1

Simulations and Validations of Rotor DynamicEccentricity Effects on Synchronous Machine

Vibrations During a Full Run UpPierre Pellerey, Vincent Lanfranchi, Guy Friedrich

Abstract—Rotor eccentricity is considered a major problemwhen working with electrical machines. Generally linked tomanufacturing tolerances and rotor balance quality, this paperanalyses the contribution of rotor dynamic eccentricities to thevibro-acoustic behavior of a synchronous machine during a fullrun up. Thanks to the coupling of electromagnetic and structuralfinite element models, we find that additional harmonics linkedto eccentricities lead to new acoustic noise components. A directlink between eccentricity and levels of vibration is achieved. Fora complete run-up, experimental results validate the numericalpredictions on a complex powertrain dedicated to automotivetraction.

Index Terms—Acoustic noise, acoustic analysis, dynamic rotoreccentricity, electrical machines, finite element methods, rotordynamics, structural dynamics analysis, synchronous machines,vibrations, wound rotor synchronous machines.

NOMENCLATURE

σr,t Airgap magnetic forces densities in the radial (r)and tangential (t) directions (Pa)

m Spatial order of the airgap magnetic force har-monic

n Time rank of the airgap magnetic force harmonicp Number of pole pairsZs Number of stator teethe Airgap thickness (mm)ed Dynamic eccentricity (mm)

I. INTRODUCTION

Besides increasing copper and iron losses and raising bear-ings stress in the machine, eccentricities and the associatedunbalanced magnetic pull tend to raise the machine’s noiselevel [1]. The impact of manufacturing tolerances on vibro-acoustic behavior of electrical machines has been considered asa key factor for several decades [2], [3]. In this paper, we usenumerical tools to analyse the impact of these manufacturingspreads and, in particular, of the rotor dynamic eccentricitieson the acoustic noise levels of the machine.

This study mainly focuses on electrical machines designedfor automotive traction that work in a large range of speedand torque. For this kind of application that considers bothvibratory and acoustic levels a comfort standard, the machineand its force harmonics potentially sweep a range of criticalfrequencies in working conditions. In this context, it is essen-tial to effectively determine and forecast the vibratory behaviorof the machine, not only on one or several working points, but

on the full working range to assure complete allowance to theuser. This kind of approach has been developed by [4], [5].

Theoretically, the main source of acoustic noise of a per-fectly symmetrical, non fractional 2p pole electrical machineis caused by the radiation of the stator radial flexion modes,whose orders are 0, 2p and its multiples (torsion modes ofthe stator and the rotor will normally be excited but are oftenless critical in terms of acoustic radiation). The presence ofirregularities or defects on the airgap shape will introduce inter-mediate spatial and temporal orders in the magnetic excitation.These new harmonics will be able to excite new modes oforder m such as m < 2p. Moreover as deformation levelsare inversely proportional to the fourth power of the spatialorder of the mode [6], these new modes, most of the time withlower frequencies, will be acoustically present and will densifythe machine’s global acoustic spectrum leading to potentiallycritical noise levels.

The machine under study is a 70 kW rated power threephases wound rotor synchronous machine (WRSM) with 2p =4 poles and Zs = 48 stator slots. To calculate the vibrationsof such a machine, we consider, neglecting the end effectsdue to the stator end winding, a coupling of the magneticforces computed with a 2D electromagnetic finite elementmodel (FEM) to a 3D mechanical FEM. Hence, we can achievean efficient and global vibro-acoustic analysis with relativelylow computation time [7], [8]. For skewed machines, 3Delectromagnetic FEM has to be used [9].

Analytical approaches to predict acoustic magnetic noiseexist and have been developed for different electric machinestechnologies, for instance for induction machines [10], [11],[12], switched reluctance machines [13], brushed DC ma-chines [14] or permanent magnet synchronous machines [15].These techniques allow fast calculations but are not suited toaccurately analyse complex systems such as the powertraindiscussed in this paper.

In this study, we are going to compute and analyse theevolution of the magnetic forces acting on both the rotor andthe stator (in the radial and tangential directions) during afull run up and various rotor dynamic eccentricities levels.We will then apply them to the structural model as nodalforces, to compute the structural dynamic responses. Thanksto spectrograms, we will analyse the evolution of the vibratoryresponse of the electrical machine with the level of rotoreccentricity. The innovation of this paper is that we willpresent the experimental validations of the previous numerical

OO’

ed

ω.t

R’

R

Stator

Rotor

ar

as

Figure 1: Geometrical definition of the dynamic eccentricity

coupling for a powertrain during a complete run up. Thepresence of rotor eccentricity in the measured structure willbe clearly demonstrated.

II. MAGNETIC FORCE COMPUTATION

A. Presentation of the simulations’ background

We seek to record the airgap magnetic pressures actingrespectively on the stator and rotor, depending on various rotordynamic eccentricity levels. These eccentricities are generallydue to bearings clearance, shaft bending, or rotor ovalization.Thus, we use a 2D magnetic finite element model in whichthe distance between the rotor center O′ and the stator centerO is set by the distance ed (Fig. 1). The rotor rotates at speedω around the stator center O with a circular orbit. Here weconsider only eccentricities in the machine plane. This meansthat we neglect 3D effects so the axes of the stator as and ofthe rotor ar remains parallel as defined in Fig. 1. The impactof such 3D eccentricities on magnetic forces has been studiede.g. by [16], [17].

In the context of a WRSM we carry on a transient analysis,which is based on a sum of quasi-static analyses using acurrent supply to record the evolution at each time step of themagnetic pressure distribution along the airgap. We neglectthe influence of the high frequencies harmonics of the current,i.e. harmonics due to PWM [18]. Thus we assume whateverthe eccentricity level is, that feeding currents are sinus andbalanced on each phase. The consequences of this hypothesishave been showed in [19], [20], where the impact of lowfrequencies current harmonics on vibrations is studied.

Since the stator winding is made of parallel branches andtaking into account the unbalanced back-emf due to eccen-tricities, there is a natural balancing due to the circulatingcurrents in the phases which equalize the flux distribution[21], [22]. Therefore, not considering this electrical couplingand its potential damping, we will be inclined to overestimatemagnetic forces. The forces due to the homopolar component[23] are also neglected.

Finally, knowing the current references linked to eachworking point, we compute the magnetic pressure distributionsfor a run up at 40 N.m from 500 rpm to 11500 rpm (Fig. 2).

Figure 2: Location of the computations over the full range

We distribute the simulations at 1000 rpm each (red stars inFig. 2). We re-iterate those calculations for three differentlevels of dynamic eccentricities such as ed/e = {0, 19, 38}%with e as the airgap length. The green stars of Fig. 2 matchwith the working points used for the experimental validationsin section V.

B. Magnetic excitation evolution

The force density distributions along the airgap can becomputed using the Maxwell tensor [19] (we will neglect theinfluence of magnetostriction). This pressure distribution canbe decomposed as a Fourier serie:

σr,t(θ, t) =

+∞∑m=0,n=−∞

σr,tm,n cos(mθ ± nωt+ ϕr,tm,n

). (1)

We find the indices r and t which belong to the radialand tangential directions of the forces, θ gives the angularposition in the airgap, ω is the rotor pulsation, m and nrespectively correspond to the spatial order and the time rankof the harmonic, whereas ϕ is the linked phase.

In order to analyse analytically and quickly how dynamiceccentricities affect the spectral content of the forces, weare going to develop a first harmonic model of the machine,neglecting magnetic saturation and slotting effects. We con-sider that, with a dynamic eccentricity, the radial surfacicpermeance of the airgap λr has an additional term λrd =e/ed cos (θ − ωt− αd). The radial air gap flux density Bris:

Br = ε. (λr + λrd) (2)

= ε0 cos (p (θ − ωt)− ϕ) .µ0

e

[1 +

e

edcos (θ − ωt− αd)

]with ε the m.m.f produced by the stator and rotor windings,ε0 the amplitude of the m.m.f wave, µ0 being the permeabilityof free space, ϕ the phase angle of the m.m.f. wave and αdthe original position of the eccentricity. We can develop theprevious equation 2:

Br = B0 cos (p (θ − ωt)− ϕ) (3)+Be cos ((p± 1) (θ − ωt)− (ϕ± αd)) .

We note Be = B0 × (e/2ed) the amplitude of the eccentricmagnetic flux density and B0 = ε0µ0/e the amplitude of theclassical magnetic flux density wave. Considering that theradial Maxwell pressure is given by σr = B2

r/2µ0, we obtain:

σr = [B20 cos (2p (θ − ωt)− 2ϕ) (4)

+ B2e cos (2 (p± 1) (θ − ωt)− 2 (ϕ± αd))

+ 4B0Be cos ((2p± 1) (θ − ωt)− (2ϕ± αd))+ 4B0Be cos (± (θ − ωt)± αd) ]/4µ0.

Thus, without any eccentricity we only have a magneticpressure wave rotating at nω = 2pω speed and with am = 2p spatial order. Adding eccentricity, we see that newspatial temporal waves appear. These terms are modulationsat 2 (p± 1), (2p± 1) and ±1 of the spatiotemporal waves.Consequently, and whatever the kind of machine, dynamiceccentricities will generate intermediate modulations of themagnetic forces that are non-multiples of the number of polepair. The conclusions would have been the same, consideringall the harmonics of the machine (slotting, winding frame ...)and the magnetic saturation [24], [25].

1) Static eccentricities: A static eccentricity could bedefined as a dynamic eccentricity in which the center of therotor O′ is fixed over time. As discussed in [2], [26], [27],there is only spatial modulation of the forces harmonics forthis type of irregularity in the air gap shape function. Thismeans that static eccentricities will not generate harmonicswith new time ranks but only with new spatial orders.However, these new spatial components will still be able toexcite new modes and, thus, to generate more acoustic noiseand vibrations. As static eccentricities are a particular case ofthe the dynamic ones, we will study these last to embrace allphenomena.

2) Dynamic eccentricities: According to eq. 1 we nowplot in Fig. 3 the time spectrums of the radial Maxwellpressures acting on the stator. These Maxwell pressures arecomputed from the finite element evaluation of the magneticfield in the middle of the airgap of the WRSM model. Thus,compared to the previous analytical study, we now consider allthe components of the magnetic field (slotting, saturation...).

Those spectrums are presented for different spatial orders,for two different values of eccentricities : ed/e = 0% anded/e = 38% and for the T = 40 N.m et N = 3500 rpmworking point. As it has already been emphasized, only lowspatial orders forces (typically with m < 5) can generatestrong acoustic noise. That is why, according to equation1, we only depicts in Fig. 3 the five first time spectrums :σrm=[0,1,2,3,4],n.

0 20 40 600

50

100

150

harmonic [n]

Am

plit

ude

dB [

Pa]

m=0

0 20 40 600

50

100

150

harmonic [n]

Am

plit

ude

dB [

Pa]

m=1

0 20 40 600

50

100

150

harmonic [n]

Am

plit

ude

dB [

Pa]

m=2

0 20 40 600

50

100

150

harmonic [n]

Am

plit

ude

dB [

Pa]

m=3

0 20 40 600

50

100

150

harmonic [n]

Am

plit

ude

dB [

Pa]

m=4

ed / e=0%

ed / e=38%

47

481224

3649

1

11 13

52

4

8

44

1014 21

27

Figure 3: Evolution of the harmonic content of the stator radialforce densities with the dynamic eccentricity level

n = 2kp 2kp± 1 2kp± 2m = {0, 4} {1, 3} 2

Table I: Spatial orders of the harmonics

Figure 3 shows that, when the eccentricity level is null(ed/e=0% on blue circles), the spectrums for spatial orderm = 0 and m = 4 contain all the harmonics of the forces.This is coherent because we are studying a 2p = 4 polesnon fractional machine, whose forces may not contain spatialorders non multiple of 2p.

Then, when the eccentricity level increases (green spectrumsfor ed/e = 38% in Fig. 3), a certain number of harmonicsemerge on the intermediate spectrums (for m = 1, 2, 3). Weremark the emergence of the symptomatic order of a dynamiceccentricity {m,n} = {1, 1} belonging to the eccentric ro-tation of the rotor. We also may notice an average smalldecrease of the original harmonic (e.g. {m,n} = {0, 48},{0, 24} or {4, 44}) when the eccentricity level increase. Thismight correspond to a dissipation of the spectrum because ofthe new modulations.

To sum up, dynamic rotor eccentricity generates forces timeranks such as n = 2kp±m (with k a natural number). In factfor m = 1 we notice that the harmonics are odd and modulated

zNij

θ

zj

Eij

θi

zNij

θ

zj

Eij

θi

Figure 4: Zoom of the mechanical mesh on one stator teethand definition of the attached coordinate system

at ±1 around multiple of 2p = 4 (e.g. n = 47 and n = 49around n = 48). The same applies to m = 2, where theharmonics are modulated at ±2 around multiple of 2p (e.g.n = 10 and n = 14 around n = 12) and for m = 3 with thesame argument (e.g. n = 21 and n = 27 around n = 24). Wesummarize this observation in the table I, in order to associateto any given n harmonic its spatial order m. Note that as2kp ± 3 = 2(k + 1)p ± 1 = 2k′p ± 1, we can classify them = 1 and m = 3 harmonics under the same modulation typein table I.

Because this first analysis only considers stator radial forcesfor one given working point, we may not draw any con-clusion about the evolution of the vibro-acoustic behaviorof the machine, when the rotor has a dynamic eccentricity[4], [28], [29]. However, the forces spectrum densificationpreviously computed points out that we will probably havenew mechanical modes to be excited, when we will introducethe eccentricity.

III. FORCES PROJECTION METHOD

Once the magnetic pressure distribution is computed forthe different points of the run up, its projection on the 3Dmechanical mesh of the stator and rotor has to be carried out.We consider that, for the machine we are studying, the stressesare constant along the axial direction (it amounts to neglectend effects). Therefore, we can apply the same forces valuesto nodes having the same angular position. Thus, the creationof an extruded regular mesh (Fig. 4) from a 2D mesh made ofshell elements simplifies the coupling process between 2D and3D FEM. Now we can associate to nodes having same θi thesame excitation. By splitting both the stator tooth and rotorpoles surfaces in equal elements, we can greatly simplify theamount of load cases that we need to apply to the structure. Forour mesh, we use isoparametric hexahedral elements (brick of8 nodes with linear shape function). The creation of the nodalforce vectors is obtained using the discrete coordinate systemconventions, defined in Fig. 4, so we can calculate the forcespectrum Fr,t(Eij , f) applied on each element Eij from:

Figure 5: Structural model of the electrical machine.

Fr,t(Eij , f) =

ˆ

Eij

Hijσr,t(θ, f)dθdz, (5)

with Hij the interpolation function of the Eij element whichis equivalent to its shape function. Since there are four nodeson the element surface where the forces are applied, andconsidering the linear evolution of Hij , we estimate that thenodal forces spectrum F (Ni, f) for each line i (that means forall the nodes Ni of this line), is the sum of the forces on theneighboring elements:

Fr,t(Ni, f) = Fr,t(Ei−1,j , f) + Fr,t(Ei,j , f). (6)

By using this technique we get the nodal loadings for eachof the three simulations (three values of ed). By a linearinterpolation, we compute the mechanical loads from 500 to11500 every 100 rpm step (111 computations).

IV. STRUCTURAL COMPUTATION

The detail of the structural computation method has beendeveloped in [19]. In Fig. 5 the complete mechanical modelused for the simulations is presented. It is made of theWRSM and its reducer. For this machine, we consideredYoung modulus of the core stacks of the stator and the rotorin the plane of the laminations Er = Eθ = 215 GPa and inthe stacks length Ez = 180 GPa. The winding are modeledas inertia elements to avoid unrealistic winding modes and tosimplify the modal basis. Finally, the damping depends on thefrequency and is identified according to measurements. Theother material properties are classic.

A. Observations about the rotor dynamics

In order to develop a more representative modeling of howmagnetic forces are applied on the mechanical structure, weadd forces on the rotor’s exterior surface. This approach isinnovative because most commonly only the stator vibrations

Frequency [Hz] Description Involved harmonics400 Stator housing covers bending n = 121200 Ovalization mode m = 2 n = 8, 124600 Flexion mode m = 4 n = 44, 52, 925700 Breathing mode m = 0 n = 48, 96

Table II: Description of the main excited mechanical mode.

are considered. Nevertheless the rotor vibrations, when con-nected to a gearbox as it is usually the case, could generatevibrations of the gearbox housings and, thus, an acousticradiation. Moreover, the eccentricities will stimulate the rotoraccording to new kind of vibrations, typically rotor bendingmodes, because of the new unbalanced forces. Thus, it isimportant to integrate this phenomenon.

To apply the forces on the rotor’s exterior surface, we usethe same methodology that we used for the application of thestator forces. Nevertheless, if the magnetic forces acting on thestator were calculated in the stationary frame, the forces actingon the rotor are now computed in the rotating frame (alwaysin the middle of the airgap). Thus, the harmonic content ofthese forces are different because of this frame change. Therotor bearings and the tri-dimensional gear teeth contacts arealso taken into account in the structural model.

However, we neglect the own dynamic of the rotor, consid-ering that its gyroscopic and centrifugal properties (linked tomass and stiffness rotor dissymmetries) are relatively weak,so that the rotor speed does not affect its eigenfrequencies.Studies integrating the impact of eccentricities on the rotordynamic exist whether they use analytical [30] or numerical[31] approaches. Unfortunately, they do not study the impactof the rotor vibrations on the stator vibrations.

Moreover, as there is no rotor movement considered in thesimulations, we also neglect the effect of the frame change inthe predicted frequencies.

B. Computed spectrograms

Spectrograms are used to present the results of the structuralcomputations made in the frequency domain. A spectrogramis a time-varying spectral representation, that shows how thespectral density of a signal changes over time. Here the timeis represented by the motor speed (constant acceleration) sothe spectrograms in Fig. 6 represent the spectrum evolution ofthe radial acceleration for one point of the stator housing withthe rotor speed for ed/e = 0% (simulation n°1, Fig. 6a) anded/e = 38% (simulation n°3, Fig. 6b).

We can observe some points worth noting. First, each lineof the spectrogram represents a n harmonic whose frequencyfn is proportional to rotor speed (fn = nω/2π with ω inrad/s). For the spectrogram Fig. 6a the most energetic ordersin terms of vibrations are n = {8, 12, 44, 48, 52, 92, 96}. Wewill further discuss some of their evolutions.

Then, the eigenfrequencies of the machine appear at fixedfrequencies independent of the machine speed. We can noticethat all harmonics excite the structure around 4700 Hz. Thisis the frequency of a particular mode of the machine (modem = 4 = 2p).

(a) Simulation n°1, ed/e=0%.

(b) Simulation n°3, ed/e=38%.

Figure 6: Simulations of the radial acceleration spectrogramsfor two different values of ed at 40 N.m.

As previously seen, we can associate to each harmonic aspatial order, e.g m = 4 for n = 44 see Fig. 3. Therefore,when the frequency and the spatial order of the harmonicmatches with the one of the mechanical mode, the vibrationlevel will be at its maximum. We summarize the mainresonances in table II.

Focusing on the differences between the two spectrograms,we see that eccentricity brings new harmonics to emerge(typically notice the huge emergence of n = 1 in Fig. 6b). Aspreviously demonstrated, these new harmonics may solicitatemodes not excited so far.

In order to have an overall view on the harmonics whosebehavior changes with the increase of eccentricity, we calculatethe average spectrum S of the radial acceleration for each

Figure 7: Illustration of the average spectrums variations.

experiment (eq. 7). This average spectrum matches with theaverage of the 111 spectrums s computed for each speed ω.Finally, we compute:

S(n) =1

111

11500∑ω=500,600,...

s(ω, n). (7)

We plot in Fig. 7 the difference of the average spectrumsresulting from experiments 2 and 3 (respectively made fored/e = 19% and ed/e = 38%) to the one of experiment 1(without eccentricity). We observe that these are the ordersn ={1, 10, 11, 13, 15} which mainly emerge in terms ofvibrations differences. It is showed in Fig. 7 that the averageincrease for n = 1 is 28 dB for simulation n° 2 and 34 dB forsimulation n° 3, compared to simulation n° 1.

C. Harmonic analysis

As it is difficult to conduct further precise analysis withthe spectrograms, we plot the evolution of some of the moreinteresting orders with the speed and the eccentricity level.These diagrams are presented in Fig. 8. The red curve belongsto ed/e = 0, the green light one to ed/e = 19% and the bluedotted to ed/e = 38%.

The level of the n = 1 harmonic increases on average bymore than 28 dB for the whole run up with the values ofeccentricity used. The modes at 15 Hz and 35 Hz (900 rpm and2100 rpm) are modes of the machine on its elastic mountings(Fig. 8).

Looking now at the n = 10 harmonic, we remark that anew mode is excited with the rotor eccentricity. This modeat 7300 rpm (1217 Hz) matches with the m = 2 ovalizationmode of the stator. As we established in table I, the harmonicn = 10 has a m = 2 spatial order, so it is coherent that thisnew mode emerge with eccentricity. For each increase of ed,we observe increases in vibration at this frequency of 18 dBand 29 dB, respectively.

0 2000 4000 6000 8000 10000 12000-30

-20

-10

0

10

20

30

40

50n = 1

Speed [rpm]0 2000 4000 6000 8000 10000 12000

-50

-40

-30

-20

-10

0

10n = 10

Speed [rpm]

0 2000 4000 6000 8000 10000 12000-50

-40

-30

-20

-10

0

10n = 11

Speed [rpm]0 2000 4000 6000 8000 10000 12000

-40

-30

-20

-10

0

10

20n = 12

Speed [rpm]

0 2000 4000 6000 8000 10000 12000-60

-50

-40

-30

-20

-10

0

10

20n = 44

Speed [rpm]0 2000 4000 6000 8000 10000 12000

-60

-50

-40

-30

-20

-10

0

10n = 45

Speed [rpm]

0 2000 4000 6000 8000 10000 12000-40

-30

-20

-10

0

10

20n = 48

Speed [rpm]

Figure 8: Harmonic analysis for different dynamic eccentrici-ties values

As the n = 11 and the n = 13 harmonics have the samebehavior, we only present in Fig. 8 the evolution of the n = 11harmonic. The average level successively increases by 17 dBand 23 dB. Even if no particular new modes emerge, we cansee that this harmonic, initially negligible, becomes important.

Considering the n = 12 harmonic, there are two newfrequencies whose responses are eccentrically dependent - at2000 rpm and 10000 rpm (respectively 400 Hz and 2000Hz).The 2000 rpm mode is a bending rotating mode of the rotorshaft coupled to a stator covers mode. As eccentricity level

increases, it is logical that the bending modes of the rotorshaft are increasingly solicited. It is interesting to notice thatthe rotor deformations are measured on the stator housingwith equivalent or more important levels than the typicalovalization mode response. For the 10000 rpm over level,it is mainly a bending mode of the rotor which is verysensitive to eccentricity coupled with an small ovalizationmode m = {2, 1} (phase shifted along the stator length).

For the n = 45 harmonic (n = 33 harmonic has the samebehavior) we observe at 3200 rpm and 3900 rpm (2420 Hzand 2970 Hz) new variations. It corresponds to the excitationof m = 3 ovalization modes of the stator. As defined in Fig.3 this harmonic is mainly of spatial order m = 3 so we canagain validate the approach.

Finally for the n = 44 and n = 48 harmonics, that are partof the most energetic harmonics, there are no big changes - asexpected in section II-B2.

Similar results were obtained by measurement in [2] forinduction machines.

V. EXPERIMENTAL RESULTS

To validate the predictions of the finite element modelscoupling, we establish a correlation between experimentalmeasures and numerical results.

We present in the top of Fig. 9 the test bench used forthe measures. The electrical powertrain is made of an electricmachine, a reducer and a power electronic converter based ona frame that is connected to the test bench via three enginebrackets (only two engine brackets are visible). Compared tothe structural model presented Fig. 5, the corresponding model(bottom part of Fig. 9) is now considering other mechanicalelements, such as the frame or the engine brackets. Thismechanical model has been updated and a special effort hasbeen made on the electric machine vibratory behavior to be asrepresentative as possible.

The experimental and simulated run-ups are both made at a160 N.m constant torque acceleration and, subsequently, at a70 kW constant power part, which is the highest rated powerof the machine (green stars in Fig. 2). From the experimentalpoint of view, the run-up from 0 to 12000 rpm last two minutesso as to consider a succession of stationary states. Fromthe numerical point of view, we compute every 1000 rpmthe magnetic forces distributions in the airgap, consideringthe measured stator and rotor currents as input (we onlyconsider the fundamentals of these signals). Then, by linearinterpolation, we get every 500 rpm, the nodal forces actingon the structural model and compute by modal superpositionthe dynamic response of the structure. We integrate roughly800 modes in the 0 -7000 Hz frequency range for the modalsuperposition.

In order to take advantage of the previous study, we computetwo different run-ups - with and without dynamic eccentricity.Thus, we will be able to analyse if the real machine presentssigns of eccentricities in its vibratory behavior. The value ofthe relative eccentricity has not been measured at this point but,considering the rotor bearing clearances, is at least of 5%.

Figure 9: Test bench used for the measurements and structuralFEM associated.

The best way to compare the measured and simulatedresults is to plot order trackings. In Fig. 10 we present theexperimental measures of the radial acceleration in red andthe numerical results for two different values of eccentricity,in blue for ed/e = 0% and in dotted green for ed/e = 19% .The point of measurement is located in the middle of the statorhousing. To avoid nodes of vibrations we always measureseveral points but this one appeared as relatively representativeof the electrical machine - reducer behavior.

In accordance with the previous analysis, we presentonly the main harmonics in terms of intensity (i.e. n =8, 12, 44, 48, 52, 92). The harmonics like n=10 or n=11 arerelatively weak compared to the background noise of themeasures to represent any interesting comparison to previousstudy.

We can observe a good agreement between the measures

Speed (rpm)

Speed (rpm)

Speed (rpm) Speed (rpm)

n=52

−50

−40

−30

−20

−10

0

10

20

Figure 10: Comparison of the measured and simulated me-chanical harmonics amplitudes (radial acceleration of the statorhousing). In red the experimental results, in blue the numericalresults for ed/e = 0 and in green the numerical results fored/e = 19%.

and the simulations made without eccentricity (the red andblue curves, respectively). The n = 44 is the most impor-tant harmonic in terms of stator housing radial accelerationintensity (which is directly a source of acoustic radiation). Itsbehavior is well predicted despite an average 3.4 dB levelshift. The 6000 rpm resonance still corresponds to a 4th

order flexion mode (table II). For the harmonics such asn = 8, 12, 44, 52, 92 the numerical predictions are precise:the mean deviation between the predicted and the measuredlevels is 4.8 dB, while the average standard deviation is5.7 dB. For the n = 48 harmonic, we are over-estimatingthe vibrations levels by 9.6 dB in average. Moreover, thefrequency of the breathing mode m = 0 at 6500 rpm is200 Hz frequency shifted. Despite that, and rememberingthe complexity of the coupled models, their representativity isconsidered very acceptable and allows being confident aboutthe previous predictions. Indeed, in the automotive domainthe typical mean deviation when predicting the vibration of aninternal combustion engine is about 10 to 15 dB.

However, if we look at the results obtained when addingeccentricity in the electromagnetic model (the green dottedcurves in Fig. 10), we observe that some of the previousvariations between the measured and simulated results werereduced. Indeed, if we look, for instance, at the 2200 rpmemergence on n = 8, or the 1600 rpm emergence on n = 12,we can see that considering eccentricity allows to improve thepredictions. This mode at 320 Hz corresponds to the 400 Hzstator housing covers bending mode of the previous structuralmodel (see table II). As previously explained, this mode ismainly excited by the rotor unbalance thus it is logic to see itincrease when adding eccentricity.

This phenomenon is almost the only one that the eccentricityconsideration allows to predict in a substantially improvedway. However, we may notice that the global predictionshave improved too. Indeed, the mean deviation between thepredicted and the measured levels was 5.7 dB for all theharmonics (n = 8, 12, 44, 48, 52, 92) and is now 5.2 dB whilethe average standard deviation decreased from 5.2 dB to 4.4dB.

Finally, the last variations between the predictions and themeasures could be explained by the structural and electromag-netic finite element model representativities, the presence ofstatic eccentricities, neglecting the magnetostriction effects orthe fact that we used a sinus current supply in our simulations[19].

VI. CONCLUSION

Using numerical simulations, we showed that dynamiceccentricities introduce magnetic force harmonics with bothintermediate spatial and temporal orders. Then, we explainedthe spatio temporal properties of these new harmonics. Thecoupling process between the electromagnetic and structuralmeshes has been detailed and dynamic computations have beenachieved.

The numerical results, presented as spectrograms and har-monic analysis, have clearly shown critical points. One ofthem is that new modes are excited. Typically, lower orderstator modes are now excited with an important and dangerousvibration efficiency. Moreover, rotor bending modes are nowalso excited and generate strong vibrations.

The last part of this study was dedicated to the experimentalvalidations. A complex electric powertrain structural modelhas been used. We only compared the vibrations levels forthe main harmonics (in terms of amplitude) and the level ofprediction has been judged as reliable. Moreover, simulationsincluding a realistic amount of dynamic eccentricity have beenable to predict better results than those without. Even if the im-provement is small, considering small amount of eccentricityin the simulation allows to have a more representative modeland thus, could be useful to take into account most part of thephenomenon.

REFERENCES

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