MANATEE SIMULATION SOFTWARE Magnetic Acoustic Noise Analysis Tool for Electrical Engineering

Transcription

1 MANATEE SIMULATION SOFTWARE Magnetic Acoustic Noise Analysis Tool for Electrical Engineering Presentation of MANATEE software (v1.05) developed and distributed by EOMYS (

2 I. PRESENTATION MANATEE is a simulation software for the optimal electromagnetic design of electrical machines including the analysis of magnetic vibrations and acoustic noise due to Maxwell forces. Two versions exist: MANATEE under Matlab (R2009b or later) without specific GUI, and pymanatee for Python (with GUI release planned summer 2016). MANATEE does not use any Matlab toolbox. Based on the hybridation of analytical, semi-analytical and finite element methods for electromagnetic and mechanical models, MANATEE represents the best compromise between accuracy and calculation time, allowing to include the variable-speed noise and vibration criteria during a fast prototyping phase or a design optimization process. 2

3 The following topologies are included in MANATEE v1.05 : Inner rotor squirrel cage induction machine (including doubly-fed operation) Inner or outer rotor surface, inset or buried permanent magnet synchronous machine

4 A few figures about MANATEE: Up 40 db acoustic noise reduction after redesign based on EOMYS consulting activities Successfully applied on both synchronous & induction machines, inner & outer rotor, from W to MW range more than 100 graphical post-processings processings ~20000 code lines (without counting comments) Our references:

5 MANATEE software contains the following multiphysic modules: Geometry and control parameters Magnet / current excitations 3D magnetic force distribution Dynamic vibrations Variable speed noise level ELECTRICAL MODULE ELECTROMAGNETIC MODULE STRUCTURAL MODULE ACOUSTIC MODULE VARIABLE SPEED MODULE MULTI-SIMULATION MODULE OPTIMIZATION MODULE 5

6 II. ELECTRICAL MODEL User defined voltage waveforms User defined equivalent circuit User defined current waveforms Machine and converter input parameters Phase voltage waveforms Phase current waveforms Option 1: Simulink PWM block Option 2: Numerically generated PWM PWM MODEL EQUIVALENT CIRCUIT PWM model with several strategies (synchronous, asynchronous, calculated, full wave)

7 III. ELECTROMAGNETIC MODEL MANATEE electromagnetic model relies on the fast calculation of the airgap radial and tangential flux density with the following modelling methods : Permeance / MMF analytical models Subdomain semi-analytical models Finite element non linear magnetostatic model (FEMM) The permeance / MMF decomposition based on winding functions allows to include PWM harmonics, skewing and geometrical asymmetries (eccentricities, non uniform airgap) and faults (broken bars, short-circuits) within a few seconds of calculation. The subdomain models also allows to include PWM harmonics and skewing within a few seconds of calculation, but does not not account for uneven airgap and eccentricity. 7

8 Permeance/ MMF Subdomain FEMM Calculation time Overall accuracy Tangential field calculation No* Yes Yes Robustness to geometry Skewing Yes Yes Yes Eccentricities & uneven airgap Yes No No Saturation Faults (e.g. short circuits, broken bar, demagnetization) Topologies *can be modelled but not included yet in MANATEE Yes No* No* BPMSM SPMSM SCIM SCIM (no-load) SPMSM IPMSM Preferred model for fast vibroacoustic analysis in healthy variable speed operation all

9 Model hybridation is possible in MANATEE such as Calculation of mmf using non-linear FEMM (e.g. rotor mmf for interior magnet machines) Calculation of permeance using non-linear FEMM (e.g. saturation effects, notch effects, magnetic wedges) Recommended models in symmetrical healthy case: SCIM SPMSM IPMSM BPMSM Permeance/mmf Subdomain Hybrid (Permance/mmf with rotor mmf calculation under FEMM) 9

10 Analytical permeance incl. geometrical assymetries (e.g. uneven airgap, eccentricities) FEA permeance incl. saturation, magnetic wedges, notches FEA mmf including non linearities Analytical mmf in linear case using winding function model Option 1: 3D ANALYTICAL PERMEANCE / MMF MODEL User defined flux distribution PROJECTION TOOL Phase current waveforms Option 2: 3D SEMI-ANALYTICAL SUBDOMAIN MODEL Airgap time and space flux distribution r=2 Harmonic magnetic forces r=3 Option 3: 2,5D FINITE ELEMENT MODEL (FEMM) Radial and tangential forces FFT2

11 The electromagnetic model includes fast AC winding design tools suitable for any type of winding (overlapping, non-overlapping, integral or fractional) Stator windings distribution (τ=1.2) Stator windings distribution (τ=6) R S T R S T curvilinear abscissa [m] Stator winding functions (τ=1.2) at t=0 s curvilinear abscissa [m] Stator winding functions (τ=6) at t=0 s R-phase WF S-phase WF T-phase WF total mmf R-phase WF S-phase WF T-phase WF total mmf mechanical angle α s [rad] mechanical angle α s [rad] 11

12 Example of all teeth wound Vs alternate teeth wound mmf harmonic content Stator windings distribution (τ=1/1.2) Stator windings distribution (τ=1/1.2) R S T R S T mechanical position along airgap in trigonometric direction [m] Stator winding functions at t=0 s mechanical position along airgap in trigonometric direction [m] Stator winding functions at t=0 s R-phase S-phase T-phase total mm R-phase S-phase W T-phase W total mm mechanical angle α s [rad] mechanical angle α s [rad]

13 The permeance / mmf and winding function model allows to make a fast analysis of the effects of skewing, rotor and stator asymmetries (e.g. tolerances, segmentation, gaps, weldings), rotor dynamic and static eccentricities, saturation, interturn short circuit, and broken bar for squirrel cage machines Example of the vibroacoustic effect of stator segmentation or rotor tolerance stator shape symmetrical deformed rotor shape 0.2 symmetrical deformed+eccentric

14 Example of the effect of additional permeance harmonics due to saturation in asynchronous machines airgap radial flux density [T] with saturation without saturation angle [rad]

15 VARIABLE SPEED NOISE CALCULATED IN 1 sec WITHOUT SATURATION Frequency f=f s (Zr/p+2) Order r=zr-zs+2p=-4 Sound Power Level at variable speed Case of a squirrel cage induction machine Zr=96 Zs=84 p=4 WITH SATURATION Frequency f=f s (Zr/p+4) Order r=zr-zs+4p=+4

16 Example of the vibroacoustic effect of a broken bar in a squirrel cage induction machine 120 Radial acceleration spectum 120 Radial acceleration spectum Acceleration level [db Re 1e-6 m/s 2 ] Acceleration level [db Re 1e-6 m/s 2 ] Frequency [Hz] Frequency [Hz] Healthy condition Broken bar

17 Example of the vibroacoustic effect of an interturn short circuit New noise & vibration line

18 Example of the vibroacoustic effect of magnetic wedges using permance /mmf model and coupling with FEMM

19 Example of the effect of skew of rotor slots (or stator slot) on the maximum acoustic noise level 110 sound power level (dba) rotor skew pitch in stator slot pitch This sensitivity study is done on the maximum noise level at variable speed as a function of the rotor skew angle. Its calculation takes less than 2 min.

20 High accuracy and fast subdomain model for synchronous machine: Finite element linear magnetostatics (FEMM) 5 min MANATEE subdomain models 0.1 s time space time space Automatic coupling with FEMM finite element software (symmetries, boundary conditions) in order to model more complex problems (e.g. shaped magnets, saturation effects) 20

21 Special algorithm based on winding functions & subdomain models to decrease CPU time: Full time and space airgap radial and tangential flux distribution due to armature field (suitable with PWM current harmonics): standard subdomain algorithm: 40s optimized algorithm: 0.8s 21

22 IV. STRUCTURAL MODEL User defined natural frequencies (e.g. experimental data) Natural frequencies automatically calculated by FEM (GetDP) on a 3D model Option 1: 2,5D ANALYTICAL CYLINDER MODEL Natural frequencies of the circumferential modes of an equivalent ring Harmonic magnetic forces Dynamic radial deflections Static radial deflections Option2: 3D FINITE ELEMENT STRUCTURAL MODEL GetDP (free) or Optistruct (commercial) S=Enveloppe fermée Dynamic radial deflections Vibration synthesis of radial deflections FRF calculation of main spatial orders of magnetic forces

23 Coupling with structural FEM tool based on open source GetDP software: Automated mesh generation using Gmsh Automated identification of coupled circumferential / longitudinal modes with different boundary conditions Modal shape selector to visualize the modes and validate the automated modal identification (4,0) (4,1) (3,0) (3,1) (2,0) (2,1) (0,0)

24 Coupling with structural FEM tool Altair Optistruct: circular lamination with any slot geometry (possibility to simplify the slot geometry to have a lighter structural model) application of physics: orthotropic properties, winding mass application of boundary conditions (e.g. clamped/clamped, free/clamped, fixed nodes) meshing based on the number of nodes in the different regions modal basis solver set-up vibration synthesis post processing

25 Resulting modal basis (simplified representation of cylindrical modes teeth rocking modes are included): CLAMPED FREE Boundary conditions FREE FREE Boundary conditions

26 V. ACOUSTIC MODEL Dynamic radial deflections ANALYTICAL ACOUSTIC MODEL Radiation efficiency of an equivalent cylinder Sound power level Sound pressure level 2D (analytical) or 3D (GetDP) spatial-averaged vibration velocity 26

27 VI. ADVANTAGES Fast variable speed vibroacoustic calculation (from <1 sec to 1 mn) based on efficient calculation methods even with 3D effects and converter harmonics High frequency acoustic calculations (up to 20 khz) within seconds, contrary to numerical approaches Several industrial validations of the vibroacoustic model Automatic vibroacoustic analysis (identification of main exciting forces) Advanced harmonic post-processings processings to understand the root cause of acoustic noise and find design improvements Possibility of decoupling electromagnetics & structural mechanics to perform vibro-acoustic optimization (pole shaping, current injection) 27

28 VII. POST PROCESSINGS & PLOT TOOLS MANATEE includes more than 100 plots accessible directly in the command line Example of Matlab 28

29 Geometry, time and space visualization, real and complex spectra for all quantities (permeance, mmf, radial and tangential flux density, force, acceleration, velocity, displacement) Magnitude [N/m 2 ] Magnitude [N/m2] 2 x Airgap radial force at t = 0 s as a function of space Angle [ ] x Airgap radial forcee force FFT over space Space harmonic [] Magnitude [T] Frequency [Hz] Spatial order [r]

30 Visual fitting tool for B(H) curve model at high excitation field

31 Magnetic harmonic forces analysis with automated identification of lines expression Airgap radial force FFT2 (PMSM) (SCIM) σ r [N/mm2] Frequency [Hz] f=22f s =2099 Hz f=20f s =1908 r=6 Hz r=3 f=16f s =1526 Hz f=14f s =1336 Hz r=-3 r=-6 f=5f s =453 Hz f=2f s =191 Hz r=3 f=4f s =382 Hz r=6 f=3f s =262 r=-3 Hz r= spatial order [r] 31

32

33 Magnetic harmonic forces analysis 33

34 Static (top) and dynamic (bottom) radial vibration spectra

35 Spectrogram of radial / tangential force harmonics for each spatial wavenumber

36 Spectrogram of radial / tangential force harmonics of a given order, including rotation direction Operational deflection shapes at a given frequency

37 Plot of tangential and radial forces per tooth in time and frequency domain r=0 r=2 r=3

38 Possibility to visualize the modal basis under Gmsh (freeware)

39 Modal contribution to acoustic and vibration spectra 39

40 Contribution of each structural mode to the acoustic noise at variable speed

41 Spectrogram and order analysis with automatic identification of main magnetic force harmonics orders and frequencies

42 Order analysis per circumferential vibration wavenumber (including rotation direction) r=0 r=1 r=-2 r=+2

43 Space vector diagram to analyze the origin of a radial or tangential force harmonic in terms of flux density waves

44 Space vector diagram to analyze the origin of a radial or tangential flux density in terms of permeance and magnetomotive force waves

45 Phasor diagrams to analyze the modal contribution to acoustic noise at a given frequency 50 Phasor diagram of SPL at frequency 3260 Hz

46 Unbalanced magnetic pull calculation (example of the slotting effect on eccentric UMP including skew of the stator)

47 Listen to your electrical machine (direct sound synthesis) resonance 250 MANATEE spectrogram 2-th 3-th 4-th SPL [dba] 60 Verification of the MANATEE synthesized sound using Audacity Supply frequency [Hz] resonance Frequency [Hz]

48 LwrA_max LwrA is_slotr is_slots is_ideal_mmfr is_mmfr is_ideal_mmfs is_mmfs Automated harmonic source analysis corr factor High correlation between maximum noise level and nominal noise level High correlation between maximum noise level and rotor slotting harmonics High correlation between maximum noise level and rotor mmf Low correlation between maximum noise level and stator winding armature spatial harmonics is m mfs al m mfs is m mfr al m mfr is s lots is s lotr LwrA wra m ax 0

49 VIII. VALIDATIONS AND DOCUMENTATION A full website is dedicated to MANATEE validations, post-processings, and tutorials: manatee?lang=en All modules are validated using special validation projects which can be run and modified by the user: >>run_manatee( EM_SPMSM_NL_001'); Validation cases are daily tested on the current version of MANATEE The input and output simulation data are stored in structures and substructures which are documented in an Excel file Tutorials for the variable speed simulation of induction machines and PM synchronous machines 49

50 EXAMPLES OF VIBRO-ACOUSTIC VALIDATIONS Case of a concentrated winding PMSM with interior magnets at partial load (blind test): TESTS MANATEE -40 db Motor B Motor A Sound level during a run-up (experiments with gearbox+watercooling+converter harmonics) Sound level during a run-up (MANATEE simulation without converter harmonics) ~5 min on a laptop

51 Case of a squirrel cage induction machine at no-load: TESTS Sound level during a run-up (experiments with PWM + gearbox +air-cooling) MANATEE Sound level during a run-up (simulation without PWM) ~2 second on a laptop 15 db reduction were obtained after redesign with MANATEE

52 IX. LIST OF MODULES Electrical modules (3) MODULE NAME FUNCTION DETAILED DESCRIPTION EL.SCIM Electrical model for inner rotor squirrel cage induction machines Calculates the stator and rotor currents based on input phase voltage waveform by calculating the equivalent circuit parameters, including skin effect and saturation effects. Some parameters (leakage and magnetizing inductance) can be evaluated with finite element (coupling with FEMM) if the module EM3 is activated. EL.PMSM Electrical model for surface, inset and buried permanent synchronous machines Calculates the stator currents based on input phase voltage waveform by calculating the equivalent circuit parameters (inductances Ld, Lq, flux linkage E), including skin effect and saturation effects. Some parameters (leakage and magnetizing inductances) can be evaluated with finite element (coupling with FEMM) if the module EM3 is activated. EL.pp Electrical post processing Plot of voltage, current, inductance waveforms and FFTs, saturation curve, skin effect factors. 52

53 Electromagnetic modules (9) MODULE NAME FUNCTION DETAILED DESCRIPTION EM1.IM Electromagnetic analytical module based on permeance / mmf and winding functions for squirrel cage asynchronous machines Calculates the airgap rotor and stator radial flux density time and space distribution based on permeance / mmf model. Includes rotor and stator skewing (any skew shape), uneven airgap and eccentricities effects, broken bar and short circuit effects, integral and fractional slot windings. EM1.PMSM EM2.SCIM EM2.SPMSM Calculates the airgap rotor and stator radial flux density time and space distribution based on Electromagnetic analytical module based on permeance / mmf model. Includes stator and rotor skew (any shape), uneven airgap, pole permeance / mmf and winding functions for inset, displacement and eccentricities effects, demagnetization and short-circuit effects, integral and surface and buried PM synchronous machines fractional slot windings. Electromagnetic semi-analytical module for inner rotor squirrel cage induction machine Electromagnetic semi-analytical module for inner rotor surface permanent magnet machine Calculates the airgap rotor and stator radial and tangential flux density time and space distribution based on subdomain models. Includes armature field with any winding type and skewing effect. Assumes semi opened slots with polar geometry Calculates the airgap rotor and stator radial and tangential flux density time and space distribution based on subdomain models. Includes armature field with any winding type and skewing effect. Assumes semi opened slots with polar geometry and tile shape magnets. EM2.IPMSM Electromagnetic semi-analytical module for inner rotor inset permanent magnet machines Calculates the airgap rotor and stator radial and tangential flux density time and space distribution based on subdomain models. Includes armature field with any winding type and skewing effect. Limited to polar geometries with semi opened slots and tile shape magnets. EM2.BPMSM Electromagnetic semi-analytical module for inner rotor buried permanent magnet machines Calculates the airgap rotor and stator radial and tangential flux density time and space distribution based on subdomain models. Includes armature field with any winding type and skewing effect. Limited to polar geometries with semi opened slots and rectangular magnets. 53

54 Electromagnetic modules (9) MODULE NAME EM3.PMSM FUNCTION Electromagnetic finite element module for inner rotor surface, inset or buried permanent magnet synchronous machines DETAILED DESCRIPTION Couples MANATEE with open-source electromagnetic software FEMM for non linear or linear magnetostatics problem (automatic drawing, meshing and post processings - torque, flux, emf, inductances). Calculates the airgap radial and tangential flux density time and space distribution, flux linkage, leakage and magnetizing inductances. EM.pp Electromagnetic post-processor module. Post-process the airgap flux density calculation to plot mmfs, permeance, magnetic force, time and space distribution or 2D FFT. EM3.SCIM Electromagnetic finite element module for inner rotorsquirrel cage asynchronous machines Couples MANATEE with open-source electromagnetic software FEMM for no-load non linear or linear magnetostatics problem (automatic drawing, meshing and post processings - torque, flux, emf, inductances). Calculates the airgap radial and tangential flux density time and space distribution, flux linkage, leakage and magnetizing inductances. Includes skewing effect. 54

55 Structural modules (4) MODULE NAME FUNCTION DETAILED DESCRIPTION SM1 Structural mechanics analytical module for modal analysis Calculates the natural frequencies of an equivalent cylinder including longitudinal modes. Calculates the dynamic radial deflections of the external structure with an equivalent 2D ring model. Calculates the wavenumber and frequency of main magnetic force harmonics. SM2 Structural mechanics finite element module for modal analysis Couples MANATEE with open-source mechanical software GetDP and open-source mesher Gmsh to calculate the mode shapes of a 3D external stator structure including winding weight. SM3 Structural mechanics finite element module for FRF calculation Couples MANATEE with open-source mechanical software GetDP and open-source mesher Gmsh to calculate the frequency response function (FRF) of the 3D structure under different magnetic force patterns and calculates the resulting dynamic deflection of the structure. SM.pp Structural mechanics post-processor module Post-process static, dynamic deflections at fixed or variable speed, and plot the corresponding FFTs (displacement, velocity, acceleration) including modal participation and order tracking analysis. 55

56 Acoustic modules (2) MODULE NAME FUNCTION DETAILED DESCRIPTION AC1 Acoustics analytical module Calculates the radiation factor of the external structure, the sound power level and sound pressure level radiated by the machine based on analytical models. AC.pp Acoustics post-processor module Post-process acoustic calculations (A-weighting, sound power and sound pressure levels, modal participation factors, sonagrams, order tracking analysis) 56

57 Control modules (4) MODULE NAME FUNCTION DETAILED DESCRIPTION CT1.SCIM Control module for squirrel cage induction machines Calculates the slip and voltage to achieve specified torque characteristics based on the equivalent circuit parameters. CT1.SM Control module for synchronous machines Calculate the current angle to achieve specified torque based on the equivalent circuit parameters according to MTPA strategy. CT2.PWM PWM module Generates 3-phase PWM voltage waveforms for asynchronous and synchronous modes, analytically or based on a Simulink model. CT2.CI Harmonic current injection module Allows to inject id or iq harmonic currents. 57

58 Variable speed & multi-simulation modules (4) MODULE NAME FUNCTION DETAILED DESCRIPTION VS1 Variable speed module Calls several fixed-speed simulations with varying input parameters depending on the control strategy (e.g. constant flux, torque/speed curve, etc) VS.pp Variable speed post-processor Post process results of variable speed simulations (spectrograms and sonagrams, outputs as a function of speed, modal contribution to noise as a function of speed). MS1 Multisimulation module Calls several times MANATEE by varying input parameters (e.g. to study the effet of the pole width or the slot numbers on noise) including correlation analysis between design variables and response variables. MS.pp Multisimulation postprocessings Display results of multi-simulations (evolution of response variables, correlation analysis, plot in design variable and response variable spaces). 58

59 Optimization modules (3) MODULE NAME FUNCTION DETAILED DESCRIPTION OP2 Multiobjective optimization module Couples MANATEE with a global optimization tool (NSGA-II) for constrained multiobjective mixed variable optimization. OP1 Sensitivity analysis module Calculates the sensitivity of a response variable with respect to design variables. OP.pp Optimizer post processor Post-processings of optimization results (Pareto front in 2D and 3D dimensions, correlation analysis of constraints, response and design variables, plot in design variable and response variable spaces). 59

60 KEYWORDS: MAGNETIC NOISE MAGNETIC ACOUSTIC NOISE ELECTROMAGNETIC ACOUSTIC NOISE ELECTRICAL NOISE AUDIBLE ELECTROMAGNETIC NOISE MAGNETICALLY-INDUCED VIBRATION ELECTROMAGNETICALLY INDUCED NOISE WHINING NOISE HIGH PITCH NOISE HUMMING NOISE TONAL NOISE MAGNETIC VIBRATION ELECTROMAGNETIC VIBRATIONS DEFLECTION MECHANICAL DEFORMATION MAXWELL FORCE MAXWELL TENSOR MAGNETIC FORCE ELECTROMAGNETIC FORCES UNBALANCED MAGNETIC PULL UMP VIBRATIONAL BEHAVIOUR NOISE MITIGATION NOISE CONTROL VIBRATION REDUCTION TOOTH WINDING CONCENTRATED WINDING FRACTIONAL SLOT WINDING DISTRIBUTED WINDING SHORTED PITCH WINDING MAGNETOMOTIVE FORCE MMF WINDING FUNCTION SKEW SLOT INCLINATION STEP-SKEW SUBDOMAIN MODEL PERMEANCE / MMF MODEL PERMEANCE / CURRENT LINKAGE MODEL PMSM IPMSM SM IM SRM SYNRM DFIG SYRM SYNCHRONOUS MACHINES PERMANENT MAGNET ASYNCHRONOUS MACHINE SQUIRREL CAGE INDUCTION MACHINE DOUBLY FED INDUCTION GENERATOR ELECTRICAL MACHINES SWITCHED RELUCTANCE MACHINES SYNCHRONOUS RELUCTANCE MOTOR BRUSHLESS AC MOTOR BRUSHLESS DC MOTOR DC SERVOMOTORS ALTERNATOR SPINDLE MOTOR ELECTRIC POWERTRAIN FOURIER TRANSFORM FORCE HARMONICS HARMONIC REDUCTION WAVENUMBER NODE NUMBER POLE PAIR NUMBER SPATIAL ORDER PULSATING ROTATING PROGRESSIVE WAVE RESONANCE NATURAL FREQUENCY STRUCTURAL MODE MODAL BASIS MAGNIFICATION DAMPING MOTS-CLEFS: BRUIT MAGNETIQUE BRUIT ACOUSTIQUE D ORIGINE MAGNETIQUE BRUIT ELECTROMAGNETIQUE BRUIT AUDIBLE ELECTROMAGNETIQUE BRUIT ELECTRIQUE BRUIT HAUTE FREQUENCE BRUIT DE SIRENEMENT BRUIT TONAL VIBRATION MAGNETIQUE VIBRATIONS ELECTROMAGNETIQUES DEPLACEMENT DEFORMATION MECANIQUE FORCE DE MAXWELL TENSEUR DE MAXWELL EFFORTS MAGNETIQUES FORCES ELECTROMAGNETIQUES BALOURD MAGNETIQUE COMPORTEMENT VIBRATOIRE REDUCTION DE BRUIT CONTRÖLE DE BRUIT REDUCTION DES VIBRATIONS BOBINAGE DENTAIRE BOBINAGE CONCENTRE BOBINAGE FRACTIONNAIRE BOBINAGE DISTRIBUE BOBINAGE A PAS RACCOURCI FORCE MAGNETOMOTRICE FMM FONCTIONS DE BOBINAGE VRILLAGE INCLINAISON DES ENCOCHES PAS DE VRILLAGE MODELE DE SOUS DOMAINE PERMEANCE / MMF MSAP MAS MRV MS MADA MACHINES SYNCHRONES AIMANTS PERMANENTS MACHINES A INDUCTION MACHINES ASYNCHRONES A CAGE D ECUREUIL MACHINE ASYNCHRONE A DOUBLE ALIMENTATION MACHINES ELECTRIQUES MACHINES A RELUCTANCE VARIABLE MACHINES SYNCHRORELUCTANTES MOTEUR BRUSHLESS SERVOMOTEUR ALTERNATEUR TRANSFORMEE DE FOURIER HARMONIQUE DE FORCE REDUCTION DES HARMONIQUES NOMBRE D ONDE NOMBRE DE NŒUDS NOMBRE DE PAIRES DE POLES ORDRE SPATIAL ONDE TOURNANTE PULSANTE PROGRESSIVE RESONANCE FREQUENCE NATURELLE MODE DE STRUCTURE BASE MODALE AMPLIFICATION AMORTISSEMENT

61 APPENDICES SOUND & VIBRATION SYNTHESIS ELECTROMAGNETIC MODEL 3D airgap time and space flux distribution HARMONIC DECOMPOSITION r=0 r=2 r=3 Tangential and radial harmonic magnetic forces (magnitude, wavenumber, frequency, phase) VIBRATION SYNTHESIS Spectrograms Vibration level Operation Deflection Shapes Modal contribution Radiating surface velocities STRUCTURAL FEA MODEL Unit harmonic loads for wavenumber r=0, ±2, ±4 STRUCTURAL FREQUENCY RESPONSE FUNCTIONS Complex FRFs (radial & tangential) for each wavenumber r Motor and frame modal basis r=0 r=2

62 VIBRATION SYNTHESIS Modal contributions from vibration synthesis SOUND SYNTHESIS Sonagrams Sound Power Level Directivity patterns Modal contributions ACOUSTIC FEA MODEL Unit radiating surface displacements ACOUSTIC FREQUENCY RESPONSE FUNCTIONS Complex FRFs for each structural mode (MATV) Acoustic Transfer Vector (ATV) m=0 m=1