ELECTRIC MACHINE VIBRATION ANALYSIS

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1 Konference diplomových prací 27 Ústav konstruování, Ústav mechaniky těles, mechatroniky a biomechaniky, FSI VUT v Brně června 27, Brno, Česká republika ELECTRIC MACHINE VIBRATION ANALYSIS student affiliation Pavel Kukula kukula.pavel@seznam.cz ABSTRACT This paper deals with magnetic vibration sources analysis in induction motors and motor structure dynamic response. The relationship between magnetic flux and mode of vibration is examined from a study of electromagnetic forces and motor's mechanical response. The flux distribution of the motor is calculated by a finite element method (FEM), and from the results of this study, flux density over the motor's radial direction is analysed in the space and time domains. Similarly, the electromagnetic forces waves are calculated by the Maxwell stress method in the space and time domain. Using Fourier analysis (FFT), harmonics of these forces are obtained, and are subsequently used as applied forces in the study of mechanical vibration. The vibration behaviour of the motor cause by electromagnetic forces is simulated using FEM for structure analysis. Finally an identification study of the main vibrations spectrum components is made. The strong influence close to natural frequency on the modes of vibration is confirmed. The influence of the machine base mechanical properties is studied. INTRODUCTION Induction motors are used in many industrial applications for years and their use has been expanding in the variable speed applications. There are the most extended electric machines ever. They are used for the different drives because they are the simplest, cheapest and the most reliable motors which demand only a low level of service. Common threephase AC supply system is sufficient for their supply. A wide increasing of induction motors leads to the mass production. This allows wide mechanization and automatization of production process and subsequently a price reduction. A large vibration phenomenon of a machine placed on a flexible support, for example a landing stage, has been appearing in the last time. This vibration does not appear within the output control in the test room because of its strong dependence on the flexible support characteristics. Problem solving after the installation at customer is economically unpleasant and damages a company s reputation. For these reasons the strong emphasis is put on computer simulation, in this time especially on the perspective finite element method. Thanks to a wide expanding of this method allowed by an expanding using of computers in engineering practise, the method has been come in useful at both its classical domain represented by static and dynamic simulation supervisor affiliation Assoc. Prof. Čestmír Ondrůšek ondrusek@feec.vutbr.cz and newly electromagnetic and thermal simulation and CFD. The computer simulation can be used already in the construction process. Interaction between designer and CA specialist leads to more perfect technical work. Vibration and noise of motors are not only dependent upon the source of vibration forces which are mainly electromagnetic, mechanical and air friction forces but also on the structure and mechanical characteristics of machine. It is well known that large vibration will be generated when the natural frequency of the machine coincides with or is close to the frequency of force applied to the stator. Calculation of mechanical characteristics is important for the analysis and reduction of vibration and noise. SOLUTION SKETCH On the basis of background research and other literature study, mainly [8] and [9], the flowchart of the whole procedure was drafted. It is shown at Fig.. Supply voltage Slip, resistance and reactance of winding Circuit model of the induction motor Stator Current Rotor Current Phase angle between stator and rotor current phasors Modal Analysis of structure using FEM Natural frequencies and mode shapes FINAL RESULTS ANALYSIS Result analysis Determine the most significant factors which cause large vibration Conclusions 2D FE model of induction motor Radial and tangential force acting on teeth distribution Note: direct data export to MATLAB using APDL Fourier Analysis Significant harmonic component inverse Fourier transform Radial magnetic force acting on teeth time behaviour Mechanical response to magnetic forces acting (Transient analysis) Displacement, velocity and acceleration time behaviour Fig. Flowchart of the whole solution procedure

2 ELECTROMAGNETIC ANALYSIS Electric machine magnetic field calculation allows determination of force field magnitude and distribution acting on parts of magnetic circuit. As was decided and discussed in previous section, the calculation was provided as static in separate time steps because dynamical consequences were not essential (noload state) or were set explicitly as external load (load state). Circuit model of induction machine is electric circuit whose equations are equivalent to asynchronous machine equations. The chart is shown on Fig. 2. Fig. 2 Circuit model of induction machine one phase model (source: [9]) Equations describing this circuit can be made up by means of the second Kirchhoff theorem: R + i X r + Z Z I U = R 2 + i X r Z Z I where R is stator winding resistance, R 2 is rotor winding resistance converted to the number of stator windings turns, X r is stator winding leakage reactance, X r 2 is rotor winding leakage reactance converted to the number of stator windings turns, Z is impedance covering parallel combination of iron losses resistance R Fe and main magnetizing reactance X m. Unknown variables are stator current phasor I and rotor current phasor converted to the number of stator windings turns I 2. Right side of this linear equation system is supply voltage phasor U. Converting relation between stator and rotor quantities is given by conversion ratio, sometimes called transformer ratio, reflecting the number of stator and rotor windings turns and winding factor much less. According to the technical report [2], the root-meansquare (RMS) value of the no-load state stator current is 47 A. This value must be converted to separate parallel branches considering the number of turns in slots. Reconstructed waveform is on Fig. 3. Similarly, the root-mean-square (RMS) value of the load state stator current is 452 A and power factor is,86. No-load state current is approximately equal magnetization current (noload state power factor equals approximately cosϕ ), so rotor current phasor magnitude I 2 can be obtained using geometrical relations as shown on Fig. 3. The RMS value of the load state rotor current is 3 643A and stator and rotor phasor angle is 2,8rad. According to the winding connection scheme, the currents time evolution was reconstructed as shown on Fig. 3. Afterwards, current density in separate slots was calculated. Phase current [A] Current in stator winding U-phase STATOR current V-phase STATOR current W-phase STATOR current Phase current [A] Current in stator and rotor winding U-phase STATOR current V-phase STATOR current W-phase STATOR current U-phase ROTOR current V-phase ROTOR current W-phase ROTOR current Fig. 3 No load (left) and load (right) currents time evolution Electromagnetic forces calculation using FEM The model was discretized by means of the PLANE53 element. This two-dimensional element contains quadratic base function and all needful degrees of freedom using in the twodimensional magnetic analysis. The basic PLANE53 element was used, only one degree of freedom magnetic vector potential was needed. (Generally three-dimensional vector reduces to one-dimensional in terms of two-dimensional analysis; one-dimensional vector perpendicular to solution plane, exactly.) Fig. 4 Used model and FE disctretization detail The element size was set to be able to achieve efficient geometry description. In the opposite side, there was endeavour to low count of elements, because a large number of solution procedures were needed to consider time evolution. The model is shown on Fig. 4. It is discontinuous along air gap there are two separate models of rotor and stator. This arrangement allows free rotor rotation to cover time evolution. To evading magnetic contact using, the rotor rotation was provided in discrete steps and coincident nodes degrees of freedom were coupled The boundary condition for vector potential is very important condition in magnetic analysis. It is necessary for its prescription at least in one node. This condition is fundamental, applied to the solution region boundary definitely There is good elasticity analogy. This solved problem is analogical to the disc fixed along its outer and inner circumference. The current density load is analogical to pressure load. Coincident nodes coupling is similar to welding of two separate discs. Primary unknown vector potential is analogical to disc deflection. This analogy enables to solve static magnetic problems using system primarily designed for elasticity problems solution. Unfortunately, the result analysis is significant problem, because neither magnetic flux density or field strength are not analogical to any mechanical stress quantity.

3 determines it and all magnetic lines of force are tangential to this boundary. The zero vector potential boundary condition was applied to all model boundary lines. These lines reflecting lamination-surrounding air interface. Such as interface represents very significant material parameters step change, so magnetic lines of force cross this interface to a very limited extend. This was confirmed by calculation of using more detail model including large surrounding piece. The rotor shaft iron was neglected because there is a wide air gap between rotor laminations and shaft. These parts are binding together through six webs. The webs shape is not significant. As was indicated in previous section, the current density load applied to the stator and rotor slots was chosen. The load is constant along a slot surface. This simplification is acceptable in the steady state solution. The current density magnitude is given by a fraction of the current value and the slot area. The current density direction was determined in according to the winding connection documentation. The magnetic field time evolution is in general a transient analysis problem. As it was written, it was decided to provide some static solutions in discrete time step with dynamical consequences considering as external load. Time axis discretization was set by taking later the Fourier Analysis into account. When spectrum with 5 Hz division is required, the maximum time value must be /5 s. Maximal spectrum frequency is determined by time step 2. As it was explained, the magnetic contact was not used and so the rotor is not able to turn a random angle. Rotation is possible is discrete steps with constant number of elements. By this way the angle is determined and after division by rotor angular velocity, time between two steps is obtained. At noload state, when rotor revolves by synchronous speed, such as time discretization is absolutely appropriate. At load state, when rotor speed is dropped due to the slip, this way of suggested time division is not optimal. However, it is sufficient. The calculation was provided during one supply voltage period (see Fig. 3). Fig. 5 Magnetic vector potential magnitude distribution (left), Flux density magnitude distribution (right) No-load state The no-load state magnetic field character and contour does not very change only revolves synchronous speed. minimum. 2 Strongly exaggerate. For the catch of waveform six points are needed at Small changes are caused by the stator s and rotor s teeth overlapping. Fig. 6 Concentrated radial magnetic force acting on stator tooth as a function of space The concentrated magnetic force radial component acting on the stator teeth as a function of space is shown on Fig. 6. The magnetic flux density magnitude along air gap is projected as a red curve for better imagination of magnetic field contour. The maximum force magnitude is under the current field pole. Concentrated radial magnetic force acting on some selected stator teeth as a function of time is shown on Fig. 7. The idea of the most significant harmonic component and mutual time delay is obvious on base of this picture. The magnetic force time evolution can provide good imagination. However, further processing before next analysis was necessary. The force time behavior was analyzed by means of the Fourier Analysis 3. The MATLAB system was used. Force [N] Concentrated radial magnetic force on a STATOR tooth as a function of time Fig. 7 Radial force acting on a stator tooth time evolution One example of obtained results is shown on Fig. 8. It was pointed out that the most significant harmonic components of the stator force are static component and Hz-component. F t i t. Subsequently, the force time evolution was reconstructed in the analytical form by using only these most significant components. According to the limited time discretization the analytical approximation is sufficient. This reflects the relation ( ) ( ) 2 3 We can t talk about the Fast Fourier Transform (FFT) because data vector has not necessary size (2 n ) needed for the FFT algorithm, for example the classical Cooley-Tukey algorithm. However, data vector is relatively small, the same size as previous static FEM analysis count, the Fourier Transform calculation according to the Discrete Fourier Tranform definition can be used with no problem. Tooth N. Tooth N.2 Tooth N.3

4 Magnitude [N] Radial magnetic force (tooth N.) MAGNITUDE Fig. 8 Magnitude spectrum of force acting on a stator tooth ROTOR STATIC DEFLECTION ANALYSIS On the basis of technical documentation [2], it was decided that rotor assembly can not be modeled as beam. The rotor-core stampings were modeled as solid. The SOLID95 element was used. Due to the simple geometry mapped mesh was able to be used. Division along perimeter was referring tooth numbers this was useful in subsequent analysis. The ventilator was, considering geometry character, modeled as shell, using the SHELL93 element. Remaining part of shaft was modeled as a beam by using the BEAM89 element. All used elements contain quadratic base function. The mutual connecting of parts modeled by using different elements was provided by way of constrain equations. These equations provide a more general means of relating degree of freedom values than is possible with simple coupling. The ring assembly is modeled as a mass point, using the MASS2 element. This element is providing to set not only mass characteristics of point, but also moments of inertia. This approach lets avoid modeling of complicated part and on the opposite site considering its inertia consequences. Bearing stiffness influence was included as spring system which stiffness was reflecting bearing stiffness. Bearing stiffness was determined with the assistance of lit. []. The displacement load dependence is able to be considered as linear in case of the roller-bearing (stiffness is constant) whereas for the ball-bearing is not. A modeling of a nonlinear behavior is disadvantageous, especially in subsequent dynamic calculation. As this is only small nonlinearity, it was decided to provide linear approximation. The equivalent approximation method was used to minimize inaccuracy. A nonlinear spring was replaced by linear spring which let the same static displacement with the same energy accumulation. Linear spring system was modeled by using the COMBIN4 element, nonlinear by using the COMBINE39 element. The most simple material model was chosen, isotropic linear with no temperature dependency. However, the rotorcore stampings material is not able to be considerated as isotropic, because the stiffness in the rotor axis direction is decreased caused by an in-stratified isolation. For this reason, the orthotropic material model was used. Relevant values were decreased to per cent. However, the rotor core packet part containing the rotor teeth is not able to be modeled neither previous approach. The tooth material stiffness is anisotropic. A load is carrying only by the teeth, not by the slots. Furthermore, the tangential load is influenced by a copper conductor-core contact which stiffness is very small due to an in-stratified isolation. For these reasons, the tangential stiffness was decreased to per cent and the radial stiffness was decreased according to slot-tooth area ratio. On the contrary, the density was increased to consider the copper conductors mass. Such approach corrects rough mesh. Deflection [mm] Mean value: Nonlinear solution: µm Linear solution: -95. µm Rotor core axis static deflection Nonlinear solution Linear solution (Ball-bearing stiffness equivalent approximation) Z-coordinate [m] Fig. 9 Rotor static deflection (left); Shape of rotor core centre line deflection (right) Important results are shown on Fig. 9. A deflection in ballbearing is 23,2 µm. The using of the mentioned Equivalent Approximation Method, a linear approximation was provided. Obtained value was 45 N/µm. Fig. 9 shows the shape of rotor core centre line deflection 4. A mean value was calculated, 95 µm, which was subsequently used as a magnetic analysis input. MODAL ANALYSIS The modal analysis, the natural frequencies and mode shapes calculation, is the basic dynamic analysis ever. Goal of this section was determine an influence of the flexible support on the natural frequency spectrum of the whole assembly. Modal Analysis Flexible support Influence Certainly, flexible support does not change the machine natural frequency and mode shape. However, the flexible support natural frequency spectrum and the machine natural frequency spectrum can merge. Further, the natural frequency spectrum of the flexible support was expected in range from Hz to 5 Hz, so in operational range (rotor speed frequency is 6,5 Hz at slip. %). By using of the FE model of the whole assembly machinesupport is not appropriate because it causes long computational time. Furthermore, many calculations were needed to provide. For these reasons, a very simplified model of machine and support was used. The machine was modeled as mass point, by using the MASS2 element (see Fig. ) with setting allows to consider moments of inertia 5. Separate parts of model were joined by no-mass extra-stiff beam and whole model structure was connected by constrain equation. 4 Data was obtained by using Path Operations module functions, exported to the MATLAB system and processed. 5 These were determined by using particular solution (the element matrix formulation) of whole machine-support assembly. The inertia tensor was obtained. The inertia tensor basic coordinate system was the same as global solution coordinate system, so no conversion was needed.

5 The flexible support was modeled as spring system with three springs in each foot-screw (one spring for each spatial coordinate). The COMBIN4 element was used. The torsion stiffness was neglected. This approach let model different support stiffness in each spatial coordinate (possibly under each foot to model machine on an asymmetric fixed console). 7 Flexible Support Natural Frequency Fig. Simple model for flexible support influence calculation Natural frequency [Hz] Support Stiffnes Coeficient [-] Fig. Flexible support natural frequencies dependence on support stiffness The dependence of the flexible support natural frequency on stiffness is shown on Fig.. The stiffness coefficient k = refers to the support stiffness using in the test-room 6. The coefficient k = is able to reflect a real support system. The curves are reflected inverse-quadratic relation. As figured, from ten-fold or fifteen-fold of using stiffness the natural frequencies are close to machine natural frequencies. So, the flexible support influence should not be neglected. Machine Modal Analysis Natural Frequencies and Mode Shape Complete model of machine-support assembly was created for natural frequencies calculation. One calculation considering absolutely stiff support and some calculations considering flexible support were provided. On the basis of technical documentation [2], a model of mechanically significant machine parts was created, shown on Fig. 2. Some parts were simplified to achieve elementary, easily describable geometry. Considering of the character of geometry, the structure was mainly modeled as shell, by using the SHELL93 element. The rotor assembly was maximally simplified was modeled as a beam and rotor stampings packet, ventilator and ring assembly as mass points. Only the stator stampings packet and outer bearing ring, considering mass of bearings and cups, were modeled as solid, using the SOLID95 element. The stator frame, front and rear shield were discretized by using the SHELL93 element. Screw connections were modeled by way of the rigid region. The rigid region diameter was referring pressure cone diameter (see Fig. 3). The master node was placed on screw centre line in thread side. 6 The manufacturer uses rubber silent-blocks. These probably have nonlinear behaviour. Used stiffness was obtained by measurement under static load. Transversal stiffness was after discussion decrease to /6 longitudinal stiffness. Fig. 2 A half of complete Fig. 3 Detail of screw machine model connection model The stator stampings packet was modeled as a simple cylinder. The stator teeth and copper conductor influence was taken into account similarly as a rotor stampings packet in previous section, by material characteristics change. The bearings were modeled as the linear spring system with stiffness determined in previous section. More than one spring for each spatial coordinate was used to better decomposition of mutual reaction. The bearing mass was reflected by increasing density of the model of the outer bearing ring. The ball-bearing axial stiffness was considered as infinite. The flexible stiffness model was the same as in case of simplified model. The idealized model of an absolutely rigid support was modeled by displacement DOF of nodes in footclamps setting to zero. Fixing would better model screwconnection, but would not able to be considered as a limit state of used flexible support model. The results are described in Tab.. Tab. Natural frequencies and mode shapes Mode shape Natural frequency Torsion vibration of rotor 46.6 Transverse rotor vibration ( st shape) 5.3 Transverse rotor vibration (2 nd shape) 8.45 Bottom frame case vibration ( st shape) 22.8 Bottom frame case vibration (2 nd shape) 29.2 Bottom frame case vibration (3 rd shape) Bottom frame case vibration (4 th shape) Stator packet vibration ( st shape) 244.9, Bottom frame case vibration (5 th shape) Upper frame case vibration ( st 254.8, 27.4, 32.8, shape) 39. Bottom frame case vibration (6 th shape) 27.8 Bottom frame case vibration (7 th shape) 294. Longitudinal rotor vibration ( st shape) 36.7 Bottom frame case vibration (8 th shape) 32.5 Upper frame case vibration (2 nd shape) 325.3, 325.4, Stator modal analysis The stator modal analysis was used to determine natural frequency and mode shape of stator stampings (in the plane perpendicular to the machine axis). The increased attention has to be given to mode shape similar to magnetic force distribution shapes.

6 The same model as used in magnetic analysis was created. Certainly, the structural PLANE82 element was used. The plane strain approach was used an infinitely long stator was considered, so stator axis direction strain was directly set to zero. No boundary condition was applied free solid vibration was solved. Two models were used one with free slots and one with copper slots. The obtained mode shapes are shown on Fig. 4. Results obtained by model N.2 are just informatively because an interface between stator stamping and copper slot is strongly nonlinear contact with stiffness strongly depending on an inserted isolation, wedge keys and wadding. Real values are probably closer to values obtained using model N.. First mode shape, see Fig. 4, was obtained by all machine structure modal analysis too. Because analysed machine was six-pole, this mode shape is not so dangerous as could be in case of two-pole machine. Mode shape frequency corresponding to distribution of magnetic forces is 2 Hz, far enough from running conditions. The static calculation of rotor loaded by magnetic forces was provided, the mean values of forces were considered. With respect to the rotor revolute, the reaction resultant direction in each time step was calculated and adds to forces caused by self-weight of rotor. Nodes of bearing rings models were loaded using these force time evolution8. The structure model was loaded by approximated time evolution of magnetic forces to time discreditaion, independent on time discreditation used in magnetic analysis, can be used. The other load was caused by self-weigh, so whole structure acceleration was set. The last one load was force system caused by rotor behaviour effect. Displacement [mm] of displacement at point N Displacement [mm] of displacement at point N Displacement [mm] of displacement at point N Magnitude [mm] Magnitude [mm].3.2. Magnitude [mm] st mode shape 27 Hz 78 Hz 5 th mode shape 962 Hz 262 Hz 2 nd mode shape 582 Hz 843 Hz 6 th mode shape 974 Hz 6 Hz 3 rd mode shape 45 Hz 377 Hz 7 th mode shape 2275 Hz 7628 Hz 4 th mode shape 54 Hz 467 Hz 8 th mode shape 2454 Hz 954 Hz Fig. 4 Stator packet natural frequencies and mode shapes (first line belongs to model N., second line belongs to mode N.2) TIME DOMAIN ANALYSIS, TRANSIENT ANALYSIS Response calculation to an external (time depend) load was provided. Model was loaded with magnetic forces. Only the stator part of machine was modelled. The rotor behaviour effect on vibration was reflected as force system. The flexible support was modelled as spring system, again. The stiffness coefficient was set to k = 5, because this value seems to be most dangerous to vibration behaviour. Used element types, material characteristics and other inputs were similar to model used in modal analysis. The forces acting on rotor are constant in case of no-load state. In case of symmetric magnetic field, total reaction forces in bearing would be zero. However, caused by rotor static deflection, the magnetic field is not symmetric and the force field resultant is not zero7. This field resultant is loading the bearing and its direction is changing with rotor revolute. 7 In fact, this force resultant is increasing rotor load, so rotor deflection is increasing and subsequently the asymmetry of magnetic field is increasing too, Fig. 5 Displacement components time evolution at point on stator frame head and their spectra Acceleration [m/s 2 ] Magnitude [m/s 2 ] of acceleartion at point N Acceleration [m/s 2 ] Magnitude [m/s 2 ] of acceleartion at point N Acceleration [m/s 2 ] Magnitude [m/s 2 ] of acceleartion at point N Fig. 6 Acceleration components time evolution at point on stator frame head and their spectra The kinematical quantities time evolution at one investigated point is shown on Fig. 5 and Fig. 6. Their frequency spectrums are on the same figures. Obviously, the considerable part of quantities include harmonic component with frequency identical the revolute frequency (about 6 Hz). and as retroaction the force resultant is increasing. This phenomenon is called magnetic thrust and has negative stiffness spring character. However, the ANSYS system (the COMBIN4 element) is not able to operate with negative stiffness, so only static deflection influence on magnetic field asymmetry was considered, with no other consequences. Dynamic behaviour is much complicated. 8 In fact, reaction magnitude distribution is not able to be consider as constant along bearing circumstance, but the maximum peak is under rotor shaft in resultant direction and the minimum peak (or zero) is on the opposite side. However, constant distribution model seems to be enough.

7 CONCLUSIONS During all work a modeling using macros was consistently used. This approach is useful always when the geometry is simple enough. This way a lot of variants can be easily saved with minimum data space requirement. At a maximum possible degree, the ANSYS and the MATLAB systems were used. The contribution of work lies in mutual data exchange without manual inserting, except several scalar values. Both systems allow generating textfiles with instruction, which are restartable in the other system. The present capacity of the computing technology allows processing of numerically obtained data by terms of the Fourier analysis. The numerically obtained data have usually much smaller size, so the DFT algorithm according to the definition can be easily used. According to future using of the DFT, it is needed to use appropriate time interval with appropriate division. The extensive spectrum can be obtained only using small time step for long time interval. The influence of the support flexibility on mechanical behavior is not able to be neglected. Absolutely rigid support model, realized as zero displacement conditions, can be considered as limit case of support with low flexibility. However, this approach is not advantageous because some mode shape combinations can appear and result analysis gets complicated. The asymmetry of the rotor packet deflection curve to stator packet probably causes moments acting on the rotor. Their static analysis requires a three-dimensional model of magnetic circuit, which computational intensity is much higher than two-dimensional model computational intensity. The complete model of the mechanical structure is not always useful to the modal analysis. It seems to be useful to model some parts separately and remaining parts influence consider as a simplified rigid body. The numerically obtained natural frequencies are adequately equal the results obtained by experimental modal analysis. The maximum stress places are in footing parts of machine. However, in comparing with static safety they are quite small. The no-load state response proves similarly as load state response. Far enough from load point the response is identical. For this reason no-load state should be consider only as simplification, but it is full load case which response can be generalized. As has been mentioned yet, provided analysis allows investigation the influence of electro-magnetic behavior on mechanic vibration. However, an inverse influence is not able to be considered exactly. But obviously, deformation, i.e. changes of parameters, is strongly influencing magnetic field shape and consequently magnetic force distribution. The contemporary capability of the FEM gives a hope that response analysis considering mutual influence of magnetic and mechanical could be provided. ACKNOWLEDGMENTS The author of this paper gratefully acknowledges the support of the research project MŠM 26358, Simulační modelování mechatronických soustav. REFERENCES [] NEVES, C. G. C., CARLSON, R., SADOWSKI, N., BASTOS, J. P. A.: A Study on Magnetic Vibration Sources Identification in Induction Motors by FEM Simulation and Experimental Produces, The IEEE Thirty-Third Industry Applications Conference Annual Meeting, Oct 998, vol. [2] NEVES, C. G. C. ET AL.: Experimental and Numerical Analysis of Induction Motor Vibration, IEEE Transactions on Magnetics, May 999, Volume: 35 Issue: 3 Part [3] ISHIBASHI, F., NODA, S., MOCHIZUKI, M.: Numerical Simulation of Electromagnetic Vibration of Small Induction Motors, IEEE Proceedings Electric Power Applications, Nov 998, [4] ZHANG, F., NINGZE, T., FENGXIANG, W.: Analysis of Vibration Modes for Large Induction Motor, Conference on Electrical Machines and Systems, ICEMS 25, Sept. 25, Vol. [5] TÍMÁR, P. L.: Noise vibration of electrical machines, Budapest, Akadémiai Kiadó, 989, 339 s. [6] ZHOU, PEI-BAI: Numerical analysis of electromagnetic fields, Berlin, Springer-Verlag, 993, ISBN , 46 s. [7] DĚDEK, L.,DĚDKOVÁ, J.: Elektromagnetismus, Brno, Vutium 998, ISBN [8] MĚŘIČKA, J., HAMATA, V., VOŽENÍLEK, P.: Elektrické stroje, Praha, Vydavatelství ČVUT 997, ISBN [9] ONDRŮŠEK. Č.: Elektrické stroje, elektronická skripta [] KRÄMER, E.: Dynamics of Rotors and Foundations, Berlin Heidelberg, Springer-Verlag, 993, ISBN [] Release. Documentation for ANSYS [2] Technical documentation provided by SEM Drásov Siemens Electric Machinery s.r.o.