ACOUSTIC NOISE AND VIBRATIONS OF ELECTRICAL TRACTION CHAINS …€¦ · NVH experimental...

© EOMYS ENGINEERING 2013- www.eomys.com

ACOUSTIC NOISE AND VIBRATIONS OF ELECTRICAL TRACTION CHAINS

Focus on electromagnetically-excited NVH for automotive applications and EV/HEV

1

LE BESNERAIS Jean

[email protected]

Note: this presentation is based on extracts of EOMYS technical traininghttps://eomys.com/services/article/formations?lang=en

Part 1 Intro & physics of electromagnetically-excited NVH

© EOMYS ENGINEERING 2013- www.eomys.com 2

I. PRESENTATION

*"Jeune Entreprise Innovante": the French government recognises that EOMYS runs significant R&D activities

• Young Innovative Company* created in may 2013

• Located in Lille, North of France (1 hour from Paris)

• Activities : engineering consultancy / applied research

• 6 R&D Engineers (electrical engineering, vibro-acoustics, heattransfer, scientific computing)

• 80% of export turnover

• Sectors : transportation (railway, automotive, marine, aeronautics), energy (wind, hydro), home appliances, industry


II. REFERENCES


• Diagnosis and solving of NVH issues on electrical systems

EOMYS can be involved both at design stage & after manufacturing of electric machines & passive components

Technical trainings on vibroacoustics of electrical systems

Multiphysic design optimization of electrical systems

III. SERVICES


IV. MANATEE software• Electromagnetic and vibro-acoustic simulation software of electrical machines (Matlab-based)

• Only software suitable for NVH assessment both in basic design (semi analytical models) or detaileddesign phase (coupling with electromagnetic FEA and structural FEA e.g. Optistruct/Ansys)

• Fault simulation (e.g. eccentricity, broken bar, demagnetization, pole displacement)

• More than 100 post processing graphs to quickly find NVH root cause

• Up to 40 dB reduction obtained on industrial cases

See more at www.manatee-software.com


• Vibroacoustic analysis of more than more than 50 electric machines

• Wide range of electrotechnical solutions (e.g. skewing, current injection, notches, pole shaping)

• Wide range of topologies: induction & PM machines, inner and outer rotor, from 5 rpm to 150 krpm, from 100 W to 3 MW

• Extended NVH measurements (OMA, EMA, ODS) focusing on magnetic noise & vibrations, including telemetry

6

V. EXPERIENCE


Combined semi-anechoic chamber & electrical lab

NVH hardware• 5 microphones (Piezotronics)• 12 miniature accelerometers (B&K)• 4 tri-axial accelerometers (B&K)• 6 mono axis accelerometers (B&K)• Impact hammer (DJB) and electrodynamic shakers (LDS)• Laser vibrometer (Ometron)• Tachometer (B&K)

VI. NVH test facilities

NVH software• Dewesoft with 16 channel dynamic acquisition system for combined

measurement of voltage, current, vibration and noise• LEA Sound Lab from Genesis

NVH post-processings• Sound Power Level • Order tracking and spatiograms• Operational Modal Analysis, Experimental Modal Analysis• Operational Deflection Shape


• Partenariats avec des laboratoires :

-

- Laboratoire Systèmes Electrotechniques et Environnement (LSEE, Béthune)

-

• Réseaux industriels & universitaires :

- cluster industriel Maîtrise Energétique des Entraînements Electriques (MEDEE)

- pôle de compétitivité en R&D Automobile et Transports publics (MOVEO)

- pôle de compétitivité des Transports Innovants (ITRANS)

• Partenaires financiers :

VII. PARTENAIRES


VIII. R&D INTERNAL RESEARCH PROGRAMME

Analysis of electromagnetically-excited NVH

▪ current injection & influence of control▪ development of fast hybrid simulation methods (FEM / semi-analytic)▪ 3D electro-vibro-acoustic simulation of rotating machines (asymmetries, skewing)▪ magnetostriction and Maxwell force simulation (GetDP)▪ psychoacoustics of electromagnetically-excited noise▪ advanced post processings of electrical machines NVH (e.g. spatiograms)▪ new characterization methods of electrical machines structural behaviour


Recent publications (see https://eomys.com/recherche/article/publications):


I. Physics of electromagnetic noise & vibrations

II. NVH ranking of electric motors for EV / HEV

III. Review of electrical actuators and passive components in EV / HEV

IV. NVH experimental characterization of electric motors

V. Modelling & simulation tools of electromagnetically-excited NVH

VI. Noise reduction techniques of passive components and electric motors

VII. Conclusions

ACOUSTIC NOISE AND VIBRATIONS OF ELECTRICAL TRACTION CHAINS

Focus on electromagnetically-excited NVH for automotive applications


PHYSICS OF ELECTRO-MAGNETIC NOISE & VIBRATIONS


ELECTRICAL SYSTEMS UNDER STUDY

• Active electrical systems: rotating electrical machines, electromechanical actuators

• Passive electrical systems: transformers, inductors, capacitors, resistors, cables

• Topologies of electrical machines:

- inner rotor / outer rotor- induction machine / synchronous machines- PMSM / WRSM- AC / DC electrical machines

• Topologies of transformers/inductors:

- planar / wound- tape wound / step-lap / laminated cores- toroidal / EI shape

• Topologies of capacitors- ceramic- plastic film (polymer)- electrolytic- metallised film

[FEEM, railway-technology.com][AVX website]

[BMW window motor]

[B48]


variable current source

What do we call electro-magnetic acoustic noise and vibrations?

noise and vibrations arising from variable electro-magnetic forces in audible range [20 Hz, 20 kHz]

Dynamic deflections due to variable magnetic field B

(Maxwell, Laplace, magnetostriction)

rotating permanent magnetor DC current source

Dynamic deflections due to variable electrical field E

(Maxwell=electrostatic, piezoelectric, electrostriction)

variable voltage source

Note: in electric machines, magnetic noise is sometimes called electromagnetic noise (because magnetic field is created by electrical currents) but it has nothing to do with electrical field

Electrical machinesTransformers, inductors

Electrical machinesCapacitors


aerodynamic sources

(e.g. fans)

electromagnetic sources

(e.g. pole/slot)

mechanical sources

(e.g. bearings, gearbox)

Noise of an electric traction machine during starting:

• Electromagnetic acoustic noise is characterized by strong tonalities compared to other noise sources

• Tonality is penalized by standards (usually 3dB per tonality)


• Magnetic forces = « forces arising from the presence of a magnetic field »

• Magnetic noise and vibration = « noise and vibrations arising from magnetic forces »

• Magnetic field can be created by current sources or permanent magnets

• As noise and vibrations are dynamic quantities field source must vary with time: magnetic noise and vibrations are due to AC current or rotating permanent magnets

• Local magnetic forces acting on the active parts can be

- magnetostrictive forces (acting inside the material) ∝ 𝐵2

- Maxwell forces (acting mainly at the interface of the material with air) ∝ 𝐵2

• Laplace forces acting on conductors ∝ 𝐼𝐵

Definitions

Force types


• Electrical forces = « equivalent forces arising from the presence of an electrical field »

• Electrical noise and vibration = « noise and vibrations arising from electrical forces »

• Electrical field can be created by electrical charges

• As noise and vibrations are dynamic quantities field source must vary with time: electrical noise is due to varyingelectrical fields, so to AC voltage

• Local electrical forces acting on the active can be

- electrostrictive forces (acting inside the material) ∝ 𝐸2

- electrostatic (acting mainly at the interface of the material with air) ∝ 𝐸2

- reverse piezoelectric ∝ 𝐸

Definitions

Force types


MAGNETIC FORCES IN ELECTRICAL SYSTEMS

• Laplace forces are mainly considered when studying winding overhang vibrations of large machines [B38]

• Laplace forces apply to conductors

• In electrical machines B is small is slots so Laplace forces can be neglected

Laplace/Lorentz forces in electric motors


• Laplace forces can be involved in cable noise issues in high current traction application

• Laplace forces generate transformer winding vibrations at load due to interaction with stray magnetic field [B39]

• Laplace forces acting on windings are generally negligible in inductors [B52]

Laplace/Lorentz forces in passive components


• Magnetostriction acts

• Magnetostrictive forces evolve with the square of the magnetic flux density

• ~1 to 10 µm/m

• Magnetostriction includes [B6]

- volume magnetostriction (isotropic): occurs for H>8kA/m, can be neglected in laminations

- Joule and transverse magnetostriction (anisotropic): what is called here by « magnetostriction »

Magnetostriction


Magnetostrictive equivalent forces in electric machines (case of an induction machine):

[B7]

• Magnetostriction forces can create radial vibrations of the outer yoke of electrical machines, thus they can potentiallycreate acoustic noise

• From EOMYS experience, magnetostriction has never been the root cause of electric machine NVH issues

[B10]

[B46]


Magnetostrictive equivalent forces in transformers / inductors

• Magnetostriction tends to enflate the magnetic core

• Depending on bounday conditions dynamic deflections of the magnetic core may be able to produce acoustic noise

• In inductors vibrations due to magnetostriction is generally negligible [B41] compared to Maxwell forces

[B41][B40] [B42] [B49]


N S

23

• Simple set-up to show the effect of Maxwell forces: tuning fork excited by a variable frequency current

• Variable frequency current creates a variable frequency magnetic field

• Magnetic field is guided by the iron of the tuning fork and creates a magnetic dipole at the fork tips

• Maxwell force is the magnetic attraction between the equivalent North and South poles of the fork tips

• The airgap tends to be reduced by Maxwell forces

Magnetic field line distribution

airgap

Maxwell forces: 1D illustration of noise and vibration


Forced excitation (AC current with fixed frequency)

• Strong tonal noise is created without any mechanical contact between coil and tuning fork



Resonant excitation (AC current with variable frequency fs)

fs=200 Hz

• A resonance (high noise level) is observed when feeding the coil at 200 Hz



What happens at resonance for fs=200 Hz?

𝐵[𝑇] ∝ 𝐼 𝑃[𝑁

𝑚2] ∝ 𝐵2𝐼[𝐴] 𝑉[𝑚

𝑠] ∝ 𝑃 𝐿𝑝[𝑑𝐵] ∝ 20𝑙𝑜𝑔10(𝑉)

current@ 200 Hz

flux@ 200 Hz

Maxwell pressure@ 400 Hz

vibration@ 400 Hz

acoustic noise@ 400 Hz

• The tuning fork natural frequency is close to 400 Hz (first bending mode)

• The match between exciting magnetic forces and tuning fork natural frequency create a resonance (high vibration level)

• Tuning fork behaves like a linear quadrupole and radiatesacoustic noise at vibration frequency of 400 Hz

http://www.acs.psu.edu/drussell/Demos/rad2/mdq.html



Airgap reluctant (Maxwell) forces

stator

rotor

Modal basis of the magnetic circuit

(400 Hz)

(2585 Hz)

(648 Hz, m=2)

(1356 Hz, m=3)

(2892 Hz, m=4)M. Boesing, « Acoustic Modeling of Electrical Drives, Noise and Vibration Synthesis

http://www.acs.psu.edu/drussell/Demos/TuningFork/fork-modes.html

Maxwell forces: 2D extension to electrical machines


Maxwell forces: 2D extension to electrical machines

[B7]

[B10]

[B46]

• Maxwell and magnetostriction forces / vibration / acoustic lines occur at same frequencies (quadratic function of the flux density)


• Maxwell forces concentrate in the region of reluctivity changes (air/iron interface) so in joint regions of step-lap transformer cores

Maxwell forces in transformers

[B44] [B43]


• Maxwell forces concentrate in the airgap of inductors

Maxwell forces in inductors

[B41] [B49]


Rotating machines

Inductors Transformers Capacitors Cables

Function Electrical to mechanical power

conversion

Magnetic energy storage (AC current filter)

Electrical power conversion (AC voltage

converter)

Electrical energy storage (AC voltage filter)

Electrical power transmission

Maxwell +(stator / rotor

attraction)

+(airgap reluctant

forces)

+ (airgaps in corners)

Laplace -(overhang / leakage

flux)

-(overhang / leakage

flux)

+ (winding / leakage flux

at load)

+ (external magnetic

flux)

Magnetostriction -(magnetic core)

+ (magnetic core)

+ (magnetic core)

Electrostatic +(armature attraction)

Piezoelectric + (ceramic capacitors)

Electrostriction -(electrostrictive

polymers)

Overview of electrical systems and main electromagnetic forces affecting their NVH behaviour


• NVH characteristics (frequency content, potential resonances) highly depend on the electrical system topology

• Ex: Maxwell harmonic frequencies are different for PMSM and SCIM

• Ex: main vibration in passive component can be at frip (current ripple frequency) or 2frip depending on the presence or absence of a DC component in the current


magnetic permeability inhomogeneitiesLorentz force per unit volumetorque if H and B are not colinear

33

• The combination of these forces can be expressed as a tensor, called « Maxwell tensor »

• The Maxwell stress experienced by the surface dS in normal and tangential directions is therefore given by

• The general formulation of the magnetic forces per unit volume in Cartesian direction i=1,2,3 is given by

Maxwell stress in electric machines - mathematical considerations


• Due to Ostrogradski (or Gauss) theorem, the global body force acting on the stator (resp. rotor) can be obtained by integrating the Maxwell stress tensor over a surface enclosing the stator (resp. rotor)

• The surfaces outside the active parts do not see any flux and do not contribute to the global force

• Therefore both rotor and stator magnetic forces are given by the Maxwell stress integration over the airgap cylinder (oriented) surface

Sso

Ss=Sso U Ssi

Ssi

B=0Sri B=0

Sro

Sr=Sro U Sri

𝒇𝑺 = ර𝑆𝑠𝑖

𝝈𝑑𝑺 𝒇𝑹 = ර𝑆𝑟𝑜

𝝈𝑑𝑺

Inner rotor topologies are only considered in this presentation except when specified

𝒇𝑺 =ම

𝑉𝑠

𝒇𝑑𝜏 =ම

𝑉𝑠

𝒅𝒊𝒗(𝑻)𝑑𝜏 = ඵ

𝑆∋𝑉𝑠

𝑻𝑑𝑺 =ර𝑆𝑠𝑖

𝝈𝑑𝑺


𝝈 =1

𝜇0

1

2(𝐵𝑟

2 − 𝐵𝜃2 − 𝐵𝑧

2) 𝐵𝑟𝐵𝜃 𝐵𝑟𝐵𝑧

𝐵𝑟𝐵𝜃1

2(𝐵𝜃

2 − 𝐵𝑟2 − 𝐵𝑧

2) 𝐵𝑧𝐵𝜃

𝐵𝑟𝐵𝑧 𝐵𝑧𝐵𝜃1

2(𝐵𝑧

2 − 𝐵𝑟2 − 𝐵𝜃

2)

In 2D assuming Bz=0, the equivalent local magnetic force dfs acting on the part of the stator dV

stator

airgap Ssi

• Taking cylindrical coordinates (global frame R linked to the stator), Maxwell stress can be written as

volume dV

dS=-dSur

𝑑𝒇𝑺 = 𝝈𝑑𝑺 = {1

2𝜇0(𝐵𝑟

2-𝐵𝜃2)(dS.ur) +

1

𝜇0(𝐵𝑟 𝐵𝜃 )(dS.u𝜃)} ur

𝜎r dS

R

dfS

+{−1

2𝜇0(𝐵𝑟

2-𝐵𝜃2)(dS.u𝜃) +

1

𝜇0(𝐵𝑟 𝐵𝜃 )(dS.ur)} u𝜃

𝜎𝜃 dS


• Other expression of Maxwell stress:

• For the airgap dS radial orientation (from outer stator or outer rotor viewpoint):

𝜎𝜃 =-1

𝜇0(𝐵𝑟 𝐵𝜃 )𝜎r=-

1

2𝜇0(𝐵𝑟

2-𝐵𝜃2)

• For the airgap dS radial orientation (from inner rotor or inner stator viewpoint):

𝜎𝜃 =+1

𝜇0(𝐵𝑟 𝐵𝜃 )𝜎r=+

1

2𝜇0(𝐵𝑟

2-𝐵𝜃2)

• The Maxwell stress can be expressed in complex form as

𝝈 = 𝝈𝑟 + 𝑗𝝈𝜃 =𝑩2

2𝜇0𝑩 = 𝑩𝑟 + 𝑗𝑩𝜃


from [B3]

𝝈 = 𝜇0

−1

2|𝑯|2 0 0

0 −1

2|𝑯|2 0

0 0 −1

2|𝑯|2

+ 𝜇0

0 0 00 0 00 0 +|𝑯|2

• Taking the local field line frame , the Maxwell tensor becomes

from [B3]

(law of maximal magnetic flux)

(law of minimal reluctance)


• The module of dfs is independent of the surface orientation dS, it corresponds to the local energy density

• Due to lower magnetic permeability, most of the magnetic energy is concentrated in the airgap

• energy cannot be stored without stress

• Energy density is homogenous to pressure, 𝜎 is in N/m2

Maxwell stress - physical considerations

• The integration of the Maxwell force over the whole rotor is the opposite of its integration over the whole stator

• A simple consequence of this is the counter torque experience on the stator side

• Similarly, all the radial force harmonics experienced by the stator are experienced by the rotor but with a frequency shift (detailed later)

• Radial and tangential stress have the same harmonic content (same pattern of null time & space harmonics in their FFT2)

• Maxwell stress can be extended to electrical field (expression of the electrostatic stress in capacitors)


« traction »attraction

« compression »repulsion

« shear »

Noise & vibrations

Noise & vibrations (also !)Torque

• For a dS in the airgap

from [B7]

1 2 3

Case 1: most common at interface iron / air in linear case


• Example of the Maxwell stress in an ideal electric machine without time nor space harmonics

𝐵𝑟= 𝐵𝑟0 cos 𝜔𝑠 𝑡 + 𝑝𝛼𝑠𝐵𝜃 = 𝐵𝜃0cos(𝜔𝑠 𝑡 + 𝑝𝛼𝑠 + 𝜑)

𝝈𝜽 =𝟏

𝝁𝟎𝑩𝒓 𝑩𝜽 =

1

2𝜇0𝐵𝜃0𝐵𝑟0 +

1

2𝜇0𝐵𝜃0𝐵𝑟0cos 2𝜔𝑠 𝑡 + 2𝑝𝛼𝑠 −

1

4𝜇0𝐵𝜃0𝐵𝑟0sin 2𝜑 sin 2𝜔𝑠 𝑡 + 2𝑝𝛼𝑠

electromagnetic static torque harmonic torque

harmonic radial forceaverage radial force

Hz

𝜎𝑟, 𝜎𝜃

0

2𝑓𝑠

𝝈𝒓 =𝟏

𝟐𝝁𝟎(𝑩𝒓

𝟐 − 𝑩𝜽𝟐) =

1

4𝜇0(𝐵𝑟0

2 −𝐵𝜃02 ) +

1

4𝜇0(𝐵𝑟0

2 −𝐵𝜃02 cos(2𝜑))cos 2𝜔𝑠 𝑡 + 2𝑝𝛼𝑠 +

1

4𝜇0𝐵𝑟0𝐵𝜃0sin 2𝜑 𝑠𝑖𝑛 2𝜔𝑠 𝑡 + 2𝑝𝛼𝑠

wavenumber2p


• 2p is called the wavenumber, it corresponds to the number of maxima of the wave (spatial frequency along the airgap)

• Space order is the normalization of the wavenumber with respect to p, wavenumber 2p means space order=2

• A fundamental field of 0.5 T creates a radial pressure at 2fs of 105 N/m2 i.e. ~10 tons per m2

• In an ideal machine the excitation at 2fs is always present, it is the strongest and it is directly linked to electromagnetic (not reluctant) torque production

• A magnetic noise or vibration issue at 2fs can therefore be very hard to tackle without degrading torque


Forced excitation (rotating magnet)

• Magnets create a fundamental flux B with 1 minium and 1 maximum (p=1 pole pair) along the airgap, but the stator deformation (so magnetic force) has 2 minima and 2 maxima

• Magnetic force F is proportional to the square of flux density and has therefore 2p=2 pole pairs (two maxima & two minima) r=2p is called wavenumber

• Quadratic relationship between B and F affects both time (cf. tuning fork) and space domains

• Ferromagnetic materials can be deformed under Maxwell stress, resulting in forced vibration and acoustic noise

2D illustration of noise and vibration due to Maxwell forces

steel tube (slotless stator)

1-pole magnet (p=1)

rotating shaft at 𝜔𝑅

𝐵 = 𝐵1 cos 𝑝𝜔𝑅𝑡 + 𝑝𝛼𝑠

𝐹 = 𝐹0 + 𝐹2 cos 2𝑝𝜔𝑅𝑡 + 2𝑝𝛼𝑠

N

S𝛼𝑠

𝜔𝑅

stator angle

frequency wavenumber


No-load flux density of a PMSM, ideal case (no mmf space harmonic, no slotting harmonic) p=12

0 200 400-0.5

0

0.5

Angle [°]

Mag

nit

ud

e [

T]

Airgap radial flux density at t = 0 s as a function of space

0 20 40 600

0.1

0.2

0.3

Mag

nit

ud

e [

T]

Space harmonic []

Airgap radial flux density FFT over space

0 100 200 300 4000

5

10x 10

4

Angle [°]

Mag

nit

ud

e [

N/m

2] Airgap radial force at t = 0 s as a function of space

0 20 40 600

1

2

x 104

Mag

nit

ud

e [

N/m

2]

Space harmonic []

Airgap radial forcee force FFT over space


-8-7-6-5-4-3-2-10 1 2 3 4 5 6 7 8

1000

2000

30000

2

4

6

x 10-12

spatial order [r]

Airgap radial force FFT2

Frequency [Hz]

r [

N/m

m2

]

{2fs,2p}{0,0}

zoom out down to 0 Hz

• An ideal machine (sine mmf, no slots, no PWM) creates a single Maxwell stress wave at twice the electrical frequency and twice the number of pole pairs


0 200 400-0.5

0

0.5

Angle [°]

Mag

nitu

de [T

]

Airgap radial flux density at t = 0 s as a function of space

0 20 40 600

0.1

0.2

0.3

Mag

nitu

de [T

]

Space harmonic []

Airgap radial flux density FFT over space

0 100 200 300 4000

5x 10

4

Angle [°]

Mag

nit

ud

e [

N/m


0 20 40 600

1

2

x 104

Mag

nit

ud

e [

N/m

2]

Space harmonic []


No-load flux density of a PMSM, real case (mmf space harmonic + slotting harmonic)


-8-7-6-5-4-3-2-101 2 3 45 6 7 8

1000

2000

3000

0

200

400

f=4fs=382 Hz

r=6

f=2fs=191 Hz

r=3

spatial order [r]

Airgap radial force FFT2

Frequency [Hz]

r [

N/m

m2

]

• Slotting and winding harmonics enrich Maxwell stress with force harmonics of lower wavenumber and higher electrical frequencies


• Airgap magnetic stress can be decomposed using Fourier transform:

• r and frequency nfs

travelling in the airgap

• r nodes

• 𝛼𝑠 is the airgap angle in the stator frame

• nfs is the electrical frequency, the mechanical rotation frequency of the force wave being nfs/r

𝜎 𝑡, 𝛼𝑠 =

𝑛,𝑟

ො𝜎𝑛𝑟 cos 2𝜋𝑛𝑓𝑠𝑡 + 𝑟𝛼𝑠 + 𝜑𝑛𝑟

Radial & tangential magnetic force waves


• Progressive (=rotating) force wave of order r=+2 at 6fs=360 Hz

F=F0cos(6wst+2αs) noted {6fs, 2}

0 100 200 300 400-500

0

500

Angle [°]

Mag

nit

ud

e [

N/m


0 20 40 600

100

200

Mag

nit

ud

e [

N/m

2]

Space harmonic []


0 100 200 300 400-500

0

500

Angle [°]

Mag

nit

ud

e [

N/m

2] Airgap radial force at t = 0.004834 s as a function of space

0 20 40 600

100

200

Mag

nit

ud

e [

N/m

2]

Space harmonic []


-1000 -5000 500 1000

-5

0

5

0

20

40

60

Spatial order [r]

Frequency [Hz]

Maxwell tensor main lines and stator natural frequencies

Mag

nit

ud

e [

dB

] {6fs,+2}

rotation in anti-trigonometric direction at 6f/r


-1000-500

0500

1000

-5

0

5

0

20

40

60

Spatial order [r]

X= 360

Y= -2

Z= 47.9588

Frequency [Hz]

Mag

nit

ud

e [

dB

]

• Progressive (=rotating) force wave of order r=-2 at 6fs=360 Hz

F=F0cos(6wst-2αs) noted {6fs, -2}0 100 200 300 400

-500

0

500

Angle [°]

Mag

nit

ud

e [

N/m


0 20 40 600

100

200

Mag

nit

ud

e [

N/m

2]

Space harmonic []


0 100 200 300 400-500

0

500

Angle [°]

Mag

nit

ud

e [

N/m


0 20 40 600

100

200

Mag

nit

ud

e [

N/m

2]

Space harmonic []


{6fs, -2}

rotation in trigonometricdirection at 6fs/r


• Standing (=pulsating) force wave of order r=-2 at 6fs=360 Hz

F=F0cos(6wst)cos(2αs) noted {6fs, +/-2}0 100 200 300 400

-500

0

500

Angle [°]

Mag

nit

ud

e [

N/m


0 20 40 600

100

200

Mag

nit

ud

e [

N/m

2]

Space harmonic []


-1000-500

0500

1000

-5

0

5

0

20

40

60

Spatial order [r]

Frequency [Hz]

Maxwell tensor main lines and stator natural frequencies

Mag

nit

ud

e [

dB

]

{6fs,+2}& (6fs,-2}

0 100 200 300 400-50

0

50

Angle [°]

Mag

nit

ud

e [

N/m


0 20 40 600

10

20

30

Mag

nit

ud

e [

N/m

2]

Space harmonic []



steady vibration nodes

• Illustration of the decomposition of a standing wave in two travelling wave of opposite direction

Acos 2𝜋𝑓𝑠𝑡 + 𝑟𝛼𝑠 + Acos 2𝜋𝑓𝑠𝑡 − 𝑟𝛼𝑠 = 2Acos 2𝜋𝑓𝑠𝑡)cos(𝑟𝛼𝑠

• At same magnitude a rotating vibration makes 3 dB more than a pulsating vibration in SWL

• The pulsating force wave has steady nodes in the stator frame only, the rotor sees two rotating waves withdifferent velocities


from [B27]

• The airgap stress waves projection on the stator and rotor give the resulting magnetic force

• Example on the 14 stator teeth of a fractional slot winding PMSM (r=0, 2, 8):


Effect of radial force harmonics on an outer stator, r=0

STATIC EFFECT OF MAGNETIC FORCES

• The radial displacements of the yoke are qualitatively given by [B8]

r=0 r=1r=0 r=2

• A radial force wave of wavenumber r=0 gives a stator radial deflections of wavenumer 0

h

R

EFU ww

100

E equivalent Young modulus of the yokeL stator lengthR yoke average radiush yoke average widthFrw magnitude of r-th order radial force wave at frequency w/(2π) – not pressure

• The thicker (high h) and the smaller (small R) is the yoke, the lower is static radial displacement


Effect of radial force harmonics on an outer stator, r=1

• A radial force wave of order 1 corresponds to an unbalanced magnetic pull which can tilt the stator, especially if the lamination is only clamped at one end

• It is also called a rigid body motion


Effect of radial force harmonics on an outer stator, r>1

3

3

4

1

h

R

ErFU wrwr

• The radial displacements of the yoke are qualitatively given by [B8]

E equivalent Young modulus of the yokeR yoke average radiush yoke average widthFrw magnitude of r-th order radial force wave at frequency w/(2π) – not pressure

• Radial displacement of due to r>1 decrease much quickly with h/R than displacement due to r=0

• Radial yoke deflection decrease in r4: only low spatial order force harmonics are of interest

• Due to quadratic relationships, high wavenumber flux density harmonics can generate high vibrations

r=2

• A radial force wave of wavenumber r gives a stator radial deflections of order r with same phase (if Zs/2

r=6


• A tangential force wave of order r=0 gives a torsional deflection of the yoke with same phase

• A tangential force wave of order r>1 gives a radial deflection of the yoke of order rwith different phase due to the teeth bending moment

from [B2]

Torsional deflection Radial deflection

from [B1]

Effect of tangential force harmonics on an outer stator


Effect of radial force harmonics on an inner rotor

• Radial force wave of order r>1 and r=0 do not create deflections due to high shaft stiffness

• Radial force wave of order r=1 (Unbalanced Magnetic Pull) generate bending of the rotor shaft

Effect of tangential force harmonics on an inner rotor

• Tangential force waves of order r=0 induce torsion of the rotor shaft

• Tangential force waves of order r>0 do not create torsion

Fy<0 net force


Case of an outer rotor with PM

• Radial force waves creates similar deflections as on the stator

• Tangential force harmonics with r>0 also create radial deflection of order r

• Standing airgap force wave {f,r}+{f,-r} is seen as two progressive force waves of different frequencies {f+/-rfR,r}

• Frequency shift is proportional to r times the rotating frequency (rfR)

see [B15]

Inner rotor Outer rotor


STRUCTURAL MODES OF ELECTRICAL MACHINES

• Structural modes of the external structure (stator or rotor) are generally characterized using an analogy with a cylinder

• Cylinder deflections can be characterized by circumferential (m) and longitudinal (n) spatial frequencies

• Each structural mode is then associated to a pair of integers (m,n)

• Note the distinction with the excitation wave number r and the structural mode order m

Structural modes of the stator lamination

(2,0)

(2,1)

(3,0)

(3,1)

Example of 3D modes


• The spatial order m and n can be linked to the number of nodes (null vibration) of the deflection

m is half the number of circumferential nodes, n is the number of axial nodes

Example of a 3D circumferential modes with clamped end

(4,0) - 1 axial node (4,1) - 2 axial nodes

Example of 2D circumferential modes with free-free BC

𝑈(𝛼, 𝑧) = 𝛾(𝑧)𝑈𝑚𝑛cos(𝑚𝛼 +𝑛𝑧

𝐿+ 𝜑𝑚𝑛)

BC modulation function


• Cylinder modes (m,n>1) are generally higher in frequency than (m,0) modes

frequency= 516Hz

mode=(2,0)

THD=0.0079

frequency= 1384Hz

mode=(3,0)

THD=0.019

frequency= 1693Hz

mode=(3,1)

THD=0.17

frequency= 2498Hz

mode=(4,0)

THD=0.029

frequency= 2539Hz

mode=(1,1)

THD=0.18

frequency= 2822Hz

mode=(4,1)

THD=0.16

frequency= 3166Hz

mode=(2,2)

THD=0.26

frequency= 3281Hz

mode=(0,1)

THD=0.19

frequency= 3315Hz

mode=(0,0)

THD=0.084

frequency= 3362Hz

mode=(3,2)

THD=0.25

frequency= 3654Hz

mode=(1,2)

THD=0.34

frequency= 3681Hz

mode=(2,1)

THD=0.14

frequency= 3761Hz

mode=(5,0)

THD=0.036

frequency= 4067Hz

mode=(5,1)

THD=0.16

frequency= 4075Hz

mode=(0,2)

THD=0.31

frequency= 4168Hz

mode=(4,2)

THD=0.24

frequency= 4856Hz

mode=(1,0)

THD=0.2

Results from MANATEE software

• Longitudinal modes importance is linked to the axial length to diameter ratio of the machine


Remarkable modes:

--> generally it has the highest frequency

- (1,0) is the bending mode excited by unbalanced pull

- (2,0) is the ovalization mode

- (0,1) is a torsional (tangential) or tilting (radial) mode(Clamped-free stator)


• When there are few teeth several modes appear at close frequencies

(0,0) tooth mode (1,0) tooth mode (2,0) tooth mode (4,0) tooth mode

• The local modes of the teeth are called tooth rocking modes [B4]

• There exist some teeth modes alone, or some coupled modes between teeth and back core

• The importance of teeth modes is linked to the tooth to yoke height ratio


(2,0) (2,1)(2,1) (2,0)


Structural modes of an inner rotor

• Only bending modes and torsional modes are of interest

• Strong electromechanical coupling can occur when rotating at the first rotor flexural frequency in large IM

n=1

n=2

From [B11]

From [B11]


DYNAMIC EFFECT OF MAGNETIC FORCES

Resonances (high vibration and noise amplification) can occur at two conditions [B8]

• time frequency match between the exciting force and the structural mode natural frequency

• spatial frequency (« order ») match between the exciting force and the structural mode shape

Generalities


Stator resonance in 2D

One pole pair (r=2 pole pairs) excitation rotating at nfs/r

nfs =f2

Elliptical mode of the stator stack(m=2) of natural frequency f2

𝐹 = 𝐹2 cos 2𝜋𝑛𝑓𝑠𝑡 + 𝑟𝛼𝑠

𝑈 = 𝑈2 cos 2𝜋𝑓2 𝑡)cos(𝑚𝛼𝑠

EXCITED STRUCTURE

EXCITATION FORCE

RESONANCE

nfs >f2

nfs<f2


• The frequency match does not hold on the mechanical frequency, but on the electrical frequency

• This is valid for the excitation of purely radial modes but also torsional modes (a torsional modal deflection has all its points moving in phase so r=0 is necessary)

• Far from resonance, the vibration pattern is given by the exciting force pattern (rotating vibration if rotatingforce, same wavenumber)

• At resonance, the vibration pattern fits with the pulsating modal shape (fixed node positions)

In 2D resonances (high vibration and noise amplification) occur at two conditions:

• match between the exciting force frequency f and the structural mode natural frequency fm:

• match between the exciting force pattern r and the structural mode shape m:

f=fm

r=m


• Far from resonance, the vibration pattern is given by the exciting force pattern (rotating vibration if rotatingforce, same wavenumber)

• At resonance, the vibration pattern fits with the pulsating modal shape (fixed node positions)

• Note that significant acoustic noise can be created without resonance


• This is the case of the ideal resonance, but a r=4 exciting force can slightly make the ovalization mode m=2 respond (lower magnification) in the presence of geometrical or mass assymetries

• The resonance speed has been proved analytically by Soedel [B8]

• Both a progressive rotating force wave and a standing pulsating force wave can resonate under resonance conditions, whatever the rotation direction

secondary resonance

main resonance

From [B9]

• In this example the secondary resonance has 10 times less magnitude (-20 dB for noise & vibration levels)


• The modal participation factor of magnetic forces can be quantified using Modal Force Matrix MFM:

[B51]


FROM MAGNETIC VIBRATION TO ACOUSTIC NOISE

Vibroacoustic transfer pathsCircumferentialwavenumber

Force direction

Transfer path Description

r>0 Radial,tangential

Air borne Radial circumferential deflection of the outer statoryoke and frame or outer rotor (rotating in forcedregime, pulsating at resonance)

r=0 Radial Air borne Radial pulsating circumferential deflection of thestator yoke and frame or outer rotor

r=0 Tangential(coggingtorque / torqueripple)

Structural borne Propagation of rotor torsional vibration to rotor shaftline and gearbox mount, or bearing sleeves and outerstator frame

r=0 Tangential(coggingtorque / torqueripple)

Air borne Deflection of the outer stator yoke and frame or outerrotor following a unbalanced torsional mode due toparticular boundary conditions

r=1 Radial(unbalancemagnetic pull)

Air borne Bending / tilting deflection of the outer stator frameor outer rotor, in particular in clamped-free conditions

r=1 Radial(unbalancemagnetic pull)

Structural borne Propagation of rotor bending vibration to rotor shaftline and gearbox mount, or bearing sleeves and outerstator frame

NA Axial Air borne Axial deflection of the end-shields


EXAMPLES

• Pure slotting noise in SCIM during sinusoidal run-up

• Asynchronous PWM noise at starting of a SCIM


• Different frame boundary conditions :

- Clamped / clamped for most horizontal motors

- Clamped / free for some horizontal motors or vertical motors

- Free / free never met under operation

- More complex boundary conditions exist on real machines (asymmetrical mounting ears)

Boundary conditions[B50]


• Real boundary conditions modify the natural frequencies, more or less depending of the type of structural mode

• Real boundary conditions generally breaks the symmetries and multiply the number of cylindrical modes (for instance two (2,0) appear at different frequencies)

• in particular fixations can introduce some order 1 in the yoke deflections which can then be excited by UMP forces, or some order 0 which will be excited by torque ripple

(2,0) (2,0)

(2,0) (2,0)

(3,0) [B37]


Structural damping

•

• Each modal deflection can be associated to a structural damping quantifying the energy dissipation due to friction

• Damping mainly comes from the winding & lamination insulation, and resin

• Depending on the VPI process the contribution of resin differs

• In practice reduced damping varies from 1 to 2% (xi/xic)

• Damping is higher for longitudinal modes such as (2,1)

• B11] and PMSM

Date post:	19-Jul-2018
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