多相电机噪声理论与计算方法(Noise-of-polyphase-electric-motors)

摘要：

本书是多相电机噪声领域权威专著，系统阐述电机噪声生成机理、计算方法与降噪技术。全书从振动、声波与噪声基础出发，拆解电磁、机械、气动三类噪声源，重点分析气隙磁场径向电磁力、谐波、偏心、磁致伸缩引发的振动噪声。深入讲解逆变器供电电机的高频谐波激励、转矩脉动与齿槽转矩特性，建立定子系统振动解析与有限元模型，结合边界元法、统计能量分析完成声学计算。书中提供噪声测试标准、数值仿真流程与主动降噪方案，覆盖感应电机与永磁同步电机，为电机低噪声设计、分析与优化提供完整理论与工程方法。

NoIsE oF PoLypHAsE ELEcTRIc MoToRs

ELECTRICAL AND COMPUTER ENGINEERING

A Series of Reference Books and Textbooks

FOUNDING EDITOR

Marlin O. Thurston

Department of Electrical Engineering

The Ohio State University

Columbus, Ohio

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

Taylor &Francis

Taylor&Francis Group

Boca Raton   London   New York

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Preface

It has been estimated that more than 65% of the electrical energy produced in de- veloped countries is consumed by electric motors. Electrical motors are the most popular machines of everyday life.  They are used either as power machines pro- viding 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 den- sity 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 con- sidered, 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 exam- ple, torque–speed characteristics.  This is because only a very s mall 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 pub- lished by Jordan in 1950.  Details of harmonic field an alysis including harmonic torques, noise, and vibration in induction motors are given in the book [87] by Heller and Hamata published in 1977.  An alysis 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 an alysis of noise and vibra- tion 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 motors.   The  demand on these motors is nowadays in second place, after the demand on induction motors.

As most books on noise and vibration an alysis 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 an alysis (SEA) [43, 230] to the prediction of noise in electrical machines. With the increase in the importance of noise an alysis and synthesis in the modern approach to the design of electrical motors, the authors have made an attempt to prepare a modern mono- graph 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, in- dustrial electromechanical drives, HVAC (heating, ventilating, air conditioning) systems, marine propulsion systems, airborne apparatus, elevators, office equip- ment, health care equipment, etc. The authors hope that this book will fill the cur- rent gap in modern treatment of the an alysis and reduction of noise in polyphase electric motors.

Jacek F. Gieras

Chong Wang

Joseph C.S. Lai

Authors

Jacek F. Gieras, Ph.D., graduated with distinc- tion in 1971 (Master’s in Engineering) from the Technical University of Lodz, Poland.   He  re- ceived 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 Trans- portation 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 scien- tific and technical papers and ten patents.  His most important books of interna- tional standing include: Linear Induction Motors, Oxford University Press, 1994, United Kingdom;  Permanent Magnet Motors Technology:  Design and Applica- tions, 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 Elec- trical Sciences, a member of numerous steering committees of international con- ferences, and cited by all Marquis Who’s Whos.

Chong  Wang,  Ph.D.,  received  B.S.  and  M.S. degrees  in  Acoustics  from  Nanjing  University, China, in  1964 and  1989,  respectively.   He  re- ceived 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 Poly- technical University, China, as an assistant lec- turer, lecturer, and an associate professor, where he lectured on several acoustics courses and con- ducted research work in noise and vibration engi- neering. In 1999, Dr. Wang joined the Institute for Research  in  Construction, National  ResearchCouncil, Canada, as a Canadian Government Laboratory Visiting Fellow.  Since 2000, he has been employed as a senior engineer with General Motors Corpora- tion, 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 mechani- cal engineering and associate dean (research) at the Australian Defense Force Academy (ADFA) Cam- pus ofthe University of New South Wales (UNSW). He was appointed in January 2004.  Prior to this ap- pointment, he was the head of the School of Aero- space 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 fiows and acoustics and vibration control.  He has published  192 papers in journals and conference pro- ceedings in these areas. His current work focuses on the aerodynamic propulsive mechanis m of fiapping wings as applied to microaerial vehicles, vibro-acoustic communication in termites, and noise and vibration control of machines, including electrical machines.

Heis a member of four International Standard Organization Acoustics Work- ing 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.

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 trans mitted 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 ofair 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 s mall-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 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 exam- ple, 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 situa- tions and exhibit:

        • refiection, 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 s mall slit or go around an obstacle;

        •  scattering, i.e., the change in direction of motion;

        • interference, i.e., mutual infiuence 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 refiected back;

        • dispersion, i.e., the splitting up of a wave depending on frequency.

Sound amplitude can be measured as sound pressure level (SPL), sound inten- sity level  (SIL),  sound power level  (SWL),  and  sound energy density  (SED) (Appendix A).

A human ear can perceive sound waves of sufficient intensity whose frequen- cies 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 differ- ent 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  corre- sponding 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 inFigure 1.2. Typical sound power levels for common sounds are also given in Table 1.1

Figure 1.1 Sound intensity and audibility zone as a function of frequency.

Figure 1.2 Comparison of some environmental noise levels.

Table 1.1 Typical sound power levels.

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 fiow 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 trans mission, elevator sheave with ropes, tooth gears, coupling, recip- rocating compressor;

        • noise due to mounting the machine on foundation or other structure.

The noise from its source is trans mitted through the medium (structure, air) to the recipient (human being, sensor) of the noise.  The process of noise generation and trans mission 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 fiux density waves in the air gap. If the stator produces Bm1 cos(⑴ 1 t +kα + φ1) magnetic fiux density wave and rotor produces Bm2 cos(⑴2 t +lα +φ2) magnetic fiux densisty wave, then their product is

0.5Bm1Bm2 cos[⑴1  + ⑴2)t + (k + l)α + (φ 1  + φ2)]

+ 0.5Bm1Bm2 cos[⑴1  _ ⑴2)t + (k _ l)α + (φ 1  _ φ2)]           (1. 1)

where Bm1 and Bm2 are the amplitudes ofthe stator and rotor magnetic fiux density waves, ⑴1 and ⑴2 are the angular frequencies of the stator and rotor magnetic fields, φ 1  and φ2  are phases of the stator and rotor magentic fiux desnity waves, k  = 1, 2, 3, . . ., and l = 1, 2, 3, . . .. The product expressed by Equation 1.1 is propor- tional to magnetic stress wave in the air gap with amplitude Pmr   = 0.5Bm1Bm2 , angular frequency ⑴r  = ⑴1 ±⑴2 , 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 distor- tion, air gap permeance fiuctuations, rotor eccentricity, and phase unbalance give  rise to mechanical deformationsand vibration. Magnetomotive force (MMF) space harmonics, time harmonics, slot harmonics, eccentricity harmonics, and satura- tion 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 2f  and high order r  = 2p 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 areproduced 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,  the voltage ripple of the rectifier is trans mitted 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 elimi- nated by increasing the distance between the impeller and the stationary obstacle.

1.4    Energy conversion process

Figure 1.4shows 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 s mall, 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.

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

                         [M]{q… }+ [C]{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   Mechanis m  of  generation   of  vibration   and  noise  in  electrical machines.

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 pA  can be related to the sound power level LWA  in dB(A), as follows

where LpA is the average sound pressure level in a free-field over a refiective 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  = 1.0m2.

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, sup- ply, load, etc. Some consequences of noise as, for example, manufacturing, mount- ing 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:

where A 1 , 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 1.2and 1.3. The enclosures of motors are an open drip proof machine (ODP) type, totally en- closed 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

Table 1.2 Maximum A-weighted sound power levels LwA  in dB(A) at no load for motors with rated speeds 1200 rpm and less according to NEMA [ 162].

Table 1.3 Maximum A-weighted sound power levels LwA  in dB(A) at no load for motors with rated speeds 1201 to 3600 rpm according to NEMA [ 162].

Table 1.4 Expected incremental increase over no-load condition in A-weighted

sound power levels  Lw A, dB(A) for rated load condition for single-speed, threephase, cage induction motors according to NEMA [162] and IEC 60034-9 Standards [93].

the internal ventilating passagesleading 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]

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

where Pmk is the amplitude of the kth harmonic, ωk = 2πk 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.,

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 s mooth 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.

Table 1.5 IEC 60034-9 limitsforsound pressure level at 1 m from machine surface,dB(A) [93].

1.6 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 an alytical 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.

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 an alytical 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 an alytical 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  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 an alysis (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  detailsto 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 an alytical ”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 an alytically, as depicted in the radiation efficiencies model in Figure 1.7b.

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