In this paper, four methods to determine the mechanical characteristics natural frequen-cies, mode numbers of electrical machine stators are developed.. Result comparison concerns analy
Trang 1222 Schlensok and Henneberger
Table 3 Maximal levels of the estimated air-borne sound-pressure
(sl: stator excitation, left-hand rotation; sr : stator excitation,
right-hand rotation; srr : stator+ rotor excitation, right-hand rotation)
f (Hz) L sl(dB) L sr(dB) L srr(dB)
by the forces amplify the axial component Therefore, it is of advantage to take the rotor excitation into account to get more exact results concerning the structure-borne sound
Acoustic simulation
The last step is to estimate the air-borne noise generated by the different excitations For this reason a boundary-element model of the entire machine structure is applied The air-borne sound-pressure is estimated on an analysis hemisphere around the machine at a distance of
d = 1 m Fig 17 shows the result for stator-rotor excitation at f = 420 Hz.
The maximum sound-pressure levels L reached for the three cases taking the stator
excitation into account are listed in Table 3
Figure 17 Sound-pressure distribution at f = 420 Hz for stator-rotor excitation and right-hand rotation
Trang 2II-7 Comparison of Stator- and Rotor-Force Excitation 223 The results show that the direction of the rotation has a significant effect on the noise
generation Except for f = 720 Hz and f = 620 Hz all orders are amplified up to L =
10 dB If the rotor-force excitation is taken into account some orders become louder and some quieter The air-borne sound-levels do not suit the acceleration measurements as well
as those of the structure-borne sound
Conclusion
In this paper the structure- and air-borne noise of an induction machine with squirrel-cage rotor are estimated
For this, different types of surface-force excitations and rotational directions are regarded for the first time In general the calculated structure-borne sound-levels suit the acceleration measurements of the industrial partner very well The acoustic-noise levels differ from those The comparison of the different excitations show, that it is necessary to take the rotor excitation into account In case of pure stator-excitation e.g the first stator-slot harmonic
at 720 Hz does not reach as significantly high levels as expected although it is one of the strongest orders measured
References
[1] C Schlensok, T K¨uest, G Henneberger, “Acoustic Calculation of an Induction Machine with Squirrel Cage Rotor”, 16th International Conference on Electrical Machines, ICEM, Crakow, Poland, September 2004
[2] B.-T Kim, B.-I Kwon, Reduction of electromagnetic force harmonics in asynchronous traction motor by adapting the rotor slot number, IEEE Trans Magn., Vol 35, No 5,
pp 3742–3744, 1999
[3] T Kobayashi, F Tajima, M Ito, S Shibukawa, Effects of slot combination on acoustic noise from induction motors, IEEE Trans Magn., Vol 33, No 2, pp 2101–2104, 1997
[4] L Vandevelde, J.J.C Gyselinck, F Bokose, J.A.A Melkebeek, Vibrations of magnetic origin
of switched reluctance motors, COMPEL, Vol 22, No 4, pp 1009–1020, 2003
[5] G Arians, Numerische Berechnung der elektromagnetischen Feldverteilung, der struktur-dynamischen Eigenschaften und der Ger¨ausche von Asynchronmaschinen, Aachen: Shaker Verlag, 2001 Dissertation, Institut fur Elektrische Maschinen, RWTH, Aachen
[6] G Arians, T Bauer, C Kaehler, W Mai, C Monzel, D van Riesen, C Schlensok, iMOOSE, www.imoose.de
[7] I.N Bronstein, K.A Semendjajew, Taschenbuch der Mathematik 25 Auflage, Leipzig, Stuttgart: B.G Teubner Verlagsgesellschaft, 1991
[8] H.D Lke, Signal¨ubertragung, Berlin, Heidelberg, and New York: Springer-Verlag, 1999 [9] I.H Ramesohl, S K¨uppers, W Hadrys, G Henneberger, Three dimensional calculation of magnetic forces and displacements of a claw-pole generator, IEEE Trans Magn., Vol 32,
No 3, pp 1685–1688, 1996
[10] Jordan, H., Ger¨auscharme Elektromotoren, Essen: Verlag W Girardet, 1950
Trang 3II-8 A CONTRIBUTION TO DETERMINE
NATURAL FREQUENCIES OF
ELECTRICAL MACHINES INFLUENCE
OF STATOR FOOT FIXATION
J-Ph Lecointe, R Romary and J-F Brudny
Laboratoire Syst`emes Electrotechniques et Environnement, Universit´e d’Artois, Technoparc
Futura, 62400 B´ethune, France
jp.lecointe@fsa.univ-artois.fr, jfrancois.brudny@univ-artois.fr, raphael.romary@univ-artois.fr
Abstract In this paper, four methods to determine the mechanical characteristics (natural
frequen-cies, mode numbers) of electrical machine stators are developed Result comparison concerns analyt-ical laws, a finite element software, a modal experimental procedure and a method based on analogies between mechanic and electric domains Simple structures are studied in order to analyze the validity
of each method with accuracy The fixation of a stator yoke allows to observe the modifications of the mechanical behavior
Introduction
The study of electrical machine noise always leads to mechanical resonance problems The noise origins are generally divided into three sources which are mechanic, aerodynamic, and magnetic [1] The noise of magnetic origin is produced by the electromagnetic radial forces between the stator and the rotor The noise resulting from these forces can be particularly severe when a force of magnetic origin is close to a natural frequency because circumferential modes of the stator are excited [2] It is particularly the case of switched reluctance machines [3] but it could be also problematic for classical alternative current machines supplied by converters That is why an accurate knowledge of the mechanical behavior of the machine— especially the natural frequencies—is important in noise and vibration prediction Most studies often use finite element software These last ones give accurate and usable results if model is well fitted to the studied structure Indeed, materials constituting the machine have to be correctly estimated; otherwise the advantages offered by FE are reduced
In this paper, different methods are studied in order to estimate which ones can give accurate values of natural frequencies of simple structures, as fast as possible in order to establish
a rapid diagnosis Four methods are performed The first one uses analytical expressions based on Jordan’s work, but the proposed laws are improved thanks to fewer restrictive hypotheses The second method uses a FE software (Ansys) The third method is original because it is based on analogies between mechanic and electrical domains Consequently, the mechanical problem is transformed into an electrical circuit resonance determination The last method is experimental: a modal hammer allows to verify the calculated values
S Wiak, M Dems, K Kom˛eza (eds.), Recent Developments of Electrical Drives, 225–236.
2006 Springer.
Trang 4226 Lecointe et al.
Table 1 Used variables
Symbol Quantity
R c Average radius of the yoke
e c radial thickness of the yoke
L e length of the cylinder
E Young modulus
P Mass density
N Poisson ratio
Usual studies consider machines in free conditions It allows the influence of different parameters to be quantified; several papers have already discussed the effects of the feet, the cooling ribs, the windings, or the end-bells [4,5] In this paper, the influence of the fixation
on a rigid chassis is studied The purpose is to evaluate the fixation impact on the natural frequencies and on the shape of the mode numbers (Table 1)
Methods of natural frequency determination
Four methods are performed to determine the natural frequencies and the associated mode number Technologies and principles for each of them are quite different The older one
is entirely based on analytic beam theory [6]–[7] whereas another one is totally numeric (finite element) The third developed method considers analogies between mechanical; electrical quantities and the identification of mechanical parameters allows to transform the mechanical problem into an electrical circuit study The experimental method uses a modal station and gives the reference results Table 1 presents the used variables
Analytical method
The presented laws have been rewritten [8] more accurately about smooth free rings Con-sidering that the stator is the most responsive part compared to the rotor, the ball bearings
or the flanges, this analytical method gives a fast determination of stator natural radial fre-quencies This method allows to determine only the frequencies in two dimensions They
are noted F i , where “i ” is the mode number:
m = 1: F1=
a + m+−√
where
= m2
++ 2
2am+− 4m2m2−
+ 4
a2− 12m2m3−
(4)
= e2c
Trang 5II-8 To Determine Natural Frequencies of Electrical Machines 227
Numerical determination with a FE software
The finite element software (Ansys) solves the conventional eigenvalue equation:
[H ] − ω2[M]
where [H ] and [M] are, respectively, the stiffness and the mass matrixes The solutions ω/(2π) and [χ] are the natural frequencies and the nodal displacements Resolution uses
the block Lanczos algorithm; values of Young modulus, mass density, and Poisson ratio are required
Equivalent electric circuit
As the first method allows to find only the natural resonances of structures in free conditions,
a second method based on analogies between mechanic and electric domains has been developed The equivalences are presented at Table 2
The stator is divided into M levels, each of them containing N cells (Fig 1) Each cell
is characterized by its mass The deformation of the structure is represented by the relative displacement of a cell compared to the others cells From a mechanical point of view, the rigid linkages can be taken into account with springs and, from an electrical point of view, with capacitors A resistor allows to take into account the energy lost in the movement by viscous friction
The equivalent scheme of the structure is presented at Fig 2 The voltages applied on the internal part of the first level represent the forces supported by the stator The voltage fluctuations at the external periphery give the evolution of the deformations Consequently,
Table 2 Equivalences
Mechanical quantities Electrical quantities
Rigid linkages K Capacitor C
Viscous friction F v Resistor R
(N, M)
(N, 1)
(1, 1) (1, 2) (2, 2)
(2, 1)
(3, 1) (3, M)
m
(1, )
m
Figure 1 Stator division.
Trang 6228 Lecointe et al.
V h,k
i h,k
u h
u h+1
R/2
R/2
R/2
C v
W h,k
Cell h,k
u h–1
Figure 2 Equivalent electric circuit.
such a model gives the possibility to model different excitations: sinusoidal or pulsed Therefore, the validity of the model can be verified with modal experimentations performed with an impact hammer
For a sinusoidal excitation characterized by a frequencyω e and a mode number m, it
becomes:
u h (t) = U mcos
ω e t − hm2π
N
(11)
where h gives the position of the force along the internal periphery Successive calculations
give the frequency value for which the vertical response is maximal and thus the radial frequency can be determined The fixation of the machine can be studied by imposing a potential zero in chosen points of the external periphery
Next step consists in determining the values of the equivalent parameters
Trang 7II-8 To Determine Natural Frequencies of Electrical Machines 229 The equivalent inductance is given by the elementary cell mass which is given by the expression:
L = π r2+1− r w
k
where r k+1and r k are, respectively, the external and the internal radius of the level k Then, the equivalence between potential and electrical energies gives the relation K =
1/C The Hooke law and the classical capacitor calculation relations allows to determine the expressions of capacitors C v and C h, according to the considered geometry:
C v= (r k+1− r k ) N (r k+1+ r k ) E y
(13)
C h = (r k+1+ r k)π (r k+1− r k ) NE y
(14)
The equivalent resistance is the most difficult to determine As it does not influence the frequency response, this coefficient is arbitrary chosen
The equations of the electric circuit lead to a second order differential system composed
of 4 × M × N lines Consequently, the response of the structure is determined in the
state space The state vector is composed of 4× M × N elements: vertical and horizontal
currents and capacitor voltages Computation is realized with Matlab⇔ and Simulink⇔ Such a process has a double advantage First, computation time is lower than FE software because the matrix size is smaller Secondly, it could be set up on any computer without any specific software
Modal experimental device: impact hammer test
The modal test using a hammer is the least expensive The examined structure is excited
by an impact given with a specific hammer (Meggitt Endevco, model 2302-5) which allows
to measure the characteristics of the shock A piezoelectric accelerometer (Bru¨el & Kjaer, model 4384) allows to observe the response of the structure A spectrum analyzer (Bru¨el & Kjaer, 2035, 2 channels) and a modal analysis software (Star SystemTM) provide the pro-cessing and the analysis of the measures The Fig 3 presents a scheme of the experimental device The frequency limit of the used hammer is around 8 kHz
Studied structures
Three elementary structures are studied Two are perfectly sleek rings (Fig 4) whereas the third one is composed of a statoric yoke made of steel which is equipped of two welded feet (Fig 5) Table 3 presents the dimensions of the structures The geometry of the cylinders
is quite different Indeed, the first one is elongated whereas the second one presents an important diameter Studying such different configurations allows the method accuracy to
be quantified A massive structure equipped of feet is deliberately studied in order to avoid the perturbations generated by coils, cooling ribs, or stack lamination In this way, the main phenomenon observed is the foot fixation
Trang 8230 Lecointe et al.
Spectrum Analyzer
Impact hammer
Data transmitted by IEEE port
Personal computer Accelerator
Figure 3 Experimental device.
Results and experimental validation
Structures in free conditions
First, all the methods are applied to the two cylinders in free conditions to check their accuracy Analytical method, FE software, and analogy method require the same parameters:
Young modulus, mass density, and Poisson ratio are, respectively, equal to E = 2, 1.1011Pa,
Figure 4 Picture of the smooth cylinders.
Trang 9II-8 To Determine Natural Frequencies of Electrical Machines 231
Figure 5 Picture of the statoric yoke.
Table 3 Dimensions
Cyl 1 Cyl 2 Statoric ring
R c(mm) 53 133.5 120.5
L e(mm) 136 104 260
Meshing and 2D deformations (cylinder 2)
3D deformations (cylinder 2)
Figure 6 FE results.
ρ = 7,850 kg/m3, andv = 0, 3 Fig 6 presents the FE meshing of the cylinder 2 and the
usual 2D deformation shape Additional 3D frequencies and bending appear
Fig 7 shows the cylinder 2 response It is obtained by the method using analogies with
M = 5 and N = 16 The excitation is sinusoidal and the current in the last vertical branch of
the circuit presents a transient state and then becomes sinusoidal The division of the cylinder
Trang 10232 Lecointe et al.
Vertical current J1,M
Time (s)
0
–0.6
–0.4
–0.2
0
0.2
0.4
0.6
0.8
1
0.005 0.01 0.015 0.02 0.025
Figure 7 Vertical current j M,Nresponse (sinusoidal excitation)
can be noticed on Fig 8 which represents the special evolution of the external surface; the different shapes correspond to the excitations (modes 2, 3, and 4) A shock simulation leads
to the response presented at Fig 9 whereas Fig 10 presents the FFT of the signal
In free conditions, the structures are suspended with rigid rubber bands or with elastic rubber bands if the weight of the structure is not too important A meshing is constituted of
192 points drawn on each external surface and shared out four planes (Fig 11) Each point
is excited with the modal hammer four times Mode shapes given by the modal software can be perfectly identified Fig 12 presents the cylinder 2 shapes
Results for each method applied to both cylinders are presented at Table 4 and hammer test is chosen as reference Results show that, independently of the considered geometry, the maximum relative error for analytical method is 2.3% for the four first modes The accuracy of such analytical process is noticeable in spite of the particular geometry of the cylinder 2 The FE method does not give results so precise but the characterization
is more complete with 3D deformations The method using analogies is not so precise
as analytical laws The advantage of this method is its future development in the third dimension
Figure 8 Shapes of deformation.