PREDETERMINATION OF THE SOUND PRESSURE LEVELS OF MAGNETIC NOISE IN MEDIUM INDUCTION MOTORS.~ ~' ".. -.~ d ~ - i* d' A dissertation submitted in partial fulfillment of the requirements for the degree of Doctoral Commit tee: Professor Edwin R. Martin, Chairman Professor Stephen S. Attwood Professor William G. Dow Associate Prof Phsor Wilfred Kaplann the Associate Professor Norman Scott 19t5 DConsul tora 1 Committee: Professor Edwrin R. Martin, Chairman Professor Stephen SI.Attwood Professor William G. Dow Associate Professor Wflfred Kaplan Associate Professor Norman R. Scott Consultant to the Committee: Mr. Philip L. Alger, Consulting Engineer General Electric Company, Schenectady

I-~~~~~~e a% ^ ^

To the Memory of Anna and Alexander Erde'lyi

PREDETERMINATION OF THE SOIU4D PRESSURE LEVELS OF MAGNETIC NOISE IN MEDIUM INDUCTION MOTORS by Edward Erddlyi Abstract Chapter I: Introduction A historical review of earlier work in the field of magnetic noise is given. It is noted that in most writings of the past the recommendations for low magnetic noise levels are intimately related with the question of slot combinatlons. The more complete single-ring theory of P. L. Alger is reviewed, and the inadequacies of this pioneering theory are discussed* The nature of the different types of noise in induction motors is described, and the origin of harmonic noise is explained. The difference between the single-ring theory and the more complete solution of the problem given in this thesis is illustrated by curves which show dynamic deflection Versus frequency. Chapter II: The Electroma netic Force Waves in the Aira Expressions are developed for the magnetomotive force distribution in the airgap of induction machines. A simple method is given for treating the influence of the variable airgap permeance. The influence of the number of slots per pole per phase on the combination of slot and permeance harmonics is pointed out. Equations for the amplitude, frequency, and mode of the force waves are derived. The influence of factors which are not in agreement with the simplified assumptions is qualitatively discussed. The theory is applied to calculate the force waves of an experimental motor. Chapter III: The Dynamic Res ponse to the Electro2manetIc Force lav.es Now the vibrating system of the stator is represented by two coupled concentric rings on a support. A brief review of the theory of Lagrange's equations and the theory of small oscillations is given. Expressions for the elastic and kinetic energies of the physical components of induction motors

are developed. Following certain simplifying assumptions the natural behavior of the experimental motor is carefully analyzed and numerical results obtained for the resonant frequencies of the stator, The forced response to the electromagnetic force waves is computed. Chaper IV: Sound Radiated by the Vibrating Stator The boundary value problem of the sound radiating motor which is represented as an infinite cylinder vibrating in a belt is solved by a Fourier transform method. The method of solution of the resulting integral is explained, and evaluated in detail for different operating conditions of the experimental motor. Chapter V: Experimental Investigation of the Special Motor The experimental arrangements and equipment for finding the natural frequencies and associated mode shapes are explained. The anechoic room and the test conditions for the recording of the motor noise on magnetic tape and the subsequent recording of sound spectra are described. The method for the determination of mode shapes under dynamic conditions is explained, and test results are recorded. Chapter VI: Conclusios A comparison between theoretical and experimental results is carried out, and it is concluded that the method of the thesis is applicable for the calculation of natural frequencies and the predetermination of the sound pressure levels of the magnetic noise in medium induction motors. A note of caution is made of the danger in transferring results of one design to that of another design. It is stressed that the analysis must be carried out individually for each type and design of machine, if significant results are desired.

ACKNOWLEDGEMENTS The author wishes to thank his Committee, especially Professor E. R. Martin, for the permission to carry out the work connected with this thesis at the General Electric Company in Schenectady, and for their assistance in discussing with him the various aspects of this problem. An expression of deep indebtedness and sincere gratitude is made to Mr. P. L* Alger, Consulting Engineer, Medium Induction Motor Department of the General Electric Company, for his suggestion of this investigation. It was he who offered both guidance and assistance during every phase of this complex assignment, and the author realizes that this work could not have been accomplished without Mr. Alger's generous help. Acknowledgement is also due to Dr. G. Horvay of the General Electric Company for introducing the author to the theories employed in Chapter 3, and to Dr. R. Plunkett of the General Electric Company for the critical review of the analysis in this chapter and for his very valuable suggestions. Mr. J. W. Ward of Carrier Corporation, Inc., Syracuse, New York, was very helpful in carrying out the preliminary computations on the IBM 605 equipment. f:*

Also the help of Dr. F.S. Beckman, and of Mr. J.F. Terlato of the Scientific Computing Service of the International Business Machine Corporation, Inc., New York, who programmed the final calculations for the IBM 701 Electronic Data Processing Machine is acknowledged. Thanks are due also to many associates at both the General Electric Company at Schenectady, and at Syracuse University. Finally the author gratefully acknowledges the assistance and forbearance of his wife, Nelly; the services of Mrs* M. Wolman in typing this dissertation; and the assistance of Miss R. Shaver of the General Electric Company, in making many time- consuming arrangements. This work was sponsored by the General Electric Company, Medium Induction Motor Department, Schenectady, New York. iv

TABLE OF CONTENTS Page ACKNOWLEDGEMENTIS ii TABLE OF CONTENTS v LIST OF TABLES viii LIST OF ILLUSTRATIONS xi INDEX OF SYMBOLS For Chapter II xiv For Chapter III xx For Chapter IV xxit I. INTRODUCTION I 1*1 Historical Review 2 1.2 The Nature of Induction Motor Noise 11 1.3 The Single Ring Theory 17 1.4 The Statement of the Problem 24 II. THE ELECTROMAGNETIC FORCE WAVES IN THE AIRGAP 30 2.1 The MMF Distribution in the Airgap due to Stator Currents 31 2.2 MMF Distribution of the Squirrel Cage Rotor 38 2.3 Rotor Current and the Reaction of the Rotor Harmonics 43 2.4 Permeance Variations of the Airgap Resulting from Open Stator Slots 56 2.5 Flux Distribution in the Airgap 62 2.6 Airgap Flux Distribution Considering Slot Openings 65 2.7 The Radial Magnetic Force Waves 70 2.8 The Tangential Force Waves in the Airgap 88 2.9 Influence of Magnetic Saturation, Skew, and Airgap Eccentricity 92 2.10 The Force Waves in the Experimental Motor 108 v

TABLE OF CONTENTS (Cont'd) Page III. THE DYNAMIC RESPONSE TO ELECTROMAGNETIC FORCE WAVES 121 3*1 Lagrangets Equations of Motion an&t the Energy Method of Rayleigh-Ritz 123 3.2 Energies in Components of Induction Motors 137 3.3 Assumptions and Notations 148 3.4 The Equations of Motion of a Vibrating Motor Stator 157 3.5 Calculation of the Natural Behavior of the Experimental Motor 168 3.6 Calculation of the Forced Response of the Experimental Motor 193 3.7 Investigation of the Convergence of Eigenvalues 200 I IV. SOUND RADIATED BY THE VIBRATING STATOR 204 4.1 Derivation of the Wave Equation 206 4.2 The Integral of the Sound Pressure 211 4.3 The Solution of the Sound Pressure Integral 220 4.4 Numerical Evaluation of the Sound Pressure 222 V. EXPERIMENTAL INVESTIGATION OF THE SPECIAL MOTOR 233 5.1 Test for Natural Frequencies and Associated Mode Shapes 234 5.2 Measurement of Motor Noise Sound Pressure Levels 242 5.3 Determination of Mode Shapes Under Operating Conditions 258 VI. CONCLUION 261 6ol 1 iscesioen of Results 261 6.2 Application of Method 265 APPENDICES I. DATA OF THE EXPERIMENTAL MOTOR 269 vi

APPENDICES (Cont'd.) Page II. ENERGIES OF THE VIBRATING RINGS (Extensional Deformations Considered) 276 III. DIAGONALIZATION OF A REAL SYTflETRIC MATRIX on the IBM 701 Electronic Data Processing Machine 278 IV. INVESTIGATION OF THE INFLUENCE OF TANGENTIAL RIBS 281 BIBLIOGRAPHY 285 vii

LIST OF TABLES Table No. Title Pae 2.7,1 Inequalities for Low Modes 86 2* 10.1 Airgap Harmonics of the Experimental Motor 109 2.10.2 Force Waves Resulting from First Order Slot Harmonics 109 2.10.3 Second Mode Force Waves in the Experimental Motor of Frequency 2+27(1-s)f Ill 2*10.4 Second Mode Force Waves in the Experimental Motor of Frequency 27(1-s)f 112 2.10.5 Summary of Calculated Force Waves 120 3.5.1 Inertia Constants of the Outer and Inner Ring 169 3. 52 Elastic Constants of the Outer and Inner Ring 169 3*5.3 Elastic Constants of the Radial Ribs 169 3.5.4 Elastic Constants of the Supports 1~0 3.5..5 Original Determinant of the Even Modes 172-173 3 5.6 Original Determinant of the Odd Modes 174-175 3.5.7 Transformed Determinant of the Even Modes 176-177 3.5.8 Transformed Determinant of the Odd Modes 178-179 3*5.9 Elgenvalues and Eigenvectors of the Transformed Determinant for Even Modes 180-181 3.5*10 Eigenvalues and Eigenvectors of the Transformed Determinant for Odd Modes 182-183 3.5.11 Eigenvalues and Eigenvectors of the Original System for Even Modes 189-190 vtii

LIST OF TABLES (Cont'd.) Table No. Title Page 3.5.12 Eigenvalues and Eigenvectors of the Original System for Odd Modes 191-192 3.6.1 Calculation of Constant K3 197 3.6.2 Summary of Constants Kr 198 3.6.3 Second Mode Deflection Amplitudes in the Cosine Axis Caused by a Rotating Force Wave of 100 lb per Radian Amplitude 198 3.6.4. Calculation of the Second Mode Deflection in the Cosine Axis Caused by a 100 lb per Radiah Amplitude Rotating Force Wave of 1,716 cps. 199 3*.6. Calculated Second Mode Cosine Axis Deflections Caused by a Second Mode 100 lb per Radian Rotating Force Wave as Function of Frequency 199 3.6.6 Summary of Calculated Deflections for 2nd Mode Force Waves of the Experimental Motor 200 3.7.1 Comparison of Eigenvalues and of Natural Frequencies af the Even Modes 203 3.7.2 Comparison of Eigenvalues and Natural Frequincies of the Odd Modes 203 4.4.1 Calculation of Integral ^ for 1,700 cps 225-226 4.4.2 Calculation of Integral 12 for 1,700 cps 227-228 4.4.3 Calculation of Integral 13 for 1,700 cps 229 4.4.4 Calculated Values of the Sound Pressure Integral for m=2; 230 4.4.5 Sound Pressure- Levels of the Experimental Motor 231 ix

LIST OF TABLES (Cont'd.) Table No. Title Pae 6.1.1 Comparison of Calculated and Measured Resonant Frequencies 262 6. 12 Comparison of Calculated and Measured Sound Pressure Levels 263 A4.1 Constants of the Tangential Ribs 282 A4.2 Second Mode Determinant Showing Effect of Tangential Ribs 283 A4.3 Third Mode Determinant Showing Effect of Tangential Ribs 284 x

LIST OF ILLUSTRATIONS Fig. No. Title ge 1.1 Dynamic Deflections of a Single Ring 22 1.2 Dynamic Deflections of an Induction Motor Stator 23 1.3 Frames of Modern Medium Induction Motors 27 2.1.1 Phasebelts of an Induction Motor 32 2.3.1 Equivalent Circuit of an Induction Motor 44 2.3*2 Equivalent Circuit of the 6th Rotor Cage 45 2.3.3 Sinor Diagram of Bar Currents 53 2.4.1 Flux Pulsation Due to Slot Openings 58 2.4.2 The Function 3 60 2.7.1 Interacting Flux Waves Differing by One Pole Pair 76 2.7.2 Deformations of One Sided-Pull 78 2.7.3 Elliptical Deformations 78 2.7*4 Polygonal Deformations 78 2.9.1 Field Shape in a Wye-Connected Stator 93 2.9.2 The Magnetizing Current in a Phase of a Delta Connected Machine 94 2.9.3 A Developed Skewed Squirrel Cage Rotor 97 2.9*4 Shift of Force Waves Created by Slot Harmonics 99 2.9.5 Force Distribution along a Generatrix in Skewed Bar Rotors 101 2.9.6 Airgap Eccentricity 3.2.1 Deformations of a Circular Are 138 xi

LIST OF ILLUSTRATIONS (Cont'd.) Fi. No. Title Pa 3.2.2 Deformation of a Radial Rib 144 3-2.3 Deformation of the Support 147 3.3.1 Simplified Cross Section of Experimental Motor 150 3.3.2 Logarithmic Decrement 154 3.3.3 System of Coordinates for the Vibrating Stator 155 3.5*1 Theoretical Second Mode Vibration of the Experimental Motor 187 3.5.2 Theoretical Third Mode Vibration of the Experimental Motor 188 4.1.1 Volume Element in Sound Field 207 4.2.1 System of Coordinates for the Vibrating Cylinder 212 4.4.1 Sound Pressure Integral for Second Mode Vibrations of the Experimental Motor 231 5.1..1 Test Arrangement for Resonant Frequencies 235 5.1*2 Second Mode Resonances of the Experimental Motor 239 5*1.3 Third Mode Resonances of the Experimental Motor 240 5.1.4 Coupling Between Frame and Stator Core 241 5.2.1 Position of Motor in Noise Measurement Laboratory 243 5.2.2 Test Arrangement for Recording of Noise 244 5.2.3 Recording of Noise Frequency Spectrum 249 5.2.4 Sound Spectrum of Experimental Motor at 40 Amp., 440 V., 60 cps. 251 xii

LIST OF ILLUSTRATIONS (Cont'd.) Fig. NO. Title Page 5.2.5 Sound Spectrum of Experimental Motor at 40 Amp., 369 V., 50 cps. 252 5.2.6 Sound Spectrum of Experimental Motor at 40 Amp., 293 V., 40 cps. 253 5.2.7 Sound Spectrum of Experimental Motor at No Load, 440 V., 60 cps. 254 5*2.8 Sound Spectrum of Experimental Motor at No Load, No Excitation 255 5.2*9 Sound Spectrum of Experimental Motor at 40 Amp., 440 V., 60 cps* 256 5.2*10 Sound Spectrum of Ambient Noise in Anechoic Room 257 A 1*1 Rotor Slot of Experimental Motor 270 A 1.2 K365 Frame of Experimental Motor 272 A 1.3 Outer Ring Cross Section 273 A 1.4 Detail of Rib Cross Section 274 xili

INDEX OF SYMBOLS FOR CHAPTER II Symbol Pa A differential area in airgap 72 A amplitude of the th harmonic 31 current layer a(x1) ordinate of fictitious current 89 distribution at xl Bx' nx. flux density at tooth center 58 Bb max. reduction of flux density 58 at slot center B1 amplitude of fundamental flux 63 distribution B0,B amplitude of 6th or pth flux 64 id'p fharmonic b(xlt) ordinate of flux distribution at 63 x1- at time t b() flux distribution of positively 94 rotating field b(e) flux distribution of negatively 94 rotating field b flux distribution of stator slot 87 harmonic br flux distribution of rotor slot 87 harmonic C3 an integer divisible by 3 82 C2 any integer 40 Cls an integer except zero 37 xiv

Symbol Page dA differential area 89 f frequency 38 fI ordinate of mmf distribution of one 33 phase f(xl,t) ordinate of mmf distribution of a 35 polyphase winding at point xl at time t k dordinate of mmf at kth tooth 51 g airgap 58 g' airgap - corrected for variable 63 permeance g" airgap - corrected for saturation 46 g(xl) variable airgap at (xl) 104 H(xl) magnetic field intensity at (xl) 105 h magnetic field intensity 50 h denotes a particular rotor bar in 40 squirrel cage I R.M.S. value of phase current 39 I fundamental stator current referred 55 to yth cage i instantaneous value of phase current 31 k. dth winding factor 33.xv

Symbol Lae th k6:h skew factor 97 kI correction coefficient 47 kC Carter correction coefficient 46 kd distribution factor 33 kp pitch factor 33 L inductance of one phase including 50 the effect of mutual inductance between phases 1i net iron length of stator 46 MI amplitude of ath harmonic of mmf 33 of one phase M~ amplitude of t mmf harmonic of a 35 polyphase winding m number of phases 31 mv mode of radial force wave 75 ns synchronous speed 38 n rotor mechanical speed 38 9 permeance function 61 p number of pole pairs Q1 total. to number of stator slots 66 xvi

Symbol Page Q2 -total number of squirrel cage bars 66 R radius of stator bore 91 R1 primary resistance 44 R2B resistance in the equivalent circuit 44 of the th cage s slip 38,s1o width of slot opening 58 t time 33 u a subscript denoting a phase 31 vs velocity of fundamental mmf 36 velocity of tharmonic m 36 Var velocity of 0 harmonic mmf 36 W magnetic energy density 71 w1 number of turns in series per 33 stator phase X1 leakage reactance of primary 44 Xls slot leakage reactance 49 X.) end connection leakage reactance 49 Xld differential leakage of stator 49 xvii

Symbol'page X1M6 magnetizing reactance of the stator 44 Xm oan harmonic X Oth X2c leakage reactance of th cage refer- 49 red to the stator X^w magnetizing reactance of the th 47 X2m harmonic with resp. to the rotor x1 denotes the position of a point on 33 rotor (x = 0 at the center of the reference phase) X2 position of a point on the rotor 38 z distance measured along the rotor 100 axis (f a simplifying function 60 r - ordinate of force wave 71 tangential component of force wave 91 r" axial component of force wave 103 ~ eccentricity of airgap 104 a coefficient in the permeance 61 function p2 a factor of differential leakage 54 Ald primary differential leakage factor 50 ^2dd <th differential leakage reactance factor of the rotor pO permeability of free space 51 F order of slot harmonic 84 distance measured from tooth center 59 xviii

Symbols Page TI force wave amplitude in lb/radian 114 p order of rotor harmonic 39 P' pp 42 cr order of stator harmonic 33 0' pa 42 ~s ~pole pitch 33 TV volume of airgap 51 rt tooth pitch 58 r skew 100 F ~angle between two consecutive bars 40 in a squirrel cage (measured in electrical radians) o circular frequency 32 I'la Cangular velocity of stadr harmonic 36 )r angular velocity of rotor 69 xix

INDEX OF SYMBOLS FOR CHAPTER III Symbol Page A,area of cross section 140 Ai amplitude of sinusoidally varying quantity 130 Am Fourier expansion coefficients 158 am Fourier expansion coefficients 158 B Fourier expansion coefficieht 158 b subscript denoting bending 142 b Fourier expansion coefficient 158 c subscript 196 cr normalizing constant IX rigidity factor 142 d subscript 195 E Young's modulus 140 E' E/(1-62) 140 F force 123 xx

Symbol p F(e) external force 123 F elastic constant 160 f rfrequency 131 f elastic constant 160 Gm inertia constant 159 gm inertia constant 159 I ssubscript denoting inner ring 149 I - moment of inertia 141 ~i subscript, summation index 123 j subscript, summation index 123 Kr constant to replace summation 195 kij coefficient, potential energy 128 I length 143 M Bending moment 140 m mass 123 m order of mode, m = 2, ~.. 158 miJ coefficient in kinetic energy 129 N normal force 140 xxi5.

$Sybol Page n number of particles 124 0 subscript, referring to outer ring 149 P shear force 145 Q generalized force 125 q generalized coordinates 124 R radius of outer ring 148;r ~radius vector 123 r radius of inner ring 148 r subscript referring to ribs 144 r superscript denotes principal mode 131 r mean radius of arc 138 s subscript referring to supports 147 s length of arc 138 T kinetic energy 127 t time t subscript denoting extensional de- 141 formation U radial displacement of outer ring 155 u radial displacement of inner ring 155 V potential energy 127 V elastic energy 141 xxii.

Symbol Page W tangential displacement of outer 15 ring We work of external force 194 w tangential displacement of inner 155 ring y normalized eigenvector component 132 z eigenvector of transformed system 184 ~ aangle of position 138 & unit elongation 139 a: a constant 146 A eeigenvalue 171 {L mass per radian 159 TF phase angle 130 Y rotation of cross section 139 W circular frequency 5 aa aordinate 138 ~ ~ a coordinate 138 xxiii

INDEX OF SYMBOLS FOR CHAPTER IV Symbol Page a radius of cylinder 212 b half width of vibrating belt 212 c sound velocity 211 d6 differential element of surface 209 F(h) a coefficient function 217 G(h) coefficient function 218 g(u) Fourier transform of f(x) 217 H (1) Hankel function of order m 216 m h separation constant 213 I sound pressure integral 219 k a constant 212 m order of mode 215 n normal 209 n separation constant 213 P pressure 210 P0 static pressure 210 xxiv

Symbol Page p instantaneous pressure 209 Q(e) a separation function 213 t ~velocity 207 R(r) a separation function 213 r a coordinate 212 s condensation 208 t time 212 V volume 210 w a variable 233 x a variable 223 y a variable 220 Z(z) a separation function 213 Zn circular cylinder function 214 z a coordinate 212 OC a constant 221 fS a constant 221 AV elementary volume 206 p density 207 0 a coordinate 212 wo circular frequency 212 a constant 221 xxv:

THE PREDETERMINATION OF MAGNETIC NOISE OF POLYPHASE INDUCTION MOTORS I. INTRODUCTION Noise - unwanted sound - has become a serious threat to the well-being of everyone of us. It is with us everywhere and all the time. Although the harmful effects of noise on the well-being and efficiency of office and industrial workers are difficult to assess and there is a conspicuous disagreement among investigators on the psychological and physiological effects of noise, there seems to be, nevertheless, substantial evidence that noise reduction has an overall beneficial effect. The general opinion seems to be that it should be possible to overcome the noise nuisance of our modern society. Many things of a practical nature have been undertaken to control injurious noises. These endeavors have lately been substantially accelerated by coordinating the efforts of workers in the sound abatement field. The training course on noise control at the University of Michigan in 1952, and the special summer program on noise reduction at M.I.T. 1 46 - I -

-2 in 1953, are examples of such concerted endeavors. Polyphase induction motors of all sizes are frequently producers and radiators of unpleasant noises. It is the aim of this thesis to contribute to the general effort to reduce noise in induction motors by evolving an analytical method for the calculation of the sound pressures in the vicinity of polyphase induction motors because of forced vibration of the motor frame caused by the radial and tangential forces. These result from interfering higher flux harmonics and produce the so-called magnetic noise. w.,1. Historical Review Early research in the magnetic noise of induction motors has been closely linked with the investigation of other effects of higher harmonic flux waves. W. Stiel systematically investigated the effects of higher harmonic fluxes on the torque-speed characteristic of polyphase induction motors and he was probably the first to recognize the relationship between slot numbers and magnetic noise. He rightly suspected magnetic radial forces were the cause of noise. Stiel also reported the effect of odd 65

-3rotor slot numbers on magnetic noise but did not give a theoretical explanation of the cause. Fritze1 analyzed Stiel's results in 1921 in a theoretical treatment, starting with an analysis of the airgap flux. He discovered that considerable vibration and magnetic noise occur in machines whose airgap flux contain two harmonic fluxes, the pole pairs of which differ by one. He attributed this fact to a resultant one-sided pull and proved that it could only occur in machines with odd numbers of rotor slots. Consequently, he concluded that only machines with odd numbers of rotor slots tend to be noisy. For a long time Fritze's work formed the foundation for the investigation of the induction-motor noise problem. His opinion regarding odd numbers of rotor slots is still held valid though considered an overstatement, as machines with even slot numbers are also known to be quite noisy and strongly vibrating. Simultaneously and independently, F. T. Chapman2 dealt with the same subject, although his paper was not published before 1923, His study is still very valuable as it contains a lucid picture of the physical phenomena 127 -2 18

- 4of the one-sided pull. He used a squirrel-cage motor with five bars in the rotor to demonstrate the resultant dissymnmetry, and gave an expression for the frequencies of the noises produced. Furthermore, he calculated the one-sided pull and provided a rule for rotor-slot numbers to be avoided. He clearly pointed out the danger of resonance of the disturbing forces occurring at the motor's critical speed, and in addition provided test results. The series of papers published by Doherty and Nickle in 1924, and the fundamental work on the performance of squirrel-cage induction motors by L. Dreyfus2, together with the paper on higher harmonic leakage of induction motors by M. Krondl3 greatly influenced the subsequent thinking of many designers. The attack on inductionmotor problems became more general.L.E. Hildebrand4 was the first to stop concentrating on the one-sided pull caused by an eccentric force wave and prescribed a more general treatment. His study was based on the noise test of some 1500 motors of 200 different designs and ratings, ranging from 1/4 to 15 horsepower. Hildebrand clearly recognized the fact that a 1 22 2 23 3.42 4 34

motor is a source of sound because some parts of it are moved by periodically varying forces. He concluded that the detailed study of induction motor noise should include an exact investigation of the parts prone to vibrate and the forces moving them. It is unfortunate that he did not carry out such an analysis. According to Hildebrand the noise producing components of induction motors can vibrate in many ways. A torsional vibration of the stator and rotor as a whole res-ults from periodic torque variations. The noise may also result from a circular vibration, similar to a dynamically unbalanced rotor; this is caused by the previously mentioned one-sided pull. The stator, Hildebrand found, may also be distorted into a rotating elliptical shape. Or, if the forces are balanced in more than two radii, the stator may be deflected in the shape of a rotating polygon. Hildebrand's study was of great importance for the fact that he put more emphasis on the effects of different factors on the field than on considerations of the field itself. He concluded that the field in the airgap and the noise of an induction motor depend on the stator and rotor winding arrangements, on eccentricities, and on saturation. He therefore strongly recommended even and concentric gaps and windings which would eliminate fields differing by one

- 6or two pole pairs. He claimed that a motor would be quieter if the number of poles, stator slots, rotor slots, and winding repetitions bore simple relationships, and the larger the common divisor of these factors the quieter the motor would be. We cannot nowadays accept all of Hildebrand's conclusions without reservations. However, the fact that he only considered the first slot harmonics as dangerous, together with his belief that most common noise frequencies were exactly or nearly twice the applied frequency, proved his reasoning erroneous. M. Krondll in a general treatise gave a very thorough discussion on the different causes of parasitic forces in the airgap of induction motors, grouping them into radial and tangential components. He describes the effects of the radial and tangential components of these forces as: synchronous and asynchronous torques, periodic deformations of the stator core, and vibrations of constructional parts. According to Krondl the last two produce the magnetic noise. It was the aim of his paper to give useful and guiding principles for the design of induction motors and fundamental data for further experimental investigation. This paper was of great importance in the analysis of the noise problem since it was the first to discuss the mechanical vibrations and the acoustical aspect of the problem. His conclusions 143

-7about some sound-producing vibrations and the individual subjective sensitivity of the ear have thrown a new light on the problem of slot-number combinations. Krondl seems also to have been the first actually to have measured sound pressures rather than to have confined himself to a mere description of subjective impressions. It was he who also warned against the application of experimental data obtained from small motors to the problems of the noise in large machines. One of the several difficulties in the development of noise theory during these early years was the lack of agreement on acoustical definitions and on sound measuring techniques and standards. The American Institute of Electrical Engineers took the initiative in starting discussions as early as 19301 Stimulated by these discussions, the American Standards Association set up a technical committee to establish standards for the description of noise, and to suggest fundamental measuring methods to determine the loudness of any noise. The preliminary standards of 1935 have greatly influenced the further development of the noise theory. At a meeting of the C.I.G.R.E. in 1933, it was again Krondl2 who contributed to the progress of the theory, 1 76 2 44

when reporting on the noise problem of electrical machinery, reviewing his previous findings, and discussing the sound propagation of electrical machines. At the same time Go Kronl tried again to derive certain rules for the number of slots. He ascribed the noise mainly to the deviations of the airgap from its average value caused by the slots. The importance of his paper lies, in addition to the explanation of hooks in the speedtorque curves, in the clever method of calculation which served as model for the further development of the theory. Morril's2 paper was noted for its clear presentation of the problem, and for providing a well organized method for the calculation of the frequency of the force waves. The subject of subharmonic waves for the rare case where the number of squirrel-cage bars is small has also been treated in this paper. JS. Gault also published a detailed study of the current wave-shapes in rotor bars resulting from voltages induced by higher harmonic fluxes. The lucid presentation helped designers to visualize the composition of the airgap field. The numerous contradictory individual experiences, and particularly test results proving that different machines with the same slot numbers produced different 41 2 52 3 28

9 - magnetic noise, firmly convinced various designers that the statement of the noise problem in the light of slot numbers alone was incomplete. They found that the problem became entirely different when machines of different ratings were considered; e.g., a slot number combination yielding excellent results in small machines showed poor results in mediumsized motors. Consequently, they realized that the concept of most opportune slot numbers makes sense only for a definite type of motor. These observations are proof that there are no magic prescriptions for the choice of slot numbers, and that the noise problem is not just an electro-magnetic one but also one of engineering mechanics and acoustics. The first to visualize the problem in its entirety was P. L. Alger, whose pioneering efforts in the conquest of the noise frontier are still leading the way for many designers. According to his theories the induction motor, as regards its mechanical vibrations, can be represented as a simple inextensional ring, and the motor's natural behavior can therefore be approximately described by the behavior of this equivalent ring. Alger provides formulae for the calculation of the resonant frequencies of inductionmotor frames, based on simple beam theory. 12;3

10 In addition to the calculation of natural frequencies, methods for calculating static deflections are given, but no extensive treatment to determine dynamic deflections at forced vibrations is provided. Alger arrived at these simple results by further assuming that the stator teeth and windings add only to the inertia forces and not to the rigidity of the structure. These last assumptions are justifiable, while the simple ring concept could be disputed. However, the first step is always the most difficult. For the acoustic behavior of the motor, Alger chose to represent the motor by an infinitely long vibrating cylinder. He provided very valuable formulae for the designer, and in addition, curves of sound intensities on the cylindrical surface caused by different modes of vibration and correction curves for sound intensities at a distance from the induction motor. The importance and influence of Alger's theories can hardly be overstated. The simplified theory demonstrates clearly how the different parameters affect the noise behavior of the induction motor. From his work it becomes obvious that the most effective remedy against noise is to prevent large radial deflections of the stator by stiffening it in medium-sized machines, and preventing certain.modes of vibrations near resonance in larger machines.

- ll - H. Jordanl in Germany simultaneously investigated the noise problem of induction motors. His simplifying concepts resemble those of Alger. Jordan develops his formulae from a partial differential equation. The acoustic behavior of the motor is approximated by a radiating sphere, and he uses Gauss' theorem to determine the sound power level at a distance from the motor. 1.2 The Nature of Induction Motor E oisee. The sound of machinery noise may be that of unpitched or of pitched noise. The unpitched noise has a more or less uniform amplitude in a broad band of nearly continuous frequency distribution. It can be traced to recurring transient disturbances. Pitched noise, on the other hand, is of a distinct frequency generally related to the operation of the machine, and the frequency of the pitched noise changes with operating conditions. As nearly all noise produced by induction motors results from forces of definite frequencies, most of the noise is pitched noise of either mechanical, aerodynamical or electromagnetic origin. Ball bearing noise is the most prominent among those attributable to mechanical causes. Continually occurring disturbances, resulting from the play of clearances in ball bearings, excite oscillations that tend to vibrate the motor. These vibrations may become more pronounced by mechanical 1 36*37

-12 unbalance of the rotor, by an eccentric airgap, or by resonance of either the motor endshield or by any parts of the bearing supports. The same causes' to a smaller extent, may also be responsible for noise in sleeve bearings. Bearing noises can be controlled by appropriately designed bearing supports, good alignment, properly chosen shaft designs, and by bearings carefully manufactured to exact specifications. Aerodynamic noise can be classified into two major groups: windage noises and the siren noises. The first are caused by fans, used to move the cooling air over rotor and stator surfaces. The frequency and amplitude of these noises depend on the number and shape of ventilator blades and on motor speed. Siren effects occur in motors when radial airducts are being used in both rotor and stator, and the number of rotor and stator slots differ by a small amount. The periodic,high-frequency interruptions of the radial airstream by the stator coils crossing the ducts occur nearly in time phase, thus causing a high-pitched noise. A pounding and cutting noise can be observed in induction motors when the high-velocity cooling air flows past a sharp edge,'or when a sudden change occurs in the airflow cross-sections. Noises of electromagnetic origin appear at different frequency ranges, depending on the origin of the

13 - driving forces. The double-frequency noise is excited by the main airgap field. The magnetic field in the airgap rotates synchronously, and as a consequence the flux density at a fixed point in the airgap varies as a sinusoidal function of time. Owing to the change in the stored electromagnetic energy at that point, a radial force of double frequency, with a tendency to mutually attract stator and rotor, is created. The distribution of this force in the airgap has a constant component, and a sinusoidal component having twice as many pairs of poles as the original field. This latter force periodically deforms the stator into the shape of a rotating polygon, and therefore excites the vibration of the stator core. The forced vibrations of the stator core are transmitted to the outer frame by structural parts, connecting the stator to the frame, which in turn vibrate and radiate sound waves from its surface to the surrounding air: the so-called airborne noise. This double-frequency noise is unavoidable, as the machine could not function without the main flux. It appears as the characteristic humming sound of induction motors. It is harmless in small or medium-sized machines with more than one pole pair. The deflections are not too large, as the resonant frequencies of the stator are generally much higher than the double supply frequency.

- 14 - For large two-pole motors, where the double supplyfrequency force wave has only two pole pairs, the vibrations may become very vigorous and the noise bothersome. In this case the problem is very similar to that of the vibration of turbine generators. The only remedy to be applied is a stiffening of the outer structure of the motor. This double supply-frequency vibration may also become a problem in small 400 c.p.s aircraft equipment. A component of the torque of polyphase induction machines working with unbalanced voltages pulsates with double supply-frequency and small changes of rotor speed occur. These pulsating torques may cause structural vibra- tions, directly transmitted to the foundations through the motor supports, and may excite noise-making resonances els$e where. This noise, called the torque noise, is very pro.-: nounced in single-phase induction motors. Furthermore, the higher harmonics in the airgap, as described below, may cause oscillating higher-harmonic torques at frequencies well in the audio-range more awkward than the double supply-frequency torque noise. The most troublesome noise of electromagnetic origin, the so-called magnetic noise, can be traced to radial and — in a much smaller measure - to tangential forces which 1 10

- 15 - occur when the airgap flux contains two harmonic fluxes with pole pairs differing by a small integer. This noise generally contains several, almost pure, whistling tones of high amplitudes. The higher harmonic fluxes in the airgap are practically unavoidable and inherent in the design of induction motors. The winding distribution introduces the socalled space harmonics, to be classified as slot- and as belt- harmonics. While the former result from the fact that each phase belt contains a limited number of slots, belt harmonics are caused by the limited number of phases. To wit, if a motor with an infinite number of phases and an infinite number of slots per phase could be designed, no such harmonics would occur. The amplitude of these space harmonics depends on the load current, and the noise caused by them increases with the load current. The non-uniform permeance of the airgap caused by the open or semi-open slots of the stator creates a whole sequence of harmonics, the so-called permeance harmonics. The order of these and their frequency distribution are the same as those of the above-mentioned slot harmonics. The magnitude of the permeance harmonics depends on the supply voltage and is independent of the load current. These

- 16 permeance harmonics are mainly responsible for the so-called no-load noise. There are two other major causes of such higher harmonic fluxes; i.e., an eccentric airgap, and the saturation of teeth, The radial and tangential forces set the core into vibration, similar to the vibration previously described at the double supply frequency. Though the polepitch of the higher harmonic fluxes is small and the radial and tangential force distribution may consist of two complete sine waves only, marked deflections may yet occur in response to small radial forces. The frequency of these forces is considerably higher than the double supply'frequency. The noise caused by these force waves is generally in the frequency range of 500 to 3,000 c.p.s., i.e., in a range where the human ear is most sensitive, which makes this noise all the more irritating. This thesis is actually intended to deal with the noise created by these radial forces. For the sake of completeness, mention should be made of the magnetostriction noise, Steel, when magnetized, expands along the axis of magnetization. The expansion depends not only on the magnetic properties of the steel, but also on flux density, and is linearly proportional to

- 7 - the length of the flux path. This flux path in the core is mostly circular, so that periodic variations of the length are taking place along concentric circles. While some parts of the core tend to expand, adjacent parts would contract, so that the average total circumference of the stator ring remains unchanged. The tangential deflections are extremely small, and magnetostriction is therefore not considered a noise producing factor in induction motors. 1.3 The Single-Ring Theory It becomes clear from the historical review that continuous attention has been given in the last decades to the reduction of the magnetic vibrations and to noise of electrical machines. Before proceeding further it seems appropriate to discuss first the methods employed to date in the endeavor to build quiet machines, together with the shortcomings of each method. The determining factor in the approach to the solution of the noise problem was generally the rating and size of the machine in question. In the case of small induction motors, usually manufactured in large numbers, it is costumary to construct, test, and improve prototype machines before undertaking quantity production. In this case the selection of an

- 18appropriate slot number is considered the crux of the problem. Until recent years, purchasers of very large machines paid little or no attention to the noise problem, but at present they insist upon quiet operation. The structure of these machines is more or less regular, and the experience with large generators is being used by the designers of motors of several thousand horsepower ratings because the problem of large motors resembles the problem of synchronous generators. Pilot models in such sizes are prohibitive. It has been suggested that full-size replicas of a representative stator section with only a fraction of the length of the machine1 be constructed. Such a frame section would then be tested by means of mechanical shakers and the natural behavior of the frame studied. Such models, relatively expensive, require undue use of production facilities, and are of practical use only for two- to four-pole machines. Models of large rotating machines scaled down to approximately 1/10 size have been used under laboratory conditions for quite some time. Using dimensional analysis, the dynamic behavior of the full-size machines could -be predicted by these models. To construct such models from 110

-19 - steel has the disadvantage of the resulting frequencies being too high for reliable measurement and correlation. Plastic models hav' been suggested and tried, and replicas constructed that were scaled down to 1/4 - 1/8 of the original machine. These models performed well and their stability with temperature, humidity, time, and frequency proved satisfactory. However, the above methods used for small and large machines become impracticable for medium induction motors. As mentioned before, the approach to these has been developed by P. L. Alger1. The slot combination still remains the nucleus of the problem. The mechanical behavior of the butt-up core is assumed to be the decisive factor. The design is based on the performance of this stator core and proper corrections made for the inertia forces of the teeth and windings. The method of noise calculation based on Alger's theories will in this thesis be referred to as the "single-ring theory". This method assumes that the stiffening effect of the frame could be neglected, an assumption which could, with reservations, be accepted for very small motors, but is not justified for medium size induction motors. Thus it can be regarded as the Achilles heel of the single-ring theory. It is hoped that the following discussion will not be construed as 1 2;3

- 20 - destructively critical, as the author fully recognizes the great importance and value of the single-ring theory in the present stage of noise prediction. The mechanical structure of an induction motor stator is schematically shown in fig. 3.3.1. The frame, which in the following discussion will be referred to as outer ring, is strongly coupled to the stator core - which in turn will be referred to as inner ring. The natural frequencies of such a structure will differ considerably from the frequencies of the inner ring vibrating alone. Furthermore, the inner ring - when considered inextensional - has but one natural frequency in each mode1, while a structure as shown in fig. 3.3.1 will have several important resonant frequencies for each mode of vibration. All these resonant frequencies must be considered if the best slot combination is to be selected to arrange mismatch between the natural frequencies of the frame and the frequencies of the driving electromagnetic forces. The main aim of the vibration analysis of the induction motor structure is to find its natural behavior and the dynamic response to impressed force waves. According to the single-ring theory- assuming a constant amplitude force wave - the deflections would gradually increase The term mode denotes the number of complete periods of a sinusoidally distributed quantity along the periphery of the stator bore.

-21with frequency until resonance is reached and beyond this dangerous point they would strongly attenuate as illustrated in fig. 1.1. Fig. 1.2 shows the behavior of the structure consisting of two rings and connected by radial ribs mounted on a support. The amplitude becomes very large at several points of the frequency range, a fact that cannot be explained by the single-ring theory. Unfortunately this is not the only objection to the single-ring theory. It can be verified that the sound waves radiated by the stator core are strongly attenuated by the outer ring. If the frame totally encloses the stator cores the attenuation may be of the order of 40 decibels. It therefore becomes obvious that the major part of the magnetic noise is caused by the vibration of the outer ring, a fact not accounted for by the single-ring theory. The following argument may illustrate the reasoning still more distinctly. Let us assume that identical stator cores and rotors are fitted into frames of different designs. According to the single-ring theory, the noise of all these induction motors would be the same. Simple tests would, however, demonstrate that this is not the case, and that the acoustic behavior of the motors in different frames will vary considerably. If the natural frequencies of small induction

22 fit I I L I I I I + i 1 1 1.1 1 i 44 i,"4 414 T 4 + l -4 IT, T I T If ---- -- 441 -4 -1-4 - tj 0 _44-ff T-, r 47, ts.1 4:1 TM - + 44-H- j- 7tI-VIVIA 1_1 i- i4 44+ 4- 403 H 4.+ 4 -1 4 - ----------'O -4. —44 +H-11- — H ++4+- 7f- + I F I-i I I i 1-1 I t H i tdl_ tu Ffi_ 4+ -41- q A 4+ -H-, 4# ti 4P 4 + j4 + 1 — T- f 4-4 4- A tz tt -T r T 44A+- If:"4 7 1114 T qN, f It 4- _J. -L J -44- fP4"- 14 4+ 4 —f-4 +;-t- r 44 4+ L i - - fi+'~.4 FFTT 0 t7 + + 4 ft T J- -fig 41 — 4 F -4 41 _44 F tit t -T' ti Tj A- — 1I4- -T 4 -ij - -4 t T -T r-T-T —''14 r J. -44 44 4- 4 44 — J -:.1 i7 4 A +4T,, I''. I - air I t'L4 I..4 JA4~4 f v fl -4i+ 47t _4 L:, 44-1 _4 4T #44i_ -.k flil #4 L

-24motors with a small number of pole pairs are computed in accordance with the single-ring theory one cannot help feeling surprised on finding that the low mode frequencies agree quite well with test results. This may be explained by the fact that the flexural rigidity and the mass of the inner ring are large compared to those of the outer ring. In medium induction motors the flexural rigidities and masses of the outer and the Inher ring are of the same order of magnitude, and the Influence of the outer ring on the complete structure vibrating as a system can not be ignored. In large multipole induction motors the flexural rigidity and the mass of the frame become larger than those of the core and will thus greatly influence the natural behavior of the induction motor structures 1.4 Thg Statement of the Problem. The purpose of this thesis is to extend the work started by P. L. Alger, and to put the predetermination of magnetic noise of polyphase induction motors on a sound theoretical basis. As the task is arduous and the complexity of the problem considerable, it often seemed presumptuaus to the author to have undertaken this analysis. As mentioned on page 20 and outlined in chapter 3,

-25 there are several resonances for each motor vibration and the designer is therefore faced with the problem of selecting slot combinations so as to avoid all these resonances. The difficulty of his task is still further increased by other considerations to be taken into account in the design of induction motors. The designer is called upon not only to design quiet motors but also to avoid higher harmonic torques which reduce starting torque; or higher harmonic flux losses which decrease the efficiency of the machineso He may have to choose a slot combination which will create force waves of a small number of pole pairs. In such cases it is of great advantage to calculate the magnetic noise of the induction motor in advance. The analysis as presented here involves three disciplines of physical science and is consequently treated in three different sections. First an analysis of the electromagnetic conditions in the airgap is developed to enable the designer to calculate the frequency, mode, and amplitude of the force waves acting on the components of the motor. The influence of saturation, eccentricity of airgap, skewing, and of uninsulated squirrel-cage bars is also described. In the following section, the vibrational aspect of the problem is given in treating the induction motor as

a continuous system, represented by two coupled concentric rings on a support. X method to find the natural frequencies indifferent modes of vibration, based on the Lagrangian equation of motion is evolved. Finally it is described how to calculate the response of the system previously determined. In the chapter on the acoustic problem, the wave equation for the sound pressures near the frame has been: solved. Here the motor is replaced by an infinitely long cylinder vibrating in a belt only. The theory is illustrated in each instance by an analysis of a medium-sized, polyphase induction motor. Design and electrical data of the motor are recorded in Appendix I. The experimental evidence at the end of this thesis is presented to verify the detailed theory. The question of why this problem had not been attacked before may arise in the reader's mind, and the following comments may serve as a reply: Fig. 1.3 shows a few representative samples of induction motor frames of modern design. The aeoustic bee havior of each of these frames with identical electrical designs must be computed individually. Considering the large number of frame sizes in each design and the different core and rotor combinations in each frame, the task of

-28treating all of this seems tremendous. Designers in need, pf quickly obtainable approximate answers were reluctant to attack this extremely complex problem, particularly at a time when noise suppression was not the critical problem it has since becomex Another answer may be found in the fact that the computational difficulties connected with the solution of the problem had been beyond the range of facilities available until recently. To solve the matrices of chapter 3 for eigenvalues with the aid of a desk computer might have taken many months, while eigenvalues and eigenvectors of the same matrices can nowadays be calculated with advanced electronic computers in a fraction of an hour. The present analysis could never have been performed without highly advanced computing facilities. It has been the author's conviction that a solution to the present problem could be found. It likewise seemed apparent that the solution of the problem could be general in principle, but that each system must be individually analyzed, owing to the many factors that play a part in the overall production of noise. While the task is somewhat simplified in the case of induction motors belonging to a particular series, because of the numerous similarities in their design, it does not

- 29 - always follow that fundamental differences within the series do not exist. In many cases, however, it might be possible to establish a single solution that is applicable to a complete line of induction motors.

IL..TB ELCTOQMGNETICFORCE WAVES 1N THE AIRAP:In this chapter a mathematical expression is developed for the distribution of the radial magnetic induction in the airgap of polyphase induction motors, as a function of position and time. This is pre-requisite to the calculation of mode, magnitude and frequency of the radial and tangential force waves in the aBrgap. The radial magnetic induction in the airgap wi11be assumed as proportional to the magnetomotive: potential.? difference radially across the airgap. The influence of the variable permeanee that results from open slots of the stator is also considered. To simplify the writing of the text, the distribution of the mgnetomotive:force plotted along the airgap will be referred to as the.MF"-: distribution. In accordance with current practices the follow..... ing assmptions. will be made: (1) the stator contains a symmetrical, 3.phase, double-layer winding with 600 phase...,''.... belts and a whole number of slots per pole per phase; (2) the stator coils may have widths shorter t a polepitch' (3) the rotor is fitted with a squirrel cage winding; (4) the airgap is concentric; (5) the stator has semi-closed slots and the rotor, fully-closed slots; and 30 -

31 - (6) the supply voltages are sinusoial and balanced. The principle of superposition is very freely applied to combine field quantities; i.e., uns aturated machines and simplified permeance cnditions are assumed in order to bring the mathematical formulation of the theory into a reasonably manageable form. It is not unusual to resort to this artifice in the solution of complicated induction motor problems. The, effects of saturation, eccentric airgap, - skewing and bar cross-currents will be discussed at the end of the chapter. 2.1,The, 1)stribution in teAra di to gtator_ currents th~e'.-~ ~ ~'~~ " *'.~~.. on if the polyphase induction motor is connected to a balanced polyphase supply, the fndaamental component of "...., thephase urrents in the stato may be expressed as iis the instantahn/.U lalue of the funda- mental component f the current at time t in the ut phase; u 0 for,the reference phase.

- 32I1 is the r.m.s. value of the phase current Is the circular frequency m is number of phases Fig. 2.1.1 shows a schematic diagram of the phasebelt location in a polyphase induction motor. /n B/1^ At Fig. 2.1.1 Phasebelts of an Induction Motor 2 C The 1MF distribution of a single phase of a polyphase winding is a pulsating stepwave. This wave is a periodic function and may be represented by a Fourier seriest (2.1,2) fI(xlt) -I sin~ t cos (6rk/) amp,

- 33 - (2.1.3) Mc = (2fi7r). (wlk/p) I1 amp. The symbols in the above equation denote f(x-lt) the value of the MF.distribution at point X1 at time t; x1 0 at the axis of the reference phase. d'is the harmonic order of a component of the stator MMi distribution* MI is the amplitude of the 6th harmonic of the IMM distribution wave of one: phase at time t = r /200 W1 is the number of turns in series per phase in the stator kf is the winding factor of the th harmonic, k6 kd 6k p where a d6* p6' kd6 is distribution factor of the d6 harmonil kp6 is pitch factor of the 6th harmonic p is number of pole pairs ~t is pole pitch in meters Although the stator windings of polyphase motors have three phases, the analysis is carried out for an mphase winding, to have more general results also valid for squirrel cage windings. The angle between two consecutive phase axes of the

-34 - tsymmetrieal m-phase winding illustrated in fig. 2*.11, will be 2r/pm msehanical radians or 2w/m electrical radians. The currents in consecutive phases have a time lag of 2w/m radians. The 6th harmonic of the MF wave of the uth phase is therefore displaced in the direction of the phase sequence from the 6th harmonic of the reference phase by 26ru/m electrical radians. The carent i of the th ut phase lags the current of the reference phase by 2T/m radians, so that the expression for the th MF wave harmonic of the uth phase can be written, (2.1.4) fdU(Xif) P sin(0t-2wr/am).eos[(6rxl/v)-(2ir/m)] Applying the relation 2 sin x. cos y = sin(xry) + sin(x-y), equation 2.1.4 can be expressed as f (xl Pt)= ( ln t-(6/x2/)]+(6-l)(2%Vm/ ) + (1/2)M (sin [Mt+( 1/t)3-(6l)(2vq/m) For negative values of ( the first term in the bracket of 2.1.5 has the same value as the second term for positive values of 6, and equations can be rewritten simply as (216) f /(x t)=(s22)Ms inCt-(xl/r)+(6-1)(21R-/mai The M distribution of the 6th harmonic resulting from all m phases will be,

-3 u5m~~~f ^~~~~~X19 ~ It) 1 = (1/2)Ml sin[cat-(6yrx1/))] cos[(-1)(2iu/m)]+ u=0 (2.1.7) au=m-l + <V2)McfCosta),t-(1rx/)] ) 7 sinnf(d-l) (2ru/m)] a=0 - The expressions behind the summation signs of equation 2.1.7 ~-i'''-, mostly equal zero, as they represent the sum of m-vectors in which consecutive vectors are displaced by (6-l)(2wr/m) radianso However, the sums will not become zero when all m-vectors are in phase; i.e., when (d-l)(2r/m) = cl2w or only for the values of 6 which satisfy the condition (2.1.8) d =clm 1 where cl is a constant, which may be any positive or negative integer or zero. Balanced 60 degree,:3-phase windings do not generate any even harmonics, or harmonics whose order number is divisible by threeo The constant cl for these windings must therefore be an even whole number, positive or negative, including zero. With this restriction, the sum of the sines in equation 2.1.7 will become zero, while the sum of the cosines will become m. Hence,

- 36(2.1.9) fn(xl,t) = M sin[wot-(Orxl/T)] amp,, where (2.1.10) = (1/2) m M< (\f2mwlk^I/p6) amp. The direction of rotation and the velocity of the higher harmonics of the MMF distribution may be calculated by assuming that (to - (rxl/c) = constant, and then differentiating this expression with respect to the variable t.. The mechanical velocity vl of - ~th the h harmonic of the stator is, (2.1.11) vl = (dxl/dt) = (ct/wr) meter per second measured on the stator bore. The velocity of the fundamental stator MMF distribution vs is vs = (27rfTr/) 2fr meter per second so that the velocity of a harmonic of the MMF distribution can! be simply expressed as, v6 = vs/6 meter per second. The angular velocity of any stator harmonic will be, (2.1.12) lo= l/6 radians per second. For negative values of the constant c, 6 will be nigativeadlso will be the velocity of the wave. This is explained by the fact that the direction of rotation of the harmonic is opposite to the phase sequence; i.e.

37opposite to the direction of rotation of the fundamental uiF distribution. In future considerations the greatest noise producing harmonies will be found to be the so-called slotharmonics which satisfy the equation, (2.1.13) 6s = lqlm where ql is the number of slots per pole per phase. Denoting with Q1 the total number of slots in the stator, the relation 2.1.13 can be rewritten as, (2.1.14) = (c1,Q1/P) + 1, where cle is a constants any positive or negative integer except zero. The amplitude of the fundamental component of the MMF wave of a distributed winding is reduced by a! few per cent only as compared with a concentrated winding. This is because the winding factor is a number only slightly below unity. The winding factor of the belt harmonies is, by contrast, a small fraction, and the amplitude of these is therefore considerably reduced. A simple evaluation would verify that the winding factor of the slot harmonies equals that of the fundamental MFL wave and their amplitudes will consequently be reduced by the same small percentage as that of the fundamental.

38 - 2.2, MF Distribution of the Sairrel Gage Rotor Before considering this topic it seems apppropriate to calculate the frequency of the currents induced in the rotor by the airgap fluxes that result from higher harmonic components of the stator MF distribution. For this purpose the stator NMF distribution is first related to the rotor periphery* Let at instant t = O point x1 0 of the stator coincide with a point x2 - 0 on the rotor. Let the slip of the rotor be, (2.2.1) s = (n-n)/n^ where, n is synchronous speed of the stator fundamental -SMe distribution in r.p.s, and n is speed of the rotor in r.p.s. The mechanical speed of the rotor will be (2.2.2) 2fz(l-s), and the position x of the rotor point x2 at time t will be, (2.2.3) x2 (t) x1 - 2fit (l-s)t x2 (t) = - [ Wi (l-s) t / ] or x1 = x2 (t) + [o (l-s)t/r ] 1 23;2

- 39 - In the following discussion x2(t) will be simply written as X2. Introducing equation 2.2*3 into equation 2*1.9 the expression for the MMF distribution will become (2.2.4) fo(x2t) = Mcsin([-6(1-s)t - (6crx2/)) On introducing (2.2.5)' s = 1 - d(1-s) equation 2.2.4 can be written as, (2.2.6) f^(x2,t) = M6 sinl8t - ( /x2/)] The velocity of the stator harmonic in respect to the rotor will therefore be, (2.2.7) 26 = (dx2/dt) = sgv/6 After this short explanation, expressions for the rotor MWE ean be developed. Consider next the most general case, i.e., that of the p harmonic of the rotor wave, induced by the 6th harmonic of the stator. The instantaneous value of the rotor cetrrent for this case can be written as, (2.2.8) 126-= 2 I26sin(s6t ) If Q2 represents the total number of bars in the squirrel cage the angle between two consecutive bars

-40-. measured in electrical radians will be = (2r/2). The th harmonic of the ht bar will be displaced from the pth harmonic of the reference bar by (2irph/Q2p radians. The phase angle of the currents of two consective bars will differ by (2rp6/Q2) radians, hence the th current in the h bar will lag the current in-the reference bar by (ar6/nQ2) radians. In a manner analogous to equation 2.*14 it may'be written that, (2.2.9) f,t)h M in [s (6 -(2rP/Q2)I. h..cos E([r/) - (-2PaQ2)hj where, (2.209a) M = (12 Q2/Prp) Using the same transformation as before, equation 2.2^9 can be expressed as, (2.2.10) fp<x2 t -M sin rh/Q )) Once again the MM1 distribution of all Q2 bars may be added and an expression similar to 2.1,7 can be obtained. The sum of the terms containing the sines will alain become zero, while the terms containing the cosines will not become zero if the following condition is fulfilled: (2.2.11) 2wrp(p-d)/Q2 = 2re2 The constant c2 may be any positive or negative integer

41 - including zero. The harmonic order of the rotor MMF distribution components will satisfy the condition, (2.2.12) p = (c2 Q2/p) + 6 The th harmonic of the IMF distribution resulting from the current in all Q2 bars will be (2.2.13) fp (x2,t) = sin [s, t-(prx2/t) ] where (2.2.14) Mp - ( Q2IO/P ) These equations are based on the hypothesis that a squirrel cage winding can be represented as polyphase winding with Q2 phases, each phase consisting of half a turn, The distribution and pitch factors -that is, the winding factors of all rotor M harmonics - will be one. An inspection of 2.2.12 suggests that the harmonic order p of the rotor M distribution component may have fractional values. To avoid these unconventional fraetional harmonic order numbers in the Fourier series of the MMF distribution, it became customary to regard the whole circumference of the airgap as the fundamental period of the MMF distribution wave in lieu of the usual double pole pitch. The fundamental of the MMB wave based on the double pole pitch period will not become the p~ ha.mo ni of the MMF distribution based on the total circumference

-42 as a period. To avoid misunderstandings, the fundamental of the double pole pitch period will in future be referred to as the main wave. Based on the new convention, all previous harmonics have to be multiplied by p and will be denoted with a prime. Equation 2.2.12 can be expressed as (2.2.12a) c2Q2 + d' where, 6' p6 and p' - pp Equations 2.2.5, 2.2.13, and 2.2.14 can now be rewritten as, (2.2.5a) s6, = 1-(b /p)(l-s) (2.2.13a) f, ( 2,t) = Mlp sin [ s6/t - (p'X/p')] (262.14a) Mp Q2 The velocity of a harmonic of the rotor MMF distibution in respect to the rotor can be calculated by the same method as in 2.1.11, and results in (2.2.15) v2p (dx/dt [ s6Zw/pr] = (lp) E 4(1-s) ] vs (2.2.15a) 2 (p/p') I - (6/p)(l-s) ] v Introducing into 2.2,13 the value of x as given in 2.2.2, an expression of the 6th harmonic of the rotor MMI distribution as a function of x'can be

- 43 - obtained: (2.2.16) fp (X1,t) M sin ([1+ [(p'-6 )( (l-ss)p ot ~p P 1,~~p' /p) th and the velocity of the ph rotor MMF harmonic follows as, (2.2.17) v O(dx/dt) = (i1/) I1 + (p 6)(l s) iv or (2.2.17a) v('2~2'7") -C^l.P)(l^(..(PI 2.3., Rotor Current and theReaction of the:. ~ -:. Rotor Harmonilg'cs For future consideration an expression for the current In the rotor bars is needed because of the higher harmonic flux waves. Simultaneously, the question of the reaction of the rotor current will be discussed. Introduci4into equation 2.2.12 zero for 2, one can readily see that every stator MMF distribution harmonic 6 creates a rotor MUF harmonic 6 p. To be more exact, te 6th harmonic of the stator MWL distril bution will induce a current I26 in the rotor of frequency sfl1. This current in turn excites rotor MMF harmonics according to equation 2.2.11, from which pth = 6th harmonic combines to give a resultant 6th harmonic in the airgap. According to Algerl and Arnold2 12 2 6.

-44the influence of the higher harmonic fluxes on the performance of induction motors is. best analyzed by adding an- additional branch for each harmonic in to the equivalent circuit, as though all the higher harmonic rotor circuits were connected in series, as shown in fig. 2.3.1. X, XL6. X1: Fig.23.1 Equivalent Circuit of an Induction Motor The symbols in this diagram denote the followings Xlmn6 is the magnetizing reactance of the stator in regard to the 6th stator harmonic R /s6 is the resistance and the load resistance of the secondary of the 6th cage referred to the primary X'2p is the leakage reactance of the 6th cage referred to the primary R1 is primary resistance X1. is leakage reactance of the primary

I126 is the current in the 6th rotor referred to the primary In this calculation it seems opportune to refer the primary current to the 6th cage and to work with secondary quantities. The 6th rotor branch represented in secondary quantities is shown in fig. 2*3*2. R2s)5 X2d X2m6 Fig. 2.3.2 Equivalent Circuit of the 6th Rotor Cage Before proceeding a review of the concept of the magnetizing and leakage reactance of an Induction motor is in orders The magnetizing reactance X1 can be defined as the impedance of each phase of the induction motor with the squirrel cage rotor rotating synchronously under the condition that no higher harmonic fluxes exist, and neglecting leakage reactance, winding resistance, and iron losses. An expression for the magnetizing reactance for the main wave or any harmonic wave can be found by the following method: Let the stator winding carry a balanced sinu

- 46 - soldal current of one amp. r.m.s. value, and express the MMF distribution in the airgap as a Fourier series. The ensuing distribution of the magnetic induction can then be determined and again expressed as a Fourier series. The numerical value of the r.m.s. voltage induced by any of the harmonic fluxes will be equal to the numerical value of the magnetizing reactance of the same harmonic flux in ohms. Using such an approach the magnetizing reactance of the main wave of the stator can be shwn to be, (2.3.1) X1 6m (w.k)2(pxl ohms th...... and the magne.tizing reactance of the 6t ha rmonic, (2.3.2) l16.6 mf ( )( (w k)( l/ghp)x.l o- hms Xn this expressionthe symbol i is the net iron length of the stator measured in meters, and g "e is the corrected airgap in meters (2.3.3) g" = kC ki-g meters In this expression g is the airgap measured between two radially opposite tooth centers; k ais a correction factor, a so-called Carter1 coefficient which corrects the value of the airgap for stator slotting; and 1 17

- 47 - ki takes care of the fact that only a part of the magnetomotive force is used up in driving the flux through the airgap while a smaller yet substantial part is consumed to establish the flux in the iron portion of the magnetic circuit. Similar considerations will lead to the magnetizing reactance of the rotor. The magnetizing reactance of the main wave of the rotor may be calculated as, (2.3.4) X2m (04Q2fl pl /g xp) x 10 ohms and the magnetizing reactance of the ath harmonic (2.3. 5) X2 ~ (0.4Q2f11j/gp6 2)xlO- ohms The magnitude of the leakage reactance is a very important prerequisite for the further analysis. The leakage reactance of induction motors consists of two components, each of an entirely different nature. One component - the so-called induction leakage - results from the simple fact that not all the flux lines of the urrentl carrying stator winding link the rotor winding, or the converse. The slot leakage and end-connection leakage are parts of the inductive leakage. The numerical value of the inductive leadge reactance is approximately the same for each higher harmonic branch of the equivalent circuit. The origin of the other component of the leak

- 48 - age reactance, the so-called differential leakage reactance, can be traced to an entirely different cause and will be clarified by the following reasoning: The instantaneous sum of all the fundamental frequency voltages in the stator must be equal to the instantaneous value of the supply voltage. The fundamental and higher harmonic fields that result from the primary current will induce just such voltages of fundamental frequencies in the stator, and therefore appear in the voltage equation of the primary. If the motor rotates, the individual higher harmonic fields of the stator will induce voltages of different frequencies in the rotor. The fundamental current of slip frequency (sf) is induced by the main field of the primary and the magnitude of the current can be determined from the slipfrequency-voltage equation of the rotor. The voltages induced in the rotor by the higher harmonic flux fields of the stator do not appear in this main rotor equation but have to be found in separate voltage equations. Consequently it appears that the higher harmonic stator fluxes induce a voltage of fundamental frequency in the stator but not in the rotor. The voltages induced in the stator by the harmonic fluxes must therefore be considered as "leakage-reactance drops. A similar observation leads also to the concept of the differential leakage of the rotor.

- 49 - The total primary leakage reactance may be expressed as (2.3.6) X1 = X13 + Xle + Xld, in which expression, Xls is the slot leakage reactance of the primary, Xle is the end connection leakage of the primary, Xld is the differential leakage of the primary. Replacing subscript 1 by subscript 2, a similar equation can be written for the main branch of the rotor circuit. Expressions for calculations of the slot leakage reactance are well known and epirical values for the end connection leakage are readily available. It now remains to develop an expression for the differential leakages Neglecting the effect of saturations stator slot opening, and rotor reaction, one can assume the differential leakage of stator and rotor to be mutually independent, making It possible to evaluate each leakage reactance separately by applying the same consideration that led to equation 2.3.1. First, the voltage induced by all stator harmonics will be calculated for the stator winding carrying one amp:e' of'curraei. In this way the numerical value of the

induced voltage will determine the total airgap reactance in ohms Xlg in the same manner as the main magnetizing reactance has been determined before. The differential leakage reactance can then be calculated as the total airgap reactance Xg reduced by the main magnetizing reactance of the fundamental. Thus (2.3.7) Xd Xlg - Xm or we may write (2.3.7a) Xld = XldXm xwhere Xld is the primary leakage factor defined as (2.3.8) ld- (Xg/Xlm) -1 The magnitude of Xg can easily be calculated from energy consideration in the airgap. The magnetic energy stored in the airgap of an induction motor with an m-phase winding is given by, W= (m/2)L 2 = (1/2) h b(dv) = (1/2) P h2(dor) where - L is the inductance of one phase including the effect of the mutual inductances of the other phases, measured in henries h is the magnetic field intensity at a point of the airgap in amp. per meter b is the magnetic induction in weber per meterZ 1 2;4

- 51 -, 1is the volume of the airgap in meter3 is the permeability of free space (4r/107) henrles/meter. The airgap reactance X can now be written, X = 2fL =(4wf/m)(W/12) x=2w (2.3.9) = (2rf/m)(g"nlp/l) J h2d(7A) x=0 The value of h can be found by dividing the magnetomotive force eat x by the corrected length of the airgap, that is, (2.3.10) h(xlt) = f(xl,t)/g where f(xlt) is given by equation 2.1.4. In the discussion following f(xl) will.be used in lieu of f(xlt). The MHF distribution with our previous assumption can be taken to be a step wave. The Integration in equation 2.3.9 can be replaced by a summation over all Q1 slots of the stator as, 1i 2 (2.3.11) Xlg I (4vrlip'/m g"I2) L (f/Q1) k= 1 where fk denotes the ordinate of the MMF distribution step wave over the kth tooth. Introducing the value of the amplitude of the main NMgF wave, as given in equation 2.1.4, into 2.3.1, the main magnetizing reactance can be expressed as,

- 52 - xim = (P-cl)(O ML)/(mMgnj 1 #. and the differential leakage factor of the fundamental calculated to be, k Q (2.3.12) Xld (2 /Q1) - 1 k= o Substituting the value of Ml from equation 2.1*10 into 2.3.11, a more convenient form of the differential leakage coefficient can be obtained: (2*3 12a) A ((rp)/mkl (fQ) -I k-o From 2.3.12 it is obvious that iXd depends on the winding configuration of the stator. Consequently, i iis necessary to establish the IMF distribution for each winding to get the value of fk As this would seem a rather cumbersome task the values of Xid have been tabulated for different windings in the reference quoted below1. The differential leakage coefficient of a squirrel cage rotor has still to be considered. Equation 2.3.12a may be set as the starting point, and slmplified to get an explicit formula for the differential leakage of the squirrel cage. The expresslon behind the summation sign can be calculated with ease from a sinor diagram of the bar currents illus1 2

53 - trated in fig 2*3*3 below. The sinor 0I represents the fundamental omrnponent I2I of the rotor current in bar 1, and T the fundamental component in bar 2, 2 0 Fig. 2.3.3 Sinor Diagram of Bar Currents and so on. The phase angle between two consecutive sinors is given by,..... - (2irp/Q2) The sinors constitute a polygon and the radius of the circumscribed circle can be expressed as R {II2,1 / 8(sin /2)1] The magnitude of the MMF potential difference f at any rotor tooth is given by the ordinate from the polygon vertex to a so-called time axis. Choosing zero radius to vertex zero for the time axis, the vertical distance -2' will be a measure of the MP along the

flux line through rotor tooth 2 2 kQ2 21fk sin (/2) )2]. (k.. <K^ J. -(4V4) 92 Cmig,1in (82)]Z (k The amplitude of the main wave of the rotor MMF distribution is given by 2.2.9a as / 2,l ({Q2 I2 1/2rp), so that - 2 (2.3.13) X2^1 21 1 where (2.3.14) 21 = (pr/Q2)/ [ sin (pr)/ Q2 or approximately (2.3.14a) 2,1 (1/3)-(p/2) The differential leakage of the rotor resulting from the rotor current induced by the main wave of the stator MIF can now be written as, (2.3.15) X2d,l = A2d,1':X2 -.. For the calculation of the rotor c-reont caused by the 6th harmonic of the stator MMF distrition the leakag rreactance referred to the th cage is needed. This leakage reactance consists of the slot, end connection, and differential leakage. The first two compo

nents are nearly equal for all harmonics but the differential leakage varies and has to be computed. The same method is used as in the case of the main wave with the difference that the 6th harmonic is now considered the main wave as though the motor had p6 pole pairs. If the IMDF distribution resulting from the current 126 does not contain a component of main period, or if it is negligibly small, the differential leakage of the rotor in respect to the 6th harmonic can be evaluated as, (2.3.16) X2d6 = X2d6* X2m6 where X2m is the magnetizing reactance of the rotor with respect to the 6h harmonic as defined in equation 2.3.5., X2d6 is the 6th differential leakage coefficient of the rotor* and is defined ast (2.3.17) i2d6'26 t, where (2.3.18>y: 26 (6'rp/Q2)/ [ sin (6rp/Q. In order to solve the circuit, an expression for 1, ( the fundamental stator current referred to the 6th cage ) has to be found, a task readily performed by the following reasoning: The dth component of the stator + M distribution resulting from current I1 has dp pole pairs. This

- 56-;harmonic can be separately considered and will induce a current in the rotor bars of the value Iid of such magnitude that the rotor MMF due to this current will have the same magnitude as the th harmonic of the stator MM.F (The current in the magnetizing branch is neglected ) This is easily accomplished by equating 2.1*10 and 2.2.14: (2.3.19) 16 = 2 (m wlk611/Q2) The current I2 and its influence on the 6th harmonic stator wave can be derived from the voltage ~th equation of the 6h cage (2.3.21) 6I2 - (-~~6 X2m6 Ii 1 )/[R2 + Js1(X2m6+X)2)] For small values of s - that is the induction motor operating at rated condition - the value of and the magnitude of R26 becomes negligible as compared to $6(X2m6 + X26) and 2.3.21 can be simply written as (2,3.22) 126- 1X2m Il/(X2m6 + X26) 2,4 Permeance Variat in of the AirgaP Besu ltinfr om ODen statr Slots The semi-open slots of the stator of an induction. motor cause a non-uniform distribution of the flux lines

- 7 - *in the aitgap. It is not difficult to understand that the ensuing permeanee of the airgap will vary as a peri dic function with a tooth pitch as a period. This permeance variation will become the cause of a whole sequence of new harmonies, the so-called permeance harmonies. The amplit:de of these harmonies depends on the permeanee functon, which in turn depends on the machine design and has to be evaluated in each case* The permeanee distribution in the airgap has in the past often been investigated, mainly for the purpose of finding the increase in the MS required for the *f-A blishment of the-iflux in the magnetic circuit. F W. Carteri was the first to arrive at a mathematical solution by the method of conformal transformations. He assumed an infinitely deep slot, and an infinitely long slot pitch. Since then many others have tried to improve on Carter's'2 work, but only Frey s paper is outstanding in this field. He published data for slots of finite depth. While these data are of great importance to designers, they can not be applied in an analytical manner for which another approach has to be made. E. Weber3 published in 1928 an approximate method of 1 17 2 26 373

58expressing the airgap permeance function in an explicit form, also applicable to this analysis. In the following, the variation of the airgap permeance is assumed to result entirely from the stator slot opening and to be unaffected by the saturation of the rotor teethe Fig. 2*4.1 illustrates the flu pulsation caused by stator slot openings. i..x'j:: >1 n tth do st ximmfu density 1at Fig. 2.4.1 The Pulsation due to Slot Openings, In the above figure, Slo is the width of the slot opening, tint is the tooth pitch g is the airgap Bmx denotes the maximum flux density at the tooth center Bb denotes the maximm. ri"duaetlo^l~ f the.f:.c dnsilty at the slot center

Bb'( ) is the flux density reduction at; i1s measured from the tooth center.'Analyzing a number of tooth pulsation: curres, Weber found that the flux distribution between the limits function, (2.4.2) b (t/slo)lo. For integer values of b the above function can be expressed by a series of finite numbers of terms: (2.4.3) sin b(a ( o2ba) - ( cos Z 2(b-1)a J] a. -.2b 0 0 os, [2 (b -2)a1 + (-1)(b 2) cos (2a5)+ + (b/(.l)(b) ).&l1,22 I a is defined as, (2.4.4) a = ^/t For non-integer values of b, the series 2.4.3 wll have an infinite number of terms. Weber suggests that the coefficient of the series first be found for integer values of b and then interpolated. The actual field distribution between two tooth

60centers, slotting considered, can,in accordance with fig. 2.4.1 and with equation 2.4.3 be expressed as, (2.4.5) B(j) - Bma - B b = [1-2sln 2b(a )] B(aX where (2.4.6) 2PB3 /maB,: The values of,/are a function of (So/g) and have been calculated by Carter. His results are shown in the graph of fig. 2.4.2 O3 _ 0o2 2 4 6 8 1 Fig.. 2.4,2 The Function /3 The expression in the square bracket of 2.4. 5 can be considered the ratio of the permeance of the slotted airgap as compared to the permeance of a. uniform airgap of the length g, or, (2.4.7) p (.). E1-2/3si 2b (a) ]3 where -;

61(u/g) the permeance of the unslotted airgap "p ( ) the permeance function of the slotted airgap Only a few terms of this permeance function are of practical value. Introducing into 2*4.7 the coefficient of the series 2.4.3, the expression for the airgap permeance will become (2.4.8) P()- (Pg)(Cl-3( +)bi) c I 2a21-3(21) scos 4a 0The term in the square bracket j11-(b)1 indicates the redution of the average value of the flux density in the airgap and is actually the reciprocal of the Carter coefficient kcKeeping in mind that 2pt= QlZ' and a = (aiQ/2pt), equation 2.4.8 can be expressed as (2 4~g~p(j) (Pogkc) 1s21 cos (vQl/pr3- 2 2(l/p/P ) where 1 = //3(b2l) / -13(2 (2.4.10) 2 /3(b- 2) / /22bl b In illustration of these equations, the coefficient for the experimental induction motor will be determined from design data as given in Appendix I't = 0.471"; Slo = 0.125";g = 0.023", hence

62(sl0/g) 5.44, and from fig. 2.4.2, (= 0.33. The value of b calculated from equation 2.4.2 is 2.8. As expected, b is no integer, and therefore the coefficient wiM ll be determined for b = 2 and 3, and then a linear Interpolation applied to find the value of 1 for b = 2.8. For b = 3; 1 = (0.33)().2 -(0.33)) 0.19 for b 2;- for b = 2; 1- (0.33)( ).23-(033)(I) 0.22'and by linear interpolation for b = 2.8; 1 0:.2. 2~ PFlux Distribution in the Airgja t (negleeting Variations of Airgap Permeanlce) The flux distribution in the irgap can be determined by multiplying the magnetomotive potential drop across the airgap at each point by the permeability,~o and by dividing these values by the corrected airgap g'o For convenience of treatment it may be advisable to determine fictitious flux distributions caused by the stator and rotor MM separately and to apply the principle of superposition. It is worth notice that the ordinates of the MMF distribution are larger than the magnetomotive potential drop in the airgap. A not inconsiderable part of the total available MMF along a flux line is used to establish the magnetic induction in the iron. To account for this in a simple manner the corrected airgap

-63g' ib multiplied by yet another correction factor ki. This correction factor varies considerably with the harmonic order of the flux wave. For the main wave ki will vary between 1.3 tad 1,9. For the harmonics of higher order it will decrease to 1, as the flux distribution of these have many poles and the flux lines close through the tooth heads of the rotor and stator. The main field is due to the resultant. MM? distribution of the primary and secondary currents of fundamental frequency or simply the field established by the magnetizing current. This field is largest at O load and only slightly reduced at full load. This main flux distribution may be expressed as, (2.5.1) bl(x1,t) = B1 sin [ Ct-(txl/) - e1)' M' where B1 is the amplitude of the fundamental flix distribution and can be expressed as, (2.5.2) B1 (pI/g")(m w to Im Ip)w eber/eter 1 is the angle by which the magnetizing current lags, the primary current that is, the reference current, and Il is the fundamental component of the magnetizing current. The angular velocity and the mechanical velocity of this field can be evaluated as in section 2.2 and obviously be the same as that of the main MMS distribution. The flux distribution of the higher stator har

-64monic MFMs can be computed in a similar fashion. It will be necessary to consider the damping effect of the rotor wave of the order p = 6 for the lower belt harmonics as described in section 2.3. For these the value of ki will be slightly less than that of the main wave. For the slot harmonics the reaction of the rotor can be neglected, and with sufficient approximation they can be written as, (2.5.3) b6(xt) B6 sin (wt-6drxl /) weber/meter2 or (2.5.3a) b(xlt) = B6 sin (mt-6'rxl/p r weber/meter2 where (2.5.4) B6 (pi/g')(m w1l Ik6/pd) weber/meter2 or (2..4a) B6 (p/g )(m2 w1 k6 1 /6' ) weber/meter2 The angular velocity and the mechanical speed of these higher harmonic flux fields will be identical with the MMF distributions causing them. The equation of the fictitious flux distributions of the rotor current is again obtained by multiplying the appropriate rotor MM? distribution by r/g" or by h/gO (2.5.5) bp (x2,t) 3 B sin s6wt - (prx2/) 3 or (2.5.5 a) bp (x2,t) B6/sln C s6t - (p'rx2/p) ] where

(2..6) Bpp (po/g'). (Q2 f I2/2rrpp) or (2.5*6a) Bp,= (po/g'). (Q2 f 12/2t/p'), Referring this wave to the stator in the manner described in section 2, the flux distribution resulting from any rotor harmonic can be expressed as, 2.5.7). bp(x1,t) = B sin l+(p-6)(l-)cot-(p r )) or (2.5.7a) b,(xlt)= Bp,sin ([l+(p-'-)(l-s)/pWJot-(p'ixl/p)) The speed of these harmonics can again be calculated as in section 2*3.. A:. 2.6 Airgap Flux Distribution considering Slot oeni Il...2. The variable permeance of the airgap considerably changes the flux distribution described in section 2.*5 It creates another series of flux harmonics, the so-called permeance harmonics. These harmonics of the flux wave are of the same order as the slot harmonics. The two series of harmonies will combine in a manner largely dependent on the quantity q. As the combination rule will be different for even and for odd values of q, it will be necessary to investigate both cases. -q is even; The expression for the permeance function has 173

- 66 - been developed assuming x = 0 at a tooth center. With an even number of slots per pole per phase, the amplitude of the main wave will also be at a tooth center, according to equation 2.1.4. For the sake of convenience the equation of the MMF distribution will be transformed to a new system of coordinate axes with the origin at the zero ordinate of the MMF wave at a time when the current in the reference phase is a maximum. Equation 2.1.4 can now be written as, MIl (xl t )=Mpcos r(x1 A)- At] -Mp os [( 5Xl7/r )+C)t]+ +M cosr(7x/)- ]t^ I-*(lm p)cosl(QlP)(x +t/PT)+t+ (2.6.1) +M(Q )cos[(Q+P) (x r/pT)-t] -... Now that the main wave and the permeance function are related to the same origin - that is, a tooth center - the equation of the first order permeance harmonics can be established by multiplying the first term of 2.6.1 by the permeance function 2*4.9 (2.6.2) bp(x1,t) = pcos[l(lrA/)-ot](o/gkc) x x fl+21lcos(x1lQ1/p)-22cos(2xirQ1/pr)] The product to the right of the equation sign will have three terms. The first term will represent the fundamental flux distri

- 67 - bution reduced by the factor kC The second term, which is our main concern, can be rewritten after a simple trtgonometr tri transformtion: (2.6.3) bp(x)X1,t)b = M Cos [(Q1+P)x1'r/pI t).(pg ) +1l 1W COS( C(Q1P)X1r/PS t).tpdrkC ) By comparing 2.6.3 with 2.6.1, the following striking observations can be mades The first term of 2.6.3 e that is, the positively rotating first permeance harmonic -has the same number of poles, direction of rotation, and angular velocity as the positively rotating first slot harmonic. Furthermore, the coefficient of the permeance harmonic has the same sign as the corresponding slot harmonic, and they will therefore add. The second term of 2.6.3 - the negatively rotating first permeance harmonic - has again the same number of poles, direction of rotation, and angular velocity as the first negatively rotating slot harmonic of the order (-Q1 +p)' However, this time the sign of its coefficient is opposite to the sign of the corresponding slot harmonic. Consequently the two waves will subtract. The third term of 2.6.2 represents the second order slot harmonics which can be evaluated in the same fashion as those of the first order. When combining slot and permeance waves of har

-68monio order (+Q1 + p), it is important to remember that the slot harmonics are produced by the WMF of the primary current, while the magnetizing current is responsible for the main field and the permeance harmonic. These two harmonics are out of phase by an angle of (01 - e) and must therefore be combined vectorially. is odd4 The origin of the MMF distribution is in this case in the middle of a slot. The expression-for the permeance function has to be related to this:rigin by the simple transformation, = + a, and we mayS:express it as(2.6.4) ^(x) ( + 21 cos (Q^I'X1/): -]'- 2t2 os ( 2Qrxl/pS t) (PJgk ) or, - - --. k; (2.6.4a) (Px) ( 1 -21 os (Qx/p t2i O$ (2Q1 Ird/p-. )Qr o/Ic)g The permeance harmonic of the fundamental may again be ealculated by multiplying 2.6*4a with the flirS- term of: 2.6.1. As before, three terms will result, with the first and4 third term unchanged, while the second ay be evaluated as::': (2.6.5) bp(x1,t) 03 cos M (Qfp)(xw/i) t ] - 11 08 O[ (Q -p)(x/,) +ot ])(pIgkc) A comparison of this expression with 2.6.1 demonstrates that in the case where q is odd, the:sign of the positively

-69 - rotating first permeance harmonic is opposite to that of the slot harmonic of the same order. By contrast, the negatively rotating slot harmonic is of the same sign as its slot harmonic counterpart in the MMF wave, and they will therefore add. The addition must again be- carried out by the parallelogram law. At the start of this chapter we assumed closed rotor slots. The saturation of the slot bridges creates the same effect as slot openings, and permeance harmonies caused by rotor slots may also occur. These are difficult to evaluate and will therefore be described qualitatively. Suppose that At time t = 0, a rotor slot center of a smooth stator induction motor is at an angle from the origin of the main IMF wave in the airgap. The permeance of the airgap resulting from rotor slotting can be then expressed as (2.6.6.,) Ip [COS2(r/P ) -Or r](P/kc) where the angular velocity of the rotor is expressed as % = %(l-s) The negative sign before 1 takes care of the fact that the origin of the MMF wave coincides with a slot center. The theoretical expression for the rotor permeance harmonics can be evaluated as before and is given for the first order harmonics as,

70 - (2.6.7) b 1)(X,t)= cos [(pxr/-t) -t J.(pgk0) which may again be expanded as previously into terms of the following form (2.6.7a) -M1 Eos( j(Q2+p)xr3 t - Q2 yo- Q2rCt) - cos ( [2(Q2p)xw/i +-wt Q2 y Q2w rt) It follows from these terms that the positively roating first permeance field adds to the rotor slot harmonic of the same order while the:ngatively rotating permeance field subtracts from it. It is rather difficult to get a numerical value for the field changes. A conservative estimate would indicate a change between O1 and 20 per lent. 2. The Radial Magnetic Pore Wave The main torque in an induction motor is the one created by the main flux distribution in the airgap and the main current distribution on the rotor. Besides its useful tangential forces, there are also parasitic forces in the airgap which do not contribute to the motor's useful operation. The radial components of these forces will be investigated as being the main cause of magnetic noise. The share of the tangential forces in noise production ir negligible, as explained in section 2.8*

- 71 - The radial forces can be readily determined from energy considerations because simplifying assumptions that the lines of force in the airgap are radial were already made. The differential area in the airgap can be expressed as A = dx.li. Now suppose that this differential area of the stator had moved towards the rotor by an amount dr. The change in magnetic energy in the elementary volume of - dW (1/2) h.b.A.dr (1/2p)b2A dr The force on the differential area resulting from the change of the magnetic energy will be P = dW/dr = (1/2p )b A, or P/A = -(1/2 p). b2 newtons/meter2 Denoting the electromagnetic pressure with r, the following equation may be writtens (2.7.1) r (xlt) = (1/2 po) b2(xlt) where bl(xl,t) is the resultant magnetic induction in the airgap. To further simplify the equations the quantity r(xl,t) will be disregarded for the time being and the quantity b2(xl,t) investigated. With the previous assumption that all magnetic lines cross the airgap radially, (2.7.2) b(xl,t) = bl(xl,t) + b2(xlt) The expression for bl(xl,t) and b2(xl,t) has been developed

-72in equations 2.5.3 and 2.5.7 and is repeated below: b1 (x1,t) =, B sin I cot - (dxl/pp) ] b2 (xl1t) ZB sin [ot [l + (P6)(l-s)/p ] - - (ptrx1/P) For the sake of simplification the frequency term in the second equation will be written, (2-7.3) o 5 (1 + U (p - 6)(l-s) I / so that the expression for the square of the flux density can be written as t 2.. (2.7.4) [bo (xt)j ( 2ZB6 sin Ct (&I/pj + bt~ (X B. sin. i where the primes denote the previously mentioned convention that the whole aacrti re circumference is considered to be the fundamental period of the Fourier series. The. order of 6' and p' for which the summation has to be carried out is given in equations 2.18 and 2.1.12a. The trigomeo"tric rsuis 2 sin2 x 1- cos 2x and 2 sin x.sin y cos(x-y) - cos (+y) will be used to represent the results of the square of 2.7.4 in an organized manner, but it seems advisable to

- 73 - discuss first the nature of the different terms in the results. As for the phase angles and fractions, they will be disregarded in the following description. Upon squaring the first summation sign, the following terms will results First term: / B~, represents a constant force uniformly 6' distributed over the whole armature circumference. This distribution does not cause vibrations. ^i o Second terms: Z B cos C[at (26'xl/p)I are the 61 so-called double frequency force waves. They pulsate with double supply frequency and are responsible for the hum characteristic of induction motors. Only the case of 6' = p' may acquire importance in 2 or 4 pole motors where the double frequency force waves have two or four pole pairs. The next smallest order of 6' 5p (the fifth harmonic) would cause a 10-pole-pair force wave in a 4pole motor of double supply frequency. The flexural rigidity of the stator structure at such a high mode is so large that the deflection caused by these force wavet is negligibly small. Furthermore, 120 c.p.s. sound pressures must be fairly large to fall into the range of audibility. Third term: (2.7.5) Z 2, B Bj cos 2.t1 - (l+6)(sni/P, where 6$#6. These are again double frequency force waves.

- 74 - They will be discussed later. Fourth terms 6$= B6l.B6 cos [(6l-62)(rxl/p) ] where 6i:f=, represents non-pulsating constant force waves sinusoidally distributed and not causing vibrations. Fifth term: ZB 2 is again a constant pressure uniformly distributed as in the first term. Sixth term: ZLB, cos [2x t-(2 p'rx/p 3 is generally a large number and the force distribution in 2 ppole pairs will cause only negligible deflections. Should p' be a small integer, BP, will become so small as to appear negligible. Seventh term: (2.7.5a) Z, -B -B'2 cos (W (co,)t - pip* WP' p~p r 1+r,r -C (pl-p2) ( /p ) where p1 p. These force waves result from:interaction of two rotor flux waves and will be investigated later. The most important term Is the cross product of the expressions behind the summation sign which yield the Eighth term: (2.7.6) 6 B' B6 c )t- (d ) where (p'+ 6) may have a low integer value in some of the terms which represent force waves of a small number of pole

-75pairs distributed over the airgap periphery. The bending moments caused by these force waves may become significant by causing considerable vibration and noise. To simplify the writing of the force wave equations, the quantities (6 j+ ~ ) in 2.7.4, (P P) in 61'' (P1 Pt2)1 in 2.7.5 and (6' + p') in 2.?.6, will be denoted my, the mode' of the force wave. This quantity represents the number of complete periods of the force distribution along the airgap; for example, mv = 2 will represent a second mode force wave. The mode of the forced vibration of the stator largely depends on the mode of the force wave and differs for ma =0, Om =I, mpv = 2 or larger than 2. The equation of the force wave for my= 0 can be written as, (2.7.7) r (xl,t) crF cos [ (J-O )t t This equation illustrates a uniform force distribution pulsating with a frequency w -,. The forced vibration of the stator structure will under this condition be similar to that of a steam pipe under periodical pressure variations. The stator rings will be subjected to extensional deformations resulting in extremely small amplitude vibrations, because of the very large extensional rigidity of the stator. The sound pressures resulting from such vibrations will be below the threshold of audibility.

76The case where mv = 1 will only occur if the interacting flux waves differ by 1 pole pair counted around the whole armature periphery. It is obviously the case of the one-sided pull. The airgap flux density will be a maximum at one point, and a minimum at a diametrically opposite point. The expression for the force wave may be written as, (2.7.8) [r(xt) [ rmax (X1 )*(rxl/pt) Fig. 2.7.1 shows the resultant flux density for the case of 6= 4, and = 3. The force wave caused by the interaction of these two fields is illustrated by dotted lines. The magnitude of the one-sided pull can be found by resolving the force at every point in the airgap into" a component parallel to the maximum force at that instant, and thiaher perpendicular to this direction. 7,/- i'. rr Fig* 2*7-1 Interacting Flux Waves differing by I Pole Pair The integral of the first component along the whole periphery represents the magnetice pull; The integral of the

- 77 - second component is zero. The one-sided pull constituting a bending load on the shaft causes deformations as illustrated in fig. 2.7.2 on page 78. The resultant one-sided force revolves around the airgap with an angular velocity (w1 + cop). If this frequency is near the critical frequency of the rotor, the vibrations caused by this one-sided pull build tp to a dangerous shaking. The amplitudes under such resonant conditions may become so large as to make the rotor grind on the stator. An excessively loud, howling noise ensues, a phenomenon familiar to designers who will have no difficulty in finding a slot combination to avoid such an occurrence. The case m = 2 will arise if there are two fields with pole pairs differing by two in the airgap. In this instance the equation is, (2.7.9) (xlt) =Vmax cos ((i)-~, )t - (2rxl/pr)) The sinusoidally distributed force wave will have two complete cycles along the airgap. There is no eccentric force on the stator or rotor and the force waves represent a bending load. Rigidity of the rotor is very large so that its deformation can be neglected. The stator will be deformed into the shape of an ellipse as shown in fig. 2.7*3 on page 78. The vibration amplitudes in this mode will become

78 - -- g+ 2*7*<7,* 2 Deformations of "One-Sided Pull" Fig. 2.7.3 Elliptical Deformatiom Polygonal -Toatig 2,74' Polgonal Deformationl l'

-79 - quite large as verified in the analysis of chapter 3. The conditions where mv is larger than two but still a small number are very similar to case - = 2. The sinusoidal force wave distributions will have a few periods around the circumference, with a tendency to deform the stator into the shape of a polygon, as shown in figures 2.7.4a, and 2.7.4b, for m 3,4, respectively. The flexural rigidity of the stator structure increases rapidly with the order of the mode m. Consequently, the deflections will decrease rapidly with increased mode orders. In most medium induction motors it will therefore suffice to consider force waves to the third mode only, with the exception of the case where the frequency of the mv f6rce wave coincides with the resonant frequency of the stator structure in the same mode. For large induction motors it will be necessary to consider also values of m, larger than 3* From force equation 2.7.6 it would appear that the force itself could become negative, denoting a repulsion force between stator and rotor. As this is physically impossible it requires clarification. In the discussion of terms it could be noted that the first term represents a uniformly attracting force Bver the whole armature. Becauase B + B2 is always larger than 2 B1B2, the total at

-80tracting force always remains positive; i.e., the negative terms only reduce the magnitude of the attracting forces. The preceding discussion indicates that there is a vast number of interfering flux waves and force waves. It is possible however to exclude a large number of the force waves by analyzing the conditions for their occurrence, though the analysis is valid only for machines with concentric airgaps. For force waves that result from the interaction of two stator fields the condition, (2.7.10) my 6l - 2 has to be satisfied. Recalling that I! 61 = p (c1 m+ ); 62 p (c2 m+ 1) the mode of the force wave can be expressed as, (2.7.10a) v = p [(c + c) m + 2 For mv = 1l this reduces to p (cl + c2)m + 2] = + 1 as c1 and c2 are whole numbers it would seem as though this condition could be satisfied only in a machine where p - 1; that is, in a two pole machine. However this will apply only to 1200 phase belt machines. For the latter case, cl and c have to be even numbers, and a one-side pull could not

occur in a modern induction motor because of interference of two stator force waves. To differ by mv, the general condition for the harmonic order of two stator waves of 60~ phase belt machines is given by (2.7.11) (Cal + c12) (m/p)-2, and this sum must be an even multiple of 3 or 0. For m = 2, it is only satisfied in two-pole machines where 1 = -5, and 62 +7, may create a twopole pair force wave. The frequency of these waves is the double supply frequency which will add to the vibrations caused by the fundamental field. Equation 2.7.11 cannot be satisfied for odd modes which means that no odd modes will occur that result from interference between two stator waves in a 60~ winding. The conditions for the rotor field to interact and thus create an mv force wave are, (2.7.12) ( ~ p ) = t > Introducing into these equations the values of Pi - 21 Q2 + 61, 2 022 Q2+6;V and for 3-phase motors, 6 = P(3cll + 1) and 6 = p(3ca2 + 1),

-82the condition for the interaction of the rotor waves can be written as two equations, Q2(e21 + 22) + 3p (ell + e12) + 2p m, or... Q2(e21 - 022) - 3P (Ol- c12) = m To simplify these equations the following constants are introduced:C2 for (21 + c22) or (e2 - 22) and Cfor 1(e 12) or (ii - 012) where C2 can be an intege except zero, while C can be any even integer divisible by 3, except zero. Condition 2.7.12 can be restated as (2.7,13a) Q2 C2 + p C3 = m and (2.7.13b) Q2 C2 + p(C3+2) mt For mf 1, Q2C2 must-be an odd number; i.eI the number of rotor bars must be odd. Introducing other values for m, similar conditions can be developed for low mode force waves caused by the interference of two rotor fields. The low mode force waves resulting from the interference of one stator and one rotor harmonic are the most frequent and dangerous ones from the viewpoint of noise production, and will have to be investigated. The

83 - condition for their occurrence is bound to the equation (2.7.14) mv p' ~+' or for 3-phase machines, (2.7.15) p(3 el2 + 1) + (3 e22Q2 = mv Introducing the same constants which led to equation 2.7.13, 2.7.15 can be split into two conditions (2.7.15a) p(C3+ 2) + C22Q2 = + m or (2.7.15b) p C + C22Q2 = + L For force waves of mode mv = 1 to occur the following equations have to be satisfied: p(C3 + 2) + 22Q2 = + 1 p 03 + C22Q2 = + 1, wherein the first terms on the left are even numbers. Force waves of the first mode can therefore only appear in machines with odd numbers of rotor bars* The conditions for mv = 2 are p(C + 2) c22Q2 = + 2 or p C3 + 22Q2= 2 Similar conditions can also be established for other low mode terms. The interaction of flux waves of slot harmonic order creates force waves belonging to this class and deserve careful consideration.

-84 The order of the stator slot harmonic is given in equation 2.1.14 6s (Cs Q1/P) + 1 or with the condition introduced on p. 36, (2.7.16) 6 = Cls Q1+ and the rotor slot harmonics by, (2.7.17) Ps = C2s Q2 + C1s Q1 + P It is worth while to recall that Cls and C2s may be any integer except zero. C1s and 2s' though both constants of the primary windingare by no means equal. The first order stator slot harmonic has Qi + P and -Q1 + p pole pairs counted on the whole armature periphery. The th order stator slot harmonic has Q1Q + P and - Q1+ p pole pairs, where f is any positive integer excluding zero. The slot harmonics of the rotor resulting from the fundamental rotor current can be calculated by assuming Cls 0 in equation 2.7.17. The first order slot harmoniec will have Q2 + P and -Q2 + P pole pairs counted on the whole armature, and those of _the 5 th order

will have Q + p and - 2 + P pole pairs also counted on the whole armature periphery. The first order slot harmonics produce the following forc waves of low modes; a) Positively rotating first order stator slot harmonic with positively rotating first order rotor slot harmonic, - = (Q1 + P) (Q2 + P) QI-Q2 b) Negatively rotating first order stator slot harmonic with positively rotating first order rotor slot harmonic,' - (-Q1 + p) + (Q2 + p) -Q1+Q2+2p c) Positively rotating first order stator slot harmonic with negatively rotating first order rotor slot harmonic, v = (Q1 + P) + (2 + P) Q1-Q2+2 - d) Negatively rotating first order stator slot harmonic with negatively rotating first order rotor slot harmonic, mV (-Q1+ P) - (Q2 + P) Q-+Q2 The higher order slot harmonics also combine to create force waves whose modes are however higher than those of the first order slot harmonics. They can be: clculated as:'v= Q + p- (Q2 + p) (Q - Q2)

- 86v = - Q1 + P 2 + P ((-Q1 Q2) + 2p v = Q1 + P - Q2 + p= ~(Q1 Q2) + 2p Table 2.7.1 shows the inequalities for IQ - Q21 to be observed for the purpose of avoiding low modes of force waves in normal induction motors. Table 2.7.1 Inequalities for Low Modes -V Q1 Q2 mode to be avoided must be unequal to 0o; (2p/:) "01 1; (2/) 2 2/; (2p ~ 2/) 3 3/3; (2p 3/i) The freuenc of the force waves is of utmost interest and can be calculated from equation 2.7.8. The frequency equation differs slightly for force waves created by fluxes rotating in the same direction from those created by fluxes rotating in opposite directions. The resultant frequencies of the former subtracts while frequencies of the latter add. (2.7.19) Af = al ~-a2 One of the frequencies resulting from the interaction of two stator waves represents constant sinusoidally distributed pressures, while the other is the 120 c.p.s. noise.

-87The frequencies of the force waves originating from the interaction of two rotor harmonics can again be determined by using equation 2.7.8. Assuming both waves to be caused by the fundamental stator MMF, the resultant frequency will be either 12+(pa+pb)(l-s)l or r(pa-pb)(l-s)]w. The frequencies of the force waves caused by the interference of one stator and one rotor harmonic are most important. In the following the harmonics are assumed to be the result of the respective fundamental currents. The frequency of the stator harmonic will be w, and that of the rotor harmonic a = C[l+(p-l)(1-s)]a. The resultant frequenP(')D. cies will therefore be of = [2+(p-l)(1-s )] ) and (2.7.20) =[(p-)(-) The frequencies of the force waves caused by the first slot harmonics deserve closer investigation. It is best to write down the equations of these harmonics and then calculate the frequency for each interaction. In the following, bss stands for stator slot harmonics and bsr for rotor slot harmonic. The + or - sign marks the direction of rotation. b ^(xitt) zos/in ~~^(, /p(l )t-(Q I)(x^/pT)x

- 88 - bsr(-)(xlt) sin l-(Q2/p)(l-s) lwt-(Ql)(xr/p)) The positively rotating first stator and rotor slot harmonic will create a force wave of frequency, (2.7.21) a(,+)= (Q2/p)(1-s) The negatively rotating first stator and positively rotating first rotor slot harmonic will create a force wave of frequency, (2.7.22) (_+) = 2+ (Q2/P)(1-s) ] The positively rotating first stator and negatively rotating first rotor slot harmonic will create a force wave of frequency, (2.7.23) (_) = [ 2-(Q2/p)(l-s) ] co The negatively rotating first stator and negatively rotating first rotor slot harmonic will have a force wave frequency of, (2.7.24) a__) = (Q2/p)(l-s) Similar formulae may be established for any harmonic combination. It is advisable to record the equation of the flux waves and determine the resultant frequency by carrying out the trigonometric pperatiot. 2.8 The Tangential Forces in the A.irRa

89 Just as the fundamental flux distributions create tangential forces, so do the higher harmonies. They are responsible for parasitic torques and also contribute in a small measure to the noise of induction motors. The tangential force on an elementary area can be calculated by Biot-Savart's law as drt = iLb. dA, or, for the case of tangential forces in the airgap of an induction motor, as, (2.8.1) dt = a(xl). b(xl) dA, In this expression a(x1) is the ordinate of the fictitious continuous current distribution of the rotor at point Xl. As the concept of a continuous current distribution is not frequently applied, the following definition may help to clarify it. Neglecting the mgnetomotive potential drop in the iron, the total MMF along the flux line can, according to Ampere's law of circuitation, be expressed as (2.8.2) H. ds f(xl,t) + f(xlt) =Z1r, where f(xlt) and f(xl,t) are the magnetomotive potential drops at the points of the airgap where the flux line crosses the airgap. I denotes the sum of the discrete currents enclosed by the flux line. Using further the theorem of the continuity of flux; that is,

- 90 2r JB d(xir/t) 0,'0 the familiar step-shaped MMF distribution along the airgap can be constructed. A In section 2.1, the Fourier series expansion of such MMF distributions has been given, as it appeared inconvenient to work with a discontinuous step wave. It is Just as unfavorable to work with discrete currents. To overcome this difficulty, the harmonic components of the MF distribution can be assumed to result from fictitious, continuous, sinusoidal current distributions on the armature, instead of discrete currents:il- slots. As the MMb distribution is the result of a summation of the discrete currents, the current distribution is necessarily the first derivative of the MMF distribution, or, (2.8.3) a(xt) df(x1,t)/dx Introducing for f(x1,t) equation 2.1.9, the eurrent distribution can be written, (2.8.4) a(xl,t) - 2m wl k I1 cos [^t-(,:-.x/=). (L/p^ and in a similar way the current distribution of a rotor harmonic is, (2.8.5) a p(xlt) = -, Q22.-cos ( l.+(p-6)(l-s)iotJ(-6/)). (3./2pv)

- 91Introducing into the above expression, (2.8.6) Ap =- 2 Q2I261/2P t equation 2.8.5 can be rewritten as, (2.8.5a) ap(xl,t) = Ai cos ( Cl+(p-)(l-s)aot- (PArxl/)) The expression for the force wave can now become, (2.8.7) [; (xt) = B Ap sin 1[wt-(6rxl/p)]..cos [1(p-6)(1-)/) The above equation may be resolved in the same fashion as equation 2.1.6. Two tangential f6rce waves of mode (< t p) and of frequency (o + a ) will be created. Comparing the resultant expression with 2.7.6, a striking similarity appears: For each radial force wave there is a tangential force wave of the same mode and frequency. The amplitude of this tangential force wave is simply related to the amplitude of the radial wave. By comparing 2.8.6 with 2.5.6a, a new equation may be formulated: Ap, = B g' p/poR Introducing this value into 2.8.7, the tangential force can now be expressed as, (2.8.8) rt(xlt) (1/2)(B6 B/o)g/)p' or Ft (x, t r) (x, t )(gR)p' newton/meter2

- 92These forces cause additional bending moments producing vibrations of the same mode as the radial forces and they will add vectorially to the deflections of the latter. The numerical value of (g'p/R) is of the order of /10th so that the tangential force becomes inconsequential and can be ignored. Eventual extensional deformations are also negligible because the extensional rigidity of the frames is large. There is very little probability of the occurrence of tangential vibrations in the stator. The frame may, however, transmit these vibratory forces to the substructure, and if any part of the foundations have resonant frequencies coinciding with the frequencies of the tangential forces, considerable vibrations could be created. 2.9 Influence of Magnetic Saturation, Skew and Airap Eccentricity At the start of our investigations, simplifying assumptions were made to bring the mathematical treatment of the conditions in the airgap within reasonable bounds. While it is very difficult to give an exact answer to the effect of non-ideal conditions encountered in practice, they still have to be dealt with. Amplitude changes may occur in some of the force waves if ideal conditions are realized, and in addition, new force waves may be created.

93Manetic saturation In the iron will havq two different effects depending on the stator winding connection. The magnetizing current in a wye-conneeted machine is sinusoidal because no triple harmonic currents can flow. Even at low saturation of the iron the flux distribution will cease to be sinusoidal as illustrated in fig. 2.9*1. B ~12 -:X. ___I__ ___ \0 -rr/2 1 r/2 Fig. 2.9.1 Field Shape in a Wye-Connected Stator The field shape is flat-topped and can be resolved into a Fourier series. The whole flux distribution r6tates synchronously and all components therefore have the same speed and direction. Thus, new rotating flux waves occur and the equation of the flux distribution may be written as, (2.9.1) b(xlt) = B1 sin[ot-(xir/A)1 - B3sin 3 tot- (x r/)]- B sin 51It-(=ir/)l-.. In a highly.saturated machine the amplitude of B3 may be 20 to 30 per cent of the amplitude of B1, and the value of B5

94 - may reach 15 per cent of B1. The conditions are somewhat different in a deltaconnected motor. The field distribution is here more or less sinusoidal, but the magnetizing currents now contain higher harmonic components with a third and fifth harmonic predominating as shown in fig. 2.9.2 I 0 - /g2 2 2 Fig. 2.9.2 The Magnetizing Current in a Phase of a Delta-Connected Machine The third harmonic components of the magnetizing currents in the three phases are in time phase, and as a consequence the resultant flux in the airgap will pulsate. In a 600 machine, the pole pitch of this alternating field will be (c/3). It can be conveniently resolved into two oppositely rotating fields: (2.9.2) b+) (xlt) (1/2) B3 sin 3It-(xl/) b( )(x,t) (1/2) B sin 3 [ot+ (x1/)

The fifth harmonic component of the magnetizing current will create a field distribution of fundamental period, but its frequency will be 5f. The fifth harmonic current is a negative sequence component of the magnetizing current and the flux field resulting from it will therefore rotate in the negative direction. (2.9.3) b(s) (xl,t) = Bs(5) C t - (i /r)] These saturation harmonies in the airgap will be considerably reduced by the damping effect of the rotor as described in section 2.3. The flux waves that result from saturation may interact with the other flux waves in the airgap, and the method for the calculation of the force waves will be the same as for other harmonics* There may also. be a secondary effect which will change the amplitude of the force waves in the airgap. The saturation of the tooth tips of the stator slot openings will effectively increase the slot openings and therefore increase the amplitude of the permeance waves. The rotor slot bridges may also become saturated, as described in section 2.6. The increase of the effective slot opening will also reduce the slot leakage and slightly increase the magnetizing reactance, which means an increased damping effect on the rotor currents. The Influence of Skewing. For the suppression of

- 96higher harmonic parasitic torques the rotor bars of small induction motors are often skewed. Skewing has two different effects on the flux distribution in the airgap. In the following discussion insulated rotor bars will be assumed. As shown in the next section by the calculation of the main noise-producing force waves, there is a large number of flux combinations which create force waves of the same mode and frequency that have to be vectorlally combined. Only a few of them have sufficiently large am — plituzes to produce any effect. The force waves created by the interaction of the two first slot harmonic force waves will have the largest amplitude. Others are caused by higher harmonic currents in the rotor. The component of the force waves resulting from the interaction of slot harmonic fluxes is not reduced by skewing. The force wave itself seems twisted to the left and to the right along the rotor axis. The magnitude of the force waves caused by a flux that results from a higher harmonic current is frequently reduced so that these components of the resultant force wave can be considered negligible. Before discussing these effects, the theory of skewing will be described. Fig. 2.9.3 shows the developed plan of a squirrelcage skewed winding. The voltage induced in the skewed bars

- 97 - -vs i Fig. 2.9.3 A developed skewed Squirrel Cage Rotor will be reduced compared with the voltages induced in straight bars, and the reaction of the seconary currents on the primary will also be different. Both:effects are best analyzed with the aid of a skew factor F5 by using the product of the winding factor k6. k, instead of the winding factor of the secondary alone, which is usually unity. With the rotor bars skewed by Zs,: th fundamental and higher harmonic voltages induced in the bal.will be reduced because the voltages of adjacent bar elements will be out of phase. The bar behaves as though it were a uniformly distributed winding as shown in fig. 2.9:.3. The ekew factor kI will therefore be of the same form as:the distribution factor of a continuous winding and can be expressed as, (2.9.4) k; = sinn [ (d^/2V) ]/(6Vs/2T) or

-98(2.9.4a) k6= sin (6 s1r/2p )/((d&r/2pT) The main and harmonic magnetizing reactance of the rotor are considerably reduced and by using the same method as for the straight bar it can be proved that equation 2.3.5 will be transformed to, (2.9.5) X2m6 (0.4 Q2fll/g, p)(k'6/6)2. 1 hms By contrast, the differential leakage will be considerably increased and equation 2.3.13, with skewing taken into account, will become (2.9.6) A2d,6 = (2,6/k) -1 The slot leakage and resistance of the bar are also increased in the ratio of the skewed bar length to the straight bar length. On calculating the rotor currents of a skewed squirrel cage, the values given in 2.9.5 and 2.9.6, have to be inserted into the equivalent circuit of fig. 2,3 for X26 and X2. With these new impedances the higher harmonic currents in the rotor bars will be considerably reduced, so that the flux waves created by them will become negligibly small. The second factor that reduces the overall effect -of slot harmonic force waves is caused by the spatial phase shift of the rotor slot harmonic flux resulting from the

- 99 - fundamental current. This will be illustrated in fig. 2.9.4. s. -....-, Fig. 2.9.4 Shift of the Force Waves created by Slot Harmonics In section 2.2, rotor MMF distribution has been related to the stator under the assumption thatat t 0, a point on the stator x1 = 0, coincides with point x2 = 0 on the rotor. On choosing x2 in the middle of the bar, traveling in one direction from the middle up, x2 will be to the left of xl, and traveling in the opposite direction along the bar, x2 will be on the right. Assuming insulated bars, there is no change in the phase andsmagnitude of the rotor current along the skewed bar, so that the MMF wave of the rotor will be shifted accordingly. This phenomenon can be taken care of by introducing in lieu of x2 in equa

1-00 - tion 2.5.7, x2-(, z/li), where z is a distance a the rotoi axis with =O at the middle of the bariland is the skew measured in meters. The equation of the flux density distrtbitien can now be expressed as, (2.9.7) b,(xl,tz) Bln ([l+(?-)(l-8)Ctl(xi ) + To offer a practical example, the forc:wave as a function of z is given for the negatively rota:tg first stator slot harmonic, and the positively rotating,1rtst rotor slot harmonic Introducing into this equation 8r [l2+(Q/p)(dr e )2+(Q -29. -'. 1_-:. - m= l(QiQ2)p'and _^rM= 2P~/Q1 and applying the trigonometric relation, cos(x+y) = os x cos y -sin x sin: the equation can be written as, F&1,t~z).axcos * C th(Q2+()z k) - f(xj, t,) -o( (XP - sj/T)co 2 (2.9-,8..

- 101 - In the above equation it is assumed that the squirrel cage.bar is skewed exactly by one primary slot pitch. This equation demonstrates that the force distribution which had been uniform along a generatric of the rotor becomes split into two parts: The first part represents a cosine-shaped force distribution along the length of the stator core with the period (li Q1)AQ2+1), and an amplitude at mid-point of the stater as illustrated in fig. 2.9.5a. This wave is in time phase with the skewed force wave. liVl/(Q2+1) 1^/(Q2+I'.. ~ —li- --. ~ i~- ~ -'l 0X, a b Fig. 2.9.5 Force llistribution along a Qeneratrix in Skewed Bar Rotors The second part represents a sine-shaped force distribution along the length of the stator core with the period Ii Q1/(Q2+I1) The origin of the sine wave is in the middle of the core as shown in fig. 2.9.5b. This wave is displaced

- 102by 900 along the armature periphery. As previously mentioned) insulated squirrel cage bars have been assumed. However, production machines do not have insulated bars and are known to have higher than calculated losses caused by currents flowing from bar to bar across the laminations. After investigating this effect, Rossmaier and Schuisky1 attributed the increased losses to the higher harmonic current components flowing in the iron from bar to bar (shown by the dotted line in fig. 2.9.3). At the end of his investigation, Schuisky regrets that he is unable to calculate the magnitude of the cross currents because no data were available for the impedance of the current path across the iron between two bars. Judging by the curves presented in Schuisky's paper, it would seem as though there were little change in the fundamental rotor current resulting from non-insulated bars, and that only the higher harmonic components of the rotor currents have changed current densities across the bars. Because most of the noise is caused by slot harmonics that result from the fundamental current, it appears that non-insulated bars would have little effect on the motor's acoustic behavior* The question of whether any vibrational axial 1 62; 63

forces are created by skewing still remains open. Bradford and Rhudy have shown that a constant non-vibrational force exists because of skewing, whose magnitude is given in, (2.9.9) F = (-rj).-tan (tlrr/+) where Pz is the axial force Vt is the tangential force, and R is the radius of the stator bore. An expression for the tangential force resulting from the higher harmonic fluxes and higher harmonic current layers was derived in section 8. This expression can be easily transformed to include the effect of skew. The tangential force distribution will again be split into a cosine and sine term in the fashion demonstrated for the radial forces in equation 2.9.8a. The resultant axial forces should contain the average value of Vt; that is, of the tangential force. The average value of the sine term will be zero, while the average value of the coslie term may yield a small remainder term depending on the ratio of (Q2 t )/Ql The resultant axial thrust on the rotor is obtained by integrating this remaihder term over the wholk bore. As there are integral numbers of tangential force1 16

- 104 - wave periods, this integral will be zero so that no vibratory axial forces will occur under the assumptions made. Influence of Airga. Eccentricity1 The airgap of medium induction motors may be a mere fraction of one per cent of the stator bore. Bearing clearances may amount to 10 per cent of the airgap, and the clearances of bearing mountings, to an equal amount. Airgap eccentricity is therefore likely to occur in induction motors and must be investigated. In the following analysis, a cylindrical stator and a balanced cylindrical rotor are assumed, but the -rotor axis is displaced from the stator center by a small distance, as shown in fig. 2*.96 below, Fig. 2.9.6 Airgap Eccentricity The flux lines are again assumed to be radial, and the magnetomotive difference in the iron, negligible. In the following analysis, the symbol g(xl) is the airgap at point x1, and ^1

10 - H(x1) is the magnetic field intensity in the airgap at point x1o Using Ampere's law of circuitation, the magnetic field intensity at point xl can be expressed as, (2.9.10) H(x1) =,H (0) go - / a(xtlt)dx: [/1g(xl) 0 where H(0) is determined by the condition 2r 211 f b (x1) dx = p H(x1)dx = 0 O 0 Introducing this condition into 2.9.10, H(0) ~can be e lressed, -- /2~ x 2;: 2 2 *9 11. ) H(O) /g (x) OdxJ [a(xtt)dx3 [1/ l/()dxlj) 0 - 0 0...Iz/g (o)3 Introducing equations 2.9.11 and 2.8.4 into; 2.9.10, and carrying out the integration, the field intensity at any point of the airgap is found by H(xlt)= [uA6g (xl)] -sin (wt)+sin [ot- (6rx/p )] +sin i(t)(2.912) 2: ~-[f^(x)lzsinlat-(x/PVrdxl i[ r d(xl]dx1 0'.0.. -... and the magnetic induction in the airgap as,

-106 - (2.9.13) b(x,t) = loM6'/g(xl)].sin [a)t-( (rxl/p)] - / 2I.;o 2w [C 0M/g(x1)]1K sin^ L t-(d x/pT l3ax.) / [/J gs(xL).dx] o Using the notation in fig. 2.9.6, the function g(x1) can be expressed, (2.9.14) g(x) = ra+8cos (xlw/p)-2cos 2(Xlx/p)-r2- e2 Because I: is much smaller than rb, the expression for. the airgap may be simplified into, (2.9.15) g(xl) = g + cos (xlT/pt) This value can be introduced in equation 2.9.13, and the result will, contain two terms of different character. The first term is, (2.9-16) b(l(x, t ) =(poM6 + cos (xl1r/pr]) sin [cot -(rx /ptg. while the second term representing pulsating fields which for'>l1 will become negligibly small and therefore ignored in the following. Since it is inconvenient to have the expression 2*9.15 in a denominator of a fraction, this difficulty can be overcome by developing the reciprocal of the variable airgap into the power series by simple division: l/1g+ /pcos (rx/p r) = (g ) (-[(A/g)os (X1/pr)] - (2.9.17) -[(E/g)2 cos2' (rX/p)]-... )

- 107 - This can be still further simplified as, (2,9.18) (l/g(x1)=(l/g)( a+al(cos (xl/pr)+a2 cos (2x/p+ +a cos ( V1X/Pt-)+'. ) Introducing this series into 2*.916, the resultant airgap induction of the 6th harmonic becomes (2.9.19) b(xl,t)(poMb'/g) 21osin t-(&xr/pr). a cos (vxl/pT) Again, this can be resolved into two fields: V (2.9,20) b6(xlt) = pOM6 //g 2 Z a sin it.-(v)(r/pt) +. + sHoY /g a] s in ot+ 6(Lv )(xTr/p) ] An inspection of 2.9.20 illustrates that an eccentric airgap will become responsible for harmonic fluxes of the order of (d'+ v) and (6'- v) pole pairs, counted along the circumference of the stator. Because v may have the value of one, (1) there may be fields in the airgap that differ only by one pole pair, counted along the circumference of the armature. Such an occurrence will originate the eccentric forces already discussed under the heading of "Onesided Pull". If the rotor shaft is not in the same line with the axis of the rotor cylinder, the variable airgap will also become a function of the rotor speed. The treatment

- 108 - of such a problem is parallel to that of the variable a-irgap just mentioned, and it is merely necessary to replace (rxl/pr) in equation 2.9.18, by (1TXi/p-ort, where 0r is the angular velocity of the rotor* 2.10 The Force Waves in the.xterimental Motor It is well to visualize first the large number of higher harmonic flux waves in the airgap with the aim of finding those flux combinations which create the greatest noise effect. All calculations will be based on data given in Appendix I. For the first part of the discussion it will suffice to know that the 4-pole motor has 60 slots and 54 rotor bars. TSble 2.10.1 lists the harmonic order of the airgap flux waves to the first slot harmonic based on equations 2.1.8 and 2.2.12. There are 44 airgap harmoAies wiin thin the range of the first slot harmonies, creating a vast number of force waves. As a first step it is best to investigate the first slot harmonics. Using equations 2*7.14 and 2.7.21 to 2.7.24, table 2.10.2 can be assembled. From this table it can be observed that the frequencies of the force waves caused by slot harmonics are well in the audible range and it can also be easily established that their amplitudes are of the same order of magnitude. But only the force wave re-sulting from the interference of the negatively rotating 29th stator harmonic, and the positively rotating 28th rotor

- 109Table 2.10.1 Airgap Harmonics of the Experimental Motor Stator Associated Rotor Harmonics - Harmonies c?.= pc p= 27+6 p=-27+6 1 -1 2 8 -26 -5:-5 22 -32 7 7 34 -20 -11 -II 16 -38 13 13 40 -14 -17 -17 10 -4 19 19 46 -8 -23 -23 4 -50 25' 25 52 -2 -29 29 — 2 -56 31 31 58 4 Table 2.10,2 Force Waves resulting from First Order Slot Harmonies Stator Harmonies -29 +31 h B^ m^ a: 6 mV - 10 0,=10 o~:'o o -26 f 27(1-s)f ff 2-27(1-s)f M m =2 ma= 6 +28 f, = 2+27(1-s)f f 27(1-s)f

- 110 - harmonic will acquire significance. In fact the force wave will be responsible for most of the pure tone noise, just because it is of a low order number. An inspection of the calculations and results in sections 3.5 and 3.6 would prove that the second mode deflection amplitudes as a response to the second mode 100 lb force wave is of the order of 10 microinches for frequencies between 1,500 and 1,900 c.p.s. A 6th mode force wave of the same amplitude would cause deflections of the order of 1/10 of a microinch in the same frequency range. As the sensitivity of the ear is logarithmic, the sound pressures caused by the vibrations of the motor in the 6th mode would be well below the threshold of hearing. There is, however, an exception to this statement. If the mth mode force wave frequency approaches the resonant- frequency of the stator structure in the mth mode, the deflection amplitudes may become considerable, and the noise output troublesome. As can be seen from table 3.5.11, part 2, the lowest resonant frequency in the 6th mode is at 4,240 c.p.s., while the 6th mode force wave frequency is 1,475 c.p.s. The same reasoning, to a still larger extent, applies to the 10th mode force wave. At a slip of 0.015, the frequency of the second mode force wave, produced by the negative 29th harmonic of stator flux and the positive 28th harmonic of the rotor flux, will in accordance with equation 2.7.24, be given by

- 111ff = 2 + 27(1-0.015)60 = 1,716 c.p.s. The motor operated at lower frequencies with the same magnetic and electric loading will slightly increase its slip. Operating from a 50 c.p.s. supply, the main noise frequency will be at 1,419 c.p.s. Operating from a 40 c.p s. supply, the main noise frequency occurs at 1,121 c.p.s. kt no-load, operating from a 60 c.p.s. supply, the motor's main noise frequency will be at 1,735 c.pos. A closer inspection of table 2.10.1 reveals the interesting fact that other harmonic flux combinations, as listed in table 2.10.3, will result in second mde force waves of the same frequency as the second mode force wave of the slot harmonics. Table 2.10.3 Second Mode Force Waves in the Experimental Motor of Frequency 2 + 27(1-s)f Interfering Flux Rotor Harmonic Remarks Waves induced by Stator Rtor Stator Harmonic -29 28 1 significant -23 22 -5 -17 16 -11 -11 10 -17 -5 4 -23 1 -2 -29 significant A detailed calculation will show that only the first and the last force waves have large enough amplitudes to be

- 112 - considered. The force waves involving flux combinations of two belt harmonics will generally be negligibly small, as the winding distribution and coil shortening reduce their amplitudes, a fact easily observed from the winding factors. In table 2.10*4,flux combinations creating second mode force waves of the same frequency as the two other slot harmonic combinations, i*.e 1,475 c*.p.s. are given. Table 2.10.4 Second Mode Force Waves in the Experimental Motor of Frequency 27(1-s) f Stator Rotor Rotor Harmonic Harmonic Harmonic Induced by Remarks Stator Harmonic 31 -32 -5th significant 7 -8 19 19 -20 7 13 -14 13 1 -2 25 significant Here again only two flux combinations aguire significanie On closer investigation one can find that in many medium induction motors slot harmonic combinations yielding a second mode force wave, and the flux combination consisting of the main flux and a second rotor harmonic, are the important As# However there may be many exceptions to this statements In the following the amplitudes of the second mode

-113force wave at 1,716 c.p.s. and 1,475 c.p.s. will be calculated. The former will be referred to as the main noise force wave, the latter as the side noise force wave. In all calculations 40 amp. load current or no-load condition are assumed. The terminal voltage is assumed to be the zero time axis. Calculation of the Main No'i Producing Foree Wave. The 29th stator harmonic flux is the resultant of the slot and permeance harmonies. As explained in section 2.6, these will add vectorially, as q is odd and the harmonies rotate in the negative direction. The amplitude of the 29th slot harmonic calculated by equation 2.5.4a, B29 (4w.lO 1 /67xlO 4) 3li (160)(0.874)(23.2)/(58r) 29,s = 0.1415 weber/meter2 The amplitude of the main airgap flux is B1 0.725 weber/meter2 The value of 1 has been calculated in section 3.4, as 01 = 02. Hence, by equation 2.6.7 the permeance harmonic flux wave will be B29p = B1. 11 (0.725)(0.2) = 0.1450 weber/meter The phase angle between slot and permeance harmonic is 68.20,so that B29 = 0.1415+(0.1450/-68.2~), and the equation of the 29th flux wave with the terminal voltage as time axis, B29(xl,t) = 0.235 sin [ wt-(61.5r/180)+29(xl/rt)]

- 114The fundamental rotor current can be ealeulated by elementary methods and is found to be 322 amp. The amplitude of the 28th flux of the rotor will, by 2.5.6, be B28 ~ = (4v.10 7/6.7x104 )(54) 2(322/44r)(1/28) = 0.1315 weber/meter2 Applying a 20 per cent correction as discussed in section 6, B28 (1.2)(0.1315) 0.1580 weber/meter2 and the equation of the flux wave B28(x,t) = 0.1580 sin 28 [ot-(ll.2T/180o-(28rxl/r)] The formula for the amplitude of the resulting force wave can be established from equations 2.7.1 and 2.7.6, as (2.10.1) (1 /2)(107/4vi)B1.B2 newton/meter2 In the numerical computations of the forced response of the experimental motor, the amplitude of the force waves will have to be inserted in pound/radian. Recalling that 1 newton/mter2 equals 1.45 x 10'4 lb/square inch, the desired force wave amplitude will be (2.10.2) II (V2)(107/4T)(1.45xl4)1R B1B2 lb/rad. Here 1s is the stator length, and R the stator bore radius, both in inches. B1 and B2 are still in weber/meter2. With the dimensions of the motor, the amplitude of the foree wave will be (2.10.3) II = (1/2)(107/4Tr)(145xO1'4)(5.75)(4.5)B1B2 = 1,495 B1B2 lb/radlan

- 115 - The amplitude of the force wave caused by the interference of the 29th stator and 28th rotor harmonic flux waves will be nI = 1,495(0.235)(O0.58) 55.4 lb/ radian and the equation of the force wave In = 55.4 cos 290t -(14/lo0)-(-x1/t) To calculate the second.significant component of the force wave, the equivalent circuit of the 29th rotor cage, as shown in figure 2.3.2, has to be established. The magnetizing reactance branch of the equivalent circuit is calculated by equation 2.3.5 X2m29 = (0.4)54(60) (0.13)(0.18)x106/ 6.7x10-4(2)(292)] = 1,98 x 10' Ohm. The differential leakage reactance of the circuit is evaluated by equations 2.3.16, 2.3.17 and 2.3.18, will be X2d2 =.([(58/54)/si (5T/54)] - 1)1.98 x 10 4*34 x 103 Ohm. The slot leakage of a bar at high frequencies is 1.ol x 104 ohm, so that the total leakage reactance is 4.45 x 103 ohm. The armature current, according to equation 3*2*22 is 12 29 = 322(1.98 x 105)/(4.45 x10-3 + 1.98 x 105) = 1.43 amp* The influence of the 29th permeance harmonic flux wave can be taken into account by increasing the current by the

116 - ratio of the resultant 29th harmonic flux to the 29th slot harmonic flux I2,29 = 1.43 (0.2350/0.1415) = 2.36 amp. The amplitude of the second harmonic flux due to this current will be, B2 = (4tr.-107/6.7 x 10 4)(54/2)(0.9)(2) x 2.36 - 0*027 weber/meter, and the equation of this flux wave is B2(x 1t) = 0.027 sint 28 [owt-(61.6r/180)+(2xl/t)]) The equation of the fundamental flux wave can be given as B1 (0.725) [t - (94.7w/180) -(rxl/)] weber/meter2 The magnitude of the force wave can be calculated to be i2 = (0.027)(0.725) x (1495) = 292 lb/radian, and its equation is n2(xlt) = 29.2 cos [29 l-(16.7r/180)+(rx1/T)lb/rad. The two force waves of 1,716 c.p.s. are 2.70 apart, and the resultant wave will be ni 155.4 + 29.2/-2.701 =84*5 lb/radian The Lo-Load Noise. At no-load, the fundamental armature current will become negligible and with it the 28th slot harmonic flux wave. Consequently the first component of the spreviously calculated force wave can be entirely ignored. The second component of the force wave caused by

- 117 - the main flux, and the second harmonic rotor flux willI although slightly reduced, be still existent. The 29th slot harmonic's flux density has to be reduced in the ratio of the no-load current to the full-load current, and will be B29, 5 (0.1415)(6.8/23.2) = 0.0415 weber/meter2 The 29th permeance harmonic flux is unchanged, but the two flux components are now in phase and add directly so that the total 29th flux will be B2 (0.1450 + 0.0415) = o.1865 weber/meter2 The amplitude of the second harmonic flux wave will be reduced, and is B2 = 0.027(0.1865/0.2350) - 0.214 weber/meter2 Finally the amplitude of the force wave responsible for the no-load noise will be n = (00214)(0*725)(1495) 23.1 lb/radian Calculation of the Force Wave creating 11475_. C#._. Sibde Noi e This second ipde force wave has the same frequency as the 6th mode force wave of the slot harmonics. From a practical point of view it is unimportant, as it will increase the total noise of the motor only by 1/20 of a decibel. Its calculation is being carried out partly as a further cheek of the theory, and partly to demonstrate the combination of a flux and slot harmonic which subtract 4;

- 118vector a lly. There is again a string of harmonic flux combinations to create this noise. As mentioned before only those involving flux waves of slot harmonic order, or the main flux wave will become significant. The first component of the force wave is the one due to the 31t stator flux and the 32d rotor flux, associated with the 5th stator harmonic. The. second component of the force wave is created by the main wave and the second rotor harmonic induced by the 2th stator harmonic. St The 31t harmonic airgap flux is again the resultant of the respective slot and permeance harmonic fluxes. The amplitude of the 31s sTIt harmonic flux will 1b~ given by 2. 24a B31 0.1325 weber/meter and the permeance harmonic B31 = 0.1450 weber/meter2 The resultant will be B31 10.1325 - (0*1450)f-68.201= 0.1570 weber/meter2 To qalculate the 32nd rotor harmonic flux wave it is again necessary to establish the equivalent circuit of the 5th harmonic rotor cage. By equation 2.3.5 2ms, = 6.68 x 10-4 ohm The differential leakage reactance calculated by using equations 2.3.16, 2.3.17, and 2.3.18, can be written

- 119X2d,5 = 11 x 10o ohm. Adding the slot leakage of 1.01 x 104 ohm, the total leakage reactance will be 2.11 x 104 ohm. The armature current caused by the 5th harmonic flux, referred to the primary, will be calculated by equation 2.*319 1,5 = [(2)(3) 160 (0.1)/54J(23.2) 41.2 amp., and referred to the secondary, according to 2.3.22 I25 (41.2)(6.68/8.79) = 31.6 amp. st Finally the flux density of the 31t harmonic, calculated by equation 2.5.6 is B2 0.022 weber/meter2 The amplitude of the force wave caused by the 31t stator and the 32nd rotor slot harmonic flux waves will be, according to equation 2.10.3 (2.10.3) II = (1495)(0.1570) (0.022) 5.15 lb/radian For the calculation of the second significant component of the force wave, the equivalent circuit of the 25th rotor cage has to be established. Using again equation 2.3.5, it can be calculated to be X2m2 = 2.67 x 10- ohm The differential leakage reactance calculated by 2.3.16, 2.3.17, and 2.3.18, is X2d,25 = 3.77 x 10 ohm

- 120Adding the slot leakage of 1.01 x 10'4 ohmsi; the total leakage reactance of the 25th cage will be t.8V1 x 10 3 ohm. Using equation 2.3.19, the prinmary urrent referred to the secondary 25th cage, will be 1125 =62*9 amp., and the secondary current by equation 2.3.22 is 12,25 0.42 amp. Using equation 2.5.6, the flux density of the second harmonic rotor flux can be now calculated as B2 = 0005 weber/meter2 and the magnitude of the force wave caused by the fundamental and the second harmonic rotor flux wave will be 1192 1495 (0.005) (0.72) = 5.42 lb/radian The two components of the force wave creating the side noise are 128.4~ apart, and the resultant force wave will therefore be IIs 4.6 lb/radian S Table 2.10*5 Summary of Calculated Force Waves Motor Operation Descript. Frequency Mode Amplitude of Noise c.p.s. my lb/radian 40 amp.,440 volt main 1,716 2 84.5 -no-load,440 volt main 1,735 2 23~1 40 amp.,440 volt side 1,475 2 4.6

1I~4 THE DYNAMIC RSP-ONSE TO ELECTROMAGNETICC FORCE WAVES The force waves -whose modes, frequencies and amplitudes have been calculated In the previous chapter - set the motor into continuous vibration: As the next step in the solution of the noise problem, the natural behavior and the forced response of the stator will be investigated. The study of the natural behavior of the stator is of threefold importance. Designers are in need of information on the resonant frequencies of the stator so that they can choose a winding combination for the mismatch of electromagnetic driving-force frequencies with the natural frequencies of the stator. Secondly, the problem of steady state vibrations can be easily solved once the eigenvalues and eigenvectors of the system are determined. And finally# the resonant frequencies can without difficulty be experimentally established and thus constitute a convenient check for the elastic equations of the system. The topic of this chapter pertains to the field of engineering mechanics, and methods used in that area of science will have to be applied extensively. Furthermore, the system under consideration is a continuous one, )the solution of which leads to systems of partial differential equations. Setting up equations for a system as complex as - 121 -

- 122 - the stator of an induction motor is not without its difficulty. recourse will therefore be taken to determine them from Lagrange's equations of motion. The energy expressions will be established with the aid of the principles formulated by Rayleigh and Ritz. Because few electrical engineers are familiar with these methods, it seems advisable to review the theorems which form the foundation of the method. The stator structures of induction motors may differ greatly in design, but they all consist of cylindrical shells, rings, radial and tangential ribs, and supports. General expressions for the elastic energy of these struetural parts will be developed first. To make the problem suseeptible to numerical solution it is essential to define the structural relation of the elements which determine the dynamic behavior of the system as a whole. Moreover, it is necessary to make assumptions based upon empirical observations, such as the effect of damping and related matters. The method itself is strictly applicable to small vibratio-s about an equilibrium position, and assumptions will have to be made pertinent to all calculations based on Lagrange's equation of motion. Finally the methods will be applied in general to a series of induction motors and the numerical calculations will be carried out for the experimental motor described in Appendix I.

- 123 -..31 LaRrange's Equation of iMotion1 and the Energy ethod of Raeih Ri As an introduction to the basic concepts of ongineering mechanics it is desirable to start with a system of particles. In the following discussion mass will be assumed constant. According to Newton's second law, the force F applied to a single particle can be expressed as (3.1.o 1) = m(d/dt) In generalizing this equation to a system of particles, one must distinguish between external and Internal forces, the latter being known as constraints. Both have to be considered, and the equation of motion can be written,.(e) (3.1.2) ji + = mi(dti/dt) mi(d i/dt2 ) where Fi(e) stands for external force, is the constraint between the jth and ith particles, and is the radius vector to the path of the particle. This restriction takes care of the fact that the individual particles are not free to move in all directions/ 1;o.29;3916

- 124 - A system consisting of n particles not conneeted can be defined by 3n quantities and denoted as a system of 3n degrees of freedom. If there are k constraints between these particles the degrees of freedom of the system will be reduced to 3n-k. The motion of the system can be fully described by using 3n-k equations of motion and k conditions of constraint. This approach leads to lengthy calculations and it is for that reason of great advantage to follow Lagrange in choosing 3n-k independent variables, so that the 3n orthogonal coordinates become functions of these new 3n-k variables called "generalized coordinates". The relation between the original and generalized coordinates can be expressed by a system of equations: (3-1.3) ri= (q lq2......, q3n-t) i=l,2,*....n The principle of virtual displacement will be most helpful when considering equilibrium conditions of a complicated system. According to this principle, the work done by all the forces on a small displacement consistent with the constraint of the system must equal zero for systems in equilibrium. (3.1.4.) 2I_.e.g0 i=102l n In the above equation, Fi represents the sum of all external and constraining forces. Restricting all further considerations to systems for which the virtual

-12 - work of constraints is zero, the same equation can simply be written as, (3-1.5) 21 =. 0 i=l,2,.. n where Fe) denotes external forces only. In the following discussion the superscript will be omitted. Equation 3.1.5 may be extended readily to dynamic conditions, using d'Alembert's principle. Equation 3.1.1 can be written as (3-1.6) F - m(d/dt) = 0 and the principle of virtual work stated thus: (3.1.7) -[fj mi(df/dt)1dbri =0 i=1,2,.....n To express the above principle in generalized coordinates it is only necessary to indroduce for i' (3.1.8) i = ( ^i/j) (b/dt), i=1,2,....n and for the virtual displacement, (3.1.9) bri =2 (bqd ) bq The first term of 3.1.7 can now be expressed as,..... iJ where QJ is the Jth component of the so-called generalized

- 126 - force corresponding to coordinate qj, that is, (31-.11) Q = Li - ($ri/Jqj) i = 1,2...n It must, however, be remembered that generalized forces by no means need to have the dimensions of force. It is solely necessary that the product of the generalized force and the corresponding generalized coordinate represent work. The second term of 3.1.7 may be transformid by the use of the complete differential that follows: d drl ari d, ^i i d a (i) dt dt qj at- 3-qj dt dt 6q but d (85) - ai and from 3.1.8 Vi = i d q i)4 i) Using these relations, the second member of 3.1.7 can be expressed as (3.1.12) Z ((d/dt) (/q)) I (l/2)mi2 ]' - i - /,q Z (1/2)(nvj2)) bq The expression behind the summation sign denotes the kinetic energy. Recalling that the virtual displacements are.independent of each other, it can be written that, (3.1.13) (d/dt)(,/434) - (YT/q) = Q)

127 Such an equation may be established for each generalized coordinate of the system; and the number of equations will equal the number of degrees of freedom. The generalized forces in 3.1.13 may be constant or variable forces, or functions of time or position, or they may even fulfill other conditions. If the external forces are conservative - that is, the work done by these forces equals the change of the potential energy of the system -then, Qj =-bV/qj, where V = V(qljq2P..qn) and equation 3.1.13 may be restated as, (3.1.14) (d/dt) (-T/q0)- (aTaq.j) + (VQV/6q) 0 If forces Q1 other than those represented by a potential act on the system, equation. 3.113 can be completely written as, (3.1.15) (d/dt)(bT/ad) - (&T/q)j) + (V/6qj) = These equations are called "Lagrange's equations of motions. Theory of Small Osciltions. Lagrange's equations of motion supply the most convenient tool for treating the problem of small oscillations about an equilibrium position needed for the development of this thesis. The potential energy of a conservative system of particles depends on the instantaneous configuration of the system, so that the potential energy can be expressed

-128 - as a function of the generalized coordinates only: v = V(q) = 1,2,......n Under such a condition the potential energy can be expanded by Taylor's theorem about the equilibrium position i J (3. 1.) V(q) _ V1(qj) + E(V1/3q~)Oq~ + + (1/2) Z (b2Vj/c)qj3q) q<iqj+ +'i,j = 1t2.......n,The term linear in qi will vanish in consequence of the condition that (3Vi/q1i)o must be equal to zero for the system to be in equilibrium. Neglecting terms: of higher than quadratic order as negligibly small, and neglecting the constant term as inconsequentialt the potenitial energy can be expressed as a function of the quadratic term only. Thus,. (3.1.17) V = 2 1 (1/2)(ki ) q q i j In this equation, kij stands for the second derivative of the expression behind the double summation sign in equation 3*1.16. This can be considered a constant because it depends only on the equilibrium values of the qis. For a system to be stable the potential energy must be a minimum in equilibrium position. Since V is zero at equilibrium, it follows that V is of a positive, definite quad

- 29 ratic form. The partial differentiation behind the double summation sign can be interchanged. Consequently, ki kji, i.e., V is symmetric. The kinetic energy is a function of the generalized velocities, =q = dq /dt and by the same reasoning as was used in the case of the potential energy it can be proved that the kinetic energy may also be expressed as a definite quadratic formt (3.1.18) T = (1/2)(mij)qi qj i j The kinetic energy is by definition always positive and the order of indices of m can be interchanged. Consequently, the kinetic energy is a positive, symmetric, quadratic form. Because no dynamic coupling exists in the system treated in this thesis, equation 3.1.18 can be simplified to (3.1.18a) T (1/2) 2 m(4i)2 Introducing equation 3.l.18a and 3.1.17 into tie generalized equation of motion, and keeping in mind that the kinetic energy is independent of the generalized coordinates, the equation of motion can be written (3.1.19) umij qJ + E ki: q3 O

- 130Substituting this value for T, Lagrangets equation gives (3.1o20) mi qi + 27 kJ q 0 j iJ The last equation represents a system of n differential equations of the same type as the one encountered in the theory of electric circuits. A solution will therefore be attempted by setting, qi = Ai sin (wt + T) (3.1*21) q'i =-a2AI sin (wt + j) On substituting the values of qi and q' into the system of equations represented by 3.1.20, n simultaneous equations will result. Lest the system have only a trivial solution, the determinant of the coefficients of the system must vanish: kllml, kl2 ~ *.. kln * k (3.1.22) =. 0 ~ 2 l1n k2n'. kn-mn The determinant will have n roots in 2 which are denoted as characteristic values or eigenvalues. The developed determinant is referred to as characteristic equation. The roots of these equations stand for the saUare of the natural frequency fr multiplied by 2wr and the frequency can easily be calculated, as'

-131(3.1.23) fr (/2r) For that reason the characteristic equation is referred to as the frequency equation, and the natural frequencies of the systems are functions of eigenvalues. On introducing the eigenvalues into equation 3.1.22, solutions for the value of Ai can be obtained. Because there are n eigenvalues, n sets of such solutions, each representing a principal mode of vibration, will result. The Ai quantities are usually referred to as eigenvectors of the system. There will be altogether n eigenvectors, each having n components. The solution of qi can be obtained by iuperposing all complementary solutions as follows: - (r): (3.-124) qi = Z At sin (ot + ) r=i (r) The set of eigenvectors of A determine the principi& rth mode of oscillation which is purely harmonic, i.e., all particles move with the same frequency and phase. Generally, there are n such modes of oscillation for a system of n particles. These modes are orthogonal, which means that v (r) (s) ZA A = 0 i3l (r) (r) (r) The magnitudes of A1, A2,...Ai... are not

132 independent and are determined only so far as their ratio is concerned. It is of advantage to choose the value so that, L (r) 2 _(A! ) - 1 which condition is called the "normalizing" condition, and to express the eigenvectors as, (r) (r)/ (r) 2 (3.1.25) y1 = A /L(A1 ) called the "normalized" eigenvectors. The relation between the normalized eigenvectors and the eigenvectors is given by, (3.1.26) A (r) r) /^ (r) 2 where cr is a constant equal to [ 1/(Ai ) The forced res onse of the system - that is, the dynamic deflections caused by vibratory forces - is of utmost interest to the problem discussed in this thesis. The equation.of motion can in this case be expressed by, (3.1.27) (d/dt(6T/Aqi) + (v/qi) = Q sin (t + ) where Qi is the amplitude of the ith component of the generalized force as defined by equation 3.1.1i. Introducing again the values for the kinetic and potential energy given in 3.1.17 and 3.1.18, the equation of motion can be expressed, (3.1.28) miqi f qr qki = Qi sin (mt+), i l...n There are n equations of the type 3.1.28 that can be estab

- 133lishted, and together they constitute a system of n nonhomogeneous equations. The solution can be found by classic methods which, however, become very lengthy for a large number of equations. The eigenvalues and eigenvectors represent the solutions to the homogeneous equations and may be used with ease to find the roots of the system of equation 3*1.28, by applying the principle of superposition and the following reasoning: Suppose the forced vibration deflections in the r mode can be expressed as n (r) (3.1.29) qi =C i cr cos (t + Y) Note that this equation I similar to 3.124, with the difference that the former contains the forcing frequency a, instead of wr as in the latter. Substituting 3.1.29 into 3.1.28, there results, (r) n2 (r) (3.1.30) -m yi cr) + r ktj Y = Qi The second term to the left of the equation sign in 3.1.30 represents the potential energy of the system, a quantity independent of time. It can be proved that the maximum potential energy is of the same magnitude as the maximum kinetic energy. Consequently, equation 3.1.30 can be simply expressed,

- 134 (r) 2 2 (3.1.31) mi Y)Cr (r - ) Qi In the following it will be courenient to represent the generalized force in the form of a series. Thus, Qi r) (3.1.32) i = br br mi yr r=l Taking into account the orthogonality properties of the eigenvectors it can be shown that, (3*1.33) Q1J yr br (mj fy(r))2 j=l J=l and on solving equation 3.1.33 for br n n insrtin Yj / j- (r)12 (3-1*34) b Q m [yj Next, inserting this value of br into 3.1.32, an expression for Qi is obtained. On substituting this expansion into 3*.131, the deflection of 3*.129 can be given as, n n n (3.1.35) Q = 2 (EQ (r) (r) 1 )7 ifyr ) J( r=l - J=l On setting g=O in the above equation, the term in the large round brackets can be written as, n n (3.1.36) q(r) _= C yQyr)y(r)/O2 mjmyr]2 is is rd t as t s ic dlein in t this is referred to as the static deflection in the rth

- 35 normal mode, and the dynamic deflection can thus be written (3.1.37) qi (r) Cq (a 2)) r=l 1 The term behind the square bracket in 3.1.37 is referred to as the resonance factor. For the sake of simplicity, the Lagrangian equations of motionused to find the natural frequencies and forced vibrations,have been developed here for a system of discretefinite number of particles. The problem treated in this thesis involves a continuous system such as can be handled as a limiting case of an infinite number of discrete 1 2 particles, as suggested by H. Goldsteinl and W. Kaplan. It will, as a rule, be difficult to express the elastic constraints between particles which in most cases can be formulated with the aid of recursion formulae, as difference equations. On carrying out the transition from finite to infinite numbers of particles, equations of motion will result containing, in addition to the derivative with respect to time, derivatives in respect to the position coordinates of all points. As a matter of fact, the position coordinates will take on the role of the subscripts i and j in the previous equations. The equation of motion of a continuous 1 29 2 38

- 136system will therefore become a partial differential equation. The summations of the discrete particle systems will be replaced in the limit by integrations. The questions of convergence occurs at such a transitional limiting process. Courant investigated this convergence with results that can be summarized as follows: If a partial difference equation converges to a partial differential equation, the solution of the difference equations converges to the solution of the differential equations in the case of equations of elliptical type. Partial differential equations of a vibrating solid body are of this type. A further aid for the solution of the vibration problem is the method of Rayleigh2, as extended by Rit3. On applying this method, assumptions are made for the deflection curve of the vibrating body which satisfy given boundary conditions. The eigenvalues and eigenvectors are, in this case, established from the consideration of the energy in the system. Truly to represent a continuous system such a method should assume a deflection curve expressed In an expanded form as the sum of an infinite set of 120 2 67 3 68

-137functions having undetermined coefficients. In general, each term of the expansion must satisfy the geometrical boundary values of the problem. As the next step the kinetic and elastic energies of the system are calculated, and introduced into Lagrange's equations of motion. The undetermined coefficients are used as generalizeid coordinateso This leads to a system of linear equations. As mentioned before, these equations have nonvanishing solutions only if the determinant of the coefficients vanishes. The vanishing of this determinant yields the eigenvalues and eigenvectors of the system. Should the set of functions used be a complete set, capable of representing the deflections at every point, the solution is in principle exact. Since, however, the deflection expansion has an infinite number of terms, the exact determinant yielding the eigenvalues is infinite. In engineering problems it becomes a practical necessity to consider finite rather than infinite determinants by assuming as negligible the higher order of the members representing the deflection curve. The result will thus be approximate only. 3.2 Energies in Components of Induction Motors EnerI y in Cylindrical Shells. The outer frame and the inner core can be considered to be cylindrical shells. The elastic

-138energy of bending and extensional deformation in the cylintical shelll will be expressed as a function of the radial displacement u, and the tangential displacement w. The effect of rotary inertia and the additional defi:etions caused by shear will be neglected. Consider an element of a circular are of a mean radius r0 in a rectangular system of coordinateo x and z as illustrated in fig, 3.2.1 hX1:Xl x wl\ DefZormatiox off a Circular Arc: ^ ^ L< ~\ ~ -.: I~ o M+dMf ^dN \ Fig. 3<2l. Deformation of a Circular Are An element of a circular arc of lengths is shown before deformation betwe'n points A and B.: After deformation, the element will have the position A:and B't and a length bs. Let A coincide with the ortgiTef the x.z, system, and B have coordinates 5x0 and o.z Suppose A' to have coordinates u and w, and B',~ and ~ 1.71.:.7 -.

-139. The position of B with respect to the circular arc shall be fixed by the coordinates ult wl. The displaemeBnt B to B' will be defined by u and w in reference to the x:z, sysis, the angle between the tagents at A and A can be expressed by the sine or cosine of the angle. The limiting value of these for 6s- O. Is given by, sin. llm (-%+ U)/bs or o b — (3.2.1) cos V = lim 0( - ow)/b] bos nO Let denote the unit elongation of the" middle fibre. Then, bs (l+. )b- From the inspection of fig 3t. 2.1, the following relations can be derived: (3.2.2a) = ~U cos (bs' o)"w1 sin (&81ro), (3.2.2b) U= sin (6s/r)+weos (CJ/o) + CSO Becaurse the angles in question are small they may be approximated by, sin (So/r-) = bS / r0 where lim (6x/r0)= o; lim (1bzbso) 1; lli- cos, s r,)=l

-140lim C(-ul+u)/bsI = 0; lim [(WL-wO)/bso = dw/ds It follows that, (3.2.3a) sin y = Cl/(l+E)].-du/ds) + (w/ro)] (3.2.3b) cos p- cl/(+le.C1+(dw/ds) + (u/a) Setting sin 0p cos p = 1, and assuming K<<I, the following equations may be written: (3.2.4) = (-du/ds)+(w/r )=(l/ro). [(-du/de)+ ] (3.2,5) e (U/ro)+(dw/ds) (l/ro). [u+(dw/de)l. The bending strain is, (3.2.6) eb [x/(ro + x)]. [d/d1e 5 (Xc/ro)(dy/de), and the combined^fibre strain is, (3.2.7) [ e= b =(i/ro ) +f (aO+ w/6) + (xdy/d) The normal force and the moment can now be expressed by displacement u and w. Recalling that, x = x (l-6)/E and x. dAI- o, the normal force may be written (A) (3.2.8) N =/6 ^dA E dAIi'A + (E'/rExdy/iE)AxdA= (E'A/ro) [(dw/de)+u] where E' = E/(1-62), and the moment expressed as, (3.2.9) M = -y/d xdA= -E'jxcdA-(E'/rO)(dl/ )J/x2 A (-EI /ro)(dV/de): [ +E'I/r2J [(d2u/d42) - (dw/de)]

- 141 - The symbols in these equations are: E' = E/(1-62) A = area of cross section I = moment of inertia in respect of the neutral axis at (ro, 8) 1-62 = Poisson's ratio In medium induction motors the cylindrical shell may be assumed as inextensional, as suggested by Rayleigh. This condition can be easily taken care of by setting 8=0. As a consequence (3.2.10) u -dw/de and (3.2.4a) p =: (l/r )[2w/dG2 )+wJ hence (3.2.9a) M (EtI/r )2 d u )+u The extensional strain energy caused by the normal force can be calculated by the formula dV (1/2) A 6f dx, where dx = rodg, and for a shell of constant cross section by (3.2.11) Vt (1/2E'a) N2 rod9 On substituting N from equation 302.8, the expression for the extensional energy in the ring can be written as, ~o5

- 1422w 2r (3.2.12) t= (E'A/2ro)- [(dw/de)+ti de o The elastic energy of bending caused by the moment can be calculated by using a formulae, (3.2.13) dVb [M2/2E'I] dx where dx is again ro de Substituting the value from equation 3.2.9, the elastic energy is, 2w (3.2.14) vb = I'ri/2r,3] d2/de2)+(w/deO de - 0 For inextensional deformation the extensional energy will l'bviously be zero, and the strain energy caused by bending deformation can besimplified to, 2w (3.2.15) Vb (B'I/2r 3) Jf d2u/dG2)+u d 0 Introducing, in agreement with Timoshenko1, the flexural rigidity of the vibrating shell, D = E'I the potential energy of bending of an inextensional shell can be written, 2w (3.2-15a) Vb = (D/2ro3)/ [(d2u/de2) + u]2 de ~1 7 o'1 o

-143The kinetic Energy of brat Shelels. Suppose a uniform vibrating mass distribution of p slugs per radian. The kinetic energy for an elementary volume can then be expressed 2 2 (3.2.16) dT = (1/2) p (V)2 de and the kinetic energy of the whole ring T (p/2) / (,)2 de 0 Substituting 2 = u2 + w2,the kinetic energy can be expressed 2T (3.2.17) T = (p2) (2 + w2) do' o.~. "l. - ~r' ~'h The Elastic Energ of the Radial Ribs. The radial ribs interconnecting the two cylindrical shells are, during vibrations, subjected to extensional and bending deformations. The tangential ribs add but little to the elastic energy of the system which will be numerically demonstrated in Appendix IV. At standstill, the rib is at A and B, as illustrated in fig. 3.2.2. Let the rib move radially during: vibration, so that its new position will be given by A1 and B1. Denoting the distances AA1 by ua, and BB' by ub the unit elongation can be expressed by, (3-2.18) ~= (ua - ub)/lr, where Ir is the length of the rib. The extensional energy during deformation can be calculated according to equation 3.2.11 as,

- 144 - (3219) Vt = (1/2 Er Ar) N2 dx, O where N = Er Ar(ua ub)lr Assuming this normal force is uniform over the whole length of the rib, the extensional energy is, (3.2.19a) Vt (1/2 1r) Er Ar (ua - Ub)2..-/?b AFg. 3,2..2 Deformation of a radial Rib For the calculation of the potential energy of bending, the rib is assumed in a condition of equilibrium, and under the action of the bending moments and shears acting at the points where the rib is built into the two cyliindrical shells. The end points A and B have moved in tangential directions, by the distances wa and wb respectively from the unrestrained position, and received rotations ~ and Yb respectively.

14 - The conditions of equilibrium require that the shear at any point x be, (3.2.20) Px P Pb and that the bending moment M at any point x be, (3.2.21) 8= Mb + (r-x) Pb The positive direction of x is from A to B. For sake of brevity, M will be written for Mb,and P for Pb. The bending energy of the rib can be xpressed as in equation 3.2.13, and for the whole rib as, (3.2.22) Vb =-/2Dr) j dx, where r ErIr/(l-62) Introducing M from 3.2*21, the expression for the bending energy can be written, ft:.. V (1/2r J (M2+2PMY-2PMx+P -2PMXx+P2)dx Carrying out thee ntegration, this can be evaluated to, (3.2,232 2 (3.2.23) Vb (1/2Dr) [+ PM1r +(P23;/3)].1 To express the strain energy as a function of the deflections and rotations, Castigliano'sl theorem can be conveniently applied: 3.2.24) v/aPb -wb Wa - 1rU (1/23Dr) lr+(2/3)4 170

-146 and (3.2.25) )Vb/cl - W a ( /2Dr)(2M + Pr ) The values of M and P can now be calculated byssolving 3.2.24 and 3.*.25 as simultaneous equations. To facilitate the solution equation 3.2.21 is written as, (3.2.26) Vb - (I + (Pl2)2 +(Pl)l2/12); d lr/2Dr Irom equations 3.2.24 and 3.2.25, it follows that, (3.2.24a) M + (2 (2/3)P1r (l/ -a W)/l ) and (3.2.25a) + (1/2)Plr = (l2ae)(b a) Subtracting equation 3.2.25a from 3.2.24a g vesi: (3.2.27) (l6)Plr = (11/2e) ([2( )/r: ( Yb)) Next,substituting 3.2.25a and 3.2.27 into 3.2.: 6,t^ the elastic energy can be expresseda.. (3..2.28) Vb 1/4(,a) 1 a r. + 12[(wb- Wa)/rl12++ gb?) Finally reverting to the original meaning of'Ci eiquation 3.2.28 can be expanded to,: (3.2.29) Vb 6(I/1)(Wb - Wa?6(Dr/1l)(1b; Wa)(Ya + )+ + 2(/i)(r:(-: + a.)

- 147Elastic Energy of the SuTorts can be derived in the same way as that of the rib. Because there is generally sufficieit play in the bolts which fasten the motor to the base plate, extensional energies will be disregarded. The notation is similar to those of the previous section and is clarified. in f ig. 3.2.3 3: \P^~-^ ~'1 It Fig. 332.3 Deformation of the Support The strain energy of bending is given by 3.2.22 as, b (i/2 D) Mx2 dx where (3.2.30) MxX= M + P (l -x) and (3.2.31) Vb = [l/2D] [1 + MP ls+(1/3)P2 i] Again applying Castigliano's theorem, the defleetion can once more be evaluated as, (3.2.32) u = (VJaP) = (1s/2D) [M ls+(2/3) I I, and the rotation,

-148(3.2.33) = (VbA/M) (ls/Ds)(M+(l/2)P ls) Solving 3.2.32 and 3.2.33 as simultaneous equations, the values of P and M can be expressed by terms of u and y. Substituting the results of the simultaneous equations and bending energy of the supports into equation 3.231 gives, (3.2.34) Vb D is/l. (6 u2/2) - (6u /l) + 21 3.3 Assumptions and Notations The theoretical elements and energy expressions developed in section 2 will be applied in section 4 in a general manner to a line of induction motors so that Lagrange's equations for the dynamic behavior of the motors may be written. The motor in Appendix I1: does in fact belong to this line of motors. Because the number of parameters is large and new concepts are being introduced, it seems advisable to elaborate on the notations and assumptions here prior to a detailed treatment in section 4. A typical cross section of the line of motors under discussion is drawn in fig. 3.3.1 on page 1t0. The cast iron frame constituting the outer shell of the motor will be named the outer ring, an expression customarily used by designers. The following symbols will be employed in connection with the outer ring: R is the radius to the centroid of a radial cross-section of the ring;

149 - 10 is the length of the outer ring; I is an area moment of inertia of the crosssectional area about an axis through it centroid parallel to the axis of the cylinders A0 is the area of a radial cross sectionThe subscript 0 will generally be used in connection with the outer ring. The stator core will be referred to asi the inner ring and the following notation applied: is the mean diameter of the inner shell. The surface described by the circle of radius r will, in accordance with Rayleigh, be named the median surface;ts 1i is the length of the inner ring; i s an area moment of inertia of the croassectional area about an axis through Xit centroid parallel to the axis of the cylinder; i AI is the radial cross section of the inner ring; I is used as subscript for parameters belonging to the inner ring. The short plates connecting the outer and the inner ring will be called ribs, and the following notation applied to the radial ribs: iL is the effective radial length of the rib;

.. -~.'....~. -;:.-.'~'.. ^1^ 1* Bil^ ^ I': "::l. l ib,:..... -.:: * 0^3~3^: simplifie r; ^c:- 0: ^Bottom Rafia; to *, -...-,';igo.. Simplified Cross Section of Experimental..~to.~

Ar is the area of the rib in a plane cut by a concentric cylinder; Ir is an area moment of inertia of the sectional area about an axis through its centroid parallel to the axis of the cylinder. The subscript r will be used in connection with the ribs. In Appendix IVN it will be shown numerically that the tangential ribs have little effect on he behavior'of the system, and for the sake of clarity the necessary notation will be provided there. The plates on both sides of the outer eylindical shell will be denoted as supports, and the follloing symbols applied: is is the length of the supports; Is is the effective moment of inertia of the supports. In order to bring the treatment into reasonable boundss the following assumptions based on empirical evidence will be made: 1) all displacements are extremely small compared with the dimensions of the system; 2) all vibrations take place in planes perpendicular to the axis of the shell; 3) the stator core made up of steel laminations will be considered a homogeneous body;

152 - 4) the cross-section determining the rigidities of the stator extends radially from the bottom of a stator slot to the outside diameter of the stator punchings. This is equivalent to the statement that the stator teeth and windings do not add to the rigidity of the stator core, but add to the mass. 5) the outer ring will be considered a complete cylinder. The local effects of plates and free edges of ventilating openings will be ignored in this treatment since they can be treated as secondary effects. 6) the mass of the ribs, supports, and other integral components of the outer frame's casting will be supposed to be uniformly distributed when calculating the kinetic energy of the outer ring; 7) the end shields will be ignored. It has been experimentally determined that the effect of the end shield on the mode shapes and frequencies of the main system is negligible. This is because the frequencies are so high and the amplitudes so small that the coupling is small in comparison to the effective masses of the two systems; 8) the motor will be analyzed as if it were a free body in space. Measurements show that the natural frequencies and modes are almost the same whether the motor is bolted to the base or supported on rubber blocks. The experiments of R. Plunkettlhave proved that a normal 56

-153base has negligible effective mass compared to that of the motor at frequencies above 500 c.p.s. The important question of damping causes difficulties in the dynamic treatment of mechanical systems. In its nature, damping may be inherent elastic damping or the result of friction between constructional parts* It will be necessary to make different assumptions for the two cases. A study of information and experimental evidence indicates that elastic damping can be considered entirely negligible if the system be not at, or very near to, resonanee. Fig. 3.3.2t shows the logarithmic decrement curve for the experimental motor at 1,140 c.p.s. resonance. From this it may be seen that the energy dissipated by damping in a cycle is about 3*7 per cent of the maximum kinetic energy. Using the analogy of an electric circuit, it can be stated that the Q of the system is large and that the damping forces may be considered negligible compared with inertia forces outside a narrow band around a resonant frequency. The decrement curve would indicate that elastic damping becomes important in a band width of approximately 2 per cent around resonance. The system of the stator is composite. The stator core is pressed into the frame and it therefore appears that non-elastic displacements between the rings may occur o

^=I l: ] l...:S t -1-1-t-; +-!2 00 _ di -0- - < i —l -! -j- -t- -— 1 —1 -— ~~ ~-'!=~ -:-~-:-i:'-0t': -"- =t "'!: —-.......... - ~i.......tr-!-' —-_ i... —............. [ ~ ~~ ~..... i' t I i | ~ -. +3 i- [ —r, i ol- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~, cl' - - i |'t —: I i I Jt I! ~ _ i;: | ~~~~~~~~~~~~~~~~~~~ 4- L fi | / t | F t t 1: 1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.............. i...............-'~'~ _.... -l-tl P L -- — t - — t - i ~;l~; 7- --- -- -~-., w0- 1-; 1 1 0!l~lil * {::lifl-l5::lrillit7 0 1 ]- l l lttt-!':'t I —- - -7 — ---- t -I —!-1 -1-1 — 1 —1-F —- 1 -1 1X~t 5! l —LL-iLXL LL[ —--— l —— t —l —-; XI L<L _,; v_,d, i~~n <h sI t J ~ v n -

- 155It has been impossible to excite the predicted modes involving opposite radial motion of the rings at the ribs. It would appear that the damping involved in such motion is so large that the modes are heavily overdamped and thus suppressed. These modes can then be ignored in practical analysis. Tte system of coordinates and the notation for the displacement of the inner and outer ring are demonstrated in fig. 3.3*3. U O -L t -— ~~- -- -- 1I Fig. 33.33 System of Coordinates for the Vibrating Stator The origin O of a polar system of coordinates is chosen at the center of the concentric rings. The argument of the radius vector will be supposed to be positive, counter-clockwise, with - = 0, at the center line of the top radial rib, and 8 = w, at the center line of the bottom rib. The radial displacement U of the outer ring and

- 56 - the radial displacement u of the inner ring will be assumed positive, directed away from the origin. The positive direction of the tangential displacement of the outer ring W, and of the inner ring w, will be supposed to coincide with the positive direction of the argument. Displacements will be measured from the unstrained position. Several concepts which need definition will be used in the, following analysis. The term mode or order of a mode will imply the number of complete periods of a.sinusoidally distributed quantity along the periphery of the ring. The quantity may be displacement, velocity, aceeleration, or force. The single uniform ring is symmetrical about any diameter. For this reason there will be two vibrations of the same frequency associated with each mode as defined above. The designation of one is arbitrary and the origin of coordinates of the other will be displaced from it by a quarter wave length. In an actual manufactured ring there are certain assymetries which fix the origin of the modes and also separate the frequencies. In the motor frame a vertical diameter is ain axis of symmetry which fixes these modes. Since the motor is made up of two such rings, there will be four frequencies associated with each such mode two each connected with each ring, modified by the coupling

157 - between them. If the sinusoidal distribution of the displacement has its amplitude at the top radial rib, as shown in fig. 3.50.1, the mode will be defined as a direct or a cosine mode. Should the zero amplitude of the sine wave displacement of the mode coincide with the center of the top rib as illustrated in fig. 3,5.1i, the mode will be defied to vibrate in a quadrature or sine mode. If two points on the same radial line, each belonging to one ring, move both outward at the same instant} the mode of vibration will be referred to as an in-tact mode. Should, however, the point of the outer ring move outward, while the point on the same radial line of the inner ring moves inward, the vibration mode will be referred to as a counter-tact mode.: 3.4 The Equations of Motion of a Vibrating Motor Stator. On the basis of the actual solution of the dynamic equations of the system it was found that the consideration of extensional energies of the outer and inner rings change the dynamic behavior of the lower modes of the medium induction motor by a negligible amount. However, the extensional energies cannot be ignored in either small or medium 2-pole motors or for the higher modes of medium sized motors. In the following, the solution will be established by supposing the ring to be inextensible. The corresponding solution for the extensional case will be provided in Appendix II. The radial displacements of the outer ring can,

- 18 - in the most general case, be represented by a Fourier series of the form (3.4.1) U = I (Am cos m + Bm sin me) m Correspondingly, the radial displacement of the inner ring may be represented by the expression (3.4.2) u =J (am cos me + bm sin me) m The values of m in expansions 3.4.1, and 3.4.2, must be chosen as 2,3,.. This will also apply to all of the subsequent expansions of this section. But as already mentioned in section 1, it is impracticable to work with infinite determinants, and it is therefore supposed that six terms in the Fourier expansion will represent the functions with a sufficient approximation for practical purposes. The condition of deformations without extension can be satisfied by requiring that the tangential and radial displacements be related by the equation (3.2.10) u =(-8w/8e) The tangential displacement of the outer ring will therefore be expressed by the series (3.4.3) W =7(l/m) (Bm cos m - Am sin mG) For the tangential displacement of the inner ring, the relationship is, (3.4.4) w =T(l/m) (bm cos me - am sin me) ^i~~r CHI -m

- 159The coefficients Am Bm, am,bm are time dependent functlons, and each is of the sinusoidal form. They can be expressed as Am(t) = Am cos t The amplitudes Am, Bm, am, bm will be the generalized coordinates of the system considered. The Kinetic Energ of the Vibrating Rings can be calculated by combining expansion 3.4.I to 3.4.4, with equation 3.2.17, and by using the solution of the following integrals 2w 27r 2w sin me sin n dO; 0 for man 0 f cos2 me d= s m sinn me de = 0 o 0 The kinetic energy of the outer ring is found to be (3-4.*) T - (Or p12)jZEm2 l)/m21.(A + B ) The kinetic energy of the inner ring is (3.4.6) T (r (/2)&2ml)m^ (a)+ b) Defining for convenience, (r p12) m2+l)/m2 - Gm the kinetic energies can concisely be written as

160 (3.4*5a) T = Gm (A2 + B) (3.4.6a) TI gm (am + ) The Potential Energy of Bending of the Vibrating R.njs may be calculated by using equation 3.2.15 which is rewriten for convenience, - -.~' 2rr -....'..~:.. ~2-.' 2 (3.2.15) Vb (D/2 r3). [(d2u/d@)-dj 2 d Using the expansion 3.4.1 and the integral relation as in the case of the kinetic energy, the potential energy of bending of the outer ring can be evilaed as (3.4.7) Vbo = (1DI2R3)E(l-m2)2. (Am2 + ) (3.47) F ar the inner ring, the potential energy is (3.4.8) 1 (1DI2rb)L^am. (a+ bm) For convenience, the following constants are introduced: for (rDI2R3)(lm2)2 = for (rD /2r3)(1-m2) fm Substituting these constants in equations 3.4*7 and 3.4.8, the bending energy of the outer ring can be expressed as (3*4.7a) VbO = 2Fm(Am2 Bm2 ) and the bending energy of the inner ring is (3.4.8a) VbI = m(a+ bm) The Bending Energy of the Supports will be evaluated by sub

- 161 - stituting the Fourier expansions 3.4.1 into equations 3.2.33. With the coordinate system as chosen in section 3, the rotation y can'be evaluated from the relation (3.4.9) = - (i)()uA/3e) The supports are at = (r/2) and 0 = (-i/2). The appropriate values of U and y at these points have to be established and substituted in equation 32.33. It follows from equation 3.4.1, subject to the proper conditions on 9, that, Ul(Ir/2) = -A2 -B3 +A4 +B5 +A6 +B7 Ue(-ir/2) A2 + B3 +A4 -aB -A6 -B7(3.4.10) Rg(w/2) 2B2 + 3A3 + 4B4 -5A5 -6B6 +7A7Re=(-r/2)- 2B2 -3A3 +4B4 + 5A5 6B6 -77A7-.. On evaluating the bending energy of both supports one finds no product-term containing a generalized coordinate of odd subscript multiplied by a generalized coordinate of even subscript. This same fact, it will be found later, will be repeated in the energy expression of the radial rib, and also holds true for both the kinetic and elastic energies of the rings. This is a very fortunate situation indeed, as it indicates that no coupling exists between the vibration modes of odd and of even order. This will make it possible to write the determinants leading to the frequency equations for even and for odd modes separately, which means that the order of these determinants will be

- 162 - reduced by one half, without impairing the approximation attempted. In the following therefore, all energy expressions will be written separately for even and for odd modes. The potential energy of bending of both supports is given by equation 3.4*11, for even modes, and by equation 3.4*12 for odd modes. These equations follow: For the even modes, V [G2D/ll 46 - 2A4 + 2A26 2A4A + + L2D I1RJ L2A2B2 4A2B4 +6A2B6 -2A4B2 +4A4B4 -6A4B6 + (3.4.11) +2A6B2 4A6B4 +6A6B]+ 4Ds/1R2 4B22 +16b2 +36B6 -16B2B4 +242B6 -48B4B6 and for odd modes, V 2D/ + B B -B3BS + 2B3B7 - 2B5B7 + + D2D5/lA~ [3A3B3 + 5AB3 -7A7B3 + 3A3B5 - A5B5 + (3.4*12) + 7AB5 - 3A3B7 + 5B - 7A7B7 ] + + 4D/1IR2t [9A1 +25A5 +49A - 30Q3A5 + 42A3A - 70A The extensional energy of the radial ritll be calculated by adapting equation 3.2.19a for the stator, by writing a = U, and ub = u. The energy of the top rib is then (3.4.13) Vt = (1/2 1,),Ar E UA o - Uo- 2 and the energy of the bottom rib is

- 163Vt - (1/2 lr) ErAr UT -,o Now there is introduced into this equation the expansions for U and u. Also the energies for both ribs are added. Since all terms containing products of coefficients of one even and one odd subscript vanish, the extensional energy of the rib can be written separately for the even modes and for the odd modes. For the even modes: vt -= arrJ/l [(A2-a2)+(A4-a4)+ A6a6) 2 [SrA~/lJ ~ +[A2 2 2 2 2 2 - [ErA/lE] 1A +A2a + a + a +- 2A2a2 2A4 - 26a6 + 2A2A4 + 2A2A6 + 2A A6 + (3 4,*14) +2a2a4 + 2a2a6 - 2a4a6 - 2a2A - 22a4 - 2a6 - 2A4a6 and for the odd modes -2A2 lt g ErAit s s r A3-a3)+(br-at) (Ach tepara [ErAr/lJ 2 +a3 +A2 +a2 +A +a -2Aa 2Aa [A3 3 5 5 7 7 3a 3 5A5~ (3.4.15) -2A7a7 +2AA5 -2A3A7 +2A +a +2aa + +2a5a7 -2a A5 -2a37 2a7 2A35 2A3 -2Aa71 The bending energy of the ribs consists of three terms, each eontaining different constants. As the expressions are vlery long it seems reasonable to treat each term separately.

- 164 - Introduce the expansion for W and w into the first term of equation 3.2.29. It can be evaluated for even modes to yield Vbrl (6D,/1) (W-W) 0 + ^ (W-iTI (3.4.16)2' (12D/l )(1/2) (B2 ) (1/4) )-(!/ 6 -b This expression is expanded, and yields for the even modes, the form: (3.4.16a) +(1/36)bB +(1/4)B2B4+(1/4)B6+(1/12)B4B + (1/ 4)bb4+ (1/6 )b26+ ( 1/26 )bb6- (1/1 ) 2b Vbr (12Dl3) 16a(1/93)B-(2+/)b 3B3+(1/9)b (1/2)B - + (1/49)bb+ (2/16)B3B+ (2/21)B3B-+ (1/12(continued on next page) The corresponding expressions for the odd modes -,is Vbrl /13) (12/3D (B3-b3)+(1/5)(B5-b5)-(./7) (B7-b7) (3*4o17) which may be expanded to the form Vbrl= (12D1 ) (1/9 )B5-(2- B + (1/9)b +(1 /25)B2 - (continued on next p2gs )

l 165 - (Equation 3.4.17a continued from previous page) +( 12D/13) (2/35)B5B7+ (2/15)b3bf (2/21)b3b7 + +(2/35)b5b7-(2/15)b3B" (2/21)b3B - -(2/35)b5B7-(2/15)b5sB3-/21 )b.7B3 - -(2/35)b7B5 The second term of 3.2.29 for both ribs can be caleulated as -6. /, +(1/3 )B2B6-(1/3 )B2b+2BB -2b-4-B4b+ -... - * +(2/3)B4B6-(2/3 )B4b6+ B3B62 + +(3/2 )B6B4(3/2)B6b B66 3 + +r~3R *2Dr/l~I *f- -bb.B2. (1/2: )b2B - (/ )b2.* + (1/3)b2B6-( 1/3 )b2b6+2B2b4-2b2-b4+B b4+ + (2/3)B 6b,-(2/3 )b 4 +3 B26 -3b b + (3/2)B6- (B6`B3/2 )B6b4-6BB6 For odd modes, the second term of 3.2.29 will be

-166Vbr2 = [12D r/I C(1/3)(B3b3)+(1/5)(B b5)+(1/7)B-b7 3. (34*19) 3((3B3/R)+(b3/r)]+5 C(B5/R)+(be/r) + (3.4.19) +7 E(B7/)-R (.' ) +7 which can be expanded to V 12Dr/1R. [ +B3+B2-B3b3+ (3/5)B3B5- (3/)B3b5 + (3/7)B3B7- 3/7)B37+ (/3 )B3B (/3 b3 -B5b ( 5/7 )B B7-( 5/7)Bb7+ (7/3 )B 3B (3.4. 19a) -( 7/3)b3B7 (7/5)(B5B7-(7/5)b7-B7b+ + 12Dr/12D.i Jb-b3 b5 b l+b b b+b(3/)bb3b (3/53)b b + (3/7)b3B7-'(3/7)b3b7+ (5<(5 )B 5/3 )b3b5 +Bb+( 5/7 )b5B7-( 5/7)b5b7+ (7/3)B3b7 -(7/3 )b3b7- (7/5)Bb7-(7/5)b57+B7b, The third term of 3.2.29 for even modes can be: expressed as Vbr3= E4D/lr V(1/R2)(2B2+4B4+6B6) 2 (1/Rr)(2B2+4B4+ (3.4.20) +6B6)(2b2+44-b) + (l/r2)(2b2+4b4+6b6 )2 which is expanded' to, Vbr3 [ C4Dr/1t21 [43&+16B 3,6B+16B2B4+24B2B6+48B4B6 ] + E+ {4D r/lRr3 [4B2b2+ 8B4b2+ 12B6b2+8B2b4+ l6b4B4+24B6b4+ -(3.4.20a) + 12B2b6+24B4b6+:36B6b6 ] + + [4D/lr2J 14b2+16b236b2+ 16b2b4+24b2b6+48b4b6 rR CO b +8B6 b~$'(r

-16 - The third term for the odd modes will be Vbr3 [4D/r/l (/R2 ) (3B3+5B5+7B7)2 (3.4.21) +(l/Rr)(3B3+5B5+7B7)(3b3+ 5b 7b7)+ +(l/r2)(3b3+5b+7b7)23 which can be expanded to br3 = 4D/lrR2J C9B+25B49B73oB3B5+42B3B7+7OB5B7 + + [4/lrRl] [9B3b3+15B5<b21B7b3+13b5+2Bb5B5 + (3.4.21a) +3 5'37b5+21B3b7+35B 749B77 + + C4D/l rr2i 9b.+2 52449b+30b3b+42b3b7+'7057b7b The above expressions for the kinetic and elastic energies should now be introduced into the equation of motion 3.1.20. Twelve equations of motions result for the even modes, and another twelve equations result for the odd modes, The coefficients of these equations will constitute the determinants, similar to3*.1.22, from which the frequency equations are obtained. To write out all 144 members of each determinant in general terms would require a great effort, many unnecessary pager and would serve no immediate purpose. Instead numerical values of the terms are used. As a representative example, Lagrange's equation of motion with respect to the generalized coordinate A2 is given below

168(d/dt XbT/Ai2 )+ (OV/A2) C-2G2+ 2F2- 24Ds13)+ (2E A/l ]A + (24Ds/l1R2 )B2+ [- ( 2ErAr/r)] a2 + E(24D5/1)+ (ErAr/Ir)l A4+ ( 3.4.22) +:-48Ds/ls B4+ ( -2ErA/lr )a + [24Ds/1 )+ (2ErA/1? )A6 + (72D8 /IR)B6+(-2pA1,/- ) a The numerical values of the constants may be, easily obtained and used for the coefficients for the undetermined equations. This work is carried out in section 5i, by using the kinetic and potential energy expressions derived in detail in this section. 3..5 Calculation of the Natural Behavior of the Experimnatal:.. Motor Both the theory and the assumptions In sections 1, 2 and 4, will be applied to the experimental iotor in order to find the eigenvalues (resonant frequencies), and eigenvectors of the system. The data of this stator are given in Appendix I, where the respective areas, moments of inertia, and mass per radian are also calculated. The following tables list the needed coefficients in accordance with these assumptions..

169Table 3.5.1 Inertia Constants of the Outer and Inner Ring Gm= (m2*l)Trpo/2m2 gm (m2+1)7rI/2m2 m 2 4 6 3 5 7 Gmxl02 7.690 6.537 6.324 6.830 6.399 6.278 gmxlO2 9.713 8.256 7.986 8.633 8.081 7.929 Table 3.5.2 Elastic Constants of the Outer and Inner Ring F= D(1l-m2)2/2R3; m= rD (1-m2)2/2r3 m 2 4 6 3 5 7 FmxlO-6 0.302 7.548 410095 2.147 19.323 154.585 fmxl1O6 1.122 28.061 152o777 7.982 71.836 287.346 Table 3.5.3 Elastic Constants of the Radial Rib (ERA/l) 12Dr/13 12D/I 2R 12Dr/12r 46.o000x1 10.465x10o 0.989x10 1.092x10i 4D/lrR2 4D/1rRr 4D/lrr2 4.200x104 4.600x104 5. 000x104

170 - Table 3 5.4 Elastic Constants of the Supports 12D (52D /1R 2 R2. 11x1o4. 80x1o4 2. 027x104 In tables'3.55 and 3.5.6, the determinants yielding the characteristic equations have been assembled. All terms except the coefficients of c2 should be multiplied by 106, though these multiplications are not recorded on the above tables for the sake of brevity. Furthermore it has been necessary to write the determinants in two sections. Part I contains columns 1-6, and Part II those of 7-12. The task of finding the eigenvalues and eigenvectors was carried out on the Electronic Data Processing Machine. The programing on the computer required: 1) the coefficient in front of the frequency term to be unity, 2) the sum of the squares of all elements of the determinants to be less than one, and 3) the determinants to be symmetric to the main diagonal. All elements of the rth row and rth column were divided by the square root of the coefficient of the rth frequency term. This procedure assures the retention of the symmetry of the original determinants, and compliance with condition 1. Tle second condition was satisfied by dividing all elements by 1010.

- HTI - In the following it will be necessary to distinguish between the determinants as given in tables 3.5.-5 and 3.5.6, and the determinants resulting from the transformations. The former will be referred to as the original determinant, while the latter will be denoted as the transformed determinant. The transformed determinants are recorded in tables 3.5.7 (even modes) and 3.5.8 (odd modes). In these X has been written for 2 3100 The data in determinants 3.5.7 and 3.5.8 have been programmed for the Electronic Data Processing lachine. The resultant eigenvalues and eigenvectors are recorded in table 3.5.9 for the even, and in table 3.5.10 for the odd modeso The method used for these calculations is explained in Appendix III. For the solution of the vibration problem it is essential to evaluate the eigenvalues and eigenvectors of the original system. The procedure for this evaluation is explained below.' To find the eigenvalues of the original system it is only required to keep in mind that X has been substituted for 2.10 10. The eigenvalues of the transformed system rmnltiplied by 010 will yield the eigenvalues of the original system. The required transformation of the eigenvectors can be obtained by the following reasoning (on page 184)

Table 3.55 Part I Original Determinant of the Even Modes, Column 1-6 1 2 3 4 5 6 A2 B2 a2 b2 A4 B4 1 A2 92.706- 2 0,112 -92.000 91.900 -0.223 W-0. 1538o 2 B2 0.*112 8.1952 -5, 012 -0.112 5.21 -0*1538W 3 a^ -92.000 94.245- 2 -0. 1943o -92 000 4 b -5.012 5.5?574 4 -5.012~-0.19432 -3.792 A 91.900 -0.112 -92.000 107.19- 01223 -o0 1307 6 B4 -0.223 5205 -3.794 0.223 19.917- -0. 1307o 7 a4 -92.000 92.000 -92.000 8 bq -0685 0.420 -0.715 9 A6 92.102 0.112 -92.000 91.000 -0.223 10 B6 0.335 592 -3*680 -10.335 3.393 11 a6 -92.000 92.000 -92,000 12 b6 1.572 -1.090 0.846

Table 3.5., Part II Original Determinant of the Even Modes, Columns 7-12 7 8 9 10 11 12 a4 b4 A6 B6 a6 b6 2 -92.000 92.102 0.335 -92.000 2 B2 -0.685 0.112 5.92 1 572 3 a2 92.000 -92.000 92.000 4 b 0.42 -3.68 -1.09 5 A4 -92.000 91.900 -0.335 -92.00 6 B4 -0.715 -0.223 3.393 -0.846 u 4 aa 148.122- -92.00 92.00 -0. 16 510a 8 4 56.32- 2 -0.896 0.101 -0.16 5;1 9 A6 -92.00 174.29- 2 0.335 -92.00 -0.1265w. 10 B6 -0.896 0.335 88.21-2 0.624 -0.1265 11 a6 92.00 -92.00 397.52-:::a6~ 3979'f~~ ~~-0.1597' 12 b6 0.101 0.624 3632 L -0.1597u

Table 3.5.6, Part I Original Determinant of the Odd Modes, Columns 1-6 1 2 3 4 5 6 A3 B3 b3 A5 B5 A3 96.691- 2 -0.167 -92.000 91.337 0.167 -0.1367a 2 B3 0.167 9.1962-2 -1.953 0.279 4.119 -0. 13670 3 a3 -92.00 1071964- 2 -92.000 -0.1727a4 b3 -1.953 16.706-2 1.673 -0.1727" 5 A5 91.337 0.279 -92.000 131.750- -0.279 -0.128w 6 B5 0.167 4.119 -1.673 -0.279 42.959 2 B5 -0.128:, 7 a5 -92.000 92.000 -92.000 8 b5 0.547 -0.335 -0.014 9 A7 92.927 -0.391 -92.00 90.45 0.391 10" B -0.167 5.004 -2.197 0.279 4.602 11 a7 -92.00 92.00 -92.00 12 b7 1.768 -0.615 1.240

Table 3*5.6, Part II Original Determinant or the Odd Modes' Columns 7-12 7 8 9 10 11 12 a5 b k B^ a^ b $5 b5 A7 B7 87A7 I A -92.000 92.927 -0.1674 -92.000 3 2 B3 0.547 -0.391 5.004 1.768 3 a3 92.000 -92.000 92.000 4 b-0.335 -2.197'0.615 5 A5 -92.000 90.450 0.279 -92.000 6 B -0.014 0.391 4.602 1.24 7 a5 235.67- -92.000 92.000 -0-1616e 2 8 b5 143.98- -0.124 02 o -0,6l 616w 9 A7 -92.000 248.753- 2 -0.391 - 0 -0. 12 56 10 B7 -0.124 -0.391 159.82- 2 0.97 a-0. 1256w 11 a, 92.00 -92.00 666.69a/7 -.209 30.586& 12 b 0.62 0.97 576.133

Table 3.5.7, Part I Transformed Determinant of the Even Modes, Columns 1-6 1 2 3 4 5 6 A2 B2 &2 b2 A4 B4 1 Ag2 06027700-.00007260 -.05323000 00000000.06480000,00015740 2 B2.00007260.00532800-.00000000 -.00290000 -.00008180.00367000 3 a2 -.05323000.00000000.04851700-.00000000 -.05773000 00000000 4.00000000 -.00290000.00000000.00286700-.000000000 -.00238100 2. ~ ~ ~ ~ ~ ~ ~ - 5 A4.06480000 -.00008180 -.05773000.00000000.08198700-.00017000 6 B4 -.00015740.00367000.00000000 -.00238100.00017000.01523000- -h 7 a, -.05774000. 00000000 05137900.00000000 -.06261900.00000000 8 B.00000000 -.00042990.00000000.00023460.00000000 -.00048670 9 A6.06603700.00008000 -.05893000.00000000.07147000 -.00017360 10B.0 00 *00424500.00000000 -.00235700 -. 00026000.00263900 a36 -.0^85/00.00000000.05224300.00000000 -.06366800.00000000 12.00000000.00100300.00000000 -.00061900.0000000.000580

Table 3.*.7, Part- II Transformed Determinant of the Even Modes, Columns 7-12 7 8 9 10 11 12 a4 b4 A6 B6 a6 b6 1 A2 -.05774000.00000000.06603700.00024000 -.05870300.00000000 2 B2.00000000 -,00042990.00008000.00424500.00000000.00100300 3 a2.05137900.00000000 -.05893000.00000000.05224300.00000000 4 b2.00000000.00023460.00000000 -00235700.00000000 -.00061900 5 A4 -.062619 00.0000000 07147000 -.00026000 -.06366800.00000000 6 B4.00000000 -.00048670 0.0017360263900.00000000.00058550 7 a4.08971000-.00000000000 -0636770 0000000.05666400.00000000 8 b4.00000000.03411000-.00000000 -.00061880.00000000.00006220 -A 9 A6 -.06367700.00000000.1377800-.00036400 -.06474000.00000000 10 B6.00000000 -.00061880.00026460.06973000-.00000000.00043910 -N 11 a6.05666400.00000000 -.06474000.00000000.24891600- 00000000 12 b6.00000000.~-00006220 0 12 b.00000000.00006220.00000000.00043910.0000000.19181000-: ^.

Table 3.5.8, Part i Transformed Determinant of Odd Modes, Columns 1-6 1 2 3 4 5 6 A3 33 a3 A B 1 A3.07072200 -.00012200 -.05988000.00000000.06907000.00012650 2 B^ -.00012200.00672600-.00000000 -.00127100.00021680.00311300 3'A 3 a3 -.05988000.00000000.06252900-.00000000 -.06190000.00000000 4 b.00000000 -.00127100.00000000.00967600-.00000000 -.00112600'-X 5 A5.06907000.00210800 -.06190000.00000000.10292900- -.00021800 6 B.00012650.00311300.00000000 -.00112600 -.00021800.03356200- 5 002000 3 60 7 a5 -.06189000.00000000 05508000.00000000 -.06398000.00000000ooooooo 8 b.00000000.00036800.00000000 -.00020000.00000000 -.00000974 9 A.07090000 -.00029800 -.06248000.00000000.07135000.00030800 10 B -.00012770.00381800.00000000 -.00149200.00022000.00363000 11 a7 -.06248000.00000000.05561000.00000000 -.0649000.00000000 12 b.00000000.00120000.00000000 -.00037200.00000000.00008705

Table 3.5.8, Part II Transformed Determinant of Odd Modes, Columns 7-12 7 8 9 10 11 12 a5 b B7 a7 b7 I A3 -.06189000.00000000.07090000 -.0001270 -.06248000. 00000000 2 B3.00000000.00036800 -.00029800 00381800 00000000. 00120000 3 a3.05008000.00000000 -.06248000 00000000.05561000.00000000 4 b3.00000000 -. 00020000.00000000 -. 00149200.00000000 -.00037200 5 A5 - 06398000.00000000.07135000.00022000 -.06459000.00000000 6 B.00000000 -.00000974.00030840.00363000.00000000.00008705 7 &a.14584000-.00000000 -.0640700.00000000.0747000.00000000 ^ -X ~0008730 8 b5.00000000.08909600 00000000 0000 8705.00000000 0038730 9 A7 -.06457500.00000000.09805000- -.00031100 -.06 9300 00000000 10 B.00000000 -.00008705 -.00031100.12725000-.00000000.00068740 11 a7.05747000.00000000 -.06519300.00000000.42034000. 0,000000 12 b.000000000.00038730.00000000.00068740.00000000. 363200007 -.

Table 3..9, lart I Eigenvalues and Eigenvectors of the Transformed Determinant for Even Modes, Col. 1-6 1 2 3 4 6 A2 B2 a2 b2 A4 B4 Elgenvalues.429799.005123.000827.000351.006608.016787 1 A2 +.334485 -.030576 +,668162 +.155638 -.518493 -.015605 2 B2 ^+.o000068 +.760054 -.140689,540286 -.073077 +.316374 3 a2 -.298358 +053920 +.706090 +.173042 +.508764 +.006378 14 ^ -.000001 -.514512 -.185667 +.808629.039123 -.209075 1 A4 +.372309 +.057456 -.022430 +.001154 +.669767 +.00O632 6 B4 -.000154 -.384095 +.012780 -.001158 +.017318 +.920991 7 a4 -.350197 -.008082 +007513 +.001144 -.110657 -006377 8 4 ~.W0000003 +.007874 -.000333 +.001153 -.000943 ~.033718 9 A6Q +.434341 +.600281 -.002440 -.000872 +.062768 +.005886 10 B6 +.000273 -.052582 -.000579 06114 +.010106 -.080075 -11 a6 -.590930 -.001274 +1001302 +.00019 -016672 -.000639 12 16 ^^+000000 -.004464 +1000099 -.000199 +.000448 -.005445

Table 3.5.9, Part II Eigenvalues and Eigenvectors of the Transformed Determinant for Even Modes, Col. 7-12 7 8 9 1011 12 a4 ~b4 A6 B6 a6 2 Eigenvalues.025204.034119.051839. 07028.152906.191822 1 A2 +.268786 +.000250 -.180848 +.000652 +.206132 +.000004 2 B2 +.000329 -.016087 -.000034 +.069938 +.000203 +.005590 3 a2 -.246328 +000000 +.160824 +.002428 -.184769 +.000000 S 4 bg -.001230 +.009404 +.000827 -.039732 -.000015 -.003455 4.)~ ^ 5 A4 +.535358 -.000395 -.253567 -.007897.+.240820 -.000005 6 B4 +.009083 -.027013 -.005053 +.054145 -.000409 +.003536 7 +4.737584 -.000531 +.505858 +.005079 -.254961 +.00000 8 b +000695 **99221 +.000569 -.018848 -.000005 +.000348 9 A6 -.186490 -000302 +.785322 +.009304 +,394668 +.000007 10 B6 +.002161 +.021905 -.011993 +.995006 +.001095 +.003932 11 a6 +.039626 +.000000 -.065886 -.001549 +.802873 +.000000 12 b6 -.000044 -.000216 +.000062 -.004625 -.000011 +.000064

Table 3.5.10, Part I EigenvaTles and Eigenvectors of the Transformed Determinant for Odd'lodes, Co 1-6 1 2 3 4 5 6 A3 B a3 b A B 3 3353 Eigenvalues.266263.005913 005357.*09922 01990 033803 1 A3.257327 +.182372.725983.004959 -.448169.00957 2 B3 -.000459 +.926376 -.224485 -.279262 +1014597.11089 3 a3 -.233463 +.150412 +1640312 +.009296.592931 -.06648 4 b +000005 +.276157 -.076870 +.956661 -.00254 -.049352 * 5 A5 +.293202 -,028169 -.071769 +.003831.65692 -003995 6 B5 +.000503 -.091069 +.017965 +079473.00996 +.991629 7 a5 -,349722 +.006227 +1023089 -.000083 -,095287 +.003063 8 h^ -,000000 -,003456 +,000818 +4003735 -.oooo000086 -.0801 9 A7 +.496062 -.003789 -.023853 -.000811 +,081742 -.00383 10 B7 -,000882 -022782 +,006434 +.018781 -,002534 -.043798 11 a7 +.651863 +.001462 +.005396 -.000019 -,019381 +,000519 12 b +,000011 -,002753 +.000657 +.001895 -.00005 -,000ooo626

Table 3 5.10, Part II Eigenvalues and Eigenvectors of the Transformed Determinant for Odd Modes, Col. 7-12 7 8 9 10 11 12 a5 b5 A7 7 A7 B7 Eigenvalues.06256.089097.107047.127541.538649.363267 3. A +.009857 +.295591 -.115421 +.001141 -.267950 -.000002 2 B3 *110089 +.002637 -.001825 +.032113 +.000149 +~003405 3 a3 -.006648 -.300152.114217 -.002083 + 240900 -.000000, 4 3 -,049352 +.000149.000207 -.013359 -.000000 -.001078:0v5 A -.003995 +*599491 -.181714 +.005333 -.290699 + 000004 6 B5 +.991630 -.006430 +.003928 +.039768 -.000160 +.000333,7?a5 +1003063 +.613679 +.636780 -.006594 +,293084 -.000000 8 b -.000801 -. 000046 - 000097 - 001920 +.000000 +.001417 9 A7 -.005383 -.288856 +.729981 -,009048 -.360881 -. oo008 10 37 -*043799 -.002619 +.012110 +.998497 +.000200 +.002979 11 a +000519 +.055227 -.045664 +.001095 + 754672 -.000007 12 b7 -. 000626 -. 2 003111 000002 00021 +.0001 +-999988

- 184 - The eigenvectors yl, y2I...Yn of the rth principal mode of the original system must satisfy the set of equations* (a11-bl )Y1+a12 Y2+. * - * *... * *h = (3.5.1) a21y1 + (a22-b2X)2..........a 2n Yn anlYl + an2 2 -' + (a-nn'b )yn= The superscripts and subscripts r were omitted. The eigenvectors Zli z2...z of the transformed matrix in the rth mode satisfy the system of [(all/bl),)J -Zl-(al2 2)Z2-'*0*(aln n ))znO (ao1/2)lb2 )-1-2 2 (a 2n/b2bn)Zn=O (3.5.2) ~b z... (anI/~ lb)Z1- (a2b'2 n)z2" - (a n )-2zJnO The superscripts and subscripts have again been omitted. By dividing the first equation of 3.5.1 by i, th b~-tio / and in the general case, by dividing the th equation {i where j = l*.nt the system of equations given below will follow. (a2l/42)y* + (a/ )22/'y ))'- yr2jY2-" (a2nf/2)' [(a/* ^1 yn= 0O.(3.5.3) (an. n, I bly+(an/4S)y2,-..+a:/(am/.)-Wn4,]

- 185 - Now multiply and divide each term of the th column byb, i = 1,2...n. Equation 3.5.3 can be rewritten as C(all/bl ) - J *lWy+ (a12/Jb*2 )Ji22+..(al bj). -\b^nyn 0 (a21/~"2 ) 6 l1+ [(a22~2 )-,_] Y272+-. (a^/b2bn)-! b n 1 ly'l+ ~ ~ ~ ~ Mn 5 ~ b -0 On comparing 3.5.4 and 3.5.2 it can be observed that (35.5) by. zhence (3.5.6) y= zj/ The eigenvectors of the original system can there-, fore be evaluated simply by dividing the th eigenvector component by the square root of the frequency term's eoefficient in the th column of the original matrix. Table 3.5.11 contains the natural frequencies, eigenvalues and eigenvectors of the experimental motor frame for the first three even modes. The resulting eigenveetors in this table were obtained by applying the transformation 3. $.6 to the eigenvectors' as calculated by the electronic data processing machine and recorded in 3.5.9. Similarly, by applying the same transformation to

- 186the data in table 3.5.10, the natural frequencies, eigenvalues and eigenvectors of the experimental motor frame in the first three odd modes are obtained. These are recorded in table 3.5.12. For better understanding, the fundamental displacement distribution of the first four principal modes of vibration have been sketched in fig* 3.5.1, for the even modes. A similar sketch is given for odd modes in fig. 3*..2. The Mode shapes in fig. 3.5.1 illustrate the following vibrations a) the counter-tact cosine resonating at 10,430 c.p.s; b) the counter-tact sine resonating at 1,140 c.ps*.; c) the in-tact cosine resonating at 460 c.p.s.; d) the in-tact sine resonating at 490 c.p.s. The lode shapes in fig. 3.5.2 demonstrat the following vibrations: a) the counter-tact cosine resonating at 8,200 c.p.s.; b) the in-tact sine resonating at 1,225 c.p.s.* c) the in-tact cosine resonating at 1,170 eip.s.; d) the counter-tact sine resonating at 1,580 c.p.s.; The real modes of vibration contain also harmonic components which complicate the shape of the displacement along the directrix of the vibrating cylinders. Similar diagrams could, without difficulty, be drawn for all other principal modes.

,L.~- -------— 7. - ~.I. itI d *ode Vibratio I:, - t E Motorl.. I.M.: ~.. z.~ *

-4 —~~~~~~~~~~~~~~~~~~~~~~~~ IdI,' >' —-41 as bw L~~l_ - - 0_r * Fig * 3.o 5,w2 =LTA -_j A: The retical Third Mod Vibrato the. * ~~~ ~ ~ ~ ~~~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~,:. i..., ~ ~~ ~ -t -~o -,- -.. -... o;r

Table 3.5.111, Part I Eigenvalues and Eigenvectors of the Original System for Even Modest Col1ns 1-6 1 2 3 4 5 A2 B2 a2 b2 A4 B Frequency f=10,430 f=15140 f-460 f 490 f=]295 f-206 Eigenvalues 4297.99x106 51.236x106 8.268106 9.110 66.08x10: 167.87 106 1 A2 0.85284 -0.07796 1.70361 0.396830 -1.322000 0039790 2 B2 O.000180 +1-93791 -0.358710 1.377560 -0*1863201 0.80666 3 a2 -0.676970 0.122340 1.602120 0.392630 1.154380 0*014470 " 4 2 -0.:0000003 -1.167430 -0.421280 1.834780 0.088770 -0.474390 4 5 A4 1.029430 0.158860 -0.062010 +0.003192 1.851910 0.5570. 6 B4 -0.000430 -1.062000 0.035340 -0.003200 0.047880 2.546540 7 a4 -0.861830 -0019890 0.018490 0.002814 -0.272330 -0.015690 8 b4 -0.0o0000 +0.019380 -0.000820 0 002840 -0.002320 0.82980 9 A6 1.221360 0.008050 -01006860 -0.002450 0.176500 O.016550 10 B6 0,000770 -0.147860 -0.001630 -0.017190 0.028420 -0.225170 11 a6 -1.47910 -0.003190 0,003260 0.000490 -0.04173 -0.001600 12 b6 0.000000 -0.011170 0.000250 -0.000500 0.001120 -0.01368 6~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Table 3.5.11, Part II Eigenvalues and Eigenveetors of the Original System for Even Modes, Columlns7-12 7 8 9 10 11 12 a4 b4 A6 B6 a6 b6 Natutral Frequency 2$60 2D40 3^60 4240 6,240 6,980 Eigenvalues 252,03x106 341.18x10 518.39x10 702.80x 1529xlf0 1918.21x110 1 A2 0,685320 0.000640 -0.461110 0.001660 0.525580 0,000010 2 B2 0.000840 -01041020 -0.000090 0.178320 0000520 0,014250 3 a2 -0.55892 0.000000 0.364900 0.005510 -0419240 0.000000 ^~ ~~~9. 02 ~... 8'' -~' 4 b2 -0.002790 0.021340 0.001880 -0.090150 0.000033 -0.007840 ^~~~~~~~~~~~~4 4 -00283 A4 1.480270 -0.001090 -0.701110 -0.021830 0.665870 -0.000013 16 B4 0.025110 -0.074690 -0.013970 0.149710 -0&001130 0.09770 0~~ t 7 a4 1.815190 -0.001310 1.244920 0.012490 -0.627460 0.000000 8 b4 0.001710 2.459080 0.001400 -0.046380 -0.000012 0.000ooo860 9 A6'-0.24410 -0.0000850 2.208320 0.026160 1.109810 0.000020 10 B6 0.060760 04061600 -0.033730 2.798000 0.003080 0.011060 11 a6 0.099180 0.000000 -0.164910 -0.003880 2.009600 0.000020 12 -0.000110 -0000540 0.000160 -0.011580 -0.000030 2.502910

TFable 3..12, Part I Eigenvalues and Eigenvectors of the Original System for Odd Modes, Columns 1-6 1 2 3 4 5 6 A3 B3 1,3 A5 B5 Natural Frequenoies e?,00 325 270 1580 2240 2940 6 6 Bigenvalues 2662.110 99.13x106 6I Bigenvalues 266x10 ~13xl0 3.53 x10 99*22x3D 199-60x10 338.03x10 1 A 0.069590 0.049320 0.196320 0.001340 0.121190 0.002670 2 B3 -0.000120 0.250510 -0o060710 -0.075520 0.003950 0.029770 3 a3 -0.06190 +0.036200 0,154100 0,002240 0.142700 -0.001600 3~~~~~~~~~~~~~~~~~~ 4 b3 0.000000 0.066460 -0.018500 0.230240 -0o.000610 -0.011880 $4 43 AA 0.081970 -0.007870 -0.020060 0.001070 0.183560 -0.001120 16 BB 0.000140 -0.025460 0,005020 0,022220 0,002790 0,277220 ^7 a5 -0.087000 0.001550 0,005740 -0,000020 -0,023700 +0.000760 8 b -0.000000 -0.000860 +0.000200 +0.,000930 -0.000020 -0.000200 9 A7 0.139970 -01001070 -0.006730 -0.000230 +0.02307 -0.00152 10 B -0.00025 -0.006430.001820 0.00030 -0, 20 -0.012360 7 11 a7 0.163700 -.0100370 0.001360 0.0000 -0,0011- -0.000130 12 b 0.000000 -0.000690 0.000690 0.000480 -0.000010 -0.000160 7

Table 3.1512, Part II Eigenvalues and Eigenvectors of the Original System for Odd Modes, Columns 7-12 7 8 9 10 11 12 a5 b5 A7 B7 a7 b7 Natural Frequencies 31950 4,750 5,200 5,580 11t650 9,780 6ul 89.716 1070 Xio Eigenvalues 6216x10 89097x1 1070x 12741x10 538649x10 3632.67x10 1 A 0.079930 -0,000000 -0.031210 0.000310 -0.07246 -0.000000 3 2 B, 0.000710 0.001240 -0,000490 0.008850 0.000040 0.000920.3 3 a^ -0,072240 +0.000000 +0.,02749 -0.000500 +0.057980 -0,000000 4 b 0.000040 -0.000630 -0.000050 -0,003220 -0.000000 -0.000260 3 N 5 A5 0,.3^759 0.000010 -0.050800 0.001490 -0.081270 0,000000 6 B -0.001800 0.000070 0.001100 0.011120 -0.000040 0.000090 7 a5 0.152660 0.000020 0.158410 -0.001640 +01072910 -0,000000 8 b -0.000010 0.248760 -0,000020 -0.000480 +0.000000 0.000350 9 A7 -0.08151 +0.000010 +0.205980 -0.002550 -0.101830 -0oo0000o 10 B -0.00740 0.000490 0.003420 0.281750 0.000060 0.000840 7 11 a7 0,013870 -0,000000 -0.011470 0,000280 0,189520 -0.000000 12 b -0.000000 -0.000360 -0.000010 -0o.o00078 +0.000000 +0.251130 7

193 3.6 Calculation of the Forced Response of the Experimental Motor There is a large number of force waves in the airgap which tend to set the motor into vibrationi While those of low order and high amplitude are the source of magnetic noise, others, of high order and low amplitude can generally be neglected. In the following discussion only the vibration amplitudes caused by single-mode, single-frequency fore waves will be treated. Any picture of the complete vibration can be constructed by using the principle of superposition. The essentials of the theory for the calculation of the forced response is given in equations 3.1.27 to 3.1*37. The rotating force wave can be represented by equation (3.6.1) P(e,t) = PM cos (mt-1'0) In this equation m' is the mode of the specific force waves and P m is the amplitude of the force wave in pounds per radian. On carrying out the tigonometric expansion in 3*6.1 one obtains (3.6.2) P(et) = Pm cos.t cos m^ + Pm sin ot sin mi9 This last equation represents two pulsating force distributions displaced by one quarter wave length along the armature circumference, and lagging by one quarter of a cycle. The maximum amplitude of each wave is the same as the amplitude

- 194 of the rotating force wave. The first term stands for a direct or cosine mode force, and the second for a quadra ture or sine mode forces The work done by the cosine force wave during deflection is given by (3*6.3) W(e) (1/2) f Pm cosm'e( a cos me 4bm sin me)de But the integral of the products containing sirnme andd all the terms containing cos m3 for mOm'will vanish. The value of integral 3.6.3 thus reduces to the expression (3.6.4) W()= (r/2) Pmramit The strain energy of deformation in the rth principal mode according to 3.130, is equal to the kinetic energy, and therefore (3.6,5) Pdir/2)a 2 2 2 2 2 (3.6.5) PI(wr/2)a2 gzm (a + b) + Gm (A; is) In both the last and the following equations the superscripts r of the generalized coordinates are being omitted. To find the static deflection An of the outer ring in the rth principal mode, both sides of equation 3.6,5: will be multiplied by A 2 222 (3.6.6) Pm' (/2)amn An + b2)+G (Aa +G ) A can now be evaluated as..'''~ ~.... (3.6-7) An PFr+/2)a mA: b~) m (Am B+) All deflections in the rth principal mode are, according to

195 - equation 3.1*25, proportional to the components of the eigenvectors. These components may be used on the right side of the equation sign in 3*6.5, as the factor of proportionality cr cancels out. The deflection An in the rth principal mode can therefore be expressed as (3.6.8) An #p'(-w2)y m( <m+im +G w it~ m For the sake of convenience in writing the constant Kr is defined as (3*6.9) K - gm(Y + bm) + Gm(2m Equation 3.6.8 may now be written as (3.6.8a) A( Pm(r/2) Yam Y- W/Kr Proceeding from the static to the dynamic deflection in l rth principal modes equation 3.6.8 has now to be multiplied by the resonance factor, in accordance with the same argument used to deduce equation 3.1.37. The dynamic deflection in the rth principal mode can thus be expressed as (3.6.9) Ar) P P(w/2) (.a 4 2 2 ) In wha' follows, the discussion will be continued by using a second-mode force wave and by considering the second mode deflection of the outer ring. The corresponding subscripts of, the eigenvectors in table 3.5.11 will be used. The deflection at the top rib caused by the second mode cosine distribution can now be written (3.6.10) (t) [p 2(r/2)y y (r)r(Y)J ct

- 196 - and the analogous expression for the deflection at the top rib as a result of the sine component of the force wave will be (3.6.11) A (t) [P(r/2)y(r)y(r) 2'K,( -W ) n 2c' L2 rY4'r In order to find the magnitude of the total displacement of the outer ring at the top rib, it is necessary first to add the deflections produced by the rth principal modes caused by the cosine component of the force wave. Then the deflections produced by the rth principal modes corresponding to the sine component of the force wave are summed separately. Finally the two resultant deflections are combined by the parallelogram law. The result is / r= n 12 r=n [J 2) (3.6.12) A~ V-A + A'2 2, 2'd' r=l r= 1 To facilitate the numerical solution for the experimental motor, the constants Kr have first been computed and then assembled.n table 3.6.2. The inertia constants given in table 3.5.1, and the eigenvalues and eigenvectors given i table1 35.1l have been used. The overdamped modes, ie. the counter-tact cosine modes as explained in section 4, have been omitted. The effect of modes 9 to 12 was found to be entirely negligible for the second mode amplitudes and is therefore not recorded here. Table 3.6.1 shows the detailed calculation of the constant K3. Table 3.6.3 illustrates the calculation for the

- 197 - static deflections and resonance factors for the displacement of a point on the outer ring at 9 = 0, caused by a second mode force wave of 100 lb per radian amplitude. Table 3.+6.4 illustrates the detailed calculations for the deflection A at the main noise frequency of the motor at full load, that is at 1,716.ep.s. The amplitude of the forcewave has again been assumed to be 100 lb per radian. Table 3.6.5 gives a summary of the calculated second mode deflections in the cosine mode caused again by a second mode rotating forcewave of 100 lb per radian amplitude for the whole frequency range between 100 and 2100 c.p.s. These calculated values have been plotted in Fig. 1.2. Finally table 3.6.6 lists the deflection amplitudes in the cosine modes for the response of the force waves listed in table 2.10*5. Table 3.6.1 Calcetulation of Constant K 3 Eigenvector y2: m m Gm o r gm Component m xO xlO1. a 2.9193 2 29 91287 3.0480 7.69 23.49 3 2.5667 4 17756 2.7442 9.71 26.654 15 1^008 60 *0.0038 6001 O00. 0051 6.54 0.033 o 0*0013 7 0.0034 0.0034 8.26 10 1 01negl. 50. 0 12. 12 313 = 8.268 x 106 1K = 8.268 x 106 x 50.12 x 10O2 = 4.144 x 106

Table 3.6.2 Summry of Constants Kr Princ. Mode 2 3 4 5 6 7 8 Kr 22.41 4.144 4.755 83.93 126.02 1 16 9.6 x 106 Table 3.6.3 Second Mode Deflection Amplitudes in the Cosine Axis caused by a Rotating Forcewave of 100 lb per Radian Amplitude Princ. /r 6 Static defl. Mode 41 x3 ^4 xlO xlO' Resonance factor o 0cos sine 2 -0.078 0.123 -1.167 22.41 5.47x10-3 -0.617 5124/(51.24,2x10l6) 3 1.704 1.602 -0.421 4.144 100.50 -26.4 8.26 /:.26-a2x10-6 ) 4 0.397 0.393 1.835 4.755 5.00 23.4 9.51. 51- 2x10- ) 6 -0.04, 0.0145 -0.474 83.93 -1.86x10-3 3.4x10-2 167.87'67.87-2x10- 6) 7 0.685 -0.559 -0.0028 126.02 -0.505 2.3x10-3 252+3/52.3 -'2x10o6) 8 0.0006 0.000 0.0213 169.6 negligible

- 199 - Table 3.6.4 Calculation of the Second Mode Deflection in the Cosine Axis caused by a 100 lb/Radian Amplitude Rotating Forcewave of 1,716 c.p.s. Defleetions by cosine component by sine component Prine. Resonance Static dynamic Static dynamic Moide Factor ps inches nhes p inches 2 -.877 -4.8x103 -0.0042 -0.54 0.475 3 -0.0744 100.5 -7.82 -26.4 1.96 4 -0.09 5 -0.45 +23.4 -2.09 6 -325 -1.86x10-9 -0.0063 3*4x10'2 0.1 7 1.85 -0.505 -0.936 2.3x109 0*004 -9.22 0.48 Total Deflection 9*23 p inches Table 3.6*5 Calculated Second Mode Cosine Axis Deflections caused by a Second Mode 100 lb/ Radian Rotating Forcewave as Function of Frequency Frequency Ampl. Frequency Ampl Frequency Ampil e.p.s. p inches c.p.s. p inches c.p*8. p inches 100 110.25 700 80.45 1700 9.25 350 251.80 900 36.6 1715 9-23 450 306.14 1000 25.4 1735 915 468 1100 2000 7.27 470 1774. 1110 26*0 2050 7.55 480 890. 1140 2070 490 1160 21.2 2100 6.9. 492 829 1200 19.80 500 651 1400 14.48

- 200 Table 3.6;* Stnumary of Calculated Deflections for 2nd Mode Foree Waves of the Experimental Motor Motor Noise -l0 Force Defl Defl. Operating fr. lb/rd Wave Defi. lb/rad A. V. cps cps p-inch Inch -cm 40 40 0 60 1,716 9.23 84.5 7.8 19.6 40 367 50 1,419 14.4 84.5 12.17 30.7 40 294 40., 121 28.2 84.5 23.9 60. 5 N.L.440 60 1,735 9.15 23.1 212 5 534 40 440 60 1,475 12.98 4.6 0,6 1.5 3_, Investigation of the Convergence of Eigenva!$ As mentioned at the start of this chapter, the exact.solution of the dynamic behavior of continuous systems is given in terms of determinants of infinite order. Also, to keep the numerical work within manageable bounds, a determinant made up of only the first few rows and columns is considered to be a sufficient approximation of the infinite determinant. It was decided to use only 24 generalized coordinates. Consequently a 24th degree h haracteristic equation resulted which, fortunately, could be factored into two 12th degree equations. It remains to be demonstra

- 201 - ted that the use of a larger number of terms in the Fourier expansion of the radial and tangential displacements would not alter the numerical results materially. With this aim in mind, the eigenvalues for even and for odd modes have been calculated, assuming that the radial and tangential displacements in both the even and the odd modes could be represented by the first four generalized coordinates. In other words, It is supposed that equation 3*4.1 which represents the radial displacement can be simplifted to read U = A2 cos 29 + B2 sin 20 (3-7.l) u = a ecos 2e + b2 sin 29 while 3*4.1 for the odd modes can be written as U = A3 cos 30 + B3 sin 30 (3*.72) u = a3 cos 3e + b3 sin 39 As a further step, the next term in the Fourier expansions has been added to 3.7.1 and 3.*72, and the eigenvalues of the 8th order determinant found. To permit ready comparison of these results, the eigenvalues of the first four principal modes, as calculated from the 4th 8th and 12 order determinants, have been assembled in table 3.7.1, for the even modes, and in 37-.2 for the odd modes. An inspection of tables 3.7.1 and 3.7.2 reveals an

202 - excellent convergence of eigenvalues and natural frequencies in the 2nd, 3rd, and 4th principal modes. The selection of only 24 coordinates for these modes seems therefore fully justified. The regrettable unruly behavior of the first principal mode is of no importance as the frequency of this over-damped mode is very high. This mode has no influence on the steady-state behavior of the vibrating stator as explained in the assumptions, and pointed out again in chapter 5. In the numerical calculation of the steady-state deflections a similar result was obtained. The effects of the 9th th to 12 principal modes on the second mode deflection were entirely negligible. The above statements apply exactly only to the first even and odd modes of vibrations. For the calculation of higher mode deflections the number of terms selected for the Fourier series should be increased. The capabilities of the Electronic Data Processing Machine are by no means exhausted by a 12 x 12 determinant. The method as explained here can therefore be extended with sufficient accuracy for the deflections in higher modes.

- 203 Table 3.7.1 Comparison of Eigenvalues and of Natural Frequencies in the Even Modes Principtal ]ode I 2 3 4. Eigenvalues xlo6 x1o6 xl " x1o6 4x4 1079.51 49.736 8.426 9.922 h 8x8 2459.91 53.027 8.275 9 535"u 12x12 4297.99 51.236 8.268 9.11 Natural Frequencies c.p.s.:s " 4x4 5,140 1,123 464 500. IIgQ 8x8 6,870 1,164 461 492 12x12 10,430 1,140 460 490 Table 3.7.2 Comparison of Eigenvalues and Natural Frequencies of th Odd Modes Principal Mode.:2- 1 2 3 4 Eiigenvalues 6 6 xlO x10 X106 x. x10 4x4 1266,45 62.373 66.218 101.482 1: 0 8x18 2861.43 59.800 64.401 99.602,:: 12x12 2662.63 59.130 63.576 99.226,N latural Frequencies c.p.s. 1. 4x4 5,670 1,247 1,290 1,605 6 ^8x8 8,540 1,233 1,272 1,590 12x12, 7,020 1,226 1,266 1,585

som^ RADIATE) BY THE VIBBTING STAYIORN The small amplitude vibrations of the stator create periodic disturbances in the medium in contact with the surface of the stator* These disturbances are characterized by alternate compressions and rarifications of the contacting air layers. The air, being an extended medium with the properties of distributed mass and elasticity, propagates the disturbances of the contact surface when excited. This results in a periodic change of the air pressure everywhere around the motor about the statie value, and in a periodic change of the velocity of the air particles above and below an average motion. This phenomenon pecifles, and is referred to as the sound waves radiated by the stator. It is the purpose of this chapter to calculate a characteristic quantity of the radiated sound waves. Since virtually all modern sound detectors respond directly to variations of the pressure in the wave disturbance, the "sound pressure at a characteristic point near the induction motor will be determined. The consideration of the elastic and inertial properties of the air leads to the partial differential equation of sound propagation, the acoustic wave equation. This will be established as the first step, assuming the air to be a continuous, isotropic medium, having the properties of - 204 -

- 205 - perfect elasticity and zero vicosity. In addition, field free conditions will be assumed. The stator of an induction motor is by no means of a shape that lends itself easily to the radiation and boundary value problem. It becomes essential therefore to make approximations without losing sight of the physical significance of the problem. The stator, as illustrated in fig. 4.2.1,will be extended on both sides by semi-infinite cylinders, so that the vibrations of the stator will not be transmitted to the extensions. This artifice can beh-y interpreted to represent an infinitely long cylinder vibrating in a belt only. The solution of the boundary value problem will be carried out by a Fourier transform method. The resulting integral for the sound pressure will be numerically evaluated. The results of the sound pressure integral will appear in dynes per square centimeter. It has become customary to express the sound pressure as a sound pressure level with respect to a reference sound pressure. For airborne sound pressures, 0.0002 dyne per square centimeter has been generally accepted as a standard reference sound pressure. On account of the sensitivity of the ears and the range of pressures it responds to, a decibel scale similar to the one in communication engineering has been universally adopt ted. This is consistent with the fact that the human ear

- 206is capable of detecting a very small increase in sound pressures when the ambient sound pressure is low. At a higher ambient level the sound pressure increases have to become considerably larger to be perceived. It has been demonstrated, in fact, that the human ear responds logarithmically to the sound pressure. To take care of this faculty of the human ear the sound pressure level is specified in terms of a unit called decibel, which by definition is equal to twenty times the logarithm to base ten of the ratio between the sound pressure in dynes per square centimeter to the reference pressure 0.0002 dyne per square centimeter. (SPL) 20 log10 (SP/0.0002) where SP and 0.0002 are in dyne/cm2 4.1 erivation of the Wave E1uation The physical theorems which form the basis of the mathematical manipulations leading to the acoustical wave equations comprise: (1) the principle of conservation of mass represented by the continuity equation; (2) the principle of conservation of momentum as given by Newton's second law; and (3) the principle of conservation of energy. In the following discussion a small element of the medium, i.e. the air, of volume AV in the vicinity of the -sound field will be considered, as shown in figure 40l.1. This particular volume has a surface S and is supposed to 1 7;32

-207q Fig. 4.1.1 Volume Element in Sound Field be fully enclosed by an ideally deformable, massless, impermeable membrane, so that no mass may escape through this membrane. This is equivalent to the statement that the mass within the volume will remain constant. In the following discussion the expression "particle" will be used to denote the elementary volume. It can be shown that the equation of continuity can'be writtent (4,1.1) p9/t +.V-ppV. = 0 In this equation f denotes the density, and q the velocity of the particle. In order to describe mathematically the conservation of mass for a specific particle considered, it is of great help to transform the continuity equation in such a way as to have the coordinates of the system attached to the particle itself. This procedure makes it possible to represent

-208the time variation of the mass of a specific volume element. In hydrodynamics this operation is commonly represented by Dp/Dt. It is not difficult to prove that (4.1.2) (Dp/Dt) = (p/t) q. vp The requirement of conservation of mass of a.pecific volume element is expressed by (4.1.3) p (V) = const Taking the time derivative at a point moving with the particle, this may be written as.(4.1.4) p [D(AV)/Dtl + (AV)(Dp/bt)= 0 N:w combine equation 4.1.2 with 4.1.3. It follows that (4.1.5) (1/A) [p(AV)/Dt + (lip) (ap/t) +. 0 It can also be shown that (4.1.6). q [= /(AV)] [D(AV)/Dt] Introducing this last equation into 4.1.5, one obtains (4.1.7) V.q + (1/p)t(, p/lt) t 4.vp] 0 Before continuing the discussion, the condensation or fractional increment of density will be defined as (4.1.8) s =(d/o) (PPo)/P hencee (4.1.9) p = o(s) where p is the density of the volume element with no sound wave present. Introducing equation 4.1.9 into 4.1.7, the latter can be expressed as (4.1.10) V. + [(L!/(1+)31 C(6s/8t) +. Vs] = 0

209 - Consider next a differential area element d6 on 0 the ideal membrane with its normal unit vector dn directed outwardly. Let the instantaneous sound pressure be p and assume no varying body forces on the medium. In consequence of Newton's second law, the total incremental force normal to the surface I jp d 6 may be equated to the acceleration force p(V) (Dq/Dt) In the last expression Dc/t again represents the time variation of velocity of this specific particle which, fora vector quantity, is given by (4.1.11) Di/Dt = aq`/t + (q V) hence (4.1.12) fJpd6 + [(c/at) + (q.V) ] p (A) - 0 Dividing this equation by AV, and taking the limit by letting the element become infinitely small, this equation is ~transformed to (4.1.13) p + [(^t) + (q*V)q] p= 0 A relationship between the sound pressure p and the condensation s has to be established from the volume changes. It can be brought about by the use of the principle of conservation of energy. As the heat conductivity of gases is low, it is safe to conclude that the heat exchange in the wave at audible frequencies is negligible, that is, that the.

- 210deformations are adiabatic. This is expressed mathematically. as PV = const. where ~ is a constant ratio of the specific heats at constant pressure and constant volume. One may state that as a direct consequence of the adiabatic expansions P/Po = where s is the condensation as defined before, and P, is the static sound pressure. Hence (4.1.14) p = PO s Restricting further discussions to small signals, and neglecting second,and higher order terms in equation 4.1.10, this can be expressed as (41.110a) V. + (^s/6t)=0 Introducing into this equation the relationship between s and pt it can be written as (4.1.15) V.4 + (1/aPo)(ap/at) = 0 and similarily, equation. 4.1.13 can, for small signals, be written as (4.1.16) Vp + po(64/t) o If now 4.1*15 is differentiated with respect to the time, the divergence operation on 4.1.16 is carried out,and remembering the interchangeatility of the operation divergence and differentiation, this equation may be rewritten as (4.1.15a) V.(}/+ t)+ (l/~P,,)(}2p/t2) 0

- 211 - and (4.1.16a) V.Vp+ eo. (^/%t) 0 The acoustic behavior equations resulting from the combination of these last two equations can be written as (4.1.17) V2p (l/c2)(62p/ t2) where 2 = (P/ o) is the square of the sound velocity in the medium. 4,2 The Integral of the Sound Pressurei To find the sound pressures in the field near the induction motor it is necessary to solve the partial differential equation represented by the wave equation for the special case of the vibrating stator. This will require two major steps: first, to find all possible solutions of the equation and secondly, to choose from these solutions the particular combination which satisfies the boundary condition of this special radiation problem. To facilitate the solution, an appropriate system of coordinates will be chosen. The boundary conditions are given on a circular, cylindrical surface. It is only natural to select a system of circular cylindrical coordinates as illustrated in fig. 4.2.1. 166

- 212 - The,vibrations of the outer stator are harmonic in time and one may, therefore, conclude that the sound pressures caused by these disturbances'are also harmonic in time, and set them as (4.2.1) p = p exp ( -jct) x r K___b_ b Fig. 4.2.1 System of Coordinates for the Vibrating Cylinder Introducing 4.2.1 into equation 4.1.17, the wave equation in its well known form will result (4.2,2) (V2 + k2)p = 0; where k2 = 2/c Substituting the Laplacian operator in cylindrical coordinates, the wave equation can be written as (4.2.3) (l/r)($/Sr) [r(~p/Sr)] + +(l/r2,(a2po2> * (a2p/az2)+k2p 0

-213The usual method of solution consists of the separation of variables by seeking a product function (4.2.4) R(r) Q(e) Z(z) which will split the partial differential equation into three ordinary differential equations. Proceeding in this manner, equation 4.2.4 is introduced into equation 4.2.3, and the whole equation divided by RQZ, which results in (4.2.5) (l/rR)(d/dr) [r(dR/dr)] +(l/r2Q)(d2Q/de2) + +(l/Z)(d2Z/dz2)+k2 0 Since all terms except the third are independent of z, the third term must be a constant and may be set equal to -h2. Thus (4.2.6) (l/Z)(d2Z/dz2) =-h2 Substituting 4.2.6 into 4.2.5, and multiplying the result by r2, it follows that (4.2.7) (r/R)(d/dR) [r(dR/dr)] + (l/Q)(d2Q/d2) + + (k2-h2)r2= 0 With the same reasoning as applied in 4.2.5, one can assume the second term to be a constant, and we now set it equal to (4.2.8) (l/Q)(d2Q/de2) _n2 Introducing 4.2.8 into 4.2.7, the latter reduces to (4.2.9) r(d/dr) [r(dR/dr)] + L(k2-h2)r2-n23R = 0 The first task to split the partial differential equation is now accomplished. The product of the solutions

-214 - of equations 4.2.6, 4.2.8, and 4.2.9, is the separated solution of the acoustic wave equation in cylindrical coordinates. The solution of 4.2.6 is given by expressions of the form (4.2.10) Z = exp (jhz) Analogously, the solution of 4.2.8 given by (4.2.11) Q exp (jne) Equation 4.2,9 Is Bessel's differential equation and its solution in series is given by (4.2*12) R = Zn(r k-h) where Zn is a circular cylinder function of the order n and argument r k 2-h. The solution of the wave equation for sound pressure is given by linear combinations of the product of 4.2*10, 4.2*11, and 4.2.12. These products are referred to as the elementary wave functions and written as (4.2.13) p = exp(jne).Zn(r \2-h2). exp (Jhz-jcot) This last equation represents a nondenumerable infinity of solutions that may be superposed in any arbitrary manner. The solution p,as givenis therefore not unique. In order to get a definite solution to equation 4*2,9 it will be necessary to express the constants in such a way as to satisfy the boundary condition, that is the prescribed behavior on the boundary. In other words, the boundary con

- 215 - dition will serve to extract from the infinity of solutions the particular solution appropriate to the specified boundary condition. The boundary conditions are partly determined by the vibrations of the cylinder and by the behavior of function p at infinity. The amplitude and frequency of the vibration of the cylindrical surface determine the velocity of the particles on the cylindrical surface when secondary reaction effects are neglected. The gradient of pressure and the time differential of velocity are correlated by Newton's second equation of motion; hence (4.2.14) p(bq/t) = -bp/6r Considering each single mode and single frequency-vibration individually, the sound pressure on the cylindrical sur* face (r=a) must, for all values of z, between +b and -b, satisfy the condition (4.2.15) ( p/~r)r=a = jAmo p exp(jme-Jmt) and be zero for any other value of z on the cylindrical surface. The distance 2b is the length of the stator. According to Sommerfeld1, wave propagation problems are not uniquely specified by their prescribed sources in the 1 64;9

- 26 - finite domain. In addition to the above boundary values, Sommerfeld's "Endlichkeitsbedingung" and "Ausstrahlungsbedingung"l as given in 4.2.16 must also be satisfied. (4.2.16) lrpl<k as r - C> r(bp/dr - jkp)-, as r -co The sound pressure distribution on the cross sectional circle of the cylinder is periodic with the period (2r/m). This condition requires the periodicity in equation 4.2.13 to be the same as in 4.2.15, that is n m. Moreover, the propagated wave is a diverging one, and must be represented by outwardly traveling waves. Also it does not include the axis of the cylinder. Hankel functions must therefore be substituted for the circular cylinder function in 4.2.12. The time dependence is given by 4.2.1, and is a function of -jot. Consequently it becomes necessary to select a function of the form (4.2.16a) H(l)(r \k2-h2) Jm(r k2h2 )+Y (r k2 h2) as the solution 4.2.12 to satisfy equation 4.2.16. With this modification the elementary wave function for the boundary problem at hand is given by (4.2.17) p = exp(jme).-H )(r \ -hexp( jhz)exp(-Jtt) If the boundary condition along the generatrix of the 1 The author is indebted to Dr. RBC, F Bartels for calling his attention to this condition.

217cylinder were periodic, the next step would require the solution for each harmonic component of the boundary condition. The final solution is obtained by the summation of all components. The boundary condition on the generatrix of the stator is here given by a step function, that is, an aperiodic function, with a continuous frequency spectrum in space from zero to infinity. Consequently the summation of discrete solutions will be replaced by an integration and the solution for the sound pressure is thus given by $+ 2 (4.2.18) p 7 exp(jm - Jcot). F(h)-H l)(r k2h2).exp(jhz)dh F(h)dh replaces the coefficients of the discrete components of the periodic case. In what followsy the term exp(Jme - ju0t) will, for the sake of brevity, be omitted, though it will be added again at the end of this derivation. The incorporation of the remaining boundary condition, that is, the finding of function F(h) to satisfy the behavior on the boundary, will be carried out by a Fourier transform method as suggested by R.F. Harringtonl and J.A. 2 Stratton * A Fourier transform pair is given by + 0*: (4.2.19) g(u) (1/12i) f(x) exp(-Jux)dx 1 31 2 66

- 218 - + O (4.2.19) f(x) = (1/ \f) g(u) exp (jux) du The existence of the transform g(u) implies certain restrictions of f(x) discussed by R.V. Churchilll. That the transform arising in the problem considered does exist can be shown from the known behavior of solutions to the wave equation. It is assumed provisionally that the integral in equation 4.2.18 is uniformly convergent, h:nce it may MI.. fferentiated under the sign of integration with respect to the parameter r. This results in the expression (4.2.20) (p/ar) f= F(h)(l/k[2-h2 1h (r ).exp(jhz) dh Hence over the surface of the cylinder where r a + - (4.2.21) (b^p/r)y f F(h)(V/) ( lk)]'.. [a ~exp(ehz) dh Now introduce the following notation (4.2.22) ({>)F(h)(1/kh) )]'(ak-7) G(h) Equation 4.2.21 can be rewritten as (4.2.21a) (/p/r) a = (1/2ir) f G(h) exp(jhz) dh r -<> If we now compare 4.2.21a with 4*2.19, it becomes obvious that the coefficients of G(h) must be the Fourier transforms of (ap/6r), that is 1. 19

219 - + o (42.2.3) G(h) = (1/2w) / (6p/r)a exp(-Jhz) dz Recalling the boundary conditions namely (ap/ar) =a 0 for Izl>b (6p/8r)a const. A for lzltb r-a the integral can be expressed as +b (4.2.24) G(h) = (1/42r) A exp(jhz) dz -b (A/ —r) [(exp(-jhb)-exp(jhb)]/(-Jh)(42.225) G(h) (2Ab/4r) Csin (hb)/hbl On introducing this value into equation 4.2.22 one obtins for (4.2.26) F(h)=(Ab/r)[sin (hb)l/hb] (l/7k )[l/LHl)1 (ak2h)] and the solution of 4.2.3 satisfying all boundary conditions can be expressed as (4.2.27) p(r,e,zt) (Ab/r) exp(jm - jat) I(mb,rz) where + -' +^o H^,)(rlk2-h2) (4.2.28) I(m b r z) = [sin (hb/b)]exp(jhz) - dh If the integral is evaluated at z=0, the middle of the vibrating belt, l) (4.2.29) I(mbr)z==2 Csin (hb)/hb I- - dh By introducing the value of the constant A in 4.2.27 the expression for the sound pressure at time t0, at a

- 220 point 9=0, and z=0, will be (4.2.30) p(r) = (m2poAm/).I(m,b,r)z=O (4.3 The Solution of the Sound Pressure Integral It would seem at first as though the integral 4*.228 might be evaluated by integration in the complex plane. Unfortunately a tabulation of the roots of derivatives of Hankel functions is not available, so that the method of residues cannot here be applied. It is therefore necessary to carry out the tedious work of numerical integration. But even this method involves some difficulties. At k2=h2 the integrand becomes indefinite. In order to overcome this obstacle the integral will be evaluated from zero to;. with 0 >)- >k. The part of the integral from zero to b will be denoted as integral I,, and the remaining part from 6 to o, as integral Ir. For the computation of Ir a new variable (4.3.1) y = a k"h2 will be introduced. Hence, (4.3.2) Jkjh= (y/a)3 h = (/a) ak2y and dh (yd/a a2k2-y2 Introducing 4.3.1 and 4.3.2 into Ir, it can be written as J. sin (oj7) l) (y) (4.3.3) 1 1 = 2 2~ dy

- 221 - The constants in this integral are defined as follows: o(= (b/a); B= (r/a); ak Before proceeding further the beha;viour of the fraction containing the Hankil functions at zero will be investigated Assuming m >0 and y <(1, the Hankel functions can be approximated by H(l)(y)= (y/2m.m )-J [2m(m-l)!/ Substituting these values for the Hankel functions, the fraction can be expressed as (Aym-JBy'm)/(Cym'_JD y-(m 1)+Eym-1 JFy' ( )) In this equation A....F are functions of m, but not of y. By multiplying the numerator and denominator by ymli it can be written as (Ay21 -JBy)/(Cy2m -j D2+Ey2m JF) The limit of this fraction as y-.O will be (O/-jF), that is, the integrand at y = 0 will be zero. The integral Ir itself wfll be split into two parts: the first part representing integration on the real axis, and the second on the imaginary axis. The integration along the real axis will be a 4k2-bz sin (a( KY2 H OH(m y) (4.3.5') 12 r dyM and on the imaginary axis

-222rsin(d\22U ) H(1) (y) (4.3.6) I3 = -2 7~V-77dy 0 o((I.y 2) To evaluate integral I3 a new variable x is introduced, with (4.3.7) y J x; dy = Jdx Using the relation Km(x) = (r/2)j(m+l) (1) Jx where Km(x) is the modified Bessel function of the second kind, integral I3 can be expressed as, (4.3.8) 3 =-4 (~. -. ~-1 dx v; (Ma^x ) lJl(x) + Kml(x) From a practical viewpoint it is not necessary to carry out the integration to infinity, but it is sufficient to evaluate it from zero to a finite upper limits and then estimate the order of the unevaluated part of the integral. In section 4 it will be demonstrated that it can be entirely neglected, so that the sound pressure integral can be written as, (4.3.8) I = 1 + 12 + 3 4.4 Numerical Evaluation of Sound Pressure. The numerical value of Integral I has to be established first before calculating the sound pressure. In this section, the detailed computations for the experimental

-223 - motor are carried out for a point on a radial line through the geometric center of the top rib, at a distance of I foot from the stator surface. The integral is evaluated for a frequency range from 1,000 to 2,000 c.p.s. The results are assembled in table 4.4.*4 and plotted in fig. 4.4*1. In the numerical work c = 34,400 cn/second; = 0.00121 gram/cm3, i.e., values at 20~C and 760 mm mercury pressure are used. The other constants, occurring in the integral, taken from fig. A 1.2, are a = 21 cm b - 13.2 cm; r 51.5 m As a representative example the detailed calculations for f 1,700 c.p.s. are assembled in tables 4.4.1 to 4.4.3. The magnitude of the remainder term is estimated below. The Remainder f theIteral can be obtained as follows: As the fraction of the Hankel functions in integral I decreases monotonically, it may be assumed to be less than 5 x 105. The unevaluated part of the integral 13 may therefore be written oo sin (0.629 6.512+x2) I3,R<(0.629)x 4(5)x10 0 9 dx R J 0o.,629x(6,.51 -x2)3J2 Now introduce = 0.629 a\6.512 + x and dx dw/0.629 The unevaluated part of the integral will be

- 2241I3,RI < 2x104 (sin w/w2) dw 5.6 <2xl104 ( /w2 dw< 2x10O4 1 /X 5 5.6 5.6 1 3,RI < 4x105, this value is entirely negligible. Using Simpson's rule the value of integral I for 1,700 c.p.s. can be calculated to be I= (4/3) [(0.02)(12.598+j 14.802)+(0.829)(-0.024-j 0.1017)+(1)(0.00292)] I 0.316 + j 0.381 - (0.495) /50.3 Finally the sound pressures of the main and the side noise of the experimental motor are calculated in table 4.45. The results in this table represent the theoretical solution.

Table 4.4.1 Part I Calculation of Integral I, for 1700 cps; K 0o310 1 2 3 4 5 6 7 8 9 10 h k 13.2h sin 13.2h Col H C 2x 9 2 _ k Col"-") 00 7 x col 6 061 8 x o0 [H'(a^ ) 1 0.3100 0 I 3~226 o068 +.315.219 + 1.016; I *219 11016 2.02.3094.264.988 3.194 0.079 +.314.252 + 1.003 4 1.008 + 4.012 3.04.3074.528.954 3.104 0.097 +.311.301+.965 2.602 + 1.930 4.06.3041.792.899 2.955 0.127 +.292.375 +.863 4 1.500 + 3.452 5.08.2995 1.056.824 2.752 0.173 +.276.476 +.760 2.952 + 1.520 6.1.2934 1.320.734 2.501 0.221 +.239.553.598 4 2.212 + 2.392 7 -12.2858 1.584.631 2.209.273 + -181.603 +.400 2 1.206 + 0.80 8.14.2766 1.848.521 1.882.319 +.087.6oo00 +.164 4 2.40 + 0o.66 9.16.2655 2.112.406 1.528.327 -0.022.500 -.034 2 1.000 - 0.068 10.18.2524 2.376.291 1.153.294 -.151.339-.174 4 1.36 - o0.696 11.20.2369 2.640.182.769.186 -.276.143 -.212 1 0.143 - 0.212 Z(Col 10) 12.598+3 14802

Table 4.4.1 Part II Ratio of Hankel Functions for 1I of 1700 cps 1 2 3 4 5 6 7 8 21k 5~1.5Jk-2 H4)(Col 3) H1 (Col 2) H (Co1 o) o1 5- Col 6 Col 4/Col 7 1 3 J 1 3 1 j 1 6.510 15.97.184-0.077 -.151-.275.038-.328 -.113 -.603.068+.315 2 6.497 15.93.180-0.085 -.154-.274 ~035-.329 -119-.603 079.314 3 6.455 15.83.172-.103 -.166-.267.022-.331 -.144 -.598.097+.311 4 6.386 15.66.152-.132 -.185-.257.000-.334 -.185 -91.127.292 o 5 6.289 15*42.118-.165 -.210-.240 -.027-.336 -.237 -.576.173.276 6 6.161 15,11.063-.195 -.242-.214 -.066-.335 -.308 -.549.221.239 7 6.002 14.72 -.016-.208 -.277-.175 -,115-.328 -.392 - 503.273.181 8 5.809 14.24 -.111-.180 -.309-.122 -.171-.311 -.480-.433.319.087 9 5.576 13.67 -.194-.095 -.336-.050 -.235-.279 -.571 -.329.327-,022 10 5.300 13.00 -.218.046 -.346.045 -.305-.223 -.651 -.178.294-.151 11 4.975 12.20 -.125.193 -.325.155 -.368-.141 -.693 +.014.186-.276

Table 4.4,2 Part I Calculation of Integral 12 for 1700 cps = a-k 6.51 c= 0.629 13 2.452 1 2 3 4 6 7 8 9 10 2 2 2 Col2 H(fy) Col 7 x Col 6 Col 8 x Col 9 s urr -s utJ s sin (Col 4) Coli I~;/y Col 4 Ool 3 --- IH^^ )'(y) 1 j 10i2 jxO'2 10 jxlO 1 0, 6.510 4,095 -0.199 -.031 2 0.829 6.457 4.061 -.196 -.030 -.0628-.041.190 +.124 4.76+.496 3 1-658 6.295 3.960 -.184 -.029.238 -.171 -.697+.501 2 -1.4 + 1.00 4 2.487 -6.016 3.784 -.18 -.026,123.326 -.323 -.857 4 -1.29- 3.43 5 3.316 5.602 3.524 - 106 -.019 -.323.151.610 -.285 2 1.22 -.S7 6 4.145 S.019 3-157 -.005 -.010 -.216 -.258 -.207 +.248 4 -.828+.992 7 4.975 4.199 2.641.182 +.043.160 -.233.692 -.889 1 +.692-.889 (Col 10) = -(2.2+j 1.5) x 102

Table 4,4.2 Part II Ratio of Hankel Functions for 12 of 1700 cps 1 2 3 4 5 6 7 8 y y H(1)H) (1) 2 0 ) 1 v H 2 y) i( ) 03 y Col5-Col 6 Col 4/Col 7 1 j 1I 1 3 1 1 3 2.829 2.033.359 -.602.380.944 -.011+10.29.369 + 9.352 -.0623-.041 3 1.658 4.066.347 +.234.575 -.310 -.080+ 1.65.495 + 1.340.238 -.171 4 2.487 6.099 -.261 +.204.499 +.145 -.215.763.284 + 0.908.123.326 5 31316 8.132 -.077 -.273.215.389 -362+.423 -.147 +.812 -.323 1 6 4.145 10.16.250.033 -.120 +.376 -*434+.131 -.554 + 07 -.216 -.258 7 4.975 12.198 -.124 +.193 -.324+.156 -.370.138-.,694 +.018.160 -.233

Table 4.4.3 Calculation of Integral 13 for 1700 cps = a. k 6.51 0= 0.629 3= 2,452 1 2 3 4 6 7 8 9 10 11 12 2+ xl2 __ s1i (CoLL3) A4..x) K (x)+ ol Col 9 Col 9 Col 2 Col 3 Col 2 + 8 x x R3(x) ~ Col 5 Col 10 02 0 xlO'2 x10'2 xlO ^ x104 0.. 1 6.587 4.143 -.203 -3.08 2.45 13.0 7.70 1.69 -5.20 4 -23.8..'-.. ~ ~. -. ~ I 2 6. 810 4.283 -.212 -3.11 4,90.596.787.76 -2.40 2 - 4.8 3 7.168 4.509 -.217 -3.03 7.36.037.162.23 -.70 4 - 2.80 4 7.640 4.806 -.207 -2.71 9.81 +.003.042.071 -.40 2-.80 5 8.209 5.163 -.175 -2.13 12.26 +.000.012.016 -.00 4 -.00 6 8.853 5.569 -.118 -1.33 14.71 +.000.003.000 -.00 1.00 Z(Col 12) =-29.2 x 104

- 230 - Table 4*4.4 Calculated Values of the Soundpressure Integral for m 2; at E = 0; z = 0 Integral I Frequency r cps. cm. Re -I Magn. Arg. 1000 -0.646 -0.163 0.67 -194.0 1100 -0.597 -0.286 0o663 -154.3 1250 515 -0.243 -4525 0.58 -105.4 1400 0.337 -0.455 o.66 - 53.4 1500 0.504 -0.2 o.542 - 21.5 1600 0.511 +0.139.5 6 15.2 1700 0.316 0.381.495 50.3 - 1800 0.029 0.469. 470 86.8 1900 -0.27 0.417 0.48 122.0 2000.-0.398 0. 23 0.46 150.0 1700 82.0 -0.242 -0.203 0.315 -140. 2

a~~~~~~~~~~~~~0 0 IoI - - I. 1 -. ii 4111 - - t:I-. —. ".~41 1II -4 -, — 4 — - I t f <j.- ~ I.ij, I< T - p~I I~.- II~,.r I -1 - -1I 211:121. 41 I+ / +4I. 124 J- II I I'I -_:,_ 12- 2ifz11- I I i I,. I I >-.Ii. K-'- II i.I tI_. I, ~ I.I —i, I -: I ~ I I -:-.- I-,~ -- - I~iI I -I-i''i ~. I~ -. - I,1 t I. I...11,: I_. - I." -~ -a-+-, -__ ____.__-_ —-_-_ -I' _m' - — I —.I. I t-fI~~~I~ — ~ ~ I.- I.II I 0.I-~I....~+++-.I:II- 0..,I I~~I II-4~~I. I.1.;I.. I!I.~~~.I..IIIIII0- - I.I? I tI I- - I..-I6 6.

Table 4.4.5 Calculated Sound Pressures of the Experimental Motor 1 2 3 4 5 6 7 8 9 Motor _Noise def l. I o2 log10 RS Operating f cm SPL.~ A, V, cps cps 106 1 C db. 40, 440 60 1,716 19.6 0.492 116.0 1.98 4.296 -85.8.9 40, 367, 50 1,419 30.7 0.60 79.0 2.43 4.385 7.7 40, 294, 40 1,121 60.5 0.650 17.8 49.6 3.47 4.540:90.8 9 N.L.,440,60 1,475 5.34 0.485 118.9.55 4.738 74.8 40, 440, 60 1,35 1.51 0.545 86.0 0.126 3.100 62.0 40, 440, 60 1,716 19.6 0.315 116.0 1.28 4.107 82.1 Col. 3 identical with last Col. of Table 3.6.6 on page 200. Col. 4 values are taken from fig. 4.4.1 on page 231. (Col. 5) bp/ 2(0.0002)1 = (13.2xl.21x10 3x103/r 2) =.17.8 Col. 7 is the product of Columns 3$4,5, and 6 The value of r is 51.5 cm in all rows,except the last row where r = 82.0 cm

V. EXPERIMENTAL INVESTIGATION OF THE SPECIAL MOTOR The aim of the experimental investigation of the special motor was to obtain a check on the calculated results, and thereby to verify the theory as developed in chapters 2 to 4. While the tests are considered auxiliaries to the theoretical work, their importance should not be underestimated. In this very complex problem- test results serve as invaluable signposts for assumptions.o be made for any future analysis. At an early stage tn the author's special assignment which led to- this analysis it became clear that the main purpose of his endeavors should be directed towards the theoretical predetermination of the sound levels of the magnetic noise, rather than towards the development of some special test equipment. Consequently only commercially produced or already existing apparatus ha4 to be used. The lack of special instrumentation did not limit the scope of the investigation to any considerable extent* The importance of the resonant frequencies for the analysis ha6 been stressed at the start of chapter 2. Because of the predominant role of the eigenvalues the tests for natural frequencies and associated mode shapes were

-234 - tests consisted of sound pressure level measurements and the search for mode shapes of vibrations. 1.! Test for Natural Frequencies and Associated Mode Shapes. Description of Equipment and Tests, In this test an external vibratory force was applied to the motor and the ensuing vibrations examined. The general arrangement for this test is shown in fig. 5.1.1 The vibratory force was supplied by an electromagnetic shaker, as illustrated on the top of the motor in fig. 5.1.1, and marked A. This shaker was a simple iron core coil whose magnetic circuit partly closed through the motor under test. A small airgap was maintained between shaker and motor in order to avoid saturation effects. The high impedance of the coil (approximately 500 ohms at 1,000 c.p.s. measured with its magnetic circuit open) limited the usefulness of the shaker to a frequency range from 20 to 2,000 c.p.s. The force exerted by the coil, an inverse function of the airgap, was of the order of 1-3 lb. In actual tests the position of the shaker was varied to drive the motpr at the amplitude of the mode shape, in order to achieve clear results. Most of the time the shaker was applied inside the stator and it became necessary to remove the two end shields and the rotor. A variable frequency supply as shown on the right of fig. 5.1.1 was used to energize the coil of the shaker.

- 236 - A Hewlett Packard Oscillator Model 200D and a 200 watt power amplifier served this purpose. The frequency of the oscillator and of the flux pulsations in the coil were the same. The attractive force of the coil, proportional to the square of the flux density in the airgap between shaker and motor, has a uniform and a second harmonic component. The latter set the stator into vibration. The vibration of the stator driven by the shaker was sensed by an inertia-operated piezo-electric pickup. The sensitive element of this device consisted of 96 per cent barjum titanate, 4 per cent lead titanate crystal of 0*25 inch diameter, and was 0.025 inch thick. The maximum outer diameter of the pickup was 0.6 inch and the height 0*8 inch. The accelerometer was mounted on a small magnet. From the electrical point of view the accelerometer may be considered as a zero impedance charge generator in series with its capacitance. The voltage sensitivity was about 0*4 millivolt on the end of a 3-ft cable* The output of the acceleration pickup was applied to an amplifier, in this case a General Radio Sound Level Meter, Type 1551-A. The amplifier in turn was connected to the Y terminals of a Cathode Ray Oscilloscope, and the X terminals of the C.R.O. to the output of the oscillator. The oscillator was gradually varied by the opera

- 237 - tor who simultaneously followed the Lissajous figure on the screen of the C.R.Oi The trace he saw was the characteristic figure eight of the superposition of two component oscillations, where the frequency of one oscillation was the double that of the other. When resonance was reached, the Y deflection on the C.R.O. was a maximum and the Lissajous figure assumed the shape of a parabola. On slightly increasing and decreasing the frequency a phase change of the trace could be observed. Now the outer and the inner ring were sensed with the pickup and the shaker moved to the position ef maximum amplitude. Keeping the shaker at a resonant frequency the pickup was moved along the outer and 01;'r ring and the phase of the Lissajous figure, which changed continuously, again observed. At a point of zero displacement on the stator, the amplitude of the Lissajous figure was a minimum, and the phase of the trace became indefiniteo Such a point Is generally referred to as a node of vibration. Moving the pickup still further a change of phase occurred. The amplitude of the Lissajous figure increased steadily until the position of another maximum amplitude was reached. The shape of the LissaJous figture at this second maximum amplitude was the mirror image of the first amplitude's trace. This procedure was continued till the whole stator had been sensed. The positions of maximum deflections and nodes

-238 give a more or less clear picture of the vibration mode. The nodes on the actual motor, as seen from the test records, were not as neatly and regularly spaced as the calculations of the simplified structure in fig. 3.3*1 would indicate. This was due to the fact that the mass distribution is not entirely symmetrical. As a consequence of irregularities in the casting and incomplete symmetry in some instances, two modes of vibration, though clearly of the same category, but differing by a few cycles were found The results of this investigation are recorded in fig. 5.1.2 for the second mode vibration, and in fig. 5.1.3 for the third mode case. The comparison of calculated and measured resonant frequencies and mode shapes is provided in the next chapter, while only the difference between tests and theoretical results is being pointed out here. The theoretical analysis yields counter-tact cosine modes at 10,000 and 1,290 c.p.s., that is in the first and the fifth principal modes. It was impossible to excite these resonances on test, which was not surprising in the case of the first principal mode as its frequency was well beyond the range of the shaker. A thorough search for the fifth principal mode was therefore made with a more powerful shakerl, again without succeeding to excite this mode I The additional tests were carried out through the courtesy of Dr. R. Plunkett, at the General Engineering Laboratory of the General Electric Company.

XB-atme^ ^ t/ I i....'- -,\i'I-. i.~:'\ -~ -.:..'i. ~~'.T, -'''~' n~f' T.~.: ~'. X -SI' — D. 1.: d -. -. I In-tact CoAinte Jt...-.. —-..'" i-g. 5.2 L | -....Secueid Mode Resonanes of,' the xperimental Motor ( "asured) n-taPt "na Motr-a-(tMau

~ ~)~~b-) L...., i~~~'' -^S~e / i.n-iact^ \ if n o ThirdI II~~MdeRsareofheEpeie aloor(a/ ln-t~cts \ /Co2 antr-taet i Cos ine si\\\ ^ ///^ Sine ^\\\ / ^ /-/ff\ incps L;;' Third Mode Resonances of the Experimental Motor (Meastrd)

-242 - of vibration. It was therefore concluded that the construction of the stator as illustrated in fig. 5.1.4 was responsible for the overdamped conditions in the countertact cosine modes, as discussed in Section 3.2. 1.2 Measurement of Moter Noise Sound Pressure Levels. This investigation consisted of two major steps. First the motor noise was recorded on a magnetic tape at different operating conditions. Later the magnetic tape was used to record the amplitude of the sound pressure levels as a function of frequency. The *Ceordin of the Motor's Noise Output was carried out in an anechoic room. An outline of the room and the position of the motor in the room are illustrated in fig. 5.2.1. Fig. 5*2*2 shows the relative position of the motor and the microphone. The air deflectors have been removed for this test. The difficulty of constructing a suitable anechoic room for as large an equipment as a medium induction motor should not be underestimated. It would be almost financially prohibitive to construct a chamber with perfect sound absorbing wallst as described by Beranek and Sleeper2 It also seems practically impossible to mount 2 11;55 13

- 243 - \4\-i Pi M24ot__ Pasition of Motor in Noise-Measurement Laboratory

-24 induction motors, weighing 200- 2,000 lb, on a suspendedt tightly stretched steel net, to conform with the design of anechoic chambers, which are usually divided into upper and lower halves* The vertical walls of the anechoic room used for this test were lined with a 6" deep and 1" thick corrugation of Fiberglass pads, and protected by a chicke wire fence. The ceiling at a height of 18' was also covered with a layer of Fiberglass pads. Unfortunately the equipment of a traveling crane and a system of pipes cluttered up the area near the ceiling. The floor of the sound room was constructed of concrete covered by Fiberglass quilts and protected by steel grids. One corner of the floor consisted of 48 steel blocks each 14 inch square, as seen in fig. 5.1l1* During tests these blocks were also covered by Fiberglass quilts. The anechoic room located in a production area had common foundations with the main building. As a consequence, all low frequency vibrations, such as are always present in factories, were directly transmitted to the sound room. The best position for the motor would have been at the geometric center of the room, but the machine was actually mounted on a 30" high base in a corner of the room. This relatively unfavorable position was required by a hydraulic absorption dynamometer which

-246 - was mounted outside of the sound room. The dynamometer served as a load to the motor and had been originally designed as a loading device for motors of 500 - 1,500 HP. It proved difficult to operate the motor at a uniform output even for the short periods of sound recording. A 30" x 24" double-glass window (which served for communication with the switchboard operator, controlling the load of the motor) located above the protruding shaft most probably acted as a sound reflector. In short, the sound measurement room left much to be desired to meet the stringent requirements of field free conditions for precise and constant measurements. However, ideal conditions for tests of large equipment are ordinarily difficult, if not impossible, to establish. A non-directional microphone of the Rochelle cry- stal-diaphragm type (as the first element of the sound recording circuit) was mounted on a tripod one foot above the center of the motor. A low capacitance line connected the microphone to a General Radio Sound Level Meter, Type 1551-A. The Sound Level Meter is basically a calibrated attenuator amplifier, including an indicating meter and a weighing network to modify the amplifier response to simulate response conditions of the human ear to pure tones at specified levels. The amplifier is stabilized by means of inverse. feedback and has a fairly flat fre

-247 - quency characteristic over the range from 20- 20,000 c.p.s. The output of the amplifier was connected to a Magnecorder Type PT-6 decade amplifier. This amplifier with a flat frequency response from 20 - 20,000 c.p.s. has a low hum and noise level and low linear distortion* The output of the amplifier was fed into a Magnecorder Type PT-6 tape recorder, a high quality instrument with a wide dynamic range and of excellent constant speed. The speed of the tape during recording was approximately 18" per second. Before, duringland after the test the whole sound recording circuit was calibrated with a General Radio Type 1552-A Sound Level Calibrator.1 The calibratorr consiated of a small stabilized loudspeaker mounted inan enclose: The enclosure is so designed that the speaker is always connected at the correct distance from the microphone diaphragm, that is fitted tightly over the microphone. The loudspeaker was supplied bya tuning-fork operated, ~ixed frequency oscillator of 400 cp.s. When the adjustable output voltage of the oscillator was set to two volts, the calibrator produced an 85 db SPL at the microphone diaphragm. The Recording of the Sound Spectrum 130

- 248 - The amplitude of the noise plotted against frequency, and referred to as a sound spectrumwas recorded with the aid of an arrangement as shown in fig. 5.2.3. First a representative sample of the tape of each recording was selected, spliced into an endless loop, and fitted to the previously described Magnecorder Type PT-6 recore r" The output of the tape recorder was supplied through the, Magnecorder Type PT-6 amplifier to a General Radio Type 736-A lave Analyzer. The wave analyzer is actually an a-c operated vacuum tube voltmeter of the heterodyne type. The intermediate frequency amplifier of this apparatus includes a highly selective quartz-crystal filter. The use of the heterodyne circuit renders the apparatus very useful for the recording of noise spectra, as it makes it possible to use a fixed frequency filter for the whole range of variable frequency. The narrow band-width of this instrument has a 4-cycle flat top and has a high rejection outside the passband. The wave analyzer's output is connected to a recorder which in turn drives the wave analyzer dial synchronously by means of a transmission link as Illustrated In fig. 5.2*3. The final element in the recording of the sound spectrum was a Bruel and Kjaer High Speed Level Recorder, Model BL-23e4. This is a graphic recording peak indicating

250 - a.c. voltmeter, designed for a frequency rge e/ ~0 to'25a' e.p.'s,. The principle of the level erdles operatrn tis that of a balanced tye of atly:1u pchsiten-:tig ~systemma de possible by a ptntiometereoazlr 0....',... -.......,.''........ mIoninggl-e dri^ing system. 11he negat le feedbaek rseoarranigement provides a high degre of stabil ity an^ damp-?g.i?~ Thie:o't:'put o:f the recorder, a rectilinear -traeu4 -is prodced- by a, sapphire stylist on opaqte waxd paer h: writting speed is 500 mm per, s-eiond.ndh t harot-. speed I mmm -per second. During the recordings:f:he motor, noise out-'ut;of Calibrator type 1552- was roo$f<r 4 deter.mit tion of the absolute s.ond level.: -':.;-:;.:' - Before preenting the4'eerds th so sure*li.,...,.... -nents a diseusio on the acctiray of the neaurm:.ms:hdO seems indicated. According to manfact ri i.tion:the aceuracy of the type 155l-ASO:nd Level er ssocirted- icrophone Is within pls- and mir4s I d:;tat of type 5I2. Sound Level Calibrit:or alSoplus::omius.l'db..'ul ep -:rnts by R4. W els sem to:indiate that the- tape.recordg my also int'oxe errors i-of 1I db i:n the frequeney range between 1 l:-:O0 d 2: e".p.s." The:type BL 23-04cecoder has an^aeura.cy of' plus- or minus 74

Th % ~ ~ L j.;:'" ~ ~ ~ ~.. ".^ Su S~1~ of- ~ p - a ir~s~ - r a-+ 4~I-4 Anp, 440' r 60cps, (Overall SPL b Li' - - ~~- 2;.: "... p

,, 4. C r 4- en: ~i n.) C 0 6., De Y` -a A.c r T r Qr.L a~t,.,. i..i _rJ4 7r~ 1, f+ q ~~~~~ i -:4 ~ ~ ~

IT l.6 Gm~ l2m ~,C~ ~~~ ~ ~ ~~~ _, 0 ~, ~".'? "i,.' C?'.,.... -c,. 4 0,^ "..4 $.6s 0 7 r...'-.3:': - -, 4.', 8 - ) E. *2? * ~: 6 2} - "-, r"'-^

So~:~ ^ xc r K Mtr >Laec"dn o4 EJ ) * (>xeer M- otor a SPL 7-I dbh ~S1. "'^'''' ~ S' ^ *. ~'. " "~:~~- ^0 5.' 7'", -~ ")~' ~.' -- i... 2.^ 2 "'~.' c-: c; ~~. )0 ". P..-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~it.

t' 4 - - Sound Spectrum of Exrernnta Totor -a No Loa", No Excitation

?u ^t 6C 80 ~"'0 -^0 20" ^^ ^^C ^>.... S~~.20.~i 5 FIg.'.2.9 Sound Spectrum of Exp~r:' mental?otor at 40 Amp, 460 V, (C cps, (Overal1 SPL 84 db Sound Spectrum of F- Ove rca I IS n La d n.15 40 45 50 ~. 6 ~ &.C?5'. Note: The microphone at this recording was 2 feet frozr the motor.

Fic. ^2,10~foa' WISH~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~t S'r" Srl t ru Ambien t NTh Anechroc Room (SFL 63 dh) 4-5 i- -'~ -.- B <sL 22 ~ ~2 1.? ~: -:' ~ P S.

28 - one per cent. As most of the recordings were at 90 db SPL it may safely be assumed that the overall accuracy of the recordings should be in the neighborhood of plus and minus 4 db. Figures 5.2.4 and 5.2.10 are the records of sound spectra taken at different operating conditions of the motor. At all recordings, except the one illustrated in fig. 5.2.9, the microphone was at a distance of 12" from the stator in a radial line through the geometric center of the top rib. The position of the microphone for the recording in fig..52.9 was in the same line at 24" from the stator. 5..3 Determination of Mode Shapes under r0eratinL Condinion This test is important for the determination of the predominant mode of vibration while the motor is operating. It serves as another check on the conclusions drawn from the theoretical analysis. The vibrating operating motor is sensed simultaneously by two vibration pickups. The output of each pickup is connected to a General Radio Type 760-B Sound Analyser. One of the sound analyzers is connected to X terminals, while the output of the other is coupled to the Y terminals of C.R.O. The sound analyzer is a battery operated, continuous spectrum instrument. It includes a 3-stage, direct

-259 coupled amplifier which is made selective by means of a tunable null circuit in a negative feedback loop. The frequency can be selected by a single dial and a set of five push-buttons. The band width is a constant percentage of the frequency to which the analyzer is tuned. This is very important as the speed of the motor under test is not absolutely constant. During the test one of the pickups was fixed to a characteristic point (on the outer ring at the center of the top rib) of the motor with the aid of a magnet. The sound analyzer was tuned to the same frequency as the motor vibration whose shape was to be determined. A second vibration pickup was placed near the first one and its sound analyzer also tuned. The trace on the screen of the C.R.O. was the characteristic Lissajous figure of two vibrations of the same frequency in phase, that is a coinciding straight line. After this adjustment the second pickup was gradually moved along a circle of the outer shell until the axes of the elliptical trace on the C*R.O. became horizontal and vertical, respectively. This point of the vibrating body is a quarter length from the fixed point of the first vibration pickup. By moving the second pickup farther away, a point was reached where the other diagonal of the rectangle enclosing the elliptical Lissajous figure appeared. This point was exactly one half wave length from the fixed

260 pickup. The predominant mode shape of the 1,716 c.p.s. main noise and the 1,475 c.p.s. side noise had been investigated and both found to be second mode vibrations.

VI. CONLUSONS The major objective of the present thesis is to establish in a general manmer,a method for the calculation of sound pressures caused by the magnetic noise of medium induction motors, at a characteristic point near the motor. This task has been performed in three steps: 1) the development of a theory for the calculation of the electromagnetic forces in the airgap of induction motors 2) a theoretical procedure for calculating the natural behavior and the forced response to electromagnetic force waves in medium induction motors 3) the development of a method to calculate the sound pressures resulting from the forced vibration of induction motor stators. The results of the theoretical analysis are given at the end of chapter, and test results in chapter 6. These results are compared prior to the discussion of the merit of the method presented In this thesis. 6*1 Discussion of Results Table 6.1.1 assembles the resonant frequencies of the experimental induction motor in different modes of vibration as calculated in section 3.5, and the experi- 261 -

Table 6.* 1 Comparison of Calculated and Measured Resonant Frequencies of the Experimental Motor lode Descrip. of Mode Calculated Measured Error eps in cps Fig. Per Cent table Counter-tact sine 1,140 3.5.11 1,220 5.1.2d 6.6 2 In-tact eosine 460 3.5.11 468 5.1.2a -.8 482.1.2b -4.4 In tact sine 490 3.5.11 473 5*1.2c +3.6 In-tact sine 1,225 3.5.12 1,240 5.1.3a -1.25 3d In-tact cosine 1,170 3*5.12 1,080 5.1.3b +8.5 1,156 5. 1.3 +1.2 Cotnter-tact sine 1,580 3.5.12 1,540 5.1.3d +2.5

Table 6.1.2 Comparison of Calculated and Measured Sound Pressure Levels of the Experimental Motor'1 2 3 24 5 6 7 8 Motor Descr. Noise Calc. Meas- Rece Error Operation of f on a SPL B PLF~f Noise Fig. A, V, cps ft cps db db Per cent 40, 440, 60 1 main 1,716 85.9 87.8 5.2.4 -2.2 40, 367,50 I main 1,419 87.7 90.0 5.2.5 26 40, 29440 1 main 1,121 90.8 94.0 526 -3.4 NL, 440,1 6 I main 1l735 74.8 74.0,52.7 +11 40, 440, 60 I side 1,475 62.0 64.0 52 -3.1 40, 440, 60 2 main 1,716 82.1 84.5 5.29 -2.9 Col. 2 denotes distance of microphone from stator Data In Col. 5 are taken from Table 4,4.5 onpage 232.

- 264mental results provided in figures 5.1.2 and 5.1.3* In table 6.1.2 the sound pressure levels of the magnetic noise of the experimental induction motor, calculated from theoretical considerations (table 4.4.5), are compared with the test results given in figures 6.*24 to 6.2.10* A study of the results reveals a good agreement of calculated and measured values. Considering the complexity of the problem, the approximations and assumptions, together with the large number of numerical operations which had to be made, the agreement between theory and experiment is better than expected. This close agreement may be partly due to unknown fortuitous circumstances. It would, as a rule, be unreasonable to expect the theoretical results to approach sound measurements by such a close margin. In view of the agreement between theoretical and test results it can be concluded that a method has been established to calculate: 1) the natural behavior and forced response of stators of medium induction motors, and 2) the sound pressure levels of the magnetic noise caused by electromagnetic radial forces in the airgap of induction motors. The author again stresses the fact,that while the

- 265 - analysis as presented here is of a general nature, solutions for specific designs of induction motors would have to be carried out separately for each class of design. A full appreciation of the method herein developed would require a comparison with results obtained by other methods. Unfortunately, however, no such complete studies accompanied by test records appear to have been published. Hence such a comparison seems impossible. This lack of data is explained by the fact that the predetermination of magnetic noise is still at an early stage of developmernts 6.2 Application of the Method The obvious question arises if this tedious and complex method involving the use of most advanced electronic digital computers is practicable, in view of a designer's busy everyday schedule. Whenever a new line of motors is under design, the advanced design staff usually has the facilities to carry out the calculations for every new frame size. Curves representing the deflections as a function of frequency and mode order for a constant amplitude force wave can be plotted in the manner illustrated in fig. 1.2. The solution of the integral necessary for the calculation of sound pressure levels can also be plotted as a function of noise frequency and mode order for a characteristic point of the induction motor. The

-266resonant frequencies and eigenvectors of each frame can be tabulated as shown in table 3*.510* The only analytical work left to be performed by the designer would be the calculation of the force waves. The remaining task would then be the selection of the right information from prepared curves and the calculation of the sound pressure level by simple slide rule operations. Clearly the total study is a greater task than can be carried out by a single individual. However, once done, it can be used readily in design practice. P. Lo Alger, who has discussed in considerable detail favorable slot combinations and other electrical design precautions to be taken in the construction of low magnetic noise induction motors, has very appropriately coined the expression "sonance design". An extension of Alger's recommendation should be underlined. Designers should realize the importance of the number of resonant frequencies in each mode, as already mentioned in the introduction, discussed theoretically in Chapter 3, and shown by test results in Chapter 5. It does not suffice to avoid low order force waves caused by slot harmonics, as belt harmonics may also cause low order force waves. These latter are of much lower amplitudes than those of the slot harmonics, and may become troublesome if their frequencies coincide with the natural frequencies of the motor in the same mode.

- 267 A strong word of caution is necessary. Generalizations by applying the results of one design to another, as often observed in design practice, may prove precarious. A few instances of false generalizations may serve as examples: 1. One of the opinions often encountered is that the magnetic noise could be considerably reduced by increasing the radial lenght of the non-slotted portion of the stator core. Although it is true that the static deflection will decrease approximately as the third power of the increase of the radial depth, it is wrong to assume that the dynamic deflections will decrease to a negligible value. In fact, the low mode natural frequencies of the stator will increase and, consequently, the resonance factor will substabeially increase. A careful estimate would prove that by increasing the radial depth of the stator core of the experimental motor by one inch, the main noise sound pressure level could be reduced by 3 or 4 decibels only. 2. From the results of Appendix IV one would tend to conclude that by replacing all radial ribs by tangential ribs, the coupling between the two rings would loosen, and the outer ring vibration amplitudes would decrease. Although this conclusion is correct, it fails to consider other essential changes. Both fig. 1.2 and the result'

- 268 - of section 3*5 reveal that the lowest mode cosine and sine frequencies are very close to each other. The designer may avoid both by one slot combination. By loosening the coupling of the two rings the two frequencies which are normally close together will be spread out, so that the designer will have great difficulty to choose an electrical design in order to avoid several natural frequencies of the stator. 3. It appears that many designers, well aware of the danger of second mode force waves, believe that the second mode vibrations could be suppressed by the selection of a frame design with six ribs. The results of the present thesis prove the fallacy of such a concept. The foregoing examples support the general view expressed in this thesis that every individual design must be separately analyzed and the sound levels of the magnetic noise separately calculated. Broad generalizations may lead to major inaccuracies.

APPENDIX I Data of the Experimental Motor Name Pate Data 30 HP, 4-pole, 440 volts, 3-phase, 1,780 r.p.m. 38 amp*, Temperature rise 40~C., Frame K365 Bjectrical Desifn DatOuter diameter of stator punchings 13.5t Diameter of stator bore 9.00" Outer diameter of rotor 8952" Inner diameter of rotor 2.625" Airgap (design) 0 024" Airgap measured, average 0.0225" Length of stator 5.75" Net iron length of stator 5.175" Flux per pole 0. 012 weber Number of stator slots 60 Depth of stator slot 1.15" Average width of stator slot 0.30" Stator slot opening 0*125" Average width of stator tooth 0225" Stator slot pitch on bore 0.470" Number of conductors in stator slot 16 Total number of conductors 960 Double-layer stator winding circuits I - 269 -

270Phases connected in delta Coilpitch 1 to 12 Number of cast aluminum straight rotor bars 54 Rotor slot shape fig. A 1.1 Cross-sectional area of cast aluminum rotor end rings 1I50 sq.in. 0.02o0 0.030 "-oJ.5i 0.z5.-.... oo- 0 - -> —-t 0.25 0.2 3 0.0 o.' -0050 Fig. A 1.1 Rotor Slot of Experimental Motor v~lyated Data for Section 2.-10 Pitch factor, fundamental 0.914 Distribution factor, fundamental 0.956 Winding factor, fundamental 0.874 Airgap 5'. XO4 meter Carter coefficient 1.175. Corrected airgap 6.7x10 4 meter Pole pitch 0.18 meter Net length of iron 0*13 meter Length of stator 0.145 meter Primary leakage reactance at normal operation 2.00 ohms

-271 - Primary effective resistance 0 75 ohm ~M Exciting susceptance -. 54x10 ohm Iron loss conductance 2.2 xl 3 ohm Secondary leakage reactance in primary terms 2.27 ohms Secondary resistance (dc) in primary terms 0.3 ohm Load resistance at 40 amp load in primary terms 19*7. fhms High frequency leakage reactance of motor bar in secondary terms 1* 01xO'1 - -hms 0 go:: Test Conditions Supply voltage at 60 c.p.s. 440 volt ( {lte: "The Supply voltage was reduced in ratio to the frequencies at tests as represented by figrets 5.2.5 and 5.2.6) Line current 4 0 amp Phase current 23.2 amp Power factor 0.895 Slip 0.015% Power output 32 HP Efficiency 0.90 No-load current 11.8 amp No-load power factor 0;12

- 273 Mechanical Design of the Stator The details of the outer ring are illustrated in fig. A 1.2. A thorough inspection showed that not all of the dimensions of the actual casting were in agreement with the drawing. The correct measured dimensions are given in fig* A 1*3 Eviluation of Data for Sections 3*4 and 3.6 OD.J5 -— ~ —-------------- ------------ 8.875" ---------- - ---— " —- -" —-— ~ — ----— ~ —-- ---......... -------. C.G. 0, 3Z5" t 7.E 9!' 7.84" _ B Fig. A 1.3 Outer Ring Cross-Section Outer Ring Parameter Inertia factor: The weight of the outer ring is 95 lb so that the inertia factor is p"o (R/2 /2)(95/21)(1/386) 6.153 x 10The centroid of the cross-section is at a radius of 7*89 inches; the moment of inertia is IO 0.5924, and the flexural rigidity factor (TD/2R3 R) ('r/2) (16x106/0.91) (0. 5924/490,49) = 3 337x10

-274 - Inner Ring The weight of the inner ring is 120 lb and the inertia factor is (irp/2) = (120/386)( /2ir)(r2/2) =7.778x102 The flexural rigidity factor of the inner ring is (rDI/2r3) = (r/2)(30x106/0.91)( 5. 175/12)(I.l/6.2)3 = 1.24716x106 Support Parameter The length of the support is 9". The effective moment of inertia is 0.176 so that the bending rigidity factor is D = 3 1x106 Rib Constants The rib is of a complicated shape. To simplifycalculations it was replaced by an effective rib as shown crosshatched in fig* A 1.4 1 — 75'1 Fig. A 1.4 Detail of Rib Cross-section

-275 - The extensional rigidity factor will be ErAr/r (16xl0O )(4/3)(3/8)(5.75) = 46 x o6 and the bending rigidity factor will be Dr ErIr/(1-62) (16x16)(V/O.91)27/512)(1/12)(5.175) = 3.8x105

APPENDIX II Energies of the Vibrating Rings (Extensional Deformations Considered) Preliminary calculations proved the influence of extensional deformations to be negligible for a low mode response of the experimental motor. However, 2-pole motors have generally very deep stator cores so that the ratio of the radial ring thickness to the median raditus is larger than 0.15. In such cases the consideration of extensional deformation may provide more accurate results. The necessary formulae for the ring energies considring these deformations will be derived here for the inner ring only* The expression for the outer ring is analogous. The radial displacement of the ring will be chosen according to equation 3.4.2, as (k 2.1) = am cos me+ b sin mO The tangential displacement is now expressed as (A 2.2) w cm cos m + dm sin mO The Kinetic Energy can now be calculated by 3.2.17 T = (p:/2) (T+f) de 157 - 276 -

- 277 as (A 2.3) TI = (p1/2) I(am d) The Bending Energy of the ring can be calculated from equation 3.2*9 by introducing into this equation -d2u/de2 = m2(a dos mO + b sin me) dw/dO = m (+dm cos me - cm sin me) The resulting expression will be 2ir (A 2.4) Vb = (Di/2r3) am+d )m cos m+ o +(tm-c)m sin me]2 de which reduces to (A 2.) TI = (irD/2r) m2m+m )2(2bm-m )2 he Extensiona Energy can be calculated by introducing into 2r (3.2.12) Vt (EIA/2r)/ [(dw/dO) + u]2 de 0 the expansions of dw/dG and u which will result in (A 2.6) Vt = (T/2r)EAI j (am +mdm) + (b -mm)2 Basically all the other expressions will remain unchanged.

APPENDIX III Diagonalization of a Real Symmetric Matrix on the IBM 701 Electronic Data Processing Machine The programing of the diagonalization of a real symmetric matrix is based on the theoreml that the eigenvalues of matrix T A T are identical with the eigenvalues of matrix A. Given an arbitrary symmetric matrix, an orthogonal matrix S of the following form may be chosen 10 1...................... 0 0 01 *0.ril*..r. I0*-***.SW.. i.......... *...... 1 00 0 where only the elements rii, rij rji, rj are different from the corresponding element in the identity matrix (of the same order as matrix A), having the property that the element in the (i,j) position of the product S A S' is zero. 1 3354 - 278 -

- 279 - The first S1 is so chosen that the first row, second column element in SiA S1 is zero. The S2 is chosen so that the first row third column element vanishes in the S2S1A S1S2. This procedure is continued, sequentially annulling each element above and below the diagonal. Nonrzero quantities may, however, reappear in off-diagonal positions at a later stage, and the matrix becomes non-diagonal when a cycle is completed, that ls$ when the nth element in the (n-l)st row has been annulled. An orthogonal transformation though, leaves invariant the sum of the squares of the matrix elements. It has also been shownl that the sum of the squares of the diagonal elements increases monotonically throughout the process. It follows that the sum of the squares of the elements off the diagonal decreases monotonically. After a cycle of operations is completed, the whole procedure is repeated, starting with the second element of the first row and continuing to the nth element of the (n-l)t row, until the sum of the squares of the off-diagonal elements decreases monotonically so close to zero as to become negligible. Approximately 30 to 40 cycles will be necessary on the IBM 701 Electronic Data Processing Machine to render the sum of the squares of the off-diagonal elements negligible. The result will 40

-280now be a matrix (SmSm-_l.S1)A (S...S m ) which is diagonal. This resultant matrix is of the form T 1A T, and its diagonal elements are therefore the eigenvalues of matrix A. It has also been shown that the 1 columns of matrix S1....S* S are the elements of the eigenvectors of matrix A. 35;40

APPENDIX IV Investigation of the Influence of the Tangential Ribs The influence of the tangential ribs can be evaluated by adding the bending energy of the tangential ribs at 9 = 600, 120~0 240~, and 3000. As the effect is one of second order, a negligible error will be introduced in the final solution by supposing that the effect due to the tangential ribs is precisely the effect due to radial ribs of the same length. With this assumption, equation 3.2*29 applies for the tangential ribs. In the following discussion only the lowest modes will be considered The first term for the second mode can be calculated, using equation 3.2.29 and values of W2, w2, at O = 600, 1200, 2400, 3000, to be Vbrl (Dr/l1)(V4)-.((B2-b2)cos 120+(A2-a2)sin 120]2 + +[(B2-b2)cos (-120)+(A2-a2)sin (-120)]2+ (A4.1) +[(B2-b2)cos 240 +(A2-a2)sin 240)]2 + +[(B2-b2)cos (-240)+(A2*a2)sin (-240)]2) which can be simplified to read - 281

-282similar consideration will yield (A4.3) Vbr = (8/3)(D/13)(B3-b3)2 The analogous evaluation of the second term in equation 3.2.29 for the second mode will be (A4.4) Vbr2 = 6(D,/lR)-(B-B2b2 + 3A-3A2a2) + + 6(Dr/l r)(B2b2-b + 3A2a2-3a) For the third mode there results (A4.5) Vbr2 24(D/lrR)(B -B3b3 )+242(D 2l r)( The third term of the bending energy as represented in equation 3.2.29, can be evaluated for the second mode, and is (A4.6) V = 8(D/lr )(B2+3A2 )+8(D/lr r)(B2+3a2)+ +8(D,/lRr) (B2b2+3A2a2) For the third mode, the contributing terms are (A4.7) Vbr3 = 72(Dr/lR2 )B3+72 (Dr/lrr )b3+72 (Dr/lRr)B3b3 The constants for the calculation of these terms, based on design data in Appendix I, are assembled in Table A4.1 Constants of the Tangential Ribs D4/1 Dlr 12 1.94 Dr,/5R Dl.9r2 D lrR 4.041 1.52 194 0.57 0.93 073 x103 The potential energies have been calculated, and added to the respective terms of tables 3.5.5, and 3.5.6.

-283The second mode determinant showing the effect of the tangential ribs is recorded in table A4.2 and for the third mode in table A4.3. The coefficients in the brackets give the corresponding terms of the same determinant, neglecting the bending energies of the tangential ribs. A close inspection reveals that the effect of the tangential rib on the behavior of the elastic system is negligible. Table A4.2 Second Mode Determinant Showing Effect of Tangential Ribs A2 B2 a2 b2 92.79- 0*112 -92.01 A2 (92.706)- (0.112) (-92.00) -,154o? 0.112 8.225- -5*022 B2 (0.112) (8.195)- (-5 012) -. 1540) -92.01 94.215a2 (-92.00) (94.245)-0. 93 u -5.022 5* 56b2 -5.013 (5.57)-0.19432

- 284 Table A4.3 Third Mode Determinant Showing Effect of Tangential Ribs A3 B3 a3 b3 96.691- -0.167 (-92) (96 691 ) -0.167 A (960611)- 2 0.167 (-92.00) 3 -0.136702 0.167 9.36- 1,93 B3 (0.167) (9962)- 19 -0. 1367a (-92.00) 107.964a3 (-92.00) (107.964)-0.1727a (-1*92) 16.767b3 (-1-93) (16.706)-0. 1727i.

BIBLIOGRAPHY 1. Alger, J. R. M., "Some Effect of Eccentric Airgaps in Induction Machines", M.S. Thesis, M.I.T., Cambridge, 1950. 2. Alger, P. L., The Nature of Polyphase Induction Machines, John Wiley & Sons, Inc, New York, 1951. 3. Alger, P.L., "The Magnetic Noise of Polyphase Induction Motors", A I.EoE Technical Per No 54-1 (1950) 4. Alger, P. L. and Oney, W.R., "Energy Flow in Induction Machines", General Electric Review, Vol. 56, (1953), No. 2, Page 56. 5* Alger, P. L. and West, H. R., "The Airgap Reactance of Polyphase Machines", A.I.E.E. Proceedings, Vol 66 (1947) Page 1. 6. Arnold, E*, LaCour, J. L. and Fraenckel, A., Alternating Current Technology (in German) Vol. V, Part I, Julius Springer, Berlin, 1909, Page 183. 7. Backhaus, H., "Vibrations of Continuous Media", (in German), Handbuch der Physik, Vol. VIII, Julius Springer, Berlin, 1927, Page69. 8. Bailey, B. F. and Gault, J. S., Alternating Current Machinery, McGraw-Hill Book Company, Inc., 1951* 9* Baker, B. B. and Copson, E. T., The- Mathematical Theory of Huygen' s Principle, Clarendon Press, Oxford, 1950, Page 25. 10 Baudry, R. A., Heller, P. A., and Curtis, L. P., "Magnetic Vibrations in Alternating Current Generator Stators", A.IE.EE. Technical Paper, No. 54-93. (1953). The references marked with an asterisk are not quoted in the text but have contributed greatly to the author's background, and it is hoped that they will be of value- to the reader. - 285 -

-28611* Beranek, L. L., Acoustic Measurements, John Wiley & Sons, Inc., New York, 1949. 12. Beranek, L. L., Acoustics, McGraw-Hill Book Companyt; Inc., New York, 19 54 13. Beranek, L. L*, Sleeper Jr., H. P., "Design and Construction of Anechoic Sound Chambers", Journal Acoustic Society of America, Vol. 18 (1946), Page 140,, 14. Bickley, W. G., et alia, "Bessel Functions", Part I, British Association of the Advancement of Science Nathematical Tables1 Vol. VI, University Press, Cambridge, 1937. * 15. Bickley, W. G., et alia, "Bessel Functions", Part II, British Association of Advancement of Science Maathematial Tables, aVol. X University Press, Cambridge,. 1952. 16. Bradford, C. E. and Rhudy, R. G., "Axial Magnetic Forces on Induction Motor Rotors", A.I.EJE. Power Apparatus Systems, Jmue 1953, Page 488. 17. Carter, F. W., "Flux Distribution in the Airgap of Blectrical Machines" Electrical World and Engineering, Vol. 38 (1901), Page 8 4. 18. Chapman, F. H., "The Production of Noise and Vibration by Certain Squirrel Cage Induction Motors" Jourral Inst. E1. Engrs., Vol. 61, (1922), Page 39. 19. Churchill, R. V., Modern 0perational Mathematics in Engineering, McGraw-Hill Book Company Inc. New York 1944. 20. Courant, R., "The Theory of Linear Partial Difference Equations", (in German) Nachrichten von der Gesellschaft der Wissenschaften zu Goettingen, 192, Vol. 1, Page 98. 21*. Den Hartog, J. P., echanical Vibrations, McGraw-Hill Book Company, Inc., New York, 1947. 22. Doherty, R. E. and Nickle, C. A., "Synchronous Machines, Part I, an Extension of Blondel's Two-Reaction Theory", A.I.E.E. Transactions, Vol. 43, (1926), 23. Dreyfus, L., The Theory of Polyphase Induction Motors with Squirrel Ca (in German) Ingeniorsvetenskatsa kademiens Handlingar, Vol. 34, (1924)

287 - 24. *Firestone, F. A., Vibration and Sound, Edward Bros., Inc., Ann Arbor, Michigan, (1939) 25. Fluegge, W., Static and Dynamics of Shells, (in German) Edwards Brothers In, Ann Arbor, Michigan, 1943, Page 49. 26. Frey, K., "The Application of Conformal Mapping to Practical Problems of Electrical Machine Design" (in German) Arbeiten Aus dem Elektrotechnischen Institute der Technischen Hochschule in Karlsruhe, Vol 4, Julius Springer, Berlin, 1925~ 27* Fritze, H., "Noise Created by Electrical Machinery", (in German) Archivfuer Elektrotechnik, Vol. 10 (1921), Page 73. 28. Gault, J. S., "Rotor Bar Currents in Squirrel Cage Induction Motors", A.I.EE. Transactions, Vol. 60, (1941), Page 784. 29. Goldstein, H., Classical Mechanics' Addison-Wesley Press, Inc., Cambrdige, Mass. 1950, Page 14, Page 347. 30. Gross, E. E., "An Acoustic Calibrator for the Sound Level Meter", General Radio Experimenter, Vol. 24, (1949) No. 7. 31. HarrIngton, R. F., Advanced Field Theoy (Mimeographed Edition of LecturesT Syracuse University, Syracuse 1952, Page 22. 32. Hartley, R. V. L., "Application of Vector Analysis to the Wave Equation", Letter to the Editor, Journal Acoustic Society of America, Vol. 22, (1950) Page 11. 33* Hildebrand, F. B., Methods of Aplied Mathematics, Prentice-Hall, Inc., New York, 1952, Page 74. 34. Hildebrand, L. E., "Quiet Induction Motors", A.I.E.E. Transactions, Vol. 49 (1930) Page 848. 35. Horvay, G., Solution of Large Euatin Systems and Eigenvalue Problems by Lanezos Mlatrix Iteration Method General Electric Company, Schenectady, New York, 1953* 36. Jordan, H., The Low Noise Electric Motor (in German), W. Girrardet, Essen (1950).

- 28837o Jordan, H., "Approximate Calculation of the Noise produced by Squrrel-Cage Motors", The Engineer's Digest, Vol. 10, (1949), Page 222 38. Kaplan, W., Advanced Calculus, Addison-Wesley Press, Inc., Cambridge, Mass., 1952, Page 616 39. Karman, von Th. and Biot, M. A., Mathematical ethods in Engineering, McGraw-Hill Book Company, Inc., New York, 1940, Page 162, Page 186. 40. Kigbetliantz, E. G., "Diagonalization of a Real Symmetric Matrix", Private Communication to the Author, 1954 41* Kron, G., "Induction Motor Slot Combinations", A.I.E.E. Transactions, Vol. 50, (1931), Page 757 42. Krondl, M., "On the Differential Leakage of Induction Motors" (in French), Revue G6ngrale de l'Electricit, Vol. 23, (1928), Page 440 and 479 43. Krondl, M., "The Parasitic Forces in Induction Motors", Bulletin Oerlikon Vol. 24, (1931), Page 654 and Vol. 25, (1931) i Page 65. 44. Krondl, M., "Noise of Electrical Machinery" (in French) Conf4rence Internationale des Grands R~seaux Electriques A Haute Tension Paris,(1933 ) Page 6 45. *Kuhlman, K., Theoretical Electrical Technology (in German) Vol. 3, Birkhaeuser Verlag, Basel, 1951 46. Knudson, V.0. and alia, The Acoustical Spectrum (Sound wanted and unwanted), University of Michigan Press, Ann Arbor, Michigan, 1952. 47. *Langsdorf, A. S., Theory of Alternating Current Machinery McGraw-HiI Book Company Inc., New York, 1935 48. *Liwschitz, M., A Treatise on Electrical Machinery, (in German), Vol.l-3, B. G. Teubner Verlag, Leipzig, 1931 49. *Love, A. E. H., A Treatise on the Mathematical Theor of Elasticity, Dover Publications, New York, 1944. 50. *McLachlan, N. W., Theory of Vibrations, Dover Publications, New York, 1951

- 28951. "Milne, W. E., Numerical Calculus, Princeton University Press, Princeton, N. J., 1949. 52. Morrill, W. J., "Harmonic Theory of Noise in Induction Motors", A.I.E.E Transactions, Vol. 59, (1940) Page 474. 53. *lorse, P. M., Vibration and Sounds McGraw-Hill Book Company, Inc., 1948. 54. Perlis, S., Theory of Matrices, Addison-Wesley Press, Inc., Cambridge, Mass., 1942, Page 169. 55. Peterson, A. P. G. and Beranek, L. L., Handbook of Noise Measurement, General Radio Company, Cambridge, Mass,, 1953. 56. Plunkett, R., "Measurement of Mobility", Journal of ApDlied Mechanics Vol. 21, (1954) Page 2'0-.. 57. Poritsky, H. and Horvay, G., "Stresses in Pipe Bundles", A.S.M.E., Applied Mechanics, Paper No. 50-A-80 58. *Prescott, J., Applied Elasticity y Dover Publications, New York, 1946. 59. Rayleigh, Lord J. W,, The Theory of Sound, Vol. I-II, Dover Publications, New York, 1945. 60. *Randall, R. Ht. An Introduction to Acoustics, AddisonWesley Press, Inc., Cambridge, Mass., 1951. 61. *Richter, R., Electrical Machinery (in German), Vol. I-V, Julius Springer, Berlin, 1924-1948. 62. Rossmaier, V., "Calculation of the Additional Losses Caused by Non-Insulated Rotor Bars.*" Elelctrotechnik und Maschinenbau, Vol. 57, (1939), Page 249. 63. Schuisky, W., "Additional Currents in Non-Insulated Squirrel-Cages" (in German) Bulletin des Schweitzer Elektrotechnischen Verbandes, Vol. 44, (1953) Page 330. 64. Sommerfeld, A. J. W., Partial Differential Eauations in Physics, Academic Press Inc., New York, 1949, Page 189.~

- 29065. Stiel, W#, "Experimental Investigation of Torques of Polyphase Induction Motors with different Number of Rotor Bars" (in German), yorschungsarbeiten auf dem Geblete des Ingenieurwesen s Vs D. I., Vol.' 212, (1919) Page 2 66. Stratton, J. A. Electromagnetic Theory McGraw- Hill Book Company, Inc., New York, 1941, Page 289, Page 349 67. Temple, G.F.J, and Bickley W. G., Raylelghs Princile and its Application to Engineering Oxford University Press, London, 1933 68* Trefftz, E., "Mathematical Theory of Elasticity" (in German), Handbuch der Physik, Vol. 6, Julius Springer, Berlin, 1928 69, Timoshenko, S. and Young, D. H., Advanced Dynamics, McGraw-Hill Book Company Inc., 1948 70. Timoshenko, S. and Lessells, J. M. Applied Elasticity Westinghouse Press, East Pittsburgh, 1925, Page 119 71. Timoshenko S., Theory of Plates and Shells McGraw-Hill Book Company Inc., New York, 1940, Page 3 72. *Timoshenko, S., Vibration Problems in Engineeri D. Van Nostrand Company, Inc., New York, 193 73. Weber, E., "The Slotting Factor in Eleutrical Machinery" (in German), Elektrotechnische Zeitschrift Vol. 32, (1928), Page 858 740 Wells, R. J., Noise Analysis Technique, Private Communication to the Author, 1954 75.*Whittaker, E. T., A Treatese on the Analytical Dynamics of Particles and Rigid Bodies Dover Publications, New York, 1944 76. Report, "Inter-Society Co-operation urged to expedite Noise Study Agreement", Electrical Enginering, Vol. 50,(1931), Page 443

UNIVERSITY OF MICHIGAN 3 9015 02826 7592