Prace Naukowe Instytutu Maszyn, Napdw i Pomiarw ElektrycznychNr 64 Politechniki Wrocawskiej Nr 64Studia i Materiay Nr 30 2010

TermsFEM calculation, magnetic stress,natural vibration, synchronous motor

Janusz BIALIK*, Jan ZAWILAK*

VIBRATION CALCULATIONS IN TWO-SPEED,LARGE POWER, SYNCHRONOUS MOTOR

This paper deals with finite element calculation of the vibrations of magnetic origin in two-speed,large power, synchronous motor. Prediction of such vibrations are very important in understanding vi-bration phenomena of electrical machines. In configuration for lower rotational speed pole numbers ofmagnetic field and numbers of excited poles of field-winding are not equal. Because of non symmetricalarmature and field windings the only one way of investigation is finite element (FE) modeling. Simula-tion are done for nominal load, for two different rotational rotor speeds: n = 500 rpm (2p = 12) andn = 600 rpm (2p = 10), and corresponding nominal active powers: P = 600 kW and 1050 kW. Thetarget of application of discussed analysis is the vibration behavior of the machine.

1. INTRODUCTION

In term to precisely modeling of electromagnetic origin vibrations of rotating ma-chines, information of loads and mechanical construction of the investigated object haveto be known. Electromagnetic forces (loads) can be calculated within time-steppingmodeling [3, 4, 7, 11]. Mechanical behavior of the construction can be obtained frommechanical calculations or vibration measurements. In many cases measurements are notpossible or it is difficult to perform them (especially with big machines). Analyticalmethods in most cases dont give results with good accuracy, especially when complexstructures are investigated [9]. Making modification of the modeled machines in designstage and looking into impact of those modifications are main advantages of numericalmethods.

Two-speed synchronous motor are examples of no symmetrical machines(asymmetric armature and field winding). Therefore investigation can be done onlywith help of finite element modeling methods [3, 4]. Those motors were built up by

__________* Politechnika Wrocawska, Instytut Maszyn, Napdw i Pomiarw Elektrycznych, ul. Smoluchow-

skiego 19, 50-372 Wrocaw, janusz.bialik@pwr.wroc.pl, jan.zawilak@pwr.wroc.pl.

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replacing stator and rotor winding with switchable windings. By switching thewindings, two different numbers of pairs of magnetic pole are obtained. Thus twodifferent speeds are obtained [1].

In this paper calculation results of the two-speed synchronous motor typeGAe1510/12p are presented. This motor has two different speeds: n = 500 rpm (2p = 12)and n = 600 rpm (2p = 10) and corresponding nominal powers P = 600 kW and 1050 kW.Determination of vibration of electromagnetic origin acting in mentioned motor is thegoal of this paper.

2. MAGNETIC STRESS CALCULATION

For investigation the motor type GAe 1510/12p is chosen, which construction isbased on convectional one-speed synchronous motor. Calculations are performed withhelp of two dimensional field-circuit model of mentioned motor [3]. Rated data of themodeled motor type GAe1510/12p is introduced in Table 1. This motor has double layerstator winding placed in 108 slots, field winding and the damper circuit, allocated in10 pole shoes. Field part of the model takes into account non-linear characteristics of themagnetic part of the motor and the motion of the rotor. Circuit part takes into considera-tion the electrical parameters of the source, damper circuit and switchable armature andfield windings. In elaborated 2D model an assumption of constant parameters of the endparts of all windings, which is of stator, rotor and damper circuit is done. Values of thereactance and resistances of the end parts are calculated according to the well-knownequations [8, 10]. To change the circular flux of the armature and field windings, whichqualify speed variation of the rotating field, the direction of the stator and rotor currentsin right section of the windings must be changed [1] (Fig. 1).

Table 1. Rated parameters of the investigated motor

Nominal power Pn kW 600/1050Nominal voltage Un V 6000Phase connection Y/YYNominal current In A 86/121Field voltage Ufn V 51/70Field current Ifn A 175/240Nominal speed nn rpm 500/600Power factor cosn 0.8 lag/0.9 leadEfficiency n % 80.0/94.2

Presented motor has unconventional circumferential distribution of phase groups ofarmature winding (Fig. 1a). Different number of magnetic poles are obtained due to

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changes of the armature current P in all stator phases A, B, C. In all groups named NPcurrents direction are unchanged, for both rotational rotor speed. Corresponding con-figuration of field winding is presented in Fig. 1b. With black color magnetic poles forthe higher rotational rotors speed are marked (convectional distribution of magneticpoles) and with grey color for lower rotational rotor speed (unconventional distribu-tion).

Fig. 1. Circuit diagram of distribution group phase of armature winding (a)and polarity of rotor (b) for both rotational speed of testing motor

In all simulation, elaborated 2D field-circuit model is used [3]. In Figure 2 numericmodel (field part) of investigated motor, together with cylindrical coordinate system(situated in air-gap, 0.3 mm below stator inner surface), and part of finite element meshis shown. Second order approximation of the magnetic vector potential is used. In termto calculate the magnetic stresses, the distribution of the radial (normal) and tangentialcomponent of the flux density in the air-gap of the motor, are determined. The time-stepping method is used, i.e. magnetic flux density for next 100 (for higher rotationalspeed; 120 for lower rotational speed) time steps, for nominal load is determined. Mag-netic field is sampled in 1024 equidistance points in air-gap. An example of such distri-bution, valid for one time moment, is presented in the Fig. 3.

For the picture clarity only space distribution for one time moment are show. Resultsare valid for both rotational speeds. Finally, collecting all calculation results (for all timesteps), matrix of flux density B(m, n) is obtained, where M is a number of space samples,and N time samples. By means of 2DFT, matrix B(m, n) can be converted into spectraldomain B(, ) [11]. In all further analysis the RMS value are taken on, both in spaceand time.

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Fig. 2. Numeric model of investigated motor

-1,2

-0,8

-0,4

0

0,4

0,8

1,2

0 60 120 180 240 300 360Angle [deg]

Bn [T] 600 rpm500 rpm

Fig. 3. Instantaneous flux density distribution (radial component)in air-gap for both rotational speeds (for rated load)

Figures 4 and 5 shows time/space distribution and modal/frequency spectrum of ra-dial component of flux density. Result are presented for both rotational speeds and forfield winding current If = 200 A.

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(a) (b)

Fig. 4. Radial component of flux density in air-gap for 600 rpm:time/space distribution (a) modal/frequency spectrum (centered) (b) nominal load

(a) (b)

Fig. 5. Radial component of flux density in air-gap for 500 rpm:time/space distribution (a) modal/frequency spectrum (centered) (b) nominal load

Maxell stress tensor components in air-gap, in 2D calculations, in cylindrical coordi-nate system, can be calculated from the well know equations [11]:

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( ) ( ) ( )( ) ( ) ( )

BnBnT

BnBnnT

0

1

22

021

=

=

(1)

where: Bn() radial component of the flux density [T], B() tangential component ofthe flux density [T], Tnn() radial component of the magnetic stress [N/m2], Tn() tangential component of the magnetic stress [N/m2], 0 absolute permeability [H/m].

In the Fig. 6 instantaneous distribution of the radial and tangential component ofmagnetic stress in the air-gap, valid for the rated load of the motor, are presented.

0

200

400

600

800

0 60 120 180 240 300 360Angle [deg]

Tnn [kPa] 600 rpm500 rpm

-400

-200

0

200

400

0 60 120 180 240 300 360Angle [deg]

Tn [kPa] 600 rpm500 rpm

Fig. 6. Instantaneous distribution of magnetic stress components in the air-gap for both rotational speeds:a) radial component at rated load, b) tangential component at rated load

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Collecting all calculation results (for all time steps), matrix of magnetic stressTx(m, n) is obtained, where M is a number of space samples, and N time samples(Tx represents either Tnn or Tn). By means of 2DFT, matrix Tx(m, n) can be con-verted into spectral domain Tx(, ) [11]. In all further analysis the RMS value aretaken on, both in space and time. In the Fig. 7 an example for time/space distribu-tions of the magnetic stress (normal component), valid for both rotational speed areshown. In addition in Fig. 8 the modal/frequency spectrum is presented. All resultsare valid for nominal load point of motor. Modal/frequency spectrum of magneticpressure should be limited only to lowest harmonic in space, because fromvibroacoustic point of view only longest waves are important [11]. In case of largepower, two-speed, silent pole, synchronous motors such approach can not be used.For all rotational speeds in modal/frequency spectrums of magnetic stresses (radialcomponent), harmonics close to 1 kHz are observed. These harmonics are con-nected with numbers of stator slots (108) and numbers of pole pairs(n = 500 rpm, p = 6 and n = 600 rpm, p = 5). For lower rotational rotor speed theharmonics number are 102 and 114, and for higher speed harmonics number 103and 113. Magnitudes of these harmonics are similar to harmonics amplitudes ofsmall order. In addition these harmonics are very close to natural frequencies of amechanical construction of two-speed synchronous motor. Therefore vibration ofsuch structure with big amplitudes can be expected.

(a) (b)

Fig. 7. Radial component of magnetic stress in air-gap (time/space distribution) for:600 rpm (a) and 500 rpm (b)

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3. FREE VIBRATIONS CALCULATION

Natural frequency analysis is done using 2D and 3D models, which are mutuallycombined [6]:

The real slotting geometry of the stator iron is analyzed within 2D mechanicalmodel. Results of such model are the base for determination equivalent cylindricalstructure of the stator core.

That equivalent structure is introduced later into 3D model together with the ge-ometry of the housing.

The two dimensional model of stator (Fig. 9) takes into account the mechanical prop-erties of the stator iron, armature winding and wedges. Impact of end-parts of armaturewinding is considered with additional mass added to the nodes of FE mesh. Results ofthe 2D model are the input data for the equivalent 3D stator core structure. Such ap-proach is forced by the computer facilities and very complicated motors structure. The2D model has about 70000 DOFs, what is equivalent to the few hours of computationtime. In 3D this numbers of the DOFs of stator core will exceed value 1000000 (450 mmaxial length of the stator iron). Adding about 500000 DOFs of the stator housing one willgive numbers of equation which will be solved more that 24 hours.

(a) (b)

Fig. 9. 2D model of the synchronous motor: a) outlook; b) part of the model with the mesh

Second reason of modeling within 2D and 3D is that too many details inside FEmodel result with very dense natural mode spectrum, which is out of practical interest.

The criterion of the equivalent cylinder to the real stator core model is the identity ofthe natural frequency of such structures.

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Table 2. Main properties of two 2D mechanical models of the synchronous motor

Full model Simplified modelOuter diameter mm 1450 1450Yoke height mm 150 64.4Young modulus Pa 2.1 1011 1.7 1011Poisson ratio 0.29 0.29Density kg/m3 7850 12125

The cylindrical model allows reducing numbers of equation in 3D space more that 50times. Table 2 shows parameters of the full 2D and simplified 2D model and Table 3shows the results of both models.

Table 3. Results of the two mechanical models

Frequency f [Hz]Full model 1488 1529 1644 1775 1897 2069 2295 2419Simplified model 1487 1527 1640 1759 1909 2062 2257 2347

Comparisons between the space natural forms of both models are shown in Fig. 10and Fig. 11. The simplified model has about 2000 DOFs.

The full 3D finite element model of a two-speed synchronous motor is made of solidelements and the total numerical size of the model is about 1 000 000 DOFs. The Fig. 12shows the outer view of the model and of finite element mesh. The longitudinal ribs,screws etc. cause a very rich spectrum of natural frequencies of a presented structure. Inthe range up to 3.5 kHz, 500 natural modes can be observed.

(a) (b)

Fig. 10. Example of a natural space mode for full 2D model (a) for r = 6 (b) for r = 12

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(a) (b)

Fig. 11. Example of a natural space mode for simplified 2 D model: (a) for r=6, (b) for r=12

Boundaryconditions(Uxyz = 0)

Terminal board(modeled with

additional masses)

Fig. 12. Mesh of the full 3D model of synchronous motor

This shows how the structure of the motor is weak. Examples of natural modes of ex-amined motor are shown in the Fig. 13 and 14.

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(a) (b)

Fig. 13. Examples of natural modes: (a) f = 92.3 Hz, (b) f = 111.8 Hz(for the picture clarity part of the model is removed)

(a) (b)

Fig. 14. Examples of natural modes: (a) f = 240.6 Hz, (b) f = 1354 Hz(for the picture clarity part of the model is removed)

Results show that dominant natural modes below 1500 Hz are: axial mode n = 1 andcircumferential mode r = 2.

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4. FORCE VIBRATION CALCULATION

In this chapter vibration calculations, caused by electromagnetic forces which actin investigated motor, are presented. Details of electromagnetic forces calculationsare described in [7]. For the linear structures the following equation can be written[2, 9]:

M + uC + Ku = F (2)where: M mass matrix, C damping matrix, K stiffness matrix, accelerationvector, u velocity vector, u displacement vector, F load vector.

Solving the equation (2) displacement values in node of FEM model are obtained.The following assumptions are taken in the vibration analysis:

displacement of the housing lugs is set to zero, zero displacement as the initial calculations condition.In all calculations the damping matrix is neglected.In Figures 15 and 16 the vibrations velocity (the RMS value) is presented (har-

monic spectrum). Results are valid for both rotational speeds, for nominal load andfor the field current 200 A. The casing vibration of the investigated motor are de-termined.

0

1

2

3

4

5

0 200 400 600 800 1000 1200 1400

f [Hz]

V [m

m/s

]

Fig. 15. Harmonic spectrum of the RMS vibrations velocity of the motorat nominal load for n = 600 rpm

Energy of the vibrations is concentrated in two ranges. First is the range of 0300 Hz,where the dominant are 200 Hz, where the investigated motor works with the higher

5.35

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speed (n = 600 rpm) and the 16,7 Hz, 50 Hz and 100 Hz where the motor works withlower rotational speed (n = 500 rpm). Harmonics 100 Hz is connected with the funda-mental wave of the magnetic field inside the motor. Harmonics 16,7 Hz and 50 Hz areconnected with the sub-harmonics. Second range of the harmonic spectrum is the range11001500 Hz, where the dominant are the 1st order of the slot harmonics. Big ampli-tudes are the results of the resonance phenomena. In the range 10001500 Hz are almost50 natural frequencies [5, 6]. Vibration amplitudes at lower speed are almost 4 timesbigger that the vibrations at higher rotational speed.

0

5

10

15

20

25

0 200 400 600 800 1000 1200 1400

f [Hz]

V [m

m/s

]

Fig. 16. Harmonic spectrum of the RMS vibrations velocity of the motorat nominal load for n = 500 rpm

5. CONCLUSIONS

In this paper results of vibrations of the magnetic origin in two-speed, large power,silent pole, synchronous motors are presented. Elaborated and described model of thetwo speed synchronous motor type GAe1510-12p allows to determine the static and thetransient characteristic as well. Presented model is useful to analyses the mechanicalphenomenon in two speed synchronous, silent pole motors. Results of a natural vibrationanalysis shows, that the structure of stator frame of examined motor is very sensitive tothe stator iron core vibration below 1 kHz can be found more that 50 natural modes.According to presented results more dense vibration spectrum on lower rotational speedn = 500 rpm can be observed.

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REFERENCES

[1] ANTAL L., ZAWILAK J., ZAWILAK T., Testing of a Two-speed Synchronous Motor, XVI Inter-national Conference on Electrical Machines ICEM 2004, Krakw 2004, pp. 793799.

[2] ANSYS Help, www.ansys.com, 2008.[3] BIALIK J., ZAWILAK J., ANTAL L., Field-circuit model of the two-speed synchronous motor,

Scientific Papers of the Institute of Electrical Machines, Drives and Metrology of the Wroclaw Uni-versity of Technology, No. 56, Wrocaw 2004, pp. 4354 (in Polish).

[4] BIALIK J., ZAWILAK J., Vibrations and electromagnetic forces in two speed, large power syn-chronous motor, Proceedings of XLI International Symposium on Electrical Machines SME 2005,Jarnotwek 2005, pp. 5564 (in Polish).

[5] BIALIK J., ZAWILAK J., Free vibration analysis of the two speed synchronous motor, Proceedingsof Exploitation of Electrical Machines and Drives PEMINE Industrial Research and DevelopmentCenter for electrical Machines KOMEL, Rytro, Mai 2008 (in Polish).

[6] BIALIK J., ZAWILAK J., Vibration modeling of the two-speed, large Power, synchronous motor,6th IEEE International Symposium on Diagnostic for Electric Machines, Power Electronics andDrives, Krakw, September 68, 2007, pp. 173177.

[7] BIALIK J., ZAWILAK J., Magnetic forces calculation in two-speed, large power, silent pole, syn-chronous motor, XLIV International Symposium on Electrical Machines, SME 2008, SzklarskaPorba, June 1720, 2008.

[8] DUBICKI B., Electrical machines, part III, PWN, Warszawa 1964 (in Polish).[9] GIERAS J. F., WANG CH., CHO LAI J., Noise of polyphase electric machines, CRS Press Taylor

& Francis Group, USA, 2006.[10] SERGEEV P.S., VINOGRADOV N.V., GORJANOV F.A., Projektirovanie Elektrieskich Main,

Energija, Moskva 1969 (in Russian).[11] WITCZAK P., WAWRZYNIAK B., Modal-frequency analysis of magnetic vibration forces in

permanent magnet machines, Proceedings of XLI International Symposium on Electrical MachinesSME 2005, Jarnotwek 2005, pp. 214219 (in Polish).

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