Model order reduction of dynamical structural simulationmodels of electric motors using Krylov subspaces

M. Schwarzer 1, E. Barti 1, T. Bein 2,1 BMW Group, Taunusstr. 41, 80807 Munich, Germanye-mail: michael.schwarzer@bmw.de

2 Fraunhofer LBF, Bartningstr. 47, 64289 Darmstadt, Germany

AbstractIn this paper the Krylov subspace method is investigated regarding its use for the vibration analysis of astator of an electric machine. It is compared to the full order direct integration method as well as commonreduction methods like the mode superposition. It is found that the Krylov subspace method provides muchbetter and faster results than the mode superposition as it is commonly used. Even if complex load casesare applied to the stator the Krylov subspace method reliably produces accurate results which in case of themode superposition are only achievable performing a residual vectors correction.

1 Introduction

The acoustic behavior of the drive train is one of the major comfort criteria of electrified vehicles and hencea central research topic in the automotive industry. Undesired noise emission of the system is primarilycaused by electromagnetic excitations that occur inside the electric motor. In order to generate a deeper un-derstanding of the relevant transfer paths that lead to the sound radiation and to efficiently develop innovativenoise reduction concepts simulations can be used. Depending on the electromagnetic design and operatingstrategy of the electric motor different load cases in wide frequency ranges have to be considered to entirelyinvestigate the acoustic behavior of the system.

In order to reduce the computational effort of structural dynamical simulations the use of model order reduc-tion techniques can be advantageous. Depending on the type of coordinates of the reduced order system thedifferent reduction techniques can be separated into three basic categories: physical coordinate reduction,generalized coordinate reduction, and hybrid coordinate reduction. While the coordinates of the physicallyreduced model represent a subset of the physical coordinates of the full model, the generalized coordinatereduction is based on a coordinate transformation into a reduced order subspace. The shape of the vectorsthat span the reduced order subspace strongly depends on the particular algorithm that was used to generatethem. Qu2004

In order to accurately calculate the force response behavior within a reduced order subspace the force exci-tation vector needs to be adequately displayed by the subspace [1]. The acoustically relevant forces withinelectric machines act on the inside parts of the structure, the stator. Depending on the electromagnetic de-sign of the machines the force vectors might show very complex spatial shapes. A frequently used subspacemethod is the modal superposition in which the dynamic behavior of the full model is represented by a su-perposition of the eigenvectors of the system and their particular dynamic behavior. The very complex loadcases that occur in electric machines excite eigenmodes that might not appear within the acoustically rele-vant frequency band and thus might not be considered in the modal subspace. In this paper it is shown thatthe force excitation is not necessarily sufficiently displayed by the given modal subspace which makes themodal superposition inapplicable for such cases. Therefore, different correction algorithm can be applied. In

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this paper the residual vector correction that extends the mode set by static pseudo-modes is introduced andanalyzed regarding its impact on the accuracy of the method.

As an alternative reduction method the Krylov subspaces method is introduced. Krylov subspaces are gen-erated based on a mathematical algorithm that considers the particular load cases of the system. In the paperall three reduction techniques, the mode superposition with and without correction algorithm as well as theKrylov subspace method, are applied to the same two structures an axially thin stator segment and a fullstator. Both structures are excited by a given electromagnetic field as it occurs in real electric motors. Fi-nally, the different methods are compared regarding the error they cause and the computational benefit theyprovide on the use of stators of electric motors.

2 Model reduction techniques for dynamical structural simulation

The transient equation of motion in matrix form for multi-degree-of-freedom systems is given by

M · x(t) + C · x(t) +K · x(t) = f(t) (1)

in which M , C, and K denote the mass, damping, and stiffness matrix of the system, f the load vector,and x, x, x the state vector of the system as well as its first and second derivative in the global coordinatesystem. In many engineering cases the input is not the load vector f(t) itself but a vector u(t) that containsthe coefficients for a linear combination of a number of load vectors which are stored in the load vectormatrix F

F (t) = [f1, f2, f3, . . . , fk] , with F ∈ <n×k (2)

so that Equation 1 can be rewritten as

M · x(t) + C · x(t) +K · x(t) = F · u(t). (3)

Often the damping matrix C of the system is assumed to be proportional which means that it can be writtenas a linear combination of the mass matrix M and the stiffness matrix K

C = αM + βK (4)

where α and β are the so called Rayleigh coefficients [2]. If damping in Equation 1 is fully neglected thesystem is simply called an undamped system. The implicit undamped system can be expressed as an explicitordinary differential equation (ODE)

x(t) = A · x(t) +B · u(t) (5)

with A = −M−1 ·K denoting the system matrix and B = −M−1 · F the scattering matrix of the explicitsystem. In order to investigate the harmonic response of the system Equation 5 is transformed into thefrequency domain by Laplace transform

ω2 · x(ω) = −A · x(ω)−B · u(ω) (6)

where ω denotes the angular frequency of the system response [3]. The transfer function of the system G(s)with s = −ω2 thus is given as

G(s) = (s · I −A)−1 ·B. (7)

From a computational perspective solving Equation 1 is very expensive since the system state vector x(t) isusually very large in size. A practical way to decrease the computational effort is to reduce the degrees offreedom (DOF) of the system by substituting the system state vector x(t) by a linear combination of a set ofvectors that are contained in a transformation matrix X

x(t) = X · z(t), with X ∈ <n×p. (8)

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Then the damped implicit ODE 1 is transformed into

M · z(t) + C · z(t) + K · z(t) = f(t) (9)

in which M = XT ·M ·X , C = XT ·C ·X , K = XT ·K ·X , f(t) = XT · f(t), and z(t) is the low-ordersystem state vector in the generalized coordinate system [4]. Many model order reduction techniques followthe scheme using the so called coordinate transformation matrix or projection matrix X ∈ <n×p to reducethe DOFs of the system from n to p with p � n [4]. The error that is generated by the order reductioncan be expressed as the difference between the state vector x(t) of the full order system and the state vectorx(t) = X · z(t) of the reduced system [5]

ε(t) = ‖x(t)−X · z(t)‖2 . (10)

3 The mode superposition

The mode superposition method is a model reduction technique which is frequently used in dynamic analy-ses. Its basic principle is to build the transformation Matrix X based on the eigenmode shapes vectors φ ofthe full order system and thus reduce the system order from nDOFs to the number of considered eigenmodesp with p� n. The modal coordinate transformation matrix X is given as

X = [φ1, φ2, φ3, . . . , φp] . (11)

Since the transformation into the modal coordinate system forms reduced system matrices that are diagonalthe mode superposition can be interpreted as substituting the response of the coupled MDOF-system by alinear combination of the responses of each individual modes [2]. Thus, the response of the dynamic systemx(ω) in Equation 1 is then given by the relationship

x(ω) =

p∑j=1

φj

φTj · F

ω2j + 2iζjωjω − ω2

(12)

in which ωj is the eigen angular frequency and ζj the damping ratio of the particular eigenmode. Hence, theparticipation of a particular eigenmode to the overall frequency response is determined by the eigen angularfrequency ωj , the degree to which the load vector matrix F excites the eigenmode shapes vector φj , and thedegree of which the eigenmode shapes vector φj influences the desired response vector coordinate xi. Basedon these three conditions the number p of considered eigenmodes within the modal coordinate transformationMatrix X can be reduced to a lower order. [1]

One of the most time-consuming tasks in performing the mode superposition is the acquisition of the eigen-pairs of the system. The computational effort to perform the modal analysis is mainly determined by thenumber of DOFs of the full order system which becomes even more relevant if damping is considered. [2]

Therefore, reducing the frequency range of the modal analysis is practical. Usually the eigenvalue solution iscut off after a certain frequency that corresponds to the frequency range of the load case that is investigatedsubsequently in the harmonic analysis. Depending on the set of eigenvectors the degree to which the loadvector matrix F excites the considered eigenmodes φj is low which might cause a significant error. In orderto reduce that error various correction mechanisms exist. One mechanism is the use of residual vectors.Residual vectors are given by the static response of the structure to a specific load case [6]. Assuming thestatic deformation is a good approximation of the dynamic deformation of the system the residual vectorsare considered as mode shapes. In order to efficiently use residual vectors within the mode superpositionthe orthonormality has to be assured. Therefore, an orthonormalization algorithm needs to be applied sub-sequently to the static analysis. The pseudo-natural angular frequency of the residual vector ωrv is givenby

ωrv =1

2π

√φT

rv ·K · φrv

φTrv ·M · φrv

. (13)

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4 The Krylov subspace method used for second order structural dy-namical problems with proportional damping

In the following section the Krylov subspaces as a model order reduction technique are introduced. A goodoverview of the basic principle and different algorithms that provide an orthonormal basis for the Krylovsubspace is given in [7]. In this section only one particular process, namely the Arnoldi process [8], will bediscussed.

4.1 Krylov subspaces

Krylov subspaces are basically all subspaces that satisfy the following

Krk = span

{v,A · v, . . . , Ak−1 · v

}, with A ∈ <n ×<n and v ∈ <n ×<1. (14)

They were first introduced in 1931 by Alexei Krylov [9]. Today Krylov subspaces are mainly used foriterative solution methods [10] or the solution of large systems of linear equations [11]. This paper focuseson the use of Krylov subspaces as the basic principle of a generalized coordinate model order reduction.

4.2 The Arnoldi algorithm

As with increasing number of vectors within the Krylov subspace the vectors get almost linear dependentdue to power iteration an orthogonalization algorithm like the Arnoldi process needs to be applied. TheArnoldi process is one of different algorithms that generate an orthonormal basis X ∈ <n × <p for a givenKrylov subspace. The main advantage of the Arnoldi algorithm over alternative approaches like the Lanczosalgorithm is the stability of the process. The Arnoldi algorithm also works for V = [v1, v2, v3, . . . , vk]with V ∈ <n×k and is then called block-Arnoldi. [7]

4.3 Pade and Pade-Type approximants

In Equation 7 the general transfer function of an undamped system G(s) was introduced. For a single input-single output system the general form of the transfer function in state space can be expressed as a rationalfunction of the state space variable s

G(s) =a(s− z1) . . . (s− zn−1)

(s− p1) . . . (s− pn)(15)

where pi are the poles and zi the zeros of the transfer function. Like any rational function the general transferfunction can be displayed by a Taylor series of s around an expansion point s0

G(s) =∞∑i=0

mi(s− s0)i. (16)

The coefficients of the polynomial mi are called moments. The idea of Pade [12] and Pade-Type approxi-mants [13] is to find a reduced order system of which the transfer function matches the first q moments ofthe transfer function of the full order system and therefore not exactly displays the transfer function for alldifferent s ∈ < but gives a good approximation of G(s) around the expansion point s0. Pade approximantsmatch the maximum number of moments, q = 2 · k, while Pade-type approximants match the first q < 2 · kmoments. The projection matrix X that is the outcome of the Arnoldi process for the right Krylov subspacebased on the system matrix A and the scattering matrix B of the explicit ODE (see Equation 5)

Krk

{(A− s0I)−1, (A− s0I)−1 ·B

}(17)

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produces a reduced system according to Equation 9 that matches the first k moments of the transfer functionand therefore provides a Pade-type approximant

mi = mi for i = 1, . . . , k. (18)

This also applies if proportional damping is used within the equation of motion [14].

5 Load cases in synchronous machines

Vibrations in electric motors are mainly induced by the electromagnetic forces caused by the air gap field.These forces have a radial and tangential component and act on the stator as well as the rotor of the machine.Hereby, the electromagnetic design of the rotor, namely the number of pole-pairs p, determines the funda-mental space harmonic of the force wave that excites the structure. Depending on further machine designparameters the force wave contains higher harmonics of the fundamental space harmonic causing the struc-ture to be excited in different frequencies [15]. The excitation frequencies are proportional to the rotationalspeed and therefore referred to as engine orders. The dynamical behavior of stators of electric machines canbe approximated by the behavior of a cylindric shell. Hence, the circumferential deformation of a particulareigenmode is given by a sinusoid [16]. Rewriting the force distribution of the stator at a particular axialposition x as a Fourier series of the space angle θ an accurate projection of one particular load f(t, θ) intothe modal subspace is assured if and only if the eigenmodes that correspond to the Fourier coefficients ai(t)and bi(t) are contained within the considered frequency range

f(t, θ) =

∞∑n=0

[an(t)cos(2pnθ) + bn(t)sin(2pnθ)] , with x = const. (19)

In this case the force f(t, θ) as well as the Fourier coefficients ai(t) and bi(t) are two dimensional vectorscontaining a radial and a tangential component. Equation 19 is based on the assumption that the time variantforce act symmetrically on the different poles. Hence, only the fundamental space harmonic as well as itshigher harmonics determine the spatial force wave f(t, θ) [15]. Assuming the deformation of the statortooth surface due to an circumferentially uneven force distribution over the surface is small the tangentiallydistributed force density on a stator tooth can be integrated and considered as a single resultant force vector.Following the Nyquist criterion, the number of slots s thus determines the maximum circumferential spatialorder of the electromagnetic force that can be displayed by the discrete system. Hence, the spatial forcef(t, θ) is given by a finite number of Fourier elements. For a system with three phases and two slots per poleand phase the spatial force waves are limited to four, namely n = 0, 1, 2, 3, for which b0, b3 = 0. [17]

The different orders of wave forms 2pn are referred to as circumferential spatial orders. The correspondingeigenmode shapes are illustrated in Figure 1 for a axially thin stator segment. According to [16] the differentmode shapes can be expressed by numbers where the first number corresponds to the circumferential spatialorder and the second number to the axial spatial order. The index r indicates a radial and t a tangentialorder deformation of the structure. Taking into account skewing of the electric machine the load case overthe axial length of the stator gets much more complex than for an unskewed machine [17]. Hence, the axialforce distribution on a stator tooth needs to be displayed by an appropriate axial discretization. According to[18] the axial eigenmode shapes of the stator are given by multiples of half-sinusoids over the stator lengthl. Therefore, the axial force distribution can be expressed by a Fourier series of the axial position x

f(t, x) =

∞∑n=0

[an(t)cos

(2πnx

2l

)+ bn(t)sin

(2πnx

2l

)]. (20)

For the circumferential spatial order Zero the different eigenmode shapes of a stator structure are illustratedin Figure 2.

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Mode 0r ,03428.3 Hz

Mode 12r1,018170 Hz

Mode 12r2,018182 Hz

Mode 24r1,044094 Hz

Mode 24r2,044140 Hz

Mode 36r ,050101 Hz

Mode 36t,06410.7 Hz

Mode 12t1,05821.4 Hz

Mode 12t2,05822.8 Hz

Mode 24t2,06375. Hz

Mode 24t1,06365.9 Hz

Mode 0t,06725.9 Hz

Figure 1: Comparison of different circumferential mode shapes and natural frequencies of an axially thinstator segment.

Mode 0,03397.7 Hz

Mode 0,12957.5 Hz

Mode 0,22907.9 Hz

Mode 0,33338.2 Hz

Mode 0,65192.8 Hz

Mode 0,76139.2 Hz

Mode 0,43701.2 Hz

Mode 0,54386.9 Hz

Figure 2: Comparison of different axial mode shapes and natural frequencies of an electric machine stator.

6 Application of model order reduction techniques on electric mo-tors

The following section focuses on the application of different model order reduction techniques for the har-monic analysis of a stator of an electric motor. The stator that is taken for the analysis is equal to the earlierinvestigated stator of Section 5. The housing and mounting of the electric motor is neglected in these consid-erations. Nevertheless, the general procedure can be transferred to a complete electric motor. All differentmodel order reduction techniques are compared to the full order model regarding the accurate projection ofthe vibrational behavior and the potential of decreasing the computational effort. The error that is causedby the model reduction is quantified based on the equivalent radiated power (ERP) which is radiated by theouter stator surface of the stator in a frequency range from 200 Hz to 10000 Hz.

The investigations of the dynamical behavior of the structure are split in two parts. In the first part anaxially thin stator segment as it was shown in Figure 1 is investigated in order to see if the circumferentialdeformation of the structure can be sufficiently projected by the reduced model. In the second part thecomplete stator with a stepwise linear skewing scheme is considered in order to analyze the axial behaviorof the full order and reduced stator structure.

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6.1 Implementation of the mode superposition method and the residual vector cor-rection

The ERP-results for the first six engine orders (12th, 24th, 36th, 48th, 60th and 72nd) for the thin statorsegment are shown in Figure 4. The mode superposition method is based on a modal subspace that containsp = 771 modes and therefore covers the frequency range from 0 Hz to 15 kHz. For the 36th as well asthe 72nd engine order both model reduction methods, the mode superposition with and without residualvector correction, show good accordance to the reference over the whole frequency band. For the 12th, 24th,48th and 60th engine order the results of the mode superposition without residual vectors show a significanterror. Especially the ERP-results of the 24th and 48th engine order deviate from the full order results.Applying the residual vector correction reduces that error significantly. The reason for the large deviationfor the mode superposition without residual vectors is that all four engine orders particularly excite higherradial spatial orders that are not contained in the considered frequency range (see Figure 1). All higherorder eigenmodes that correspond to the the radial component of the spatial force are not located within theconsidered frequency range. Using residual vectors leads to a significant reduction of the error caused by theinsufficient modal subspace.

In Figure 5 the results for the harmonic analysis of the whole stator using the mode superposition with andwithout residual vectors are compared to the results of the full order model. The mode superposition isbased on 3653 modes that covers the frequency range from 0 Hz to 15 kHz. The stator is linearly skewed insix discrete steps. Again the error for the 36th and the 72nd engine order is small for both order reductionmethods. The multiple local maxima of the curve indicate that different axial eigenmodes are excited by thesame engine order. Indeed, the frequencies of the maxima match the natural frequency of the different axialeigenmodes that are illustrated in Figure 2. The error that is generated by the mode superposition for the12th, 24th, 48th and 60th engine order again is large as it was already detected within investigation of the thinstator segment for the same engine orders. Especially for the 24th and 48th engine order the deviation is largewhich indicates that the load vector matrix for these particular load cases again is insufficiently displayed bythe subspace.

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or[d

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60th 72nd36th 48th12th 24th

MORp=50M.Sup. Res.Vec. MORp=100 MORp=20 MORp=10

Figure 3: Maximum error of the mode superposition with and without residual vector correction and theKrylov subspace method for different numbers p of reduction vectors for the skewed stator

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Figure 4: Simulated ERP-results for the first six engine orders of a thin stator segment using the full integra-tion method, the mode superposition, and the mode superposition including residual vectors.

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Figure 5: Simulated ERP-results for the first six engine orders of a skewed stator using the full integrationmethod, the mode superposition, and the mode superposition including residual vectors.

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Figure 6: Simulated ERP-results for the first six engine orders of a thin stator segment using the full inte-gration method and the Krylov subspace method based on different numbers of projection vectors p per loadvector.

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Figure 7: Simulated ERP-results for the first six engine orders of a skewed stator using the full integrationmethod and the Krylov subspace method based on different numbers of projection vectors p per load vector.

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6.2 Implementation of the Krylov subspace method

In this section the Krylov subspace method is used as a model order reduction method. Since the load vectorcontains two independent complex components, namely the tangential and the radial part, the block-Arnoldiprocess needs to be applied. The complex force vector is decomposed into two separate load vectors. Hence,four different independent load vectors are considered. The expansion point s0 is set to so = 0 in eachinvestigation which denotes a frequency equal to Zero. Again the first step of the investigation is the analysisof an axially thin stator segment.

Using the Krylov subspace method no guideline of how many vectors need to be used to produce an accuratereduced model exists. Therefore the number of reduction vectors per load vectors is varied in four discretesteps p = 1, 5, 10, 20. Figure 6 shows the ERP-results for the first six engine orders (12th, 24th, 36th, 48th,60th and 72nd) for the thin stator segment. While the 72nd engine order is accurately projected by only a fewreduction vector at least 20 Krylov vectors per load case are needed to generate a good approximation forthe 36th engine order. For the 12th, 24th, 48th and 60th the reductions based on 10 and 20 Krylov vectorsshow good results up to the resonance frequency at about 5820 Hz and fairly good results up to 10 kHz.The reduction based on a lower number of vectors causes significant errors especially at higher frequencies.Around the expansion point frequency f = 0 all engine orders show good accordance for the minimumnumber of Krylov vectors p = 1 which matches the theory of Section 4.3.

In order to accurately display the dynamic behavior of a skewed stator over a frequency range of 10 kHza larger number of Krylov vectors needs to be used. Hence, the minimum number of reduction vectors ischanged to p = 10. Furthermore p = 20, 50 and 100 are investigated. Since the phase angle of the complexforce vectors of two neighboring stator segments does not vary over time only four independent load casesare needed to build the load vector matrix. Again for all engine orders the dynamic behavior of the statoraround the expansion so = 0 up to 2 kHz is sufficiently displayed by the reduced system based on theminimum number of Krylov vectors p = 10. For higher frequencies the transformation matrix based 10 and20 Krylov vectors per load case causes significant errors. Increasing the number of reduction vectors to 50provides fairly good results for all engine orders over the total frequency range.

In Figure 3 the maximum error values of the different model reduction techniques for the skewed stator aresummarized. The mode superposition without residual vector correction as well as the reduced order modelsbased on 10 and 20 Krylov vectors show significant deviation to the reference model. With increasingnumber of Krylov vectors the error decreases. The maximum error for the 60th engine order using 50 Krylovvectors lays around 4 dB and for 100 Krylov vectors around 1.5 dB. As mentioned before the residual vectorcorrection reduces the maximum error caused by the mode superposition significantly.

In Table 1 the computation times for the different analyses are listed. For all model order reduction techniquesthe time that is needed to compute the reduced model is quantified. For the full model only the time for thedirect integration is listed. Each case of model reduction leads to a reduced solution time of the harmonicanalysis compared to the full order model. Especially using the Krylov subspace method the computationaltime can be reduced significantly. Even if the effort to perform the transformation into the Krylov subspaceis taken into account the method reduces the overall computational time for a harmonic analysis of the thinstator segment by the factor 20 and for the whole stator by eight (based on the maximum considered numberof Krylov vectors).

7 Conclusion

In the paper different model reduction techniques, namely the mode superposition and the Krylov subspacemethod, were introduced and implemented for the simulation of the dynamical structural behavior of a statorof an electric machine. The reduction techniques were compared regarding the accuracy of the results and thecomputational effort. First the different reduction methods were applied to one stator segment and later on afull skewed stator in order to investigate the influence of the axial force distribution on the system behavior.

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Stator segment StatorModel reduction Harmonic analysis Model reduction Harmonic analysis

time [s] time [s] time [s] time [s]

Full 1029 5558.6M. Sup. 829.5 567.5 4936.1 4525.2Res. Vec. 1002.4 575.6 15575 1950.81

MORp=1 6.63 0.35MORp=5 15.1 1.12MORp=10 24.5 2.30 90.8 3.05MORp=20 44.2 3.78 151.7 5.43MORp=50 356.5 10.0MORp=100 707.7 23.3

Table 1: List of the computational time of the different model reduction approaches and harmonic analysesfor the simulation of the thin stator segment and the skewed stator (1Due to restrictions of the FEM-softwareonly 2000 instead of 3653 normal modes were considered within the mode set)

The most frequented model reduction method that is applied in engineering cases is the mode superpositionmethod. It was shown that the electromagnetic forces that act on the stator teeth in radial as well as tangentialdirection excite distinct circumferential eigenmodes that are not necessarily contained in the set of modestaken for the mode superposition. Hence the mode superposition method causes significant simulation errorsfor different load cases. Therefore, multiple correction algorithm can be applied. In this paper the residualvector correction that extends the mode set by static pseudo-modes was investigated. Considering residualvectors within the mode superposition the solution accuracy was improved significantly. However, usingresidual vectors the computational time to generate the reduced model increased considerably.

As an alternative approach the Krylov subspace method based on the Arnoldi process was applied to reducethe structural model. As the number of reduction vectors that leads to good results was not known differentnumbers were analyzed. For the stator segment a sufficient number of Krylov vectors was found to be 20for the skewed stator an increased number of 100 Krylov vectors per load case was found to be a goodnumber as the maximum error for all engine orders laid below 2 dB. The model reduction based on Krylovsubspaces and the solution of the reduced system led to an impressive reduction of the overall computationaltime. For the application of a single stator segment the overall computational time was reduced by the factor20 for the skewed stator by eight. Based on the significant acceleration of the computational performancethat is generated without suffering accuracy the model reduction method using Krylov subspaces is stronglyrecommendable for the structural dynamical analysis of stators of electric machines.

References

[1] J. H. Ginsberg, Mechanical and structural vibrations: theory and applications, Wiley New York (2001).

[2] J. Tedesco, W. McDougal, C. Ross, Structural dynamics: theory and applications, Addison-Wesley.

[3] A. Jeffrey, Advanced engineering mathematics, Academic Press (2001).

[4] Z.-Q. Qu, Model Order Reduction Techniques with Applications in Finite Element Analysis: With Ap-plications in Finite Element Analysis, Springer (2004).

[5] A. Dutta, C. Ramakrishnan, Error estimation in finite element transient dynamic analysis using modalsuperposition, Engineering Computations, Vol. 14, No. 1, (1997), pp. 135–158.

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[6] R. Alvarez, J. Benito, On the use of residual shapes in modal analysis, Earthquake Engineer 10thWorld, Vol. 7, (1992), pp. 3915.

[7] E. B. Rudnyi, J. G. Korvink, Review: Automatic model reduction for transient simulation of mems-based devices, Sensors Update, Vol. 11, No. 1, (2002), pp. 3–33.

[8] W. Arnoldi, The principle of minimized iterations in the solution of the matrix eigenvalue problem, 9,Quart. Applied Math (1951).

[9] A. Krylov, On the numerical solution of the equation by which in technical questions frequencies ofsmall oscillations of material systems are determined, Izvestija AN SSSR (News of Academy of Sci-ences of the USSR), Otdel. mat. i estest. nauk, Vol. 7, No. 4, (1931), pp. 491–539.

[10] O. Nevanlinna, Convergence of interations for linear equations, Springer (1993).

[11] Y. Saad, Iterative methods for sparse linear systems, Siam (2003).

[12] G. Baker, P. Graves-Morris, Pade approximants (Encyclopedia of Mathematics and its Applications vol59), Cambridge University Press, Cambridge (1996).

[13] C. Brezinski, Pade-type approximation and general orthogonal polynomials, Springer (1980).

[14] R. Eid, B. Salimbahrami, B. Lohmann, E. B. Rudnyi, J. G. Korvink, Parametric order reduction ofproportionally damped second-order systems, Sensors and Materials, Vol. 19, No. 3, (2007), pp. 149–164.

[15] L. Timar, P, A. Fazekas, J. Kiss, A. Miklos, G. Yang, S, Noise and vibration of electrical machines,Elsevier (1989).

[16] F. Fahy, Sound Intensity, Elsevier Applied Sciences (1995).

[17] J. Gieras, C. Wang, J. Lai, Noise of polyphase electric motors, CRC Press Taylor & Francis Group(2006).

[18] J. Yang, S, Low-noise electrical motors, Clarendon Press Oxford (1981).

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