.e., µ0 L i T =− g




2π

[F1 (α, t) + F2 (α, t)]

0

∂F2 (α, t) dα. ∂

(4.5)

Harmonic torques are calculated on the basis of Equation 4.5 putting F1 and F2 in form of the Fourier series. A torque can be produced only by harmonics having an equal number of pole pairs [87].

4.2

Numerical methods of instantaneous torque calculation

The instantaneous torque can be numerically calculated by using • virtual work method

  ∂w(, θ )  ∂w  (i, θ)  T =− =   ∂θ ∂θ =const i=const

(4.6)

where w, w  , , i and θ are the magnetic energy, coenergy, flux linkage vector, current vector and mechanical angle, respectively. • Maxwell stress tensor 

 1   1 2 F = B( B · n)  − B n d S; T = r × F (4.7) µ0 2µ0  T = Li

1 µ0



 r Bn Bt dl

(4.8)

l

where n,  L i , l, r , Bn and Bt are the normal vector to the surface S, stack length, integration contour, radius, radial (normal) component of the magnetic flux density and tangential component of the magnetic flux density, respectively. • Lorentz force theorem  
π/(2m 1 p)  T = i j (t) 2 pN r L i B(θ, t)dθ j=A,B,C

=

−π/(2m 1 p)

1 [e f A (t)i 1A (t) + e f B (t)i 1B (t) + e f C (t)i 1C (t)] m

(4.9)

where p, θ, m and N are the number of pole pairs, mechanical degree, mechanical angular speed and conductor number in phase belt, respectively. The products e f A (t)i 1A (t), e f B (t)i 1B (t), and e f C (t)i 1C (t) represent electromagnetic power per phase.

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4.3

79

Electromagnetic torque components

The instantaneous torque of an electrical motor T (α) = T0 + Tr (α)

(4.10)

has two components (Figure 4.1), i.e.: • constant or average component T0 ; • periodic component Tr (α), which is a function of time or electrical angle α, superimposed on the constant component. The periodic component causes the torque pulsation also called the torque ripple. There are many definitions of the torque ripple. The torque ripple is caused by both the construction of the machine and power supply. The torque ripple can be defined in one of the following ways: Tmax − Tmin Tmax + Tmin Tmax − Tmin tr = Tav Tmax − Tmin tr = Tr ms [torque ripple]r ms Trr ms tr = = Tav Tav

tr =

(4.11) (4.12) (4.13) (4.14)

where the average torques in Equations 4.12 and 4.14
α+T p
Tp 1 1 Tav = T (α)dα = T (α)dα Tp α Tp 0

T Tr T0

Time Tp

Figure 4.1 Constant and periodic components of the torque.

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(4.15)

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and the r ms or effective torque in Equation 4.13
Tp 1 Tr ms = T 2 (α)dα. Tp 0

(4.16)

The r ms torque ripple Trr ms in Equation 4.14 is calculated according to Equation 4.16 in which T (α) is replaced by Tr (α). The time T p in Equations 4.15 and 4.16 is the period of the torque waveform. For sinusoidal waveform the half-cycle √average value is (2/π )Tm , where Tm is the peak torque and the r ms value is Tm / 2. For the waveform containing higher harmonics, the r ms value of the torque ripple is

Trr ms = Trr2 ms1 + Trr2 ms2 + . . . + Trr2 msν . (4.17)

4.4

Sources of torque pulsations

There are three sources of the torque ripple coming from an electrical machine: (a) cogging effect (detent effect), that is, interaction between the rotor magnetic flux and variable permeance of the air gap due to the stator slot opening geometry; (b) distortion of sinusoidal or trapezoidal distribution of the magnetic flux density in the air gap; (c) the difference between permeances of the air gap in the d and q axis. The cogging effect produces the so-called cogging torque, higher harmonics of the magnetic flux density in the air gap produce the field harmonic electromagnetic torque, and the unequal permeance in the d and q axis produces the reluctance torque. The causes of torque pulsation coming from the supply are: (d) current ripple resulting, for example, from PWM; (e) phase current commutation.

4.5

Higher harmonic torques of induction motors

The theory of higher harmonic torques of induction machines due to stator and rotor slots have been developed by G¨orges (1896), Arnold (1909), Punga (1912), Dreyfus (1924), Krondl (1926), Lund (1932), Heller (1947), Alger (1954), Jordan (1962), Oberretel (1965), and others [87]. Harmonic torques of induction motors will not be considered in this book. Any induction motor may be imagined as a series of mechanically coupled asynchronous and synchronous motors having a different number of poles [87]. An asynchronous torque is created if a certain stator MMF harmonic of the order νp produces in the spectrum of the rotor MMFs a harmonic of the same order

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νp = µp [87]. To reduce asynchronous torques due to higher space harmonics, the recommended numbers of slots should be selected according to the following unequalities: s2 ≤ 1.25s1 (4.18) s2 < s1

(4.19)

|s2 − s1 ± p| ≥ 4

(4.20)

where s1 is the number of stator slots and s2 is the number of rotor slots. A synchronous torque is produced if the spectra of the stator and rotor MMF harmonics contain harmonics of the same order νp and if the rotor harmonic of this order is produced by a stator harmonic of another order νp =  µp. To reduce synchronous torque the following number of slots should be avoided: • to reduce synchronous torques at standstill s2 = 2m 1 pk

k = 1, 2, 3, . . .

where

(4.21)

s2 =  s1 − 2m 1 p

(4.22)

s2 =  s1 − 4 p

(4.23)

• to reduce synchronous torques in the motoring region s < 1 s2 =  2m 1 pk + 2 p

(4.24)

• to reduce synchronous torques in the braking region s < 1 s2 =  2m 1 pk − 2 p.

4.6

(4.25)

Cogging torque in permanent magnet (PM) brushless motors

The frequency of the cogging torque is f c = s1 n s = s1

f p

(4.26)

where s1 is the number of stator slots, n s = f / p is the synchronous speed, p is the number of pole pairs and f is the input frequency. There are three sources of the torque ripple coming from the construction of the magnetic and electric circuit of a machine (Section 4.4). The most comprehensive literature review and comparison of methods of the torque ripple reduction is given in [26]. Analytical methods of cogging torque calculation usually neglect the magnetic flux in the stator slots and magnetic saturation of the stator teeth [1, 16, 18, 30, 46, 76, 83, 253, 256]. Cogging torque is derived from the magnetic flux density distribution either by calculating the rate of change of total energy stored in the air gap with respect to the rotor angular

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position [1, 16, 18, 46, 76, 83] or by summing the lateral magnetic forces along the sides of the stator teeth [253]. Many cogging torque calculations are specific to a particular method of reducing the cogging effect [48]. The proposed anlytical method uses the energy of the magnetic field in the air gap and classical equations for the air gap magnetic flux density distribution in a.c. machines with slotted stators [40, 51, 87, 236].

4.6.1

Air gap magnetic flux density

The magnetic flux density distribution in the air gap of a PM machine can be analyzed in a similar way as that of an induction machine with slotted stator and smooth rotor [40, 87]. The normal component of the magnetic flux density in the air gap is a sum of the mean flux density bmean (x ) and periodical component with the period equal to the slot pitch t1 , i.e.:    2π b f (x ) = bmean (x ) + (4.27) Bk cos k t1 k where [40, 87] Bg . (4.28) kC Carter coefficient kC is expressed using the well known Equation 2.72 [87]. Considering the air gap magnetic field distribution only in the interval of one pole pitch τ , the flux density B0 = Bg where Bg is the flat-topped value of the magnetic flux density in the air gap. Fourier series coefficient Bk in Equation 4.27 depends on the approximation of the magnetic flux density distribution over the slot opening. For skewed slots Bk can take the following form (see Equation 2.71) bmean (x ) =

Bk = −2γ1

g 2 2 B0 kok ksk t1

(4.29)

where kok is the coefficient of the slot opening according to Equation 2.64 and ksk is the skew factor. Thus, Equation 4.27 can be brought to the form    ∞ 1 2π g 2 2 b f (x ) = B0 − 2γ1 k k cos k x . (4.30) kC t1 k =1 ok sk t1 In order to include the influence of the finite width of PMs or pole shoes on the cogging effect, the flux density B0 must be expressed as a periodic function of the pole pitch τ , i.e., B0 = b P M (x), where b P M (x) is the distribution of the PM magnetic flux density normal component in a machine without slots along the x-axis. Thus, the normal component of the magnetic flux density distribution in the air gap (Figure 4.2) of a PM brushless motor with slotted stator core can simply be expressed as: b P M (x) b f (x) = + bsl (x). (4.31) kC

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bf (x)

83

1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6

−1.0

−0.8 −1

0 0

0.1

0.2

0.3

0.4

x

0.5 2⋅p⋅τ

Figure 4.2 Distribution of the normal component b f (x) of the magnetic flux density in the air gap (Tesla vs. meter) of a 10-pole, 36-slot PM brushless machine according to Equations 4.31, 4.32, and 4.33. The x coordinate is in the direction of rotation. The magnetic flux density component excited by the rotor PMs is a sum of higher space harmonics µ, i.e.,

π   b P M (x) = (4.32) Bg bµ ksµ cos µ x τ µ and the magnetic flux density component due to stator slots is approximated as

bsl (x) = −2γ1

  ∞

π  2π g 2 2 kok ksk cos k x Bg bµ ksµ cos µ x . t1 k=1 t1 τ µ

(4.33)

Equation 4.33 describes the distribution of the normal component of the magnetic flux density excited by PMs (slotless machine) for B0 = b P M (x). For periodic variation of the PM flux density    ∞

π  1  g 2 2 2π b f (x) = Bg bµ ksµ cos µ x − 2γ k k cos k x . τ kC t1 k=1 ok sk t1 µ (4.34) The coefficient of Fourier series bµ for higher space harmonics µ = 2m 1 k ± 1 in Equation 4.34 depends on the shape of the normal component of the magnetic flux density distribution in the air gap. It can be expressed by Equation 2.105 [73]. The pole width b p –to–pole pitch τ ratio for PM brushless motors αi = b p /τ = 0.65 to 0.92. The parameter α in Equation 2.105 describes the shape of the magnetic flux density waveform under the PM or pole shoe. For a constant (flat-topped) value of the magnetic flux density in the range −0.5b p ≤ x ≤ 0.5b p as shown in

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Figure 4.2 the parameter α = 0. For the magnetic flux density curve changing underneath the pole according to hyperbolic cosine law (concave curve) the parameter 0 < α ≤ 1. Equation 4.34 results from the classical theory of a.c. electrical machines, e.g., [40, 51, 87]. In the above Equations 4.31, 4.32, 4.33, and 4.34, kC is the Carter coefficient of the air gap taking into account slot openings, Bg is the flat-topped value of the periodical magnetic flux density waveform excited in the air gap by the rotor PMs, k = 1, 2, 3, . . ., µ = 2m 1 k ±1 are the rotor higher space harmonics, ksµ is the rotor PM skew factor, τ is the pole pitch, γ1 is the parameter depending on the slot opening b14 and the air gap g, t1 is the stator slot pitch, k0k is the stator slot opening factor and ksk is the stator slot skew factor. In general, both PMs and stator slots can be skewed. The skew factors are defined by the following equations: • the rotor PM skew factor ksµ

  sin µπ b f s /(2τ ) = µπ b f s /(2τ )

(4.35)

• the stator skew factor for k = 1, 2, 3, . . . ksk =

sin [kπ pbs /(s1 t1 )] sin [kπ bs /(2τ )] = . kπ bs /(2τ ) kπ pbs /(s1 t1 )

(4.36)

The stator slot opening factor is defined by Equation 2.64 [87]. The skew of PMs is b f s and the skew of stator slots is bs . For most PM configurations the air gap g is to be replaced by an equivalent air gap g  ≈ g + h M /µrr ec where g is the mechanical clearance, h M is the radial height of the PM (one pole), and µrr ec = 1.02 to 1.1 is the relative recoil permeability of the PM material [75].

4.6.2

Calculation of cogging torque

If the magnetic saturation and armature reaction are negligible, the cogging torque is independent of the stator current. The frequency of the fundamental component of the cogging torque is f c = s1 n s , where s1 is the number of the stator slots and n s is the rotor speed in rev/s. Assuming that the total energy W of the magnetic field is stored in the air gap, the cogging torque is expressed as Tc (x) = −

D2out d W dW =− d 2 dx

(4.37)

where D2out ≈ D1in is the rotor outer diameter, D1in is the stator inner diameter and the x axis is in the direction of rotation. The mechanical angle =

Copyright © 2006 Taylor & Francis Group, LLC

2τ . D2out

(4.38)

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The rate of change of the air gap coenergy is Li g W (x ) = 2µ0


b2f (x )d x

(4.39)

where L i is the effective length of the stator stack, µ0 is the magnetic permeability of free space, g is the air gap, and b f (x ) is the air gap magnetic flux density distribution according to Equation 4.31. Substituting Equations 4.31 and 4.39 into Equation 4.37 and assuming that the maximum energy change is in the interval X + a ≤ x ≤ X + b, the cogging torque equation becomes L i g D2out ∂ Tc (X ) = − 2µ0 2 ∂ x




X +b X +a



2

b P M (x ) + bsl (x ) kC

d x.

(4.40)

According to [46], the maximum energy change occurs for a = 0.5t1 and b = 0.5b14 + ct where ct = t1 − b14 . Taking into account only fundamental space harmonics µ = 1 and k = 1, the expression in the square bracket under the integral will have the following form 

2

b P M (x ) + bsl (x ) kC



1 = B cos(βx ) − AB cos(αx ) cos(βx ) kC

2

where g 2 A = 2γ1 kok k2 ; t1 =1 sk=1 α=

2π ; t1

B = Bg bµ=1 ks µ=1

(4.41)

π . τ

(4.42)

β=

With a stationary stator, only the magnetic flux density excited by the rotor PMs depends on the rotor position with respect to the coordinate system fixed to the stator [87]. Magnetic flux density waveforms are visualized in Figure 4.3. It is easier to take the integral over b f (x)2 assuming that the rotor is stationary and the stator moves with synchronous speed, that is, only stator slots expressed by the term A cos αx change their position. Thus,  
L i g D2out X +b 1 Tc (X ) = − 2 B cos(βx) − AB cos(αx) cos(βx) . 2µ0 2 kC X +a ×(−ABα sin αx cos βx)d x (4.43) After performing integration with respect to x, the cogging torque equation takes the following form: Tc (X ) = −

L i g D2out [I1 (X ) + I2 (X )] 2µ0 2

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(4.44)

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bf (x) N

bsl(x) x

S

ns

Rotor bPM(x)

bPM (x)

bf (x)

x

S

N

Rotor

ns

Figure 4.3 Magnetic flux density waveforms in the air gap: b P M (x) excited by the rotor PMs, bsl (x) due to stator slots and b f (x) resultant. Only b P M (x) moves with the rotor.

where   2α − AB 2 sin(αx) cos2 (βx) d x kC X +a      2  a+b AB a−b 2 sin α X + = sin α kC 2 2    α a+b + sin (α + 2β) X + α + 2β 2    a+b × sin (α + 2β) 2    α a+b + sin (α − 2β) X + α − 2β 2    a−b × sin (α − 2β) 2


I1 (X ) =

X +b

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(4.45)

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  2α A2 B 2 sin(αx ) cos(αx ) cos2 (bx ) d x X +a     a+b 2 1 = AB sin 2α X + sin[α(a − b)] 2 2    α a+b + sin 2(α + β) X + × sin[(α + β)(a − b)] 4(α + β) 2     α a+b + sin 2(α − β) X + × sin[(α − β)(a − b)] . 4(α − β) 2 (4.46)

I2 (X ) =

X +b

Since I1 (X ) >> I2 (X ), the following shorter form of cogging torque equation can be used L i g D2out Tc (X ) ≈ − I1 (X ). (4.47) 2µ0 2 Figure 4.4 clearly shows that the fundamental frequency of the cogging torque (first integral) is f c = s1 n s , i.e., there are s1 pulses per one revolution. The fundamental waveform is modulated by another waveform with frequency f cp = 2 pn s , i.e., 2 p pulses per revolution. The second integral indicates that there is negligible frequency f c2 = 2s1 n s , i.e., 2s1 pulses per revolution.

4.6.3

Simplified cogging torque equation

Putting b P M (x) = Bg in Equation 4.40, the cogging torque will become a waveform with predominant frequency f c = s1 n s and constant amplitude. Such a simplified equation, which does not take into account the finite width of rotor poles, has been derived in earlier publications by the first author [75,76]. For b P M (x) = Bg , µ = 1 and k = 1 the simplified cogging torque equation has the following form 2
X +b  Bg L i g D2out d − ABg cos(αx) d x 2µ0 2 d X X +a kC
X +b  2 d Bg 2 2 2 2 2 − ABg cos(αx) + A Bg cos (αx) d x d X X +a kC kC

Tc (X ) = − =−

L i g D2out 2µ0 2

=−

L i g D2out [K 1 (X ) + K 2 (X )] 2µ0 2

(4.48)

where      4 a+b a−b 2 K 1 (X ) = − ABg sin α X + sin α kC 2 2    a+b K 2 (X ) = A2 Bg2 sin 2α X + sin[α(a − b)]. 2

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(4.49)

(4.50)

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Noise of Polyphase Electric Motors 0.2 0.16 0.12 0.08 I1(x) 0.04 0 I2(x) −0.04 −0.08 −0.12 −0.16 −0.2 −0.2 0.2

0 0

0.1

0.2

0.3

0.4

0.3

0.4

x

0.5 2⋅p⋅τ

(a) 0.005 0.005 0.004 0.003 0.002 0.001 0 I2(x) −0.001 −0.002 −0.003 −0.004 −0.005 −0.005

0 0

0.1

0.2 x

0.5 2⋅p⋅τ

(b)

Figure 4.4 Comparison of the first and second term in Equation 4.44: (a) integrals I1 (X ) and I2 (X ) according to Equation 4.45 and 4.46; (b) integral I2 (X ). Calculation results for a PM brushless motor with embedded magnets, s1 = 36 stator slots, 2 p = 10 poles, g = 1 mm, L i = 198 mm, t1 = 14 mm, τ = 50.3 mm, b14 = 3 mm, b f s = 0, bs = 14 mm, Bg = 0.76 T, n s = 11 rev/s. Since K 1 (X ) >> K 2 (X ) (Figure 4.5), Equation 4.48 can also be written in a shorter form: L i g D2out Tc (X ) ≈ − K 1 (X ) (4.51) 2µ0 2

4.6.4

Influence of eccentricity

The variation of the air gap for static eccentricity can be expressed analytically as [87]    π g(x) = g 1 −  cos x (4.52) pτ where g is the mean air gap,  = e/g is the relative static eccentricity and e is the displacement between the stator and rotor axis. See also Equation 2.83 in which

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Torque Pulsations 0.1 0.08 0.06 0.04 K 1(x) 0.02 0 K 2(x) −0.02 −0.04 −0.06 −0.08 −0.1 −0.1

89

0.1

0 0

0.1

0.2

0.3

0.4

0.5 2⋅p⋅τ

0.3

0.4

0.5 2⋅p⋅τ

x (a)

0.005 0.004 0.003 0.002 0.001 0 K 2 (x) −0.001 −0.002 −0.003 −0.004 −0.005 −0.005 0.005

0 0

0.1

0.2 x (b)

Figure 4.5 Comparison of the first and second term in Equation 4.48: (a) integrals K 1 (X ) and K 2 (X ) according to Equations 4.49 and 4.50; (b) integral K 2 (X ). Simulation has been done for a PM brushless motor with the same parameters as specified in Figure 4.4 caption. α is according to Equation 2.30. Since the air gap is now the function of the x coordinate, the magnetic energy change depends on the variation of the air gap with the x coordinate. It is difficult to analyze Equation 4.40 with the air gap g(x) in the square bracket (second term) dependent on the x coordinate. It is more convenient to use the simplified Equation 4.48 in which A = 2γ1

g(x) 2 k ksk=1 . t1 0k=1

(4.53)

Thus, Tc (X ) = −

L i g D2out [K 1 (x) + K 2 (X ) + K 3 (X ) + K 4 (X ) + K 5 (X )] 2µ0 2 ≈−

L i g D2out [K 1 (X ) + K 3 (X )] 2µ0 2

Copyright © 2006 Taylor & Francis Group, LLC

(4.54)

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where K 1 (X ) and K 2 (X ) are according to Equations 4.49 and 4.50, respectively, and the remaining terms are K 3 (X ) = 2

       1 a+b a+b sin ( + α) ABg2  sin ( + α) x + kC 2 2

       a+b a−b + sin ( − α) x + sin ( − α) 2 2

(4.55)

       a−b a+b K 4 (X ) = −A2 Bg2  2 sin  x + sin  2 2       a+b a−b + sin ( + 2α) x + sin ( + 2α) 2 2

(4.56)

       a+b a−b + sin ( − 2α) x + sin ( − 2α) 2 2  K 5 (X ) = −A2 Bg2  2

   a+b 1 sin 2( + α) x + sin[( + α)(a − b)] 4 2

   1 a+b + sin 2( − α) x + sin[( − α)(a − b)] 4 2    1 a+b + sin 2γ1 x + sin[γ1 (a − b)] 2 2

(4.57)

    1 a+b + sin 2α x + sin[α(a − b)] . 2 2 Figure 4.6 shows that K 3 (X ) >> K 4 (X ) and K 3 (X ) >> K 5 (X ). Including the finite width of poles the cogging torque equation eventually be Tc (X ) = −

L i g D2out [I1 (x) + I2 (X ) + K 3 (X ) + K 4 (X ) + K 5 (X )] 2µ0 2 ≈−

L i g D2out [I1 (X ) + K 3 (X )]. 2µ0 2

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(4.58)

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0.002 2 ⋅ 10−3 K 3 (x) 0.001 K 4 (x) K 5 (x)

0 −0.001

−2 ⋅ 10−3 −0.002

0 0

0.1

0.2

0.3

0.4

x

0.5 2⋅p⋅τ

(a) 1 ⋅10−4 1 ⋅10−4

5 ⋅10−5

K 4 (x)

0 −5 ⋅10−5

−1 ⋅10−4

−1 ⋅10−4

0 0

0.1

0.2

0.3

0.4

0.3

0.4

x

0.5 2⋅p⋅τ

(b) 1 ⋅10−6 1 ⋅10−6

K 5 (x)

5 ⋅10−7 0 −5 ⋅10−7

−1 ⋅10−6 −1 ⋅10−6

0 0

0.1

0.2 x

0.5 2⋅p⋅τ

(c)

Figure 4.6 Comparison of the third, fourth, and fifth terms in Equation 4.54: (a) integrals K 3 (X ), K 4 (X ) and K 5 (X ) according to Equations 4.55, 4.56, and 4.57; (b) integral K 4 (X ); (c) integral K 5 (X ). Simulation has been done for a PM brushless motor with the same parameters as specified in Figure 4.4 caption and relative eccentricity  = 0.2.

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4.6.5

Calculation and comparison with measurements

The results of analytical calculation of the cogging torque according to Equation 4.44 which includes the finite width of the rotor magnet are shown in Figure 4.7a while the results of calculations according to the simplified Equation 4.48 are shown in Figure 4.7b. The calculated and measured cogging torque produced by the same motor 2 p = 10, s1 = 36 and the same air gap magnetic flux density Bg = 0.76 T is shown in Figure 4.8. To capture the cogging torque pulsations, a torque transducer, oscilloscope and data acquisition system have been used. It can be seen that the measured cogging torque waveform is modulated by a sinusoid

1

1 0.8 0.6 0.4 0.2

T c (x)

0 −0.2 −0.4 −0.6 −0.8 −1 −1

0 0

0.1

0.2

0.3

0.4

0.5 2 ⋅p ⋅ τ

0.3

0.4

0.5 2 ⋅p ⋅ τ

x (a)

1.0

T c (x)

1 0.8 0.6 0.4 0.2 0 −0.2 −0.4

−0.6 −0.8 −1.0 −1

0

0.1

0

0.2 x (b)

Figure 4.7 Cogging torque calculation results (Nm vs. meter): (a) according to Equation 4.44; (b) according to Equation 4.48. Design data of the PM brushless motor are the same as specified in Figure 4.4 caption, relative eccentricity  ≈ 0.2.

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Calculations Measurements

1 0.8 0.6

Torque, Nm

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1.2

0

0.1

0.3

0.2

0.4

0.5

x. m

Figure 4.8 Comparison of calculated cogging torque according to Equation 4.58 and measured cogging torque for one full rotor revolution. Design data of the PM brushless motor are the same as specified in Figure 4.4 caption, relative eccentricity  ≈ 0.2.

with the period equal to one revolution 2 pτ , ie.:   π K (x) = −M sin x pτ

(4.59)

This effect is due to the misalignment of the rotor and transducer shafts. The small shaft misalignment is difficult to avoid. The constant M in Equation 4.59 is proportional to the degree of misalignment. In comparison with the finite element method (FEM), the presented analytical method gives immediate results and allows for fast analysis of different magnetic circuit geometries of PM brushless motors. It can easily be implemented in the design procedure of commercial PM brushless motors and new products. Equation 4.44 can capture the effect of the finite width of PMs while the simplified Equation 4.48 can only predict the fundamental frequency of the cogging torque and its approximate amplitude. Equation 4.48 can only be recommended for 2-pole machines with large pole shoe–to– pole pitch ratio αi ≥ 0.9. The effect of eccentricity is included by K 3 (x), K 4 (x) and K 5 (x) and can be predicted with the aid of Equation 4.54. For the detailed study of the cogging torque the best results can be obtained using Equation 4.58.

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There is a certain discrepancy between the analytical prediction and measurements caused by the misalignment of the rotor and torque transducer shafts. On the other hand, cogging torque measurements are difficult and the accuracy of measurements is always questionable.

4.7

Torque ripple due to distortion of EMF and current waveforms in PM brushless motors

Torque ripple due to shape and distortion of the electromotive force (EMF) and current waveforms is called the commutation torque. Assuming no rotor currents (no damper, very high resistivity of magnets and pole faces) and the same stator phase resistances, the Kirchhoff’s voltage equation for a three-phase machine can be expressed in the following matrix form: 

    v 1A i 1A R1 0 0  v 1B  =  0 R1 0   i 1B  0 0 R1 v 1C i 1C      i L L L e d  A B A C A   1A   f A  L B A L B LC B i 1B + e f B . + dt L i 1C efC C A LC B LC

(4.60)

For inductances independent of the rotor angular position the self inductances L A = L B = L C = L and mutual inductances between phases L AB = L C A = L C B = M are equal. For no neutral wire i a A + i a B + i aC = 0 and Mi a A = −Mi a B − Mi aC . Hence,     i 1A v 1A R1 0 0  v 1B  =  0 R1 0   i 1B  v 1C i 1C 0 0 R1 



     L1 − M 0 0 i 1A ef A  d  i 1B  +  e f B  . 0 L1 − M 0 + efC 0 0 L 1 − M dt i 1C

(4.61)

The instantaneous electromagnetic torque is the same as that obtained from Lorentz Equation 4.9, i.e., Td =

1 [e f A i 1A + e f B i 1B + e f C i 1C ]. 2π n

(4.62)

) At any time instant, only two phases conduct. For example, if e f A = E (tr f , (sq) (sq) ) , i 1B = −I1 and i 1C = 0, the instantaneous e f B = −E (tr f , e f C = 0, i 1A = I1

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electromagnetic torque according to Equation 4.62 is [28, 177] (sq)

Td =

) 2E (tr f I1

(4.63)

2π n

(sq)

) where E (tr are flat-top values of trapezoidal EMF and square wave current f and I1 (Figure 4.9). For constant values of EMF and currents, the torque (Equation 4.63) does not contain any pulsation [177]. Since e f = ωψ f = (2π n/ p)ψ f where ψ f is the flux linkage per phase produced by the excitation system, the instantaneous torque (Equation 4.62) becomes

Td = p(ψ f A i 1A + ψ f B i 1B + ψ f C i 1C )

(4.64)

To obtain torque for periodic waves of the EMF and current, it is necessary to express those waveforms in forms of the Fourier series. The magnetic flux on the basis of Equation 4.32
wc  f (t) = L i b P M (x) cos(µωt)d x 0

=

4 τ Li Bg 2 π S

∞ 

1 π sin(µS)ksµ [1 − cos(µ w c )] cos(µωt) 3 µ τ µ=1,3,5,...

(4.65)

where w c is the coil pitch, S is according to Figure 4.9a and ksµ is according to Equation 4.35. The EMF induced in the phase A is calculated on the basis of the

Ef (tr)

e fA S

el. degrees 0

60

120 180

240

300

360

(a) i aA S1

I1(sq) el. degrees

0 60 S2

120

180 240

300

360

(b)

Figure 4.9 Idealized (solid lines) and practical (dash lines) waveforms of: (a) phase EMF; (b) phase current.

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electromagnetic induction law e f A (t ) = −N1

=

d [ f (t )] dt

∞ 

π   8 τ Li 1 Bg f N1 1 − cos µ w c sin(µωt ). (4.66) sin(µS )k s µ π S µ2 τ µ=1,3,5,...

The EMFs e f B (t ) and e f C (t ) induced in the remaining phase windings are shifted by 2π/3 and −2π/3, respectively. For phase A, the current rectangular waveform expressed as Fourier series is i 1A (t ) =

∞ 4 (sq )  1 I1 cos(ν S1 ) sin(νωt ). π ν ν=1,3,5,...

(4.67)

Harmonics of the magnetic flux and current of the same order produce constant torque. In other words, the constant torque is produced by all harmonics 2m 1 k ± 1 where k = 0, 1, 2, 3, . . . of the magnetic flux and armature currents. Harmonics of the magnetic flux and current of different order produce pulsating torque. However, when the 1200 trapezoidal flux density waveform interacts with 1200 rectangular current, only a steady torque is produced with no torque pulsations [177, 178]. In practice, the stator current waveform is distorted and differs from the rectangular shape. It can be approximated by the following trapezoidal function i 1A (t ) =

∞  1 4Ia(sq ) [sin(ν S1 ) − sin(ν S2 )] sin(νωt ) π(S1 − S2 ) ν=1,3,5,... ν 2

(4.68)

where S1 − S2 is the commutation angle in radians. The EMF wave also differs from 1200 trapezoidal functions. In practical motors, the conduction angle is from 1000 to 1500 , depending on the construction. Deviations of both current and EMF waveforms from ideal functions result in producing torque pulsations [28, 177]. Peak values of individual harmonics can be calculated on the basis of Equations 4.66, 4.67, and 4.68. Using the realistic shapes of waveforms of EMFs and currents as those in practical motors, Equation 4.68 takes into account all components of the torque ripple except the cogging (detent) component. Figure 4.10 shows trapezoidal EMF, trapezoidal current and electromagnetic torque waveforms, while Figure 4.11 shows the same waveforms in the case of sinusoidal EMFs and currents. The torque component due to phase commutation currents Tcom = Td − Tav where Tav is the average electromagnetic torque. The frequency of torque ripple due to phase commutation is f com = 2lm 1 f where l = 1, 2, 3, . . .. For sinusoidal EMF and current waveforms the electromagnetic torque is constant and does not contain any ripple. Torques shown in Figure 4.11 are not ideally smooth due to a slight current unbalance.

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97

300 200 efA(ωt) ef B(ωt)

0

efC(ωt) −200 −300

0 0

5

10

15

20

25

30 2⋅π⋅p

20.94

26.18

31.42 2⋅π ⋅p

4

5

6 2⋅π

ωt (a)

25 25 12.5 i1A(ωt) i1B(ωt)

0

i1C(ωt) −12.5 −25

−25

0 0

5.24

10.47

15.71 ωt (b)

150 150 100 Td(ωt) Tcom(ωt)

50 0 −50

−100 −100

0 0

1

2

3 ωt (c)

Figure 4.10 Electromagnetic torque produced by a medium power PM brushless motor (m 1 = 3, 2 p = 10, z 1 = 36): (a) trapezoidal EMF waveforms; (b) trapezoidal current waveforms; (c) resultant electromagnetic torque Td and its commutation component Tcom . The current commutation angle is 50 .

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Noise of Polyphase Electric Motors 300 300 200 efA(ωt) 100 ef B(ωt)

0

efC(ωt)

−100 −200

−300

−300

0 0

5

10

15 ωt

20

25

30 2 ⋅ π ⋅p

(a) 25

i1A(ωt) i1B(ωt)

25 20 15 10 5 0

i1C(ωt) −5 −10 −15 −20 −25 −25

0 0

5

10

15 ωt

20

25

4

5

30 2⋅π⋅p

(b) 150 150 100

Telm(ωt) Tcom(ωt)

50 0 −50

−100 −100

0 0

1

2

3 ωt

6 2⋅π

(c)

Figure 4.11 Electromagnetic torque produced by a medium power PM brushless motor (m 1 = 3, 2 p = 10, z 1 = 36): (a) sinusoidal EMF waveforms; (b) sinusoidal current waveforms; (c) resultant electromagnetic torque Td and its commutation component Tcom .

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4.8 Tangential forces vs. radial forces For an accurate analysis of magnetic forces in electrical machines it is necessary to include both radial and tangential forces. The radial magnetic pressure obtained on the basis of Maxwell stress tensor, is expressed by Equation 2.114. The tangential component of the magnetic pressure can also be obtained with the aid of Maxwell stress tensor, i.e., pt (α, t ) =

1 b(α, t )bt (α, t ) = b(α, t )a (α, t ) µ0

N/m2

(4.69)

where b(α, t ) is the normal component of the magnetic flux density waveform according to Equation 2.110 or Equation B.6, bt (α, t ) is the tangential componet of the magnetic flux density distribution and a (α, t ) is the line current density. The radial and tangential components of the magnetic pressure for an 8-pole PM brushless motor are plotted in Figures 4.12 and 4.13. Every harmonic of the magnetic flux density in the air gap corresponds to a certain harmonic of the line current density. As a result of interaction of rotating harmonics of the magnetic flux density in the air gap and harmonics of the line current density, rotating tangential forces varying in the space and time are produced. Those forces can also be found on the basis of Biot-Savart law. The following boundary equalities for tangential components of the magnetic field can be written • at the stator core–air gap boundary bt1 (α, t ) = ±µ0 a1 (α, t )

(4.70)

Radial Magnetic Pressure, Pa

600000 500000 400000 300000 200000 100000 0 −100000

0

100

200 300 400 500 Circumference, mm

600

700

Figure 4.12 Distribution of the radial magnetic pressure along the air gap circumference for a PM brushless motor with 2 p = 8 and s1 = 36 as shown in Figure 2.1b as obtained from the 2D FEM simulation.

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Tangential Magnetic Pressure, Pa

300000 200000 100000 0 −100000 −200000 −300000

0

100

200 300 400 500 Circumference, mm

600

700

Figure 4.13 Distribution of the tangential magnetic pressure along the air gap circumference for a PM brushless motor with 2 p = 8 and s1 = 36 as shown in Figure 2.1b as obtained from the 2D FEM simulation. • at the rotor core–air gap boundary bt2 (α, t) = ±µ0 a2 (α, t)

(4.71)

where the line current density waveforms can be obtained by differentiating the MMF waveforms • for the stator

or

√

m 1 2 N1 kw1ν dF1ν (x, t) π  a1ν (x, t) = =− I1 sin ωt ∓ ν x dx π pτ τ 

π = Am1ν sin ωt ∓ ν x (4.72) τ a1 (α, t) =

∞ 

Am1ν sin(νpα ± ωt)

(4.73)

ν=1

• for the rotor of an induction motor √ 

m 2 2 N2 kw2µ dF2µ (x, t) π a2µ (x, t) = =− I2 sin ωµ t ∓ µ x + φµ dx π pτ τ 

π (4.74) = Am2µ sin ωt ∓ ν x τ or ∞  a2 (α, t) = Am2µ sin(µpα ∓ ωµ t + φµ ). (4.75) µ=1

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101

For the rotor of a PM synchronous motor it is recommended to use the tangential component bt2 (α, t) of the rotor magnetic flux density instead of the line current density a2 (α, t). Every harmonic of the tangential magnetic force, similar to the radial force, has a certain frequency fr and an order r which describes its distribution around the stator periphery. The tangential magnetic pressure due to the stator and rotor magnetic flux density components is obtained by putting Equations 4.70 and 4.71 into Equation 4.69. Three groups of the tangential pressure can be identified as a result of interaction: • harmonics of the stator magnetic flux density and stator line current density ptν (α, t) = b1ν (α, t)a1ν (α, t)

(4.76)

• harmonics of the stator line current density and rotor magnetic flux density ptνµ (α, t) = b2µ (α, t)a1ν (α, t)

(4.77)

• harmonics of the rotor magnetic flux density and rotor line current density ptµ (α, t) = b2µ (α, t)a2µ (α, t).

(4.78)

Frequencies and orders of tangential magnetic forces are expressed by the same Equations 2.125, 2.126, and 2.127 as for radial magnetic forces. The stator and rotor MMF  waveforms are expressed by Equations 2.107 and 2.108 in which ∞ F1 (α, t) = ∞ F (α, t) and F (α, t) = F (α, t). The peak values of 1ν 2 2µ ν=1 µ=1 the stator and rotor line current densities Am1ν and Am2µ are expressed by Equations 2.54 and 2.59. Note that the peak values of line current densities obtained from the MMF waveforms contain stator and rotor winding factors kw1ν and kw2ν for higher space harmonics. Thus, the following relationships between the MMF and line current densities amplitudes for the νth space harmonics can be written • for the stator F1mν =

1τ Am1ν ; νπ

Am1ν =

1 τ Am2µ µπ

Am2µ =

νπF1mν τ

(4.79)

µπ F1mµ . τ

(4.80)

• for the rotor F2mµ =

Assuming the magnetic saturation of the stator and rotor core to be negligible, on the basis of Ampere’s circuital law the peak values of the magnetic flux densities as functions of MMFs are • for the stator B1mν = µ0

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F1mν kC g

(4.81)

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Noise of Polyphase Electric Motors • for the rotor

F2m µ (4.82) kC g The ratio of the radial–to–tangential magnetic pressures created by the stator and rotor magnetic flux density components are: B2m µ = µ0

• for the product of the stator harmonics of the same number pr ν (α, t ) b1ν (α, t ) τ = = cot(νp α ∓ ωt ) pt ν (α, t ) 2µ0 a1ν (α, t ) 2νπkC g

(4.83)

• for the product of the stator ν harmonics and rotor µ harmonics pr νµ (α, t ) 2b1ν (α, t )b2µ (α, t ) τ = = cot(νp α ∓ ωt ) pt νµ (α, t ) 2µ0 b2µ (α, t )a1ν (α, t ) νπkC g

(4.84)

• for the product of the rotor harmonics of the same number pr µ (α, t ) b2µ (α, t ) τ = = cot(µp α ∓ ωµ t + φµ ) pt µ (α, t ) 2µ0 a2µ (α, t ) 2µπ kC g

(4.85)

Since τ >> g, the magnitude of the radial magnetic pressure is much higher than that of the tangential magnetic pressure. Consequently, the tangential forces producing electromagnetic torques are much smaller than radial forces. Peaks in Figure 4.13 have been obtained due to inaccuracy of the finite element method (FEM) model. Equations 4.83, 4.84, and 4.85 can be used for calculation of the tangential magnetic pressure on the basis of radial magnetic pressure.

4.9

Minimization of torque ripple in PM brushless motors

The torque ripple can be minimized both by the proper motor design and motor control. Measures taken to minimize the torque ripple by motor design include elimination of slots, skewed slots, special shape slots and stator laminations, selection of the number of stator slots with respect to the number of poles, decentered magnets, skewed magnets, shifted magnet segments, selection of magnet width, direction-dependent magnetization of PMs. Control techniques use modulation of the stator current or EMF waveforms [20, 59].

4.9.1

Slotless windings

Since the cogging torque is produced by the PM field and stator teeth, a slotless winding can totally eliminate the cogging torque. A slotless winding requires increased air gap, which in turn reduces the PM excitation field. To keep the same air gap magnetic flux density, the height h M of PMs must be increased. Slotless PM brushless motors use more PM material than slotted motors.

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4.9.2

103

Skewing stator slots

Normally, the stator slot skew bs equal to one slot pitch t1 can reduce the cogging torque practically to zero value — Equations 4.35 and 4.36. Sometimes, the optimal slot skew is less than one slot pitch [136]. On the other hand, the stator slot skew reduces the EMF which results in deterioration of the motor performance. Skewed slots are less effective in the case of rotor eccentricity.

4.9.3

Shaping stator slots

Figure 4.14 shows methods of reducing the cogging torque by shaping the stator slots, i.e., (a) (b) (c) (d)

bifurcated slots (Figure 4.14a), empty (dummy) slots (Figure 4.14b), closed slots (Figure 4.14c), teeth with different width of the active surface (Figure 4.14d).

Bifurcated slots (Figure 4.14a) can be split into more than two segments. The cogging torque increases with the increase of the slot opening. When designing

(a)

(b)

(c)

(d)

Figure 4.14 Minimization of the cogging torque by shaping the stator slots: (a) bifurcated slots; (b) empty slots; (c) closed slots; (d) teeth with different width of the active surface.

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Noise of Polyphase Electric Motors

closed slots (Figure 4.14c), the bridge between the neighboring teeth must be properly designed. Too thick a bridge (thickness measured in radial direction) increases the stator slot leakage to an unacceptable level. Too narrow a bridge can be ineffective due to high saturation. Because closed slots can only accept “sewed” coils, it is better to close the slots by inserting an internal sintered powder cylinder or to separate the stator yoke and tooth-slot section. In the second case, stator slots are open externally from the frame side.

4.9.4

Selection of the number of stator slots

The smallest common multiple Ncm between the slot number s1 and the pole number 2 p has a significant effect on the cogging torque. As this number increases the cogging torque decreases [83]. Similarly, the cogging torque increases as the largest common factor between the slot number and the pole number increases [30, 83].

4.9.5

Shaping PMs

PMs thinner at the edges than in the center (Figure 2.2b) can reduce both the cogging and commutation torque ripple [75]. Magnets may require a polygonal cross section of the rotor core. Decentered PMs together with bifurcated stator slots can suppress the cogging torque as effectively as skewed slots with much less reduction of the EMF [175].

4.9.6

Skewing PMs

The effect of skewed PMs on the cogging torque suppression is similar to that of skewing the stator slots. Fabrication of twisted magnets for small rotor diameters and small number of poles is rather difficult. Bread loaf shaped PMs with edges cut at a slant are equivalent to skewed PMs.

4.9.7

Shifting PM segments

Instead of designing one long magnet per pole, it is sometimes more convenient to divide the magnet axially into K s = 3 to 6 shorter segments. Those segments are then shifted one from each other by equal distances t1 /K s or unequal distances the sum of which is t1 . Fabrication of short, straight PM segments is much easier than long, twisted magnets.

4.9.8

Selection of PM width

Properly selected PM width with respect to the stator slot pitch t1 is also a good method to minimize the cogging torque. The magnet width (pole shoe width) is to be b p = (k + 0.14)t1 where k is an integer [136] or including the pole curvature b p = (k + 0.17)t1 [92]. Cogging torque reduction requires wider pole shoes than the multiple of slot pitch.

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4.9.9

105

Magnetization of PMs

There is a choice between parallel, radial, and direction-dependent magnetization. For example, if a ring-shaped magnet of a small motor is placed around the magnetizer poles without an external magnetic circuit, the magnetization vectors will be arranged similar to the Halbach array. This method also minimizes the torque ripple.

4.9.10

Creating magnetic circuit asymmetry

Magnetic circuit asymmetry can be created by shifting each pole by a fraction of the pole pitch with respect to the symmetrical position, or designing different sizes of North and South magnets of the same pole pair [18].

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5 Stator System Vibration Analysis In previous chapters the magnitudes, frequencies, and orders of radial and tangential magnetic forces have been analyzed. Analytical and numerical methods of computations of these forces have been discussed. The noise and vibration of the motor structure are the direct responses of the excitation by these forces. For example, if the frequency of the radial magnetic force is close to one of the natural frequencies of the stator system and the force order r is the same as the circumferential vibrational mode m of the stator system, significant vibration and acoustic noise can be produced. In this chapter, the modal vibration behavior of the stator system is discussed. Several analytical approaches for the calculation of natural frequencies and vibrational modes of cylindrical stators with frames (enclosures) are given.

5.1

Forced vibration

The stator and frame of a motor structure are basically cylindrical shells. A complete analytical analysis of the vibration of the system may be difficult. However, if only the stator is the concern, or if the overall vibration behavior of the motor is attributed to the stator, a circular cylindrical shell can be employed as a simplified model of the stator for an analytical approach. The vibration of a continuous structure due to the force excitation is discussed in Appendix D. For a cylindrical shell, although the equation of motion is of the same form as Equation D.32, the differential operator is more complicated than that for the beam and the plate [190]. In fact, due to the curvature of the shell, the vibration in the three orthogonal directions, radial, axial, and tangential, are coupled to each other. A result of this coupling is that any excitation in one direction would cause vibrations in all three directions. Such effects are more prominent at low frequencies. This makes it difficult to solve the equations directly, and various 107 Copyright © 2006 Taylor & Francis Group, LLC

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Noise of Polyphase Electric Motors

approximations/simplifications have to be made in the analytical solution [190]. A detailed and accurate analysis may have to reply on the numerical approach such as the finite element method (FEM). The stator vibration is primarily induced by the electromagnetic force acting on the inner surface of the stator core. As described in previous chapters, the electromagnetic force has radial and tangential components. Both components, strictly speaking, may excite the vibration in the radial direction which is the primary concern in acoustic radiation. But since the radial component is approximately an order of magnitude larger than the tangential component in general, it might be reasonable to neglect the contribution from the tangential component in the electromagnetic force, which means that the coupling in the vibration between the radial and tangential directions could be neglected. The forced vibration response of a cylindrical shell under the excitation of the electromagnetic force in the radial direction thus should be of the form as given in Equation D.37. As an example, a stator could be simplified as a circular cylindrical shell with both ends free of constraints. It is subject to the electromagnetic force as expressed in Equation 2.118. If only the circumferential modes of the stator are of interest, it can be shown that the force component of the order r only excites the mode of the same order for the stator, i.e. r = m. The amplitude of vibration displacements of mode m can be derived as Fm /M Am =
2 (ωm − ωr2 )2 + 4ζm2 ωr2 ωm2

(5.1)

where M is the mass (kg) of the cylindrical shell, ωm is the angular natural frequency of the mode m, ωr is the angular frequency of the force component of the order r , and ζm and is the modal damping ratio. The amplitude of force Fm = π D1in L i Pmr , in which D1in is the inner diameter of the stator core, L i is the effective length of the stator core, and Pmr is the magnitude of the magnetic pressure of the order r according to Equations 2.115, 2.116, or 2.117. It can be seen that when the frequency of the excitation force is close to the natural frequency of the stator; the vibration reaches a maximum and excessive noise may be generated. This result indicates that among all the electromagnetic force components, only those of which the frequencies are close to the natural frequencies of the stator modes are important to the stator vibration response. By rearranging Equation 5.1 a magnification factor can be defined as hm =

Am 1 =
. 2 2 Fm /Mωm [1 − ( fr / f m ) ]2 + [2ζm ( fr / f m )]2

(5.2)

The h m value calculated by Equation 5.2 is plotted against ( fr / f m ) and different damping ratios ζm in Figure 5.1. The magnification factor increases as f = | fr − f m | decreases and the damping factor ζm decreases. Particularly at fr = f m , the vibration amplitude is controlled by the mechanical damping in the structure. Analytical determinations of the mechanical damping are generally difficult. Experimental approaches are usually adopted. The mechanical damping

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109

Magnification factor hm

10 xm = 0.05

8 6

xm = 0.1 4 xm = 0.2

2 0

xm = 0.3 0 0

1

0.5

2

1.5

2.5

3 3

fr/fm

Figure 5.1 Magnification factor h m as a function of ( fr / f m ) for different damping factors ζm . of the stator is frequency dependent and very much affected by winding and the lamination of the stator core. An empirical expression for the small and medium sized electrical machines is suggested as [248] ζm =

1 (2.76 × 10−5 f m + 0.062). 2π

(5.3)

This relationship is plotted in Figure 5.2. For a more accurate result, the modal damping ratio ζm can be extracted directly from modal testing [223]. The conversion between various damping parameters, such as the damping ratio, logarithmic decrement, and the loss factor, and so forth, can be found in Table 10.1. 0.06 0.06 0.05 0.04 ξ m(f ) 0.03 0.02 0.01 0

0

0 0

2000

4000

6000 f

8000

1 ⋅ 104 10000

Figure 5.2 Variation of damping ratio ζ with frequency [248].

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Noise of Polyphase Electric Motors

It should be noted that Equation 5.1 is derived for a single circumferential mode. The total response of the stator subject to the electromagnetic force excitation should be the superposition of the contributions from all the modes, as discussed in Appendix D. However, if the vibration modes are well separated over the frequency domain, Equation 5.1 can be employed around the natural frequency of each mode. Expressed as a function of the magnetic pressure magnitude Pmr , Equation 5.1 becomes, Fm π D1in L i Am = hm = Pmr h m . (5.4) M ωm2 M ωm2 The amplitude of the vibration velocity for mode m is, Vm = ωr Am = 2π fr

5.2

π D1in L i Pmr h m . M ωm2

(5.5)

Simplified calculation of natural frequencies of the stator system

Calcultation of natural frequencies of the stator system is essential in the vibration analysis of electric machines. In classical approaches [87, 119, 200, 248], the stator system, that is, the stator core, winding, and frame (enclosure) is considered as a single thick ring loaded with teeth and winding. Circumferential vibrational modes of a thin cylinder can be visualized in a similar way to force orders as shown in Figure 2.9. The natural frequency of the stator system of the mth circumferential vibrational mode can be expressed as:  Km 1 fm = (5.6) 2π Mm where K m is the lumped stiffness (N/m) and Mm is the lumped mass (kg) of the stator system. For a stator core with the thickness h c , mass Mc , and the mean diameter Dc , the lumped stiffness and lumped mass in the breathing mode, that is, the circumferential vibrational mode m = 0 is [119, 248] K 0 = 4π

Ec hc L i ; Dc

M0 = Mc kmd = π Dc h c L i ρc ki kmd

(5.7)

where ρc is the mass density of the stator core, ki is the stacking factor, and kmd is the mass addition factor for displacement defined as [248] kmd = 1 +

M t + M w + Mi . Mc

(5.8)

In the above equation, Mt is the mass of all stator teeth, Mw is the mass of stator windings, Mi is the mass of insulation, and Mc is the mass of the stator core cylinder (yoke).

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111

Equations 5.6 and 5.7 give the following formula for the natural frequency of m = 0 circumferential mode  Ec 1 f0 = . (5.9) π Dc ρc ki kmd Typically, for laminations, the elasticity modulus E c = 200 × 109 Pa = 200 GPa, mass density ρc =
7700 kg/m3 , and stacking factor ki = 0.96. This will give the phase speed cc = 200 × 109 /7700 = 6096 m/s. For the circumferential vibrational mode m = 1 (bending mode) K 1 = 4π  F1 =

Ec hc L i ; Dc

M1 =

2 ; 1 + κ 2 kmr ot /kmd

Mc kmd M0 = 2 F12 F1

(5.10)

hc κ=√ . 3Dc

(5.11)

In Equation 5.11, kmr ot is the mass addition factor for rotation [248]      hc M w + Mi 1 s1 ct L i h 2t hc 2 kmr ot = 1 + + 1+ + π Dc I c Mt 3 2h t 2h t =1+

s1 ct L i h 2t π Dc I c



M w + Mi Mt

1+

 (4h 2t + 6h c h t + 3h 2c )

(5.12)

where

h 3c L i (5.13) 12 is the area moment of inertia about the neutral axis parallel to the cylinder axis, s1 is the number of stator teeth (slots), ct is the tooth width, and h t is the tooth height. Ic =

Then, the natural frequency for the circumferential mode m = 1 is [119, 248]   Ec 2 1 f1 = = f 0 F1 . (5.14) π Dc ρc ki kmd 1 + κ 2 kmr ot /kmd Generally, for circumferential vibrational modes m ≥ 2 [119, 248] K m = 16π Mm = Mc

kmd m 2 + 1 kmd m 2 + 1 = π D h L ρ k c c i c i Fm2 m 2 Fm2 m 2

 Fm =

E c Ic 2 (m − 1)2 ka2 Dc3

κ 2 (m 2 − 1)[m 2 (4 + kmr ot /kmd ) + 3] 1+ m2 + 1

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−1/2

(5.15)

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where the coefficient ka > 1 accounts for end bells and frame support (foot or flange mounting). The natural frequency of the stator system for circumferential modes m ≥ 2 can be derived as   Ec Ic m(m 2 − 1) 2 1 m(m 2 − 1) √ fm = ka Fm = f 0 κ √ ka Fm . 2 π Dc ρc ki kmd h c L i m2 + 1 m2 + 1 (5.16) Equation 5.16 is in fact a modified equation for circumferential vibration of a ring derived by Hoppe [90]. Most monographs on noise analysis of electrical machines, for example, [87, 119, 200, 248], recommend Equations 5.9, 5.14, and 5.16 for calculating natural frequencies of stator systems. However, Equations 5.9 to 5.16 do not guarantee a good accuracy of the calculation since the stator system is a complex structure which consists of the laminated stack with yoke and teeth, winding distributed in slots, encapsulation (potting), and frame (enclosure or casing). Equations 5.9 to 5.16 can only be used to estimate the natural frequency of the stator core alone, without any frame and end bells and with only partial influence of the winding and teeth.

5.3

Improved analytical method of calculation of natural frequencies

The analytical approach presented in this section is an attempt to consider the effects of winding, teeth, and particularly the outer frame on the natural frequencies of the stator system. In this method, the stator core, winding, and teeth, and the frame are modeled separately. Then a formula is suggested for the system.

5.3.1

Natural frequency of the stator core

If the stator length–to–mean diameter ratio is L i /Dc ≤ 1, the stator core can be considered as a ring and the following classical formula for m ≥ 2, first published by Hoppe in 1871 [90], can be used 1 m(m 2 − 1) √ fm = 2π m2 + 1



E c Ic 2 m(m 2 − 1) √ = ρl (0.5Dc )4 π Dc2 m 2 + 1

 E c Ic ρc L i h c

(5.17)

where the area moment of inertia Ic about the neutral axis parallel to the ring axis is given by Equation 5.13, h c is the stator yoke thickness, and the mass per unit circumference ρl =

1 Mc = ρc (π Dc )L i h c = ρc L i h c . π Dc π DC

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(5.18)

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113

According to Equations 5.6 and 5.17 the lumped stiffness and mass of the finite length cylinder are, respectively, Km =

16π E c Ic m 2 (m 2 − 1)2 Dc3 m2 + 1

Mm = Mc = πρc Dc L i h c .

(5.19) (5.20)

However, Hoppe’s equation does not give good results if L i /Dc ≥ 1. Better results are obtained if the stator of an electrical machine is regarded as a cylindrical shell of infinite length. In most mechanical engineering textbooks, the natural frequency of an infinitely long shell for circumferential vibrational modes m ≥ 0 is expressed as [135]  Ec

m fm = (5.21) π Dc ρc (1 − νc2 ) where νc is the Poisson ratio for the stator core. On the basis of Donnel-Mushtari theory [135], the parameter m , that is, the roots of the second order characteristic equation of motion are: • for circumferential mode m = 0

0 = 1 • for circumferential modes m ≥ 1
1

m = (1 + m 2 + κ 2 m 4 ) ± (1 + m 2 + κ 2 m 4 )2 − 4κ 2 m 6 ; 2

(5.22)

(5.23)

• where the nondimensional thickness parameter κ2 =

h 2c 3Dc2

(5.24)

To express Equation 5.21 in the form of Equation 5.6, the lumped stiffness can be written as Km =

4 2m π L i h c E c . Dc 1 − νc2

(5.25)

Meanwhile, Mm = πρc Dc L i h c , which is actually the stator core mass Mc . It is recommended to use Equation 5.21 even if L i /Dc < 1. The stator core (yoke) behind the teeth of 2-pole machines (2 p = 2) is thicker in the radial direction than that of machines with 2 p > 2. Therefore, the stiffness (Equation 5.25) of 2-pole machines is higher and deflection of the stator core is smaller than in machines with a larger number of poles. Two-pole machines can often be less noisy than multipole machines.

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5.3.2

Natural frequency of a frame with end bells

A frame (enclosure) with end bells behaves as a circular cylindrical shell with both ends constrained mechanically. Mathematically, both ends can be approximately described as being either simply supported or clamped. With the circumferential modal patterns being illustrated in Figure 2.9, typical axial modal patterns for circular cylindrical shells with simply supported ends are shown in Figure 5.3. The axial vibrational modes are n = 1, 2, 3 . . .. For a closed circular cylindrical shell of finite length L f with simply supported condition at both ends, the characteristic equation of motion has the following form [135]

6mn − (C2 + κ C2 ) 4mn + (C1 + κ C1 ) 2mn − (C0 + κ C0 ) = 0.

(5.26)

This is actually a third order equation with respect to the nondimensional frequency parameter 2mn . Three sets of roots correspond to the vibrations in three orthogonal directions, in which the smallest real root is associated with the natural frequency of the flexural vibration of the frame. According to Donnell-Mushtari theory, the constants in Equation 5.26 are [135] 1 C2 = 1 + (3 − ν f )(m 2 + λ2 ) + κ 2 (m 2 + λ2 )2 (5.27) 2 

1 2 2 2 2 23 − νf 2 2 2 2 C1 = (1−ν f ) (3 + 2ν f )λ + m + (m + λ ) κ (m + λ ) (5.28) 2 1 − νf C0 =

λ = nπ

Rf ; Lf

  1 (1 − ν f ) 1 − ν 2f λ4 + κ 2 (m 2 + λ2 )4 2 C2 = C1 = C0 = 0 R f = 0.5(D f − h f );

κ2 =

(5.29) (5.30) h 2f

12R 2f

(5.31)

where ν f is the Poisson ratio, R f is the mean frame radius, D f is the external frame diameter, h f is the frame thickness, and L f is the frame length. When both ends of the frame are clamped, the characteristic Equation 5.26, is still valid, and the constants in the equation are the same as shown in Equations 5.27 n=1

n=2

n=3

Figure 5.3 Nodal patterns for a circular cylindrical shell supported at both ends by shear diaphragms.

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to 5.30. However, Equation 5.26 should be solved with the parameter λ in Equation 5.31 replaced by λe , which according to Arnold and Warburton is [135] λe = nπ

Rf ; L f − L0

L0 = L f

0.3 n + 0.3

(5.32)

where L 0 is the correction to the length of the “clamped-clamped” shell, which depends on the axial modal number n. For either a “clamped-clamped” shell or a shell simply supported at both ends, the circumferential vibrational modes m = 0, 1, 2, 3, . . . should be calculated for each of the axial vibrational mode n = 1, 2, 3, . . .. The axial mode n = 0 does not exist. Thus, the natural frequency of the frame is   Ef K mn 1 1 mn f mn = = (5.33) 2π R f 2π Mf ρ f (1 − ν 2f ) where E f is the elasticity modulus of the frame material, and ρ f is the material density. The lumped stiffness for the frame therefore is K mn =

2mn E f V f 2 2mn π L f h f E f = Rf R 2f 1 − ν 2f 1 − ν 2f

(5.34)

and M f is the mass of the frame, i.e., M f = ρ f V f = ρ f (2π R f )L f h f ;

5.3.3

V f = 2π R f L f h f .

(5.35)

Natural frequency of a stator core–frame system

Most electrical machines have stator cores pressed into frames (enclosures). A frame increases both the mass and stiffness of the stator. If the stator core is short, an external structure produces a pressure which is transmitted uniformly throughout the stator core. Since the resultant moment of inertia of coaxial cylinders is equal to the sum of moments of inertia of each cylinder, the natural frequency of the statorframe system is to be calculated under the assumption that the lumped stiffness of (f) the stator core K m(c) and frame K mn are in parallel, that is, the equivalent stiffness (c) (f) is K m + K mn . The same assumption has been made for equivalent masses of the stator core and frame, that is, the resultant mass of the stator-frame system is Mc + M f . On the basis of these assumptions, the natural frequency of the stator-frame system can be expressed as1 :  (f) K m(c) + K mn 1 f mn ≈ (5.36) 2π Mc + M f (f) where K m(c) can be determined based on Equation 5.19 or 5.25, and K mn is according to Equation 5.34. Lumped masses Mc and M f are calculated from Equations 5.20 and 5.35.

1 See also S. Huang, M. Aydin, and T. A. Lipo, Electromagnetic vibration and noise assessment for surface mounted PM machines, IEEE PES Summer Meeting, Vol. 3, pp. 1417–1426, Vancouver, BC, Canada, 2001.

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In large machines the deflection of the stator core which produces noise is about ±0.25 × 10−4 mm [220]. On the other hand, the manufacturing tolerances between the outer diameter of the core and inner diameter of the frame are from ±0.02 to ±0.1 mm. This means that the mechanical coupling between the core and frame is small for low amplitude deflections [220]. Small machines with stamped ring laminations are more likely to produce some mechanical coupling than segmental laminations of large machines. The mechanical coupling between core and frame is increased due to radial expansion after heating. In the calculation of noise and vibration for most constructions of electrical machines, the • frame may be neglected only in large machines; • frame and its moment of inertia cannot be neglected in small machines.

5.3.4

Effect of the stator winding and teeth

The winding is located in slots which are separated by steel teeth. The toothslot zone with the winding can be regarded as an additional ring internal to the stator core (yoke). Thus, the natural frequency of the stator-frame system with the winding will be  f mn ≈

(f)

K m(c) + K mn + K m(w) Mc + M f + Mw

1 2π

(5.37)

where K m(w) is the lumped stiffness of the tooth-slot zone including the winding and Mw is the mass of the teeth, winding, and insulation. The lumped stiffness of the tooth-slot zone can be determined based on the method given in Section 5.3.1. • for short length winding (finite length cylinder) K m(w) =

m 2 (m 2 − 1) 2π E w Iw m2 + 1 Rw3

(5.38)

• for long length winding (infinitely long cylinder) K m(w) =

2m E w Vw Rw2 1 − νw2

(5.39)

The nondimensional frequency parameter m is according to Equations 5.22 and 5.23 for Rw = Rc − h sl , where h sl is the slot depth, Vw = 2π Rw h t (L i + 2h ov ) is the volume of the winding considered as a cylinder, L i is the effective length of the stator stack, h ov is the axial length of a one-sided end connection, E w is the equivalent elasticity modulus for winding and insulation, and νw is the Poisson ratio for the winding with insulation. The elasticity modulus for copper winding is 110 to 120 GPa and for polymer insulation 3 GPa. The equivalent elasticity modulus for copper and insulation assuming the slot filling factor 0.5 is approximately 9.4 GPa [241]. The Poisson ratio for copper is 0.33 to 0.36 and for polymer insulation exceeds 0.36. The mass of winding Mw = m 1 ρw N1ltur n aw a p sw

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(5.40)

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where m 1 is the number of phases, ρw is the mass density of conductors (8890 kg/m3 for copper, 2700 kg/m3 for aluminum), N1 is the number of turns per phase, ltur n is the average length per turn, a p is the number of parallel current paths, aw is the number of parallel wires, and sw is the conductor cross section area. Similar to the stator core, Equation 5.39 gives better results than Equation 5.38 even if (L i + 2h ov )/(2Rw ) < 1.

5.3.5

Analytical calculation of natural frequencies for a stator core-winding-frame system

A cylindrical stator system (core, winding, frame, end bells) of a 10 kW a.c. machine with s1 = 36 slots will be considered. The mass of the stator core (yoke) is 8.58 kg, mass of teeth is 10.27 kg, mass of the winding is 11.45 kg, mass of the insulation is 0.57 kg, and mass of the aluminum frame is 6.24 kg. The dimensions of the stator system are: • • • • • • • • •

stator core inner diameter D1in = 0.16 m stator core outer diameter D1out = 0.233 m effective length of the stator core L i = 0.1975 m stator tooth width ct = 5.87 mm stator slot opening b14 = 3 mm thickness of stator yoke (core) h c = 8.42 mm length of winding overhang h ov = 48 mm diameter of frame D f = 0.246 m length of frame L f = 0.359 m

Material parameters are specified below: • • • • • • • • •

specific mass density of laminations ρc = 7700 kg/m3 specific mass density of copper ρw = 8890 kg/m3 specific mass density of frame (aluminum) ρ f = 2700 kg/m3 modulus of elasticity of laminations E c = 200 × 109 Pa modulus of elasticity of winding with insulation E w = 9.4 × 109 Pa modulus of elasticity of frame (aluminum) E f = 71 × 109 Pa Poisson ratio of laminations νc = 0.3 Poisson ratio of copper νw = 0.35 Poisson ratio of frame (aluminum) ν f = 0.33

Natural frequencies of the stator core The mean radius of the stator core (yoke) is Rc = 0.5(D1out − h c ) = 0.5(0.233 − 0.00842) = 0.112 m and the parameter κ according to Equation 5.24 is κ 2 = h 2c / (12Rc2 ) = 0.00842/(12 × 0.1122 ) = 0.000469. Since the stator core is regarded as an infinite shell, the parameter m and natural frequencies f m of the core are only a function of circumferential modes m = 0, 1, 2, 3, . . .. The parameter m is calculated with the aid of Equations 5.22 and 5.23, e.g., m = 0.08 for m = 2. Natural frequencies of the stator core calculated on the basis of Equation 5.21 for circumferential modes 0 ≤ m ≤ 8 are given in Table 5.1.

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Table 5.1 Natural frequencies of the stator core for 0 ≤ m ≤ 8 calculated on the basis of Equation 5.21. 0 7572

1 116

2 464

circumferential mode m 3 4 5 6 1313 2517 4070 5971

7 8218

8 10,811

Natural frequencies of the stator winding The stator winding distributed in slots is regarded as a homogenous cylinder with an equivalent modulus of elasticity E w = 9.4 × 109 Pa (copper, insulation, encapsulation) and specific mass density equal to that of copper, that is, ρw = 8890 kg/m3 . The mean radius of the stator winding is Rw = 0.5(D1in +h t ) = 0.5(0.16 + 0.028) = 0.094 m and the parameter κ according to Equation 5.24 is κ 2 = h 2c /(12Rw2 ) = 0.0028/(12 × 0.0942 ) = 0.0074. Since the stator winding is regarded as an infinite shell, the corresponding parameter m and natural frequencies of the winding are only a function of the circumferential modes m = 0, 1, 2, 3, . . .. The parameter m is calculated with the aid of Equations 5.22 and 5.23, e.g., m = 0.307 for m = 2. Natural frequencies of the stator winding calculated on the basis of Equation 5.21 with E c , ρc , νc , and Rc replaced by E w , ρw , νw , and Rw , respectively, for circumferential modes 0 ≤ m ≤ 8 are given in Table 5.2.

Natural frequencies of the frame The mean radius of the frame is R f = 0.25(D f + D1out ) = 0.25(0.246+0.233) = 0.12 m and the paremeter κ according to Equation 5.24 is κ 2 = h 2f /(12R 2f ) = 0.000246 where the thickness of the frame h f = 0.5(D f − D1out ) = 0.5(0.246 − 0.233) = 0.0065 m. A cylindrical frame with end bells can be regarded as a “clamped-clamped” shell for which the equivalent wavelength λe as a function of axial vibrational modes n = 1, 2, 3, . . . is expressed by Equation 5.32. Putting λe into Equations 5.27, 5.28, 5.29, and 5.30, the roots of the third order characteristic Equation 5.26

Table 5.2 Natural frequencies of the winding for 0 ≤ m ≤ 8 calculated on the basis of Equation 5.21 in which E c = E w , ρc = ρw , νc = νw , and Rc = Rw . 0 1857

1 113

2 570

circumferential mode m 3 4 5 6 1360 2471 3901 5647

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7 7708

8 10,080

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Table 5.3 Smallest polyroots 2mn of Equation 5.26 for 0 ≤ m ≤ 8 and n = 1, 2, 3. axial mode n 0 1 2 1 0.65 0.251 0.079 2 0.902 0.599 0.3 3 0.937 0.787 0.538

circumferential mode m 3 4 5 6 7 8 0.049 0.082 0.174 0.344 0.623 1.048 0.176 0.17 0.254 0.433 0.731 1.182 0.379 0.34 0.414 0.603 0.926 1.414

can be easily found with the aid of, for example, Mathcad technical calculation tool. For m = 0 and n = 1 the polyroots are 201 = 0.65, 0.837, and 2.02; for m = 2 and n = 1 the polyroots are 221 = 0.079, 2.318, and 6.516; for m = 4 and n = 1 the polyroots are 241 = 0.082, 6.372, and 18.729, and so on. Table 5.3 contains the smallest polyroots 2mn for 0 ≤ m ≤ 8 and 1 ≤ n ≤ 3. The natural frequencies f mn of the aluminum frame with end bells found on the basis of Equation 5.33 are given in Table 5.4.

Natural frequencies of the stator system Natural frequencies of the stator system (core with teeth, winding with insulation, and frame) have been calculated using simplified Equation 5.37. The stiffness of the core K m(c) is given by Equation 5.25, the stiffness of the winding K m(w ) is given (f) by Equation 5.39, and the stiffness of the frame K mn is given by Equation 5.34. The mass of the core with teeth is Mc = 8.58 + 10.27 = 18.85 kg, mass of the winding with insulation is Mw = 11.45 + 0.57 = 12.02 kg, and mass of the frame is M f = 6.24 kg. The natural frequencies of the stator system for 0 ≤ m ≤ 8 and 1 ≤ n ≤ 3 are given in Table 5.5. It is always recommended to verify the natural frequencies of the stator system obtained from analytical Equations 5.17, 5.21, 5.33, 5.36, and 5.37 with the 3D FEM structural modeling.

Table 5.4 Natural frequencies of the frame with end bells for 0 ≤ m ≤ 8 and 1 ≤ n ≤ 3 calculated on the basis of Equation 5.33. axial mode n 1 2 3

circumferential mode m 0 5819 6856 6990

1 3616 5589 6406

2 2033 3952 5294

3 1604 3027 4444

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4 2072 2976 4210

5 3010 3638 4644

6 4233 4751 5606

7 5697 6173 6947

8 7392 7850 8584

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Table 5.5 Natural frequencies of the stator system for 0 ≤ m ≤ 8 and 1 ≤ n ≤ 3 calculated on the basis of Equation 5.37. axial mode n 1 2 3

5.4 5.4.1

circumferential mode m 0 4722 4897 4921

1 1299 2002 2293

2 995 1568 2010

3 1720 1950 2270

4 3039 3133 3309

5 4775 4831 4940

6 6904 6947 7028

7 9419 9458 9526

8 12,316 12,352 12,414

Numerical verification FEM modeling

Numerical approach in engineering computations is typically used when a system is far too complex to analyze analytically. One of the most popular generalpurpose 3D FEM analysis software packages is Ansys. In the FEM, a complex system is divided into very small pieces called elements, the shape and size of which are decided by the user. The accuracy of the FEM results generally depends on the number of elements in the model. The forces or loads on each element are calculated one by one using differential equations. The results then can be presented in numerical or graphical form. The following generic steps are taken to solve a problem in Ansys or other FEM software packages: • Building the geometry. 2D or 3D representation of the object or system is constructed. • Defining material properties. A library of the necessary material properties that compose the object or system is created. • Mesh generation. Most software packages can generate the mesh automatically. However, it must be defined how the modeled system should be broken down into finite elements. • Application of loads. Constraints such as physical loadings or boundary conditions are assigned. • Solution. Solver module solves numerically differential equations. The FEM software needs to understand what kind of solution is expected, that is, steady state, transient, and so forth. • Postprocessing. The results can be presented in form of graphs, contour plots, or tables. Structural analysis of machine housings, machine parts, and machine tools is the most common application of the FEM. The term “structural” implies not only civil engineering structures but also naval, aeronautical, and mechanical structures. The following types of structural analyses can be performed with Ansys: • Static analysis. It is used to determine displacements, forces, and stresses under static loading conditions.

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• Modal analysis. It is used to calculate the natural frequencies and mode shapes of a structure. • Harmonic analysis. It is used to determine the response of a structure to harmonically time-varying loads. • Transient dynamic analysis. It is used to determine the response of a structure to arbitrarily time-varying loads. • Spectrum analysis. This is an extension of the modal analysis to calculate stresses and strains due to a response spectrum or a power spectral density (PSD) input (random vibration). • Buckling analysis. It is used to calculate the buckling loads and determine the buckling mode shape. Nonlinearities in static, dynamic, and buckling analyses can include plasticity, stress stiffening, large deflection, large strain, hyper elasticity, contact surfaces, and creep. The modal analysis is necessary to determine the vibration characteristics, that is, natural frequencies and mode shapes of a structure or machine component. The natural frequencies and mode shapes are important parameters in the design of a structure for dynamic loading conditions. They are also necessary in a spectrum analysis, mode superposition harmonic analysis, and transient analysis. In vibrational and acoustic analysis of electrical rotating machinery, it is necessary to compare the frequencies and orders of radial magnetic forces with natural frequencies and vibrational modes of the stator system. All frequencies of magnetic forces that are close to natural frequencies of the stator system for the same force order as vibrational mode can produce dangerous vibration and excessive noise. Figure 5.4a shows the Ansys element plot of the stator laminated core with 36 skewed axial slots. The Ansys model consists of 3D solid elements and surface to surface contact elements taken from the Ansys element library. The laminated stator is modeled as a continuous solid. The material properties assigned to the stator laminations are orthotropic in nature. The orthotropic material properties enable a difference in stiffness between the radial and axial directions. The stator

(a)

(b)

Figure 5.4 Element plots: (a) stator laminated stack; (b) stator winding.

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n=2

n=3

Figure 5.5 Circumferential modes m = 2 for axial modes n = 1, 2, and 3 for an aluminum frame with end bells (outer diameter 0.246 m, inner diameter 0.233 m, and length 0.359 m) obtained from the 3D FEM analysis. stack is connected to the frame (enclosure) using a bonded contact pair. Since the stator is pressed into the frame, the contact pair is also used to calculate the prestress effect of the interference fit on the natural frequencies of the system. The volume of material representing the stator winding is modeled also using solid elements. The stator axial slots are considered fully filled with the winding material. The winding parameters (specific mass density, elasticity modulus, Poisson ratio) are an equivalent representation of copper, insulation, and epoxy resin. Winding elements are shown in Figure 5.4b. The Ansys model of the aluminum frame is shown in Figure 5.5. For the circumferential mode m = 2 three axial modes, i.e., for n = 1, 2, and 3 has been plotted. It has been assumed that the frame is furnished with end bells (“clampedclamped” shell).

5.4.2

Comparison of analytical calculations with the FEM

First, a simple thin circular square solid ring with its diameter 0.508 m, thickness h c = 0.00127 m, axial length L i = 0.00127 m, modulus of elasticity E c = 206.8 × 109 Pa, and specific mass density ρc = 7955 kg/m3 will be considered. The natural frequencies f 0 for m = 0 and f 2 for m = 2 of radial vibration will be determined. The frequency f 0 can be calculated with the aid of Equation 5.9 for ki = 1 (solid homogeneous ring) and kmd = 1 (no mass addition), i.e.,  1 206.8 × 109 f0 = = 3194.6Hz. π 0.508 7955

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Table 5.6 Comparison of natural frequencies for 0 ≤ m ≤ 4 and n = 1, 2, 3 obtained from the 3D FEM and analytical calculations for an aluminum frame with end bells. Values calculated analytically are in parentheses. axial mode

m=0

m=1

n=1

-

2927 2927 2919 2924 (3616) (3563) 5184 5095 5134 5093 5139 (5589) (5516) 6311 6031 6176 6045 6201 (6406) (6331)

n=2

n=3

(5819) (5693) 6175 6220 6157 6218 (6856) (6772) 6865 6612 6794 6637 6818 (6990) (6914)

circumferential mode m=2 m=3 m=4 1884 1841 1833 1827 1829 (2033) (1985) 3626 3519 3524 3506 3526 (3952) (3889) 5182 4918 4980 4906 4996 (5294) (5222)

1687 1533 1520 1513 1517 (1604) (1518) 2986 2772 2764 2743 2764 (3027) (2950) 4491 4129 4148 4087 4157 (4444) (4364)

2298 1966 1941 1924 1933 (2072) (1963) 3202 2782 2773 2727 2767 (2976) (2876) 4504 3923 3939 3843 3937 (4210) (4114)

method ANS N1 N2 N3 N4 A(5.33) AWL ANS N1 N2 N3 N4 A(5.33) AWL ANS N1 N2 N3 N4 A(5.33) AWL

The frequency for m = 2 can be calculated according to Hoppe’s formula (Equation 5.17), i.e.,  2(22 − 1) 206.8 × 109 × 2.168 × 10−13 2 √ f2 = = 12.373Hz π × 0.5082 22 + 1 0.013 where ρc L i h c = 7955 × 0.00127 × 0.00127 = 0.013 kg/m and the moment of inertia according to Equation 5.13 is Ic = 0.001273 × 0.00127/12 = 2.168 × 10−13 m4 . The same frequencies obtained from Ansys are f 0 = 3226.4 Hz and f 2 = 12.496 Hz. The convergence is good. For thin square solid ring analytical equations give slightly smaller values of natural frequencies. In the next example natural frequencies of a frame of an electrical machine will be calculated. Tables 5.6 and 5.7 show the comparison of natural frequencies of an aluminum frame with end bells (outer diameter 0.246 m, inner diameter 0.233 m, length 0.359 m) obtained from the 3D FEM and analytical calculations. Two different FEM packages, namely Ansys and Nastran have

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Table 5.7 Comparison of natural frequencies for 5 ≤ m ≤ 8 and n = 1, 2, 3 obtained from the 3D FEM and analytical calculations for an aluminum frame with end bells. Values calculated analytically are in parentheses. axial mode n=1

n=2

n=3

m=5

circumferential mode m=6 m=7 m=8

method

3404 2891 2836 2804 2819 (3010) (2897) 4080 3425 3410 3323 3392 (3638) (3526) 5177 4309 4343 4178 4326 (4644) (4535)

4826 4136 4023 3968 3990 (4233) (4118) 5395 4516 4470 4335 4434 (4751) (4636) 6351 5186 5231 4979 5193 (5606) (5491)

ANS N1 N2 N3 N4 A(5.33) AWL ANS N1 N2 N3 N4 A(5.33) AWL ANS N1 N2 N3 N4 A(5.33) AWL

6510 5656 5447 5355 5387 (5697) (5582) 7025 5939 5825 5631 5762 (6173) (6055) 7893 6446 6474 6123 6405 (6947) (6829)

8436 7446 7087 6937 6988 (7392) (7275) 8920 7654 7421 7149 7317 (7850) (7731) 9731 8027 7991 7527 7881 (8584) (8465)

been used. In order to show the convergence of the results with the number of elements, two mesh sizes have been examined in the Nastran analysis. Results obtained from different element types are also listed in Tables 5.6 and 5.7. The following abbreviations have been used: ANS = Ansys (cube elements), N1 = Nastran 2003 14 × 40 HEX8 elements, N2 = Nastran 2003 30 × 60 HEX8 elements, N3 = Nastran 2003 14 × 40 CQUARD4 elements, N4 = Nastran 2003 30 × 60 CQUARD4 elements, A(5.33) = analytical method according to Equation 5.33, AWL = analytical method according to Wang and Lai [226]. Both analytical methods give reasonable estimations on the natural frequencies except for the low order modes, such as m = 1, n = 1. Figure 5.6 shows circumferential modes m = 2 and 3 and axial mode n = 1 obtained from the 3D FEM Ansys for the stator with aluminum frame equipped with end bells (outer diameter 0.246 m, inner diameter 0.233 m, length 0.359 m), laminated core, and copper winding. The diameters are as follows: stator inner

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Stator System Vibration Analysis m=2

125 m=3

Figure 5.6 (See color insert following page 236.) Circumferential modes m = 2 and 3 and axial mode n = 1 for the stator (aluminum frame with end bells, laminated core, and copper winding) obtained from the 3D FEM Ansys. (teeth) diameter 0.1601 m, stator core (yoke) inner diameter 0.1769 m, stator outer diameter 0.233 m, frame outer diameter 0.246 m. The stator core length L i = 0.1975 m, stator winding length with end connections 0.2935 m, stator frame length L f = 0.359 m. The natural frequencies obtained from analytical calculations (Equation 5.37) and Ansys 3D FEM software package for the stator system (aluminum frame with end bells, laminated core, copper winding) have been compared in Table 5.8. Although, the accuracy of Equation 5.37 is reasonable, it can be found that as the system becomes complicated, the accuracy of analytical solutions deteriorates. For a complete electrical machine system, it is always recommended to use the FEM analysis for accurate results. Table 5.8 Comparison of natural frequencies obtained from analytical calculations and Ansoft 3D FEM software package for the stator system (aluminum frame with end bells, laminated core, copper winding). Values calculated analytically are in parentheses. axial circumferential mode m mode n 0 1 2 3 4 5 6 7 8 1 4289 − 1009 2089 3332 4541 5785 − − (4722) (1299) (995) (1720) (3039) (4775) (6904) (9419) (12,316) 2 − − 1789 2388 3401 4553 − − − (4897) (2002) (1568) (1950) (3133) (4831) (6947) (9458) (12,352) 3 4934 − − − − − − − − (4921) (2293) (2010) (2270) (3309) (4940) (7028) (9526) (12,414)

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6 Acoustic Calculations The sound radiated from structures is an impact of the structural vibration on the environment. From the vibro-acoustic point of view, it is a result of interactions between the structure and the ambient medium. Generally speaking, more vibrations may produce more sound. However, the relation between the two is not simple, especially for machinery structures. The sound radiation generally depends on the structure dimensions, boundary conditions, material properties, external excitations, and ambient medium conditions, and so on. In acoustics, in order to facilitate the discussions on the mechanism of sound generations from various structures, simple structural elements, such as plates, spheres, and cylinders, are treated as basic sound radiators because of their frequent occurrences in practical applications [160]. For more than a century, numerous papers and books have been published on almost every aspect of the topic. It is, therefore, not wise and impossible to cover all the materials in a document of limited size. The intention of this chapter is to summarize the outcome that may be related to the sound radiation from electric motor structures, and to present readers with the most recently updated results and techniques to benefit their relevant studies and practices. For this purpose, plane and cylindrical radiators are discussed directly without much emphasis on the basic acoustics theories. Readers might need to refer to Appendix A and [124, 176] for prerequisite knowledge for the contents of this chapter.

6.1

Sound radiation efficiency

As it is known, a vibrating structure produces sound waves in the ambient medium. The sound pressures at any specific location in the medium space, and the sound power radiated by the structure are the two parameters used to quantify the local and global acoustic effects in the field. They are generally functions of vibration levels on the structure surface, which are further determined by the boundary conditions and the external excitations. For a specific structure, higher vibration levels mean higher sound pressures in the field and higher radiated sound power. 127 Copyright © 2006 Taylor & Francis Group, LLC

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In many cases, however, it is not the absolute value, but the transfer function from the structural vibration to the acoustic response that is of greatest interest. A properly defined vibro-acoustic transfer function could fundamentally reveal the energy transfer mechanisms and efficiently eliminate the duplications of the tedious derivation/calculation in accounting for the interactions between the structure and the ambient medium, thus simplifying the vibro-acoustic analysis. One of the most common definitions used in vibro-acoustic analysis is the sound radiation efficiency, as given by, σ =


 ρ 0 c0 S v 2

(6.1)

where ρ0 , c0 are the density and the sound speed in the ambient medium (air) respectively, S is the area
ofthe radiating surface,  is the sound power radiated from the structure, and v 2 is the spatial averaged mean square velocity over the structure radiating surface. This expression relates the radiated sound power to the spatial averaged vibration level. Physically, it indicates the capability of a structure in sound radiation in comparison with a piston with the same surface area and same vibration levels under the condition that the piston circumference greatly exceeds the acoustic wavelength [176]. However, since the vibration distribution is generally affected by the excitation, the spatial averaged mean square velocity could be quite different for different excitation configurations. For example, a point force excitation might result in different sound radiation efficiencies from a distributed force or point moment excitations for the same structure. As a result, the modal radiation efficiency (σm ) associated with each vibration mode (m) is normally introduced in theoretical analyses [37], m
 σm = (6.2) ρ0 c0 S v m2
 where m is the sound power radiated by the mth mode of the structure, and v m2 is the spatial averaged mean square velocity of the mth mode. The modal radiation efficiency is generally independent of the external excitation and, hence, reflects the inherent characteristics of the structure in the energy transfer. Since the total sound power  is the sum of the power radiated from each vibration mode m , i.e.,  = m , the relationship between Equations 6.1 and 6.2 is [37], m


 σm v m2 m σ = 
 . v m2 

(6.3)

m

As Equation 6.3 takes into account the contribution from each vibration mode, the result is also called the modal averaged radiation efficiency. In practice, the modal averaged radiation efficiency is what can be directly measured.

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On the other hand, since the radiated sound energy is a loss to the structural vibrational energy, the radiation loss factor can be defined to describe the extent to which the vibration is damped due to the radiation of power from the structure to the ambient medium, i.e.,  ηrad = (6.4) ωE where ω is the angular frequency, E is the total vibrational energy of the structure. For a uniform structure with
a spatially uniform vibration distribution over the structure surface E = M v 2 , where M is the mass of the structure [37]. The relationship between the radiation efficiency and the radiation loss factor is, σ =

M k0 ηrad ρ0 S

(6.5)

where k0 = ω/c0 is the acoustic wavenumber. Compared to the radiation efficiency, the radiation loss factor is a small number, especially for the radiation into the air. Since the radiation loss factor is also related to the spatial averaged mean square velocity over the structure surface, the modal averaged radiation loss factor (ηrad ) can be expected to have the same relationship to the corresponding modal averaged radiation efficiency (σ ) as that shown in Equation 6.5. Another parameter often used in the sound radiation analysis is the acoustic radiation impedance, defined on the radiating surface of the structure as,  p( r )S  ZR = (6.6) v r=r 0 where p( r ) is the sound pressure in the field, v is the vibration distribution over the structure, S is the radiating surface area, and the subscript r = r0 denotes the location on the surface of the structure. The real part of the impedance represents the energy radiated into the ambient medium, while the imaginary part of the impedance represents the energy stored in the near field of the source [124]. This expression is convenient to use when the vibration amplitude is uniform over the structure; examples are, a point source, a piston radiator, or a plate vibrating in one of its modes. The relation between the acoustic radiation impedance and the radiation efficiency can be found as, σ =

e[Z R ] ρ 0 c0 S

(6.7)

where e denotes the real part of a complex number.

6.2

Plane radiator

Plane structures are very common in industries. Substantial research has addressed the importance of the sound radiation from plane structures. For electric machine structures, plates are often found as accessory components that are necessary for the assembly and the installations; for example, the end-shields and the support

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which in some cases are found important in the total sound power contributions. In this section, the sound radiation from an infinite plate is briefly reviewed to benefit the analysis of finite plates. Then the finite-sized plates are discussed in regard to their modal radiation and modal averaged radiation efficiencies.

6.2.1

Infinite plates

In order to have a better understanding of the sound radiation phenomena of practical plane radiators, it is helpful to have a simple review on the sound radiation from an infinite plate (Figure 6.1). For simplicity, it will be assumed an infinitely extended plate in the y = 0 plane is vibrating transversely with a velocity distribution given as (omitting the time component e jωt , without loss of generality), v(x) = v 0 e− jk p x

(6.8)

where k p is the wavenumber of the plate in bending motion. This equation actually represents a one-dimensional approaching wave propagating in the +x direction with the wavelength and the wave speed being λ p and c p , respectively. Since the vibration is uniform in the z direction (−∞, +∞), and so will be the acoustic field, the acoustic wave motion equation can be simplified as ∂ 2 p(x, y) ∂ 2 p(x, y) + + k02 p(x, y) = 0 ∂x2 ∂ y2

(6.9)

where p(x, y) is the radiated sound pressure in the field. The solution of Equation 6.9 may be generally written as p(x, y) = p0 e− jkx x e− jk y y

(6.10)

y

l0 ky

k0 kx

kp

q

x lp

Figure 6.1 Radiation from an infinite plate.

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where k x and k y are the two components of the acoustic wavenumber k0 in the x and the y directions, respectively, and k02 = k x2 + k 2y . This expression must satisfy the acoustic boundary condition at y = 0, which is  1 ∂ p(x, y)  v(x) = − jωρ0 ∂ y  y=0 then,



kx = k p



p0 = v 0 ρ0 c0 k0 /k y = v 0 ρ0 c0 /

(6.11)

1 − k 2p /k02 .

The radiated sound pressure in the field is therefore given as √2 2 v 0 ρ0 c0 p(x, y) =  e− jk p x e− j k0 −k p y 1 − k 2p /k02

(6.12)

From Equations 6.6 and 6.12, the radiation impedance per unit area of the plate can be obtained as  p(x, y)  ρ 0 c0 ρ 0 c0 ZR = = = (6.13) v(x)  y=0 2 2 1 − k /k 1 − λ2 /λ2 p

0

0

p

Equation 6.13 indicates that for an infinite plate, the sound radiation has two fundamentally different behaviors. If the wavelength in the plate λ p is greater than the acoustic wavelength λ0 in the air (or equivalently, the wave speed in the plate c p is faster than the speed of sound in the air c0 , namely the supersonic effect), the radiation impedance of the plate is a real number, indicating that the energy is transmitted into the ambient medium. The sound radiation efficiency per unit area of the plate can be written as σ =

e(Z R ) 1 1 = = , or simply ρ0 c0 1 − k 2p /k02 1 − λ20 /λ2p

σ =

1 . (6.14) cos θ

If the structural wavelength is smaller than the wavelength of the sound wave in the air (or equivalently, the structural wave speed is slower than the speed of sound in the air, namely the subsonic effect), the radiation impedance is a pure imaginary number, which means that the sound pressure is 90◦ out of phase with the plate velocity. In this case, the energy will be exchanged between the plate and the acoustic medium with no sound energy radiated. The sound radiation efficiency of the plate is therefore zero, and the sound pressure simply decays along the y direction in the form as √2 2 jv 0 ρ0 c0 p(x, y) =  (6.15) e− jk p x e− k p −k0 y . k 2p /k02 − 1 An important phenomenon that must be addressed here is, when the acoustic wavelength is matching the structural wavelength, i.e., λ p = λ0 , Equation 6.12

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Radiation Efficiency, σ

100

10

1

0.1 1

0.1

10

f/fc

Figure 6.2 Radiation efficiency as a function of frequency for an infinite plate.

gives an infinity solution for the sound pressure in the field. This is because the radiated sound wave, under this condition, is propagating in the direction parallel to the structural wave in the plate, both waves having the same wavelengths and speeds. As a result, the sound wave will be enhanced continuously along the propagation path and the energy will be accumulated to infinity. In practice, due to the finite size and the damping in the plate, the radiated sound pressure always remains finite. However, large radiation still can be observed. Note that the wavelengths of both structural and acoustic waves are frequency dependent. A frequency, namely the critical frequency of the plate f c , at which the structural wavelength is equal to the acoustic wavelength (or equivalently, the structural wave speed is equal to the speed of sound in the air) exists, i.e., [55], cp = and therefore,

 1/4 √ ω Eh 2 = ω· kp 12ρ(1 − ν 2 ) √

3c0 fc = πh



ρ(1 − ν 2 ) . E

(6.16)

In Equation 6.16 h is the thickness of the plate. E, ρ, and ν are the Young’s modulus, the density and the Poisson’s ratio of the plate material respectively. Figure 6.2 shows the variation of the sound radiation efficiency vs. the frequency. It is seen that for an infinite plate, sound radiation only occurs in the frequency range above the critical frequency f c . The radiation efficiency approaches to unity quickly as the frequency increases, indicating that the plate is an efficient radiator at high frequencies.

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6.2.2

133

Finite plates in bending motion

When a plate is finite in size, its vibration will have modal behaviors. The modal parameters, such as natural frequencies, mode shapes, and so on, are inherent dynamic characteristics of a structure, depending on the geometry, material properties, and boundary conditions of the structure. Theoretically, a vibration of a structure can be described by a superposition of all the vibration modes with different participation amplitudes that are determined by the external excitations (see Appendix D). Consequently, the sound radiation due to the vibration of the structure can and will be related to the structural modal behavior. In order to discuss the effect of the finite size on the sound radiation from a plate, it is instructive to consider a plate strip of width l in the x direction embedded in an infinite baffle with simply supported boundary conditions at both edges, as shown in Figure 6.3. With the baffle, the sound radiated from each side of the plate does not interfere with each other, such that only the sound field on one side of the plate is considered here. Again, assume that the vibration in the z direction is uniform (−∞, +∞), the velocity distribution for a particular mode m over the y = 0 plane is,

v m (x) =

v 0 sin(mπ x/l) 0 < x < l 0 0 > x > l.

(6.17)

By transforming v m (x) to the wavenumber domain,  Vm (k x ) =

l

v m (x)e jkx x d x = v 0 ·

0

mπ [(−1)m e− jkx l − 1] · l k x2 − (mπ/l)2

(6.18)

where Vm (k x ) and v m (x) satisfy, v m (x) =

1 2π



∞

−∞

Vm (k x )e− jkx x dk x .

(6.19)

Equation 6.19 indicates that the modal vibration distribution in the x direction can be decomposed into a series of waves propagating along an infinite plate in the x direction with different wavenumbers and magnitudes. The sound radiation from each of these waves thus can be treated as the radiation from an infinite plate with the velocity distribution as Vm (k x )e− jkx x . Based on Equation 6.12, the sound

y

lpm = 21/m x l

Figure 6.3 Modal behavior of a plate strip in a baffle.

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pressure produced on the plate surface associated with each wave component is ρ 0 c0 k 0 Pm (k x )| y=0 = Vm (k x )  . k02 − k x2

(6.20)

Correspondingly, the total sound pressure due to all the wave components can be written as,  ∞ 1 ρ0 c0 k0 − jkx x pm (x)| y=0 = Vm (k x )  e dk x . (6.21) 2π −∞ k02 − k x2 With the known vibration velocity distribution and its induced sound pressure on the plate surface, the time-averaged sound power radiated per unit length (in the z direction) of the plate is,    l   ∞ 2 c k |V (k )| 1 ρ 0 0 0 m x  m (ω) = e e  pm (x)| y=0 v m∗ (x)d x = dk x  2 4π 0 −∞ k2 − k2 0

x

(6.22) where “∗ ” denotes the complex conjugate. Note that the term within the integral is real only for −k0 < k x < k0 . By using Equation 6.18, Equation 6.22 can be rewritten as  k0 m 2l 2 k x l − mπ 2  m (ω) = πρ0 c0 k0 v 0 dk x . (6.23) sin2 2 −k0 (k 2 l 2 − m 2 π 2 )2 k02 − k x2 x Although Equation 6.23 cannot be evaluated in a closed form, approximate solutions corresponding to the two distinguished regions discussed for the infinite plate can be obtained. • If k0 mπ/l, i.e., the acoustic wavelength is greater than the structural wavelength λ pm (or equivalently, in the subsonic region where the structural wave speed is slower than the speed of sound), Equation 6.23 becomes [37]  2  1 ρ0 c0 k0 v 02l 2 k0 ρ0 c0 k0 v 02 l  m (ω) ≈ dk x = (6.24) 2m 2 π 3 2 mπ −k0 k02 − k x2 then the corresponding modal radiation efficiency is,  2 m 2k0 l 4m
= σm = = . l mπ ρ0 c0lv 02 ρ0 c0l v m2

(6.25)

• If k0  mπ/l, i.e. the acoustic wavelength is shorter than the structural wavelength λ pm (or equivalently, in the supersonic region where the structural wave speed is faster than the speed of sound), one gets m (ω) ≈

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 ρ0 c0 k0lv 02 /[4 k02 − (m π/l )2 ] ≈ ρ0 c0lv 02 /4 . The modal radiation efficiency is, m k0
≈ σm = ≈ 1. (6.26) 2 2 k0 − (m π/l )2 ρ0 c0 l v m Compared to the infinite plate, the most prominent feature of a finite plate in its sound radiation is that the modal radiation is not zero in its subsonic region. Given an explanation, a finite plate with simply supported edges in a particular mode is considered in Figure 6.4. For a higher order mode of which structural wavelengths λ px , λ py along the two directions are smaller than the acoustic wavelength λ0 in the air, as shown in Figure 6.4a, the adjacent subsections of which the vibration is 180◦ out of phase are separated by less than a wavelength in the surrounding air. The air displaced outward by one subsection moves to occupy the space left by the adjacent subsection, without being disturbed much in the far field, and thus very little power is radiated. In other words, the sound radiated from adjacent subsections cancel each other. At the corners, however, such “cancelation” is not complete with four monopole sources remaining and making primary contributions to the total sound power radiated from the plate. If only one of the structural wavelengths is smaller than the acoustic wavelength, for example λ px < λ0 as shown in Figure 6.4b, the adjacent subsections are separated by less than an acoustic wavelength in the l x direction, while the adjacent subsections in the l y direction are beyond an acoustic wavelength. In this case, the cancelations only happen along one direction as shown in the graph with two edges remaining as effective radiators. Along the l y direction, the two

Uncanceled Corners

+ ly l0 l py

Uncanceled Edges

− + +

−

+ Typical Region of Cancelations

+

−

+

l0

lpy

+

ly − +

l0

(a)

−

Typical Region of Cancelations − + −

lx

lx lpx

−+

−

lpx

l0 (b)

Figure 6.4 Sound radiation from: (a) uncanceled corners; (b) uncanceled edges of a finite plate.

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subsections radiate sound independently. Thus, edges normally radiate more power than corners in the frequency range where they both occur. In the supersonic region, the structural wavelengths in both directions are greater than the acoustic wavelength, i.e. λ px , λ py > λ0 . All subareas radiate sound independently and make their contributions in the total sound power. If the plate is large compared to the acoustic wavelength, the boundary effect is negligible and the acoustic behavior of a finite plate approaches to that of an infinite plate. Figure 6.5 shows the modal radiation efficiencies of a series of modes associated with a baffled finite plate with simply-supported boundary conditions. As expected, all the modal radiation efficiencies generally increase as the acoustic wavenumber k0 in the subsonic region, and reach maximums around k0 = k p  (k p = (nπ/l x )2 + (mπ/l y )2 for a simply-supported plate, where n, m are the mode numbers along the l x and l y directions, respectively). In the supersonic region, the radiation efficiencies are unity as that of the infinite plate. An interesting and important feature that may be concluded from Figure 6.5 is that, for each mode (n, m), the frequency at which the acoustic wavenumber k0 matches the structural wavenumber k p is different. In the frequency domain, lower order modes reach unity in their sound radiation efficiencies at lower frequencies, thus being more efficient in sound radiation than higher order modes. Apparently, an accurate calculation of the sound power radiated from a finite plate depends on a detailed analysis of the plate vibration from which the participation amplitudes of each mode in the total response can be determined.

10

Modal Radiation Efficiency

1 0.1 Mode (1, 1)

0.01

Mode (1, 3) Mode (3, 3)

0.001

Mode (1, 2) Mode (2, 3)

0.0001

Mode (2, 2) 0.00001 0.01

0.1

1

10

k0/kp(m, n)

Figure 6.5 Modal radiation efficiencies of a finite plate in a baffle [221].

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However, if the plate is mechanically excited by a random noise, the vibration response within a frequency band  f is primarily dominated by the modes of which the natural frequencies are within this band. Assume that the modal energies within this band are equally distributed, general expressions for the modal averaged radiation efficiency of the plate might be derived. Since each mode generally takes effect around their own natural frequencies because of a relatively large participation amplitude in the total vibration response, it has been shown that statistically a maximum that could exceed unity can still be observed in the modal averaged sound radiation efficiency around f c for a finite plate [55]. Two specific plate configurations that were considered in the analysis are that the plate is embedded in an infinite baffle and that the plate is unbaffled with the edges being acoustically free. Baffled plates Specifically, at low frequencies, the behavior of the plate is dominated by the first mode, normally the piston mode, of the plate. In this case, the radiation efficiency of the plate embedded in an infinite baffle can be calculated using the following equation [176, 212] σ =1−

2J1 (4k0 S/π ) 4S f 2 ∼ 2 8k0 S/π c0

for

λ0 

√

2S

(6.27)

where J1 is the Bessel function of the first order. Ver and Holmer [212] suggested this equation generally applies below the first natural frequency of the plate, that is, f < [c02 /(2S f c )][P 2 /(8S) − 1] , where S is the area of one side of a plate, and P is the plate perimeter. When the baffled plate is in the multimodal region, a formula for the modal averaged radiation efficiency based on the simply supported boundary conditions is given in [146], i.e.,  σ cor ner + σ edge f < fc     √  σ = (6.28) l x /λc + l y /λc f = f c      √ 1/ 1 − f c / f f > fc where σ cor ner , and σ edge are the modal averaged radiation efficiencies associated with the radiation regions around corners and along the edges of the plate respectively. They were given as, σ cor ner

8 λ2 = 4· c × π S

σ edge =



√ (1 − 2α)/(α 1 − α 2 ) f < f c /2 0 f > f c /2

1 Pλ2c (1 − α 2 ) ln[(1 + α)/(1 − α)] + 2α  · · 2 4π S (1 − α 2 )3

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(6.29)

(6.30)

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√ where α = f / f c . It was suggested that for the clamped boundary conditions, 3 dB should be added to the radiation efficiency in the range f < f c [212]. Figure 6.6 compares the modal averaged radiation efficiencies of two plates calculated analytically and by the boundary element method (BEM). In BEM calculations, the baffled plate was excited mechanically by a point force. To simulate the equally distributed modal energies, the BEM results were averaged over three excitation locations randomly chosen on the plate. It is seen that for small plates, the piston mode could dominate the acoustic behavior even higher than the critical frequency. As the plate becomes bigger, and thinner, the piston mode may only work at very low frequencies. However, Figure 6.6b shows that the piston mode region is wider than that suggested for simply supported and clamped boundary conditions. This is simply because the first mode of the plate (piston mode) is more efficient in sound radiation than higher order modes. Equation 6.28 works well in the multimodal region, which is above the frequency corresponding to the lower one of the natural frequencies f 21 and f 12 , i.e., f > [c02 /(2S f c )][P 2 /(8S) + 3l y /(2l x ) − 1] for a rectangular plate with l x and l y being the long and short edges of the plate, respectively. For baffled plates with completely free boundary conditions, it was shown that rigid motions, one piston and two rotation motions, may predominately contribute to the low frequency sound radiation. As a result, it seems difficult to derive a general expression for the modal averaged radiation efficiency because, depending on the excitations, the sound radiation efficiency may behave quite differently due to the contribution from one or some of these rigid motions [15]. However, studies have shown that if there is any localized constraints along which the two rotation motions are restricted, Equations 6.27 and 6.28, though developed based on simply supported boundary conditions, might give reasonable results in the sound radiation efficiency. Unbaffled plates As mentioned above, the baffle around the plate separates the space into halves, in which the sound radiation considered is only from one side of the plate. In practice, a typical example of this is a plate backed with a large cavity enclosed by rigid walls. The sound radiation from the two sides are isolated from each other or their interactions can be ignored. However, in many places, plane structures are found unbaffled. In this case, the sound radiation from both sides of the plate and their interactions should be considered. An instant effect of removing the baffle in the sound radiation discussions is that the radiating area in the definition of the sound radiation efficiency (Equation 6.1) is twice that of the baffled plate. For an unbaffled plate, the radiation from both sides may exchange the energy due to the oscillating inertial flow around the edge. Since the vibration velocities on both sides are 180◦ out of phase relative to their normal directions (into the ambient medium), the overall sound pressure field is of different characteristics from that of a baffled plate. Specifically, at low frequencies where the piston mode dominates the plate behavior, the unbaffled plate becomes a dipole source

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10 BEM, Simply-Supported Plate

Radiation Efficiency, σ

BEM, Clamped Plate 1

Equation 6.27

0.1

0.01

0.001 1000

100

10

10000

Frequency (Hz) (a) 10

Radiation Efficiency, σ

BEM, Simply-Supported Plate BEM, Clamped Plate Equation 6.28

1

Equation 6.27

0.1

0.01

0.001 1000

100

10

10000

Frequency (Hz) (b)

Figure 6.6 Modal averaged radiation efficiencies of a baffled plate: (a) 0.17 m×0.19 m×0.01 m aluminum plate, 9 × 10 elements in the BEM model; (b) 0.6 m×0.7 m×0.005 m steel plate, 40 × 46 elements in the BEM model. acoustically. Asymptotic solutions for the radiation efficiency of an unbaffled circular plate with radius a were derived by Silbiger [188], i.e.,

σ =

8(k0 a)4 /(27π 2 ) k0 a 1 1 k0 a  1.

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(6.31)

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On the basis of Equation 6.31, a general expression for an arbitrarily shaped plate with the unbaffled acoustic boundary condition can be obtained as,

σ =

√ 4S 2 f 4 /c04 λ0 > √2S 1 λ0 ≤ 2S

(6.32)

where S is the area of one side of a plate. In the multimodal region, below the critical frequency, instead of pulsating sphere and cylinder sources radiating into the half space, corners and edges of an unbaffled plate will behave more like oscillating sphere and cylinder sources, which are less efficient in sound radiation [124]. As a result, the modal averaged radiation efficiency of an unbaffled plate with simply supported boundary conditions can be written as [172] σ = F plate (Fcor ner σ cor ner + Fedge σ edge )

for

f < fc

(6.33)

where F plate , Fcor ner , Fedge are the correction factors due to the unbaffled condition. Particularly, F plate is for the effect of the inertial flows that surround the plate when the acoustic wavelength is greater than the plate dimensions, while Fcor ner , Fedge account for the localized inertial flow effects associated with corner and edge radiations. Semiempirical expressions for these correction factors were given as, F plate =

53 f 4 S 2 /c04 ; 1 + 53 f 4 S 2 /c04

Fcor ner =

1 13α 2 ; 2 1 + 13α 2

Fedge =

1 49α 2 . 2 1 + 49α 2 (6.34)

It can be seen that these factors take effects in different frequency ranges corresponding to different multimodal radiation behavior of the plate. Above the critical frequency, since the acoustic wavelength is smaller than the structural wavelength, the radiation from each small subsection on the plate will not cancel each other and is making contributions independently in the total radiated sound energy. Although the corners and edges may still be less efficient due to inertial flow effects if the acoustic wavelength is greater than the thickness of the plate, the effect on the total radiation is limited and therefore may be ignored. As a result, the modal averaged radiation efficiency of an unbaffled plate is still unity above the critical frequency. Figure 6.7 compares the modal averaged radiation efficiencies of two plates calculated analytically and by the BEM. The BEM results were also averaged over three excitation locations. It can be seen that Equation 6.32 gives a good estimation for the sound radiation from the piston mode. In the multimodal region, there is about 3 dB difference in the sound radiation efficiency between simply supported and clamped boundary conditions below the critical frequency. Again, Equation 6.33 is valid in the range f > [c02 /(2S f c )][P 2 /(8S) + 3l y /(2l x ) − 1] for a rectangular plate.

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10

Radiation Efficiency, σ

1

0.1

0.01 BEM, Simply-Supported Plate BEM, Clamped Plate

0.001

Equation 6.32 0.0001 100

1000 Frequency (Hz)

10000

(a) 10 BEM, Simply-Supported Plate BEM, Clamped Plate

Radiation Efficiency, σ

1

Equation 6.33 Equation 6.32

0.1 0.01 0.001 0.0001 0.00001 10

1000

100

10000

Frequency (Hz) (b)

Figure 6.7 Modal averaged radiation efficiencies of an unbaffled plate: (a) 0.17 m×0.19 m×0.01 m aluminum plate, 9 × 10 elements in the BEM model; (b) 0.6 m×0.7 m×0.005 m steel plate, 40 × 46 elements in the BEM model.

6.3

Infinitely long cylindrical radiator

Cylindrical shells are one of the most widely used structures in industries, for example, the casing of electric machines, and pipes used in various constructions. Due to the geometrical curvature, the vibration and the sound radiation is

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of different behaviors from that of plane structures. Acoustically, an elementary cylindrical radiator is an infinitely long circular cylindrical shell. By employing the cylindrical coordinates (Figure 6.8), the acoustic wave equation can be written as (omitting the time component e jωt , without loss of generality), ∂ 2 p(r, θ, z) 1 ∂ p(r, θ, z) 1 ∂ 2 p(r, θ, z) ∂ 2 p(r, θ, z) + + + + k02 p(r, θ, z) = 0. ∂r 2 r ∂r r2 ∂θ 2 ∂z 2 (6.35) The general solution of this equation is [160] p(r, θ, z) = p0 (Ae jkz z + Be− jkz z ) cos mθ Hm(2) (kr r )

(6.36)

where p0 , A, and B are constants that are determined by the acoustic boundary conditions on the cylindrical shell surface, m is the circumferential modal number of the cylindrical shell, Hm(2) (·) is the second kind of Hankel function of the m th order, kr and k z are the two components of the acoustic wavenumber k0 in the r and the z directions respectively, i.e. k02 = kr2 + k z2 . Assume that the vibration velocity distribution on the surface of the infinite length shell (r = a) is taking the form as, v(θ, z) = v 0 e− jksz z cos mθ (6.37) where v 0 is the amplitude, ksz is the z component of the structural wavenumber ks , as shown in Figure 6.8. This equation represents a structural wave traveling along the +z direction of the cylindrical shell. With the acoustic boundary condition on the surface of the shell, i.e.,  1 ∂ p(r, θ, z)  v(θ, z) = −  jωρ0 ∂r r =a

ksz

ks

ksq = n/a

rr z

aa

q

kz k0 kr

Figure 6.8 Cylindrical shell coordinates, structural and acoustic wavenumbers.

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one obtains, A = 0;

B = 1;

k z = ksz ;

p0 = −

jv 0 ρ0 c0 k0 (2)

m (kr a) kr d Hd(k r a)

.

(6.38)

The circumferential distribution taking the same form in Equations 6.36 and 6.37 is also a result of complying with the acoustic boundary condition. Thus, Equation 6.36 can be rewritten as p(r, θ, z) = −

jv 0 ρ0 c0 k0 (2)

m (kr a) kr d Hd(k r a)

cos mθ Hm(2) (kr r )e− jkz z

(6.39)

 where kr = k02 − k z2 . In the far field, i.e., kr r >> 1, an asymptotic solution of the radiated sound pressure can be obtained as  jv 0 ρ0 c0 k0 2 j 2 (1+ 1 ) p(r, θ, z) = − e π 2 cos mθe− jkr r e− jkz z . (6.40) (2) d Hm (kr a) π k r r kr d(kr a)

The above Equation 6.40 indicates that for a particular axial structural wavenumber ksz (= k z ), the sound radiation from an infinite length cylindrical shell only happens under the condition k0 > k z . In the range k0 < k z , the pressure is 90◦ out of phase with shell surface velocity. No sound wave is propagating into the far field and the sound pressure near the shell surface decays along the radial direction. Based on Equation 6.40, the sound power radiated into the far field per unit axial length is  2π  2π l 1 v 02 ρ0 c0 k02 = | p(r, θ, z)|2r dθdz = cos mθdθ.  (2) 2  m (kr a)  0 2ρ0 c0l 0 0 πlkr3  d Hd(k  r a) (6.41) The mean square velocity on the shell surface is given by,  2π l 
 v 02 2π 1 2 2 v = |v(θ, z)| adθdz = cos mθdθ. (6.42) 4πal 0 0 4π 0 The radiation efficiency per unit length thus has a simple and closed form as,  k0 ≤ k z 0   (2) 2  σ = (6.43)   m (kr a)  k0 > k z .  2k02 / πakr3  d Hd(k r a) Equation 6.43 clearly indicates that an infinite length cylindrical shell does not radiate sound below a certain frequency where the acoustic wavenumber matches the axial component of the structural wavenumber. In this frequency range, the acoustic wavelength is greater than structural wavelength in the axial direction, and the sound generated from the two adjacent subsections with 180◦ out of the phase movement will interfere with each other, resulting in a complete cancelation along the axial direction of the shell. Beyond this frequency range

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when the acoustic wavelength is smaller than the structural wavelength in the axial direction, the sound generated from adjacent subsections is independent of each other and the total sound radiation from the shell becomes more efficient. As frequency increases, an asymptotic value of unity can be expected as for a plane radiator. For simplicity, Equation 6.43 is sometimes employed to calculate modal radiation efficiencies of finite length cylindrical shells by setting k z equal to n π/l, which are the modal wavenumbers in the axial direction of a cylindrical shell of length l and with simply-supported boundary conditions at both ends, where n is the axial mode number. According to Equation 6.43, for each vibration mode, there exists a cut-off frequency, at which the corresponding acoustic wavenumber k0 is equal to k z = n π/l. Below this cut-off frequency, the modal radiation efficiency is zero. Above this frequency, the modal radiation efficiency increases as frequency until it reaches unity where the acoustic wavenumber k0 is equal to the structural wavenumber (n π/l )2 + (m /a )2 . This result is not obvious because if one uses infinite flat plates as an analogy, the modal radiation efficiencies  of infinite length cylindrical shells would be expected to be zero when k0 < (n π/l )2 + (m /a )2 due to the intercell cancelation. However, for cylindrical shells, the structural wave speeds in the axial and the circumferential directions have dispersion effects, the wave in the axial direction increasing faster with frequency than that in the circumferential direction due to curvature effects below the ring frequency. Therefore, the axial wave would catch up the acoustic wave much faster than the circumferential wave as the frequency increases. Hence, there is a frequency range within which the axial component of a vibration mode is supersonic while the circumferential component is still subsonic. The modal radiation efficiency would continue to increase with frequency until both components are supersonic. The frequency at which the axial component becomes supersonic is the cut-off frequency of this mode. Generally, the cut-off frequency increases with number n. Figure 6.9 shows the radiation efficiencies for a series mode associated with a simply supported shell calculated by Equation 6.43. Particularly, if the axial vibration distribution is uniform, i.e. k z = 0, Equation 6.43 gets a simpler form as [200] σ (m , k0 a ) =

=

2 2  (2)   π k0 a  d Hdm(k0(ka0)a ) 

(6.44)

(k0 a )2 [Ym (k0 a )Jm +1 (k0 a ) − Jm (k0 a )Ym +1 (k0 a )] [m Jm (k0 a ) − (k0 a )Jm +1 (k0 a )]2 + [mYm (k0 a ) − (k0 a )Ym +1 (k0 a )]2

where Jm (k0 a ) and Jm +1 (k0 a ) are the first kind Bessel functions of the m th , (m + 1)th order respectively, Ym (k0 a ) and Ym +1 (k0 a ) are the second kind of Bessel functions (Neumann functions) of the m th , (m + 1)th order, respectively. The sound radiation efficiencies of various mode number m vs. k0 a are plotted in Figure 6.10. Equation 6.44 allows for a simple calculation of the sound radiation efficiency of the pure circumferential modes m = 0, 1, 2, 3, . . ..

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Modal Radiation Efficiency

10

1

0.1 Mode (1, 2) 0.01

Mode (2, 2) Mode (2, 1)

0.001

Mode (3, 2) Mode (2, 3)

0.0001 0

0.5

1

1.5

2

k0 /ks(m, n)

Figure 6.9 Modal radiation efficiencies of a circular cylindrical shell calculated by the infinite length model l/a = 3.

6.4

Finite length cylindrical radiator

In practice most circular cylindrical structures are finite in length. As the length becomes shorter, it can be expected that the end effects would become more important, and the results obtained by using an infinite length model could be in severe error. By reviewing the literature, the techniques used in the analysis could be categorized into two groups, namely the statistical approach and the deterministic analysis. A statistical approach [150, 198] is employed when the resonant frequencies are densely packed in a frequency band, and the acoustic properties are lumped together so that individual frequency analysis becomes impractical. Obviously, this method is only valid at high frequencies or for relatively thin shells with high modal densities. The most successful work about a cylindrical source was carried out by Szechenyi [198] who developed a set of general, but approximate equations for predicting the radiation efficiencies of cylindrical shells having large length to thickness and radius to thickness ratios within a given frequency bandwidth. These results have been verified by experiments except for low frequencies [198]. In deterministic analysis, specific vibration velocity distributions and their acoustic response, which could be affected by the geometry and the boundary conditions of a cylindrical shell at each frequency, are considered. Since at low frequencies where the modal density is low, the behavior of each individual mode becomes important, the corresponding modal averaged radiation efficiency may have different characteristics from that at high frequencies. Various studies using deterministic analysis have been reported. The early contribution to the acoustic

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Modal Radiation Efficiency

2

1.5

1

0.5

m=1

m=2

m=4

m=5

m=3

0 0

1

2

3

4

5

6

7

8

9

10

k0a

Figure 6.10 Radiation efficiency factor curves according to Equation 6.44 for a cylindrical radiator and m = 0, 1, 2, . . . 5.

radiation from finite length cylindrical shells was made by Williams [238] who analyzed the radiation of finite length cylinders with a uniform radial vibration velocity profile and compared the sound field with that of the infinite length model. It was found that infinite length models may overestimate the acoustic radiation from finite length cylinders, and the error would increase as the length of the cylinder becomes shorter. Then, Schenck (1968) presented a surface Helmholtz integral formulation for obtaining approximate solutions of acoustic radiation problems for an arbitrary surface, which improved the accuracy of Williams, results in the axial direction. In order to take the reaction of the fluid inside the cylindrical shell into account, Sandman [182] examined the model of a baffled finite length cylindrical shell. It was found that the model is a reasonable approximation in determining the acoustic radiation from the surface of finite length cylindrical shells, and is applicable to arbitrary radial vibration distribution. Based on this model, Stepanishen [192] discussed the radiation impedance of different vibration modes of a finite length cylindrical shell, and Zhu [255] examined the “relative sound intensities” for different vibration modes. Also, Stepanishen [193] and Laulagnet et al. [134] investigated the effects of fluid on the acoustic radiation of finite length cylindrical shells. In this section, the sound radiation from acoustically thin shells is first introduced briefly. Then the baffled finite length cylindrical shell model is employed to calculate the modal radiation efficiencies and modal averaged radiation efficiencies for different boundary conditions. The limitation of the infinite length model and the effects of boundary conditions on the modal radiation efficiencies of finite length cylindrical shells are discussed last.

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6.4.1

147

Acoustically thin shells

The acoustically thick and thin shells are distinguished in terms of shell curvature effects on the acoustic radiation behavior. For cylindrical shells, the ring frequency is an important parameter used to indicate the frequency range in which curvature effects are important. The ring frequency fring is defined for the condition when the wavelength of extensional waves in the shell is equal to the shell circumference (compare Equation 5.21),  E 1 fring = (6.45) 2πa ρ(1 − ν 2 ) where a is the radius of the cylindrical shell, ρ is the density of the material, E is the Young’s modulus, and ν is the Poisson ratio. Above the ring frequency, the structural wavelength is so short that the structural wave propagation is very much controlled by the local stiffness. The curvature effects of the shell disappear, and the structure will behave like a plate. The sound radiation, as a result, will be governed by the compliances between the total structural wavenumber ks and the acoustic wavenumber k0 , as discussed in Section 6.2. In this frequency range, the critical frequency defined for the plate f c (Equation 6.14) is of physical significance because of no dispersion effects for the structural wavenumbers. Below the ring frequency, the curvature changes the stiffness in the axial direction, making the structural wave speed in this direction faster than that in the circumferential direction. Then the sound radiation is primarily determined by the behavior of each individual mode in the total response. The physical significance of the critical frequency defined for the plate f c becomes questionable, as will be seen in Section 6.4.2. Given these two different acoustical features, the cylindrical shells are consequently defined as acoustically thin for fring / f c < 1, and thick for fring / f c > 1 [89]. However, it should be emphasized that a shell can be geometrically thin (thickness h >>radius a) but still acoustically thick. For example, for an aluminum (Young’s modulus, E = 71 × 109 Pa, density ρ = 2700kg/m3 and the Poisson ratio, ν = 0.33) or steel (E = 210 × 109 Pa, ρ = 7850 kg/m3 and ν = 0.3) circular cylindrical shell, by using Equations 6.42 and 6.14, one obtains, fring h ≈ 67 . fc a

(6.46)

If a / h is smaller than 67, the shell would be considered to be acoustically thick but still geometrically thin. In practice, a typical example of acoustically thin shells is an airplane fuselage. For acoustically thin cylindrical shells, because of the high modal density, statistical analysis can be employed [150, 198]. By measuring the area ratio of the supersonic and subsonic regions that vibration modes possibly fall into in the wavenumber contour map, Szechenyi [198] generated a family of curves of the sound radiation efficiency for various cylindrical shell configurations, as shown in Figure 6.11. It is seen that for acoustically thin shells, that is, when

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Noise of Polyphase Electric Motors 0 −2 Radiation Efficiency, 10 log10σ

−4

0.60

−6

0.55 0.50

−8

0.52

−10

fring /fc

2.0 5 1.1 .0 1 70 0. 0 0.5 0 0.3 0 0.2 .15 0 0 0.1 5 0.0

−12 −14 −16 −18 −20

0.20

0.15 0.10

0.51

2

0.0

−22 −24 0.02 0.04

0.30

0.1

0.4

1.0

2.0

4.0

10

f/fring

Figure 6.11 Modal averaged radiation efficiency of finite length acoustically thin cylindrical shells [198]. f c > fring , the radiation efficiency of cylindrical shells has three distinguished features corresponding to three frequency ranges. Below the ring frequency, the radiation efficiency increases at a rate of 3 to 6 dB per octave to a maximum at the ring frequency. Above the ring frequency, the curvature effects are no longer important and the cylindrical shells will vibrate like flat plates. Therefore, in the frequency range between f ring and f c , the radiation efficiency would first decrease, then increase, as the frequency approaches f c in a manner similar to flat-plate radiation. Above the critical frequency, the radiation efficiency then maintains a value of unity. In the analysis, Szechenyi [198] concluded that below the ring frequency, the modal averaged radiation efficiency is only a function of h/a, independent of the length and the boundary conditions. The influence of the length and the boundary conditions for thin shells is limited to the region f ring < f < f c being in a way similar to that in flat-plate radiation. Although analytical expressions could not be given for the full frequency range, there are specific cases where analytical calculations are possible.  √ f  0.346 1 − ν 2 ffc ≤ 0.48  fring   √   √   0.241 1 − ν 2 f fring 0.48 < f ≤ 0.83 fc fring σ = 2 1/2 1/2  (ha/l ) ( f / f ) f fc c  > 1 and fring > 10   1.86π(1−ν 2 )1/4 (1− f / f c )3/2 fring    f 1 >1 fc

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(6.47)

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149

Acoustically thick shells

According to Equation 6.46, most practical cylindrical shells encountered in industries are acoustically thick, for which Szechenyi’s results cannot be applied. For acoustically thick cylindrical shells, the physical significance of f c is complex because curvature effects play an important role in determining the flexural wave speed and the acoustic radiation behavior. By analyzing the sound transmission through the pipe walls, Holmer et al. [89] showed that the radiation efficiency of an acoustically thick cylindrical shell under acoustic excitation normally increases smoothly with frequency until it reaches a constant value of about unity at high frequencies, and the coupling between the acoustic modes and the structural modes has to be considered. However, as pointed out by Fahy [55], the acoustic behavior of thick shells is strongly influenced by the nature of the excitation. For example, mechanical excitation of pipes can produce quite different radiation efficiencies from those associated with the excitation by fluid flow or sound waves in the internal fluid. The acoustic radiation of a cylindrical shell subjected to a point force excitation was studied by Laulagnet and Guyader [134] theoretically. They found that for light fluid (in which the ratio of its specific impedance to the angular frequency is negligible compared to the shell mass per unit area), the coupling between the acoustic modes and the structural modes is negligible. As an example, Figure 6.12a shows the modal averaged radiation efficiencies of an acoustically thick steel cylindrical shell, 200 mm long, 1.6 mm thick and 63.5 mm in radius, with three different boundary conditions under mechanical point excitations, calculated by the BEM. The critical and ring frequencies of this cylindrical shell are 7392 Hz and 13,190 Hz, respectively. The number of nodes and elements of the BEM model is 2480 and 2400, respectively, so that the number of elements per acoustic wavelength below 8 kHz is greater than 6. A point force was applied to the surface of the shell in the radial direction at a point 6.6 mm away from one end. It is seen that depending on the boundary conditions the radiation efficiency reaches unity at a frequency much lower than the critical frequency f c . On the other hand, Szechenyi’s results [198] for acoustically thin shells are independent of the boundary conditions. Although Szechenyi’s results display a general trend similar to the BEM results, there are significant quantitative differences. Figure 6.12b shows the results for another two steel cylindrical shells, each 3 mm thick, and 19.5 mm in radius, but 20 mm and 60 mm in length respectively, with free-free boundary condition under a point excitation. The critical and the ring frequencies for these two shells are 4036 Hz and 41.9 kHz, respectively. The number of elements for the two models is 1000 and 3000, respectively. It can be clearly seen that for a thick shell, different lengths could result in significant differences in acoustic radiation. In contrast, Szechenyi’s results, which are only valid for acoustically thin shells, are independent of the length. In order to facilitate the discussions on sound radiation from each individual vibration mode in details, a steel acoustically thick circular cylindrical shell,

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Modal Averaged Radiation Efficiency

100

10

1 fc

0.1 Simply-Supported

0.01

Clamped-Clamped Free-Free

0.001

Szechenyi’s result 0.0001

0

1000

2000

3000 4000 5000 Frequency (Hz)

6000

7000

8000

(a)

Modal Averaged Radiation Efficiency

10

1

0.1

0.01 BEM results, l = 20 mm 0.001

BEM results, l = 60 mm Szechenyi’s result

0.0001

0

1000

2000 3000 Frequency (Hz)

4000

5000

(b)

Figure 6.12 Modal averaged radiation efficiency of acoustically thick cylindrical shells calculated by BEM: (a) a = 63.5 mm, h = 1.6 mm, l = 200 mm; (b) a = 19.5 mm, h = 3 mm. ◦◦◦ Equation 6.70, l = 20 mm;  Equation 6.70 l = 60 mm [227].

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200 mm long, 1.6 mm thick, and 63.5 mm in radius, is chosen as an example. A simplified frequency-wavenumber characteristic equation for a cylindrical shell with neglecting in-plane deflections [50] is given by the following equation [169]: ω2 =

4 D K (1 − ν 2 ) ksz 2 · (ksz + ks2θ )2 + · 2 + k 2 )2 ρh ρha 2 (ksz sθ

(6.48)

where D = Eh 3 /[12(1 − ν 2 )], K = Eh/(1 − ν 2 ). It was shown [226] that this relationship is independent of the boundary conditions. For finite length cylindrical shells with different boundary conditions at both ends, the axial component of the structural wavenumber ksz takes different forms as discussed by Wang and Lai [226]. Based on Equation 6.48, a series of wavenumber curves associated with different frequencies obtained are presented in Figure 6.12. It can be clearly seen from Figure 6.13a that, unlike flat plates, the vibration modes of shells  having

2 + k2 ; the same natural frequencies could have different wavenumbers ks = ksz sθ e.g., the wavenumbers at 13 kHz in the graph. Only above the ring frequency would the wavenumbers at the same frequency approach the same value. This phenomenon is due to the curvature effects which modify the wave speed in the axial direction below the ring frequency. To illustrate curvature effects on sound radiation, three structural wavenumber curves and three corresponding acoustic wavenumber curves are plotted in Figure 6.13b. From the acoustic point of view [55], the vibration modes could be categorized to be acoustically fast modes and acoustically slow modes, sometimes called supersonic and subsonic modes. Acoustically fast modes refer to those of which the structural wavenumbers are smaller than the corresponding acoustic wavenumbers, and the modal radiation efficiencies of these modes are unity. Acoustically slow modes are inefficient in acoustic radiation because structural wavenumbers are greater than the corresponding acoustic wavenumbers, which would lead to some cancelation in the radiation. For isotropic and flat plates [55, 37], the unique demarcation for these two cases is the critical frequency. However, for cylindrical shells, from Figure 6.13b, it can be seen that, below the critical frequency of 7.33 kHz, the structural wavenumber curve will always intersect the acoustic wavenumber curve, such as point A, at a given frequency. This point of intersection in the wavenumber domain changes as the frequency changes. When the frequency is the critical frequency, the curves are tangent to each other as shown by point B in Figure 6.13b. When the frequency is greater than the critical frequency, the two curves will no longer meet. This result indicates that at any frequencies below the critical frequency, acoustically fast modes and slow modes could exist simultaneously, and the demarcation depends on the frequency. Therefore, for cylindrical shells, it is impossible to define a unique “critical frequency” for describing the acoustic properties as for flat plates. Figure 6.13b shows that the critical frequency defined for plates indicates the condition for all possible vibration modes of cylindrical shells to be supersonic. Thus, the radiation efficiency of cylindrical shells should be unity above the critical frequency because the modal radiation efficiencies of all modes are unity.

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200

19 kHz 16 kHz 150 kzn

13 kHz (ring frequency)

100

ks

9 kHz ks

6 kHz

50

ks

3 kHz 0

0

50

100

200

150

250

kθm (a) 200 Acoustic Wavenumber f = 9 kHz 7.33 kHz (critical frequency)

150

kzn, k

Structural Wavenumber

100 9 kHz

5 kHz

f = 7.33 kHz

50

A 5 kHz

0

B 0

50

100 kqm, k

150

200

(b)

Figure 6.13 Wavenumber diagram for a cylindrical shell: kθm — circumferential structural wavenumber; k zn — axial structural wavenumber; k — acoustic wavenumber [227].

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Below the critical frequency, since both supersonic and subsonic modes could exist simultaneously, the overall radiation efficiency could depend on the number and the types of modes dominating the vibration response. For example, if the mode is supersonic, the modal radiation efficiency is unity, and if it is subsonic, the modal radiation efficiency is less than one. Three points associated with the determination of the radiation efficiency below the critical frequency can be concluded here. First, the demarcation for each mode, i.e. whether the mode is supersonic or not, has to be determined. Second, the modal averaged radiation efficiency could be unity if supersonic modes dominate the response. Third, for a given cylindrical shell, the modal averaged radiation efficiency is dependent on the excitation and boundary conditions because the excitation determines the modal participation amplitudes and any changes of boundary conditions could make some supersonic modes subsonic or vice versa. By comparison, for flat plates, the boundary conditions are not likely to fundamentally affect the contribution of the vibration modes because statistically, the dominated modes occurring below the critical frequency are basically subsonic and the modes above the critical frequency are always supersonic [55, 37]. Therefore, it can be expected that the variation of the modal averaged radiation efficiency of cylindrical shells due to the change of boundary conditions could be substantial compared with flat plates. To explain the results of sound radiation efficiencies shown in Figure 6.12, the behavior of each individul mode of the cylindrical shell has to be examined. The natural frequencies of the cylindrical shell can be calculated based on the method described in [226]. Corresponding to each natural frequency, there must exist a structural wavenumber curve to which the vibration mode belongs and an acoustic wavenumber curve of the same frequency, as shown in Figure 6.13b. Note that at each point of the intersection of the two curves, the structural wavenumber is equal to the acoustic wavenumber. So one can compare the structural wavenumber of this mode ks with the acoustic wavenumber k0 to determine whether this mode is supersonic or not. For this purpose, an index is defined as follows, ωmn  2 L (m , n ) = k0 − ks = − ksz (n ) − (m /a )2 . (6.49) c0 Obviously, this index depends on the natural frequency ωmn and the structural wavenumbers of the vibration mode. If L = 0, the acoustic wavenumber is greater than the structural wavenumber, indicating that the mode is supersonic, and if L < 0, the mode is subsonic. In Figure 6.14, L of all the vibration modes associated with three different boundary conditions (simply supported, free, and clamped) is plotted against their own natural frequencies. It can be seen that the critical frequency ( f c ) of the equivalent flat plate is an indicator showing whether all the vibration modes of the cylindrical shell are supersonic or not. Below the critical frequency, corresponding to each supersonic mode in Figure 6.14, there is a peak in the modal averaged radiation efficiency in Figure 6.12a, for example, the peaks around 2300 Hz for simply supported and clamped conditions, 2800 Hz for free condition. According to Figure 6.14, although subsonic modes exist below the critical frequency, the number of supersonic modes increases as the frequency

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DL (m−1)

60 40 20 0 −20

fc

−40 −60

0

1000 2000 3000 4000 5000 6000 7000 8000 fmn(Hz)

Figure 6.14 The index L of each vibration mode of the cylindrical shell: + + + simply supported; • • • free–free; ✷✷✷ clamped–clamped [227]. increases, so that the modal averaged radiation efficiency could reach unity at a frequency much lower than the critical frequency, as shown in Figure 6.12a.

6.4.3

Modal radiation efficiencies of acoustically thick shells

For acoustically thick shells, the modal radiation behavior of a particular mode could dominate the overall performance. The estimation of modal radiation efficiencies of cylindrical shells thus becomes necessary. Although the infinite length model could be employed for this purpose, errors will be introduced due to ignoring the end effects. To take the finite length into account, a model of an infinite length circular cylindrical shell with finite length vibration distribution, as shown in Figure 6.15, is adopted here. In the model, the sound field inside the cylindrical shell and the effects of internal sound field on the structural vibration and the outer x r q

a

z

y l

Figure 6.15 Finite length cylindrical shell model.

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sound field are not considered. For cylindrical shell structures, this may be acceptable when the shell is excited mechanically and the fluid inside the cylindrical shell is light, such as air. As a result, the surface normal displacement of each vibration mode can be written as, (omitting the time component e jωt )

u mn (θ, z) =

u 0 γn (z) cos mθ 0 < z < l 0 l
(6.50)

where u mn is the modal amplitude in the radial (r ) direction for mode (m, n). Here the mode shape in the axial direction γn (z) depends on the boundary conditions at z = 0 and l. Note that generally the vibration and corresponding acoustic solutions in the two orthogonal directions (z and θ ) can be treated separately and that the vibration displacement of a circular cylindrical shell in the circumferential direction (θ) always takes the cosine form. Then by transforming γn (z) to the wavenumber domain [55, 37],  n (k z ) =

l

γn (z)e jkz z dz

(6.51)

0

where n (k z ) and u mn (θ, z) satisfies  u0 ∞ u mn (θ, z) = n (k z )e− jkz z dk z cos mθ 2π −∞

(6.52)

The above two Equations 6.51 and 6.52 indicate that the finite vibration distribution can be decomposed into a number of waves propagating along an infinite length cylindrical shell with different wavenumbers and different amplitudes and phases. Therefore, the present problem is reduced to that of the sound radiation from an infinite length cylindrical shell with the vibration displacements being of the form u 0 n (k z )e− jkz z cos mθ. Associated with each vibration component, the sound pressure produced in the free field is, Pmn (k z z) = AHm(2) (kr r ) cos mθe− jkz z

(6.53)

and the corresponding acoustic particle velocity is Vmn (k z z) = −

j Akr d Hm(2) (kr r ) 1 ∂ Pmn (k z z) = cos mθe− jkz z jωρ0 ∂r ωρ0 d(kr r )

(6.54)

 where kr = k02 − k z2 , and A is a constant depending on the acoustic boundary conditions. By applying the boundary condition at the structure surface, the sound pressure on the surface of the cylindrical shell associated with each wave component is given by the following equation [55] Pmn (k z z)|r =a =

ω2 ρ0 u 0 Hm(2) (kr a) n (k z ) cos mθe− jkz z . d Hm(2) (kr a) kr d(kr a)

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(6.55)

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By transforming it back to the time domain, the sound pressure produced by the vibration mode (m, n) on the vibrating surface can be obtained pmn |r =a =

1 2π



∞

ω2 ρ0 u 0 Hm(2) (kr a) n (k z )e− jkz z dk z cos mθ. d Hm(2) (kr a) kr

−∞

(6.56)

d(kr a)

Then the acoustic power radiated by this vibration mode is found as  2π l 1 mn (ω) = e − jωpmn u ∗mn adzdθ (6.57) 2 0 0    2π l  ∞ 3 (2) 1 − jω ρ u H (k a) 0 0 r m  = e n (k z )e− jkz z dk z cos mθ · d Hm(2) (kr a) 8π 2 kr 0 0 −∞ d(kr a)  ∞     · u 0 n∗ (k z )e− jkz z dk z cos mθ adzdθ −∞

Taking the integration over z first, Equation 6.57 can be further reduced to    2π   ∞ 3 (2) − jω ρ u H (k a) 1 0 0 r m 2 2   mn (ω) = e |n (k z )| dk z cos mθadθ . d Hm(2) (kr a) 4π kr −∞ 0 d(kr a)

(6.58) Since in practice the sound power is expected to be real, using the relationship Jm (x)

2 d Nm (x) d Jm (x) − Nm (x) = dx dx πx

where Jm (x) and Nm (x) are the Bessel and Neumann functions respectively, Equation 6.58 becomes  mn (ω) =

k0

−k0

ω3 ρ0 u 20 |n (k z )|2 (2) m (kr a) 2 π kr2 | d Hd(k | r a)

dk z ×

1 n=0 1/2 n =  0.

(6.59)

From Equation 6.59, it can be seen that if the mode shapes in the axial direction (which depend on the boundary conditions) are known, the corresponding acoustic power radiated by each vibration mode can be predicted. From the assumed vibration displacement for the vibrating surface, the spatial averaged mean square velocity for mode (m, n) can be determined from


 ω2  2π  l 2 = v mn u 20 |γn (z)|2 cos2 mθadzdθ 2S 0 0

2 2  l π ω u0 1 n=0 2 = |γn (z)| adz × 1/2 n =  0 S 0

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where S is the surface area of the finite length cylindrical shell. Then the modal radiation efficiency of finite length cylindrical shells can be found as σmn (ω) =

mn (ω) 1 k
 = 02 ·  l 2 π 2 ρ0 c0 S v mn 0 |γn (z)| adz



|n (k z )|2

k0

−k0

(2)

m (kr a) 2 kr2 | d Hd(k | r a)

dk z . (6.61)

In Equation 6.61, modal radiation efficiencies depend on the mode shapes in the axial direction. According to [226], the mode shapes of cylindrical shells in the axial direction can be described with good approximations by using the beam functions. As Equation 6.61 involves the ratio of two integrals, it is unlikely that a simple expression like that for infinite cylindrical shells can be obtained. However, for cylindrical shells simply supported at two ends, the mode shapes in the axial direction are given exactly by the following equation: γn (z) = sin

nπ z l

(6.62)

Using Equation 6.62, the parameter n (k z ) can be found as n (k z ) =

nπ (−1)n e jkz z − 1 · 2 . l k z − (nπ/l)2

(6.63)

Substituting Equations 6.62 and 6.63 into Equation 6.61, one obtains  σmn (ω) =

k0

−k0



2k0l π 2 akr2 |

d Hm(2) (kr a) d(kr a)

|2

nπ/l k z + nπ/l

2

sin2 [(k z − nπ/l)l/2] dk z . [(k z − nπ/l)l/2]2 (6.64)

Generally, Equations 6.61 and 6.64 have to be solved numerically.

6.4.4

Modal averaged radiation efficiency

Although the modal radiation efficiencies of a finite length cylindrical shell have been obtained above, in practice it is the modal averaged radiation efficiency, rather than the modal radiation efficiency, that relates the total radiated sound power and the vibration levels over the structural surface. Generally, to obtain the modal averaged radiation efficiency, one has to work out the vibration distribution on the structure due to specific excitations. According to Soedel (1993) if the distributed loads on the cylindrical shell in the three directions are assumed as Fz e jωt , Fθ e jωt , Fr e jωt (N/m2 ), the steady-state harmonic vibration velocity response of mode (m, n) in the radial direction (r ) is v mni (θ, z) =

jωFmni Umnri (θ, z) 2 − ω2 ) + 2 jζmni ωmni

2 (ωmni

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where Fmni =

1 ρh Nmni



2π



0

l

(Fz Umnzi + Fθ Umnθi + Fr Umnri )adzdθ

0



Nmni =

2π

0

 0

l

2 2 2 (Umnzi + Umnθi + Umnri )adzdθ.

In here, subscript i denotes that there are three sets of solutions corresponding to the excitations in the three directions respectively. For every m, n combination, there are three sets of natural frequencies ωmni (i = 1, 2, 3) corresponding to the three vibration modes (longitudinal, torsional, and flexural); ζmni is the corresponding modal damping ratio of the three vibration modes, and Umnzi , Umnθi , Umnri are the mode components of the longitudinal, torsional, and flexural vibrations of the cylindrical shell, respectively (see [190] for details). Normally these mode shapes have the forms Umnzi (θ, z) =

Amni αn (z) cos mθ Cmni

Umnθi (θ, z) =

Bmni βn (z) sin mθ Cmni

(6.66)

Umnri (θ, z) = γn (z) cos mθ where αn (z), βn (z), and γn (z) are the mode shapes of the three vibrations along the axial direction (z) of the cylindrical shell, Amni /Cmni , and Bmni /Cmni are the coupling ratios between the longitudinal and flexural vibrations, and between the torsional and flexural vibrations respectively, as given in [190]. For simplysupported ends, αn (z), βn (z), and γn (z) have exact solutions as αn (z) = cos

nπ z nπ z nπ z , βn (z) = sin , γn (z) = sin , n = 1, 2, 3, · · · (6.67) l l l

For other boundary conditions, αn (z), βn (z), and γn (z) can be approximately obtained from a longitudinally vibration beam, and a transversely vibrating beam of the same corresponding boundary conditions respectively [190]. For example, the mode shapes of flexural vibration for free-free ends are [135] γ0 (z) = 1,

γn (z) = cosh(k zn z)+cos(k zn z)−

k zn ≈

γ1 (z) =

1 z − l 2

(6.68)

cosh(k zn l) − cos(k zn l) [sinh(k zn z)+sin(k zn z)] sinh(k zn l) − sin(k zn l)

π nπ − , l 2l

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n = 2, 3, 4, · · ·

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and for clamped-clamped ends are, γn (z ) = cosh(k zn z )+cos(k zn z )−

cosh(k zn l ) − cos(k zn l ) [sinh(k zn z )+sin(k zn z )] sinh(k zn l ) − sin(k zn l )

π nπ − , n = 1, 2, 3, · · · (6.69) l 2l Since the relationships between the natural frequencies and the wavenumbers for the three vibrations are independent of the boundary conditions, therefore so are the coupling ratios. For all the boundary conditions, the coupling ratios obtained from simply-supported conditions can be directly used by substituting the appropriate natural frequencies. If only Fr exists, although vibrations in the three directions would be excited, the flexural vibration would dominate the vibration response (i=3). By using the definition of modal averaged radiation efficiency, the modal averaged radiation efficiency of the cylindrical shell can be obtained from,
  ∞ ∞ σmn |Fmn3 |2 Nmn3 2 m ,n σmn v mnr m n (ω2 −ω2 )2 +4ζ 2 ω4 mn3 mn3 mn3  = ∞ ∞ σ (ω) = 
. (6.70) |Fmn3 |2 Nmn3 2 2 m n (ω −ω2 )2 +4ζ 2 ω4 m ,n v mnr mn3 mn3 mn3 k zn ≈

With Equation 6.70, the modal averaged radiation efficiencies of the steel circular cylindrical shells discussed in Section 6.3.2 have been calculated and compared with the BEM results in Figure 6.16 and 6.12b. In the BEM calculation, the modal damping was set as 0.001. It can be seen that for simply-supported, and free-free conditions, the results obtained from Equation 6.70 and BEM agree reasonably well. For clamped-clamped condition, at low frequencies the curve obtained by Equation 6.70 is shifted slightly to higher frequencies. This is because the natural frequencies predicted by using the beam function are higher than the numerical results [226]. Nevertheless, Figures 6.16 and 6.12b indicate that Equation 6.70 can predict the modal averaged radiation efficiency of circular cylindrical shells with reasonable accuracy. In Figure 6.16b, the BEM and analytical results by Equation 6.70 are compared to measured modal averaged radiation efficiency. Equation 6.70 clearly indicates that the contribution of each vibration mode to the response and thus the modal averaged radiation efficiency depends on the external excitation. In principle, different modal averaged radiation efficiencies would be expected for different type of excitations, such as mechanical and acoustic excitations, and even for different load distributions such as point and surface forces. It seems, therefore, impossible to obtain a unique modal averaged radiation efficiency for a given cylindrical shell. However, if all the vibration modes are excited in a structure, at each natural frequency, the vibration response is usually dominated by the vibration mode corresponding to this frequency. Thus, the modal averaged radiation efficiency at ω = ωmn may be approximated by the value of the modal radiation efficiency of mode (m, n) at ω = ωmn , as shown in Equation 6.71. σ |ω=ωmn ≈ σmn (ωmn ) (6.71)

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Modal Averaged Radiation Efficiency

100 10

1

0.1

0.01

BEM Equation 6.70

0.001

0.0001

Equation 6.71

0

1000

2000

3000 4000 5000 Frequency (Hz)

6000

7000

8000

(a) 10

Radiation Efficiency

1

0.1

0.01

Measured

0.001

BEM Equation 6.70

0.0001

0

500

1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency (Hz) (b)

Figure 6.16 Comparison of analytically calculated radiation efficiency with BEM and measured results for a = 63.5 mm, h = 1.6 mm, l = 200 mm: (a) simply supported cylindrical shell; (b) free–free cylindrical shell [227].

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Since Equation 6.71 only presents discrete values of modal averaged radiation efficiencies at each natural frequency, it can be expected that the larger the separation between the two adjacent natural frequencies, the more information will be lost in that region. Therefore, the accuracy of Equation 6.71 would improve if the modal density is high. By using Equations 6.61 and 6.71, the modal averaged radiation efficiencies of the cylindrical shell model with three different boundary conditions are calculated at each natural frequency and plotted in Figure 6.16. It is seen that except at low frequencies, say below 4000 Hz, where the number of vibration modes is less, the results obtained by Equation 6.71 are good approximations. However, if the structure is excited coincidentally so that only one vibration mode exists in a wide frequency range, Equation 6.71 may incur some errors because the assumption that each mode would dominate the response at its natural frequency is not valid. In this case, the modal averaged radiation efficiency (σ (ω)) would be equal to the modal radiation efficiency (σmn (ω)) of that particular mode. For example, in the frequency range below the first natural frequency of the shell, the response is very much determined by the corresponding vibration mode. For a very thick shell such as the stator core, the first natural frequency could be relatively high so that the modal radiation efficiency of the first mode could cover a wide frequency range at low frequencies.

6.4.5

Validity of using an infinite length model

As discussed previously, the modal radiation efficiency is the most elementary factor describing the acoustic radiation properties. Though normally it does not indicate the overall performance of a practical structure directly, the corresponding computations are essential in the sound radiation analysis. Particularly as shown in Equation 6.71, the modal averaged radiation efficiency even can be represented approximately by the modal radiation efficiency at each natural frequency. Equation 6.61 presents a general expression for the modal radiation efficiency of a cylindrical shell. In engineering, however, the infinite length model is often adopted in the computation, simply because analytical solutions are available (Equation 6.43) and easy to apply. As a result, it would be beneficial to discuss the condition for which the results of infinite length models can be employed with reasonable accuracy. Note that the modal radiation efficiency generally depends on the boundary conditions. For convenience, a finite length shell with two ends simply supported is considered in this section. Figure 6.17 shows some modal radiation efficiencies of various vibration modes of a finite length cylindrical shell with both ends simply supported. It is seen that, like an infinite length cylindrical shell, all vibration modes would become supersonic when the acoustic wavenumber is greater than the structural wavenumber, and the corresponding modal radiation efficiency is of the order of unity. When the vibration modes are subsonic, the modal radiation efficiencies are less than one. This result supports the use of the index L for determining the acoustic behavior of each vibration mode. However, by comparing to Figure 6.9, it is found that differences in the modal radiation efficiencies between the finite

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and the infinite length models are substantial, especially in the subsonic region in which the finite length model generally gives higher values as expected. In order to find out a condition under which a finite length cylindrical shell could be treated by using an infinite length model, the modal radiation efficiencies calculated using Equation 6.64 are compared with those obtained using Equation 6.43 in Figure 6.18a through 6.18e for five different modes (1,2), (3,2), (5,2), (3,4), and (5,4) respectively. It can be seen that for low l /a, severe errors would occur from applying the results of infinite length model, especially in the region near k0 /ks ≤ 1. This result can be interpreted by analogy with finite/infinite plates. For finite length cylindrical shells, there are no cut-off frequencies because the intercell cancelation does not occur at the ends. Thus, the radiation efficiency of a finite length model is higher than that of the infinite model below and around the cut-off frequency for the infinite model. Around the demarcation of subsonic and supersonic modes, as the acoustic wave speed and structural wave speed are almost the same, there should be more energy accumulated along the infinite length cylindrical shell than that along the finite length model just like for flat plates [55]. Consequently, the radiation efficiency of an infinite length model is higher than that of a finite length model in that region. All these results in Figure 6.18 show that the larger the length/radius ratio, the smaller the difference between the two results given by Equations 6.43 and 6.64. Furthermore, the length/radius ratio for good agreement between infinite and finite length shells decreases as the circumferential mode order increases, and as the axial mode order decreases. Thus, the condition

10

Modal Radiation Efficiency

1

0.1

Mode (2, 1)

0.01

Mode (3, 1) 0.001

Mode (2, 2) Mode (2, 3)

0.0001

Mode (4, 3) 0.00001

0.1

1

10

k0/ks (m, n)

Figure 6.17 Modal radiation efficiencies of a circular cylindrical shell for l/a = 3.

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1

0.1 Finite Model, l/a = 10 Infinite Model, l/a = 10 Finite Model, l/a = 20

1

0.1 Finite Model, l/a = 3 Infinite Model, l/a = 3

0.01

Finite Model, l/a = 7 Infinite Model, l/a = 7

Infinite Model, l/a = 20 0

0.5

1

1.5

0.001

2

0

0.5

k/ks (m, n)

1.5

2

k/ks (m, n)

10

10

(d) Mode (3, 4) Modal Radiation Efficiency

(c) Mode (5, 2) Modal Radiation Efficiency

1

14:37

0.001

1

0.1 Finite Model, l/a = 2 Infinite Model, l/a = 2

0.01

Finite Model, l/a = 4

1

0.1 Finite Model, l/a = 7 Infinite Model, l/a = 14

0.01

Finite Model, l/a = 7 Infinite Model, l/a = 14

Infinite Model, l/a = 4 0.001 0.0

0.5

1.0 k/ks (m, n)

1.5

2.0

0.001 0.0

0.5

1.0

1.5

2.0

k/ks (m, n)

163

Figure 6.18 Comparisons of the modal radiation efficiencies of infinite and finite length cylindrical shells: (a) mode (1, 2); (b) mode © 2006 Francis LLC (3,Copyright 2); (c) mode (5, Taylor 2); (d) &mode (3,Group, 4) [228].

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(b) Mode (3, 2)

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(a) Mode (1, 2)

0.01

Acoustic Calculations

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for applying the results of an infinitelength model to a finite length cylindrical shell can be stated as [228]. l m · ≥ constant (6.72) n a Obviously, the larger the constant, the closer the two results. Based on Figure 6.18, this constant can be chosen as 10. As Equations 6.43 and 6.64 indicate, the modal radiation efficiencies are only influenced by the length l, radius a, and mode numbers m, n, the conclusion that (l /n ) · (m /a ) ≥ 10 should apply to all circular cylindrical shells. Generally, when n increases, the areas of the “edges” and “corners” would decrease so that the radiation efficiency of a finite structure would decrease approaching the results of an infinite length model. However, for finite length cylindrical shells, increasing n not only would reduce the radiation efficiency, but also would increase the cut-off frequency of the corresponding infinite length model. Thus, the differences between the results obtained using an infinite length model (Equation 6.43) and a finite length model (Equation 6.72) become greater as n increases. In fact, Equation 6.72 basically implies that when the wavenumber in the circumferential direction is much greater than that in the axial direction, the acoustic behavior of a cylindrical shell is mainly determined by the wavenumber in the circumferential direction. Under this condition, the length of a cylindrical shell is no longer important, and the results of the two models would tend to be the same.

6.4.6

Effects of boundary conditions on the radiation efficiency

Equation 6.61 shows that the modal radiation efficiency of a finite length cylindrical shell depends on the Fourier transform of the mode shape in the axial direction, which indicates that boundary conditions at both ends of the cylindrical shell would have effects on the modal radiation efficiencies. In Figure 6.19, modal radiation efficiencies obtained by using Equation 6.61 for clamped and Equation 6.64 for simply supported ends for modes (2,1), (2,2), (2,4), and (8,1) have been compared. It is seen that the modal radiation efficiencies of the clamped-clamped shell are normally lower than those of the simply supported shell in the region k0 /ks (m, n) < 1, which is due to the effect of near fields at the two ends. Generally for a clamped end, the vibration amplitude near the end would be smaller than that for a simply supported end because of the additional zero bending moment condition. As a result, in the subsonic region, the acoustic power radiated from a clamped-clamped shell, which is actually due to the radiation from the cells at both ends, would be smaller than that from a simply-supported shell, thus resulting in lower radiation efficiency. Furthermore, it can be observed that when m increases, the differences between the two boundary conditions increase. This is because as m increases, the intercell cancelation for simply supported condition is more complete so that the end effects for clamped condition become more important. In the region k0 /ks (m, n) ≥ 1 , all the modal radiation efficiencies approach unity as expected. Here, although only the clamped-clamped and simply-supported boundary conditions are examined, they have been demonstrated to remarkably

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Modal Radiation Efficiency

2.0

1.5

1.0

Simply-Supported, Mode (2, 1) 0.5

Clamped, Mode (2, 1) Simply-Supported, Mode (8, 1) Clamped, Mode (8, 1)

0.0 0.0

0.5

1.0 k/ks (m, n)

1.5

2.0

(a)

Modal Radiation Efficiency

2.0

1.5

1.0

Simply-Supported, Mode (2, 2) 0.5

Clamped, Mode (2, 2) Simply-Supported, Mode (2, 4) Clamped, Mode (2, 4)

0.0 0.0

0.5

1.0

1.5

2.0

k/ks (m, n) (b)

Figure 6.19 Comparisons of modal radiation efficiencies of simply supported and clamped-clamped cylindrical shells for l/a = 1.5: (a) mode (2, 1) and mode (8, 1); (b) mode (2, 2) and mode (2, 4) [228].

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affect the acoustic behavior of a cylindrical shell. Hence, Equation 6.61 generally has to be used for finite length cylindrical shells with appropriate beam functions. It is also observed from Figure 6.19 that for the clamped-clamped boundary condition, and k0 /ks (m, n) < 1, the modal radiation efficiency decreases as m increases. This is because modes with small m are normally more efficient in acoustic radiation than those with larger m. It has been argued in Section 6.3.5 that provided that (l/n) · (m/a) ≥ 10, the circumferential waves would dominate the overall acoustic performance and the length of a cylindrical shell is not so important. Since the boundary conditions at both ends of a cylindrical shell only affect the wavenumber (k z ) in the axial direction, the effects of boundary conditions can be neglected provided that (l/n) · (m/a) ≥ 10. In Figure 6.19a, the radiation efficiencies of mode (8,1), which satisfies Equation 6.72 for simply-supported and clamped boundary conditions, are plotted. It can be seen that the two curves agree with each other well. Thus, it can be concluded that Equation 6.72 is also a condition for which the effects of boundary conditions are not important for modal radiation efficiencies. It should be pointed out that although sometimes the effects of boundary conditions on modal radiation efficiencies can be neglected, the effects of boundary conditions on the modal averaged radiation efficiency cannot be neglected. According to Equation 6.70 the difference of natural frequencies due to different boundary conditions would also cause some difference in the modal averaged radiation efficiency.

6.5

Calculations of sound power level

The sound power radiated from a structure can be calculated directly from a vibroacoustic analysis by solving the vibration/acoustic motion equations and boundary conditions. For machinery structures, a thorough analytical vibro-acoustic analysis may not be possible. But the finite element method/boundary element method can be generally applied for low frequency solutions. In engineering, an alternative approach is to employ the sound radiation efficiency. With a known sound radiation efficiency, the calculation of sound power from a vibrating structure only requires vibration levels on the structure surface as the input. The spatial averaged mean square velocity can be either calculated or measured. Since the sound radiation efficiency generally requires a uniform vibration distribution and only simple geometrical structures have solutions for their sound radiation efficiencies, a machinery structure may have to be substructured into several components. The assumption involved in this substructuring scheme is that the sound radiation from each component is statistically independent. This means errors may be introduced at low frequencies where the interference between sound waves from each component may occur. In this section, a simple example of calculating the sound power radiated from a structural component, a stator, at a particular frequency is given first. Then the calculation of the total sound power of electromagnetic noise from an induction motor over a wide frequency range, based on the sound

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radiation efficiencies of simple structure components, is described. The sound power radiated from permanent magnet brushless motors is discussed last.

6.5.1

Sound power radiated from a stator

Sound radiated from the stator is of particular interest when the frequency of space/time harmonics in the electromagnetic force is close to a natural frequency of the stator vibration modes. At this frequency, the vibration behavior of the stator is dominated by this particular mode, and the sound radiated from this mode plays an important role in the total sound power. In this case, the modal radiation efficiency can be directly used in the calculation of the total radiated sound power according to the discussion made in Equation 6.71. Based on Equation 6.1, the radiated sound power from the outer surface of the stator for mode m can be calculated from the following equation, ω Am  ≈ m = ρo co ( √ )2 σm S f (6.73) 2 where S f is the outer surface of the stator, and Am is the radial vibration displacement given by Equation 5.1 or 5.4. The modal radiation efficiency of the stator core σm is expressed generally by Equation 6.61. As an example, a 7.5 kW PM brushless motor with its frame outer diameter D f = 0.248 m and frame length L f = 0.242 m has been considered. An electromagnetic force harmonics at fr = 2792.2 is the concern, and this harmonics excites the circumferential mode m = 2, n = 0 of the stator, making the primary contribution to the total sound power radiated from the motor. The amplitude of surface vibration displacement at this frequency has been determined as 0.8 × 10−6 mm. In calculating the radiated sound power, the air density ρo = 1.188 kg/m3 , sound speed in the air co = 344 m/s, both at 200 C and 1000 mbar. The angular frequency is ω = 2π f = 2π ×2792.2 = 17, 543.91 radians. The external surface of cylindrical frame S f = π D f L f = π 0.248 × 0.242 = 0.1885 m2 . The modal radiation efficiency σm =2,n =0 at this frequency can be calculated by Equation 6.61, which is a general expression for the finite length cylindrical frame. However, based on Equation 6.72, the infinite length model, Equation 6.44, can be used instead. The wavenumber k0 = ω/c0 = 17, 543.91/344 = 51m −1 , the radius of the outer frame surface a = D f /2 = 0.248/2 = 0.124 m and the product k0 a = 51 × 0.124 = 6.324. Either from Figure 6.10 or Equation 6.44, the sound radiation efficiency for this mode at ka = 6.324 is found as σ ≈ 1.2. The radiated acoustic power according to Equation 6.73    2 ω Am 2 17, 543 × 0.8 × 10−9 √  = ρ o co √ σm S f = 1.188×344 ×1.2×0.1885. 2 2 = 9.104 × 10−9 W

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3

2

1

4

(b)

(a)

Figure 6.20 A 2.2 kW induction motor: (a) general view; (b) casing and base plate. 1, 2, and 3 — three portions of the casing in which 2 is attached to the stator, 4 — supporting base plate. The sound power level according to Equation A.27 L w = 10 log10

6.5.2

9.104 × 10−9 = 39.59 dB. 10−12

Total sound power of an induction motor

Basic electromagnetic components of an induction motor are the stator and rotor. But for a vibro-acoustic analysis, all parts that may contribute to total sound power should be considered. The motor studied here is a three-phase, 50 Hz, 2.2 kW induction motor manufactured by Fasco, Australia. At rated output, the supply voltage, current and the speed are 415 V, 4.6 A and 1500 rpm, respectively. The motor assembly is shown in Figure 6.20. It has a rotor, a stator which was pressed fit into an outer casing, two end-shields, and a supporting base plate. This motor was driven by a pulse width modulation (PWM) inverter for variable speed control. If only the noise due to the electromagnetic origin is of interest, the overall sound power can be evaluated by employing the sound radiation efficiencies of the primary acoustic radiators, which are the outer surface of the casing and the base plate in this case, together with measured spatial averaged mean square velocities because the electromagnetic noise is due mainly to the vibration of the motor structure, in which the stator is subject to the electromagnetic force wave. For this purpose, the 1/3 octave noise spectra in the audible frequency range is usually evaluated. The calculation of sound radiation efficiency of the base plate is straightforward by using the expressions for the unbaffled plate. The dimension of the base plate (aluminum plate) was 0.17 m × 0.19 m × 0.01 m. Therefore, Equation 6.32 can be employed in the frequency below 1.35 kHz. Since the sound radiation efficiency reaches unity around 1.3 kHz, the high frequency radiation efficiency thus can be easily determined as unity, as shown in Figure 6.21. The outer casing, however, has to be divided into three sections for the sound radiation efficiency calculations because the modal testing of this motor structure

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Sound Radiation Efficiency

10 1 0.1

0.01

Casing-1 Stator

0.001

Casing-2 Base Plate

0.0001 0

2000

4000

6000

8000

10000

Frequency (Hz)

Figure 6.21 Sound radiation efficiencies of the four substructures [233].

has shown that the outer casing behaves differently in vibration for the portion attached to the stator and the two sections on both sides [223]. For each portion, the technique described in Section 6.4 can be applied. Note that the calculation of the modal averaged radiation efficiency generally requires the knowledge of the force distribution which, however, may not always be available. A point force acting on the structure, for which the response is averaged over all excitation locations, may be used to simulate the random force excitations. Equation 6.70 thus can be simplified as [37], ∞ ∞ 2 2 2 2 4 −1 m n σmn [(ωmn3 − ω ) + 4ζmn3 ωmn3 ]  σ (ω) ≈  . (6.74) ∞ ∞ 2 2 4 2 2 −1 m n [(ωmn3 − ω ) + 4ζmn3 ωmn3 ] In Equation 6.74, the modal averaged radiation efficiency only depends on the modal radiation efficiencies and the natural frequencies, implying that the force components associated with each vibration mode are equal at any frequencies. For induction motors, this might be acceptable only for frequency band averaged results, such as 1/3 octave frequency bands. Figure 6.22 shows the modal averaged radiation efficiencies of the casing calculated by Equation 6.74. In the calculations, the stator was treated as a circular cylindrical shell and the boundary conditions at both ends were assumed to be simply supported. The vibration measurements were taken in situ. The motor was running at 450 rpm and 1500 rpm with no load. An accelerometer was moved around over the surface of the casing and the base plate to record the vibration levels at a total of 130 positions (30 for portion 1, 40 each for portion 2 and 3, and 20 for the plate). Those results were averaged to obtain the spatial averaged mean square velocity for each substructure. As a reference, the sound power radiated from the motor was measured in an anechoic room by the sound intensity technique.

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Noise of Polyphase Electric Motors 70 60

Sound Power (dB)

50 40 30 20 10 0 100

250

630 4000 1600 1/3 Octave Center Frequency (Hz)

10000

(a) 80 70

Sound Power (dB)

60 50 40 30 Experiment 20 Estimated 10 0 100

Aerodynamic Noise 250

630 1600 1/3 Octave Center Frequency (Hz)

4000

10000

(b)

Figure 6.22 Sound power radiated from an induction motor driven by a PWM inverter at no load: (a) 450 rpm; (b) 1500 rpm.

The sound power due to the aerodynamic and the mechanical origins was also measured with the motor uncoupled from the supply and driven by a separate d.c. motor [229]. In Figure 6.22 the sound power estimated by applying the sound radiation efficiencies and the measured vibration levels is compared with the experimental

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Acoustic Calculations

171

70

Sound Power (dB)

60 50 40 30

Casing 1

20

Stator

10

Casing 2 Base Plate

0 100

250

4000 630 1600 1/3 Octave Center Frequency (Hz)

10000

Figure 6.23 Sound power radiated from major components at 1500 rpm. results. At high frequencies (greater than 1250 Hz), the agreement is quite well. At 450 rpm, there seems to be a shift between the estimation and the experiment. This could be because the vibration, used for the estimation, and the acoustic power, used as the reference here, were not measured simultaneously and, thus, there might be some variations in the actual motor operating speed. At low frequencies, some discrepancies can be observed, which could be caused by the interference between sound waves radiated from each component. Besides, the measured total sound power includes the aerodynamic noise which was not considered in the calculations. In Figure 6.22, the measured aerodynamic and mechanical noise is shown generally important at low frequencies, and for high speeds. Figure 6.23 shows the contribution from each component. It is seen that for this motor structure, the base plate and the stator are the major contributors in the acoustic radiation below 3 kHz. This information is useful in the noise source allocation and the implementation of noise control retrofit plan.

6.5.3

Permanent magnet synchronous motors

The sound power level of PM synchronous motors can be calculated similar to that radiated by induction motors. Figure 6.24 shows the sound power level (SWL) calculated analytically for a high speed, 90 kW, 4-pole, 36-slot PM synchronous motor at no load. The stator inner diameter is D1in = 95 mm, stator outer diameter D1out = 146 mm, effective length of the stator stack L i = 120 mm, air gap (mechanical clearance) g = 1.1 mm, and total nonmagnetic gap including the retaining sleeve gt = 4.2 mm. In high speed machines PMs must be protected against centrifugal forces by a nonmagnetic retaining sleeve (carbon graphite, glass fiber, titanium alloys, or stainless steels). The stator tooth width is ct = 4 mm, slot

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50 40 30

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SWL, dB

60

60 50 40 30

20

20

10

10 0

1000

2000 3000 4000 5000 6000 7000 8000 9000 Frequency, Hz

0

1 ⋅104

0

1000 2000 3000 4000 5000 6000 7000 8000 9000 1 ⋅104 Frequency, Hz

(a)

(c) 100 90

80 1800 Hz

70

9000 Hz

40

2700 Hz

30

5400 Hz 8100 Hz

20

7200 Hz

10 0

70 SWL, dB

4500 Hz 3600 Hz

60 50 40 30 20 10

0

1000 2000 3000 4000 5000 6000 7000 8000 9000 1 ⋅104 Frequency, Hz (b)

0

0

1000 2000 3000 4000 5000 6000 7000 8000 9000 1 ⋅104 Frequency, Hz (d)

Figure 6.24 Effect of rotational speed on the sound power level spectrum of a 90 kW, 4-pole, 36-slot PM synchronous motor at no load: (a)Copyright 13,500 rpm, 450Taylor Hz; (b) 900 Hz. © 2006 & 27,000 Francis rpm, Group, LLC

Noise of Polyphase Electric Motors

SWL, dB

80

6300 Hz

60 50

14:37

0

noise

70

172

100 90

80

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173

0.8 0.6 0.4 b(X)

0.2 0 −0.2 −0.4 −0.6 −0.8

0

0.1

0.05

0.15 x, m

0.2

0.25

0.2

0.25

(a) 3 ⋅ 105

Pr, N/m2

2 . 5⋅ 105 2 ⋅ 105 1 . 5⋅ 105 1 ⋅ 105 5 ⋅ 104 0

0

0.05

0.1

0.15 x, m (b)

1 ⋅ 104

Force, N

1 ⋅ 103 100 10

1

0

1000

2000

3000

4000 5000 Frequency, Hz

6000

7000

8000

(c)

Figure 6.25 Results of electromagnetic calculations of a 90 kW, 4-pole, 36-slot PM synchronous motor at no load and n s = 27, 000 rpm: (a) distribution of the normal component of the magnetic flux density in the air gap along the pole pitch; (b) distribution of the magnetic pressure; (c) radial magnetic force spectrum.

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opening b14 = 3mm, and length of the one-sided stator overhang h ov = 37 mm. The stator single-layer, full pitch winding parameters are as follows: number of turns per phase N1 = 18, conductor diameter d = 0.405 mm, number of parallel paths a p = 1, number of parallel wires aw = 63, and winding factor kw1 = 0.96. The distribution of the normal component of the magnetic flux density in the air gap along the pole pitch is shown in Figure 6.25a. Analytical calculations of the sound power level spectrum have been performed on the basis of equations given in Chapters 2 and 5, and Equation 6.44 given in Chapter 6, which means that only the stator and its circumferential modes are considered. The natural frequency of the stator system including winding and frame are: 8301 Hz for circumferential mode m = 0, 10,272 Hz for m = 1, 2079 Hz for m = 2, 4071 Hz for m = 3, 7157 Hz for m = 4, 11,102 Hz for m = 5, 15,900 Hz for m = 6, 21,557 Hz for m = 7, and 28,076 Hz for m = 8. For the rotational speed n s = 13, 500 rpm (Figure 6.24a) the frequency of the stator current is f = 450 Hz, for 27,000 rpm (Figure 6.24b) the frequency of the stator current is f = 900 Hz, for n s = 32, 400 rpm (Figure 6.24c) the frequency of the stator current is f = 1080 Hz, and for n s = 40, 500 rpm (Figure 6.24d) the frequency of the stator current is f = 1350 Hz. The rated speed is n s = 27, 000 rpm ( f = 900 Hz). It can be seen (Figure 6.24) that predominant frequencies of the sound power level are frequencies fr = f , 2 f , 3 f , 4 f , . . . (Table 2.6). The rotational speed affects both the frequencies and magnitudes of the sound power level.

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7 Noise and Vibration of Mechanical and Aerodynamic Origin Mechanical vibration is caused by rotor unbalance, journal ovality, bent shaft, couplings, sliding contacts, bearings, their defects, and so on. The rotor should be precisely balanced as it can significantly reduce the vibration. The rotor unbalance causes rotor dynamic vibration which in turn results in noise emission from the stator, rotor, and rotor support structure. Oval bearing journals contribute to the rotor eccentricity that also produces vibration and mechanical noise. Both rolling and sleeve bearings are used in electrical machines.

7.1

Mechanical noise due to shaft and rotor irregularities

The shaft and rotor produce vibration and noise at the following frequencies as a result of: • the rotor unbalance, bent shaft, eccentricity, and rubbing parts f unb = kn m , k = 1, 2, 3, . . .

(7.1)

characterized by sinusoidal vibration at a frequency of once per revolution (number of revolutions per second n m ) or a multiple of the number of revolutions per second; • oval bearing journals and double stiffness of the rotor (frequency of twice per revolution) f ov = 2kn m 175 Copyright © 2006 Taylor & Francis Group, LLC

(7.2)

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Noise of Polyphase Electric Motors • due to shaft misalignment f sh = 2n m

(7.3)

• loose stator laminations which result in vibration frequencies flam equal to double the line frequency with frequency sidebands f sdb approximately equal to 1000 Hz, i.e., flam = 2 f

f sdb ≈ ±1000 Hz

(7.4)

• mechanical looseness in either motor mounts or bearing end bells which in turn results in directional vibrations with a large number of harmonics; • gear problems characterized by frequencies of the number of teeth per revolution, usually modulated by the speed (see Section 7.3); • resonance (natural frequencies of shaft, machine housing or attached structures are excited by speed or speed harmonics) which results in a sharp drop of vibration amplitudes with small change in speed; • thermal unbalance which results in a slow change in vibration amplitude as the motor heats up.

7.2

Bearing noise

The increase in vibration and noise level of bearings when the rotational speed changes from n 1 to n 2 can be expressed as [81] L n = 20 log

n2 n1

dB

(7.5)

The increase in radial loading from Q 1 to Q 2 has less effect on vibration level than change in speed [81], i.e., L Q = 0.3 log

Q2 Q1

dB

(7.6)

Vibration and misalignment greatly accelerate bearing wear and increase operating temperature. Vibration levels should be monitored on a regular basis as increased vibration is often indication of impending bearing failure. Motors operating above 1500 rpm should have amplitudes of vibration of 0.025 mm or less.

7.2.1

Rolling bearings

According to SKF Group, a rolling bearing does not generate noise by itself. Vibrations generated directly or indirectly by the bearing and transmitted to other neighboring parts or structure produce an audible effect that is called bearing noise. The noise of rolling bearings depends on: • the type of bearing and its construction; • accuracy of bearing parts;

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Noise and Vibration of Mechanical and Aerodynamic Origin • • • • • • • • •

177

tolerances; alignment; waviness of parts; running speed; number of rolling elements carrying the load; mechanical resonance frequency of the outer ring; contamination and presence of dirt particles; lubrication conditions; temperature.

An increase in the speed produces an approximately proportional increase in the noise level up to a saturation speed. Beyond the saturation speed no further increases in the noise level occur. The noise level increases slightly with the load. As the viscosity of the lubricant increases, the noise level significantly decreases. Figure 7.1 shows comparison of vibration levels produced by single-row and double-row ball bearings. Rolling bearings generate mechanical impulses when the rolling element passes the defective groove. This causes a small radial movement of the rotor. Even if the frequency of those mechanical impulses is different than the natural frequency of the stator-frame system, it can result in significant vibration of the electrical machine. The ball pass frequency or characteristic frequency is the frequency at which the defect in the outer or inner race causes an impulse when the ball or roller passes the defected area of the race. The ball pass frequency on the outer race is 
db Nb f or = nm 1 − cos α (7.7) 2 D

Vibration Level, dB

2

1

Rotational Speed, rpm

Figure 7.1 Vibration level as a function of speed for: (1) single-row ball bearing; (2) double-row ball bearing [81].

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Noise of Polyphase Electric Motors a

D

db

Figure 7.2 Cross section of a ball bearing. or f or = 0.4Nb n m

(7.8)

where D is the pitch diameter, n m is the rotational speed in rev/s, Nb is the number of rolling elements (balls), db is the ball diameter and α is the contact angle of rolling element (Figure 7.2). The ball pass frequency on the inner race is [81] 
db Nb f ir = nm 1 + cos α (7.9) 2 D or f ir = 0.6Nb n m

(7.10)

The ball spin frequency is [81] f bs

 
2 D db 2 = nm 1 − cos α 2db D

(7.11)

and the cage fault frequency is expressed as
 db 1 fc f = nm 1 + cos α (7.12) 2 D Other frequencies of vibration and noise generated by rolling bearings are [200]: • due to irregularities in the ball cage di (7.13) di + do • due to irregular shapes of rolling elements, while turning about their own axis f bc = n m

fr e = n m

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di do dr (di + do )

(7.14)

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Table 7.1 Predicted frequencies of noise and vibration of mechanical origin for an electric motor with ball bearings at 1500 and 1800 rpm. Source Frequencies due to the rotor unbalance and eccentricity, f unb , Hz Frequencies due to oval bearing journals, and shaft misalignment f sh , Hz Ball pass frequency on the outer race, f or , Hz Ball pass frequency on the inner race, f or , Hz Ball spin frequency, f bs , Hz Cage fault frequency, f c f , Hz Frequency due to irregularities in ball cage, f bc , Hz Frequency due to irregular shape of rolling elements, fr e , Hz Frequencies due to variation of bearing stiffness, f st , Hz

1500 rpm

1800 rpm

25, 50, 75, . . . 50, 100, 150, . . .

30, 60, 90, . . . 60, 120, 180, . . .

163.2

195.8

236.8 64.6 14.8

284.2 77.5 17.8

10.2

12.2

64.5

77.5

162.6, 325.2, 487.8 . . .

195.1, 390.2, 585.3 . . .

• due to variation of the bearing stiffness f st = Nb kn m

di di + do

(7.15)

where di is the diameter of the inner contact surface, do is the diameter of the outer contact surface, and k = 1, 2, 3, . . . is a positive integer. No typical frequency pattern is produced due to contamination. The vibration level depends on the amount, size, and composition of the overrolled dust particles. Predicted frequencies of expected vibration and noise of mechanical origin for two speeds 1500 and 1800 rpm are given in Table 7.1. The diameter of the inner contact surface of the bearing is di = 101.6 mm, the diameter of the outer contact surface of the bearing is do = 148.3 mm, the pitch diameter D = 125 mm, the diameter of the ball is db = 23.35 mm, the number of balls is Nb = 16 and the contact angle α = 10o . According to Equation 7.5, the increase in vibration level when the speed increases from 1500 to 1800 rpm (from n 1 to n 2 ) is L n = 1.584 dB.

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7.2.2

Sleeve bearings

Sleeve (journal) bearings consist of two cylindrical surfaces with oil distributing grooves, one moving in relation to the other. The inner surface can be babbittlined, bronze-lined, or lined with other materials generally softer than the rotating journal. In self-lubricated bearings the bearing surface contains a lubricant, usually solid, that is liberated or activated by friction in the bearing. In small sleeve bearings the lubricating oil is impregnated into the porous bronze bearing and is fed to the shaft via the small porosity openings in the bearing bore. Rotation causes a wedge or film of oil to build up on which the shaft rides. In a perfect bearing this oil film prevents metal-to-metal contact and eliminate almost all bearing noise. Because the shaft and bearing have rough surfaces, at least on a microscopic level, the bearing can create a scraping or grinding sound. Also, due to forces such as unbalance and motor driving frequencies, vibration can cause the shaft to rock in the bearing and make contact at the bearing ends. This type of contact causes a knocking or rattling sound. Another source of noise is from the thrust washers which must slide relative to each other, thereby creating a rubbing sound. Noise from a sleeve bearing is usually broad band in nature and somewhat intermittent. The sound pressure level of sleeve bearings is lower than that of rolling bearings. Most sleeve bearings are very quiet and stay quiet until they begin to run out of oil. The vibration and noise produced by sleeve bearings depends on the roughness of sliding surfaces, lubrication, stability, and whirling of the oil film in the bearing, manufacture process, quality and installation. The most important exciting forces are produced at the following frequencies due to: • oval or uneven journal f ov = kn m , k = 1, 2, 3, . . .

(7.16)

f gr = N g n m

(7.17)

• axial groves where n m is the speed of the rotor in rev/s and N g is the number of grooves. • oil whirl in bearings, characterized typically by frequencies of 0.43 to 0.48 times the number of revolutions per second. i.e., f ow = (0.43 . . . 0.48)n m .

7.3

Noise due to toothed gear trains

Some electric motors are equipped with toothed gear trains. Transmission of power through two toothed gears produces intensive noise of mechanical nature. Main causes of mechanical noise are: • impact of teeth; • friction between moving surfaces of teeth;

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• vibration of air or oil pressed from interteeth space; • vibration of gears due to transmitted forces; • variable stiffness of the tooth zone as a function of rotation angle; In addition to the broadband noise and vibration, the following frequencies of noise and vibration can be identified: • harmonics of rotation of shafts fr 1 = kn 1 ;

fr 2 = kn 2 ;

k = 1, 2, 3, . . .

(7.18)

 n 2 ). where n 1 and n 2 are speeds in rev/s of two shafts (n 1 = • tooth harmonics f t = Nt1 n 1 = Nt2 n 2

(7.19)

• combination of tooth and rotation harmonics f c = k f t ± k1 n 1 ± k2 n 2

(7.20)

where k1 = 1, 2, 3, . . . and k2 = 1, 2, 3, . . .. In general, the noise level of toothed gears depends on the type of the gear, its geometry, construction, profile of teeth, pressure angle, engagement factor, backlash, face of gear, materials, accuracy of fabrication, smoothness of surfaces, static, and dynamic run-out. The sound power level can be found using the following equations: • for gears rated below 50 kW L w = 55 + 20 log10 P

dB

(7.21)

• for gears rated over 50 kW L w = (40 . . . 50) + 23 log10 v

dB

(7.22)

where P is the power in kW and v is the peripheral speed of wheels in m/s.

7.4

Aerodynamic noise

The basic source of noise of an aerodynamic nature is the fan. Any obstacle placed in the air stream produces a noise. In nonsealed motors, the noise of the internal fan is emitted by the vent holes. In totally enclosed motors, the noise of the external fan predominates. According to the spectral distribution of the fan noise, there is a broadband noise (100 to 10, 000 Hz) and siren noise (tonal noise). The siren effect is a pure tone being produced as a result of the interaction between fan blades, rotor slots,

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or rotor axial ventilation ducts and stationary obstacles. The frequency of pure tone due to the fan blades is f p = k Nbl n m

(7.23)

where k = 1, 2, 3, . . ., Nbl is the number of fan blades and n m is the circumferential speed of the fan in rev/s. The frequency of the pure tone due to the cooling air passing through rotor axial ducts (if they exist) is f r d = k Nd n m

(7.24)

where Nd is the number of ducts. Siren noise can be eliminated by increasing the distance between the impeller and the stationary obstacle. The sound power level from fans depends on the machine power and the capacity of the fan, i.e., the static pressure and/or the discharged volume (flow rate). The sound power level from fans can be approximately estimated using the following empirical equations: L w = 67 + 10 log10 (Pout ) + 10 log10 ( p)

(7.25)

L w = 40 + 10 log10 (Q) + 20 log10 ( p)

(7.26)

L w = 94 + 20 log10 (Pout ) − 10 log10 (Q)

(7.27)

where Pout is the motor rated power in kW, p is the fan static pressure in Pa, and Q is the flow rate in m3 s−1 . To avoid a noisy operation of a fan, the discharge velocity should be kept within certain limits, as shown in Table 7.2. Small motors with shaft driven fans radiate much less noise when the speed is reduced below the rated speed. The fan noise level drops by about 10 dB at 75% of the rated speed, 20 dB at 50% of the rated speed, and 0 dB at 25% of the rated speed according to the following equation [161]: √ L p = 24.4 3 v

dB(A)

(7.28)

Table 7.2 Discharge velocity from fans. Application Sound studios, churches, libraries Cinemas, theaters, ballrooms Restaurants, offices, hotels, shops

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Maximum discharge velocity, m/s Supply System Exhaust System 4-5 5-7 6-8

5-7 6-8 7-9

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Participation Rate in Total Sound Power Generation, %

Noise and Vibration of Mechanical and Aerodynamic Origin

183

100

80

60

Total Sound Power

40

Electromagnetic Contribution Aerodynamic and Mechanical Contribution

20

0 Rotational Speed at Constant Power, rpm

Figure 7.3 Typical relative contributions to total sound power radiation as functions of speed [137]. where v is the peripheral speed of blades in m/s. Equation 7.28 is only valid for totally enclosed fan cooled (TEFC) motors (Chapter 1). As the speed increases, the cooling fan becomes the primary source of noise. For two pole induction motors fed from solid state inverters at frequencies in excess of the standard 50 or 60 Hz, the dominating noise is always the noise of aerodynamic origin [137]. Noise source curves vs. the rotational speed are schematically plotted in Figure 7.3 [137]. On the other hand, for motors with separately driven fans, no improvement in airflow noise is achieved when the speed is reduced [161]. According to Bies and Hansen [11], the sound power radiated by the fan discharge and the fan inlet can be estimated with reasonable accuracy on the basis of the following formula: 1 L w = C F + 10 log10 Q + 20 log10 p + η p − 48 dB (7.29) 3 where C F is an empirical coefficient given for each octave band in Table 7.3, Q is the flow rate in m3 s−1 , η p is the percentage of peak efficiency, and p is the static pressure in Pa. It is recommended to assume η p = 99 when the fan operating point efficiency is unknown. The reference sound power level in Equation 7.29 is 10−12 W. The octave band in which the blade passing frequency occurs should be corrected by adding the blade frequency increment i b f that is given in Table 7.3 [11]. The blade passing frequency is given by Equation 7.24 in which k = 1. The following accuracy is claimed: ±2 dB for the octave bands from 250 to 4000 Hz, ±4 dB for the 125 Hz band, and ±8 dB for the 63 Hz band.

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Table 7.3 Specific sound power levels C F (dB refers to 10−12 W) and blade frequency increments i b f for various types of fan [11]. Fan type Centrifugal airfoil, backward curved, backward inlined: • over 0.9 m • under 0.9 m forward curved radial blade pressure blower: • over 1 m • 0.5 to 1.0 m • less than 0.5 m Vanexial • over 1 m • under 1 m Tubeaxial (no vanes) • over 1 m • under 1 m

Octave band center frequency, Hz 63 125 250 500 1000 2000 4000 8000 i b f

32 32 36 38

31 36

29 34

28 33

23 28

15 20

7 12

3 3

35 39 55 48 63 57

42 48 58

39 45 50

37 40 44

32 38 39

30 37 38

29 37

8 8 8

39 36 37 39

38 43

39 43

37 43

34 41

32 38

22 32

6 6

41 39 40 41

43 47

41 46

39 44

37 43

34 37

27 35

5 5

These values are the specific sound power levels radiated from either the inlet or outlet of the fan. If the radiated total sound power level is desired, it is necessary to add 3 dB to the above values.

Example A backward curve blade centrifugal fan has the diameter Dbl = 0.35 m, Nbl = 33 blades, spins at 1800 rpm, operates with a peak efficiency η p = 92% and delivers a volume flow rate Q = 0.210 m3 s−1 against a static pressure of p = 125 Pa. Find the sound power level. The blade passing frequency according to Equation 7.23 is f p = 33 × 1800/60 = 990 Hz. Table 7.3 shows that the blade passing frequency f p = 990 Hz lies in the 1000 Hz bands and blade frequency increment i bl = 3 dB must be added. The sound power levels according to Equation 7.29 are given in Table 7.4

7.5

Mechanical noise generated by the load

An oil injected rotary screw compressor will be considered as an example of the load. Screw compressors are usually driven by constant speed or variable speed cage induction motors. A single rotary screw compressor consists of parallel male and female screw rotors that mesh together.

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Table 7.4 Sound power levels for 33-blade, 1800-rpm backward curve blade centrifugal fan. Parameters c F , dB i b f , dB L w , dB

63 36 – 52.3

Octave band center frequency, Hz 125 250 500 1000 2000 38 36 34 33 28 – – – 3 – 54.3 52.3 50.3 52.3 44.3

4000 20 – 36.3

8000 12 – 28.3

The fundamental frequencies of noise and vibration are determined by the following equations: f m = Nm n m ;

fn = N f nm

(7.30)

where Nm is the number of teeth of the male rotor, N f is the number of teeth of the female rotor, and n m is the speed of the rotor in rev/s. There are also frequencies of noise and vibration produced by the operating speed and interaction between male and female rotor teeth [66], i.e.,

fo = nm ;

f om =

Nm Nm nm = fo ; Nf Nf

Copyright © 2006 Taylor & Francis Group, LLC

fo f =

Nf Nf nm = fo . Nm Nm (7.31)

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8 Acoustic and Vibration Instrumentation In this chapter, the basic principles for the operation of acoustic and vibration instrumentation will be described. Various methods for determining the sound power level of a noise source will be introduced. Sound power results obtained by using the sound intensity technique for a test motor driven by three different controllers will be used to illustrate the effects of operating speed, inverter strategies, and load on the emitted noise.

8.1 Measuring system and transducers In order to assess the severity of noise and vibration, objective scientific measurements have to be made. As shown in Figure 8.1, a typical acoustic/ vibration measurement system consists of a measuring transducer (accelerometer, vibrometer, displacement transducer), a preamplifier and an analysis system (which may be a frequency analyzer using the fast-fourier-transform [FFT], digital or analogue filters or a simple root-mean-square [rms] voltmeter). In the following sections, the principles underlying the operation of pressure and vibration measuring transducers will be explained, followed by a discussion of the various practical aspects in the use of the transducers. International Standards IEC61672-1 [96], ISO 2954 [98], and ISO 5348-3 [110] specify the requirements of acoustics and vibration instrumentation. A transducer is a device which converts changes of mechanical quantities into changes of other physical quantities, usually an electrical signal proportional to the mechanical parameter of interest. If the transducer requires the supply of electrical energy to convert changes of mechanical energy to changes of electrical energy, it is called a passive transducer. Otherwise, it is considered an active (or self-generating) transducer. If the input mechanical energy is converted into output electrical energy via an intermediate quantity (such as optical), the conversion is 187 Copyright © 2006 Taylor & Francis Group, LLC

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Noise of Polyphase Electric Motors Tape Recorder

Input

Transducer Displacement Velocity Acceleration Pressure

Detector/Indicator Preamplifier Analogue/Digital Filters FFT Analyser

Figure 8.1 Typical acoustic/vibration measuring system.

Electromagnetic Direct Conversion

Electrodynamic Piezoelectric

Self-Generating Indirect Conversion Transducers

Photoelectric Resistive Potentiometer Semiconductor Strain Gauge Electrolyte

Direct Conversion

Capacitive Inductive Variable Reluctance Mutual Inductance Differential Transformer

Passive

Optical-Electronic Indirect Conversion

Acoustic Radioactive Photoelectric

Figure 8.2 General classification of transducers [85].

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Acoustic and Vibration Instrumentation

189

considered indirect; otherwise, it is considered as direct. A general classification of transducers employing various means to operate is shown in Figure 8.2. Piezoelectric transducers are, by far, the most often used.

8.2 8.2.1

Measurement of sound pressure Choice of microphones

Listed below are some of the factors that need to be considered in selecting the type of microphones to be used: • Expected properties of the sound field •

Free field for closed chamber;

•

Important range of sound pressure level;

•

Important frequency range.

• Desired precision of measurement •

Sensitivity tolerance;

•

Frequency distortion tolerance;

•

Phase distortion tolerance;

•

Nonlinear distortion tolerance;

•

Self-noise tolerance.

• Environmental conditions

8.2.2

•

Background noise level;

•

Temperature;

•

Humidity;

•

Atmospheric pressure;

•

Wind;

•

Strong electromagnetic fields;

•

Mechanical shock;

•

Mass and space limitations.

The sound pressure sensor–condenser microphone

As it is the pressure variation acting on the eardrum that is diameter causing the sensation of hearing, the universal means for detecting the presence of sound is through pressure sensing. There are numerous types of pressure transducers and in particular, microphones. A good review of the types of microphones and their development has been presented, for example, by Bauer [12]. Only the condenser type microphone will be considered here. Specifications of condenser microphones are given in [34].

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Insulator

Base Plate

Hole for Static Pressure Equalization

Figure 8.3 Schematic diagram of a condenser microphone.

Principle of operation Figure 8.3 shows schematically the components of a condenser microphone. Basically, it consists of a thin metal diaphragm with a rigid back plate, forming an air dielectric capacitor. A constant charge is applied to the capacitor by a DC voltage (known as the polarization voltage) so that any change in capacitance due to a change in sound pressure on the diaphragm will be transformed into voltage variation. A static pressure equalization vent is normally provided to ensure the same static pressure inside and outside the cartridge cavity. A preamplifier is normally coupled physically next to the microphone, in order to increase the sensitivity of the microphone by reducing its stray capacitance. Effect of the microphone as a disturbance to the sound field Owing to the presence of a cylinder with its axis aligned with the direction of a plane wave propagation, the pressure measured at the end of a cylinder increases with decrease in wavelength as shown in Figure 8.4 [157]. It is, therefore, expected for a microphone, the diaphragm of which is normal to the direction of propagation of a plane wave, interference effects would become important at frequencies when the microphone diameter is of the same order as the wavelength. Frequency response of a microphone Figure 8.5 shows the typical frequency response of a half-inch free field microphone with the sound wave propagating at different incident angles to the microphone. It can be seen that at zero incident angle, the response is virtually independent of frequencies for frequencies below 20 kHz and such a microphone

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Pressure without Cylinder

Pressure with Cylinder

4

3 D

2

1 .01

10

1

.1 D/λ

Figure 8.4 Interference effect due to the presence of a cylinder in the sound field [157]. D is the cylinder diameter and λ is the wavelength. The pressure refers to that on the end of the cylinder at its center.

is known as a free-field microphone. However, the response will be attenuated at high frequencies with the increase in incident angle (directional effects). Hence, if a microphone with a ‘free-field’ response is used in a diffuse field, a ‘randomincidence’ correction has to be applied so that the response in that case is virtually independent of the frequency. A microphone can also be made to give a flat frequency response for a 90 degree incident angle, in which case it is known as a pressure microphone and is useful for the measurement of road traffic noise.

Relative Response, dB

25

0°

20

60°

15

R 90°

10

q°

5 0

30

50

100

R = Random Incidence Response

500 1000 2000 Frequency, Hz

10,000

20,000

Figure 8.5 Typical response of a free-field microphone with incident angles [6].

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The frequency response of a microphone is mainly determined by the following factors: (a) diaphragm stiffness; (b) diaphragm mass; (c) mechanical damping of the diaphragm due to viscous resistance to air movement between the diaphragm and back plate; (d) interference and diffraction effects at frequencies when the microphone diameter is of the same order of magnitude as the wavelength (see Figure 8.4). Above the resonance frequency of the diaphragm, the motion of the diaphragm is mainly determined by its mass so that the sensitivity decreases with increasing frequency. Around the resonance frequency, interference and diffraction effects become important and a microphone can be made to give a free-field response by increasing its internal damping. Below the resonance frequency, the motion of the diaphragm is primarily determined by its stiffness. The upper frequency limit of the response of a microphone is determined by the diaphragm diameter as seen in Figure 8.4, whereas the lower frequency limit is determined by the volume of the cavity behind the diaphragm and whether the pressure equalization vent is outside and inside the field as the air exchange between the cavity and the outside affects the total stiffness of the system. As a rule of thumb, halving the size of a microphone reduces its sensitivity by 10 dB but doubles the frequency at which directional effects become important. Hence, small microphones are more suitable for high sound pressure measurements and for maintaining flat frequency response at very high frequencies. Dynamic range The lower limit of the dynamic range is determined by the electrical noise or the inherent thermal noise and the upper limit is determined by the sound pressure level which will result in nonlinear behavior of the diaphragm (that is, distortion effects). Since large diaphragm requires much less sound pressure to produce nonlinear behavior, the upper limit of small microphones is higher. Long-term stability The sensitivity of a microphone will increase slightly over a long period of time, due to the creep of the diaphragm which reduces the diaphragm tension. The rate of change of sensitivity is most severe at high temperatures and a typical rate at 150◦ C is of an order better than 1 dB over 2 hours. Effect of temperature The relative sensitivity decreases with increasing temperature which affects the diaphragm tension and the density and viscosity of air. Within the normal operating temperature range (i.e., 0◦ C to 50◦ C), the change in sensitivity is negligible.

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Effect of ambient pressure Since the response of a microphone at low frequencies is determined by stiffness, its sensitivity increases with decrease in ambient pressure. At high frequencies, the response is determined by the mass of the diaphragm. Since the air in the gap between the diaphragm and the back plate, which adds mass to the diaphragm, is affected by static pressure variation, the sensitivity at high frequencies also changes with varying ambient pressure. Within the normal day-to-day variation of the ambient pressure, the change in relative sensitivity rarely exceeds 1 dB. Effect of humidity Humidity change generally has negligible effect on the sensitivity of a microphone. The change in sensitivity of a microphone at 20◦ C is of the order of ± 0.1 dB over a range of relative humidity from 20 to 90%. Effect of vibration Vibration has a maximum effect when it is applied normal to the diaphragm. A root-mean-square vibration amplitude of 1 m/s2 applied normally to a half-inch microphone is equivalent to a sound pressure level of about 60 dB. Effect of wind Wind-induced noise due to air circulation generated either naturally, or by machines operating, is usually of low frequency. At wind speeds below 20 km/h (i.e. 5 m/s), a windscreen, which will reduce the wind-induced noise by about 20 30 dB, should be used in outdoor measurements. However, at higher wind speeds, outdoor measurements should not be performed. Windscreens tend to modify the frequency response at moderate and high frequencies and, therefore, should not be used in those indoor measurements which do not involve air circulation.

8.2.3

Sound level meter

A sound level meter is a basic instrument for measuring the sound pressure level. As shown in Figure 8.6, a typical sound level meter consists of the following components: (a) (b) (c) (d)

microphone preamplifier overload detector processing unit • weighting network • filters • amplifier

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Preamplifier

Time constants "F", "S", "I" Weighting Networks

Amplifier

Display RMS Detector

65.6

Microphone

Filters

Hold Circuit

Figure 8.6 Schematic of a typical sound level meter. • r ms detector (with a fast “F” and a slow “S” time constant) (e) output and display Sound level meters are classified according to their precision and tolerance into two performance categories: class 1 and class 2 [96]. Class 2 meters have greater tolerance limits than class 1 meters. Frequency bandwidths As the acoustic intensity I of most sounds is distributed over a range of frequency, this distribution within a band with lower and upper frequencies f 1 and f 2 can be described by the spectral density I( f ) so that
I =

f2

I( f )d f

(8.1)

f1

where I( f ) = I / f , and I is the intensity within the frequency interval  f = 1 Hz. The intensity spectrum level (ISL) L I s is defined as L I s = 10 log[I( f )/Ir e f ]

(8.2)

where Ir e f is the reference intensity level (10−12 W/m2 for air). If the intensity spectrum level is constant over a bandwidth  f , then the intensity level L I is related to the intensity spectrum level L I s by L I = L I s + 10 log( f )

(8.3)

Similarly, the sound pressure level L p is related to the sound pressure spectrum level L ps by L p = L ps + 10 log( f )

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(8.4)

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Frequency analysers or filter network used in the sound level meter may be based on either constant bandwidth or proportional bandwidth devices. The constant bandwidth device is basically a tunable narrow-band filter with constant bandwidth  f = f u − fl where fl and f u are the lower and upper half-power frequencies. The proportional (i.e., constant percentage) bandwidth device consists of a series of relatively broadband filters such that f u / fl = constant

(8.5)

The center frequency f c of a proportional bandwidth filter is given by fc =



fl f u

(8.6)

Most common proportional bandwidth filters are in the form that satisfies f u / fl = 2x

(8.7)

where x = 1 for octave band filter, x = 1/3 for 1/3-octave band filter, and so on. Figure 8.7 shows the characteristics of a typical octave band filter. The preferred center frequencies for octave band filters are given in Table 8.1. For details, see [96]. Sound level meters are usually equipped with either a 1/3-octave or an octave filter for frequency analysis of the sound pressure spectrum. Weighting network Since the human ear is most responsive to frequencies in the range 500 Hz - 5 kHz than to other frequencies, a weighting function is normally applied to the sound pressure levels so that the response of the sound level meter will approximate the

Effective Bandwidth

(Amplitude)2

Ideal Filter

Practical Filter

3 dB Point

fl

fc

fu

Linear Frequency Scale

Figure 8.7 Response of a typical octave filter.

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Table 8.1 Preferred center frequencies for octave band filter and frequency weighting characteristics. (IEC 61260 Standards Ed.1.0 [1995] Electroacoustics— Octave-Band and Fractural-Octave-Band Filters.) Nominal frequency Hz 16 31.5 63 125 250 500 1000 2000 4000 8000 16000

Relative free-field frequency response, dB A-weighting -56.7 -39.4 -26.2 -16.1 - 8.6 - 3.2 0 + 1.2 + 1.0 - 1.1 - 6.6

B-weighting -28.5 -17.1 - 9.3 - 4.2 - 1.3 - 0.3 0 - 0.1 - 0.7 - 2.9 - 8.4

C-weighting -8.5 -3.0 -0.8 -0.2 -0.0 -0.0 0 -0.2 -0.8 -3.0 -8.5

D-weighting -22.6 -16.7 -10.9 - 5.5 - 1.6 - 0.3 0 + 7.9 +11.1 + 5.5 - 0.7

loudness contours in Figure 1.2. There are four types of weighting that can be applied: 1. A-weighting – This function weighs a signal in a manner which approximates an inverted 40-phon loudness contour and is intended for measuring sound pressure level below 55 dB. 2. B-weighting – This function corresponds to an inverted 70-phon loudness contour and is intended for measuring sound pressure level between 55 and 85 dB. 3. C-weighting – This function has an approximately flat response except at very low and very high frequencies and is intended for use above 85 dB. 4. D-weighting – This function has been standardized for use in aircraft noise measurements. The above four weighting functions are listed in Table 8.1 for the octave band frequencies and are graphically represented in Figure 8.8. Most sound level meters are equipped with the A and C weighting networks and the A-weighting network is the most widely used because the B and C weighted measurements do not correlate well with subjective tests.

8.2.4

Acoustic calibrator

It is important to check the performance of a sound level meter before and after measurements with an acoustic calibrator, which provides a known sound pressure level. An acoustic calibrator usually produces a discrete frequency in the range of

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Weighting Factor (dB)

20 0 −20 A Weighting −40

B Weighting

−60

C Weighting

−80

D Weighting 10

100

1000

10000

Frequency (Hz)

Figure 8.8 Frequency weighting functions. 250 Hz to 2000 Hz and may be of the “pistonphone” type or of the “sound level calibrator” type. If precision type sound level meters (such as class 1) are used for accurate measurements, then a pistonphone should be used for checking the calibration. For general purpose sound level meters (such as class 2), a sound level calibrator, consisting of an electronic oscillator driving a miniature loudspeaker, is suitable for checking the calibration. The relevant international standard for acoustic calibrator is IEC 60942 [95].

8.2.5

Level recorder

If the time history of the sound pressure level is required (such as in some ambient sound level measurements or in reverberation time measurements), then the output of the sound level meter may be connected to a level recorder to generate a hard copy. A level recorder may be in the form of an analogue chart recorder, or a personal computer with a data acquisition facility may be used.

8.3

Acoustic measurement procedure

For detailed guidance on the use of portable sound level meters, please refer to Australian Standard AS 2659.1-1988 [6].

8.3.1

Effect of the operator on measurement results

It has been found that reflections from an operator close to a measurement point may cause errors of 2 to 3 dB in the frequency range 200 Hz to 4000 Hz with an error of up to 6 dB around 400 Hz. Hence, it would be preferable to use an extension cable or an extension rod between the microphone and the sound level meter, so that the influence of the presence of an operator on a measurement may be minimized. Where it is not practicable to use an extension cable or rod, the sound level

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meter should be held in front of an operator at arm’s length with the predominant sound waves approaching the operator from the side, as shown in Figure 8.9 [6].

8.3.2

Measurement position

For OH&S measurements, the measurement position is usually near the operator’s ear. It should be noted that nearby reflective surfaces can often increase the sound pressure level to which an operator is exposed. Hence, it is not recommended to extrapolate results obtained at locations influenced by reflective surfaces to locations without the influence of reflective surfaces.

8.3.3

Standing waves

In any reverberant sound field, ‘standing waves’ may occur due to the interaction of sound waves of the same frequency traveling in opposite directions. The ‘standing wave’ phenomenon is particularly prominent in the low frequency range, such as 125 Hz, where maximum and minimum sound pressure levels can be identified by slowly moving a sound level meter in the room.

8.3.4

Measurements of ambient sound pressure levels

For the measurements of ambient sound pressure levels, a number of microphone positions should be used so that a spatial-averaged sound pressure level may be obtained. Precautions relating to the use of a microphone and measurement locations as described above should be observed.

Predominant Directional Component

Predominant Directional Component

(a)

(b)

Figure 8.9 Correct position for meter operator: (a) measurement of a free-field with a microphone having a flat normal incidence response; (b) measurement of a free-field with a microphone having a flat grazing-incidence response. The operator’s shoulders are in line with the direction of the propagation of the sound being measured. The sound level meter is held directly in front of the operator [6].

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Table 8.2 Correction for background sound pressure levels. Difference between sound pressure level measured with sound source on and background sound pressure level with sound source off, dB 5 6 7 8 9 10

8.3.5

Correction to the sound pressure level measured with sound source operating to obtain level from source alone, dB -1.7 -1.3 -1.0 -0.8 -0.6 -0.4

Corrections for background sound during source measurements

When measurements are made in the presence of background noise, corrections to the results have to be made if the sound pressure level in the presence of background noise is between 5 and 10 dB higher than the background noise alone, as given in Equation 8.8, Table 8.2, or Figure 8.10. If the sound pressure level exceeds

Subtract from source noise measured in the presence of background noise (dB)

10 9 8 7 6 5 4 3 2 1 0 0 10 5 1 2 8 7 3 4 6 9 Difference between source noise measured in the presence of background noise and background noise alone (dB)

Figure 8.10 Graph for corrections for measurements made in the presence of background noise.

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the background noise by 10 dB, then the error in measurement attributable to background noise is less than 0.4 dB. If the sound pressure level is less than 5 dB above the background noise, then a satisfactory measurement cannot be made.   correction = 10 log 1 − 10 exp [−0.1(L pr − L n )]

(8.8)

where L pr = sound pressure level with the source operating, L n = background noise level with the sound source off.

8.3.6

Polar plots

The directionality of a noise source can be obtained by measuring its radiated sound pressure level at a fixed distance from the center of the source for different angles and presenting the results in a polar plot as shown in Figure 8.11a. Alternatively, contours of the overall sound pressure level can be plotted as a function of distance from the noise source in a polar plot as shown in Figure 8.11b. The minimum distance from the source at which measurements should be taken is twice the largest dimension of the source or one wavelength of the lowest frequency of interest, whichever is larger. For example, if the source’s largest dimension is 2 m and the lowest frequency of interest is 125 Hz (corresponding wavelength ≈ 2.75 m), then the measurement distance from the source should be 5.5 m.

8.4

Vibration measurements

Since vibration is basically concerned with oscillatory motion, it can be quantified by measuring the peak-to-peak level, the rms value and crest factor of the displacement as well as the frequencies of its occurrence. As displacement,

330°

0°

30°

300°

330° 60°

270°

70

90

240°

110 dB

120° 210° 180°

150°

(a)

0°

30°

300°

60°

270°

2

4

6 8 m 0°

240°

120° 210° 180°

150°

(b)

Figure 8.11 Example of a polar plot for a noise source: (a) at 5 m from the source; (b) 85 dB(A) contour.

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0 Acceleration

Attenuation (dB)

−20 −40

Velocity

−60 −80

Displacement

−100 −120 −140 −160 100

101

102 Frequency (Hz)

103

104

Figure 8.12 Attenuation introduced by integrating the acceleration signal. velocity, and acceleration are related to each other for sinusoidal signals, velocity, and acceleration can also be used as parameters to describe vibration levels. If only rms values are required, then displacement and velocity levels can be obtained by integrating the acceleration signal, the effect of which is shown in Figure 8.12. For frequency analysis of vibration signals, the parameter that gives the flattest frequency spectrum is normally chosen so that the signal-to-noise ratio can be improved. In practice, displacement is seldom measured except perhaps for balancing rotating machinery, because large displacements normally occur at low frequencies. In general, vibration severity is best indicated by measuring the velocity for the frequency range of 10 Hz to 1kHz. For higher frequency range of interest, the parameter to be measured is normally acceleration. The measured rms vibration levels a may be expressed on a decibel (dB) scale as 20 log(a /ar e f ), where the reference value ar e f is 10−12 m for displacement, 10−9 m/s for velocity, and 10−6 m/s2 for acceleration.

8.4.1

Theory of operation of vibration-measuring transducer

A vibration-measuring transducer may be classified into the following two types according to the type of frame of reference used: • Fixed-reference transducer in which the motion to be determined is measured relative to a fixed point in space. • Seismic transducer in which the motion of a mass is measured relative to the point whose motion is to be determined. A seismic transducer is the most commonly used and is basically a single degree of freedom system consisting of a mass (M), spring (with stiffness K ), and damper (with damping coefficient C) as shown in Figure 8.13.

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K

x y

M C

Figure 8.13 Seismic transducer.

Let the displacement of the seismic mass and the vibrating body relative to a fixed frame of reference in space be x and y, respectively. Then the displacement z of the seismic mass relative to the transducer case is given by z=x−y

(8.9)

By Newton’s second law, the equation of motion of mass M is given by d2z dz + Kz = 0 +C dt 2 dt Assume that the motion of the vibrating body is sinusoidal so that M

(8.10)

y = Y sin ωt where ω = 2π f is the angular frequency in rad/s and f is frequency in Hz. Then Equation 8.10 can be rewritten as d2z dz + K z = Mω2 Y sin ωt. +C 2 dt dt The steady-state solution to Equation 8.11 is given by M

(8.11)

z = Z sin(ωt − φ) where Z β2 β2 = = Y hm (1 − β 2 )2 + (2ζβ)2

(8.12)

2ζβ and the phase angle φ is given by φ = tan−1 1−β 2 in which β = ω/ωm = f / f m is √ the frequency ratio, ωm = K /M is the undamped natural frequency, ζ = C/Cc √ is the damping ratio, and Cc = 2 M K is the critical damping coefficient.

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Figure 8.14 Amplitude response of the relative displacement of a seismic transducer. Seismometer (vibrometer) — low natural frequency Figures 8.14 and 8.15 show the amplitude and phase of the response of the seismic mass. It can be seen that for high frequency ratio, that is, for low natural frequency of the transducer and high frequency of the vibration, the relative displacement Z approaches that of the displacement Y of the vibratory body, virtually independent of the damping ratio. Under this condition, the seismic mass remains stationary in space while the transducer case attached to the vibratory body moves up and down with it. Such a transducer is known as a seismometer. Note that from Figure 8.15, there is a phase difference of 180◦ between the input and output motion. A seismometer can be implemented in practice by using the electrodynamic principle as shown in Figure 8.16. The seismic mass is constructed from a permanent magnet (PM) attached to the case through a spring-damper system and for frequencies above the natural frequency of the transducer, the motion of the mass relative to the coils fixed to the case will generate an output voltage e proportional to the velocity ωY of the vibratory body as given by Equation 8.13. e = −ωBlY

(8.13)

where B = magnetic flux density, and l = length of the conductor. A seismometer typically has a natural frequency between 1 and 5 Hz and the sensitivity (of the order of 100 mV per 10 mm/s) is quite high particularly at low frequencies. The impedance of the coils is usually quite low so that the

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phase angle

204

β

Figure 8.15 Phase response of the relative displacement of a seismic transducer. output voltage can be measured directly with a high-impedance voltmeter. As the relative motion of the seismometer is of the same order of magnitude as that of the vibration to be measured, its usefulness is limited by its size and mass. Accelerometer — high natural frequency Equation 8.11 can be rewritten in terms of the acceleration ω2 Y of the vibration to be measured: Z 1 1 = 2 2 2 ω Y ωn (1 − β )2 + (2ζβ)2

Figure 8.16 Seismometer.

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(8.14)

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ζ=0.1

ζ=0

4.0 3.8 3.6 3.4 3.2 3.0 2.8 2.6 2.4 ζ= 0.25 ωn2 Z 2.2 2.0 2 ω Y 1.8 1.6 1.4 ζ= 0.5 1.2 1.0 0.8 ζ=1.0 0.6 0.4 ζ=2.0 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 β

Figure 8.17 Amplitude response of Z /ω2 Y . The response as given by Equation 8.14 is shown in Figure 8.17 for various values of the damping ratio. It can be seen that if the natural  frequency of the transducer is high, β approaches 0 so that the denominator (1 − β 2 )2 + (2ζβ)2 approaches 1. Hence, the relative motion Z is proportional to the acceleration of the vibratory body as given by Equation 8.15. Z=

1 2 ω Y ωn2

(8.15)

In order to ensure flat frequency response over the frequency range of interest, either the natural frequency of the transducer has to be very high (such as in piezoelectric crystal accelerometers) or a damper with a damping ratio of about 0.707 has to be used to extend the operating frequency range (such as in electromagnetic type accelerometers). In order to reproduce without distortion a periodic waveform containing harmonics, the phase angle must be zero (such as in an undamped transducer for β < 1) or all the harmonic components must be phase-shifted equally by using ζ = 0.707 in which case φ = πβ/2 for β < 1. Figure 8.18 shows the construction of a piezoelectric accelerometer based on the compression design where the piezoelectric element acts as a spring. Briefly, due to the vibration, the inertia force of the seismic mass causes a mechanical strain in the piezoelectric element, which produces a charge proportional to the acceleration of the motion. The sensitivity is generally expressed in pC per g (acceleration due to gravity) or m/s2 . Since the capacitance is normally very small, the output impedance is very high, so that a charge amplifier is required. The

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Seismic Mass

Piezoelectric Discs

Base

Figure 8.18 Compression type accelerometer.

signals from a compression-type accelerometer can be greatly affected by base bending, and thus it is only used for measurements of very high vibration levels where the error is small compared with the signal itself. Piezoelectric shear type accelerometers (such as the one shown in Figure 8.19) are the most commonly used because they • provide a wide operating frequency range; • possess good linearity over a wide dynamic range; • are durable (no moving parts); • are compact;

1

2

3

4

Figure 8.19 Shear type accelerometer. (1) triangular center post; (2) seismic mass; (3) preloading ring; (4) base [19].

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• have a high sensitivity to mass ratio; • are virtually unaffected by base bending and temperature fluctuations.

8.4.2

Characteristics of piezoelectric accelerometers

Since velocity and displacement can be obtained by integrating the acceleration signals, most vibration measurements are made with piezoelectric accelerometers, and the charge amplifiers are normally supplied with a built-in integration circuit. Sensitivity The sensitivity of an accelerometer is defined as the electrical output per unit of input acceleration. As seen from Equation 8.14, the sensitivity generally increases with the mass of the transducer. For a piezoelectric accelerometer, it can be expressed as charge sensitivity (pC/ms−2 ) or voltage sensitivity (mV/ms−2 ). As electrical equipment has to be connected to an accelerometer through cables, additional capacitance known as shunting capacitance Cs is introduced. The equivalent circuits for a charge source and a voltage source are shown in Figure 8.20. For the charge equivalent circuit (Figure 8.20a), the charge sensitivity remains unchanged. For the voltage equivalent circuit (Figure 8.20b), the voltage sensitivity will be modified by a factor Cc /(Cc + Cs ), where Cc is the capacitance of the accelerometer. For this reason, charge preamplifiers are preferred to voltage preamplifiers. Ideally the transverse sensitivity of an accelerometer (that is, the sensitivity along an axis at right angles to the mounting axis) should be zero. However, owing to the manufacturing process and irregular characteristics of the transducing element, the axis of maximum sensitivity makes an angle θ to the mounting axis so that the ratio of the transverse sensitivity to the sensitivity along the mounting (main) axis is tanq. In practice, the maximum transverse sensitivity is kept below 5% of that of the main axis. Linearity limits Piezoelectric accelerometers generally have a very wide dynamic range (typically more than 100 dB) over which the response is linear. The lower limit of linearity

Q

CS

CC

(a)

CC E

CS

EO

E

(b)

Figure 8.20 Equivalent circuits (a) charge; (b) voltage.

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is determined by electrical noise from the preamplifier, cable, and the environment, while the upper limit is determined by the design of the transducer.

Operating frequency range

4 5

10 −

Lower Frequency Limit D Ma isp xi la mu ce m m en t

6

10 − Acceleration,

1

Upper Frequency Limit

10 10

n

io at

10−5

2

ler ce Ac

10−4

10

um im

in

10−3

3

10−2

10

Operating Range

10−1

4

1

10

um n im tio ax ra M cele Ac

10

M

Velocity Amplitude, in./Sec

102

10 − 1 0−

3

103

Peak-to-peak Displacement, IN.

2

10 −

10 −

1.

0

1

The operating frequency range of an accelerometer is defined as the frequency range over which its sensitivity is within certain specified limits of the rated sensitivity. The lower frequency limit of an accelerometer is generally determined by the frequency response of preamplifiers. The upper frequency limit of an accelerometer is determined by the natural frequency and the damping of the transducer. For a given stiffness of the piezoelectric element, the upper frequency limit can be extended by reducing the mass of the seismic mass which, however, would reduce the sensitivity of the transducer. From Equation 8.14, for an accelerometer with negligible damping, the maximum vibration frequency which can be measured within 10% of its rated sensitivity is approximately 0.3 f m , where f m is the mounted resonance frequency. This frequency is referred to as the 10% frequency limit. A 3 dB frequency limit corresponds approximately to 0.5 f m . The linear operating range in amplitude and frequency can be specified by an operating envelope as shown in Figure 8.21 with velocity amplitude as the ordinate and frequency as the abscissa. Lines of constant displacement amplitude

1

102 103

10

104

105

Frequency, HZ

Figure 8.21 Operating range of a transducer [85].

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(6 dB per octave) and constant acceleration amplitude (−6 dB per octave) are also shown. Phase response As shown in Section 8.4.1, in order to ensure negligible distortion of the waveform, the damping ratio of the accelerometer must be either 0 or approximately 0.707. For other damping coefficients than these two values, the phase shift will be nonlinear, resulting in distortion of the waveform. Transient response If the accelerometer and preamplifier are operated outside their useful operating ranges, waveform distortion in the signal will result, particularly in measuring transient vibrations. Owing to the finite leakage time constant of the accelerometer, leakage effects have to be considered for measuring transient vibrations. For a square wave transient, the lower limiting frequency of the preamplifier has to be set to less than 0.008/T while for a half-sine transient, it should be set to less than 0.05/T , where T is the period. Ringing is the waveform distortion observed when the resonance of the accelerometer is excited with high frequency vibration components. For measurement error to be within 5%, the mounted resonance frequency of an accelerometer should be higher than 10/T , where T is the length of the transient. Waveform distortion due to ‘ringing’ can be reduced by introducing damping, or by using a mechanical filter for mounting the accelerometer, or by low-pass filtering the signal to compensate for the increase in sensitivity near the mounted resonance frequency. Environmental effects Temperature. The sensitivity, natural frequency and damping of an accelerometer may be affected by the temperature and its fluctuations. The piezoelectric accelerometer has a wide operating temperature range, typically −100◦ C to 250◦ C for an error of 1 dB. Temperature fluctuations produce charges in the piezoelectric elements (known as the pyroelectric effect), which in turn give rise to low frequency noise in addition to the acceleration signal. Shear-type accelerometers are almost 100 times less sensitive to temperature transients than compression types. If required, local heating or cooling can be applied to the transducer to reduce the effects of temperature on its performance. Nonuniform thermal expansion or compression may produce strain on the piezoelectric element, hence producing an electrical charge. Humidity. Most accelerometers are hermetically sealed so that they can be operated under extremely humid conditions. Acoustic Noise. Compression-type accelerometers generally are much more sensitive to acoustic noise than shear types. Commercially available shear-type

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accelerometers typically have an acoustic sensitivity of the order of 0.001 m/s2 for a sound pressure level of about 150 dB. Mass-loading effects The local vibration characteristics of a structure may be changed by the attachment of an accelerometer. The added mass M due to the attachment of an accelerometer becomes important when the accelerometer mass is more than one-tenth of the mass M of the structure. For frequencies up to 0.9 f n (mounted resonance frequency of the accelerometer), the natural frequency f n of the structure will be reduced by  f n according to the following equation:  fn −  fn M (8.16) = fn M + M Effects of different mounting techniques on frequency response The dynamic range and operating frequency range of an accelerometer can be significantly affected by different mounting techniques. International Standard ISO 5348-3 [8.3] recommends some mounting methods and the effects of these different mounting methods on the sensitivity and operating frequency range of an accelerometer have been reproduced in Figure 8.22. The advantages and disadvantages of various mounting methods have been discussed by Serridge and Licht [186]. 40

30

d a

e

b

dB

20 c 10

0

−10 0.1

0.5

1

5

10

20

kHz

Figure 8.22 Effects of different mounting techniques on frequency response of an accelerometer [24]. (a) hand-held probe; (b) thin double-sided adhesive; (c) magnet; (d) oil film and stud; (e) bees wax.

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Do Not Stress Vibrating Surface

Do Not Stress

The Cable Is Fixed to the Vibrating Surface

Figure 8.23 Attachment of accelerometer cable [110].

Cable noise Triboelectric charges will be produced when two dissimilar materials are rubbed against each other. Accelerometer cables should, therefore, be so attached and secured to minimize any excessive relative movement. Examples shown in International Standard ISO5348-3 [110] are reproduced in Figure 8.23. Cables should also be shielded from electromagnetic fields as a voltage can be induced across them. Ground loops Extraneous noise in the signals can be produced due to ground loops. Precautions should be made to ensure that the instruments are grounded at only one point.

8.4.3

Other vibration transducers

Displacement pickup There are various types of displacement pickups according to the principles used; for example, inductive, resistive, capacitive, and so on. Figure 8.24 illustrates a capacitive displacement pickup which offers the advantage of noncontact, high sensitivity and a wide frequency range. The vibrating surface, however, must be conductive and the dynamic range is normally limited. The capacitance between two parallel plates is inversely proportional to the distance between them. Impedance head Mechanical impedance is defined as the ratio of input force to output velocity. Impedance measurements are made basically to determine the natural frequencies,

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Figure 8.24 Capacitive type displacement pickup. (Bruel & Kjaer, Master Catalogue) stiffness, damping and mode shapes of a structure. An impedance head is used to make an impedance measurement at the driving point. It combines a force transducer and an accelerometer in one unit and is normally made of piezoelectric material. Figure 8.25 illustrates a typical piezoelectric impedance head.

Mounting Surface

1 2

3 4

5

9

4

6

8 1

7

Figure 8.25 Impedance head: (1) tapped hole; (2) housing (titanium); (3) seismic mass; (4) two piezoelectric disks; (5) force output socket; (6) silicone rubber; (7) driving point; (8) driving platform (stainless steel); (9) acceleration output socket. (Bruel & Kjaer, Master Catalogue)

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Exciters In order to determine the dynamic characteristics of structures, and machinery, etc. vibration exciters are used to provide the excitation so that vibration measurements can be made. The most commonly used exciter is an electrodynamic exciter, which is based on the principle of operation of a moving coil generator similar to that of a loudspeaker. A typical electrodynamic exciter is shown in Figure 8.26. Basically, a moving coil is attached to the driving spindle operating in a magnetic field set up by either a permanent magnet or electromagnet. The force produced at the driving spindle is proportional to the current supplied through the moving coil.

8.5

Frequency analyzers

Sound level meters equipped with octave filters offer a portable means for analysing the noise spectrum in field measurements. However, a more detailed analysis of the noise/vibration spectrum would require filters of much narrow bandwidth. Such frequency analyzers may be analogue or digital. Analogue frequency analyzers are essentially tunable heterodyne band-pass filters. Digital analyzers are either based on Fast-Fourier-Transform, i.e., constant bandwidth, or Digital Filter, i.e., constant percentage bandwidth algorithms. For a practical description of using

2

3

1

4

8

5

7

6

2

Figure 8.26 Electrodynamic exciter: (1) table; (2) threaded hole; (3) rubber dirt shield; (4) drive coil; (5) magnetic structure; (6) PM; (7) rubber foot; (8) flexures. (Bruel & Kjaer, Master Catalogue)

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various types of frequency analyzers, see [180]. A theoretical treatment can be found in [14].

8.6

Sound power and sound pressure

A basic measure of the acoustical output of a sound source operating in a particular medium is its sound power  which is independent of the distance from the source. It is defined by Equation 8.17 as follows:


I · d S =

= S





Ir d S = S

pu r d S

(8.17)

S

where I is the intensity vector (rate of energy flow per unit area), S is the hypothetical surface enclosing the source, Ir is the component of the intensity vector normal to S, p is the sound pressure (Pa), u r is the particle velocity normal to S (m/s), and the overbar represents the time-average. On the other hand, sound pressure, which is a scalar quantity often measured, gives an indication of the human response to noise. Unlike sound power, the level of sound pressure measured for a given sound source is dependent on the acoustic environment such as the amount of absorption and the characteristics of other noise sources existing in the operating environment. Hence, it is important to be able to determine the sound power of a source, as this data can be used to predict the sound pressure level produced by the source in different operating environments, or to compare its noise-producing capacity with other sources. There are various methods in determining the sound power of a noise source and these methods can basically be classified into two types, namely, indirect and direct methods. In the indirect methods, the sound power of a noise source is determined by measuring the sound pressure level in some particular acoustic environment such as in anechoic or reverberant rooms. The International Standards ISO 3740-3747 [99, 100, 101, 102, 103, 104, 105, 106] cover methods are in this category. With the advent of digital signal processing technique, the sound power of a source can now be determined directly by measuring the component of sound intensity normal to a given hypothetical control surface area enclosing the source, as given by ISO 9614 [107, 108, 109]. It must be pointed out that the sound power of a sound source has generally been assumed to be independent of the acoustic environment. However, Norton [168] has shown that depending on the appropriate sound power model used for the source under consideration, the radiated sound power, though independent of the distance from the source, is affected by the presence of reflective surfaces. Table 8.3, extracted from [168], shows the variation in sound power for different sound power models. The behavior of most sources encountered in practice is between a constant power and a constant volume model.

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Table 8.3 Variation in sound power for different sound power models. Sound power model Directivity Constant power Constant volume Constant pressure Source position Dθ  = 0  = 0 D θ  = 0 /Dθ Free space 1(+0 dB) 0 0 0 Center of a large 2(+3 dB) 0 20 (+3 dB) 0 /2 (-3 dB) flat surface Intersection of two large flat 4 (+6 dB) 0 40 (+6 dB) 0 /4 (-6 dB) surfaces Intersection of three large flat 8 (+9 dB) 0 80 (+9 dB) 0 /8 (-9 dB) surfaces

8.7 Indirect methods of sound power measurement 8.7.1

Determination of sound power in an anechoic/ semianechoic room

An anechoic room is a room in which free-field condition is simulated by lining the inside of the room with highly absorbent material so that the sound pressure is inversely proportional to the square of the distance from the sound source. The sound power of a source can be determined by measuring the average sound pressure level over a hypothetical closed surface in an anechoic room, since the intensity I is related to the rms pressure p in a free-field by Equation 8.18, so that the sound intensity level L I is equal to the sound pressure level L p . I =

p2 ρ0 c0

(8.18)

where ρ0 c0 is the characteristic impedance of the medium. The hypothetical enclosing surface can be spherical (in the case of a freefield) or hemispherical (in the case of a free-field over a reflecting plane. Figure 8.27 shows the microphone positions for measurements in a free-field over a reflecting plane (i.e., semianechoic condition) where the microphones at a given elevation are spaced 90◦ apart, so that the microphone positions are associated with equal surface areas. The sound power level L W is then calculated from the equation [101]. L W = L p + 10 log S

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(8.19)

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Source

Figure 8.27 Microphone positions for sound power measurements in a free-field over a reflecting plane.

where L p is the average surface sound pressure level in dB. L p = 10 log

N 1  0.1L pi 10 N i=1

where L pi is the band pressure level (dB) resulting from the ith measurement, N is the number of measurements, and S is the area of the test surface (m2 ) which is 4πr 2 for a test sphere of radius r in an anechoic room and 2πr 2 for a test hemisphere of radius r in a semianechoic room. Uncertainties in measurements under various conditions are specified in International Standards ISO 3744, 3745 and 3746 [102, 103, 104].

8.7.2

Reverberation room

The average sound absorption coefficient of all the surfaces of a reverberation room should not exceed 0.06 in order to ensure an adequate reverberant field. The minimum volume of the room increases with decreasing frequency of interest. For example, for 125 Hz octave band, a volume of 200 m3 is required. In a diffuse field, the intensity I is related to the characteristic impedance ρ0 c0 by I =

p2 . 4ρ0 c0

(8.20)

The sound power level L W of a source can be determined by measuring the average sound pressure level L p inside a reverberation room and calculated from L W = L p − 10 log T60 + 10 log V + 10 log[1 + (Sλ0 /8V )]

(8.21)

where T60 is the reverberation time of the room (s), V is the volume of the room (m3 ), S is the total surface area of the room (m2 ), and λ0 is wavelength at the center frequency of the frequency band (m). The relevant International Standard for sound power determination in a reverberation room is 3741 [100].

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8.7.3

217

Juxtaposition principle using a reference sound source

In this method, the sound power level Lw of a noise source is determined by the measuring the sound pressure levels produced by the noise source and by a reference sound power source using the following equation: L W = L W r + L p − L pr

(8.22)

where L W is the sound power level of noise source under test, L W r is the sound power level of reference source, L p is the sound pressure level due to noise source, and L pr is the sound pressure level due to reference source. The sound pressure level due to noise source is first measured. With the reference source switched on, its sound power output is increased until the sound level meter reading is 3 dB higher than that due to the noise source alone. The sound power output from the noise source is then equal to that from the reference sound source. The relevant International Standard for sound power determination in a reverberation room is 3747 [105, 106].

8.8 8.8.1

Direct method of sound power measurement — sound intensity technique Historical perspective

Although the first device for sound intensity measurements was patented by Olson [170] in 1932, it was only until 1977 when Fahy [54] and Chung [32] independently applied digital signal processing techniques to sound intensity theory that commercial systems became available. The sound intensity technique enables sound power measurements to be made directly instead of the traditional method conducted under controlled acoustic environment (such as anechoic, semianechoic or reverberant rooms) via sound pressure level measurements. It also offers other advantages such as external noise suppression capability, noise source identification and noise ranking.

8.8.2

Theoretical background

The theory of the sound intensity technique has been given elsewhere, see for example, Chung [32], Fahy [58], and Gade [69, 68]. To enable subsequent discussions, the theory is briefly outlined as follows. The equation governing the motion of the fluid particles is the Navier-Stokes equation, for example, Schlichting [185] which is described by ∂ U

1 + (U · ∇)U = ∇ P + γ ∇ 2 U

∂t ρ

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(8.23)

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where U is the instantaneous velocity vector, P is the instantaneous pressure ∇ is the ‘del’ operator, t is the time, and γ is the kinematic viscosity. By considering small perturbations so that P = Pa + p U = U 0 + u + high order terms where Pa is the ambient pressure, U 0 is the mean velocity vector in the medium, p is the fluctuating pressure, and u is the fluctuating velocity vector (due to turbulence and acoustic velocity), and neglecting viscous effects, then Equation 8.23 yields the following linearized Equation 8.24 of the first order: ∂ u

1 + (U 0 · ∇)

u + (

u · ∇)U 0 = ∇ P ∂t ρ

(8.24)

In the absence of mean flow (i.e. U 0 = 0) and turbulence, Equation 8.24 is reduced to the following form for direction r : ∂u r ∂p =− (8.25) ∂t ∂r Equation 8.25 can be approximated by measuring the pressures p A and p B detected by two microphones closely spaced at a distance r (as shown schematically in Figure 8.28) along the direction r and applying finite differencing as given in the following equation:
1 ur = (8.26) ( p B − p A )dt ρr ρ

Let P( f ) and U ( f ) be the Fourier transform of p(t) and u(t) respectively, then 1 [PA ( f ) + PB ( f )] 2

P( f ) = U( f ) =

1 [PB ( f ) − PA ( f )]. jωρr

(8.27)

∆r

r A

Spacer

Figure 8.28 Sound intensity microphone pair.

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B

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The intensity Ir (r, f ) is then given by the expected value of P ∗ ( f ) and U ( f ), where P ∗ ( f ) is the complex conjugate of P( f ) so that m(G AB ) 1 (8.28) +j (G A A − G B B ) ωρr 2ωρr √ where Ir is the active intensity, Jr is the reactive intensity, j = −1, m is the imaginary part, G AB is the cross-power spectrum between signals A and B, G A A is the auto spectrum of signal A, G B B is the auto spectrum of signal B, and ω is the circular frequency in rad/s. Hence, the active intensity spectrum is related to the imaginary part of the cross-power spectrum between the two microphone signals. Ir + j Jr = −

Ir = −

8.8.3

1 m(G AB ) ωρr

(8.29)

Sound intensity probe

The microphones in a sound intensity probe can be mounted side-by-side, or backto-back, or face-to-face. Research shows that a face-to-face mounting configuration is desirable in most situations. Typical characteristics of the directional response of a sound intensity probe at two different frequencies are shown in Figure 8.29. The response follows basically a ‘cosine’ variation (approximating a figure of ‘8’), so that if the measuring axis is perpendicular to the sound intensity vector, the response is minimum. This property is widely-used as a quick means of identifying noise source (null-search method) and is carried out as follows. The intensity probe is swept with its axis parallel to the plane of suspected noise source until the sign

°

QP 510

2

0° 15

180°

0°

0° 15

21

° 180°

Brüel & K jo e r

120

Brüel & K jo e r

(a)

°

10 811420

0° 24

2

0°

10 0

2

21

20

QP 510

QP 510

0° 24

50 30 270° 40

120

20

10 0

50 90° 40 30

20

∆r = 12 mm 8 kHz 10

∆r = 12 mm 500 Hz

20

2

300

50 90° 40 30

° 60

QP 510

5 dB

° 60

50 30 270° 40

30 °

°

°

330

30 °

330

5 dB 300 °

Brüel & K jo e r

0°

0°

Brüel & K jo e r

811353

(b)

Figure 8.29 Directional response of sound intensity probe: (a) 500 Hz; (b) 8 KHz.

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PB

10 20

PA 30 40

1 dB

PB PB — PA

50 60 70 80

0

2

4

6

8

10 kHz

(a) dB PB

10

PA

20 30 40 50

PA

60

1 dB

PB

70 80

PB — PA 0

2

6

4

8

10 kHz

(b)

Figure 8.30 Frequency response of a sound intensity microphone pair: (a) probe aligned with the direction of sound propagation; (b) probe perpendicular to the direction of sound propagation [69]. of the intensity measured changes. The location of sign change corresponds to the location of noise source because sound incident at 85◦ and at 95◦ will give the intensity of opposite sign. The typical frequency response of the two microphones in the probe to sound incident at 0◦ and 90◦ is shown in Figure 8.30.

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8.8.4

221

External noise suppression

According to Gauss theorem, if there is no source or sink within a measurement volume, then the sound power as given in Equation 8.16 is zero. This property offers a significant advantage in using sound intensity technique for sound power determination, as external noise will not contribute to the sound power of the noise source under test provided that the external noise is stationary and there is no absorption within the measurement volume. Thus in situ sound power measurements with high accuracy can theoretically be achieved. In practice, the accuracy is limited by the dynamic capability of the system which will be discussed later.

8.8.5

Error considerations

Possible sources of error in applying the sound intensity technique have been given by Fahy [58]. The errors arising from sound intensity measurements are mainly due to: • low frequency limit determined by phase mismatching between two channels; • high frequency limit due to finite difference approximation used in evaluating particle velocity; • near field limitations; • random error. Low frequency limit It has been shown [58] that the active and reactive intensity are given respectively by Ir = −

p 2 ∂ ρck ∂r

p2 ∂ Jr = − ρck ∂r



| p| ln p0

 (8.30)

where ∂/∂r is the phase gradient of the sound field, and k is the wavenumber. Hence, it can be seen that the low frequency limit of a sound intensity system is related to the phase mismatching of the microphones. Figure 8.31 shows the typical phase matching characteristics of a pair of 1/2-inch microphones. For an accuracy to within 1 dB, the phase change of the sound field over the microphone separation distance r has to be greater than 5 times the phase mismatch.

High frequency limit On the other hand, the high frequency limit of a sound intensity system is due to the finite difference approximation described by Equation 8.31 as can be readily seen from Figure 8.32. For a plane sinusoidal wave, the ratio between the measured intensity I and the true intensity I is given by

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1°

0

20

50

100 Hz

200

500

Figure 8.31 Phase matching of a typical sound intensity microphone pair [68].

I sin(kr + φ) = (8.31) I kr where φ is the phase mismatch. Hence the high frequency limit is determined by kr  1. For an accuracy to within 1 dB, 6r < λ

(8.32)

where λ is the wavelength.

Near field limitation The accuracy of sound intensity measurements in the near field of a source is limited by the intensity change along the probe. For a spherical wave from a monopole source,

  −1 I sin(kr ) 1 r 2 = . 1− I kr 4 r For good accuracy, r/r  1. For an accuracy within 1 dB, • 1.1r < r monopole

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(8.33)

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223

∆P ∆r

∂P ∂r

∆P ∆r

∂P ∂r

Figure 8.32 Accuracy of finite difference approximation. • 1.6r < r dipole • 2.3r < r quadrupole Random error It has been shown, for example, [229] that the random error is given by 1 1 − γ2 random ≈ √ 1+ 2γ 2 sin2  BT

(8.34)

where B is the bandwidth, T is averaging time, γ is the coherence function between 2 signals, A and B, and  is the phase difference between signals A and B.

8.8.6

Dynamic capability and pressure-intensity index

The pressure-intensity index L K is defined as the difference between the sound pressure level L p and the sound intensity level L I and is related to the phase change  over the spacer distance r for a wavelength λ by  

ρc  λ φ L K = L p − L I = 10 log . (8.35) + 10 log r 3600 400 The relationship expressed in Equation 8.35 has been displayed as L I − L p in an intensity index nomogram in Figure 8.33 [229]. Owing to the phase-mismatch between the two channels, the intensity measured is not zero, even if the two channels receive an identical signal and the pressure-intensity index measured under this condition is known as the residual

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∆r 6 200 50 12 mm mm mm mm 0 10

−10 0

−10

dB

L – L = L = –10 log I P K

0.2 0.5° ° 0.1 0.0 ° 5° 0.0 2 0.0 ° 1 0.0 ° 05° 0.0 0 0.0 2° 01°

2°

1°

5°

k∆r Φ

10°

20°

50°

100 200 ° °

0 dB, ∆r = 200 mm 0 dB, ∆r = 50 mm

200 0 dB, ∆r = 12 mm ° 100 ° 0 dB, ∆r = 6 mm 0 −10 50° −20 2 0° −10 10° −10 −20 5° −30 2° −20 1° −20 −30 0.5° −40 0.2° 0.3° Maximum −30 0.1° Phase Mis-match in Type 3360 0.0 0.00 0.01 0.0 5° 0.00 0.00 2° 5° 2° 1° ° 2.5 Hz 5 5 10 20 10 20 40 80 160 315 630 1.25 2.5 16 4 8 16 31.5 63 125 250 500 1 kHz 2 4 8 3.15 6.3 12.5 25 800 1.6 3.15 6.3 12.5 50 100 200 400 0

Figure 8.33 Intensity index nomogram. (Bruel & Kjaer Catalogue, Techn. Rev., No. 4, 1985, pp. 3–31.) pressure-intensity index L K O . The dynamic capability L D of the sound intensity measuring system is therefore defined as LD = LKO − K

(8.36)

where K = 7 dB for an accuracy within 1 dB. Figure 8.34a shows the effect of the direction of the sound field on the phase measured, while Figure 8.34b displays the residual reactivity index and dynamic capability of a typical sound intensity measuring system. By using the definition for L K and L K O , the error L e due to phase mismatch between the two channels can, therefore, be expressed as L e = 10 log[1 + 100.1(L K −L K O ) ].

(8.37)

The relevant International Standards for measurement procedures to determine the sound power level using the sound intensity technique are ISO 9614 [107, 107, 109].

8.9 8.9.1

Standard for testing acoustic performance of rotating electrical machines Background

As described in Section 8.6 and by Wang et al. [229], sound power is the proper and more appropriate quantitative measure of the acoustic performance of an electric machine. International Standard IEC 60034-9, 4th edition [94] specifies the

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25 Residual Pressure-Intensity Index LK0 ∆ rcosθ λ θ

∆r

φ=

360° Level (dB)

φ=

∆r 360° λ

20 Dynamic L D Capability 15

10

7 dB

250 500 2k 4k 1k 1/3 Octave Center Frequency (Hz)

125

Figure 8.34 Pressure-intensity index: (a) pressure-intensity index and phase; (b) residual pressure-intensity index and dynamic capability. maximum allowable overall A-weighted sound power level at no-load and rated load for electric motors, based on power rating and cooling classifications (see Table 1.6). The current International Standard for determining the acoustical performance of rotating electrical machines in terms of the sound power level is ISO 1680:1999 [112]. As pointed out by Lisner and Timar [137], this standard is primarily for testing machines under steady state conditions at the nominal speed, nominal voltage magnitude and nominal load (generally no-load). The standard does not really specify how to determine the acoustic performance of electrical machines with variable speed operations (in particular, inverter-driven machines) and load torque variations. The following recommendations to improve the current standard for motors operated at variable speeds have been made by Lisner and Timar [137]: • motors are to be tested over the full operating speed range by sweeping the supply across the frequency range at a rate sufficiently slow enough for narrow resonant peaks to be identified; • motors are to be tested at a number of different test speeds to assess the effects of the inverter’s switching frequencies; • motors are to be tested at nominal voltage +10% as the worst-case scenario for saturation harmonic effects; and • motors are to be tested by slowly sweeping the loading torque from no-load to full-load. Basically, ISO 1680 [112] describes how to set up a motor for the sound power level to be determined. The guidelines for using International Standards for the determination of sound power levels are given in ISO 3740:2000 [99]. As described in Section 8.7, the sound power level of a noise source can be determined using sound pressure measurements in either an anechoic room [103], a semianechoic room [102, 103, 104] or a reverberation room [100, 101]. Alternatively, a

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sound intensity measurement system could be used to determine the sound power level in any environment where the background noise is stationary, as described in Section 8.8. Determination of sound power levels could be made using sound intensity measurements at discrete points [107] or by scanning [108, 109]. There are also less accurate methods based on sound pressure measurements using a reference sound power source [105, 106].

8.9.2

Acoustic tests on an induction motor

The sound intensity method is particularly useful to determine the sound power level of an electric motor when it is loaded by another electric machine, as the technique can discriminate against stationary background noise produced by the load motor. Figure 8.35 shows the experimental set up for determining the sound power level of a 2.2 kW induction motor using the sound intensity technique by scanning in an anechoic room [94]. The test motor was mounted over a reflective aluminum plate and was separated from the load motor (1.2 kW DC motor) by a 10 mm thick aluminum plate to further reduce the influence of the noise generated by the load motor. The wire frame marks out the four surfaces, over which the sound intensity probe was scanned. The test motor was driven by three different controllers: a Benchmark controller (with almost purely sinusoidal output), a pulse width modulation (PWM) inverter and a PWM inverter with random modulation. Details of the three controllers are given in [94]. As shown in Figure 8.36 the voltage spectra supplied by the Benchmark controller and the random modulation inverter are predominantly

Wire Frame

Load Motor

Test Motor

Torque Transducer

Figure 8.35 Laboratory set-up for load test of a 2.2-kW induction motor using the sound intensity technique in an anechoic room [229].

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Acoustic and Vibration Instrumentation (a)

227

20

Voltage, dB

0

−20 −40 −60 −80

(b)

20

Voltage, dB

0

−20 −40 −60 −80

(c)

20

Voltage, dB

0 −20 −40 −60 −80 0

400

800 Frequency, Hz

1200

1600

Figure 8.36 Voltage spectra supplied by three different controllers: (a) Benchmark; (b) PWM inverter; (c) PWM inverter with random modulation [229].

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Aerodynamic & Mechanical Noise Benchmark

Sound Power Level (dB)

60

40

20

0 100

1000 Frequency (Hz)

10000

(a)

PWM

Sound Power Level (dB)

60

Random

40

20

0 100

1000

10000

Frequency (Hz) (b)

Figure 8.37 Measured sound power spectra for the motor driven by different controllers at 450 rpm and no load: (a) Benchmark, aerodynamic, and mechanical; (b) PWM and random modulation inverters [229].

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80

Sound Power Level (dB)

PWM 70 Random Modulation 60 Benchmark 50 Aerodynamic & Mechanical Noise 40 400

500

600

700

800

900

1000 1100 1200 1300 1400 1500

Rotor Speed (rpm)

Figure 8.38 Variation of sound power level with rotor speed at no load [229].

80

Sound Power Level (dB)

PWM full load 75 PWM no load 70 Random Modulation full load

65

60 Random Modulation no load 55 400

500

600

700

800 900 1000 1100 1200 1300 1400 1500 Rotor Speed (rpm)

Figure 8.39 Comparison of the total sound power level between full load and no load for two different inverters [229].

sinusoidal with the harmonics at least 30 dB below the fundamental whereas the voltage spectrum of the PWM inverter contains a number of harmonics with magnitude even higher than the fundamental. The sound power spectrum due to aerodynamic and mechanical noise was first measured with the motor uncoupled from the supply and driven by the 1.2 kW DC motor. The measured sound power spectra for the motor driven by the benchmark controller at 450 rpm and no load is

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compared with that due to the aerodynamic and mechanical noise in Figure 8.37a. As expected at this low speed, the total sound power level is not dominated by the aerodynamic and mechanical noise. Figure 8.37b shows that the sound power spectrum for the motor driven at 450 rpm and no load by the PWM inverter contains numerous harmonics compared with that driven by the random modulation inverter and is higher in level by 20 dB on average for frequencies between 1000 and 5000 Hz. On the other hand, it can be seen from Figure 8.37 that there is very little difference between the sound power spectra for the motor driven by the benchmark controller and the random modulation inverter. The variation of the total sound power level with the rotor speed for the test motor driven at no load by three different controllers is compared with that due to aerodynamic and mechanical noise in Figure 8.38. It can be seen that the sound power level due to aerodynamic and mechanical noise increases by 12 dB per doubling of speed, and that it becomes more significant as it approaches the rated speed of 1500 rpm. The effects of harmonics in the supply on the sound power level are much more significant at low rotor speeds (such as below 1000 rpm) where electromagnetic noise dominates. Even at higher rotor speeds such as at the rated speed of 1500 rpm, the difference in the test motor can still be rather significant, being 6 dB between being driven by the Benchmark controller and that driven by the PWM inverter. The difference in the total sound power level between full load and no load at various rotor speeds for two different inverters is shown in Figure 8.39. As have already pointed out by Lisner and Timar [137], it does not necessarily follow that the sound power level at full load is always higher than that at no load. This is because the effects of load on the emitted sound power level depend on a large number of factors including the motor resonant frequencies, the motor speed, the harmonics in the supply and load-induced slip.

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9 Numerical Analysis 9.1 Introduction As described in Chapter 2, the ability to quantify the electromagnetic field and hence the electromagnetic force in an electric motor is essential to the prediction of acoustic noise from the motor. The process for predicting the acoustic field due to the operation of an electric motor can be illustrated in Figure 9.1. This process involves developing three models, namely, an electromagnetic model for estimating the electromagnetic force distribution in the motor, a structural model for describing the vibration behavior of the motor structure, and an acoustic model for predicting the acoustic response as a result of the electromagnetic excitation. All three models can be developed analytically. For example, the classical magneto-motive force and permeance wave theory has been used traditionally to analyze the distribution of the electromagnetic force in induction motors, and the effects of various parameters such as current harmonics, magnetic saturation, rotor eccentricity on its distribution. While the frequencies of harmonics due to different sources in the force spectrum could be clearly identified using the classical theory [31, 200, 248], it would be quite difficult to predict the magnitudes of these harmonics with accuracy. Analytical models can often provide good insight into the physics of acoustic noise generation. For example, by using an analytical approach, the lowest limit of the acoustic noise radiation for a variable speed induction motor can be determined [201]. With the advent of computer technology and numerical techniques, numerical methods such as the finite element method (FEM) have been widely used in modeling the detailed distribution of the electromagnetic force in the air gap, the vibration, and acoustic response of the motor structure [113, 153, 152, 163, 209, 232, 230, 246, 247, 252]. Numerical methods enable the assessment of the effects due to intricate structural details such as the tooth shape, slot depth, end shields, which cannot be otherwise made with analytical models.

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Noise of Polyphase Electric Motors Voltage, Current

Electromagnetic Model (FEM) Electromagnetic Force Structural Model (FEM)

Acoustic field or Acoustic Power Measurements

Validate Model Testing Model Update

Vibration Velocity Acoustic Model (BEM)

Validate

Sound Radiation Efficiency Measurements

Acoustic Field & Acoustic Power

Figure 9.1 Procedure for predicting acoustic power from an electric machine (BEM: boundary element method) [232]. In order to establish confidence in the modeling procedure using numerical methods, the development of each model generally requires validation by experimental measurements. While it is not straightforward to validate the electromagnetic force distribution with measurements, experimental modal testing is usually used to validate and update a structural model, and sound radiation efficiency measurements are used to validate an acoustic model. Recently, the vibration response due to electromagnetic forces was determined only from a 2D electromagnetic force model and a 2D structural model using the finite element method [113, 163]. As a 2D structural model cannot capture the vibration modes in the stator’s axial direction and the influence of the end-shields, validation of simulation results using actual measurement data can be misleading. For virbo-acoustic studies, it would therefore be imperative to use both a 3D structural model and a 3D acoustic model for the prediction of the vibration response and the acoustic radiation. The process for developing each model as shown in Figure 9.1 with the ultimate objective to enable evaluation of the acoustic performance of an electric motor is described in detail in this chapter.

9.2

FEM model for radial magnetic pressure

There have been numerous studies using the finite element method for modeling the electromagnetic force distribution in a motor, for example, [252]. There are many commercially finite element codes that can be used, for example, ANSYS

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and OPERA. Despite the rapid increase in computer speed and numerical techniques, modeling the electromagnetic field distribution in a motor remains a challenge especially for 3D and transient/nonsteady analysis because the memory and computing time required is prohibitively large for it to be incorporated in a routine design cycle. Consequently, most FEM electromagnetic models have been limited to 2D models [3, 152].

9.2.1

Induction motor

In this section, the process for modeling the electromagnetic force distribution in a variable speed induction motor is described. While there are many approaches in modeling the electromagnetic force distribution for an induction motor driven by an inverter, they would require applying currents to the finite element model and solving the electromagnetic field governing equations in the time domain. This procedure is illustrated with the following example for a Brook Crompton Betts three-phase, 4-pole, 415 V, 50 Hz, 2.2 kW induction motor with 44 rotor slots and 36 stator slots driven by a pulse width modulation (PWM) inverter. The sinusoidal PWM inverter was type MSC2000 manufactured by Zener Electric with the switching frequency being proportional to the test speed of the motor at a rate of 21 Hz/Hz. At the rated speed of 1500 rpm for the motor, the switching frequency was 1050 Hz [3]. The example motor is shown in Figure 9.2. The objective was to calculate numerically by the finite element method the time and spatial distribution of the electromagnetic force in the motor for a given time history of input voltage and current conditions. The method used here requires the rotor currents to be externally calculated and imposed on the rotor in the

Casing

Isolator

Stator (inside)

Support

End-Shield

Slots

Rotor Shaft

Base Plate

Figure 9.2 Four-pole, 2.2 kW, 50 Hz induction motor [224].

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+

R1

L1

R2 + +

V1

i1

−

i2

Lm

jw y 2 −

−

Figure 9.3 Equivalent circuit of an induction motor.

finite element (FE) model separately by prescribing the measured stator currents. The stator referred rotor currents including harmonics could be calculated using a multiparameter state space model. An example of a four parameter state space model is shown in Figure 9.3. The effective rotor to stator turns ratio was found by adjusting it so that the torque-speed curve calculated by the FE model agreed with that estimated by the state space model, and measured experimentally for a given operating condition. Quasistatic analysis in the time domain was used to allow for the rotation of the rotor and the interaction between the force harmonics due to the saturation. In this model, since the rotor referred stator current was not estimated, the stator slot harmonics induced in the rotor slots could not be calculated. The 2D FEM model was developed using a commercial finite element code ANSYS. This 2D model would be a good approximation provided that the variation of the force in the air gap along the axial direction is not significant. Because of symmetry and by applying odd periodicity conditions, only a quarter section of the 4-pole stator and rotor was modeled. Care must be taken to ensure the meshes in the air gap overlap properly. The frequency range simulated is determined by the number of nodes in the air gap and the smallest simulation time step is determined by the time taken between nodes in the air gap. The minimum number of nodes in the air gap for this model was 198 and there were 8432 nodes for the rotor and 12661 nodes for the stator. The rotation of the rotor can be implemented using the “moving band” method [16]. However, this method, implementing a layer of elements which distort between the two layers of nodes, requires the whole motor to be modeled. By using an overlapping layer of nodes from the stator and rotor subject to constraints relating the potential of these nodes, the effect of rotation can be simulated without physically rotating the meshes. There are four distinct cases that need to be considered for one full electrical cycle as shown in Figure 9.4. For each time step, these relationships have to be updated. As the nonlinear properties of the iron material were modeled, the solution of the magnetic field potential has to be iterated until a convergence criterion has been satisfied. The flux distribution from an ideal current supply with the motor operating at 1500 rpm is shown in Figure 9.5.

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235 −

−

−

−

+

+

+

−

−

+

+ (a) Case 1

+

+

+

+

(b) Case 2

−

−

−

+

+

+ (c) Case 3

−

+ +

(d) Case 4

Figure 9.4 Relationship between overlapping nodes to simulate the effect of rotor rotation.

In the calculations, the time period analyzed was chosen as a quarter of one rotor cycle, and divided into 198 time steps. Thus, the bandwidth of the force spectrum data is 30 Hz and 100 Hz for 450 rpm and 1500 rpm, respectively, corresponding to an upper frequency limit of 3 kHz and 9.9 kHz respectively. For an ‘ideal’ sinusoidal inverter, the measured stator current data for the PWM inverter were low-pass filtered so that only the fundamental component was input to the four-parameter state space model shown in Figure 9.3. By applying fast Fourier transform to the calculated radial force distributions, the spatial distributions of the force spectrum for an ideal sinusoidal and a PWM inverter are shown in Figure 9.6. The rotor harmonics are clearly identifiable in Figure 9.6b for the PWM inverter. A global comparison of the effect of the controller can be made by examining the total radial electromagnetic force acting on the stator under no load for the ideal sinusoidal drive and the PWM drive. By integrating the spatial distributions of the

Figure 9.5 Flux distribution showing the rotation of the rotor at 1500 rpm [232].

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Figure 9.6 (See color insert.) Spatial distribution of radial electromagnetic force spectrum for a motor operating speed at 1500 rpm and no load with two different controllers: (a) sinusoidal; (b) PWM. force spectra in Figure 9.6, the total force spectra of the ideal sinusoidal and that driven by the PWM inverter
at 1500 rpm are shown in Figure 9.7. Here, the total force F is defined as |F | = m | f m |2 , where f m is the radial force component at each node along the circumferential direction. It is clear that the total force for the PWM drive is an order of magnitude higher than that of the ideal sinusoidal drive. While these force results could not be validated by experimental measurements, they do show a qualitatively correct trend and were used as the input to the structural

1000 ideal sinusoid, no load PWM inverter, no load

Total Force (N)

100 10 1 0.1 0.01 0

2000

4000 6000 Frequency (Hz)

8000

10000

Figure 9.7 Comparison of the total force spectra between an ideal sinusoidal drive and a PWM drive at 1500 rpm and no load.

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acoustic models in the later sections for predicting the acoustic noise radiation. These calculated acoustic noise radiation results were validated by measurements.

9.2.2

Permanent magnet synchronous motor

The mechanical structure of the permanent magnet (PM) synchronous motor can be described by the modal analysis technique [213]. Any mechanical system with N degrees of freedom can be described by the matrix Equation 1.2 in which {F(t)} is the force vector. The displacement in a set of N degrees of freedom can be expressed by the vector {(t)}i where the subscript i is the number of the mode shape. When the set of M modal shapes {}i is known for a certain frequency range, the stator structural deflection shape can be expressed as [213] {(t)} =

M 

{}i · qi (t)

(9.1)

i=1

where the general coordinate qi must comply with the following equation m i q¨i + ci q˙i + ki qi = i (t).

(9.2)

The modal parameters m i , ki , and ci of the mode shape {}i in the Equation 9.2 can be found experimentally from the real structure. The function i (t) is the generalized force acting on the mode shape i. The constant M in Equation 9.1 is the number of considered eigenmodes. Using the mass matrix [M] and stiffness matrix [K ] in Equation 1.2, and natural frequency ωi of the eigenmode i, the eigenshapes {}i satisfy the following set of equations {}T · [M] · {}i = 1

(9.3)

{}T · [K ] · {}i = ωi2

(9.4)

The mode shape is normalized to the mass matrix. Thus, the modal parameters m i , ci , and ki can be rewritten as m i = 1;

ci = 2ξi ;

2 ki = ωdi + ξi2

(9.5)

where ωdi is the damped eigenfrequency and ξi is the damping coefficient obtained from the experiment. The generalized force i (t) in Equation 9.2 can be mathematically expressed as i (t) = {}iT · {F(t)}. (9.6) Physically, the generalized force i (t) stands for the work done by the external force vector that deforms the stator system according to the mode shape i. The FEM can be used to calculate i (t). The PM excitation system can be replaced by conductors carrying the fictitious current I f that produces the same

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magnetic field as PMs. For stator currents I1A , I1B , I1C and fictitious rotor current I f , the magnetic energy can be expressed as Wmagn (I1A , I1B , I1C , I f ) =

3, f   k=1

ψk

idψ.

(9.7)

0

Superimposing small currents i 1A , i 1B , i 1C on the currents I1A , I1B , I1C , Equation 9.7 expresses an increase in the magnetic linkage flux [213], i.e., Wmagn (I1A + i 1A , I1B + i 1B , I1C + i 1C , I f + i f ) =

3, f   k=1

ψk +ψk

idψ.

(9.8)

0

Since the current i f is an induced current, for a PM synchronous motor i f = 0 and for an induction motor i f =  0. The currents i 1A , i 1B , i 1C are small, so that the magnetic energy equation becomes [213] Wmagn (I1A + i 1A , I1B + i 1B , I1C + i 1C , I f ) = Wmagn (I1A , I1B , I1C , I f ) +

3  ∂ Wmagn k=1

∂ Ik

ik .

(9.9)

The change of the energy in the system as a result of interaction
of the fundamental field and small injected currents is given by the last term 3k=1 (∂ Wmagn /∂ Ik )i k . Thus,

Wmagn (i 1A , i 1B , i 1C ) = ψ A i 1A + ψ B i 1B + ψC i 1C (9.10) where ψ A , ψ B , ψ A C represent phase flux linkages excited by the fundamental field. For constant currents and the mode shape i the generalized force i (t) given by Equation 9.6 can be found as i (t) = i 1A

∂ψ A ∂ψ B ∂ψC + i 1B + i 1C ∂ fi ∂ fi ∂ fi

(9.11)

where f stands for the deformation, i.e., the partial derivation ∂U/∂ f expresses the change in U due to geometric deformation of the stator defined by the modal deformation f ∗ {} [213]. If the mode shape i is known, the generalized force i (t) given by Equation 9.11 can be found on the basis of the FEM. Using the modal analysis concept, the real magnetic forces do not need to be calculated. A standard FEM approach and magnetic vector potential can be used to find flux linkages ψ A , ψ B , ψ A C and magnetic field distribution in the cross section of a PM synchronous machine with a deformed stator [213]. An accurate numerical analysis of magnetic field requires the assumption that the magnetic field is not only created by PMs, but also by the eccentric motion of the rotor through the variation of the air gap [115]. Also the coupling effect between the magnetic and mechanical field needs to be considered. The full set

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of equations to be solved should contain: • equations of magnetic field; • equations of external electric circuit; • mechanical equations of motion. These cross-coupled equations can be solved simultaneously. The PM synchronous and brushless motors are inverter-fed motors. Thus, Maxwell’s equations describing the magnetic field are supplemented by voltage equations [115], i.e.:

× ∇



1

∇×A µ



= J ext +

Vi − VN = Ri I1i + L i



1

∇×M µ

dψ d I1i + dt dt

 (9.12)

(9.13)

is the magnetic vector where µ is the magnetic permeability of the material, A

is potential, Jext is the current density supplied by the external power source, M the magnetization vector of a PM, dψ/dt is the electromotive force (EMF), and subscript i denotes the phase A, B, or C. The following Newton-Euler’s equation determines the mechanical motion of a PM synchronous motor for six degrees of freedom of a general rigid rotor [115]: 2 

magn + F

centr + F

bear ) = m d r G (F (9.14) dt 2 

(T magn + T bear ) =

G dH dt

(9.15)

G is the angular momentum with respect to where r G is the displacement vector, H

is the force, and T is the moment. The force and mothe mass center of the rotor, F ment have three components: unbalanced magnetic, centrifugal Fcentr = me2 , and bearing reaction. The magnetic field can be determined by the nonlinear timedependent method. The unbalanced magnetic force and torque can be calculated with the aid of the Maxwell stress tensor. Mechanical motion Equations 9.14 and 9.15 can be solved numerically using the Runge–Kutta method.

9.3

FEM for structural modeling

Early vibro-acoustic studies of a motor structure were focused on the stator core of the motor. Many analytical methods based on the vibration theory of shells were applied to calculate the natural frequencies and modal sound radiation efficiencies of simplified single stator models, as described in Chapters 5 and 6. While this

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B&K2032 Analyzer

Charge Amplifier Charge Amplifier HP-300 Computer

Shaker Impedence Head

Motor

Accelerometer

Figure 9.8 Schematic diagram of experimental set-up for modal testing [224].

type of vibration analysis has been extended to a stator structure consisting of a thin frame and a thick laminated core, these models are still far too simple to account for the complicated structural details of a modern motor. Since the 1970s, the finite element method has been used to investigate the effects of the motor structural details in the modal analysis [166, 224, 235, 247]. The acoustic power radiated from an axisymmetric motor model including the end-shields was calculated using the FEM [252]. The motor structure is excited by the electromagnetic force generated in the gap between the rotor and the stator. While a 2D FEM electromagnetic model might be adequate if there is no rotor skewing, a FE structural model must be three-dimensional in order to account for the vibration modes in the axial direction, and other significant effects due to structural details such as casing, the slots in the casing, end-shields, and support. In developing a proper modeling procedure, a structural model should be validated using experimental modal testing measurements because the structural dynamic behavior not only depends on the geometry, but also on the material properties, and the coupling conditions at the boundary. Details of experimental modal testing have been described by Ewins [53]. The setup for modal testing of the example motor using an electrodynamic shaker driven by a random noise signal is shown in Figure 9.8 and the measurement grid is given in Figure 9.9. The FEM structural modeling was developed using ANSYS with a step-bystep approach [18]. As the stator is a dominant noise source, the stator model initially included the intricate details of all the teeth as shown in Figure 9.10a. The stator core, made of laminated steel, was 92 mm long, 160 mm in diameter, and 12.35 mm thick with 36 teeth 20 mm deep. Detailed results of this stator model compared with that of a cylindrical shell model without teeth indicate that for this particular stator in which the total mass of the teeth is much smaller than the total mass of the stator and windings, a single cylindrical shell model is a reasonably good approximate model of the stator [224] by treating

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Figure 9.9 Measurement grid for experimental modal testing [224].

the teeth and windings as extra mass distributed throughout the stator in the model. The next step was then to model the stator and the casing with slots in the casing using a 2-cylindrical-shell model as shown in Figure 9.10b with a total number of 3041 nodes and 2127 elements. Following [72], the stator was modeled as an orthotropic structure with the Young’s modulus in the axial direction being 2% of that in the circumferential direction. Results obtained for mode numbers greater than 4 with this FEM model compare very well with the experimental modal testing results obtained by suspending the motor without the rotor and isolators Figure 9.11. As shown in Figure 9.12, the calculated mode shapes agree quite well with those obtained by experimental modal testing. Finally, the motor structure was modeled as two concentric cylindrical shells, one for the casing with 1128 quadrilateral shell elements and the other for the stator with 720 solid elements. The full model as shown in Figure 9.13 has a

(a)

(b)

Figure 9.10 FEM structural models: (a) stator stack; (b) enclosure [224].

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Rubber Bands

Stator

Figure 9.11 Test set-up of the motor without the rotor and isolators.

total number of 4480 nodes and 3423 elements respectively. Comparison between results obtained with this FE model and experimental modal testing indicate that the rotor and isolators are only important for a low frequency response. Furthermore, as shown in Table 9.1, both the natural frequencies and the mode shapes for the first 20 modes of the motor structure can be predicted reasonably well. Except for the modes that were not captured, the average relative error in natural frequencies was 7.5%, better than that reported in [163]. As the validation of an FE model is dependent on the experimental modal testing, it is important to ensure that the modal testing results are of good quality. The radiated sound power spectra of the motor driven by an ideal sinusoidal inverter, a PWM inverter (inverter 1), and a PWM inverter with random modulation (inverter 2) have been measured using the two-microphone technique described in Chapter 8. As can be seen in Figure 9.14, for almost each peak in the sound spectra, a vibration mode has been identified, indicating significant noise being caused by the coincidence between the harmonics of magnetic forces and structural natural frequencies and the reliability of the modal testing results. It is rather difficult to determine the contact pressure between laminations in the stator and hence the Young’s modulus in the axial direction. In our example, we used an estimate of 2% as determined by Garvey [72]. Recently, a systematic and ‘almost automated’ numerical approach, known as model updating, has been developed to use experimental modal testing results to produce improved FE models [208] by adjusting model parameters (such as Young’s modulus in the axial direction in this case) within reasonable bounds. One of the major criteria used in this approach is the so-called MAC (Modal Assurance Criterion) which is basically a correlation of the numerical and experimental mode shape. The model updating procedure, as shown in Figure 9.15, is an iterative procedure and commercial software such as “FEM Tools,” which work with most FEM software, are now available.

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(a)

(b)

(c)

Figure 9.12 Comparison of some experimentally determined mode shapes (lefthand side) with the FEM calculated mode shapes (right-hand side) for the example motor with no rotor and end-shields: (a) mode 10, f = 1369 Hz; (b) mode 15, f = 1605 Hz; (c) mode 21, f = 2099 Hz [224].

9.4 9.4.1

BEM for acoustic radiation Governing equation and boundary conditions

Although commercial finite element software has been available for more than three decades for engineering applications such as stress analysis, it is only since the 1990s that commercial software utilizing advanced numerical techniques, such

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Figure 9.13 Full 2-cylindrical-shell model with end-shields for the example motor [224]. as finite element and boundary-element methods (primarily more suitable for low frequencies), have been made available for acoustic calculations. With the use of computational methods for acoustics, it is now possible to incorporate ‘numerical’ acoustic analysis in the design process, so that the effects of noise radiation can be analyzed even before prototypes are built and innovative engineering techniques in reducing noise radiation may be examined. The propagation of sound waves in a medium is governed by the familiar wave equation, e.g. [124]: ∇2 p −

1 ∂2 p =0 c2 ∂t 2

(9.16)

70 Ideal Inverter 2

Sound Power Level (dB)

60

Inverter 1 Modal Testing

50 40 30 20 10 0 0

500

1000

1500

2000

2500

3000

Frequency (Hz)

Figure 9.14 Comparison of sound power spectra with modal testing results [224].

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Table 9.1 Comparisons of natural frequencies between measurements and the FEM calculations [224]. Mode 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Measurement 404.28 584.40 668.39 736.68 833.75 966.83 1043.97 1097.55 1179.73 1279.94 1418.64 1551.53 1636.95 1742.19 1723.14 1870.07 1924.96 2041.52 2167.28 2239.56

Damping (%) 3.1 1.2 0.9 0.5 1.4 1.4 0.9 0.7 0.7 0.9 0.6 0.8 0.8 0.8 0.5 0.6 0.6 0.7 1.0 0.9

FEM 375.1 – 485.8 695.6 – – 1121.6 1123.8 1097.5 – 1353.1 1559.6 – 1682.1 1932.6 1897.0 – 2167.0 2320.4 2476.2

where c is the speed of propagation of sound waves; p is pressure, t is time; and ∇ is the Laplace operator. By assuming a steady-state harmonic√motion of the form p(x, y, z, t) = p(x, y, z)e jωt , ( j is the imaginary number −1), Equation 9.16 can be reduced to Helmholtz equation, i.e., ∇2 p + k2 p = 0 (9.17) where k is wavenumber = ω/c and ω is the circular frequency in radians/sec. The above Helmholtz equation can be solved by imposing appropriate boundary conditions on some boundary surfaces which involve prescribing usually (a) the surface pressure ( ps ); (b) the normal surface velocity (v n ); or (c) the normal surface admittance (An ) or impedance as follows: p = ps ∂p = − jρωv n ∂n ∂p = − jρω An p ∂n

on surface S1

(9.18)

on surface S2

(9.19)

on surface S3

(9.20)

where n is the coordinate normal to the surface.

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Noise of Polyphase Electric Motors Selection of Parameters, Responses, and Convergence Criteria

Mode Shape Pairing Sensitivity Analysis Parameter Estimation

Database Updating

Use Internal Solver? Yes

No

External Solver

Internal Solver

Correlation Analysis

Convergence?

No

Yes Database and Results Output

Figure 9.15 Flow chart for model updating procedure.

Furthermore, for external radiation problems, the acoustic field vanishes at points farther than c(t − t0 ) because a wave disturbance initiated at time (t0 ) would not have reached that distance in the time (t) of interest. This condition is known as the Sommerfeld radiation condition [176] and can be expressed in spherical coordinates as   ∂p lim r α + jkp → 0 (9.21) r →∞ ∂r where r is the distance from the surface of source excitation and α = 1/2 for 2D problems and α = 1 for 3D problems.

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The boundary-valued problem as described by Equations 9.17 to 9.21 is difficult to solve analytically except for very simple geometries and boundary conditions. Approximate analytical solutions can be obtained for high and low frequencies using perturbation methods [38]. Consequently, numerical methods have to be sought for general problems. Numerical methods available for solving of Equation 9.17 are finite difference, finite element, and boundary-element methods. While finite difference techniques have been used for some acoustic problems [55], they are basically only suited for simple and bounded geometries and require large number of calculation points compared with finite element methods. Hence, only finite element and boundary-element methods will be considered here.

9.4.2

FEM

Finite element methods have been applied to interior problems which involve applying the weighted residual formulation to incorporate the field and boundary residuals into a set of weighted equations, for example [38]. The whole solution domain has to be discretized as shown in Figure 9.16a. The system of discretized equations can be written as a matrix equation of the following form: {[K ] − jρω[C] − ω2 [M]}[ p] = jρω[F]

(9.22)

where [C] is an acoustic damping matrix; [F] is an acoustic forces vector; [K ] is an acoustic stiffness matrix; [M] is an acoustic mass matrix; and [ p] is the unknown acoustic pressure vector. Eigenfrequencies and eigenmodes can be obtained from Equation 9.22 by setting [F] = [0] and the response at a given frequency can be expressed as a linear combination of the eigenmodes. The finite element method has been well established and Equation 9.22 can be solved efficiently due to the banded nature of the matrices. For acoustic radiation studies, however, since the far field boundary was difficult to be treated by FEM (perhaps with the exception of the use of infinite elements and wave envelope elements) which requires prohibitively large number of elements for calculating the radiation into a space, the BEM is more suitable.

Node

Node

Element Element

Figure 9.16 Discretization of solution domain.

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BEM

A good description of various boundary-element methods is given by Ciskowski and Brebbia [33]. As an example, the direct boundary integral formulation based on Kirchoff-Helmholtz theorem [176] is given here. The pressure at a field point X in the interior (Vi ) or the exterior (Ve ) volume can be expressed in terms of the pressure p (Y ) and the normal velocity v n at a boundary point Y on the related closed boundary surface S as given in Equation 9.23.  ∂G (X , Y ) C (X ) p (X ) = { p (X ) − jρωv n (Y )}d S(Y ) (9.23) ∂n Y S where n Y is the normal at the boundary point Y directed into the fluid domain; G (X , Y ) is a Green’s function satisfying the Helmholtz equation; and C is a coefficient dependent on the location of the field point X . For interior problems, C = 1, 0.5 and 0 for X inside the volume Vi , on the surface S and outside the volume Ve , respectively. For exterior problems, C = 0, 0.5 and 1 for X inside the volume Vi , on the surface S and outside the volume Ve , respectively. Unlike the finite element method, only the boundary surface of a boundaryelement model has to be discretized as shown in Figure 9.16b. The resulting discrete set of equations relate nodal pressures [ p] to nodal velocities [v] by: [A][ p] = [B][v]

(9.24)

While boundary-element methods are most suited for solving interior or exterior radiation problems, the matrices [A] and [B] are fully populated and sometimes nonsymmetric. Furthermore, for exterior problems, the solution may not be unique at frequencies that correspond to the interior cavity resonant frequencies, but this can be overcome by using an overdetermination procedure [33, 35].

9.4.4

Radiating sphere

In this example, the radiation efficiency of a spherical shell with radius a of 1 m and thickness of 200 mm was calculated for three different types of excitation, namely (a) uniform radial velocity pulsation; (b) velocity pulsation on one meridian plane; and (c) force excitation on two meridian planes, as shown in Figure 9.17a to c, respectively. The direct boundary-element method has been used for the calculations and the problem has been assumed to be axisymmetric with 46 elements and 47 nodes, giving an upper frequency limit of about 900 Hz. It has been shown analytically [176] that the radiation efficiency (σ ) of a pulsating sphere is given by σ =

(ka)2 . 1 + (ka)2

(9.25)

The sound radiation efficiency, as defined in Chapter 6, describes the capability of a structure in energy transformation from vibration to noise, and is dependent on the material properties, geometry, and the boundary conditions of

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v=1 a

v=1

249 f=1

v=1

v=1

v=1

f=1

f=1

t

v=1 v=1

v=1

f=1

Figure 9.17 A spherical shell subject to three different types of excitation.

the structure, and the properties of the ambient medium. The variation of the radiation efficiency with ka for the three types of excitation is shown in Figure 9.18 for air as the medium and for water with type (a) excitation. Here k is the wavenumber. It can be seen that the numerical results for type (a) excitation with air or water as the medium agree very well with the analytical solution given in Equation 9.25. There are, however, significant differences between the radiation efficiency results for types (b) and (c) excitation compared with type (a) excitation, thus indicating that the radiation efficiency of a radiating structure is dependent not only on the geometry, but also on how the excitation is introduced. This example serves to illustrate the importance of being able to model the excitation source properly, which in the case of an electric motor is the electromagnetic force distribution.

1.5

Radiation Efficiency

Analytical 1.0 Type (a)-air Type (b)-air Type (c)-air 0.5 Type (a)-water

0.0 0

2

4

6

8

10

12

14

16

ka

Figure 9.18 Radiation efficiency of a spherical shell.

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9.4.5

Noise of Polyphase Electric Motors

Application to the prediction of radiated acoustic power from an inverter-fed induction motor

The prediction of radiated acoustic power from the example inverter-driven induction motor was made with a commercial BEM code, SYSNOISE (Version 5.3A) developed especially for vibro-acoustic analysis [197]. The boundary element mesh of the motor structure, produced by skinning the ANSYS FE structural model in SYSNOISE, consists of 4056 nodes and 3711 elements. Based on 6 elements per acoustic wavelength, the upper frequency limit for this mesh is 2.8 kHz. As the boundary conditions to be imposed on the boundary-element (BE) acoustic model are the vibration distributions over the structure due to an external excitation, the modal base of the motor structure obtained by ANSYS was imported into SYSNOISE to evaluate the corresponding vibration response. The solver used for the solution of the wave Equation 9.2 was the indirect coupled BEM analysis [197]. In order to validate the BE acoustic model as shown in Figure 9.1, the sound radiation efficiency of the motor structure subject to a point force excitation was calculated and compared to the experimental data [232]. Although the sound radiation efficiency is an inherent property of a structure, it could be affected by the distribution and the type of the excitation forces, as has already been described in the radiating sphere example in Section 9.4.4. In determining experimentally the sound radiation efficiency of the motor structure, the motor was excited using an electromagnetic shaker, as shown in Figure 9.19. The average vibration level over the structure surface, as well as the sound power radiated from the motor structure, was measured respectively with an accelerometer and a two-microphone sound intensity probe in an anechoic room. The sound radiation efficiency σrad of the motor structure was calculated based on Equation 6.1, in which the radiated

B&K2032 Analyzer

Power Amplifier

SI Probe

Shaker

HP-300 Computer

Charge Amplifier

Accelerometer Motor Anechoic Room

HP3569A Analyzer

Figure 9.19 Schematic diagram of the experimental set-up for measuring the sound radiation efficiency [232].

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Sound Radiation Efficiency

10

1

0.1

0.01 Experiment-horizontal excitation 0.001

BEM-horizontal excitation

0.0001 0

500

1000

2000 1500 Frequency (Hz)

2500

3000

Figure 9.20 Comparison of the sound radiation efficiency of the motor structure calculated by BEM and experiment [232].

acoustic power was determined by the two-microphone sound intensity technique. As shown in Figure 9.20, there is reasonable agreement between the sound radiation efficiency calculated by SYSNOISE and the measurements. At low frequencies around 500 Hz, the discrepancies are caused by the error in estimating the vibration response by the FE structural model in which vibration isolators and the rotor are neglected. Although the FE model could be improved by incorporting the isolators and the rotor and by using a finer FE mesh, compromise usually has to be made between the extent of the details to be modeled and the computing resources required. For this example model, about 64 and 150 Mbytes RAM for ANSYS and SYSNOISE were required to run in-core respectively. It took about 4 CPU hours to obtain the modal base from ANSYS and 2 CPU hours to calculate the sound pressure field for one frequency step using SYSNOISE on a SUN SPARC 20 workstation. The radiated acoustic power from the example inverter-driven induction motor can now be calculated using the validated FE structural model and BE acoustic model and the electromagnetic force distributions obtained from the FE electromagnetic model. There were 72 nodes on each circumference of the BE mesh, but there were 198 nodes within a quarter of the stator in the FE electromagnetic model. Thus, the electromagnetic force to imposed on each node on the inner side of the stator mesh in the BE model was synthesized from 11 nodes on the FE electromagnetic model. Although the FE structural model does not have teeth, the FE electromagnetic model does, as shown in Figure 9.5. Since the FEM electromagnetic model was two-dimensional, but the BE acoustic model

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was three-dimensional, the electromagnetic force was applied uniformly along the axial direction of the stator, which might cause some error in the final results. BEM indirect coupled analysis with the modal base calculated by the ANSYS FE structural model and the damping values extracted from the modal testing as given in Table 9.1. The acoustic power radiated from the motor due to electromagnetic forces for different operating conditions was calculated. The results for the motor driven by the PWM inverter at 450 rpm with no load, and by the ideal sinusoidal inverter at 450 rpm and 1500 rpm with no load, together with the corresponding experimental results [30], are compared in Figures 9.21 and 9.22, respectively. In order to facilitate the comparison, the measurement data were synthesized into the same bandwidth with that of the force data. The experimental results include the electromagnetic noise and the aerodynamic and mechanical noise and the measured sound power spectra due to aerodynamic and mechanical origin at 450 rpm and 1500 rpm are given in Figure 9.23. The measured aerodynamic and mechanical noise have been added to the BEM results shown in Figures 9.21 and 9.22 for comparison with the experimental results. Generally, there is reasonably good agreement between the predicted and measured acoustic power spectra. By comparing Figure 9.21 with Figure 9.23, it is quite clear that for the motor driven by the PWM inverter with no load at 450 rpm, the predicted electromagnetic noise dominates the total acoustic power and the effect of the aerodynamic and mechanical noise is negligible because of the presence of harmonics in the supply. By comparing Figure 9.23 with Figure 9.23, it can be seen that for the motor driven by the ideal sinusoidal inverter at 1500 rpm, the total acoustic power is dominated by the aerodynamic and mechanical noise while the electromagnetic noise is important for frequencies

90 Measured BEM

Sound Power Level (dB)

80 70 60 50 40 30 20 10 0 0

500

1000

1500 2000 Frequency (Hz)

2500

3000

Figure 9.21 Comparison of the acoustic power calculated by BEM with experimental results: PWM Inverter, no load, 450 rpm [232].

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70 1500 rpm 450 rpm Measured Predicted

Sound Power Level (dB)

60 50 40 30 20 10 0 0

500

1000

1500 2000 Frequency (Hz)

2500

3000

Figure 9.22 Comparison of the acoustic power of the motor driven by the ideal sinusoidal inverter at no load for two operating speeds [232]. around 1000 Hz. These predicted acoustic power results are consistent with those of the experimental study (in Section 8.92). In the calculations, as the total electromagnetic force in a frequency bandwidth (30 Hz for 450 rpm, and 100 Hz for 1500 rpm) was assigned to the center frequency, the predicted acoustic response at this frequency might be artificially amplified because of the concentration of all the electromagnetic force into this

70 450 rpm 1500 rpm

Sound Power Level (dB)

60 50 40 30 20 10 0 0

500

1000

2000 1500 Frequency (Hz)

2500

3000

Figure 9.23 Measured sound power spectra due to aerodynamic and mechanical origin [232].

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SYSNOISE – COMPUTATIONAL VIBRO – ACOUSTICS

Model Mesh (1) (C): Displacement at 507.440 Hz (dB/Y component) Field Point Mesh (2) (C): Pressure at 507.440 Hz (dB)

Y Z

X

Model 2 Displacement −7.000E+01

Pressure 4.000E+01

−8.000E+01

3.3 1 2 E+01

−9.000E+01

2 . 6 2 5 E+01

−1.000E+02

1 . 9 3 8 E+01

−1.100E+02

1 . 2 5 0 E+01

−1.200E+02

5 . 6 2 5 E+00

−1.300E+02

−1.2 5 0 E+00

−1.400E+02

−8.1 2 5 E+00

−1.500E+02

−1.5 0 0 E+01

(a) SYSNOISE – COMPUTATIONAL VIBRO – ACOUSTICS Model Mesh (1) (C): Displacement at 1134.300 Hz (dB/Y component) Field Point Mesh (2) (C): Pressure at 1134.300 Hz (dB)

Y Z

X

Model 2 Displacement −7.000E+01

Pressure 4.000E+01

−8.000E+01

3.3 12 E+01

−9.000E+01

2 . 6 2 5 E+01

−1.000E+02

1 . 9 3 8 E+01

−1.100E+02

1 . 2 5 0 E+01

−1.200E+02

5 . 6 2 5 E + 00

−1.300E+ 02

−1.2 5 0 E+00

−1.400E+02

−8.12 5 E+00

−1.500E+02

−1.5 0 0 E+01

(b)

Figure 9.24 (See color insert following page 236.) Sound pressure field produced by the motor calculated by the BEM for ideal sinusoidal inverter at no load: (a) displacement and pressure at 507.44 Hz; (b) displacemnet and pressure at 1134.3 Hz [232].

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frequency. Therefore, for the results shown in Figures 9.21 and 9.22, it would be more appropriate to concentrate on the general trend instead of the details relating to each peak. The effect of changing inverter strategies on the radiated noise can be readily identified in Figures 9.21 and 9.22. Both the measurements and predictions reveal a significant increase in the radiated sound power level by changing from the ideal sinusoidal inverter to the PWM inverter. In addition to the total acoustic power radiated from the motor, the sound pressure field produced by a motor is also of practical interest. In SYSNOISE, the sound field at a particular frequency can be analyzed and visualized. For convenience, three image planes around the motor structure have been created. Each plane of which the dimensions are 0.3 m × 0.4 m is located 0.15 m away from the axis of the stator core, as shown in Figure 9.24. The contour maps of the distribution of the sound pressure levels on the image planes and the vibration levels (of the Y component) on the motor driven by the Benchmark inverter at two different frequencies, thus, can be displayed. It can be clearly seen from Figure 9.24 that the major contributors to the total acoustic noise at low frequencies are the base plate and the stator. It can be expected, therefore, that the low frequency noise of such a motor can be efficiently controlled by reducing the vibration levels of the base plate and the stator.

9.5

Discussion

While the FEM/BEM numerical approach described in this chapter gives a reasonable assessment of the acoustic noise radiated from an inverter-driven induction motor, there are still a number of issues that need to be considered before the approach can be incorporated routinely within a design/test cycle. The example calculations were made with a 2D FEM electromagnetic model, a 3D FEM structural model, and a 3D acoustic model. It has been shown that the low frequency vibro-acoustic response of the motor structure is very much affected by the vibration isolators and the rotor [224, 232]. The low frequency results could be improved by including these components in the FEM structural model. Alternatively, these effects might be included by applying appropriate structural damping to the FE structural model. The 2D FEM electromagnetic force model was verified indirectly by comparing the estimated and the measured acoustic power radiated from the motor structure. Such comparison, however, is not sufficient for the electromagnetic force model because the discrepancies in the final results are the accumulation of the effects from three models rather than the force model itself. Therefore, it would be helpful to have an independent approach to verify the correctness and the accuracy of the FEM electromagnetic force model. As the mechanical damping of the motor structure plays an important role in the vibration response and thus the acoustic response, it would be necessary, in addition to the modal testing,

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to examine the vibration response of the FEM structural model due to a point force excitation so that the modal damping of the structure could be determined reasonably well. In the example given, the modal damping was assigned in the calculation based on the modal testing results [224]. As the numerical prediction of the vibro-acoustic response of a structure depends on the structural and acoustic models as well as an accurate determination of the distribution of the excitation force, it would be desirable to use a 3D electromagnetic model especially when a skew rotor is used. In order to better assess the interaction between the current harmonics and the structural resonance in details, the resolution of the electromagnetic force calculation in the frequency domain needs to be refined. This may require developing a reasonably quick algorithm for calculating the electromagnetic forces with narrow frequency bandwidth, because the FEM electromagnetic force model used here is very time-consuming (of the order of 100 CPU hours for each run with 30 Hz bandwidth for 450 rpm and 100 Hz bandwidth for 1500 rpm). A 3D FEM electromagnetic model for determining the electromagnetic force distributions for transient/unsteady operations is still very much a challenge even with today’s computing power [152].

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10 Statistical Energy Analysis 10.1

Introduction

Analysis of the vibration and acoustic behavior of a complex structure, such as an electric motor or other machinery structures, is very useful but difficult. According to the classical vibration and acoustics analysis principles, the motion equations, and the corresponding boundary conditions for each structural element have to be established [160]. By solving these equations, the vibration displacement, sound pressure, and other relevant parameters can be obtained. As discussed in previous chapters, this approach is usually suited for the detailed analysis of the vibration and acoustic responses at low frequencies. When the structure is very complex, especially at high frequencies where a large number of vibration modes are involved in the calculations, the computing time and the resources required can be so demanding that renders the problem almost impossible to solve. For example, the prediction of the acoustic noise radiated from a full model of a small motor may take hours to solve for only one frequency step by the finite element method (FEM) and boundary-element method (BEM). Furthermore, for a complex structure, it is an arduous task to establish the motion equations and boundary conditions correctly for each structural element. This is because currently not all the practical structural elements have their equations of motion fully established and their boundary conditions mathematically described. Therefore, in pursuing the vibration and acoustic analysis for a complex structure with high frequency solutions, a statistical approach, namely the statistical energy analysis (SEA), is usually adopted. SEA was originally developed by Lyon in the 1960s for the analysis of random vibrations of complex structural and fluid systems [139, 141]. According to Lyon [141], “SEA has been described as a point of view in dealing with vibration of complex structures, and as such it employs a series of analytical and experimental ‘methods,’ most of which pre-date the identification of SEA. The view-point is statistical because the systems under analysis are presumed to be drawn from 257 Copyright © 2006 Taylor & Francis Group, LLC

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populations with random parameters; energy is the independent dynamical variable chosen because, by using it, distinctions between acoustical and mechanical systems disappear; and analysis emphasizes that SEA is an approach to problems rather than a set of techniques as such.” In SEA, a complex structure is required to be divided into several simple, identifiable mechanical and/or acoustical subsystems. The behavior of each subsystem is characterized in a statistical sense. By solving the energy balance equations which involve the expressions for the power flowing from one subsystem to another, the distribution of the vibration energy among these coupled subsystems can be estimated. Since the time and spatial averaged vibration energy of each subsystem is normally the primary variable in SEA, SEA is not able to describe the vibration details inside each subsystem. Nevertheless, in noise and vibration control engineering, such statistical results are sufficient in most cases to quantify vibro-acoustics effects, especially at high frequencies. Since the day that SEA was developed, applications of SEA in scientific research and engineering have drawn great attention within the noise and vibration community. So far, it has been widely used in analyzing the sound transmission in buildings [39], noise and vibration distributions in large scale structures, such as aircrafts [155], ships [207], vehicles [174], and the noise radiated from machinery structures [181], and so on. SEA has already been proven to be an effective approach dealing with the vibration and acoustics problems of complex structures. Several major benefits that are considered in favor of applying SEA to practical structures include: • The SEA model is relatively simple since the dimensions of the system are dramatically decreased due to the averaging required in the spatial and frequency domain, making the model updating and sensitivity analysis very straightforward and inexpensive. SEA is, therefore, very useful in early design studies; • Rather than a detailed description of the system, only a few gross parameters associated with each subsystem are required in building up a computational based SEA model; • SEA provides an excellent framework under which the energy transmission paths within the system can be identified and investigated either analytically or experimentally, such that the system response could be modified or even optimized in an efficient and economic way. However, questions associated with applications of SEA to practical structures still remain. For example, currently it is still difficult, in fact, there is no well established procedure or guidance to estimate the confidence level in SEA results; in the low frequency range or for heavy structures, SEA cannot provide good results; for a real complex structure, no methods/procedures are available for subsystem identifications such that the corresponding SEA model could provide accurate results. Consequently, building an accurate SEA model nowadays still requires an experienced engineer who has sufficient knowledge of the SEA basic theory, hypotheses, and its limitations.

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In this chapter, the basic theory of the power flow between linearly coupled oscillators is reviewed first, in order to obtain a good understanding of SEA theory and its hypotheses. Then, the basic SEA theory including the subsystem identifications, SEA parameter determinations is introduced and limitations of the classical SEA approach are discussed. Experimental SEA approach and some recent developments based on the SEA framework are also introduced in this chapter. An application of SEA particularly to electrical motors is given last.

10.2 Power flow between linearly coupled oscillators In SEA, the primary concerns are those systems that may be subdivided into some subsystems that are directly excited by the external forces, and other subsystems that are excited indirectly due to structural couplings. For instance, a running electric motor, mechanically, can be thought as a single structure excited by the electromagnetic force. But in the sense of SEA, it may be treated as a collection of subsystems, in which the stator is directly actuated by the electromagnetic force waves, the casing, and end-shields are excited through their various couplings to the stator. Apparently, such couplings are the mechanism by which the vibrational energy is transferred from directly excited subsystems to others. If the modal description is applied to the system, the structural coupling can be described as the intermodal couplings with a certain type of coupling mechanisms associated with the structural connections. As shown in Appendix D, the very basic mechanical system is a mass-spring-damper oscillator. The vibration behavior of any continuous system can be mathematically described as a superposition of a series of such oscillators with different mass, stiffness, and damping properties. Before we move onto the general SEA theory for continuous systems, it is helpful to focus on the interaction of oscillators first in regard to their energy sharing and power transferring.

10.2.1

Two coupled oscillators

Figure 10.1 shows a system consisting of two simple linear oscillators coupled by a stiffness element kc . Since the system has two degrees of freedom, corresponding two equations of motion for masses m 1 and m 2 can be established as


m 1 x¨ 1 + c1 x˙ 1 + k1 x1 + kc (x1 − x2 ) = f 1 . m 2 x¨ 2 + c2 x˙ 2 + k2 x2 + kc (x2 − x1 ) = f 2

(10.1)

Assume that the external forces are statistically independent, that is, a broadband noise with a flat frequency spectrum within a frequency band. The time averaged power flow within the frequency band, that is, the transmitted energy per unit time, between the two oscillators P12  can be written as P12  = x˙ 1 · kc (x1 − x2 )

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(10.2)

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c1

f2 x2

m1

m2 kc

k1 c1x1 k1x1

c2

k2

kc(x1 − x2)

m1

f1 kc(x1 − x2)

m2

k2x2 c2x2 f2

Figure 10.1 Two linear oscillators coupled by a spring element kc . where · denotes the time average. Since for the steady-state processes, x˙1 · x1  = 0, Equation 10.2 then becomes, P12  = kc · x˙ 1 x2  .

(10.3)

By evaluating the statistics x˙ 1 · x2 , the following equation can be obtained [139] P12  = g12 (E 1  − E 2 )

(10.4)

where E 1  and E 1  are the time averaged total vibrational energies of the two oscillators respectively. Besides, in Equation 10.4, g12 = κ 2 (1 + 2 )/[(ω12 − ω22 )2 + (1 + 2 )(1 ω22 + 2 ω12 )] ω12 = (k1 + kc )/m 1 , ω22 = (k2 + kc )/m 2 √ 1 = c1 /m 1 , 2 = c2 /m 2 , and κ = kc / m 1 m 2 . Equation 10.4 establishes a simple relationship between the time averaged power flow and the time averaged vibrational energy of each oscillator. It has been proven that for the coupling other than the stiffness kc , such as inertial and gyroscopic elements, Equation 10.4 still holds for arbitrary coupling strengths, but with a more complicated expression of g12 [141]. This is useful because it makes it possible to calculate the power flow either from input power to the system or from measured vibration levels. Moreover, several important characteristics, which form the basis for the analysis of the energy flow in complex mechanical and acoustical systems, can be observed from Equation 10.4 as follows: • The time averaged power flow is proportional to the difference in actual time averaged vibrational energies of the two oscillators, the constant of proportionality g12 being only a function of oscillator parameters and coupling strength, irrespective of external excitations.

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• Since the parameter g12 is positive definite, the time averaged power flow is always from the oscillator of greater to that of less vibrational energy. This is in contrast to the sinusoidal excitations in which, though Equation 10.4 still holds, the time averaged power flow could be from the low-energy oscillator to the high-energy one with a corresponding negative g12 , depending on the excitation frequency and the oscillator parameters. • Parameter g12 is sensitive to the difference of the resonance frequencies of the two oscillators (|ω12 −ω22 |), being a very large number when ω1 = ω2 , but very small otherwise. This fact indicates that vibrational energies exchange easily between the two oscillators if their natural frequencies are within a resonant bandwidth of each other. • Parameter g12 is symmetrical, that is, g12 = g21 , and thus the power flow is reciprocal. Given as an example showing the significance of Equation 10.4 in analyzing the energy sharing between the oscillators, assume that only oscillator 1 is excited by a broadband source. The net power transmitted out of oscillator 1 must be equal to that dissipated by the damping in oscillator 2 in order to achieve a power balance. Therefore, P12  = 2 E 2  = g12 · (E 1  − E 2 ). (10.5) This equation can be rewritten as, E 2  g12 . = E 1  2 + g12

(10.6)

With oscillator 1 excited with a broadband noise, the time averaged energy ratio between the two oscillators depends on the coupling strength and the damping value in oscillator 2. For small damping in oscillator 2, the vibrational energy tends to distribute equally in the system. But since 2 is always positive, the time averaged energy in oscillator 2 will not be greater than that in oscillator 1.

10.2.2

Three series coupled oscillators

Systems of interest in practice are much more complicated than the two coupled oscillators. One may hope that a similar relationship between the time averaged power flow and the total vibrational energies could be established in a multimodal system. Simply extending the findings from two oscillators model to multimodal systems appeared to be straightforward. It was argued, however, that it might not be appropriate to do so because it has to be assumed that the power flow between two directly coupled oscillators is not affected by the oscillators that are coupled indirectly. An alternative, actually an equivalent, description of this requirement is that the coupling is generally ‘weak’ or ‘light’ so that the influence of indirectly coupled oscillators could be ignored. Though the validity of this assumption was not very clear in the early days of the SEA development, it appeared to be inevitable in applying SEA principles to a more complex system. Apparently, judging the fairness of this assumption depends upon a study on a more sophisticated model

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c1 m1 k1 c1x1 k1x1

m1

f3 x3 c 3

f2 x2 m3

m2

k2

kc

c2

kc(x1 − x2) f1 kc(x1 − x2) c2x2

k3

m2

k2(x2 − x3) f2

k2(x2 − x3)

m3

k3x3 c3x3 f3

Figure 10.2 Three linear oscillators coupled serially by spring elements [231].

than the coupled two oscillators. Recently, a model of three series coupled oscillators, shown in Figure 10.2, was analyzed in [195]. Compared to Figure 10.1, this model has oscillator 3 directly connected to oscillator 2 through k2 but indirectly coupled to oscillator 1. The corresponding equations of motion were established as,   m 1 x¨ 1 + c1 x˙ 1 + k1 x1 + kc (x1 − x2 ) = f 1 m 2 x¨ 2 + c2 x˙ 2 + kc (x2 − x1 ) + k2 (x2 − x3 ) = f 2 . (10.7)  m 3 x¨ 3 + c3 x˙ 3 + k3 x3 + k2 (x3 − x2 ) = f 3 By following the same analyzing techniques as the two-coupled oscillators, the net power flow from oscillators 1 to 2 can be derived as [195], P12  = β12 · (E 1  − E 2 ) + γ13 · (E 1  − E 3 )

(10.8)

where E i  still is the time averaged total vibrational energy of oscillator i (i = 1, 2, 3), while β12 , and γ13 are constants of proportionality, both depending on the parameters of all oscillators. This equation can be easily justified by assuming a very large m 3 with asymptotic solutions of β12 , and γ13 expected to be g12 , and 0, respectively. Equation 10.8 concludes that the power flow between three series coupled oscillators generally consists of two parts, the direct power flow Pd  = β12 · (E 1  − E 2 ) between the oscillators which are physically coupled; and the indirect power flow Pi  = γ13 · (E 1  − E 3 ) between the oscillators which are not physically coupled. Although it was shown that reciprocity still holds for either direct or indirect power flow, that is, β12 = β21 , γ13 = γ31 , the constant of proportionality β12 appears to be different from g12 , simply because β12 is a function not only of the directly coupled oscillators and coupling parameters, but also that of the indirectly coupled oscillators and coupling parameters.

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The dependence of the constant of proportionality β12 on all oscillator parameters adds difficulties in identifying the key parameters affecting the modal behavior and characterizing the power flow in the system explicitly. However, numerical studies have shown that a maximum direct power flow between oscillators 1 and 2 can still be observed when a resonance occurs between the two oscillators. More importantly, when ω1 = ω2 , the ratio of indirect power flow to the direct power flow reaches a minimum, the indirect power flow between oscillators 1 and 3 being orders of magnitude less than the direct power flow between oscillators 1 and 2, as shown in Figure 10.3. A more general conclusion that can be reached here is that, between the two directly coupled oscillators, the direct power flow is associated with resonant transmissions; while the indirect power flow is with nonresonant transmissions. This result is significant as it shows that the indirect power flow may not be important when a resonance occurs between directly coupled oscillators. In this case, the three series coupled oscillator could be simplified to the two coupled oscillators model, or in other words, effects of the oscillators other than the directly coupled two on the direct power flow could be ignored. Moreover, it is interesting to note that decreasing k2 has similar effects as increasing m 2 on the constants of proportionality β12 , and γ13 . Therefore, when k2 is small, β12 and γ13 should be approaching g12 and 0, respectively, indicating that the direct power flow may not be affected if the indirect coupling strength is weak. Given as a conceptual illustration, Figure 10.4 shows three typical cases associated with the coupling of oscillators in energy sharing. If the natural frequencies

1 a

10−1

b

Pi /Pd

10−2

c d

e

10−3

f

10−4 10−5 10−6 10−7

0.01

0.1

1

10

100

1000

w12/w22

Figure 10.3 Ratios of direct power flow to input power vs. ω12 /ω22 for selected values of m 3 for m 1 = 1.5 kg: (a) m 3 = m 1 ; (b) m 3 = 10m 1 ; (c) m 3 = 100m 1 ; (d) m 3 = 1000m 1 ; (e) m 3 = 10000m 1 ; (f) m 3 = 100000m 1 [195].

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Oscillator 1

Oscillator 1

f

f

f

Oscillator 2

Oscillator 2

Oscillator 2

f

f

f

Oscillator 3

Oscillator 3

Oscillator 3

f Df

Oscillator 1

f Df

f Df

Figure 10.4 Coupling between oscillators [231].

of oscillators 1 and 2 are within the frequency band  f , the resonant transmission occurs between oscillators 1 and 2, and the corresponding power flow is so dominant that the power flow between oscillators 1 and 3 can be neglected. If the resonance between oscillators 1 and 2 does not happen within the frequency band, the power flow between indirectly coupled oscillators 1 and 3 might not be ignored. In this case, depending on whether a resonance occurs between the oscillators 1 and 3 (oscillator 2 acts as a coupling element), the indirect power flow may also be characterized as resonant or nonresonant transmissions accordingly. However, unlike the direct power flow which is always dominated by the resonant transmission between directly coupled oscillators (especially for small damping values), the nonresonant transmission between indirectly coupled oscillators can play an important role in the indirect power flow.

10.2.3

Energy exchange between groups of oscillators

Figure 10.5 shows a system with two groups of oscillators. Each group of oscillators comprises a so-called subsystem. Assume a white noise excitation and in a given frequency band, the number of oscillators (modes), in each subsystem is N1 and N2 respectively. The time averaged power flow from the m th 1 mode in subsystem 1 to the m th 2 mode in subsystem 2 is similar to that of two coupled oscillators. Thus, from Equation 10.4, Pm 1 m 2  = gm 1 m 2 · (E 1,m 1  − E 2,m 2 ).

(10.9)

In here, three hypotheses have to be made to proceed further: • The time averaged total vibrational energy in a subsystem is equally distributed over all of the modes in the frequency band, and the modal responses are incoherent so that energy sums apply;

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• Since at resonance, the magnitude of parameter gm 1 m 2 is primarily controlled by the damping of the oscillators, damping values are assumed to be similar for all the modes within a subsystem and frequency band, so that the constant of proportionality gm 1 m 2 is similar for all the modal pairs; • The modes are many and equally distributed in the frequency band. With the first assumption, 

       E 1  E 1,1 = E 1,2 = · · · = E 1,m 1 = · · · = E 1,N1 = , N1

and



       E 2  E 2,1 = E 2,2 = · · · = E 2,m 2 = · · · = E 2,N2 = (10.10) N2 where E 1 , and E 2  are the time averaged total vibrational energy of subsystem 1, and 2, respectively. As a result, the time averaged power flow due to the coupling of N1 modes in subsystem 1 to the m th 2 mode in subsystem 2, subject to the second assumption made above, can be written as         P1m 2 = gm 1 m 2 N1 N1 · ( E 1,m 1 − E 2,m 2 ) (10.11) where · N1 denotes the average over N1 modes in subsystem 1. Including the contributions from all the modes in subsystem 2, we have,       P12  = gm 1 m 2 N N N1 N2 · ( E 1,m 1 − E 2,m 2 ) (10.12) 1 2 where · N1 N2 denotes the average over N1 and N2 modes in subsystems 1 and 2 respectively. By defining,   gm 1 m 2 N1 N2 N2 η12 = (10.13) ω

1

1

2

2

m1

m2

N1

N2

Subsystem 1

Subsystem 2

Figure 10.5 Modal coupling between two subsystems.

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Equation 10.12 can be rewritten as,  P12  = ωη12 N1 ·

E 1  E 2  − N1 N2

 (10.14)

Note that gm 1 m 2 is symmetrical, as concluded in Section 10.2.1. Therefore, N1 η12 = N2 η21

(10.15)

Thus, Equation 10.14 can also be written as, P12  = ωη12 E 1  − ωη21 E 2  .

(10.16)

For directly coupled groups of oscillators, Equations 10.14 and 10.16 present a similar relationship as that for the two coupled oscillators (Equation 10.4) between the average total power flow and and the average total vibrational energies of coupled groups, thus laying the foundation of the classical SEA theory for continuous structures. Although, the validity of Equations 10.14 and 10.16 appears to be independent of the third assumption above for direcly coupled oscillator groups, it cannot be generally extended to coupled multigroup oscillators because of the existence of the indirect power flow, which is associated with the nonresonant transmission through the intermediate oscillators for which the energy sum might not apply. However, for a system with multigroups of oscillators, by assuming that each group of oscillators complies with the three assumptions made above, and in addition, any pairs of directly coupled groups have their modes coupled in the way as illustrated in Figure 10.5, and most importantly, resonances should spread over the frequency band  f between any pair of the modes in directly coupled groups to ensure that the indirect power flows within the system are negligible, then Equation 10.16 can be applied for any directly coupled oscillator groups based on Equation 10.8 and the discussions made in Section 10.2.2. This result thus forms the basis of the classical SEA theory in which only resonant transmissions between directly coupled oscillators are considered [141]. Apparently, the third assumption made above is necessary for this resonance to occur within the frequency band. Quantitatively, the additional resonance condition introduced here requires that there are at least N =  f /( f η) modes equally distributed within the frequency band  f , where f , η respectively are the averaged natural frequency and damping loss factor of the modes within this band, and f η therefore is the half-power bandwidth of a resonance [37]. An equivalent expression can be written as n f η  1, where n = N / f is the modal density in the subsystem. For practical structures, this condition is relatively easy to satisfy at high frequencies where the modal density is generally higher, indicating that SEA generally works well at high frequencies. For multigroup systems, if the resonance condition is not fully complied, indirect power flows may exist [165]. For indirect resonance transmissions, by

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generally assuming that the modal responses are statistically independent, an expression similar to Equation 10.16 for the indirect power flow might be obtained as, P13  = ωη13 E 1  − ωη31 E 3  (10.17) where subscript “13” denotes an indirect transmission path, and the indirect cou  pling loss factor η13 is defined as η13 = γm 1 m 3 N1 N3 N3 /ω, in which N3 is the number of oscillators in the group indirectly coupled to group 1. Also, N1 η13 = N3 η31 .

(10.18)

A typical example of energy transfer due to indirect resonant transmissions is a limp panel (no stiffness, thus no resonance) installed between two large acoustic cavities with rigid walls, the two cavities being the resonant subsystems. For the indirect nonresonant transmission, however, Equations 10.17 and 10.18 might not hold because the modal responses are likely coherent, for which the energy sums may not apply. For practical structures, this normally happens at low frequencies where very few or even no modes are within the frequency band of interest.

10.3

Coupled multimodal systems

10.3.1

General SEA equations

In order to explain the SEA theory clearly, a system with three subsystems, as shown in Figure 10.6, is taken as an example. Here P i , (i = 1, 2, 3), is the averaged total power input to the subsystem i. E i is the averaged total vibrational energy of the subsystem i. For each subsystem, the modal density n i and the internal loss factor ηi are the two parameters used in SEA. P i j denotes the time averaged power transfer from subsystem i to subsystem j, and P id is the time

P1

P2 n1

E1 h1 P1d

P12

P3 n2

E2 h2

P21

P2d

P13

P23 P32

n3

E3 h3 P3d

P31

Figure 10.6 SEA model with three subsystems.

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averaged power dissipated in the subsystem i due to internal loss. According to the energy conservation theorem, for each subsystem, the following energy balance equation can be written as: P i = P id +

3

(P i j − P ji )

(10.19)

j =1 j= i

Equation 10.19 indicates that the power input to subsystem i is equal to the power dissipated in subsystem i and the net power transmitted out of subsystem i. The dissipated power can be written in terms of the time and spatial averaged vibration energy [37]: P id = ωηi E i (10.20) For P i j , it has been shown in Section 10.2 that the power transmitted from one subsystem to another is proportional to the difference of the modal vibration energy of the two subsystems:  P i j = ωη12 n i ·

Ei Ej − ni nj

 (10.21)

where ηi j is the coupling loss factor between subsystems i and j, indicating the part of the vibrational energy transmitted in one cycle of the vibration. Therefore, similar to the internal loss factor, it describes the capability of the power transmission between the two subsystems. In SEA, the determination of the coupling loss factors for different coupled structures is the major crux. By using Equations 10.20 and 10.21, the energy balance equations for the system shown in Figure 10.4 can be written as,      P1 E 1 /n 1 η11 n 1 −η21 n 1 −η31 n 1  P 2  = ω  −η12 n 2 η22 n 1 −η32 n 2   E 2 /n 2  (10.22) −η13 n 1 −η23 n 1 η33 n 3 P3 E 3 /n 3 where ηii = ηi +

3

ηi j ,

i = 1, 2, 3.

j=1 j= i

It can be seen that if the input power, the internal loss factor, the modal density of each subsystem, and the coupling loss factors between each pair of subsystems are known, the vibration energy of each subsystem can be solved from Equation 10.22. Consequently, the vibration displacement and sound pressure corresponding to the structural and acoustic subsystems can be derived, respectively. Since the power flow between two subsystems satisfies the theorem of reciprocity, it follows that, n i ηi j = n j η ji (10.23)

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Equation 10.23 is very useful for simplifying SEA calculations. For example, for a structure with N subsystems, the energy balance equations can be rewritten as      P1 E1 η11 −η21 · · · −ηi1 · · · −η N 1  P2   −η12 η22 · · · −ηi2 · · · −η N 2   E 2        ..   .. ..   ..  .. ..  .    .  . ··· . ··· .  .      (10.24)  P i  = ω  −η1i −η2i · · · ηii · · · −η N i   E i        .   .. .  .  . ..  ..   . . · · · .. · · · ..   ..  −η1N −η2N · · · ηi N · · · η N N PN EN where ηii = ηi +

N

ηi j ,

i = 1, 2, 3, · · · , N .

j=1 j= i

No modal densities appear in this equation. Equation 10.24 is often employed in the determination of the coupling loss factors by measurements which will be discussed in Section 10.4. As a simple example, consider a two subsystem model in which subsystem 1 is excited by external forces while subsystem 2 is driven through the coupling. The power balance equations are given by,
P 1 = ωη1 E 1 + ωη12 E 1 − ωη21 E 2 . (10.25) 0 = ωη2 E 2 + ωη21 E 2 − ωη12 E 1 Then the energy ratio between the two subsystems is E2 η12 η12 = = . η + η η n /n E1 21 2 12 1 2 + η2

(10.26)

This equation indicates that, though the modal energy of subsystem 2 will never exceed that of subsystem 1, the total vibrational energy of subsystem 2 could be greater than that of subsystem 1, if the internal damping loss factor is much less than the coupling loss factor, i.e. η2 η21 , and the modal density of subsystem 1 is greater than that of subsystem 2, i.e. n 1 > n 2 . The extreme situation of this case is that the vibrational energies in the two subsystems are equipartitioned, and the power flow between the two subsystems vanishes when their modal energies are at the same level. However, if the coupling between the two subsystems is weak, that is, η2  η21 and η12 , the vibration level of subsystem 2 will always be less than that of subsystem 1, and inversely proportional to the internal damping in subsystem 2.

10.3.2

SEA model establishment

It should be noted that the SEA energy balance equations defined in Section 10.3.1, though significant, only present a mathematical description for the power flow between subsystems. No information or even indications are given with regard to the

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subsystem breakdown for a particular structure. It has been shown by many studies that an appropriate breakdown of a system into subsystems in accordance with SEA principles is essential for a good SEA model and meaningful results. Three major issues that are usually considered in the SEA model establishment are subsystem definitions, coupling identifications, and the form of external excitations, among which defining subsystems is the most crucial one. A good subsystem definition may help to define appropriate couplings within the system, leading to meaningful coupling parameters and yielding reasonable model outcomes. Currently, subsystem definitions for a SEA model still very much rely on a priori knowledge of the substructuring, or in other words, on the experience of the SEA model builder. A general principle [141] for defining a subsystem in SEA was to identify a group of similar modes within a physical component of a system which would likely result in a uniform response over the subsystem. The physical components of a system are often identified as the relatively uniform portions of simple elements such as beams, plates, and cylindrical shells. An element, other than those simple ones, could also be identified if its modal behavior is fully understood. The groups of similar modes in each physical component are identified as those having resonance frequencies in a given frequency band and having similar mode shapes so that they will likely have similar values of damping, coupling parameters, and therefore similar modal energies. Based on this principle, the mode groups in SEA should be expected as those that relate to possible wave types in the components such as compression, shear, and flexural waves. Giving an example, a L-shaped beam system is shown in Figure 10.7. The system consists of two straight beam components connected with the right angle. Corresponding to the three types of wave motions in each of the beam components, three groups of modes, bending, longitudinal, and torsional modes, could be identified in each of the two beams as SEA subsystems. As a result, a general SEA model for the L-shaped beam system could have six subsystems with nine possible coupling combinations between each pair of the subsystems in the two beams, respectively. However, depending on the excitations, not all the mode groups are necessary to be considered in the SEA model. For instance, if beam

2

vL(2) FB vB

(2)

FL vL(1)

vB(1)

1

Figure 10.7 L-shaped beam system.

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271 FB

Flexural Vibration

FL Flexural Vibration (1)

Longitudinal Vibration (2)

Flexural Vibration (2)

(a)

hLB

Longitudinal Vibration(1)

hBL (b)

Figure 10.8 SEA models for the L-shaped beam system: (a) beam 1 is excited laterally; (b) beam 1 is excited longitudinally.

1 is excited by a lateral force FB as shown in Figure 10.7, the bending modes of beam 1 will be easily excited and a fairly large amount of vibrational energy in beam 1 will be stored in those bending modes. Those bending modes may primarily be coupled with the bending modes, and the longitudinal modes in beam 2 with less, maybe negligible, energy coupled to the torsional modes in beam 2 and even less energy coupled back to the longitudinal, and torsional modes in beam 1. The corresponding simplified SEA model, thus has three subsystems as shown in Figure 10.8a. On the other hand, if beam 1 is excited longitudinally, only longitudinal modes in beam 1 and bending modes in beam 2 need to be considered. A SEA model with only two subsystems Figure 10.8b is thus a good representation of the system. These simplifications, are based on one’s prior knowledge and understanding of the system dynamic behaviors. Although, the natural boundaries of simple elements in a system provide a clue in identifying the SEA subsystems and such substructuring scheme has been shown quite efficient and successful in many reported applications, it is not always correct and easily implemented for a real complex structure. Figure 10.9 shows a vehicle floor structure. The ribs and stiffeners on the top and the bottom

(a)

(b)

Figure 10.9 Vehicle floor structure: (a) top view; (b) bottom view.

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sides of the floor make it very hard to give a reasonable guess on the modal similarity over the floor structure to facilitate the SEA sub-structuring. In fact, since the effects of those ribs/stiffeners change with frequency, the modal behaviors that are characterized in accordance with the SEA modal grouping principles would change with frequency. For example, at low frequencies, the floor tends to vibrate as a whole structure due to the ribs/stiffeners, while at high frequencies, the similar modes associated with specific local floor vibrations may be grouped as one subsystem. As a result, the number of subsystems may not be constant over the whole frequency range. Actually, due to the low modal density of the floor structure at low frequencies, SEA generally requires a coarse division of the structure so that each subsystem could have enough modes within a frequency band for SEA to be fully functioning. As the modal density increases with the frequency, the number of subsystems may be expected to increase with frequency to correctly capture the gradients of the energy level along the propagation path and its variation with the frequency [70].

10.3.3

SEA parameters

SEA power balance equations establish a link between the power flow and the vibrational energy among subsystems. A major difference between the classical deterministic approaches and SEA is that deterministic approaches are concerned with field variables such as displacement, velocity, acceleration, and pressure at specific locations and frequencies, while in SEA the primary concerns are average global variables with no phase information included. A SEA variable normally involves the frequency and spatial average in a certain frequency bandwidth and over the extent of a subsystem respectively. The choice of the frequency bandwidth should ensure that the basic SEA assumptions be satisfied. Generally, it is recommended that each frequency band should contain at least five modes for SEA to give reasonable results [56]. The spatial average is essential in describing a global behavior. Statistically, if a frequency band contains enough modes, the variation in the response over a subsystem is small so that the average behavior could be obtained from limited locations [126]. In engineering, the ensemble average is a notion of practical significance because it provides the statistical mean and the confidence level for populations of nominal system assemblies. It was argued that SEA subsystem responses are much more representative in the sense of ensemble averages which are taken over “similar” subsystems with physical parameters, such as material properties and structure dimensions, drawn from statistical distributions other than from a single sample. For example, a good SEA model predicts the mean level of the radiated sound power for a population of induction motors out from an assembly line. However, at present, we generally lack the detailed knowledge about the statistical behaviors of each SEA related variable and, for a complicated structure, the variances in the SEA responses due to all possible uncertainties associated with the material properties and the manufacturing process. An approximate equivalency between the ensemble statistics and the frequency statistics is usually accepted in most

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current SEA applications. The basic idea of such equivalency is that the random nature in structural changes can be expected to modify the positions of the natural frequencies, so that the average taken at a single frequency across the ensemble is equivalent to averaging over several modes along the frequency axis. Although it was shown that frequency averages are not exactly equal to ensemble averages [145], in practice, many systems have enough inherent complexities such that the variation in the response over frequency and location could represent the ensemble statistics adequately. This is particularly true if the modal overlap of a system is relatively high [132]. Apart from the average power flow and the average vibrational energy, a few basic parameters, which are all defined in the average sense, are often used in SEA and introduced below. Internal loss factor ηi The internal loss factor ηi of subsystem i is defined based on Equation 10.20 as, ηi =

P id . ωE i

(10.27)

Physically, it indicates the amount of vibrational energy dissipated by the damping in the subsystem in one cycle of the vibration. The major damping mechanisms that are considered in SEA for a particular subsystem include the material damping, additional damping treatment, acoustic radiation damping (an energy loss to the structure) and the equivalent damping due to the friction at the boundaries. In the literature, there are several different parameters used to characterize the damping of a structure, such as the loss factor, the modal damping ratio, the decay rate, and the quality factor, all of which can be converted from one to another. Table 10.1 gives the relationship of these parameters with the damping loss factor. In SEA, the internal damping loss factors of each subsystem are often determined

Table 10.1 Damping relationships Dissipation Descriptor Relation to η Quality factor (Q) η = 1/Q Damping ratio (ζ ) η = 2ζ Reverberation time in a cavity (T60 ) η = 2.2/( f · T60 ) Decay rate (D R) η = D R/(27.3 f ) Logarithmic decrement (δ) η = δ/π Damping coefficient (C)∗ η = C/(2π f M) Damping bandwidth (BW ) η = BW/ f    Complex Young’s modulus ( E) η = m( E)/e( E) Averaged sound absorption coefficient in a cavity (α)∗ η = c0 Sα/(4π f V ) * M- mass, V - cavity volume, S- cavity surface area, c0 - speed of sound

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experimentally simply because analytical expressions are generally not available. Three commonly used experimental approches include the modal testing, decay rate measurement, and the power injection method. Detailed discussions regarding the testing theroies, procedures, and the results can be found in [17, 37, 53, 141].

Coupling loss factor ηi j Unlike internal loss factors for which a lot of work has been done outside SEA, the coupling loss factor is uniquely defined for SEA. According to Equation 10.21, the coupling loss factor is defined as ηi j =

 P i j  ωE i  E j =0

(10.28)

Similar to the internal loss factor, it indicates how much vibrational energy transmitted out of the subsystem in one cycle of the vibration. Because the energy transmission from one subsystem to another is very much dependent on the geometrical details of the connections, it is quite difficult, if not impossible, to describe every single joint condition for practical structures mathematically. However, in the statistical sense, it may be reasonable and acceptable to categorize all possible joint types into point, line, and area junctions. For example, one end of a beam connected to another beam or plate exhibits a point junction; a plate connected to another plate along the edge by welding is a line junction case; an area junction could be, for an electric motor, the structural coupling between the stator and the portion of casing that is directly attached to the stator. The evaluation of the coupling loss factors for various types of junctions certainly lies at the heart of the engineering application of SEA. Under the assumption that the coupling loss factors between the two directly coupled subsystems are not affected by the existence of indirectly coupled subsystems, the coupling loss factors associated with various junctions could be studied in a generic way by only examining each individual junction independently, irrespective of the whole structure system. By reviewing the literature, there are three major approaches for deriving coupling loss factors between the subsystems that are directly coupled, namely the wave, modal, and mobility approaches [57]. In the wave approach, the vibration fields of the two directly coupled subsystems are represented by a superposition of a series of traveling waves, and the wave intensities are related to the vibrational energy of a subsystem. Considering the wave reflection and the wave transmission at the junction, the power transfer coefficient for each incident wave between the two subsystems may be evaluated. By assuming that the traveling waves in each of the subsystems are statistically uncorrelated, the averaged power transfer coefficients and thus the coupling loss factors between the two subsystems can be obtained. Since this method involves no direct descriptions of the modal behaviors, such as mode shapes, natural frequencies, dampings, and modal distributions, it is generally favored over other methods in engineering

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applications. However, though generally being functions of subsystem damping, the coupling loss factors derived from the wave approach are independent of the damping in the two subsystems. Extensive studies showed that the wave approach presents reasonable results when the modal overlap factors are generally greater than unity (meaning a well resonant system), where the coupling loss factors appear not to be so sensitive to the damping in the subsystems. For the systems with low modal overlap factors (generally less than unity), the dependency of the coupling loss factors on the damping should be considered and the wave approach might not apply in this case because the behavior of each individual mode may become predominant. Successful applications of the wave approach for point, and line junctions can be found in [37, 129, 141]. In the modal approach, the equations governing the dynamic behaviors of the subsystems have to be established explicitly. Mathematically, the vibration response of the system can be expressed in terms of the modal series of uncoupled subsystems of which the boundary conditions are described based on the physical nature of the subsystem. Then the interaction between the subsystems can be derived based on the uncoupled modal series, and the corresponding multimode power transmission coefficients may be obtained subject to certain assumptions regarding the modal distributions and the coupling strengths, in order to smear out the specific modal behavior and generalize the solutions within the SEA framework. In principle, the modal approach provides more accurate solutions than the wave approach because the modal approach is based on the detailed and exact mathematical descriptions of the system. However, due to the difficulties in establishing the vibration governing equations and corresponding boundary conditions for each subsystem mathematically, the modal approach is generally hard to apply directly for practical complex structures. Examples for beam and plate coupled systems could be found in [42, 49, 129, 145]. The mobility approach is to utilize the concept of dynamic mobility or impedance to express the result of interaction between coupled subsystems. The mobility and the impedance, which are the inverse of each other, are the transfer functions relating the force input to the velocity response at the same locations [37]. Corresponding to different types of force inputs associated with the junctions, the point, line, and the surface mobilities or impedances are the common forms employed in the discussions. Since the power transmitted through the junction is the time averaged product of the force and the velocity, the coupling loss factors can be generally expressed in terms of the mobility or impedance at the junction interface. The advantage of this approach is that the coupling loss factors might have simple and unique expressions for each type of the junctions. Therefore, discussions regarding the statistical distributions of the subsystem parameters and their effects on the coupling loss factors can be made to the corresponding mobility or impedance at the subsystem level. This approach was reviewed in detail in [151, 25]. The determination of coupling loss factor by experiments will be discussed in Section 10.4.

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Modal density n i The modal density of a subsystem is defined as the number of modes per unit frequency interval. It is related to the total number of modes in the subsystem Ni ( f ) by d Ni ( f ) ni ( f ) = . (10.29) df In SEA, the modal density may not necessarily appear in the power balance equations as shown in Equation 10.24. However, since it indicates how many resonant modes are available to store and transfer energy, which is critical for a SEA model to yield reasonable solutions, it is an important parameter that in most cases needs to be examined in the SEA analysis. The modal density of a subsystem can be determined either by experiment or analytical expressions. A useful expression in the derivation of modal densities is found as [141], n i ( f ) = 4Mi G i ( f ) (10.30) where Mi is the mass of the subsystem, and G i ( f ) is the frequency and spatial averaged real part of the mechanical drive point mobility of subsystem i. For simple structural elements, such as beams, plates, cylindrical shells, and acoustic cavities, accurate analytical expressions can be found in literature. A few examples are given below. Beams. In SEA, beam subsystems are those structural elements only carrying one-dimensional waves. Three types of modes associated with three types of wave motions are normally of interest, flexural, longitudinal, and torsional modes. For a beam in pure bending motion, in the frequency range f < cl /(4π κ) (the bending wavelength is much smaller than the thickness), the modal density can be calculated by [141] l ni ( f ) = √ (10.31) 2π f κcl √ where cl = E/ρ is the longitudinal wave speed in the material with E and ρ being the Young’s modulus and density of the material respectively, l is the length of the beam, and κ is the radius of gyration of the √ beam cross section. For a homogenous beam of uniform thickness h, κ = h/ 12; for a pipe of average √ radius a, κ = a/ 2. For longitudinal modes, a vibration in the direction along the beam axis, the modal density is 2l (10.32) ni ( f ) = . cl The modal density of torsional modes can be calculated by [141]  ρ n i ( f ) = 2l (10.33) 4G Ix I y where G is the shear modulus of the material, Ix and I y are the cross section area moments of inertia about the two perpendicular axes.

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Plates. For an isotropic plate in pure bending motion, the modal density can be written as [141] √ 3S ni ( f ) = (10.34) hcl where S is the surface area, and h is the thickness of the plate. This equation is only valid for the flexural wavelength much greater than the plate thickness, that is, in the frequency range f cl /(4π κ). For an orthotropic plate of which the stiffness of the plate in the two orthogonal directions are different, the total modal count can be approximately calculated by using the geometric mean of the mode numbers associated with the two principal directions [141]. As a result, the modal density of the plate in bending motion can be obtained as √ 3S ni ( f ) ≈  (10.35) (1) (2) h cl cl where cl(1) and cl(2) are the longitudinal wave speeds in the two orthogonal directions, respectively. Circular cylindrical shells. Owing to the coupling between flexural, longitudinal, and torsional waves in curved shell structures, three sets of modes, corresponding to each of three waves, always exist in the structure. However, depending on the excitations, one type of modes may dominate the response. Assume that only flexural modes are of interest. The modal density of the cylindrical shell is given as [141],  4 −1/4  3l π/2 ni ( f ) = + 1+ √ h fr α + α 3.5 /2 √

 √ 4 −5/4  5  √ π/2 3lα π/2 √ + ( α + 3.5α 3.5 ) 1+ √ πh α + α 3.5 /2 α + α 3.5 /2 (10.36) where α √= f / fring , l is the length, a is the radius, h is the thickness, and fring (= E/ρ/(2πa), in which E, and ρ are Young’s modulus and material density respectively, is the ring frequency of the cylindrical shell respectively. It is worth mentioning that for any curved shell structures, their vibro-acoustic behaviors approach that of a flat plate with the same surface area and thickness in the frequency range f > fring (the ring frequency). For a shell structure with a large radius such that the corresponding ring frequency fring is lower than the frequency range of interest, the structure can be treated directly as a flat plate in√SEA. Particularly, if the cylindrical shell is long enough so that l  2πa a/ h, the shell is basically in bending motion, regardless √ of the boundary conditions at both ends, in the frequency range below fl (= 3/5 fring h/a). In this case the

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modal density of the cylindrical shell can be calculated by Equation 10.31 with the appropriate radius of gyration κ for the shell cross section [228]. For an orthotropic circular cylindrical shell, of which the stiffness along the axial and the circumferential directions are different, the modal density might be approximately  calculated by Equation 10.36 with the ring frequency fring replaced with

(1) (2) (1) (2) fring fring , where fring = cl(1) /(2πa), and fring = cl(2) /(2πa)

with cl(1) and cl(2) being the longitudinal wave speeds in the two orthogonal directions respectively. Modal overlap Mi The modal overlap of a subsystem is defined as Mi = ωηi n i .

(10.37)

Physically, this parameter denotes the average number of modes lying within a typical modal half power bandwidth. A high modal overlap factor means that neighboring resonance peaks largely merge together, giving a high probability for a system to be characterized as resonant, which is essential to SEA applications. Although the modal overlap is a parameter that might not directly appear in the SEA equations, it is often used to justify the appropriateness of employing the wave approach for evaluating coupling loss factors within the system, and also might serve as one of the indicators showing whether the resonance occurring in a subsystem reaches a level for SEA to be able to give reasonable solutions. Generally, it is assumed that a system with its subsystem modal overlap factors less than unity is not amenable to an SEA approach. However, one should be clear that the actual number of modes being excited within a system depends on the initial condition of the excitations, not the modal overlap. The modal overlap only gives the maximum number of modes (in the average sense) that are possibly excited.

10.3.4

Limitations of SEA

As seen from the discussions above, several assumptions that are critical to the validity of SEA foundations (Equation 10.21) were made inevitably when extending the results for simple coupled oscillators to coupled multimodal systems. Due to the complexities and varieties of structure assemblies, these hypotheses were expressed quite generically in the modal space with no explicit math descriptions being given. As a result, the implications and relationships of these hypotheses to the apparent vibro-acoustic behavior of continuous structures are not very clear until now and are still today’s hot topics of interest to engineers and researchers. Currently, the most widely recognized condition, though not clearly defined, for a valid SEA analysis is that the coupling between subsystems should be weak. A criterion that was given in the early days of the SEA development is that the coupling would be weak if the ratio of the coupling loss factor to the internal loss factor is substantially less than unity, i.e., ηi j ηi , η j [189]. Later, the modal overlap was introduced as an indicator of the coupling strength between

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subsystems. For a plate system, it was suggested that a modal overlap greater than unity and more than 5 modes in the frequency band are the conditions for the SEA theory to be appropriate to apply [56]. Recently, Wester and Mace [237] argued that the coupling strength between two plates (a, and b) may be described as weak if T02 (ka d)(kb d)(Ma Mb )/(4π 2 ) 1, where T0 is the transmission coefficient of the junction for a normally incident plane wave; d is the junction length; ka , kb are the bending wavenumbers; and Ma , Mb are the modal overlaps of the two plates respectively. However, various studies showed that all these criteria appeared to be the necessary, but not the sufficient conditions for a valid SEA analysis. Through the study on the spatial and frequency averaged Green functions of coupled systems, Langley [129] even argued that the weak coupling condition does not guarantee the validity of a classical SEA analysis, in which the coupling only exists between directly coupled subsystems, unless the wave field in each subsystem is also reverberant. Given as an example, a serially coupled five-plate structure, each plate coupled to adjacent ones with right angles by exhibiting a line junction is shown in Figure 10.10. The dimensions of the plates are, from left to right, 0.7 m × 0.6 m × 0.001m, 0.3 m × 0.6 m × 0.001m, 0.8 m × 0.6 m × 0.001m, 0.5 m × 0.6 m × 0.001m, and 0.4 m × 0.6 m × 0.001 m, respectively. For the steel material and with the internal damping loss factors assumed to be 0.05, the modal overlaps of all five plates were found to exceed unity above 400 Hz and the coupling strengths defined in [237] for all four junctions were less than unity above 500 Hz. Besides, above 500 Hz, the modal counts within each 1/3 octave bandwidth are greater than 5 for all the plates. All these indicate that the coupling between plates is weak in general at high frequencies. A classical SEA model which consists of five subsystems, each for one plate, with only direct coupling paths considered are, therefore, straightfoward and can be generated without much difficulty. In order to provide a benchmark result, the finite element method was employed in the analysis. The number of nodes for each plate is 2793, 1225, 3185, 2009, and 1617 correspondingly. In order to generate incoherent modal responses in the plates, a point force was applied separately in the normal direction at five positions chosen randomly over plate 1 and the spatial and frequency averaged vibration response of each plate was further averaged for the five different excitation positions. As a result, the upper limit of the valid frequency range

Plate 5

Plate 1

Plate 2

Plate 3

Plate 4

Figure 10.10 FEM model of the five-plate structure.

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E2 /E1

Energy Ratio (dB)

−10

−20

−30

E5/E1 FEM

−40

- - - - SEA EFM

−50 100

1000 Frequency (Hz)

10000

Figure 10.11 Comparisons of energy ratios of plates calculated by SEA (- - -), FEM (—), and energy flow method (EFM) (•).

for this model was around 3000 Hz. The response of each plate, with only plate 1 being excited, was calculated by FEM and compared with the outcome from the classical SEA model in Figure 10.11. It is seen that although the classical SEA model predicts the response well for plate 2, which is adjacent to the excited plate 1, it severely underestimates the response of plate 5, which is further away from the excited plate even in the frequency range that the suggested weak coupling conditions were well satisfied [251]. The reason is that this particular structure construction acts as a spatial filter that favors a particular set of waves or modes in the energy transfer. A result of this is that the modal energies tend not to be evenly distributed among the modes within the subsystem as the energy transmits away from the excitation. In other words, the wave fields in those subsystems are not diffuse or reverberant enough to comply with the classical SEA requirements. In this case, the indirect power flows are expected to exist in the system [129]. At present the characteristics of the indirect power flow within the system is not well understood yet and the estimation of indirect coupling loss factors is difficult. The failure of the classical SEA in this example reveals that, for a particular structure, we still generally lack the knowledge and effective means to pre-justify which of the assumptions that are necessary for a classical SEA analysis are reasonable, which of them may be violated and if so, how much error it may result in the predictions. Nevertheless, by reviewing the research development, Langley [132] summarized the basic assumptions that appear to be critical to the applications of

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the classical SEA theory and the conditions associated with the apparent vibroacoustic behaviors that may help to promote the assumptions to be real. The two basic assumptions were, • there must be an adequate number of modes within a subsystem in the frequency band. • there must be equipartition of energy among the modes within a subsystem, and the modes must be uncorrelated. This statement is equivalent to that the response in each subsystem must be a diffuse/reverberant wave field in which the waves are uncorrelated and carry almost the same amount of energy in each direction. The conditions have been described as, • high frequency excitations to generally ensure that more modes exist in the frequency band; • distributed excitation which is sometimes stated as “rain on the roof” excitation to ensure the equipartition of modal energy and the uncorrelated modal response; • excitations in multi-subsystems to promote diffuse wave fields in all subsystems; • an irregularly-connected pattern between subsystems so that each subsystem receives energy from different directions, promoting a diffuse wave field; • weak couplings between subsystems to promote uncorrelated behavior between coupled subsystems (in contrast to the “strong coupling” for which the modes in subsystems tend to be correlated, that is, the “global” mode behavior); • high modal overlaps to promote the multimodal behavior and the equipartition of modal energy. In practice, one may expect that one or more of the conditions may not be fulfilled completely in a particular application. However, it should be noted that it is the two assumptions rather than the conditions listed here that are essential to a valid classical SEA analysis.

10.4

Experimental SEA

For various complex structures, a thorough analytical SEA analysis on the power flow within the system and the vibro-acoustic responses due to specific excitations is not always feasible. This is not only because of the difficulties involved in determining the SEA parameters, such as the coupling loss factors and internal loss factors, by purely analytical approaches, but also due mainly to the lack of knowledge about the actual energy exchanging and sharing scheme within the system and their compliances with the classical SEA hypothesis or requirements. This gives little knowledge about confidence levels of analytical predictions, especially for a complex structure. The experimental SEA was first developed to provide a means to acquire the parameters necessary for an accurate SEA model in situ. The

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measured parameters such as coupling loss factors and internal loss factors not only incorporate the uncertainties/variations that are neglected or impossible to consider in the analytical approaches, but also accommodate the effects associated with the particular substructuring scheme chosen in the test. As the mechanisms of indirect coupling loss factors and negative loss factors found in the measurements being revealed in the investigations, the technique has been extended beyond the original scope and playing an important role in studying the actual power flow behaviors within complex systems. A more general notion, namely the energy flow analysis which can be pursued either experimentally or analytically/numerically, has been developed based on, but not necessarily limited to the classical SEA framework. Nowadays, the experimental SEA or energy flow analysis has been recognized as an efficient tool to diagnose noise/vibration problems in regard to power flow paths within the system and to explore various engineering solution strategies by extrapolating the model parameters. In this section, the basic experimental SEA principle is discussed first, then recent developments within the experimental SEA framework are introduced briefly.

10.4.1

General theory

The basic idea of experimental SEA is, by measuring the vibrational energy levels of various subsystems for various power input configurations, to inversely yield the SEA parameter matrix. Since in SEA all the relevant parameters are frequency band and spatially averaged, the corresponding measurements are relatively robust but do not require detailed descriptions of the system. For a system with N subsystems, if only subsystem 1 is excited, a SEA power balance equation like Equation 10.22 can be written as,      E 11 η11 −η21 · · · −η N 1 P1  0   −η12 η22 · · · −η N 2   E 21       (10.38)  ..  = ω   .   .   · · · · · · . . . · · ·   ..  −η1N −η2N · · · η N N 0 E N1 where E i1 is the average total vibrational energy of subsystem i when subsystem 1 is excited. If each of the N subsystems is excited in turn, N sets of SEA power balance equations in the form of Equation 10.38 can be established. Assuming that all of the coupling and internal loss factors are independent of external excitations, that is, the coupling and internal loss factors keep unchanged when the excitations change from one subsystem to another. The combined matrix equation for coupling and internal loss factors can be written as,     P1 0 0 · · · 0 η11 −η21 −η31 · · · −η N 1  −η12 η22 −η32 · · · −η N 2   0 P2 0 · · · 0     1  −η13 η23 η33 · · · −η N 3      =  0 0 P3 · · · 0  ·  .. ..  ω  .. .. .. . . ..  .. . . ..  .  . . . . .  . .  . . −η1N −η2N −η3N · · · η N N

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0 0 0 ··· PN

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

E 11 E 12  E 21 E 22   ·  E 31 E 32  .. ..  . . E N1 E N2

E 13 E 23 E 33 .. . E N3

−1 · · · E 1N · · · E 2N   · · · E 3N   ..  .. . .  ··· ENN

(10.39)

where E i j represents the average total vibrational energy of subsystem i when subsystem j is excited. It is seen that with all the subsystems being excited in turn, all the internal and the coupling loss factors can be solved from Equation 10.39. However, it was found that the solution of Equation 10.39 is very sensitive to a small variation (for example, less than 10%) in measured energy levels, making the energy matrix easily ill-conditioned. One of the consequences is that Equation 10.39 may yield negative coupling and internal loss factors which are not physically meaningful. To resolve this problem, Lalor [128] has proposed to split the calculation of coupling loss factors and internal loss factors into separate matrices. Thus, from Equation 10.39, N sets of matrix equations for the coupling loss factors ηri (relating to the ith subsystem) can be obtained as,     −1  E 11  − EE 1i · · · EEr 1 − EEri · · · EEN 1 − EEN i   1 E i1 ii i1 ii η1i  i1 . ii  . .    . .. .. ..  ..    .. .         .. ..  .. ηri = P i  E ri E rr · .    . . −   ωE   . E ir E rr ii   ..     . .. .. . ..   .. . . .        ηN i 1 E 1N E 1i Er N E ri ENN E Ni · · · · · · − − − E E E E E E iN

ii

iN

ii

iN

ii

i = 1, 2, · · · , N

(10.40)

In this equation, the energy matrix has a smaller dimension, (N − 1) × (N − 1) for N > 3, than that in Equation 10.39, and therefore tends to be better conditioned. The internal loss factors ηi can be obtained by substituting Equation 10.40 into Equation 10.39 to give, E ji Pi − ηi j + η ji . ωE ii E ii j=1 j=1 N

ηi =

j= i

N

(10.41)

j= i

In fact, internal loss factors can be evaluated directly by considering the power balance of the whole system, that is, the input power to any one subsystem must be equal to the total power dissipated in all the subsystems. Thus, we have 

η1  η2   ..  .





E 11 E 21   E 12 E 22 1   =   ω  ··· ··· ηN E 1N E 2N

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−1   · · · E N1 P1   · · · E N2    P2     . .. . · · ·   ..  ··· ENN PN

(10.42)

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However, the use of Equation 10.40 still cannot guarantee that all the coupling loss factors obtained are positive. Note that in the energy matrix of Equation 10.40, the energy ratio E rr /E lr , l = r will be much greater than unity since the energy of a directly driven subsystem must always be greater than that of a subsystem either directly or indirectly connected to it if the weak coupling condition is satisfied (the weak coupling condition in here refers to ηi , η j  ηi j ). Lalor [128] has simplified Equation 10.40 further by assuming that the diagonal terms in the energy matrix are much greater than the off-diagonal terms such that the off-diagonal terms could be neglected. The inverse of the energy matrix can be approximately written as       E E 1i E ri E N1 E Ni Er1 11 − − − · · · · · · E i1 E ii E i1 E ii  E i1 . E ii  .. ..   . .   . .     .. .   E ri E rr .  → . . −E   E ir rr   .. .. ..   . . .       E 1N − EE 1i · · · EEr N − EEri · · · EEN N − EEN i E iN

ii

iN

      →    

ii

iN

 0 ··· ··· 0  .  0 .. 0   .. ..  E rr . . .   E ir  .. . 0  0  ENN 0 ··· ··· 0 E

ii

E 11 E i1

(10.43)

iN

As a result, the coupling loss factor between any of the two subsystems has a simple form as 1 E ji P j . (10.44) ηi j ≈ ω E ii E j j Since all the terms in this equation must be positive, the coupling loss factor obtained from Equation 10.44 is always positive, thus eliminating the negative coupling loss factors completely. Another benefit of using Equation 10.44 is that it does not require inverting a large matrix, which makes the measurement even more simple because the calculation of a coupling loss factor is only related to the response of the two subsystems of interest, regardless of how many other subsystems exist within the system. However, great attention should be paid when using Equation 10.44 for a complex structure system because the assumption that the diagonal terms in the energy matrix are much greater than the off-diagonal terms may not be true for some subsystem pairs. For example, a diagonal term E rr /E lr , l = r could be smaller than an off-diagonal term E ir /E jr , i = r, j = r, i = j if subsystem j is further away from the driven subsystem r while subsystem i is close to the excitation. Though E ir , E jr and E lr all have smaller numbers compared to E rr , the

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ratio E ir /E jr could be greater than E rr /E lr . Apparently, due to the complexities of real structures, it is rather difficult to give a full discussion on the validity of Equation 10.44. By taking a three subsystem SEA model as an example, Lalor [128] showed, however, that the assumption made here is quite reasonable and Equation 10.44 should apply with sufficient accuracy for subsystem pairs that are directly connected. In fact, it can be observed that Equation 10.44 implies that for any subsystem pairs considered, there is only one direct power flow path existing, other power flow paths related to these two subsysytems being neglected. Since Equation 10.44 is subject to the weak coupling assumption, it only applies to the subsystems that are directly connected, that is, the direct coupling loss factors. For indirect coupling loss factors, Equation 10.44 might be acceptable for the subsystem pairs separated by only one or two subsystems, depending on the coupling strength, and the damping values of the system.

10.4.2

Recent developments

Experimental SEA principles so far have inspired extensive studies and investigations that have reached the field beyond the traditional SEA territory. New findings and techniques that help to understand the fundamental pros and cons of classical SEA theory and enhance the modeling capabilities especially in the midfrequency range have been reported world widely in recent ten years. The midfrequency problems are referred to those in the frequency range beyond the upper limit of the FEM due to unacceptable computing resources required and the lower bound of the classical SEA because of its fundamental flaws as discussed above. With FEM and SEA having different concerns and strategies in extending their frequency limits into the midfrequency range, a better understanding of the characteristics of the energy flow in the mid to low frequency range is essential to impose new connotations and values onto SEA. Negative coupling loss factors Negative coupling loss factors found originally in many experimental SEA studies were believed to be related to the testing errors in energy ratios and the corresponding matrix inversion (Equation 10.39). Although using Equation 10.44 ensures positive coupling loss factors, the weak coupling assumption invoked in the approach introduces constraints and uncertainties in applying the method to strongly coupled systems and complex systems at low frequencies where the couplings tend to be strong in general. Woodhouse [243] proposed an iterative procedure to produce a symmetric matrix with positive loss factors. The approach was described later by other researchers as ‘massaging’ the system responses into the SEA framework, although fundamentally the system might not be of SEA behavior. Recent studies showed that although it is true that the derived coupling loss factors are sensitive to small variations in the responses due to matrix inversion, some aspects related to the uncertainties that appear to be fundamental to the structure or the approach itself should not be ignored. One of the arguments is that the

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measurement, which is normally carried out on a specific structure, is only one sample of the assemblies, of which the results, from the statistical point of view, may have large deviations off the assembly mean. Given the dynamic behaviors of structures, such deviation is directly related to the modal overlap factors of the system. Greater modal overlap factors generally yield smooth, less varied solutions with relatively high confidence levels [62, 91]. At low frequencies where the modal overlap is normally low, large deviations can be expected and the negative loss factors might be observed. Moreover, based on a two-coupled-oscillators model, Sun et al. [195] investigated the relationship between the exact loss factors defined in the model and the derived loss factors in accordance with the experimental SEA principles. It was found that derived loss factors, including coupling loss factors and internal loss factors, could be different from the exact values defined in the model if there were energy dissipation mechanism in the path of power transfer, for example the damping at the interface of the junctions. In this case, the effect of the damping at the junction was distributed to subsystems as additional internal energy dissipations, thus making the derived internal loss factors as equivalent ones. This equivalency generally becomes obvious when the coupling is strong, which could trigger negative loss factors to appear at low frequencies in the derived solution. The implication of this finding for continuous structures is that the substructuring scheme chosen for a specific structure plays a critical role in deriving the negative loss factors. As discussed previously, for a practical coupled structure, how those oscillators are actually grouped and coupled, how the energy actually dissipates, and whether the substructuring scheme complies with the SEA requirements are not always clear. Although for relatively simple structures, subsystems may be reasonably determined in terms of the physical boundaries. It is very likely for a complex system that the substructuring scheme employed based on one’s prior experience does not describe the nature of the power flow topology in the structure adequately. Also in most cases, simplifications may have to be made in the SEA model by ignoring some structural details or components, which, however, may dissipate vibrational energies under specific conditions. By applying this inappropriate model, the behavior of the structure is forced to fit the corresponding energy balance equations. All the derived parameters are therefore equivalent ones which may no longer have actual physical significances. The negative loss factors, which strictly speaking should not be called loss factors in this case, are simply a result of such inappropriate substructuring scheme, not even mentioning the testing errors and the ensemble variations. Quasi-SEA or SEA-like analysis As discussed above, a classical SEA analysis which is subject to specific requirements is not always applicable to a specific structure. Although, the necessary conditions are favored toward a successful application of the approach as the frequency of interest goes higher and higher, care still should be taken as an inappropriate substructuring scheme may affect the results. Nevertheless, recent studies have shown that if a system can be generally divided into N subsystems

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which are excited by random, steady, and distributed forces, under the assumptions that the system is linear and the excitations applied to different subsystems are uncorrelated, the average total energies of all subsystems are related to the average input power in a general form as, {P} = ω{C}{E}

(10.45)

where ω is the angular frequency, {P} is a (N × 1) vector of input powers, {E} is a (N × 1) vector of vibrational energies, and {C} is generally a (N × N ) matrix relating the power inputs and the vibration responses. Depending on the characteristics of the matrix {C}, three types of analyses, the classical SEA analysis, the quasi-SEA or SEA-like analysis, and the energy flow analysis, all of which fall in the scope of Equation 10.45 but have different valid conditions and governing territories, were summarized in [145], i.e.,: • The classical SEA only considers resonant transmissions between directly coupled subsystems. No indirect coupling paths exist in the model. All the power flow paths are reciprocal, and thus matrix {C} is symmetric with the elements being loss factors as defined in Equation 10.24. The coupling loss factors and internal loss factors are positive with their physical significances as defined. • Quasi-SEA or SEA-like analysis includes direct coupling paths and indirect coupling paths as well because the requirements for the classical SEA analysis are not being fully satisfied. The reciprocity relationship is still valid for all power flow paths, indicating that only resonant transmissions, either direct or indirect, are considered. Thus matrix {C} is still symmetric. However, negative loss factors might occur in the model due to the equivalencies in the parameters caused by the substructuring scheme that is not fully compliant with the physical nature of the power flow topology within the system. • The energy flow analysis basically includes all the power transmission paths, not only the resonant but also the no-resonant transmissions, in the model. Thus matrix {C} is generally full and not necessarily symmetrical, indicating that there is no reciprocity relationship in the system. The elements in the coupling matrix are generally energy flow coefficients and could be positive or negative. Energy flow method (EFM) Given as an example of the energy flow analysis, the energy flow method (EFM) developed recently is introduced briefly here [250, 144]. Assume that two coupled linear subsystems i and j are excited by uncorrelated stationary random forces. Then the averaged vibrational energies E i and E j in the two subsystems can be written as      cii ci j Ei Si = (10.46) c ji c j j Ej Sj

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where ci j are the energy influence coefficients (EIC), and Si is the auto spectral density of the excitations which are related to the power fed into the two subsystems in the form as      d 0 Pi Si = ii (10.47) 0 djj Pj Sj where di j are the power influence coefficients (PIC). By assuming the rain-on-the-roof excitation, that is, the subsystem subjected to numerous uncorrelated point excitations proportional to modal mass matrix, the EIC’s and PIC’s can be derived as [250, 144] c ji =

k, p

djj =

kp

kp

mi m j

Bkp mk m p

(10.48)

m kk j Jk ωk π k

(10.49)

where m k , ωk are the modal mass and the natural frequency of the k th mode kp respectively in the coupled system, m i is a modal parameter that relates to the mass matrix and the modal vectors of each subsystem [251]. For a random noise with ω1 and ω2 being the lower and the upper bounds of a frequency band of interest, the cross-mode coupling terms Bkp and term Jk also have closed forms as √ √  √ √ −z k (ω2 − ω1 ) −z k −z p (ω2 − ω1 ) −z p 1 Bkp = e tan−1 + tan−1 π z p − zk ω1 ω2 − z k zk − z p ω1 ω 2 − z p   ω2 − ω2 ω2 − ω2 1 Jk = (10.50) tan−1 2 2 k − tan−1 1 2 k 2 ηk ωk ηk ωk where z k = ωk2 (1 − jηk ), and ηk is the modal damping loss factor of the k th mode. Assume that the system responses are generally dominated by the resonant modes within the frequency band. Further simplifications to Equation 10.50 can be made. Bkp ≈

(ω + ω p )2 (ηk ωk + η p ω p )  2k , 4 (ωk − ω2p )2 + (ηk ωk2 + η p ω2p )2

Jk ≈

π 2

(10.51)

The results obtained here are actually valid for any subsystem pairs in a multicoupled system. With the modal analysis done by the FEM, all the information needed for calculating the EICs and PICs is available in the FEM database. The spatial and frequency averaged response of each subsystem thus can be constructed by Equations 10.46 and 10.47. Since the actual modal information is considered in the calculations, the local as well as the global behaviors of the system are captured. Although the method still assumes the rain-on-the-roof excitations to facilitate the math derivations, no assumptions such as diffuse wave fields and high modal densities as necessary for a classical SEA analysis are required. As seen from Equations 10.46 and 10.47, the terms associated with the indirect coupling in the

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system generally exist to represent the actual power flow in the system. The method is therefore reckoned to be effective in the mid-frequency range where the classical SEA is not valid. In Figure 10.10, the results calculated by the energy flow method for the five-plate system show good agreement with the standard FEM analysis. However, since the assumption of incoherent modal behavior is still necessary in this method, the predominant peak performance associated with individual modes that is usually seen at low frequencies should not be expected in the EFM results. This technique has already been implemented in MSC/NASTRAN v2004 as a postprocessing tool. Other methods that are currently under investigation by researchers include the fuzzy structure method [191], the energy finite element method [218], and the hybrid SEA/FEM method [131]. Readers may refer to these publications for details. Automatic substructuring It is worth noting that all the methods mentioned here still reply on the prior knowledge of the substructuring. As discussed above, inappropriate subsystem definitions may lead to matrix inversion problems, yielding nonphysical solutions for the model derived parameters. In trying to reduce the level of “human effect” involved in the model substructuring so as to improve the model quality, Gagliardini et al. [70] developed an automatic substructuring scheme based on, but not limited to, the finite element analysis. The basic idea was to identify the subsystems by performing an optimization searching and grouping scheme over the calculated or measured point to point frequency response functions (FRF) over the structure system, assuming that a subsystem is an energetic entity that must exhibit at the same time significant energy level difference with other subsystems, and minimized internal heterogeneity. The advantages of this approach are that it is independent of the user expertise, and it can be performed in each frequency band, which may lead to different substructuring schemes as necessary as required in SEA. This technique was shown quite effective and promising for complex structures, such as the vehicle floor. Variances in SEA responses It was argued that SEA results are more meaningful in the sense that the average is taken across a population or a ensemble of similar structures. In statistics, the variance of a data set, in addition to the mean value, is another important measure, from a different aspect, for a random process. For SEA, the significance of knowing the variance in the response is not limited to understanding the dispersion and the confidence level of the predictions. Information such as the probability that a given product will meet the specified requirement, the expected upper bounds of a vibro-acoustic environment, and the sensitivity of the mean and the variance to a particular structural component, all of which is valuable at the early stage of the product design, may become available. Recently, Langley and Cotoni [133]

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presented a method to predict the variances in SEA responses. The method only considers the uncertainties caused in the dynamic properties of subsystems, for example, the local mode shapes and natural frequencies. Two basic assumptions were made to facilitate the analysis: the SEA model provides good estimations of the statistical mean which means that the SEA is built correctly and all standard SEA requirements are met; and in each subsystem, the wave fields are sufficiently random and the frequency of interest is above a transition frequency when the random variation in an individual natural frequency exceeds the mean spacing between two adjacent natural frequencies. It was further argued that the natural frequencies of a subsystem conform to the Gaussian Orthogonal Ensemble (GOE) statistics. A result of this was that the variance in the dynamic properties of a subsystem was shown not to depend on the detailed causes of uncertainty, which thus makes it possible to predict the variance without knowing the detailed differences from one product to the next. Two key factors that directly affect the variances in the SEA responses were found as the variance arising in the external input powers and that in the coupling powers between the subsystems. By introducing two new parameters associated with the loading and the coupling types, an analytical expression was developed for the variances in the SEA predicted total energy responses of each subsystem. It was concluded from the study that the variances for a built-up structure primarily depends on the modal overlap of each subsystem, the frequency band over which the response is averaged, the type of the excitation and the way in which the subsystems are coupled together. Generally, high modal overlaps and wide frequency average bands yield less variance in the SEA responses. R This method has been implemented in AutoSEA2version 2004 as the variance module.

10.5

Application to electrical motors

As seen in previous chapters, owing to the complexities involved, a thorough analytical vibro-acoustic analysis is almost impossible for a motor structure which normally consists of a stator, a rotor, and a casing. In the literature a lot of vibroacoustic analysis work was conducted only on the stator as it was believed that the stator would dominate the total vibro-acoustic response of the whole motor structure. This is only true, however, for large motor structures. For small and medium sized motors, as shown in previous chapters, all the structural details, such as the casing, end-shields and the support, should be taken into account in the analysis. Currently, numerical methods, such as the FEM and the BEM, are usually employed if the vibro-acoustic behavior of an electric motor is of interest at the assembled level. As described in Chapter 9, one may need an electromagnetic FEM model to calculate the force acting on the stator, and a structural FEM model to analyze the structural vibration behavior of the motor structure. Then an acoustic BEM model may be used to evaluate the acoustic response due to the electromagnetic force excitations. Although such numerical schemes seem to work

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well, there are quite a number of limitations for it to be applied in practice. One of major concerns is that finite element/boundary element methods, by their nature, are limited to low frequencies. A large number of elements that are necessary for extending the upper frequency limit would require sufficient computing resources that may be expensive or even prohibitive. In this sense, SEA may hold promise as a tool for the vibration and acoustic analysis of complex motor structures, especially for high frequency responses. Compared to the applications to aerospace, automotive, and building structures, it is a new attempt to apply SEA to a motor structure and at the time of preparing this book, very few works have been reported. Delaere et al [43] have measured the internal loss factors of the stator and the cast iron frame, and the coupling loss factor between the stator and the frame by using the experimental SEA technique. Wang and Lai [229] developed a SEA model for a 2.2 kW induction motor to estimate the radiated sound power due to the electromagnetic origin. The work conducted at the Australian Defence Force Academy [229] is introduced as an example.

10.5.1

Subsystems of a motor structure

For a typical motor structure, there are four basic components, the stator, the rotor, the casing, and the end-shields. In addition, a motor may have other structural elements because of installation or specific requirements. Figure 10.12 shows the motor structure in the study. The stator was press fitted into the casing, and the stator/casing was supported by a plate which was mounted on four isolators. The modal testing results showed that the vibration of the casing behaves differently in three areas, the part attached to the stator and the other two parts on both sides of the stator [224]. Therefore, it is reasonable to divide the casing and the stator into three subsystems: the stator and the part of the casing attached to the stator; and the part of the casing on each side of the stator. The base plate is another subsystem. Since the end-shields only provide the boundary conditions to the casing, and their vibration and acoustic radiation are not so important, they are

Input Power

Stator Casing 1

Base Plate Casing 2

Figure 10.12 Motor structure for the SEA model [233].

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neglected in our SEA model. Also, isolators and the rotor are neglected in the model because they are only important at low frequencies [224]. These simplifications may cause some error at low frequencies. Figure 10.11 shows the tear-down model of the motor structure. It can be seen that this model only involves the structural subsystems, implying that all the mechanical input power would be dissipated in the structure. Since the acoustic power is usually very small compared with the structural vibration power, this simplification is not expected to cause much error in the final results. Normally, the acoustic power radiated from the motor can be directly calculated by using the sound radiation efficiency and the estimated vibration energy of each substructure, as shown in Section 6.5.1. The SEA power balance equation therefore is, 

  E1 η11  E2   −η12 1  =   E 3  ω  −η13 −η14 E4

−η21 η22 −η23 −η24

−η31 −η32 η33 −η34

−1   0 −η41  P2  −η42    . −η43   0  η44 0

(10.52)

Here, only the stator (subsystem 2) is subject to the mechanical power input due to the electromagnetic force excitation.

10.5.2

Internal and coupling loss factors

The determination of internal and coupling loss factors is crucial for establishing the SEA model. Normally, internal loss factors of each subsystem have to be measured and the coupling loss factors could be either measured or predicted. In this study, the experimental approach developed by Lalor [128] has been employed to determine the internal and coupling loss factors. The experimental set up for the energy ratio and input power measurements is shown in Figure 10.13. The motor was excited by a B&K 4810 shaker driven

B&K2032 Analyzer

Power Amplifier Shaker Charge Amplifier Charge Amplifier

HP-300 Computer

Impedence Head

Charge Amplifier

Accelerometer Motor HP3569A Analyzer

Figure 10.13 Schematic diagram of the experimental set-up for measuring loss factors [233].

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by a B&K 2706 power amplifier. The B&K 8001 impedance head was inserted between the shaker and the excitation point. The force and acceleration signals, via two B&K 2635 charge amplifiers, were input to a B&K 2032 analyzer for cross spectra processing. The imaginary part of the cross spectrum was then processed on a HP series 300 microcomputer to obtain the mechanical input power. In the experiment, each of the four subsystems was excited in turn. In order to obtain the uncorrelated modal behavior as required by SEA, two excitation positions, with the normal force applied in the horizontal and vertical directions respectively, were tested for subsystem 1 to 3 and corresponding responses were averaged. The vibration responses of each subsystem for every excitation set-up were measured by a B&K 4383 accelerometer at a total of 130 points (30 for subsystem one, 40 each for subsystems two and three, and 20 for subsystem four). A HP 3569A dual channel frequency analyzer was again used to store and process the data for these 130 points. By transferring the data to a PC, the spatial averaged vibration levels for each subsystem were obtained, and were processed into 1/3 octave bands from 100 Hz to 10 kHz. Figure 10.14 shows that some of the measured internal and coupling factors are high, even greater than unity. This is due mainly to the weak coupling assumptions made in Equation 10.44. Generally, the coupling strength is strong at low frequencies, and the energy tends to distribute uniformly over the whole structure. At high frequencies, since the couplings between subsystems are becoming weak as shown in Figure 10.14 the four subsystem SEA model appears to be reasonable.

10.5.3

Input power to the stator

In order to use Equation 10.52 to predict the vibration response of the motor structure, the mechanical power input to the stator has to be calculated from the electromagnetic force determined for the operating conditions. In vibro-acoustics, the mobility is a parameter that is often used to describe the ability of a structure in responding to a force excitation. The transmitted power can be easily derived if the mobility and the total force are known [37]. For an arbitrary structure, the mechanical mobility associated with a point excitation, namely the point mobility, is determined by the modal density and the mass of the structure, as shown in Equation 10.30. However, for a surface load, such as the electromagnetic force acting on the stator in induction motors, since the vibration response which strongly depends on the force distribution is quite different from that of a point force excitation, the mechanical mobility associated with a surface load, namely the surface mobility, has to be investigated. Generally the surface mobility may not have a unique definition because different force distributions on the surface may lead to different behavior of the surface mobility. However, for a particular case, such as induction motors in which the surface load has common characteristics, it might be meaningful to derive an equivalent surface mobility with a specified equivalent force. As a result, the input power P 2 is related to the real part of the

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Noise of Polyphase Electric Motors 100

Internal Loss Factor

10 1 0.1 0.01 0.001

0.0001

100

160

h1

h2

h3

h4

250

400

630

1000 1600

2500 4000 6300 10000

1/3 Octave Center Frequency, Hz 10

Coupling Loss Factor

1

h 12

h 13

h 14

h 23

0.1 0.01 0.001 0.0001

100

160

250

400

630

1000 1600

2500 4000 6300 10000

1/3 Octave Center Frequency, Hz

Figure 10.14 Measured loss factors [233]. surface mobility M S of a structure subjected to an equivalent exciting force F by: P2 =

1 2 |F| e(M S ) 2

(10.53)

The electromagnetic noise of an electric motor is basically caused by the interaction of electromagnetic forces acting on the rotor and the stator. Since this force generally has a very large radial component (approximately an order of magnitude larger than the tangential component), the vibration of a motor structure is dominated by the flexural vibration. For the vibration response of the stator subject to electromagnetic force wave excitation, Yang [248] argued that axial vibration modes are not as important as circumferential modes, and, therefore, presented a formula for estimating the amplitude of the radial velocity of a stator core with the

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circumferential vibration modes m ≥ 2, which is,   12ωFa a 3 vr |ω=ωm = 4 m ES h

(10.54)

where a is the radius of the stator, h is the thickness, S is the inner surface area of the stator, E is Young’s modulus, and F is the amplitude of the total force acting on the stator. Since the relationship between the circumferential mode number m and the corresponding natural frequencies can be expressed as [233].  ωm h h2 m 2 (m 2 − 2)2 ∼√ = · for m ≥ 2 (10.55) m2 2 ωring 12a m2 + 1 12a where ωring is the angular ring frequency of the stator core. At high frequencies, the modal overlap is high, that is, the natural frequencies of adjacent vibration modes are close to each other. As a result, the vibration response at an excitation frequency ω may be approximated by that of the nearby natural frequency ωm . By substituting Equation 10.55 into Equation 10.54, the real part of the surface mobility is approximately given by e(M S ) ≈

vr |ω=ωm 1 ≈ F Mω

(10.56)

where M is the total mass of the stator. Equation 10.56 indicates that under the excitation of the electromagnetic force wave, the real part of the equivalent surface mobility of a circular cylindrical shell, at high frequencies, is inversely proportional to the mass of the structure, and the frequency. This result supports the discussion made in [229] that for reducing the acoustic power radiated from the motor structure, the random modulation technique may not be effective when the switching frequencies are set high. This is because at high frequencies, the total mechanical power input to the motor is only related to the mass of the motor structure according to Equation 10.56. A change in the force spectrum content rather than the magnitude would not make any changes to the input power and the radiated acoustic power. It should be noted that although Equation 10.56 has been obtained from Equation 10.55 which is only applicable for circumferential modes m ≥ 2, the natural frequencies for a cylindrical shell with both ends free are zero for circumferential modes m = 0 and 1 [226]. In this study, the electromagnetic force for specific operating conditions was calculated by FEM. The corresponding total forces acting on the stator inner surface are shown in Figure 10.15.

10.5.4

Sound power radiated from the motor structure

As described above, the SEA model does not include the acoustic environment. The sound radiation cannot be obtained directly from Equation 10.52. However, with the vibration energy of each subsystem obtained from Equation 10.52, the

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Noise of Polyphase Electric Motors 1000 ‘ideal’ sinusoid, no load ‘ideal’ sinusoid, full load PWM inverter, no load

Total Force, N

100

10 1

0.1 450 rpm 0.01 0

500

1000

1500 2000 Frequency, Hz

2500

3000

(a) 1000 ‘ideal’ sinusoid, no load PWM inverter, no load

Total Radial Force

100

10

1

0.1 1500 rpm 0.01 0

2000

4000 6000 Frequency, Hz

8000

10000

(b)

Figure 10.15 Total radial force acting on the stator calculated by FEM: (a) frequency from 0 to 3000 Hz; (b) frequency from 0 to 10,000 Hz [233]. acoustic power radiated from each subsystem could be determined if the sound radiation efficiencies of each subsystem are known. In Section 6.5.1, the modal averaged radiation efficiencies for each of the four subsystems were estimated analytically. It can be seen from Figure 6.20 that at low frequencies, the stator and base plate are more efficient in acoustic radiation than the casing.

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80

Sound Power Level, dB

70 60 50 40 30 20

Experiment

10 0

SEA 0

500

1000

1500 Frequency, Hz

2000

2500

3000

(a) 60 Experiment

Sound Power Level, dB

50

SEA

40 30 20 10 0

0

500

1000

1500 2000 Frequency, Hz

2500

3000

(b)

Figure 10.16 Comparison of the acoustic power calculated by SEA with experimental results: (a) PWM inverter, no load, 450 rpm; (b) Benchmark controller, no load, 450 rpm. (Continued on next page.) With the total force shown in Figure 10.15 as the input, by employing Equations 10.52, 10.53, and 10.56 and the sound radiation efficiencies of each subsystem shown in Figure 6.20, the acoustic power radiated from each subsystem, and hence, from the whole motor structure can be obtained. The results for the motor driven under various conditions are shown in Figure 10.16. It can be seen that the total sound power obtained from the statistical model agree fairly well with measured results in general. Note that at high motor speed, the aerodynamic noise dominates the total acoustic power, while SEA model here was only for the electromagnetic origin. At low speed, since the force calculation was

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Noise of Polyphase Electric Motors 70 Experiment

Sound Power Level, dB

60

SEA

50 40 30 20 10 0

0

500

1000

1500 Frequency, Hz

2000

2500

3000

(c)

Sound Power Level, dB

70 60

Experiment

50

SEA

40 30 20 10 0

0

2000

4000 6000 Frequency, Hz

8000

10000

(d)

Figure 10.16 (Continued): Comparison of the acoustic power calculated by SEA with experimental results: (c) Benchmark controller, full load, 450 rpm; (d) Benchmark controller, no load, 1500 rpm [233]. done only up to 3 kHz [3], the SEA calculations were also made up to 3 kHz. For the Benchmark inverter, the agreement at high frequencies is quite promising. At low frequencies, the variation in the SEA sound power prediction is directly related to that in the total force. Note that the FEM electromagnetic force model is two-dimensional [231]. A better correlation might be expected with improved electromagnetic force simulations. However, since the primary concern in SEA is the averaged response, specific peaks caused by prominent structural resonances,

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Statistical Energy Analysis 60 50 40 30 20 10 0 −10 −20 −30

299

Experiment

SEA 0

500

1000 1500 2000 Frequency (Hz)

2500

3000

Sound Power Level (dB)

(a)

40 30 20 10 0 −10 −20 −30

Experiment

SEA 0

500

1000 1500 2000 Frequency (Hz)

2500

3000

(b)

Figure 10.17 Effects of the load and harmonics on the total acoustic power calculated by SEA: (a) increment of sound power due to PWM inverter compared to Benchmark controller at no load and 450 rpm; (b) increment of sound power with full load compared to no load at 450 rpm for Benchmark controller [233].

particularly at low frequencies, are not able be detected by this statistical approach, for example, the peak around 2.5 kHz in Figure 10.16. In Figure 10.17, the effects of the load and the harmonics in the current wavefront introduced by the PWM inverter on the radiated acoustic power are also shown. The agreement between the SEA prediction and the testing results are reasonably good.

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11 Noise Control 11.1 Mounting 11.1.1

Foundation

A rigid foundation is essential for minimum vibration and proper alignment between the motor and load. Usually, a rigid and solid sole plate or common bed is used as a foundation (Figure 11.1). Reinforced concrete makes the best foundation for large motors and large driven loads (Figure 11.2). A heavy, poured concrete slab is the best foundation design. The concrete foundation should weigh at least 3 to 5 times the combined weight of the motor, coupling, and load machine. If the mass is sufficient, it provides rigid support that minimizes deflection and vibration of the whole system. The concrete foundation can be located on the soil, structural steel, or building floors. A fabricated steel base (sole plate) between motor feet and foundation is recommended. The foundation changes natural frequencies of the stator system of the machine. There will be four points of mounting for an electric motor installation, one at each corner of the mounting base. The best anchor bolts are L- or T-shaped, and should be set in pipe sleeves approximately 50 mm larger than the anchor bolt diameter. Electric motors with aluminum feet require suitable heavy flat steel washers (shim pack) between the foot and the mounting fasteners to spread the bolt-clamping load out over a suitable area. All mounting points of the motor and load machine must be on exactly the same plane to ensure the proper alignment between the motor and level. Grouting is the process of firmly securing equipment to the concrete base. The “grout” is a plastic filler which is poured between the motor sole plate and the foundation upon which it is to operate [204]. The base is a continuation of the main foundation designed to damp any machine vibration present and prevent the equipment from shaking loose during operation. A serviceable and solid foundation can be laid only by careful attention to proper grouting procedure [204].

301 Copyright © 2006 Taylor & Francis Group, LLC

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Noise of Polyphase Electric Motors 1

2

3

4

Figure 11.1 Common bed as a foundation: (1) motor; (2) coupling; (3) load machine; (4) common bed.

A one piece polymer concrete shell can replace the base plate, foundation, anchor bolts, and grouting system (Figure 11.3). Small motors may incorporate a rigid mount, with the frame welded directly to a steel plate formed to match the shape of the frame and incorporating mounting holes. Special mounting brackets are also used.

4 3 2

1

Figure 11.2 Concrete foundation: (1) L-shaped concrete bolt; (2) pipe or sheet metal sleeve; (3) grout under entire base angle; (4) shim pack.

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(a)

(b)

Figure 11.3 (See color insert following page 236.) Polymer concrete foundation Polyshield ® : (a) foundation; (b) electric motor on a polymer concrete foundation. (Photo courtesy of Structural Preservation Systems, Inc., Houston, TX, U.S.A.)

11.1.2

Principles of vibration and shock isolation

Vibration isolation is used to prevent or limit the amount of variable force transferred to the supporting structure by a motor in operation. Machine isolation can also be used to minimize the transmission of vibration to the motor. Variable forces are undesirable, as they may be distracting or destructive to a process. In an extreme case, vibration can be dangerous enough to destroy the supporting structure. Equipment is supported on vibration isolation devices such as springs and pads using different methods, depending on the equipment mounted and rigidity of the supporting structure.

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(a)

(b)

(c)

Figure 11.4 Isolation bases: (a) rails and channels; (b) welded base; (c) concrete inertia base. (Courtesy of J.P. America, Hampshire, IL, U.S.A.) The vibration isolation rails are used for smaller motors Figure 11.4a. They add stiffness to the mounting platform of the motor [217]. The vibration isolation base is usually supplied to either support a piece of equipment with a frame which is not sufficiently stiff to be isolated independently, or to unitize the mounting of a piece of equipment and the motor [217]. It should be mentioned that bases are not sufficiently rigid to be mounted on isolation. The vibration base is constructed of structural members of sufficient depth to

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(a)

(c)

(b)

(d)

Figure 11.5 Vibration isolators: (a) rubber-in-shear mount; (b) free-standing open spring mount; (c) housed spring mount; (d) restrained spring mount. (Courtesy of Greenheck, Schofield, WI, U.S.A.) provide rigidity for the isolation installation, typically channel, I-beam, or angle (Figure 11.4b). The concrete filled vibration isolation base also called the inertia base is stiffer and more stable than the steel vibration isolation base (Figure 11.4c). This type of base is preferable for load machines that are coupled directly to the motor, as the stiffness helps to maintain the coupling alignment [217]. There are several types of vibration isolators also called vibration mounts used in vibration isolation installations (Figure 11.5). The vibration isolator is a resilient member that acts both as a time delay and a source of temporary energy

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Noise of Polyphase Electric Motors Table 11.1 Vibration limits. Type of electrical machine Motors 1000 to 2000 rpm Motors over 2000 rpm Generators

Balance condition peak-to-peak displacement mm

Overall peak velocity 10 to 1000 Hz mm/s

Overall peak acceleration g 0 to 5000 Hz

0.05

5

0.5

0.025 0.05

5 5

1.0 0.5

storage. The main purpose of isolator damping is to reduce or to attenuate the vibration as rapidly as possible. A good vibration mount slows the equipment response to the force or motion disturbance (time delay), absorbs temporarily the energy, and, to some extent, dissipates or damps the energy. When a well-designed vibration mount is installed, the natural frequency of the equipment with the mount is substantially lower than the frequency of the vibration source. The rubber-in-shear mount (Figure 11.5a) consists of two load plates of steel which are embedded in a rubber pad [217]. The equipment to be isolated is bolted to the top load plate, and the bottom load plate is attached to the supporting structure. The open spring (Figure 11.5b), is the simplest of the spring mounts. It must be laterally stable without using a housing. This requires a horizontal stiffness to vertical stiffness ratio from about 0.75 to 1.25. The spring usually provides approximately 50% overload capacity [217]. The housed spring (Figure 11.5c), is similar to the open spring, but is contained in some type of enclosed housing. Often, this housing is telescoping, with rubber pads of some sort between the housing halves, providing some snubbing action for horizontal loads [217]. The restrained spring (Figure 11.5d), is the same in design as the open spring, but a housing or frame is included to restrain the vertical and/or horizontal motion of the spring.

11.1.3

Vibration limits

Table 11.1 is a general guide for acceptable vibration on electrical machines. This table was compiled from industry standards, some published specifications, from manufacturers, and from field experience.

11.1.4

Shaft alignment

The motor shaft and the driven shaft should be aligned within the tolerance of ±0.03 mm in both angular and parallel alignment [171]. This is necessary to prevent mechanical vibration due to the shaft misalignment with frequency given by Equation 7.3, i.e., f sh = 2n m where n m is the shaft speed in rev/s. The angular misalignment is the amount by which the center lines of the motor and driven shafts are skewed (Figure 11.6a). The parallel misalignment is the amount by which the centerlines of the motor and driven shafts are out of parallel (Figure 11.6b).

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2

3 (a) 1 2

3 (b)

Figure 11.6 Shaft alignment: (a) angular; (b) parallel. (1) indicator; (2) indicator base; (3) coupling hubs. (Courtesy of Teco Westinghouse Motor Company, Round Rock, TX, U.S.A.)

11.2

Standard methods of noise reduction

Rotor unbalance noise reduction The noise caused by the rotor unbalance is reduced when the dynamic rotor unbalance is within an appropriate limit. There are several rotor balance criteria in common use which are based on the practical experience of electrical machine manufacturers [248]. A “rigid” rotor is a rotor whose operating speed n rig is much less than the first critical speed n cr . A practical flexible rotor operates below its first critical speed, if [248]
   n rig 2 e f > erig 1 − (11.1) n cr where e f is the allowable mass eccentricity for the practical flexible rotor and erig is the allowable mass eccentricity for a “rigid” rotor.

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Table 11.2 Design measures for minimization of bearing vibration and noise. Measure

Effect

Reduction of vibration due to uncontrolled Coil springs as an motion of loose rolling elements. Modification axial preload on the bearing of the elastic properties of the bearing. Elastic damping elements Reduction of the noise and vibration at in the housing support structure frequencies above several hundred Hz. Bearings with cages made from More quiet operation at high speeds. synthetic plastic-type materials or solid machined cages Bearings with shields or seals Prevent ingress of dirt and foreign matter. Selection of different natural Mismatching the natural frequencies of frequencies of the bearing outer those two parts with an important ring and the bearing support mechanical and electromagnetic exciting structure force frequencies. Avoiding the imperfection Minimization of the bearing of bearing housing positions angular misalignment. Taking into account the effect Suitable clearance group (grade) of machine temperature rise to minimize the bearing induced vibration. on clearance between the rolling elements Precision bearings For machines with special low noise and vibration requirements. Selection of suitable grease Minimization of grease noise. A factory balanced machine may exhibit unacceptable noise and vibration on site when it is coupled to other machines because of: • coupling misalignment or unbalanced coupling; • unbalance of the coupled equipment. It is recommended to carry out balancing jobs in situ, that is, to balance the complete rotating system under actual operating conditions. Motor bearing noise reduction In addition to the proper choice of the bearing type and size, it is recommended to apply the design measures listed in Table 11.2 [248]. Stator stack noise reduction Loose stator laminations and mechanical stresses as a result of stacking and stator assembly contribute to the motor noise. The vibration and noise frequency due

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to loose laminations is expressed by Equation 7.4 and is equal to double the line frequency with frequency sidebands approximately ±1000 Hz, This mechanical noise can be reduced by • impregnation (encapsulation) of the stator stack; • good stator stack and frame assembly. Reduction of the air gap magnetic flux density According to Equation 2.114 the amplitude of the radial force pressure is proportional to the stator and rotor amplitudes of the magnetic flux density (MFD) waves. Reduction of the MFD reduces the radial force waves causing the acoustic noise. On the other hand, the electromagnetic torque developed by the motor decreases. The air gap MFD can be reduced in one of the following ways: • enlarging the air gap g because the magnetic flux density wave b(α, t) is inversely proportional to the air gap, i.e., b(α, t) =

F(α, t) g

(11.2)

• increasing the number of the stator turns N1 per phase or extending the length of the stator stack L i , i.e., k E V1 Bmg = √ 2 2 f N1 kw1 τ L i

(11.3)

where Bmg is the peak value of the normal component of the air gap magnetic flux density, k E is the phase electromotive force (EMF)–to–phase voltage V1 ratio, f is the input frequency, kw1 is the winding factor for the fundamental harmonic, τ is the pole pitch, and L i is the effective length of the stator stack. Reducing the pulsating noise Homopolar flux waves are responsible for producing pulsating noise and vibration in 2-pole machines [248]. Those flux waves are produced by static and dynamic rotor eccentricity and magnetic permeance variations in the stator and rotor cores (Appendix B). The following measures are needed to reduce pulsating noise from 2-pole machines [248]: • the use of a dynamically balanced rotor; • minimization of the backward-rotating field; • minimization of magnetic permeance variations in the stator and rotor cores. Skewing Skewing of the stator slots introduces phase angles between the radial magnetic forces at different axial positions [248]. As a consequence, the average radial force, vibration, and noise level is reduced by skewing.

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For small values of the stator slot skew bs , both the amplitude of the radial force (2.120) and amplitude of the radial displacement (5.1) is proportional to the slot skew factor ksk , i.e., Pmsk = ksk Pm

Amsk = ksk Am

(11.4)

where the stator skew factor for the kth harmonic is given by Equation 4.36, i.e., ksk = sin(k πbs /(2τ )/[k π bs /(2τ )], bs ≈ t1 = π D1in /s1 . Skewing is a successful method of noise reduction for short core machines, i.e., L i ≤ 300 mm [220]. For larger machines with longer cores skewing is not satisfactory and cannot be regarded as an effective way of noise reduction [220]. Reduction of sound radiation efficiency An electric motor with a cylindrical frame will be considered. The relative sound radiation factor also called the relative sound radiation efficiency varies with [248]: • the magnetic force order (circumferential mode), r ; • the effective radius of surface, ka = ωa /c, see Equation 6.44; • the length–to–diameter ratio of the machine, L f /D f , whre L f is the frame length and D f is the frame diameter. Figure 6.10 shows the variation of the relative sound radiation efficiency with ka for force orders (circumferential mode numbers) r = 0 to 6 and infinitely long cylinder, neglecting the length–to–diameter ratio. Finite length of the cylinder and consequently end effects cause that the radiation efficiency of a practical machine is different than that shown in Figure 6.10. The relative sound radiation factor of a practical machine is a function of the frame length L f and frame diameter D f (see Equations 6.44 and 6.61). The radiation efficiency for the given mode r of vibration increases as the ratio of the frame length L f to its diameter D f increases [200, 248]. This effect can be utilized for reducing the radiated acoustic power (6.73), (A.12). The machine with shorter length L f and larger diameter D f (lower L f /D f factor) radiates less accoustic power and is less noisy than that with longer length and smaller diameter. Reduction of dynamic vibration of the machine surface The reduction of the dynamic deflection of the stator-frame system surface requires a very accurate analysis of the natural and exciting frequencies. The following methods can be used to reduce the dynamic vibration: • Mismatching the natural frequencies of the machine structure and frequencies of important exciting forces. It is necessary to determine with a very high accuracy the natural frequencies of the stator-frame structure and rotor-shaft

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system and then to compare the mechanical natural frequencies with frequencies of the important forces. • Adjustment of the mechanical coupling between the stator core laminations and the frame. By adjusting the stiffness of the connection between the frame and the core, stiffness of the frame and mass of the frame, the outer frame vibration can be reduced to only a small fraction of the stator core. • Increasing the stiffness of stator core (yoke) by increasing its thickness. Approximately, if the core thickness increases by 50%, the sound power level is reduced by 10.5 dB. Owing to the cost increase, this method is only recommended for small machines. • Increasing the damping capacity of the machine structure by incorporating damping materials in the machine, for example, sealing the gap between the stator core and frame with varnish or epoxy resin or inserting shear damping rings in the motor yoke. • Increasing the exciting force order r (Figure 2.9). The stator vibration decreases significantly when the force order r increases.

11.3

Active noise and vibration control

11.3.1

Principles of active noise control

The active noise control (ANC) is a technique in which unwanted noise is canceled by introducing controllable secondary sources. The outputs of the secondary source are arranged to interfere destructively with the noise from the primary source. The performance of an ANC system is monitored by error sensors that measure the residual noise. In contrast to passive techniques, ANC systems are small, portable, adjustable to different environments, and less costly. Although the idea of ANC is not new, its practical application had to wait for the recent development of sufficiently fast electronic control technology. An ANC system can be effective across the entire noise spectrum, but it is particularly appropriate at low frequencies, typically from 20–500 Hz, where passive methods using absorptive materials or noise partitions become inefficient or expensive. Most efficient noise cancelation is usually achieved by the combination of active and passive methods. ANC is based on the principle of superposition of sound waves. In most cases, to cancel a sound wave traveling in space, a second sound wave, the socalled antinoise wave, having the same amplitude but opposite phase to the primary sound wave, is created. Fundamental issues in choosing the control methods are the physical control strategy and the control approach [52]. The integral part of the ANC system is

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2

1 Reference Signal

Error Signal

Control Signal 3 (a)

2

Error Signal

Control Signal

3 (b)

Figure 11.7 Basic ANC systems: (a) feedforward system; (b) feedback system [84].

the signal processing unit which produces the output signal on the basis of the input signals [52]. Figure 11.7 shows basic ANC systems for electric motor driven fans installed in a confined space, for example, air duct of industrial air handling system or office building air conditioning system [84]. The most common is an adaptive feedforward ANC system (Figure 11.7a). This control system consists of a reference sensor, control source, error sensor, and electronic controller (including the control algorithm) [84]. A reference sensor (microphone) samples the incoming signal, which is then filtered by the electronic controller. The electronic controller produces output power signal for the control source (loudspeaker). The error sensor provides a signal for the control algorithm to adjust the controller output signal in such a way as to minimize the sound pressure detected by the error microphone. In Figure 11.7b the electronic controller is an adaptive filter and algorithm (for adaptive system) or consists of a fixed low pass filter and amplifier (nonadaptive system). Although still in the development stage, ANC is receiving considerable attention for various applications from active hearing protectors to active attenuation of aircraft fuselage noise. It is also used in industrial apparatus, dynamic systems, automobiles, and building appliances as HVAC (heating, ventilation, and air conditioning).

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11.3.2

313

Induction motor acoustic noise reduction

The acoustic noise of magnetic origin can be reduced by generating in the air gap the antinoise MFD with the same amplitude as the primary MFD wave, but shifted in space by 180◦ . Let us assume that the space harmonic ν = 7 produces excessive noise. According to Equation 2.111, the 7th harmonic stator MFD wave can be expressed as b17 (α, t ) = Bm7 cos(7 p α − ωt ). (11.5) The wave Equation 11.5 gives the rise to the following radial magnetic pressure due to the 7th harmonic of the stator MFD pr 7 (α, t ) =

1 b17 (α, t )2 . 2µ0

(11.6)

If the following anitinoise stator MFD can be generated b17a (α, t ) = Bm7a cos(7 p α − ωt − π )

(11.7)

and magnitudes Bm7 = Bm7a are equal, the total radial magnetic pressure produced by the two waves (Equation 11.5) and (Equation 11.7) is equal to zero, i.e., pr 7t (α, t) =

1 [b17 (α, t) + b17a (α, t)]2 = 0. 2µ0

(11.8)

Figure 11.8 illustrates a partial cancelation of the magnetic pressure produced by the 7th space harmonics ν = 7 of the primary and antinoise MFD waves (t = 0) with amplitudes Bm7 = 0.18 T and Bm7a = 0.7Bm7 in a 2-pole (2 p = 2) induction motor. Even if the MFD waves differ in amplitudes (Figure 11.8a), the magnitude of the reduced radial magnetic pressure is more than ten times lower than that produced by the primary wave (Equation 11.5), i.e., 12,890 N/m2 vs. 1160 N/m2 . This principle can be applied to a.c. machines for low frequency ANC of magnetic origin. A special hardware and software is required. In [13] an ANC of induction motor by injection into the stator winding higher current harmonics with controllable amplitude, phase, and frequency has been proposed. The induction motor is fed with a voltage source PWM inverter at constant voltage–to–frequency ratio V1 / f = const. The phase current signal is transferred to the synchronization system which in turn provides a signal (proportional to the frequency) to the constrol system. The control system adjusts the voltage harmonic parameters for the inverter [13]. The maximum magnetic noise of and electrical machine is for frequencies of radial magnetic forces which are the same or very close to natural frequencies of the stator system. Avoiding the excitation of stator system resonance frequencies can be a key factor in ANC strategy of induction motors [86]. A VVVF inverter-fed induction motor operates at constant voltage–to–frequency ratio, i.e., V1 / f = const. The acoustic feedback can be used to alter the V1 / f ratio in order

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314

Noise of Polyphase Electric Motors 0.2 0.2 0.1 b17 (α, t) b17a (α, t)

0 −0.1

−0.2 −0.2

0 0

1

2

3 α (a)

4

1

2

3 α

4

5

6 2⋅π

1.5 ⋅104 1.5 ⋅104 1 ⋅104 pr7 (α, t) pr7t (α, t)

5000

0 0

0 0

5

6 2⋅π

(b)

Figure 11.8 Cancelation of radial magnetic pressure due to the 7th space harmonic of the MFD (Equation 11.5) by generation of the antinoise wave (Equation 11.7) for Bm7 = 0.18 T, Bm7a = 0.7Bm7 , 2 p = 2, and t = 0: (a) MFD waves according to Equations 11.5 and 11.7; (b) radial magnetic pressures according to Equations 11.6 and 11.8. to achieve a given torque–speed operating condition using a range of different inverter frequencies [86]. In order to maintain the same torque–speed operating point, it is necessary to vary V1 / f which results in motor operation at higher and lower magnetic flux than the rated flux. The torque–speed characteristic can be effectively changed. The excitation frequencies range from the nominal frequency f n to 1.1 f n to avoid sharp resonances which may have bandwidths of only 2 to 3 Hz [86]. The motor can maintain the required loading conditions and, at the same time, keeps away from the excitation at resonant frequencies. In this way, large noise peaks in the noise spectrum are suppressed. The ANC of an induction motor is shown in Figure 11.9. The controller responds to the feedback through a minimization strategy while the accurate

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315 Input Noise Signal

IM

Comparator

Current Voltage

Controller

Figure 11.9 Schematic diagram for ANC of an induction motor [86]. excitation frequency and corresponding voltage are adjusted to minimize the vibroacoustic response at the operating speed and load [86].

11.3.3

Active vibration isolation

The structure-borne noise is produced by vibrating machine parts as, for example, engine crank shaft, gear box, compressor, pump, electric motor, and so on. The basic vibro-isolation method is to interrupt the propagation of vibration on the route from the source to the object [125]. Therefore, active vibration mounts and active shock absorbers have been developed. Those active parts are stiff enough to carry the static load and are dynamically resilient to avoid transmission of vibration. The vibro-isolated object is fitted with an appropriately controlled actuator which exerts a force that counteracts the forces producing vibration. For active vibration isolation the following actuators have found applications: electropneumatic actuators, electrohydraulic actuators, piezoelectric actuators, and electromechanical actuators (voice coil motors, moving magnet motors, linear motors). As active shock absorbers magnetorheological dampers with piston, magnetic circuit, and magnetorheological fluid are used. The dynamic vibration absorber is designed to reduce the influence of a force, the excitation frequency of which is nearly coincidental with the natural frequency of the system. An ideal undamped dynamic vibration absorber consists of an auxiliary mass and a stiff spring, tuned to the excitation frequency necessary to cause the steady amplitude of the system to be near zero at that frequency [29]. The so-called reaction mass actuator (RMA) combines an electricallypowered actuator with a reaction mass in order to generate time-variable forces [78]. An RMA can be added to existing equipment and deployed either at the source of vibration or, alternatively, at critical locations where vibration must be small. Smart magnetic materials, for example, magnetostrictive materials (Figure 11.10a) or linear electric motors (Figure 11.10b) can be used as RMA. Magnetostrictive

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(a)

(b)

Figure 11.10 Reaction mass actuators: (a) magnetostrictive actuator; (b) linear induction motor. (Photo courtesy of SatCon, Cambridge, MA, U.S.A.) materials offer extremely high force density ratio at frequencies above 400 Hz, for example, canceling helicopter gearbox vibration and noise [78]. Linear motors offer cancelation at low frequencies, limited only by the stroke capability, and can be used for example, as electromagnetic shakers for flight flutter testing [78]. As an example, a high speed elevator will be considered. Small deformations or misalignments in the elevator guide rails can make them vibrate from side-to-side

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317 Linear Motor

Figure 11.11 ANC of passenger elevator car using a high performance permanent magnet (PM) linear motor. (Photo courtesy of Mitsubishi, Japan.) making passengers feel uncomfortable. Mitsubishi in Japan have come up with a new technique which uses active vibration control and PM linear motors to stabilize the passenger compartment as it moves (Figure 11.11). The system uses accelerometers to sense the minuscule lateral vibrations. This information is fed to a small electronics package, which controls a pair of active roller guides attached to the bottom of the lift. Linear motors consume less than 20 W of power. The system will allow elevators to travel at speeds more than 5 m/s, with a lateral vibration acceleration level less than half the usual level, that is, 10 cm/s2 .

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Appendix A Basics of Acoustics A.1

Sound field variables and wave equations

In acoustics, sound waves are not specifically limited to the sensation of hearing, but are broadly defined as the small fluctuations in the pressure, the medium density, the temperature and the velocity in the wave-carrying medium, such as air, which is caused by a disturbance such as a solid vibrating surface, or jet flows, and so on. Such fluctuations in the medium, spatially, constitute the sound field. In acoustics, the sound field is normally characterized by the variable (perturbation) values of the pressure and velocity in the medium, namely the sound pressure p and the medium particle velocity v. An equation, namely the acoustic motion equation or the wave equation, which governs the wave propagation in a homogeneous medium is given as [124] ∇ 2 p(r , t) −

1 ∂ 2 p(r , t) =0 c02 ∂t 2

(A.1)

where c0 is the sound wave speed or the phase speed of the wave in the medium (for air, c0 = 344 m/s at 20◦ C). Given as an example, assume one-dimensional sound waves (plane waves) traveling along the x direction. The corresponding wave equation becomes ∂ 2 p(x, t) 1 ∂ 2 p(x, t) − 2 = 0. 2 ∂x ∂t 2 c0 The general solution of this equation is of the form

 x x p(x, t) = g1 t − + g2 t + c0 c0

(A.2)

(A.3)

where g1 (·) and g2 (·) are arbitrary functions, with g1 (t −x/c0 ) generally representing a wave propagating in the +x direction and g2 (t + x/c0 ) in the −x direction. For a plane wave varying sinusoidally with time and traveling in the +x direction, 319 Copyright © 2006 Taylor & Francis Group, LLC

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the solution can be written as,




x p(x, t) = p A cos ω t − c0

 (A.4)

where p A is the wave amplitude in Pa, ω = 2π f , and f is the frequency of the wave in Hz. Often, the complex-number representation is used for the convenience in theoretical studies [124], i.e., p(x, t) = p A e j (ωt−k0 x)

(A.5)

where k0 = ω/c0 is the acoustic wavenumber. The acoustic wavelength λ0 , which is inversely proportional to the acoustic wavenumber, is expressed as a function of c0 and f , c0 2π 2π c0 λ0 = = m. (A.6) = k0 ω f The particle velocity in the sound field for this plane wave can be derived from the sound pressure as [124] v(x, t) = −

p A j (ωt−k0 x) 1 ∂ p(x, t) = e jωρ0 ∂ x ρ0 c0

(A.7)

In acoustics, the ratio between the sound pressure and the particle velocity is defined as the specific impedance of the wave or the wave impedance, i.e. zs =

p v

Ns/m3

(A.8)

The specific impedance is generally a complex number, indicating the capability of energy propagating to the adjacent medium particles along the traveling direction. For a plane wave of which the propagation is thorough, we have z s = ρ0 c0 . Particularly, ρ0 c0 is called the characteristic impedance of the medium. Any discontinuities of the wave impedance or mismatches between the wave impedance and the characteristic impedance along the propagation path would cause sound reflections or even energy dissipations [124]. The energy conveyed by the sound waves is usually described by the sound intensity and the sound power. The sound intensity is defined as the energy transmitted through a unit area perpendicular to the wave propagation direction per unit time.  Tp 1 I = pvdt W/m2 (A.9) Tp 0 where T p is the period of the signal. When the sound pressure and the particle velocity are described by the complex-number representation, the sound intensity for a plane wave can be expressed as, I =

1 p 2A e( pv ∗ ) = 2 2ρ0 c0

(A.10)

where e(·) and the superscript ∗ represents the real part and the conjugate of a complex number, respectively.

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Appendix A

321

The sound power is the total energy through the area S p perpendicular to the wave propagation direction per unit time, which is  = IdS W. (A.11) Sp

The radiated sound power from a source has a general expression that relates the surface vibration velocity to the characteristic impedance of the ambient medium.   = ρ 0 c0 S v 2 σ (A.12)   where S is the area of the radiating surface, v 2 is the spatial averaged mean square velocity over the structure radiating surface, and σ is the sound radiation efficiency of the radiator. The radiation efficiency of different radiators is discussed in Chapter 6. The sound energy density (SED) per unit volume of a medium is D=

p 2A p2 = A 2 γ P0 ρ0 c0

(A.13)

where P0 is the static pressure of the medium, and γ = 1.4 for diatomic gases, for example, air.

A.2

Sound radiation from a point source

Assume a spherically symmetric source, for example a pulsating sphere with radius a and the surface velocity u(t), for which the sound field is expected to have no angular dependence. By utilizing spherical coordinates in which the source is located at the origin point, the wave equation can be written as, ∂ 2 p(r, t) 2 ∂ p(r, t) 1 ∂ 2 p(r, t) − 2 + =0 2 ∂r r ∂r c0 ∂t 2

(A.14)

where the sound pressure in the field p(r, t) is only a function of the radial distance r and the time t. Assume that the sphere surface is pulsating sinusoidally with time as u(t) = u 0 e jωt . The sound pressure response in the field will also have sinusoidal variations with time as p(r, t) = p A (r )eωt . The wave equation can be rewritten as, d 2 p A (r ) 2 d p A (r ) + k02 p A (r ) = 0. + (A.15) dr 2 r dr The general solution of Equation A.15 is of the form, p A (r ) =

A − jk0 r B e + e jk0 r r r

(A.16)

where A, and B are two constants. The first term represents an outgoing wave from the source, and the second term represents a wave propagating inward to the

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origin. In a free-field where the inward going wave does not exist due to the acoustic unbounded boundary condition at infinity, we have B = 0. At the sphere surface, the acoustic boundary condition requires that the surface normal velocity u 0 be equal to the corresponding component of the medium particle velocity, which is  1 d p A (r )  v A (r )|r =a = − = u0. (A.17) jωρ0 dr r =a Thus, constant A can be determined from Equation A.17, and the sound pressure in the field can be obtained as, p A (r ) =

jρ0 c0 k0 Q 1 − jk0 a − jk0 r e 4πr 1 + k02 a 2

(A.18)

where Q = 4πa 2 u 0 is called the source strength, which is the volume velocity at the sphere surface. Equation A.18 indicates that the amplitude of the sound pressure generated by this spherical source in a free-field is inversely proportional to the distance between the receiver and the source, and also depends on the source strength, frequency, source dimension, and medium characteristics. Particularly, at low frequencies where the source radius is much smaller than the acoustic wavelength, i.e. k0 a 1, there is | p A | = (ρ0 c0 au 0 )(k0 a)/r ; and at high frequencies where the source radius is much greater than the acoustic wavelength, i.e. k0 a 1, we have | p A | = (ρ0 c0 au 0 )/r . It is seen that for a given vibration amplitude u 0 , the sound pressure in the field has smaller amplitudes at lower frequencies, indicating a less efficient sound radiation. Normally, a large source (a ↑) or large source vibration amplitudes (u 0 ↑)are necessary in order to have the low frequency acoustic response be at the same level as high frequency responses. An example of this is the speaker system. In particular, the source that satisfies the condition k0 a 1 is called the point source in acoustics, for which the sound pressure in the field has a simple expression as, jρ0 c0 k0 Q − jk0 r p A (r ) = e . (A.19) 4πr The medium particle velocity can be obtained as 
pA 1 1 d p A (r ) v A (r ) = − = . (A.20) 1− jωρ0 dr ρ0 c0 jk0r The specific impedance of the wave there is 
p jk0r z s = = ρ 0 c0 · . v jk0r − 1

(A.21)

It is seen that in the so-called near field where the field point is very close to the source compared to the wavelength, i.e. k0r 1, the particle velocity and the sound pressure are 90◦ out of phase such that a large amount of energy exchanges between the source and the medium, as if a layer of medium is moving with the

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Appendix A

323

source [124]. In the far field, i.e., k0r 1, the particle velocity and the sound pressure are oscillating in phase like a plane wave, such that the specific impedance is equal to the characteristic impedance of the medium. The energy is thus conveyed by the medium and propagating via medium particle movements to infinity. The sound intensity (in the +r direction) at any point in the field can be obtained as, 1 ρ0 c0 k02 Q 2 e( p A v ∗A ) = . (A.22) 2 32π 2r 2 Often, the total energy radiated from the source is of interest because it represents the impact of the source on the environment. Therefore, the total sound power, which is the total energy transmitted through a spherical surface enclosing the source per unit time, is,  2π  π ρ0 c0 k02 Q 2 = . (A.23) I r 2 sinθdθdϕ = 8π 0 0 I =

As expected, the total radiated sound power is a constant irrespective of the radial distance.

A.3

Decibel levels and their calculations

In engineering, the sound pressure is measured by the root-mean-square (rms) value pe , i.e.,   Tp 1 pe = [ p(t)]2 dt N/m2 . (A.24) Tp 0 Also, the logarithmic scale is adopted for quantifying the variations in the sound pressure because the human hearing response varies from the threshold of hearing of 20µPa to the threshold of pain of 200 Pa (i.e., a range of 10 millians). The sound pressure level (SPL) in decibels (dB) is defined as pe = 20 log pe + 94 pr e f

L p = 20 log10

dB

(A.25)

where pr e f = 2 × 10−5 Pa is the reference sound pressure in the rms value. Correspondingly, the sound intensity level (SIL) is defined as, L I = 10 log10

I = 10 log I + 120 Ir e f

dB

(A.26)

where Ir e f = pr2e f /(ρ0 c0 ) = 10−12 W/m2 is the reference sound intensity. The sound power level (SWL) is defined as L W = 10 log10

= 10 log + 120 r e f

dB.

(A.27)

The reference sound power r e f = 10−12 W is corresponding to the reference sound intensity transmitted through a unit area. For a plane wave or a wave that

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can be treated as a plane wave over an area S p , the relationship between the sound power level and the sound pressure level is, L W = L p + 10 log10 S p

dB.

(A.28)

Equation A.28 indicates that for a source in the free field, the dB increments in the sound pressure level and the sound power level are the same. For example, an increase in the source output such as the sound pressure in the sound field doubled in the amplitudes would lead to a 6 dB increment in both the sound pressure level and the sound power level. However, the increment ratios in the sound power and sound pressure are different. Table A.1 shows the dB conversions of various sound power and sound pressure ratios. The plus and minus dB values correspond to the ratio greater and less than unity, respectively. For a sound field with multisources, the total sound pressure in the sound field should be the linear superposition of the sound pressure (with phase information included) generated by each of the sources. Particularly, if the sources are incoherent, that is, the sources are either operating at different frequencies or having random phases between their generated waves, the total sound pressure is added on a linear energy basis, which is  pt2 = pi2 (A.29) i

where pt is the total sound pressure, and pi is the sound pressure associated with each source. Correspondingly, the relationship between the total sound pressure level L pt and each individual sound pressure level L pi is of the form L pt = 10 log



10

L pi 10

.

(A.30)

i

Based on Equation A.30, two incoherent waves with the same amplitudes will yield 3 dB higher in the total sound pressure level than either one of the waves. Given as Table A.1 dB conversions. Sound Power Ratio 1 1 1.25 0.8 1.6 0.63 2 0.5 2.5 0.4 3.15 0.315 4 0.25 5 0.2 6.3 0.16 8 0.125

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dB 0 ±1 ±2 ±3 ±4 ±5 ±6 ±7 ±8 ±9

Sound Pressure Ratio 1 1 1.12 0.89 1.25 0.8 1.414 0.708 1.59 0.63 1.78 0.56 2 0.5 2.24 0.45 2.5 0.4 2.82 0.36

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Appendix A

325

an example, assume that there are three incoherent sources, each generating 90 dB, 87 dB, and 87 dB at a specific point in the sound field separately. When three sources are operating together, 87 dB plus 87 dB yields 90 dB. With another 90 dB, the total sound pressure level at this point is 93 dB.

A.4

Spectrum analysis

Frequency spectrum is the representation of a signal in the frequency domain. The signal is broken into multiple periodic signals, each with an amplitude and phase. A tunable narrowband filter has a constant bandwidth w = f u − fl where f u and fl are the upper and lower half-power frequencies. The center frequency of the band is defined as f c = f u fl . (A.31) Spectrum analysis uses filters whose bandwidth is a fractional ratio of the center frequency f c of the filter. For example, a 1/3 octave filter centered at 1000 Hz would have a bandwidth of 260 Hz. Bandwidth (relative to a normalized center frequency of 1) is computed as 2x − 1, where x = 1, 1/3, 1/12 and 1/24 for octave, 1/3 octave, 1/12 octave and 1/24 octave respectively, which are the typical bandwidths used primarily for acoustical and vibration analysis. The ratio of the upper to lower half-power frequencies is f u / fl = 2 for the octave-band filter, f u / fl = 21/3 for the 1/3 octave-band filter, and f u / fl = 21/10 for the 1/10 octave-band filter. The preferred center frequencies and the corresponding logarithmic bandwidths for modern octave-band and 1/3 octave-band filters are shown in Table A.2.

Table A.2 Center frequencies and bandwidths for the preferred octave and 1/3octave-band. ISO band numbers 11, 12, 13 14, 15, 16 17, 18, 19 20, 21, 22 23, 24, 25 26, 27, 28 29, 30, 31 32, 33, 34 35, 36, 37 38, 39, 40 41, 42, 43

Octave-band center frequency f c Hz 16 31.5 63 125 250 500 1000 2000 4000 8000 16,000

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One-third octaveband center frequencies Hz 12.5, 16, 20 25, 31, 40 50, 63, 80 100, 125, 160 200, 250, 315 400, 500, 630 800, 1000, 1250 1600, 2000, 2500 3150, 4000, 5000 6300, 8000, 10,000 12,500, 16,000, 20,000

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Appendix B Permeance of Nonuniform Air Gap B.1

Permeance calculation

A space containing a quasistationary magnetic field may be partitioned into flux tubes or geometrical figures in which all lines of flux are perpendicular to their bases and no lines of flux cut their sides [173]. The permeance of a flux tube is G = µ0 µr

S = S l

(B.1)

where µ0 = 0.4π 10−6 H/m is the magnetic permeability of free space, µr is the relative magnetic permeability, S is the cross-sectional area, l is the total flux tube length, and  = µ0 µr /l is the relative permeance per unit area. The relative permeance of the smooth air gap of a machine with ferromagnetic stator and rotor cores and magnetic saturation being neglected is g0 = µ0

1 g

(B.2)

where g  is the equivalent air gap thickness which includes • the effect of slots for induction machine g  = gkC

(B.3)

• the effect of slots and permanent magnet (PM) equivalent thickness for PM brushless machines hM g  = gkC + (B.4) µrr ec In the above equation g is the air gap thickness (mechanical clearance), h M is the PM thickness (height), kC is Carter’s coefficient according to Equation 2.72, and 327 Copyright © 2006 Taylor & Francis Group, LLC

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Noise of Polyphase Electric Motors 3 ⋅ 10−4 2.5 ⋅ 10−4 2 ⋅ 10−4 Λg 1 ⋅ 10−4 5 ⋅ 10−5 0

0

0.1

0.2

0.3

0.4

0.5

x

Figure B.1 Distribution of the air gap relative permeance according to Equation B.5 for PM synchronous motor with s1 = 36, 2 p = 10, b14 = 3 mm, g = 0.95 mm, e = 0.1 mm, ed = 0.1 mm. µrr ec is the relative recoil permeability of the PM. To take into account the increase in the air gap due to the stator slot opening, the mechanical clearance g is to be replaced by an equivalent air gap g  = gkC . The relative air gap permeance with stator slot openings and eccentricity effect taken into account is g (α, t) = g0 λg1 (α)λg2 (α)λge (α)λged (α, t)

(B.5)

where g0 is the permeance of smooth air gap according to Equation 2.69, λg1 (α) is the relative value of harmonic permeance according to Equation 2.70 that includes the stator slot openings, λg2 (α) is the relative value of harmonic permeance according to Equation 2.80 that includes rotor slot openings, λge (α) is the relative permeance given by Equation 2.92 that includes the static eccentricity and, λged (α, t) is the relative permeance given by Equation 2.93 that includes the dynamic eccentricity. The relative permeances λge (α) and λged (α, t) have been derived below. The resultant permeance g (α, t = 0) according to Equation B.5 is plotted against the circumferential direction x = ατ/π in Figure B.1.

B.2

Eccentricity effect

The magnetic flux density distribution in an a.c. machine with nonuniform air gap can be described as follows: 1 b(α, t) = µ0 Fm cos( pα − ωt − φ) = ge (α)Fm cos( pα − ωt − φ) g(α)kC (α) (B.6) where g(α) is the air gap varying with the angle α, kC (α) is Carter’s coefficient depending on the angle α and φ is the angle between the field axis and axial plane from

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Appendix B

329

R

α

r

O′

O

e

Figure B.2 Magnetic circuit of an electrical rotating machine with eccentricity. which the angle α is measured. According to Figure B.2 for small eccentricities e < 0.5g the air gap can be expressed as a function of the angular displacement α, i.e., g(α) = R − r − e cos α. For uniform air gap the eccentricity e = 0 and the air gap g(α) = g = R − r . Defining the static relative eccentricity according to Equation 2.84, the variation of the air gap with the angle α can be expressed as g(α) = g − e cos α = g[1 −  cos α].

(B.7)

As the air gap decreases, Carter’s coefficient kC in Equation B.6 increases (Figure B.3). Assuming kC (α) = kC = constant (as for a uniform air gap), Equation B.6 takes the form b(α, t) = µ0

1 cos( pα − ωt − φ) cos( pα − ωt − φ) = Bm Fm gkC 1 −  cos α 1 −  cos α

= ge (α)Fm cos( pα − ωt − φ) = Bm cos( pα − ωt − φ)λge (α) where Bm = µ0

Fm gkC

(B.8) (B.9)

and 1 1 1 = µ0 λge (α). (B.10) gkC 1 −  cos α gkC Equations B.8 and B.10 do not include the variation of the air gap permeance with slot openings. The relative permeance due to static eccentricity λge in Equation B.10 can be resolved into Fourier series as follows ge (α) = µ0

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330

Noise of Polyphase Electric Motors 1.5 1.4 b14 = 5 mm 1.3 4 mm

kc

3 mm 1.2 2 mm

1 mm

1.1 1

0.002

0.001

0

0.003

0.004

0.005

g

Figure B.3 Carter’s coefficient kC according to Equation 2.72 as a function of the air gap for the stator slot openings b14 = 0.0001 m = 1 mm, 2 mm, 3 mm, 4 mm, and 5 mm. λge (α) =


1 = λ0e + [λck cos(k α) + λsk sin(k α)] 1 −  cos α k

where • the constant coefficient λ0e

1 = 2π



π

−π

1 1 dα = √ 1 −  cos α 1 − 2

(B.11)

• the magnitudes of the first harmonics √  1 − 1 − 2 cos α 1 π λc1 = dα ≈ 2 √ (B.12) π −π 1 −  cos α  1 − 2  sin α 1 π λs1 = dα = 0. (B.13) π −π 1 −  sin α Figure B.4 shows the values of coefficients λ0e and λc1 plotted against the relative eccentricity . In general, including only the first two terms of the Fourier series √ 1 1 − 1 − 2 λge ≈ λ0e + λc1 cos α ≈ √ +2 √ cos α (B.14) 1 − 2  1 − 2 In practice, the eccentricity  < 0.5, so that on the basis of Figure B.4 λ0e ≈ 1 Since √

1 1−

2

1 ≈ 1 + 2, 2

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the term

(B.15) λc1 ≈ 

(B.16)

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Appendix B

331

l 0e l c1

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

l 0e

l c1

0

0.2

0.1

0.3

0.4 e

0.5

0.6

0.7

0.8

Figure B.4 Coefficients λ0e (Equation B.11) and λc1 (Equation B.12) plotted against the relative eccentricity . Thus, including only the first two terms of Fourier series λge (α) ≈ 1 +  cos α

or

ge (α) =

µ0 (1 +  cos α) gkC

(B.17)

Similar to Equation B.14 can be written for the dynamic eccentricity. The magnetic flux density in the case of nonuniform air gap is described by the following equation resulting from Equation B.8 b(α, t) = Bm cos( pα − ωt − φ)λge (α, t) = Bm [λ0e + λc1 cos(α + ω t)] cos( pα − ωt − φ) = Bm λ0e cos( pα − ωt − φ) +

Bm λc1 Bm λc1 cos[( p + 1)α − (ω + ω )t − φ] + cos[( p − 1)α + (ω − ω )t + φ] 2 2 ≈ Bm cos( pα − ωt − φ)

  +Bm cos[( p+1)α−(ω+ω )t −φ]+Bm cos[( p−1)α+(ω−ω )t +φ]. 2 2

(B.18)

In Equation B.18 the following trigonometric identities cos x cos y = 0.5[cos(x + y)+cos(x − y)] and cos(−x) = cos(x) have been used. Owing to the eccentricity, in addition to the fundamental harmonic of the magnetic flux density, there are also space harmonics with wave numbers ν = p − 1 and ν = p + 1. For a machine with 2 p > 2 the total flux around the periphery from 0 to 2π is zero [41], i.e.,  2π

0

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b(α, t)dα = 0.

(B.19)

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In 2-pole machines ( p = 1) the integral taken from 0 to 2π over b(α, t) with respect to α is 

2π

Bm cos(α − ωt − φ)[1 +  cos(α + ω t)]dα = π  Bm cos[(ω − ω )t + φ].

0

(B.20)

Equation B.20 clearly shows that a 2-pole machine with eccentricity ( > 0) produces a homopolar axial magnetic flux hom (t) = π  Bm cos[(ω − ω )t + φ]. This flux passes through the shaft, bearings, and end bells and in addition to the unbalanced magnetic pull can induce bearing currents. Those induced currents can shorten the lifetime of bearings.

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Appendix C Magnetic Saturation Equation B.5 expresses the resultant relative permeance of the air gap. The saturation effect due to the main (linkage) magnetic flux can be included by introducing an additional permeance of the air gap sat according to Equations 2.102 and 2.103. Thus, magnetic saturation of the ferromagnetic core reduces the permeance of the air gap λg0 . This is equivalent to increasing the air gap by the coefficient ksat > 1, i.e., • for an induction machine λg0 = µ0

1 1 = µ0 g gkC ksat

(C.1)

• for a permanent magnet (PM) synchronous machine λg0 = µ0

1 1 = µ0  g gkC ksat + h M /µrr ec

(C.2)

 where geq is the equivalent air gap increased by the equivalent thickness h M /µrr ec of the PM, h M is the PM thickness, g is the air gap (mechanical clearance), kC is Carter’s coefficient according to Equation 2.79, µ0 is the magnetic permeability of free space, and µrr ec , is the relative recoil magnetic permeability of the PM. The saturation factor of the magnetic circuit is defined as

• for an induction machine ksat = 1 +

2V1t + V1c + 2V2t + V2c 2(Bg /µ0 )gkC

(C.3)

• for a PM synchronous machine ksat = 1 +

2V1t + V1c + V2c 2(Bg /µ0 )gkC

333 Copyright © 2006 Taylor & Francis Group, LLC

(C.4)

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where Bg is the air gap magnetic flux density, V1t is the magnetic voltage drop (MVD) along the stator teeth, V2t is the MVD along the rotor teeth,V1c is the MVD along the stator core (yoke) and V2c is the MVD along the rotor core. MVDs V1t , V2t , V1c , and V2c are calculated as products of magnetic field intensity in each portion of the magnetic circuit times the length of the magnetic circuit. Magnetic field intensities are read on the B–H magnetization curve of the magnetic circuit material for calculated values of the magnetic flux densities. In addition to the air gap permeance, magnetic saturation due to main (linkage flux) affects also the parameter α in Equation 2.105 which expresses Fourier coefficient bµ of the rotor magnetic flux density (Equation 2.104). The magnetic saturation slightly reduces the parameter α as follows αsat ≈

1 1/3 ksat

α.

(C.5)

There is also an influence of the magnetic saturation due to the stator leakage flux on the width of the stator slot opening b14 . As the stator tooth heads become saturated, the relative magnetic permeability of the tooth heads is very small. This effect is equivalent to an increase in the stator slot opening width. Magnetic saturation due to the stator leakage flux can be approximately included according to Norman’s method [167]. According to Norman, the magnetomotive force (MMF) of a single stator slot of an induction motor is expressed as: Fsl = 0.707Ia




Nc w 1 ap

0.75

wc s1 + 0.25 + k p1 kw1 τ s2

 (C.6)

where Nc is the number of conductors in a coil, w 1 is the number of stator winding layers, a p is the number of stator winding parallel paths, w c is the stator coil pitch, τ is the pole pitch, k p1 is the stator winding pitch factor for fundamental, kw1 is the stator winding factor for fundamental, s1 is the number of stator slots, and s2 is the number of rotor slots. For a synchronous motor s2 = 2 p, where p is the number of pole pairs. For the MMF Fsl a fictitious leakage magnetic flux density in the air gap that saturates the tooth head is calculated, i.e., Bfg =

1.6 ×

106 g



Fsl



0.64 + 2.5

g t1 +τ



(C.7)

where t1 is the stator slot pitch and τ is the rotor pole pitch, in PM synchronous machines equivalent to the rotor slot pitch. The fictitious magnetic flux density in the air gap B f g can take very large unrealistic values, up to 50 T. The saturation factor κsat due to leakage flux is a function of B f g and can be approximated as follows 5.25 κsat ≈ + 0.25. (C.8) 7 + B 2f g

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Appendix C

335

The increase in the slot opening due to magnetic saturation 
t1

b14 = − b14 (1 − κsat ). 3

(C.9)

The slot width with magnetic saturation taken into account b14sat = b14 + b14 where b14 is the slot opening of an unsaturated stator.

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(C.10)

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Appendix D Basics of Vibration D.1

A mass–spring–damper oscillator

A very basic vibration system is the classical harmonic oscillator which consists of two basic elements, a mass and a spring [122]. With an initial disturbance, such a system will vibrate freely and maintain a steady-state vibration without any decay in the amplitude since no energy dissipation mechanism is included in the system. In practice, however, all vibration systems have various energy dissipation mechanisms, such as friction, and material viscosities. This means that unless there is extra energy fed into the system continuously, the vibration will ‘die’ out eventually. Given as an example, a mass–spring–damper system subject to a steady-state force excitation is shown in Figure D.1. The motion equation of the system is, M

d 2 X (t) d X (t) + K X (t) = F(t) +C 2 dt dt

(D.1)

where M is the mass, K is the stiffness of the spring, and C is the damping coefficient. Assume that the system is subject to a sinusoidal force F0 e jωt . For steady-state solutions, we can expect the displacement of the mass being of the form X 0 e jωt . Then Equation D.1 becomes, X 0 (−ω2 M + jωC + K )e jωt = F0 e jωt

(D.2)

The displacement of the mass is thus yielded as X0 =

K−

F0 ω2 M

+ jωC

=

ω




F0 e jϕ (K /ω − ωM)2 + C 2

where tan ϕ =

K /ω − ωM . C 337

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(D.3)

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338

Noise of Polyphase Electric Motors F(t) = F0e jw t X(t) = X0e jw t M

C

K

Figure D.1 Mass–spring–damper oscillator. It is seen that the system response, including the magnitude and the phase angle, is generally frequency and system parameter dependent. Since the velocity of the mass is related to the displacement as V (t) = jωX 0 e jωt , the impedance of the system can be obtained as,   F(t) K Z (ω) = =C+ j − ωM . (D.4) V (t) ω By setting the imaginary part of the impedance to zero, the resonant frequency of √ the system can be obtained as ω0 = K /M. This frequency is corresponding to the free vibration of a mass-spring oscillator with no damping mechanism considered. Then the solution of Equation D.1 can be rewritten as X (t) =

F0 e jωt 2 M (ω0 − ω2 ) + jηω0 ω

(D.5)

where η = C/(ω0 M) is the loss factor of the system. It is interesting to examine the energy dissipation within the system. The time averaged input power is Pi  =

1 1 1 e(F V ∗ ) = e(Z )|V |2 = Cω2 |X 0 |2 . 2 2 2

(D.6)

It is seen that the input power is directly related to the damping C of the system. It is not difficult to derive that the energy consumed by the damping C is equal to the power fed into the system, i.e., Pd  = Pi . This is not surprising though because the power balance is the inevitable consequence of a steady-state vibration process. The time averaged total vibration energy of the system is the sum of the time averaged kinetic and potential energy, i.e., E =

1 1 1 Me(V V ∗ ) + K e(X X ∗ ) = (ω2 M + K )|X 0 |2 . 4 4 4

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(D.7)

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Appendix D

339

When ω = ω0 , i.e., the system is at resonance, the time averaged kinetic energy is equal to the time averaged potential energy. Then, E =

1 1 K |X 0 |2 = ω02 M|X 0 |2 . 2 2

(D.8)

The well-known relationship between the dissipated power and the vibrational energy of the system thus can be obtained, Pd  =

D.2

1 Cω02 |X 0 |2 = ω0 η E . 2

(D.9)

Lumped parameter systems

The lumped parameter system generally refers to a simplified modeling process in which the spatial dependence of the variable under consideration can be neglected. For mechanical structures, the distributed mass or inertia is replaced by a finite number of lumped masses or rigid bodies which are connected by massless elastic and damping elements, such as the spring and resistance. The basic principle involved in such simplifications is that a change in the variable, such as vibration displacement or velocity, is considered equal and simultaneous for each element. This, therefore, indicates that the approach is only valid at low frequencies where the wavelength is much longer than the dimension of the structure, so that the spatial variations in a variable are small. Depending on the complexity of the structure and the modeling objectives, the lumped parameter model can be as simple as a mass–spring–damper oscillator, or may have many mass/rigid body, spring, damper elements connected in a complex way. Mathematically, the minimum number of the coordinates necessary to describe the motion of each lumped mass or rigid bodies defines the number of degree of freedom of the system. Corresponding to each degree of freedom, there is one ordinary differential equation, as discussed in Section D.1. It can be seen that one of the advantages for the lumped parameter model is that, instead of solving the partial differential equation for a continuous system, it only deals with a set of ordinary differential equations. The motion equations of a lumped parameter system can be generally obtained from Newton’s second law of motion. Assuming that xi and θi are the generalized coordinates of the system representing the transverse and rotational movement respectively, the motion equations are of the form,   m i x¨ i = j Fi j (for mass m i )  (D.10) Ji θ¨i = j Mi j (for rigid body of inertia Ji )   where j Fi j denotes that sum of all forces acting on mass m i , and j Mi j indicates the sum of moments of all forces acting on the rigid body of mass moment of inertia Ji . Other methods used to establish the motion equations include Lagrange’s equation, and the electromechanical analogy technique. Based on the

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340

Noise of Polyphase Electric Motors

Stator

Winding

End-Shield Fan Rotor

Foot

(a) L q1

Rotor (m1)

y1 kb

kb

ks

ks

Stator (m2)

x1

q2 y2 x2

kf

kf

(b)

Figure D.2 Induction motor and its lumped parameter model: (a) motor; (b) model. motion equations, eigenvalues and eigenvectors of the system can be yielded, from which the natural frequencies and the mode shapes of the system can be extracted. A simple example of using a lumped parameter model for the dynamic analysis of a mechanical structure is given here for a typical medium-sized motor structure, as shown in Figure D.2a. A lumped parameter model which only considers the two rigid motions, namely the transverse and rotational motions of the rotor and the stator in the x y plane can be established and shown in Figure D.2b. In the model, the motor feet are simplified as two identical stiffness elements k f ; and the stator and the rotor are represented by two lumped mass elements m 1 , and m 2 . In the real motor structure, both ends of the rotor are sitting on the rotor bearings which are supported by the end-shields. In the lumped parameter model, the two

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Appendix D

341

serially coupled spring elements kb and ks are connecting the rotor and the stator, representing the bearing stiffness and the in-plane stiffness of the end-shields. Since each of the rigid element is allowed to have the transverse movement in the x direction and rotations in the z direction, two variables are thus needed to describe the motion for either the rotor or the stator, meaning this is a system with four degrees of freedom. The motion equations of the whole system can be established as, 

m1  0   0 0

0 m2 0 0

0 0 J1 0

  x¨ 1 0  x¨ 2  0  + 0   θ¨1  J2 θ¨2



    2k1 0 −2k1 0 0 x1 −2k1 2(k1 + k2 ) x2 0 0 0  =  +  0 −k1 L 2 /2 θ1 0 0 k1 L 2 /2 0 0 0 −k1 L 2 /2 (k1 + k2 )L 2 /2 θ2

(D.11)

where k1 = ks kb /(ks + kb ), k2 = 2k f . It can be seen in Equation D.11 that the transverse and rotational motions are not coupled. This is because the system, for simplicity, is modeled symmetrically in geometry. As a result, the transverse and the rotational vibrations can be solved independently. For example, the motion equations for the transverse motions are,

m1 0 0 m2





x¨ 1 2k1 x1 0 −2k1 + = . −2k1 2(k1 + k2 ) x2 0 x¨ 2

For harmonic solutions,



x1 x2

=

A1 e jωt A2 e jωt

(D.12)

(D.13)

where A1 , and A2 are the vibration amplitudes of the rotor and the stator respectively. Substituting Equation D.13 into Equation D.12,

2k1 − ω2 m 1 −2k1 −2k1 2(k1 + k2 ) − ω2 m 2



A1 A2

=

0 . 0

(D.14)

For nonzero solutions of A1 , and A2 , the determinant has to be zero, i.e.,    2k1 − ω2 m 1  −2k1   = 0. (D.15)  −2k1 2(k1 + k2 ) − ω2 m 2  The roots of this equation are, therefore, the eigenvalues of the system. 2 ω1,2

=

B±

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2 2 B 2 − 4ω11 ω22 2

(D.16)

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Noise of Polyphase Electric Motors

2 2 2 2 where B = ω11 + ω22 + 2k1 /m 2 , ω11 = 2k1 /m 1 , ω22 = 2k2 /M2 . The two natural frequencies of the system can be obtained as, 
2 2 B ± B 2 − 4ω11 ω22 ω1,2 = . (D.17) 2

For the rotational motions, it can be easily shown that the two natural frequencies 2 2 take the same expressions as Equation D.17 but with B = ω11 +ω22 +k1 L 2 /(2J2 ), 2 2 2 2 ω11 = k1 L /(2J1 ), ω22 = k2 L /(2J2 ).

D.3

Continuous systems

Strictly speaking, all mechanical systems are continuous, being with the mass and stiffness distributed evenly over the system. Due to the complex relationship between the stress and the strain of elastic materials, various types of waves, such as longitudinal, transverse, torsional, and flexural (bending) waves may exist in the system [37]. Consequently, the motion equations that govern the vibration behaviors might be different for different types of vibrations. However, all the motion equations are of the type [190],   ∂ ∂ ∂ ∂ 2ξ λ·L , , ξ +ρ 2 =0 (D.18) ∂ x ∂ y ∂z ∂t where ξ is the vibration displacement, ρ is the material density, and L(·) is a differential operator, λ is a stiffness related parameter—both of which takes different forms for different wave types. For example, for a beam in a longitudinal vibration along the x direction,   ∂ ∂ ∂ ∂2 λ·L , , (D.19) =E 2 ∂ x ∂ y ∂z ∂x where E is Young’s modulus of the material. If this beam is in bending motion,   ∂ ∂ ∂ E I ∂4 λ·L , , (D.20) = ∂ x ∂ y ∂z S ∂x4 where S is the area of the cross section of the beam, and I is the second moment of area of the cross section. For a plate embedded in the x y plane in bending motion,  λ·L

∂ ∂ ∂ , , ∂ x ∂ y ∂z



Eh 2 = 12(1 − µ2 )



∂2 ∂2 + ∂x2 ∂ y2

2 (D.21)

where h is the thickness of the plate, and µ is Poisson’s ratio of the material. For a cylindrical shell, the differential operator will take a form that is even more complicated due to the coupling between different types of vibrations.

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It is not always an easy task to solve the motion equations for a particular structure. Only simple and regular structures, such as a beam, a round or a rectangular plate might have analytical solutions. Here, a beam in bending motion is given as an example. E I ∂ 4 ξ(x, t) ∂ 2 ξ(x, t) + ρ = 0. S ∂x4 ∂t 2

(D.22)

For sinusoidal solutions, ξ(x, t) = ξ0 (x)e jωt . The motion equation can be rewritten as, d 4 ξ0 (x) + k 4 ξ0 (x) = 0 (D.23) dx4 where k = ω1/2 (E I /ρ S)−1/4 is the wavenumber of the bending wave in the beam. It is seen that unlike the sound wave in the air, the speed of the bending wave (c = ω/k) in a beam depends on the frequency. The general solution of this differential equation is of the form, ξ0 (x) = Ae− jkx + Be jkx + Ce−kx + Dekx

(D.24)

where A, B, C, D are constants depending on the boundary conditions at both ends. Generally, the first two terms represent two approaching waves traveling in the opposite direction, while the last two terms represent the fields that decrease exponentially with distance from the two ends, thus namely the “near field” solutions. There are three typical boundary conditions in the vibration analysis, namely simply supported, clamped, and free. For a beam, these boundary conditions can be described mathematically as,  d 2 ξ0 (x)  ξ0 (x)|x=x0 = 0, = 0 for a simply supported end d x 2 x=x0  dξ0 (x)  ξ0 (x)|x=x0 = 0, = 0 for a clamped end d x x=x0   d 3 ξ0 (x)  d 2 ξ0 (x)  = 0, = 0 for a free end. d x 3 x=x0 d x 2 x=x0 These mathematical descriptions are associated with the behavior of four typical field variables for bending waves, the transverse displacement ξ0 (x), angular displacement dξ0 (x)/d x, bending moment ∼ d 2 ξ0 (x)/d x 2 , and the shear force ∼ d 3 ξ0 (x)/d x 3 . With the simply supported condition at both ends of the beam (x0 = 0, l), we have, A = −B,

C = D = 0,

sin kl = 0.

(D.25)

This equation indicates that in order to comply with the boundary conditions, the wavenumber has to take a set of discrete numbers, namely the modal wavenumbers, as, nπ kn = , n = 1, 2, 3, · · · . (D.26) l

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344

Noise of Polyphase Electric Motors

The corresponding set of discrete frequencies, namely the natural frequencies of a simply supported beam, are   E I 1/2  nπ 2 ωn = , n = 1, 2, 3, · · · . (D.27) ρS l Associated with each natural frequency ωn , the vibration displacement takes a particular pattern ξ0n (x) = sin(nπ x/l), namely the mode shape functions. In vibration, a mode shape function together with its corresponding natural frequency, and modal wavenumber comprise a so-called “vibration mode.” The vibration response of the beam is, therefore, the linear superposition of these vibration modes. ∞ ∞  nπ    ξ0 (x) = An ξ0n (x) = An sin x (D.28) l n=1 n=1 where An are a set of constants determined by the external excitations. If the two ends of the beam are clamped, the mode shape of the beam can be derived as, cosh(kn l) − cos(kn l) [sinh(kn x) + sin(kn x)] sinh(kn l) − sin(kn l) (D.29) where the modal wavenumbers kn are the solution of, ξ0n (x) = cosh(kn x) + cos(kn x) −

cosh(kn l) · cos(kn l) = 1.

(D.30)

The roots of this equation can be shown as 4.73, 7.85, 11.0, 14.14, · · ·, which can be represented approximately by, kn ≈

π nπ − , l 2l

n = 1, 2, 3, · · · .

(D.31)

The results shown here clearly indicate that the modal behaviors, for example, natural frequencies, mode shapes, of a structure very much depend on the boundary conditions. Since the modal behaviors are generally independent of the external excitations, they are the inherent characteristics of a structure in terms of the structural dynamics. In most cases, the forced vibration is of greatest interest. A more general equation of motion with the energy dissipation mechanism considered by introducing the complex stiffness parameter is given as,   ∂ ∂ ∂ ∂ 2ξ λ(1 + jη) · L , , (D.32) ξ +ρ 2 = f ∂ x ∂ y ∂z ∂t where f is the distributed force density over/within the structure. For steady-state harmonic solutions ( f = f 0 e jωt , ξ = ξ0 e jωt ), it becomes   ∂ ∂ ∂ , , ξ0 − ρω2 ξ0 = f 0 λ(1 + jη) · L (D.33) ∂ x ∂ y ∂z

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Appendix D

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 Note that the general solution of this equation is of the form ξ0 = ∞ n=1 An ξ0n where ξ0n are the mode shape functions which are determined by boundary conditions and satisfy the relationship,   ∂ ∂ ∂ λ·L , , (D.34) ξ0n − ρω2 ξ0n = 0. ∂ x ∂ y ∂z Equation D.33 can be rewritten as, ∞ 

An ρ[ω02 (1 + jη) − ω2 ]ξ0n = f 0 .

(D.35)

n=1

By mutilplying Equation (D.35) with ξ0m and taking integration over the solution domain (τ ), all terms on the left-hand side vanish due to orthogonality replations among the mode shape functions, except the term for which n = m. Then,   2 An [ω02 (1 + jη) − ω2 ] ρξ0n dτ = f 0 ξ0n dτ. (D.36) τ

τ

The solution to Equation D.32 thus can be obtained as, ξ=

∞  n=1

ξ0n e jωt 2 n [(ω0 − ω2 ) + jηω02 ]

 τ

f 0 ξ0n dτ

(D.37)

 2 where n = τ ρξ0n dτ is called “norm.” It can be found that the response of a continuous system is a linear superposition of all the modes. Each mode is of the type of a simple oscillator as described in Equation D.5. The contribution of each mode to the total response, also namely the participation factor ( An ), primarily depends on the distribution of external forces with regard to the mode shape functions, and how far away the corresponding natural frequency is from the exciting frequency.

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Symbols and Abbreviations A Am Amr a ap aw  B Bm b bs b f sk bp bsk C c cE c0 cp cT D E Ef Ei e ed ef F F f fc fr fring G g

line current density, A/m peak value of the line current density, A/m magnitude of radial forced vibration displacement radius of cylindrical shell number of parallel current paths of the stator winding number of parallel conductors of the stator winding vector magnetic flux density, T peak value of the magnetic flux density, T instantaneous value of the magnetic flux density; width of slot skew of stator slots skew of PMs pole shoe width skew of stator slots damping coefficient wave velocity; tooth width armature constant (EMF constant) sound velocity in the air structural wave speed torque constant diameter EMF, r ms value; Young modulus (elasticity modulus) EMF per phase induced by the rotor without armature reaction averaged vibrational energy of the subsystem i instantaneous EMF; static eccentricity dynamic eccentricity instantaneous EMF excited by PMs in the stator phase winding force magnetomotive force (MMF) frequency critical frequency; center frequency of the band natural frequency of the r th order ring frequency pearmeance, H; shear modulus air gap (mechanical clearance) 347

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348 

g H h hM hm I I1 I2 i J j1 K k kC kd1 k f skµ ki k p1 ksat ksk ksµ kw1 k0 L LI Li LP LW l1e lM M m m1 m2 N N1 N2 Ncm n nm P Pelm

Symbols and Abbreviations equivalent air gap magnetic field intensity, A/m height height of the PM magnification factor current; sound intensity; moment of inertia stator current rotor current instantaneous value of current moment of inertia current density in the stator winding, A/m2 lumped stiffness, N /m coefficient (general symbol) Carter’s coefficient stator winding distribution factor for fundamental space harmonic PM skew factor stacking factor of laminations stator winding pitch factor for fundamental space harmonic saturation factor of the magnetic circuit due to the main (linkage) magnetic flux stator slot skew factor referred to the pole pitch τ rotor PM skew factor referred to the pole pitch τ stator winding factor kw1 = kd1 k p1 for fundamental harmonic acoustic wave number k02 = k x2 + k 2y in the air inductance; length sound intensity level (SIL), dB effective length of the stator stack sound pressure level (SPL), dB sound power level (SWL), dB length of the one-sided end connection axial length of PM mutual inductance; lumped mass; modal overlap mass; circumferential mode number of stator phases number of rotor phases number of turns number of stator turns in series per phase number of rotor turns in series per phase smallest common multiple number of a higher time harmonic; rotational speed in rpm or rev/s; axial mode; modal density mechanical speed of the rotor (shaft) in rpm or rev/s; active power; acoustic pressure electromagnetic power

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Symbols and Abbreviations Pi P p pr Q1 q1 R R1 r S SM s s1 s2 T Tc Td T0 Tp Tr Tsh t t1 tr u V V1 v W wM X Z α αi β γ d η  λ µ µ0

349

averaged input power to subsystem i active power losses number of pole pairs; sound pressure radial magnetic force (magnetic pressure) of the r th order per unit area number of stator slots per pole number of stator slots per pole per phase resistance; radius stator winding resistance magnetic force order apparent power; surface cross section area of PM S M = w M L M or S M = b p L M slip for fundamental harmonic number of stator teeth or slots number of rotor teeth or slots torque cogging torque developed torque constant of avarage component of the torque period periodic component of the torque shaft torque (output or load torque) time stator slot (tooth) pitch t1 = π D1in /s1 relative torque ripple vibration displacement electric voltage; volume stator input voltage per phase instantaneous value of electric voltage; linear velocity magnetic field energy, J width of PM reactance √ impedance Z = R + j X ; | Z |= Z = R 2 + X 2 electrical angle α = π x/( pτ ); coefficient of approximation of the distribution of the magnetic flux density above the rotor pole effective pole arc coefficient αi = b p /τ overlap angle of pole constant dependent on slot opening and air gap relative static eccentricity relative dynamic eccentricity efficiency; damping loss factor relative permeance ( = G/S), H/m2 relative specific permeance (dimensionless); wavelength number of the rotor µth harmonic magnetic permeability of free space µ0 = 0.4π × 10−6 H/m

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350 µr µr ec µrr ec ν  ρ σ σf τ  f φ  ζ  m ω

Symbols and Abbreviations relative magnetic permeability recoil magnetic permeability relative recoil permeability µrr ec = µr ec /µo number of the stator νth space harmonic; Poisson’s ratio sound power, W specific mass density radiation efficiency; electric conductivity form factor to include the saturation effect pole pitch τ = π D1in /(2 p) magnetic flux, Wb excitation magnetic flux; PM flux, Wb power factor angle; phase angle of the magnetic flux density wave flux linkage  = N  modal damping ratio (see also η = damping loss factor) angular speed  = 2πn mechanical angular speed of the rotor m = 2π n m ; roots of characteristic equation angular frequency ω = 2π f Subscripts

A, B, C phase A, B, or C elm electromagnetic f field, frame g air gap in inner l leakage M magnet m peak value (amplitude) n number of time harmonic; normal components out output, outer s slot; synchronous sat saturation t tooth; tangential components x, y, z Cartesian coordinate system eccentricity 1 stator; fundamental harmonic 2 rotor Superscripts (e) (tr ) (sq)

element trapezoidal square wave

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Symbols and Abbreviations Abbreviations ANC a.c. BEM CAD CPU DFT DSP d.c. EMF FEM FFT HVAC MFD MMF MVD ODP PM PMBM PSD PWM RMA SEA SIL SED SPL SSC SWL TEFC VVVF WPII

active noise control alternating current boundary-element method computer-aided design central processor unit discrete Fourier transform digital signal processor direct current electromotive force finite element method fast Fourier transform heating, ventilating, and air conditioning magnetic flux density magnetomotive force magnetic voltage drop open drip proof permanent magnet permanent magnet brushless motor power spectral density pulse width modulation reaction mass actuator statistical energy analysis sound intensity level sound energy density sound pressure level solid state converter sound power level totally enclosed fan cooled variable voltage variable frequency water protected type II

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