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Abstract — This paper addresses the accuracy of most encountered analytical models used for the determination of stator natural frequencies, for specific small industrial radial flux permanent magnet motors. External diameters are in the range of 30 to 90 mm. Two particularly different stators are investigated using ring and cylindrical shells analytical theories. Results are compared to finite elements and then validated thanks to measurements. Height of teeth and housing made of bearing shields (flange-mounted motor) without outer cylindrical frame are responsible for loss of accuracy between calculated and measured values.

I. INTRODUCTION

Electric motors manufacturers are highly concerned with low noise and vibration emissions in vehicular or home applications where motors are close to the end-user. Noise of magnetic origins is increased when resonance occurs between natural frequencies of the machine and excited frequencies of the magnetic force, typically multiples of 2 where is the rotational frequency in Hertz and the pole pair number [1]. Accurate determination of natural frequencies is thus one of the key issues to design low noise and vibrations motors. The only method that can take into account any details of the machine, such as end-windings, bearing shields or fixations, is finite element analysis (FEA). However, even if the level of geometric details is only related to the user commitment, material properties of steel sheets anisotropy or assembled winding wires, especially with temperature dependence, have to be approximated. These assumptions also apply to analytical models but their fast computation speed benefits optimization processes or at least parametric analyses in the early steps of design. As a result, multi-physics analytical modeling is widely used in the determination of optimal machine design [2]. Thus, accurate natural frequencies analytical models suitable for various machine dimensions are necessary.

Most encountered analytical methods used for the fast derivation of machines natural frequencies are based on homogeneous shapes such as rings, thin cylindrical shells and thick cylinders. Even in recent publications, very approximated ring formulas are still being used [3]-[6] and although they can give good results in specific cases, they should not be used in general analytical simulations tools such as optimization methods for example [7]. Thus, Hoppe, Jordan and, often quoted from [8], Frohne equations give poor results

for short-length machines compared to alike time-consuming more recent formulas derived by acoustic specialists [9]. Formulas taking into account teeth, windings and housing other than with modified mass are for example shown in [10].

Natural frequencies of low-manufacturing cost small industrial permanent magnet synchronous motors are investigated in this paper. Typically, such motors have their phase copper wires wounded on plastic bodies, which are then inserted on straight teeth [11]. The number of teeth is rather low (≤ 12) which leads to large open slots. In case of alternately-wound teeth, half of the teeth are left free and behave as cantilevers. Also, the machine’s active length can be much smaller than their external diameter which leads to a high influence of the frame, constituted by bearing shields, on natural frequencies. Two motors of different ratios of length over diameter are considered to illustrate the models and the influence of bearing shields is assessed.

This paper is structured as follow. Analytical models and their limits are presented in section II. In section III, natural frequencies of a motor with a long axial length are computed using analytical and FEA models and then measured. In section IV, the same analysis is performed on a shorter motor. Concluding remarks are then expressed.

II. ANALYTICAL MODELS

A. Ring models As a first bi-dimensional (2-D) approximation, a radial flux

electric machine without rotor can be represented as a thick ring (thickness not negligible compared to radius). First works strictly applied to electric machines began using ring theories, originating from Hooke’s law of elasticity. The behavior of a thin ring (thickness negligible compared to radius) has been investigated by Hoppe [12] and applied to the case of an electric machine by Carter [13]. Shear stress and moment of inertia are neglected in these equations, which thus provide inaccurate results for high vibrational modes. These effects were taken into account by Jordan [14]. Corrective terms were added by Voronetskii and Lubcke in order to fit their measures, as quoted by [15]. In order to give general rules, authors like Pavlovsky tried to take into account slots, first without windings and lately with windings considered as an additional mass [15]. The model developed by Frohne in 1959

Natural Frequencies Analytical Modeling of Small Industrial Radial Flux Permanent Magnet Motors

G. Verez1, C. Espanet1 1Moving Magnet Technologies, 1 rue Christiaan Huygens, 25000 Besançon, France

E-mail: guillaume.verez@movingmagnet.com

and presented in [8] uses these considerations. It is still employed in modern works [3]-[5] et al. with a precision from 3 to 18% versus experimental results (ratio of axial length, L, over radius, R, of 1.7) on first circumferential modes [16].

The influence of the frame has been modeled as a second ring linked to the first, using Rayleigh-Ritz method which is based on Lagrange equations of the second kind [15][17]. This method has also been developed in 2-D in [18][19] and in 3-D in [18][20]. For a short machine (L/R = 0.8), results obtained with these models are within a precision of 0.7 to 1.5% compared to experimental values [15]. For different thicknesses, errors ranging from 0.3 to 5.5% have been obtained in [20]. The thinner the ring, the closer are simulation results to experimental ones. However, simulation time is not explicit and clearly is higher than for fast computation analytical formulas, as ones given in [8].

B. Thin cylindrical shells models As a tri-dimensional approach, a radial flux electric

machine can be represented by a hollow, thick, circular cylinder of finite length (radius not negligible compared to axial length). The study of natural frequencies of such cylinders has been enabled by the study of thin cylindrical shells. Shell theory originates from elasticity theory in which some hypothesis are made [21], such as the fineness of the shell that justifies constant stress along its thickness [22]. Shell vibrations have been studied by well-known scientists such as Euler, Poisson, Lamé, Bernoulli, Germain, Lagrange, Kirchhoff, Aron, Rayleigh and Love to name few. A detailed history is given in [23] and with more hindsight in [24]. Love laid the foundations of thin shell vibration theory. It is based on elasticity and the resolution of Newton’s law of motion thanks to forces and resultant moments obtained from small structural distortions [25]. Numerous authors then proposed hypotheses on stress and strain for vibrations of infinite or finite length thin cylindrical shells. Leissa [25], in his book now considered as a reference by many [26][27], described theories of Byrne, Flügge, Goldenveiser, Lur’ye, Novozhilov, Love, Timoshenko, Reissner, Naghdi, Berry, Sanders, Donnell, Mushtari and Vlaslov. Every boundary condition that can be applied on a cylinder ends have been studied using Flügge theory [28][29]. In practice, for a high number of circumferential waves, the influence of boundary conditions on natural frequencies is negligible and it is possible to consider simply supported conditions [29].

If a smooth and thin cylindrical frame of an electric machine can almost stand within the limits of thin shell theory, a stator yoke is too thick to fit. Though, thin cylindrical shell theory is used in recent works using decomposition proposed in [9], originating from works by [2]. The stator is separated in two [2] or three [9] parts in a stator core considered as a cylindrical shell of infinite length, since is it not constrained at its ends, teeth and windings considered as another cylindrical shell of infinite length and a frame considered as a cylindrical shell of finite length with boundary conditions. Each cylindrical shell has natural frequencies of particular circumferential modal shapes for infinite length cylinders and

also longitudinal modal shapes for finite length ones. Equivalent natural frequencies for pairs of circumferential and longitudinal modes for the entire machine can then be derived [10]. The precision of such a fast computation model is within 1.5 to 20% depending on the type of mode [9].

C. Thick cylinders models Infinite length thick cylinders can be studied using 3-D

elasticity theory as for thin shells [30]. An approximate approach called ‘3-modes’ is based on decomposing displacement into power series and retaining only first terms representing radial, longitudinal and axial strains [31]. For simply supported conditions of a finite length thick cylinder, exact solutions do not exist and it is necessary to use approximate methods [32]. Thus, hollow and solid, free or simply supported, cylinders have been studied using FEA in [32]. A solution involving similar displacement decomposition as in [31] for a free cylinder has been developed in [33] and gives similar results to those obtained by [32].

Rayleigh-Ritz method has been used for vibrations of 3-D rings of any cross-section in [18]. This preliminary work helped calculating potential and kinetic energy of an electric machine including core, teeth, windings and fins [34]. Good accuracy using this method have been verified experimentally [35][36]. Indeed, for a thick isotropic steel cylinder (L/R = 0.8, R/h =3), natural frequencies are simulated between 0.2 and 3.3% for first circumferential and longitudinal modes. The case of a thick and finite length cylinder has been studied using Rayleigh-Ritz method in [26] and simplified in [37].

Vibrations of laminated structures are of particular interest in electric machines and have to take into account shear stress [25]. If some authors have studied the influence of each laminated sheet, calculations are not achievable for a stator core. The global effect of a plate lamination has been studied in [38]. Formulations for thick laminated simply-supported cylinders have been proposed in [39]. Other boundary conditions have been studied in [40].

III. ANALYSIS OF A LONG MOTOR In this section, a motor whose specifications are given in

Table I, and represented in Fig. 1, is studied. This machine has 12 teeth and 6 coils, wounded on bobbins [11]. It is flange-mounted with aluminum bearing shields. The proposed analysis of this motor is divided into natural frequencies study of the lamination stacks, without windings and then with windings and epoxy resin impregnation. Impregnation is shown in Fig. 2. Finally, natural frequencies of the entire machine, stator with bearing shields and rotor, are computed.

Pure radial (or circumferential) modes are due to pure radial motion. The stator’s cylinder keeps a constant cross-sectional shape along its axial length [27]. As only natural frequencies of pure radial modes are of interest in this article, all materials are considered isotropic. Material mechanical parameters are given in Table II. Copper windings with insulation equivalent properties are given in [9].

TABLE I LONG MOTOR PARAMETERS

Quantity Value (Unit) Internal stator radius 19.8 (mm) Parallel teeth radial thickness 17.9 (mm) Coil radial thickness 14.7 (mm) External stator radius 42.5 (mm) Parallel teeth air gap opening 22 (°) Non-parallel teeth air gap opening 6 (°) Coil circumferential length 17.4 (mm) Stator axial length 62.0 (mm) Front flange axial length 11.3 (mm) Rear flange axial length 17.0 (mm)

A. Stator laminations The stator core is a cylinder than can be approximated as a

thin cylindrical shell. Natural frequencies of the core are computed using FEA (Salome and Code_Aster software) and compared to analytical shell theories in Table III. Boundary conditions are free at both ends. In the following, shell theories are abbreviated using SX where X is from 1 to 13 in this order: Donnell-Mushtari, Love-Timoshenko, Goldenveizer-Novozhilov, Houghton-Johns, Biezeno-Grammel, Flügge, Sanders, Reissner-Naghdi-Berry, Vlasov, Epstein-Kennard, Simplified Kennard [25], Love-Wang [41] and Donnell-Soedel [24]. For the stator core, all theories provide results with an average error for the six first modes around 2%, except for Donnell-Mushtari, Flügge and Donnell-Soedel theories. Thus, the stator core can be approximated by a thin shell.

The application of thin cylindrical shell theories to the stator yoke (core and teeth) is however not relevant. Considering the yoke as a single shell of thickness equal to the one of the yoke leads to a huge stiffness of the cylinder and natural frequencies are highly increased. However, in reality the opposite effect happens since teeth only provide mass without stiffness and natural frequencies must be decreased. In the same manner, it is worthless to try to apply cylinder separation [2] since another unslotted cylinder inside the stator core, with same material also results in higher natural frequencies since it increases stiffness more than the mass. However, it is possible to apply cylinder separation with the stator core and a cylinder representing teeth with real mass density and Poisson ratio but a null Young modulus.

Fig. 2. Impregnated stator.

TABLE II

MATERIAL MECHANICAL PARAMETERS Part Parameter Value (Unit)

Steel stator sheets Young modulus 205 (GPa) Poisson ratio 0.32 Mass density 7850 (kg.m-3)

Aluminum flanges Young modulus 69 (GPa) Poisson ratio 0.33 Mass density 2700 (kg.m-3)

Copper windings with insulation

Young modulus 9.4 (GPa) Poisson ratio 0.35 Mass density 8890 (kg.m-3)

Plastic bobbin Young modulus 12.3 (GPa) Poisson ratio 0.3 Mass density 1550 (kg.m-3)

Epoxy resin impregnation

Young modulus 0.5 (GPa) Poisson ratio 0.34 Mass density 1860 (kg.m-3)

Thus, natural frequencies are mainly impacted by the effect

of teeth mass. Results are given in Table IV. Donnell-Mushtari theory provides best results, with a mean error of 22.4%. Especially, modes 2, 3 and 4 are calculated with errors under 5%. However, Love-Wang and Donnell-Soedel theories are not suitable in this case.

B. Stator with windings Coils are wounded on plastic bobbins. Using FEA on a

non-impregnated stator, a 1.4% average decrease of natural frequencies is reported for six first circumferential modes

(a) (b)

Fig. 1. Long motor (a) exploded view and (b) front view of stator sheets and windings.

when bobbins are not modeled compared to a model where bobbins are. Thus, for simplification, bobbins will not be modeled anymore. Natural frequencies of the stator with coils and without impregnation are computed. Results are given in Table V. Ring theories are abbreviated using RX where X is from 1 to 3 in this order: Hoppe [42], Frohne [8] and an energetic approach from the beam theory [43]. As in the previous section, shells theories using cylinder separation were computed with a null Young modulus for the inner cylinder representing teeth and windings. Shell theories are giving errors of around 30% except for Love-Wang and Donnell-Soedel theories that are giving poor results. The model of a thick ring loaded with teeth, windings and rotor expressed by Frohne equations is giving best results with a mean error of 14.8%.

Impregnation stiffens the stator and makes a link between all teeth, wounded or not. In this condition, Young modulus of the internal cylinder has to be taken into account. For the stator with impregnation, experimental measurements have been achieved using a shock hammer and an accelerometer. Modal shapes were not assessed but resonance frequencies stood out on the vibration spectra and correlated very well with FEA results as can be seen from Table VI. In this table, natural frequencies are also calculated using previous analytical models. Ring theories provide same results since they are not impacted by Young modulus of windings. As a consequence, the error becomes larger with these models. However, thin shell theories are impacted by Young modulus of windings and in particular, Love-Wang theory gives an average error of 10.8%. As a result, for non-impregnated stator, ring theory of Frohne is the most suitable whereas for an impregnated stator, shell theory of Love-Wang with cylinder separation is the most suitable, with an acceptable mean error of 10 to 15%.

C. Machine with impregnated stator, flanges and rotor The machine being flange-mounted, without external

cylindrical frame, the three (frame/core/teeth) cylinder decomposition depicted in [9] cannot be applied with success. If thickness of the frame cylinder is null with correct axial length, frequencies cannot be calculated. If thickness of the frame cylinder in the radial direction is equal to thickness in the axial direction, stiffness of the machine is exaggerated to a point where natural frequencies are three times larger than measured. However, it is possible to use a two (frame/core) cylinder decomposition where the frame is a thin cylindrical shell which radii are those of the stator core and which length is equal to the entire machine’s axial length. In this case, analytically calculated natural frequencies are reported in Table VII, and compared to experimental measurements and FEA results. Again, FEA results are in good agreements with measures, especially for first two modes

Discrepancies between shell theories and measures of 30% are acceptable for the necessary level of approximations. Donnell-Soedel theory provides the lowest mean error equal to 25%. Presented ring theories used in Tables V and VI are not modified in Table VII and provide in these three cases same results. If the entire machine is considered, Hoppe theory gives as good results as shell theories. However, as with Hoppe

theory natural frequencies do not vary with frame or flanges dimensions, results are unexpected.

TABLE III FIRST NATURAL FREQUENCIES FOR THE LONG STATOR CORE

Circumferential modes Theory 2 3 4 5 6 Error FEA 1424 4022 7696 12409 18120 Ref.S1 35,3% 14,3% 8,6% 6,4% 5,5% 14,0%S2 1,5% 1,6% 1,9% 2,3% 2,8% 2,0%S3 1,5% 1,6% 1,9% 2,3% 2,8% 2,0%S4 4,4% 0,8% 1,7% 2,2% 2,8% 2,4%S5 1,5% 1,7% 1,9% 2,3% 2,9% 2,0%S6 43,5% 17,5% 10,4% 7,7% 6,6% 17,1%S7 1,5% 1,6% 1,9% 2,3% 2,8% 2,0%S8 1,5% 1,6% 1,9% 2,3% 2,8% 2,0%S9 1,5% 1,7% 1,9% 2,3% 2,9% 2,0%S10 1,8% 1,9% 2,2% 2,6% 3,2% 2,3%S11 1,5% 1,7% 1,9% 2,3% 2,9% 2,1%S12 1,4% 1,5% 1,7% 2,0% 2,5% 1,8%S13 51,3% 20,5% 12,0% 8,5% 7,0% 19,8%

TABLE IV FIRST NATURAL FREQUENCIES FOR THE LONG STATOR YOKE

Circumferential modes Theory 2 3 4 5 6 Error FEA 1194 2974 4896 6297 6853 Ref.S1 0,3% 4,5% 5,5% 29,6% 72,4% 22,4%S2 25,2% 15,0% 1,0% 24,6% 68,0% 26,8%S3 25,2% 15,0% 1,0% 24,6% 68,0% 26,8%S4 29,4% 15,7% 1,2% 24,5% 68,0% 27,8%S5 25,2% 15,0% 1,0% 24,6% 68,1% 26,8%S6 5,8% 1,8% 7,3% 31,1% 74,2% 24,0%S7 25,2% 15,0% 1,0% 24,6% 68,0% 26,8%S8 25,2% 15,0% 1,0% 24,6% 68,0% 26,8%S9 25,2% 15,0% 1,0% 24,6% 68,1% 26,8%S10 24,9% 14,8% 0,7% 25,0% 68,7% 26,8%S11 25,2% 15,0% 1,0% 24,6% 68,1% 26,8%S12 115,2% 140,0% 152,3% 157,6% 199,9% 153,0%S13 234,0% 201,6% 225,8% 295,7% 423,6% 276,1%

TABLE V FIRST NATURAL FREQUENCIES FOR THE LONG STATOR

WITH WINDINGS AND WITHOUT IMPREGNATION Circumferential modes

Theory 2 3 4 5 6 Error FEA 1021 2347 4021 5140 5713 Ref.S1 5,9% 9,9% 16,6% 44,1% 87,8% 32,9%S2 20,6% 2,2% 9,4% 38,6% 83,0% 30,8%S3 20,6% 2,2% 9,4% 38,6% 83,0% 30,8%S4 25,1% 3,0% 9,2% 38,5% 83,0% 31,8%S5 20,6% 2,2% 9,5% 38,6% 83,1% 30,8%S6 12,3% 13,0% 18,6% 45,9% 89,7% 35,9%S7 20,6% 2,2% 9,4% 38,6% 83,0% 30,8%S8 20,6% 2,2% 9,4% 38,6% 83,0% 30,8%S9 20,6% 2,2% 9,5% 38,6% 83,1% 30,8%S10 20,3% 2,0% 9,8% 39,0% 83,7% 30,9%S11 20,6% 2,2% 9,5% 38,6% 83,1% 30,8%S12 183,0% 242,0% 243,3% 247,0% 289,1% 240,9%S13 340,1% 330,9% 347,0% 446,4% 608,0% 414,5%R1 10,9% 6,9% 15,5% 40,3% 76,8% 30,1%R2 25,8% 15,4% 13,7% 1,1% 18,2% 14,8%R3 2,4% 16,2% 27,6% 59,2% 197,2% 60,5%

TABLE VI

FIRST NATURAL FREQUENCIES FOR THE LONG STATOR WITH WINDINGS AND WITH IMPREGNATION

Circumferential modes Theory 2 3 4 5 6 Error Exp. 2261 5611 8449 10759 14125 Ref.FEA 1,5% 1,4% 4,7% 3,9% 2,9% 2,9%S1 3,0% 1,5% 18,0% 43,2% 47,9% 22,7%S2 21,8% 10,8% 17,0% 58,0% 63,3% 34,2%S3 21,8% 10,8% 17,0% 58,0% 63,3% 34,2%S4 26,4% 11,2% 17,7% 57,1% 62,5% 35,0%S5 21,8% 9,4% 21,4% 55,6% 61,2% 33,9%S6 11,2% 6,1% 43,6% 60,1% 64,8% 37,2%S7 21,8% 10,8% 17,0% 58,0% 63,3% 34,2%S8 21,8% 10,8% 17,0% 58,0% 63,3% 34,2%S9 21,8% 9,4% 21,4% 55,6% 61,2% 33,9%S10 5,2% 23,4% 27,3% 43,7% 48,9% 29,7%S11 21,5% 8,9% 23,3% 55,0% 60,7% 33,9%S12 24,7% 15,5% 1,4% 6,0% 6,3% 10,8%S13 16,2% 5,4% 24,4% 52,7% 67,4% 33,2%R1 59,8% 55,3% 45,0% 33,0% 28,5% 44,3%R2 66,5% 64,6% 58,9% 52,8% 52,2% 59,0%R3 55,9% 51,4% 39,3% 24,0% 20,2% 38,1%

TABLE VII

FIRST NATURAL FREQUENCIES FOR THE LONG MACHINE WITH IMPREGNATED STATOR, FLANGES AND ROTOR

Circumferential modes Theory 2 3 4 5 6 Error Exp. 2732 3169 4771 5245 9753 Ref.FEA 0,3% 0,5% 6,7% 6,1% 5,7% 3,9%S1 46,4% 4,9% 13,6% 63,3% 27,7% 31,2%S2 55,8% 15,2% 6,6% 56,8% 24,3% 31,7%S3 56,1% 15,3% 6,5% 56,8% 24,3% 31,8%S4 57,7% 16,0% 6,3% 56,7% 24,3% 32,2%S5 56,0% 15,2% 6,6% 56,9% 24,4% 31,8%S6 48,1% 6,7% 12,4% 62,2% 27,2% 31,3%S7 56,1% 15,3% 6,5% 56,8% 24,3% 31,8%S8 55,8% 15,2% 6,6% 56,9% 24,3% 31,8%S9 56,0% 15,2% 6,6% 56,9% 24,4% 31,8%S10 56,0% 15,1% 6,9% 57,5% 25,0% 32,1%S11 55,3% 14,9% 6,7% 57,0% 24,4% 31,7%S12 57,6% 22,8% 3,5% 41,5% 11,4% 27,4%S13 43,1% 8,7% 6,2% 50,5% 16,3% 25,0%R1 66,7% 20,9% 2,6% 37,4% 3,5% 26,2%R2 72,3% 37,4% 27,3% 3,1% 30,8% 34,2%R3 63,5% 14,0% 7,6% 56,0% 74,1% 43,0%

IV. ANALYSIS OF A SHORT MOTOR In this section, a motor whose specifications are given in

Table VIII, and represented in Fig. 3, is studied. This machine has 6 teeth and 3 coils, wounded on bobbins [44]. It is flange-mounted with aluminum bearing shields, each being almost equal to the stator axial length. The stator is, this time, not impregnated.

A. Stator with coils on bobbins In this section, the stator is modeled with coils on bobbins.

Experimental measures could not be achieved due to accelerometers being too big compared to the machine’s axial length. As a matter of fact, only FEA and analytical results are compared. Results are given in Table IX. Coils represent 140% of the stator yoke mass but they do not provide stiffness since they are inserted on teeth and they do not link from one teeth to another. Thus, they greatly decrease natural frequencies of the assembly. When they are calculated using thin cylindrical shell decomposition, this decrease in natural frequencies is not high enough, even when considering a null Young modulus for the shell made of teeth and windings. Thus shell theories overestimate natural frequencies by more than 28%. Frohne equations provide best results with a mean error of 11.3%. Results are thus comparable to those obtained for the long stator machine without impregnation or flanges of section III and shown in table V.

B. Machine with stator, flanges and rotor The two bearing shields represent 63% of the stator yoke

mass and 26% of the stator yoke with coils mass. Their influence on natural frequencies is dominating as can be seen in Table X where experimental measures have been reported. Circumferential mode 2 natural frequency was previously 1705 Hz and is now 18010 Hz due to flanges. Modes over 3 could not be measured since they were out of range (OOR) of measuring devices. Mode 4 was computed using FEA. Thin shell theories were utilized with a two (frame/core) cylinder decomposition where the frame is a thin cylindrical shell

(a) (b)

Fig. 3. Short motor (a) exploded view and (b) front view of stator sheets and windings.

TABLE VIII SHORT MOTOR PARAMETERS

Quantity Value (Unit) Internal stator radius 7.0 (mm) Teeth radial thickness 9.4 (mm) Coil radial thickness 8.0 (mm) External stator radius 18.0 (mm) Teeth air gap opening 22 (°) Coil circumferential length 10.0 (mm) Stator axial length 6.0 (mm) Front flange axial length 5.0 (mm) Rear flange axial length 5.0 (mm)

TABLE IX

FIRST NATURAL FREQUENCIES FOR THE SHORT STATOR WITH WINDINGS AND WITHOUT IMPREGNATION

Circumferential modes Theory 0 2 3 4 5 6 Error FEA 22812 1705 3493 8940 12921 14270 Ref. S1 4,5% 47,5% 71,8% 22,0% 33,3% 74,9% 41,6%S2 4,5% 10,7% 52,8% 14,5% 28,3% 70,6% 29,5%S3 4,5% 10,7% 52,8% 14,5% 28,3% 70,6% 29,5%S4 4,5% 4,3% 51,6% 14,3% 28,2% 70,5% 28,2%S5 4,5% 10,7% 52,8% 14,6% 28,3% 70,7% 29,5%S6 4,5% 56,5% 76,7% 24,1% 35,0% 76,8% 44,9%S7 4,5% 10,7% 52,8% 14,5% 28,3% 70,6% 29,5%S8 4,5% 10,7% 52,8% 14,5% 28,3% 70,6% 29,5%S9 4,5% 10,7% 52,8% 14,6% 28,3% 70,7% 29,5%S10 4,5% 11,2% 53,3% 15,0% 28,8% 71,4% 30,0%S11 4,5% 10,7% 52,9% 14,6% 28,3% 70,7% 29,5%S12 4,5% 10,5% 52,6% 14,3% 27,8% 69,8% 29,2%S13 1,0% 65,0% 81,2% 25,8% 36,0% 77,4% 47,6%R1 1,0% 24,0% 66,7% 20,4% 29,1% 63,5% 34,0%R2 1,9% 4,7% 36,7% 4,5% 1,2% 20,9% 11,3%R3 1,0% 42,4% 200,3% 38,0% 51,6% 55,2% 64,6%

TABLE X FIRST NATURAL FREQUENCIES FOR THE SHORT MACHINE WITH

NON-IMPREGNATED STATOR, FLANGES AND ROTOR Circumferential modes

Theory 2 3 4 Error Exp. 18010 21100 OOR Ref. FEA 0,3% 0,6% 29172 0,4% S1 38,7% 17,6% 0,2% 18,9% S2 43,9% 23,6% 6,6% 24,7% S3 44,4% 23,9% 6,5% 24,9% S4 45,5% 24,1% 10,5% 26,7% S5 44,2% 22,1% 9,0% 25,1% S6 38,0% 10,8% 12,8% 20,5% S7 44,3% 23,8% 6,5% 24,9% S8 44,3% 23,8% 6,5% 24,9% S9 44,2% 22,1% 9,0% 25,1% S10 37,4% 11,2% 0,6% 16,4% S11 43,6% 21,0% 8,6% 24,4% S12 50,3% 56,7% 52,8% 53,3% S13 32,3% 13,3% 5,4% 17,0% R1 88,3% 72,4% 63,1% 74,6% R2 90,1% 77,4% 70,7% 79,4% R3 86,5% 50,3% 57,7% 64,8%

which radii are those of the stator core and which length is equal to the entire machine’s axial length. Among shell theories, Donnell-Soedel proves to be the most efficient as it was the case for the long motor. However, compared to Table VII, ring theories are not suitable for this kind of motor. Also, for the short motor, shell theory of Epstein-Kennard is the

most precise with an error of 16.4%. With this theory, natural frequencies increase more slowly with the number of modes than with other shell theories.

V. CONCLUSION Analytical models derived for the fast computation of

natural frequencies of common electric machines have been utilized for small industrial motors. In this kind of machine, coils are externally wound on bobbins and then inserted on teeth while the machine is flange-mounted, without external cylindrical frame. Thin cylindrical shell theories and ring theories applied to electric machines have been compared to finite element and experimental results.

In order to calculate natural frequencies of the stator core alone, thin shell theories are accurate. However, the influence of long teeth acting as free cantilevers makes shell theories poorly precise, even with parallel cylinder decomposition. When windings are added, without impregnation, the model of Frohne, originating from ring theory, gives best results for a short and a long motor. Indeed, thickness of the cylinder made of teeth and coils is largely above the limits of validity of thin cylindrical shells. Also, coils provide additional mass and no stiffness to the machine which can poorly be accounted with shell theories. However, if the stator is impregnated, stiffness and thus natural frequencies increase and make shell theories more suitable. Finally, bearing shields influence, especially for the short motor in which they account for a fourth of the mass, is hardly modeled. In that case, Donnell-Soedel thin shell theory is the most suitable.

This work aimed at providing motor designers insights on which analytical models to use for their motors, especially when they are not made with distributed windings linking teeth together. Also, this type of frame without an external cylinder is not commonly studied and a methodology has been proposed. Additional motors could have been studied to further verify models choices.

REFERENCES

[1] Y. S. Chen, Z. Q. Zhu, and D. Howe, “Vibration of PM brushless machines having a fractional number of slots per pole,” IEEE Transactions on Magnetics, vol. 42, no. 10, pp. 3395–3397, Oct. 2006.

[2] S. Huang, M. Aydin, and T. A. Lipo, “Electromagnetic vibration and noise assessment for surface mounted PM machines,” 2001, vol. 3, pp. 1417–1426.

[3] M. Hecquet, A. Ait-Hammouda, M. Goueygou, P. Brochet, and A. Randria, “Prediction of the electromagnetic noise of an asynchronous machine using experimental designs,” Mathematics and Computers in Simulation, vol. 71, no. 4–6, pp. 499–509, Jun. 2006.

[4] R. Islam and I. Husain, “Analytical model for predicting noise and vibration in permanent-magnet synchronous motors,” IEEE Transactions on Industry Applications, vol. 46, no. 6, pp. 2346–2354, Nov. 2010.

[5] J. Le Besnerais, A. Fasquelle, M. Hecquet, J. Pelle, V. Lanfranchi, S. Harmand, P. Brochet, and A. Randria, “Multiphysics modeling: electro-vibro-acoustics and heat transfer of PWM-fed induction machines,” IEEE Transactions on Industrial Electronics, vol. 57, no. 4, pp. 1279–1287, Apr. 2010.

[6] S. Bujacz and J. Nieznanski, “Estimation of acoustic noise of P.M. motor by multi-physical model,” presented at the ISIE, 2011, pp. 597–600.

[7] G. Verez, H. Tiegna, G. Barakat, O. Bennouna, G. Hoblos, and Y. Amara, “Analytical study of vibrations of electromagnetic origins in

short permanent magnet synchronous motors,” presented at the ELECTRIMACS, 2014.

[8] P. L. Timar, A. Fazekas, J. Kiss, A. Miklos, and S. J. Yang, Noise and vibration of electrical machines. Amsterdam: Elsevier, 1989.

[9] J. F. Gieras, J. C. Lai, and C. Wang, Noise of polyphase electric motors. Boca Raton, FL: CRC/Taylor & Francis, 2006.

[10] J. F. Gieras, C. Wang, C. S. L. Joseph, and N. Ertugrul, “Analytical prediction of noise of magnetic origin produced by permanent magnet brushless motors,” presented at the IEMDC, 2007, pp. 148–152.

[11] D. Prudham, “Polyphase electric motor especially for driving pumps or ventilators,” WO 2007/113436 A1, 11-Oct-2007.

[12] R. Hoppe, “Vibrationen eines ringes in seiner ebene,” Journal für die reine und angewandte Mathematik, vol. 73, pp. 158–170, 1871.

[13] F. W. Carter, “Magnetic noise in dynamo-electric machines,” Engineering, vol. 134, 1932.

[14] H. Jordan, Geräuscharme elektromotoren: lärmbildung und lärmbeseitigung bei elektromotoren, W. Girardet. 1950.

[15] [15]A. J. Ellison and S. J. Yang, “Natural frequencies of stators of small electric machines,” Proceedings of the Institution of Electrical Engineers, vol. 118, no. 1, p. 185, 1971.

[16] N. Bracikowski, M. Hecquet, P. Brochet, and S. V. Shirinskii, “Multiphysics modeling of a permanent magnet synchronous machine by using lumped models,” IEEE Transactions on Industrial Electronics, vol. 59, no. 6, pp. 2426–2437, Jun. 2012.

[17] E. Erdelyi, “Predetermination of sound pressure levels of magnetic noise of polyphase induction motors,” Transactions of the American Institute of Electrical Engineers. Part III: Power Apparatus and Systems, vol. 74, no. 3, Jan. 1955.

[18] M. Endo and O. Taniguchi, “Flexural vibrations of a ring,” Bulletin of JSME, vol. 14, no. 74, pp. 761–771, 1971.

[19] J. Kirkhope, “Simple frequency expression for the in-plane vibration of thick circular rings,” The Journal of the Acoustical Society of America, vol. 59, no. 1, p. 86, 1976.

[20] R. K. Singal, K. Williams, and H. Wang, “Effect of radial thickness on the in-plane free vibrations of circular annular discs,” Journal of Vibration and Acoustics, vol. 113, no. 4, p. 455, 1991.

[21] J.L. Sanders, “Nonlinear theories for thin shells.” Technical report no. 10 of the United States Office of Naval Research, 1961.

[22] H. R. Hamidzadeh and R. N. Jazar, Vibrations of thick cylindrical structures. 2010.

[23] A. E. H. Love, “The small free vibrations and deformation of a thin elastic shell,” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 179, no. 0, pp. 491–546, Jan. 1888.

[24] W. Soedel, Vibrations of shells and plates. New York: Marcel Dekker, 2004.

[25] A. W. Leissa, Vibration of shells. American Institute of Physics, 1993. [26] R. K. Singal and K. Williams, “A theoretical and experimental study of

vibrations of thick circular cylindrical shells and rings,” Journal of Vibration Acoustics Stress and Reliability in Design, vol. 110, no. 4, p. 533, 1988.

[27] H. Wang and K. Williams, “Vibrational modes of thick cylinders of finite length,” Journal of Sound and Vibration, vol. 191, no. 5, pp. 955–971, Apr. 1996.

[28] K. Forsberg, “Influence of boundary conditions on the modal characteristics of thin cylindrical shells,” AIAA Journal, vol. 2, no. 12, pp. 2150–2157, Dec. 1964.

[29] G. B. Warburton, “Vibration of thin cylindrical shells,” Journal of Mechanical Engineering Science, vol. 7, no. 4, pp. 399–407, Dec. 1965.

[30] D. C. Gazis, “Three-dimensional investigation of the propagation of waves in hollow circular cylinders. I. Analytical foundation,” The Journal of the Acoustical Society of America, vol. 31, no. 5, p. 568, 1959.

[31] H. D. McNiven, “Axially symmetric waves in hollow, elastic rods: Part I,” The Journal of the Acoustical Society of America, vol. 40, no. 4, p. 784, 1966.

[32] G. M. L. Gladwell and D. K. Vijay, “Natural frequencies of free finite-length circular cylinders,” Journal of Sound and Vibration, vol. 42, no. 3, pp. 387–397, Oct. 1975.

[33] J. R. Hutchinson and S. A. El-Azhari, “Vibrations of free hollow circular cylinders,” Journal of Applied Mechanics, vol. 53, no. 3, p. 641, 1986.

[34] R. Girgis and S. Verma, “Method for accurate determination of resonant frequencies and vibration behaviour of stators of electrical machines,” Electric Power Applications, IEE Proceedings B, vol. 128, no. 1, 1981.

[35] S. P. Verma, R. K. Singal, and K. Williams, “Vibration behaviour of stators of electrical machines, part I: Theoretical study,” Journal of Sound and Vibration, vol. 115, no. 1, pp. 1–12, May 1987.

[36] R. K. Singal, K. Williams, and S. P. Verma, “Vibration behaviour of stators of electrical machines, part II: Experimental study,” Journal of Sound and Vibration, vol. 115, no. 1, pp. 13–23, May 1987.

[37] J. So and A. W. Leissa, “Free vibrations of thick hollow circular cylinders from three-dimensional analysis,” Journal of Vibration and Acoustics, vol. 119, no. 1, p. 89, 1997.

[38] J. N. Reddy, “A simple higher-order theory for laminated composite plates,” Journal of Applied Mechanics, vol. 51, no. 4, p. 745, 1984.

[39] K. Y. Lam and W. Qian, “Free vibration of symmetric angle-ply thick laminated composite cylindrical shells,” Composites Part B: Engineering, vol. 31, no. 4, pp. 345–354, Jun. 2000.

[40] G. R. Buchanan and C. B. Y. Yii, “Effect of symmetrical boundary conditions on the vibration of thick hollow cylinders,” Applied Acoustics, vol. 63, no. 5, pp. 547–566, May 2002.

[41] C. Wang and J. C. S. Lai, “Prediction of natural frequencies of finite length circular cylindrical shells,” Applied Acoustics, vol. 59, no. 4, pp. 385–400, Apr. 2000.

[42] S. J. Yang, Low-noise electrical motors. Oxford; New York: Clarendon Press�; Oxford University Press, 1981.

[43] J. Boisson, F. Louf, J. Ojeda, X. Mininger, and M. Gabsi, “Analytical approach for mechanical resonance frequencies of high-speed machines,” IEEE Transactions on Industrial Electronics, vol. 61, no. 6, pp. 3081–3088, Jun. 2014.

[44] D. Prudham, P. D. Pfister, and T. Richard, “Three-phase electric motor with a low detent torque,” WO 2010/130894 A1, 18-Nov-2010.