mathematical models and nonlinear dynamics of a linear ......3 motor’s physical interactions 3.1...

Nonlinear Dyn (2018) 94:377–396https://doi.org/10.1007/s11071-018-4365-0

ORIGINAL PAPER

Mathematical models and nonlinear dynamics of a linearelectromagnetic motor

Jakub Gajek · Jan Awrejcewicz

Received: 26 January 2018 / Accepted: 22 April 2018 / Published online: 23 May 2018© The Author(s) 2018

Abstract The aim of this paper was to construct amathematicalmodel of a drive systemwith ironless per-manent magnet synchronous linear motor and to ani-malize the influence of the magnetic field distributionfunctionon themodel’s accuracy and ease of simulationcomputation. The studied motor employs an U-shapedstationary guideway with permanent magnets placedperpendicularly to the motor’s direction of motion anda forcer with three sets of rectangular coils subjectedto alternating external electrical voltage. The system’sparameters are both mechanical (number of magnetsand coils, size of magnets, distances between magnets,size of coils) and electromagnetic (auxiliary magneticfield, permeability, coil’s resistance). Lorentz forceallows for the transition from electromagnetic parame-ters tomechanical force, andFaraday’s lawof inductioncreates a feedback between the forcer’s speed and coilsvoltage. An Ampere’s model of permanent magnet isused to determine the function of auxiliary magneticfield distribution throughout the stator. Two simplifieddistribution functions are introduced and studied. Dur-ing validation, external current function is applied toeach coil (serving as excitation), while the displace-ment of forcer in time is the output function. Modelparameters are found via genetic algorithms such thatthe numerical solution of themodel best fits experimen-

J. Gajek · J. Awrejcewicz (B)Department of Automation, Biomechanics andMechatronics, 1/15 Stefanowski Street (building A22),90-924 Łódź, Polande-mail: [email protected]

tal data. Several cases of motor operation are comparedagainst simulation results showing good coincidencebetween computation and experiment.

Keywords Linear motor · Magnets · Coils · Jacobi-function · LuGre model

1 Introduction

It is well known and documented that linear machines,including motors, generators, and actuators, generatemotion/translation directly without rotation and trans-mission conversion devices, and due to their compact,simple and relatively cheep structure with a simulta-neous high dynamic efficiency and performance, theyare extensively applied in various branches of industry.They are employed in manufacturing [1], transporta-tion [2], aerospace industry [3], automation, and roboticindustries [4,5], bioengineering, and other.

However, the proper design of the linear machinesproducing large forces and operating at high speedsrequires advanced mathematical modelling in order toachieve precise control to carry out required perfor-mance. On the other hand, a proper mathematical mod-elling and derivation of the reliable governing dynamicequations requires a comprehensive and self-containedknowledge that span from mechanics, applied mathe-matics, electrical, engineering, and mechatronics witha particular emphasis on nonlinear effects.

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378 J. Gajek, J. Awrejcewicz

In what follows, we briefly describe the state of theart of the problem by exhibition of the derived andemployed simple mathematical models based on fun-damental physical/mechanical lows and supported byexperimental validation.

Karnopp [6] studied linear electromagnetic motorscomposed of coils of a copper wire interacting withpermanent magnets serving as mechanical dampers.In particular, novel coreless design for use with highenergy magnets was proposed.

Compter andFrissen [7] presented an innovative pla-nar motor levitating and propelling platform. It wascomposed of a simple structure with four three-phaseamplifiers. Full 6-DoF single-input-single-output con-trol was developed, and a stable controlled electrody-namical planarmotionwith a long strokewas obtained.

Holmes et al. [8] designed, fabricated, and tested anaxial-flux permanent magnet electromagnetic genera-tor. The latter consisted of a polymer rotor with embed-ded permanent magnets sandwiched between two sil-icon stators. The effects on performance of designparameters (number of layers in the stator coils, therotor stator gap, soft magnetic pole pieces on the sta-tors) were investigated using the finite element simula-tions. The employedmicrofabricated axial-flowmicro-turbine was used to produce a compact power conver-sion device for power generation andflow sensing oper-ations.

Lemarquand and Lemarquand [9] considered thestructures of permanent magnet pickups for an electricguitar with an emphasis put on the nonlinear dynamicsgenerated by the induced electromotive force yieldedby the string, magnet, and coil interactions. The studywas supported by the analytical computations using theCoulombian model of magnets.

Yatchev and Ritchie [10] compared two approachesfor simulation of dynamics of a permanent magnetlinear actuator, i.e. the so-called full coupled model(standard) and the decoupled model (proposed by theauthors). It was illustrated that the decoupled modelof a linear actuator with moving permanent magnetemploying bicubic spline approximations yielded onlythe required accuracy but demonstratedmore flexibilityin comparison with the standard approach.

Hamzehbahmani [11] derived amathematicalmodelfor a singleside short-stator linear induction motorwith the end effects. MATLAB/SIMULINK-basedsimulations were compared with the experimentalresults. The latter included measurement of the equiv-

alent circuit parameters, both free acceleration andlocked primary experiments.

The voice coil motor consisted of a wounded coilin the centre covered with Halbach magnet array withiron yoke was mathematically modelled with Lorentzforce principle by Jeong et al. [12]. Relation betweencoil-passing area and suppressing parasitic motion wasderived and the motor equation was expressed by geo-metric dimensions of coil and magnets, which openeda possibility for its design.

Yao et al. [13] designed based on the mathemati-cal modelling a novel tubular linear machine with 3Dhybrid permanent magnet arrays and multiple movers.The governing equations were derived using the sourcefree property and Maxwell equations. The magneticfield distribution was analytically formulated usingBessel functions and harmonic approximation to themagnetization vector. The obtained analytical solutionswere validated by numerical simulations. In addition,it was demonstrated that the variation of magnetic fluxdensity in the linear machine was consistent with themagnet patterns.

Mathematical modelling and dynamic analysis ofa novel moving magnet linear actuator were carriedout by Hassan et al. [14]. The stator was composedof two reversely wound coils being electrically excitedwith single-phase AC power. The time-dependent cur-rent model of the stator winding was proposed, and thestroke, velocity, and acceleration of the armature werederived. The spring, damping, inertial and magneticforces were estimated. The system dynamics were ana-lytically approximated and experimentally validated.

As it has been already pointed out, the linear motorsare taking on bigger and bigger shares of the mar-ket for precise positioning systems. They provide acostly but dynamically superior alternative to standarddrives such as feed screw conveyors. In order to cor-rectly project a desired motion profile, every position-ing system has to equip a specific closed-loop con-troller between the motor and mains. The quality ofsuch system, and therefore the quality of motion pro-jection, depends on, amongst other things, the precisionof the system elements’ mathematical models used onthe programing stage of building the controller. Mostindustrial controllers today use simplifiedmodels. Thispaper focuses on construction of a novel model for anexemplary positioning system which is different fromstandard available in the existing literature and hashigher level of complexity.

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Mathematical models and nonlinear dynamics 379

This paper is organized in the following way. Sec-tion 3 deals with basic electromagnetic interactionsbetween elements of themotor. It presents a basemodelof a single winding in a vicinity of a C-shaped magnetand reports the values of Lorentz force acting on it aswell as the value of electromagnetic induction. It thendescribes the magnetic distribution function of a singlemagnet with use of Ampere’s model and Biot–Savartlaw. In Sect. 4, a model of a three-phase motor, high-lighting the impact of choosing a proper magnetic fielddistribution function for a guideway with numerousmagnets, is proposed. Two functions and therefore twomodels are presented. The first on simplifies the dis-tribution to a sine function, while the second attemptsfor a more true to life distribution. Flaws of both arenamed, and for the simplified function, a scope of use islisted and explained. Section 5 presents the method ofmodels’ validation introducing final changes to mod-els’ equations. A resistive force of mechanical guide-ways is defined, and friction model is chosen to bestsuit the free-wheeling operations. Then, both models’parameters are identified using genetic algorithm andverified against experiment. Section 6 contains con-cluding remarks.

2 Validation platform

A base for model construction was a laboratory stand,with HIWIN’s coreless linear motor and Copley Con-trols’ servo drive. The same platform is also used formodel validation. The stand’s forcer (inductor), shownas “1” in Fig. 1, allows the load to move at speeds ofup to 5 meters per second and generate a continuousand peak force of 45N and 180N, respectively. Thelength of three magnetic U-shaped guideway (“2” ofFig. 1) allows for a theoretical maximum stroke of 830mm. For safetymeasures, the actualmaximum stroke islimited to 780 mm. Two high-quality linear guideways(denoted by “5” in Fig. 1) provide for a reliable andquiet motion. The position and velocity measurementis done by a Renishaws analogue optical linear encoder(see “3” in Fig. 1). It is able to read its position with aresolution of 0.1 µm, and the entire system’s position-ing error is not trailing far behind. The stand programallows it to work both automatically based on a signalfrom external devices and via manual input. This standhas been designed and originally built [15].

The servo drive allows for a multitude of options tofine tune the motor’s control loops. Amongst them isthe “scope” function, which gives the option to subjectthe motor and the entire system to several functionsof current, velocity, and position including sinusoidalfunction, square function and microstepping. It alsomeasures the actual time graphs of said quantities andallows exporting them to a csv file. In this paper, the“scope” function is used to obtain data employed formodel validation.

3 Motor’s physical interactions

3.1 Base model

In the most basic model of a linear motor, the induc-tor can be represented as a single, perfectly rectangularconductor loop. This loop allows for a certain voltagefunction Ug to be applied to it and cause a current ofvalue I to flow through it. The exactmeans of this appli-cation is unimportant for theworkings of themodel andtherefore will be omitted.

The magnetic guideway provides a source of mag-netic field. A typical U-shaped guideway is composedof several pairs of strongmagnets. Each pair’smagneticfield distribution can be modelled as that of a single C-shaped magnet in a way depicted in Fig. 2

Assuming that the loop cannot deform or rotate andcan only move along the direction of y-axis, it shallalways remain a rectangle with a centre placed on y-axis and with shorter sides parallel to the direction ofmotion.

The describedmodel is a basemodel for the paper. Itshould be noted that the actual linear motor bears manysimilarities to the base model and the only differencesare number and displacement of magnets, number ofconductor loops wound together within a single wind-ing and number and displacement of windings.

This section focuses on the interactions within thebase model, noting the differences and appropriatechanges in the equationswhere the entiremotorsmodelis considered.

3.2 Lorentz force

Every charged particlemoving through amagnetic fieldwith a given speed is subjected to aLorentz force,whichworks perpendicular to both the velocity vector of the

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Fig. 1 The laboratory standused for model verification(1—motor’s inductor,2—magnetic guideway,3—linear encoder, 4—cable chain, 5—mechanicalguideways)

Fig. 2 Model of a singlewinding

particle and magnetic distribution vector. Same forceworks on a conductor placed within a magnetic field.In such a case, the force is perpendicular to the vectorof the current flowing through the conductor [16].

For the base model, only the y-component of theLorentz force is relevant (where y is the axis ofmotion).It can be obtained by calculating a closed-loop integralalong the conductor path. For a specific case of a rect-angular shape, the closed-loop integral can be rewrittenas a sum of two definite integrals in the following form

F = I

⎛⎜⎜⎝

lx2∫

− lx2

B

(x, y0 −

ly2

)dx −

lx2∫

− lx2

B

(x, y0 +

ly2

)dx

⎞⎟⎟⎠ ,

(1)

where B(x, y) is the function of the magnetic field’s z-component distribution, lx , ly are the dimensions of therectangular conductor loop in x an y axes, respectively,

and y0 is the position of the loop relative to centre ofthe magnet [17].

A singlemotorwinding consists ofn of such conduc-tor loops placed a top each other along z-axis. Assum-ing that B ′ distribution has the same values for each ofthe loop, the force for an entire single winding can becalculated as

FM = nI

⎛⎜⎜⎝

lx2∫

− lx2

B

(x, y0 −

ly2

)−

lx2∫

− lx2

B

(x, y0 +

ly2

)⎞⎟⎟⎠ .

(2)

For a motor model, each coil’s magnetic field also actson adjacent windings; however, the sum of this forcesfor the entire motor is equal to zero. The stiffness ofthe inductor disallows movement of windings relativeto each other, and therefore, these forces are omitted inthe model.

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Fig. 3 Distribution ofC-shaped permanentmagnet’s magnetic field(z-axis coefficient) on a lineran parallel to y-axis inbetween the magnet’s poles(highlighted part marks theposition of magnet’s poles)

Fig. 4 Magnetic guidewaymodel

Fig. 5 3D plot of motor’sguideway magnetic fielddistribution in yk-plane(where k = σy

χ)

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Fig. 6 Contour plot ofΔ(k, n) (approximatedabsolute error functiondifferenced over k) with redline highlighting its zerovalues. (Color figure online)

Fig. 7 Elliptic thetafunction distribution for realvalues of z and q

3.3 Electromagnetic induction

When a closed conductor loop moves through densemagnetic field, themagnetic flux inside it changes. Thiscauses anEMF (electromagnetic force) to induce insidethe loop. The value of EMF for the base model can becalculated as follows

�i = − ddt

ly2∫

− ly2

lx2∫

− lx2

B(x, y)dxdy. (3)

where �i is the EMF and lx , ly are the dimensions of therectangular conductor loop in x an y axes, respectively.

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Fig. 8 Shape of theKBΘ(y) function with aconstant magnetsdisplacement χ and varyingk

Fig. 9 Laboratory stand’svelocity profile duringfree-wheeling operationswith three different pusherweights

Similarly for a rectangular winding with n loops it canbe recast to the following form

�i = −n ddt

ly2∫

− ly2

lx2∫

− lx2

B(x, y)dxdy. (4)

The electric current I flowing through the rectan-gular coil with n conductor loops is the result of theexternal voltage function UG(t), the electromagneticforce �i induced in the loop due to external magneticflux change and the electromagnetic force caused byself inductance. It can be calculated by solving the fol-lowing ODE

İ = 1LUG(t) − nL

d

dt

ly2∫

− ly2

lx2∫

− lx2

B(x, y)dxdy − REL

I, (5)

where Re is the electric resistance of the coil and Lis its leakage inductance. For a model with a set ofcoils in close proximity additional EMF resulting fromwindings’ mutual inductance must be taken into con-sideration.

3.4 Magnetic field distribution

In accordance with reference [18], the value of mag-netic field of a permanent magnet can be approximated

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Fig. 10 Laboratory stand’sdisplacement profile duringfree-wheeling operationswith three different pusherweights

Table 1 Parameters obtained during resistive force model iden-tification

Parameter Symbol Value

Coulomb force value FC 1.720 [N]

Static friction value FS 3.198 [N]

Average bristles stiffens σ0 11.69 [N/m]

Damping coefficient of the bristles σ1 92.14 [Ns/mm]

Viscous coefficient σ2 2.517 [Ns/mm]

Stribeck velocity vs 2.993 [mm/s]

Forcer mass m 0.588 [kg]

by using Amperes model, that is, by assuming that amagnets magnetic field is the same as that of a per-fect, tightly wound solenoid, with current IS flowingthrough it. Then, Biot–Savart law can be applied to cal-culate the exact value of magnetic field at any point Pin the vicinity of the magnet via the following formula

B(r) = μ4π

∫

L

ISdl × r′∣∣r′∣∣3

, (6)

where

r′ = r − l, (7)and r is the distance between point P and the centreof the magnet, l is the distance between the magnetscentre and the infinitely small length of the solenoiddl, whileμ is themagnetic permeability of themagnetsenvironment.

In case of a C-shaped magnet, the field can be cal-culated as a resultant fields of two bar magnets placed

perpendicular to each other with opposing poles facingeach other. For a single barmagnet, the field can be thencalculated by solving the following integral equation

B(xp, yp, z p

) = μi4π

(B1

(xp, yp, z p

) + B2(xp, yp, z p

)

+B3(xp, yp, z p

) + B4(xp, yp, z p

)), (8)

where

B1(xp, yp, z p

)

=σz2∫

− σz2

σx2∫

− σx2

(z − z p

)2ŷ + (yp − σy2

)2ẑ

√(xp − x

)2 + (yp − σy2)2 + (z p − z

)23dxdz,

(9)B2

(xp, yp, z p

)

=σz2∫

− σz2

σx2∫

− σx2

(z − z p

)2ŷ + (yp + σy2

)2ẑ

√(xp − x

)2 + (yp + σy2)2 + (z p − z

)23dxdz,

(10)B3

(xp, yp, z p

)

=σz2∫

− σz2

σy2∫

− σy2

(z − z p

)2x̂ + (xp − σx2

)2ẑ

√(xp − σx2

)2 + (yp − y)2 + (z p − z

)23 dydz,

(11)B4

(xp, yp, z p

)

=σz2∫

− σz2

σy2∫

− σy2

(z − z p

)2x̂ + (xp + σx2

)2ẑ

√(xp + σx2

)2 + (yp − y)2 + (z p − z

)23 dydz,

(12)

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Fig. 11 Six resistive force model solutions (black line), compared with down-sampled experimental data (blue points). (Color figureonline)

and(xp, yp, z p

)are the coordinates of point P,

(σx , σy,

σz)are the bar magnet dimensions in their respective

axes, x̂, ŷ, ẑ are the unit vectors of axes x, y and z, andi is the current density over the sides of the coil.

Figure 3 shows the C-shaped magnets magneticfields z-component distribution in a in y-axis (forz = 0) calculated with Eqs. (8)–(12) for a samplemagnet.

For the ease of use and because of the good coinci-dence with actual values, the C-shaped magnet’s mag-netic fields distribution on a xy-plane (coordinate sys-tem set as in Fig. 2) will be approximated with 2DGaussian function in the following form

BS = B0e−

(x2

σx 2+ y2

σy2

)

, (13)

where B0 is the magnet’s given constant.

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Fig. 12 Measured current values during experiment

Fig. 13 Interpolated current functions used as model excitationduring identification

Table 2 Parameters obtained during sine model identification


Force constant K f −133.8 [N]Guideway density factor ω 0.222 [1/mm]

Windings displacement factor αp 0.5368 [-]

Shift parameter αS 1.936 [-]







4 Stand’s model

The laboratory stand described in Sect. 2 can be mod-elled as an 4-DOF oscillator in which the pusher’s dis-placement and phase currents are the output functionsand three external voltages serve as excitation. A three-phase coreless linear motor translates the voltages into

pushing force and the guideways and air friction createa countering resistive force. The oscillator is governedby the following ODE

mÿ =3∑j=1

FMj(UGj , ẏ, y + ( j − 2)yp

)

−R (ẏ, y) . (14)

where y, ẏ, ÿ are, respectively, the position, velocity,and acceleration of the pusher, R is the resistive forceof the guideways, FMj are the force functions of eachof themotor’s winding governed by Eqs. (2) and (3), y jis the distance between two adjacent phases, and UGjare the voltages on each winding.

As mentioned, the stands magnetic guideway con-sists of several dozen C-shaped neodymium magnets.They are placed in such a way that each two neighbour-ing magnets have their poles placed oppositely and thatthe distance between magnets is the same for each pairand equal to χ (see Fig. 4).

For such a guideway, the magnetic field distributionfunction B(x, y) in Eq. (2) is a superposition of eachindividual magnet’s field and can be calculated as asum of magnetic distribution functions of every mag-net. It should be note that the physical pusher, due tomechanical limitations, can only move in such a waythat all the windings are enclosed inside the guideway.Therefore, the model will not loose any of its accuracyif an infinitely long guideway model is used, bearingin mind that the actual stands stroke is still limited andevery responses with a pushers displacement greaterthan the stroke of the stand is physically unachievablein real life.

The infinitely long magnetic guideway field distri-bution can be calculated as a sum of distribution frominfinitely many magnets, that is

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B(x, y) =∞∑

i=−∞(−1)i BS (x, (y + iχ))

=∞∑

i=−∞(−1)i B0e

−(

x2

σx 2+ (y+iχ)2

σy2

)

. (15)

However Eq. (15) is cumbersome and provides sig-nificant complications for use in the stands ODE, mak-ing it highly nonlinear and difficult for numerical sim-ulations. It can, however, be casted and simplified toa more computation friendly form. This paper willpresent two twinmotormodels, built based on Eq. (15).

4.1 Motor with sinusoidal field distribution

For different values of σy and χ , the shape of the curvedescribed by Eq. (15) changes significantly along they-axis. The x-axis distribution is only dependent on theσx and B0 parameters. Let the distribution function befixed in x = 0. In such case, the one argument functionof distribution is given as follows

B(y) = B(0, y) =∞∑

i=−∞(−1)i B0e

−(

(y+iχ)2σy2

)

. (16)

Let there be a correlation between σy and χ givenin a form of a dimensionless parameter k such that

k = σyχ

. (17)

In such case, the shape of themagnetic field distribu-tion depends solely on the value of k. Figure 5 presentsthe magnetic field distribution in y-axis for varying k.The obtained plot affirms that B is periodic for everystudied value of k (where k = 0.5 means that thereis no free space between adjacent magnets and k = 0means that magnets are infinitely apart.) For lower kvalues, magnetic distribution from individual magnetsis more prominent and typical Gaussian distributionpeaks can be observed with large ranges of y for whichB ≈ 0 in between. Larger k produces a distributionwhere individual magnetic fields are distorted by adja-cent magnets producing a net distribution similar to asine wave.

It can easily be seen that for a certain value of k thedistribution of B can be approximated with satisfactoryprecision by a cosine function in the form of

BC (y) = B0 cos(

π

χy

). (18)

The inaccuracy of such an approximation, at a givenpoint along the guideway, is the square of differencebetween values of magnetic distributions given by Eqs.(16) and (18) and is therefore a function of y and k. Asboth functions rely on the magnitude factor B0 in asimilar fashion, the relative error in the form of

�R(y, k) =⎛⎝

∞∑i=−∞

(−1)i e−(

(y+iχ)2k2χ2

)

− cos(

π

χy

),

⎞⎠2

(19)

can be used to measure the quality of cosine approx-imation in any given point. The absolute error of thecosine approximation is a function dependent solelyon k coefficient. It can be obtained by calculating thesum of errors from each point along the guideway. Forcontinuous function �R and a theoretical infinitely longguideway, this sumbecomes an improper integral in theform of

�A(k) =∞∫

−∞�R(y, k)dy. (20)

In this paper, we assume that the scope of use forcosine approximation is reserved only to the motormodel instances where �A achieves a minimum value.The absolute error is theoretically a function of both kand χ ; however, in case of the latter it can only achieveminimum when χ approaches infinity, i.e. for a singleinfinitely longmagnet. Because of that, only the impactof k on the error function is analysed.

The �A function takes minimal values for k = k0,where k0 is a value for which the first derivative of �Ais equal to 0 and the second derivative is greater then0. Using the Leibniz integral rule [19, Chapter 8], thesum rule for integration and the value for the square ofsum, the value of the first derivative can be obtained inthe form of

ΔI (k) = 2χ∞∑

i=−∞�A2(k)

+χ∞∑

i=−∞

∞∑j=−∞

�A3(k, i, j), (21)

where

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Fig. 14 Comparisonbetween simulation andexperimental data duringmodel identification

�A2(k) = 12e−

14 kπ

2(k2π2 − 2

)√π, (22)

�A3(k, i, j) = (−1)i+ j e−(i+ j)22k2

[(i − j)2 + k2

]√π2

.

(23)

It can be noted that for nonzero χ , as is the case withphysical linear motors, the roots of ΔI do not depend

on its value. The sums in Eq. (21) do not convergeto a finite value; however, the result of Eq. (21) can beapproximated numerically after substituting the infinitesummations to a finite ones with boundaries of−N andN , where N is a natural number. The larger the N, thelonger the computation timeneeded to approximate andthemore accurate the approximation. For every N thereexists a k = kN meeting the following criteria

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Δ(kN , N ) = 2N∑

i=−N�A2(k)

+N∑

i=−N

N∑j=−N

�A3(k, i, j) = 0. (24)

With the results of several numerically computedkN for different N , an interpolation functions kI (n),having all positive real values in its domain, can beconstructed. The interpolated function has to have theproperty that

∧N∈N kI (N ) ≈ kN and limn→∞ kI (n) =

kI0 ≈ k0. Figure 6 presents a contour plot for aΔ(n, k)function (with the values for n /∈ N interpolated fromtwo adjacent point for the sake of graph clarity). Thered line on the graph joins points whereΔ reaches zero,i.e. points with coordinates (kN , N ). As best fitting theresults, the kI (n) function should take an exponentialform of

kI (n) = a0 − a1ea2na3 , (25)where the values of parameters a0, a1, a2, a3 can beobtained with the least squares method. They are equalto: a0 = 0.4594, a1 = 0.1547, a2 = 0.4118, a3 =0.4077. The limit of the exponential interpolation func-tion is equal to limn→∞ kI (n) = a0; therefore, thescope of use for sine model is motors with the correla-tion factor k equal to 0.4594.

The same can be proven true for a 2D function ofmagnetic distribution given by Eq. (15) and a 2D cosineapproximation function in the form of

BC (x, y) = B0e−(

x2

σx 2

)cos

(π

χy

). (26)

The absolute error of the 2D approximation, equalto

�2A(k) =∞∫

−∞

∞∫

−∞B0

2e−

(x2

σx 2

)

×⎛⎝

∞∑i=−∞

(−1)i e−(

(y+iχ)2σy2

)

− cos(

π

χy

)⎞⎠

2

dydx,

(27)

after integrating over x and dividing over the constants,reduces to the form given by Eq. (20). Therefore, theapproximation is valid for the same values of k.

For motors matching that criteria, the integrals inEqs. (2) and (3), i.e. values of the Lorentz force and

EMF, can be analytically calculated as

FM = K f sin(

πy

χ

)I, (28)

�i = K f sin(

πy

χ

)ẏ, (29)

where K f is the winding’s force constant, equal to

K f = 2nB0σx√π erf(

lx2σx

)sin

(lyπ

2χ

). (30)

By introducing Eq. (28) to the stand’s model’smotion’s equation (Eq. (14)) and calculating the val-ues of currents on each coil with Eqs. (5) and (29), thefollowing system of equations for the laboratory standis obtained

mÿ =3∑j=1

K f sin (ωg y + αpj )I j − R (ẏ, y) ,

İ j = KUUGj (t) − KV sin (ωg y + αpj )ẏ− 1

KeI j for j ∈ {1, 2, 3} .

(31)

In Eq. (31),UG j is the external voltage function andKV , Ke, KU , ωg and αpj are the motor’s back EMF,electrical time and voltage constants and guidewaydensity, windings displacement factors, respectively,defined as

KV =K fL

, Ke = LRE

, KU = 1L ,

ωg = πχ

,

αpj =πypχ

( j − 2) = αp( j − 2) for j ∈ {1, 2, 3, },

(32)

where L and RE are the leakage inductance and elec-trical resistance of each winding.

4.2 Motor with elliptic theta function field distribution

Another method of using Eq. (15) in laboratory standsODE is to insert it directly, but after enclosing the infi-nite sum of Gaussian distributions inside a single, morecomputational friendly function. This can be done byfirst ridding Eq. (15) of the (−1)i and instead split-ting the sum into the difference of two other sums,each of them representing the resultant field distribu-tion of same faced magnets in the guideway. The resul-tant sums can be recast to a form in which a singleJacobi theta functions of the third kind is used for eachof them.

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Fig. 15 Validation of identified model against experimental data

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Fig. 16 Comparisonbetween simulation andexperimental data duringmodel identification

This paper follows the Whittaker and Watson [20,Chapter 22] notation of elliptic functions; therefore, theϑ3 symbol will be defined as

ϑ3(z, q) =∞∑

n=−∞qn

2e2n i z . (33)

Although the elliptic theta is a complex function,while both the argument z and nome q are complexnumbers (with q restrained only to the upper half-

plane), it becomes real if the argument and nome arereal and when q ∈ (−1, 1). Figure 7 presents theplot of ϑ3 for real z values and for different values ofq.

Equation (15) can then be presented in the form

B(x, y) = KBB0e−x2

σx 2 (ϑ3 (z0y, q0)

−ϑ3(z0y − π

2, q0

)), (34)

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where

KB =√

π

2|k|,

z0 = π2χ

,

q0 = e− 14 k2π2 .

(35)

It can be seen that the nome is the same for both thetafunctions and is a function of the correlation parameterk [as defined in Eq. (17)]. The z0 parameter, whichaffects the frequency of the Jacobi functions, is alsoequal for those functions. The π2 component acts as aphase parameter shifting the negatively signed functionby exactly half of its period. An auxiliary function Θcan be introduced to the equation for greater clarity.This function is defined as follows

Θ(z, q) =(ϑ3 (z, q) − ϑ3

(z − π

2, q

)). (36)

Figure 8 presents the plot of KBΘ(z0y, q0) func-tion for different values of correlation parameter k andconsequently different values of q. It bares a very closeresemblance to Fig. 5 as, indeed, mathematically theonly difference between Eqs. (16) and (36) is in thefield’s magnitude parameter B0.

Similarly, aΘ0(z, q) function, used in stand’smodelconstruction to incorporate the size of coils, can bedefined as

Θ0(z, q) = Θ (z − zL , q) − Θ (z + zL , q) , (37)where zL = ly z0/2 is a winding length factor for amotor with windings’ sizes of lx , ly .

By introducing Eqs. (34) and (37) to laboratorystand’s equation of motion (Eq. (14)) and the currents’equation (Eq. (5)) as well as defining the motor’s thetaforce constant, theta back emf constant, and windingsdisplacement factor in the form of

K ′f = B0KBn√

πσx erf

(lx2σx

),

K ′V =K ′fL

,

z j = z0yp( j − 2) f or j ∈ {1, 2, 3, },

(38)

the governing ODE take the form of

mÿ =3∑j=1

K ′f Θ0(z0y + z pj , q0

)I j − R (ẏ, y) ,

İ j = KUUGj (t) − K ′VΘ0(z0y + z pj , q0

)ẏ

− 1Ke

I j for j ∈ {1, 2, 3} .

(39)

5 Model identification and validation

Following their construction, both models were vali-dated against the physical motor. However, the stand’sservodrive disallows subjectingmotors to voltage func-tions directly and instead provides current functionsas the source for external excitation. Because of that,the identification and validation was not done using aknown voltage function (such as a sinusoidal or squarewave), but instead by measuring both the currents ineach phase [I j in Eqs. (31) and (39)] and the dis-placement of motor y. The models were then suppliedwith the measured current functions, and their param-eters were found, using genetic methods, such that thedifference between the values of displacement for themodel and physical motor was minimal. With param-eters found, several numerical simulations and exper-imental measurements were taken to further validatethe models. It has to be noted here that the modelsneed additional shift parameter to include the differ-ence between the position y = 0 for the servo driveand for magnetic field distribution functions Eqs. (34)and (18).

5.1 Resistive force

Prior to identifying the model’s parameters, the natureof system’s resistive force should be discerned. In orderto do so, the free-wheeling method was employed. Thestand’smotorwas cut from any external excitation volt-age and windings were kept electrically separated so toavoid the induction of EMF. The pusher was then man-ually accelerated to a fixed speed and released allowingit to move solely on inertia. By incorporating the standsservo drive in the measurement process, a precise val-ues ofmotor’s free-wheeling times as well as speed anddisplacement functions during that time were obtained.To further enhance the accuracy of collected data, thesame process was repeated for different initial speedsand with additional weights added to the pusher. Fig-ures 9, and 10 show a sample velocity and positiongraphs for three free-wheeling operations with differ-ent pusher weights.

It can be noted that unlike the position graph thevelocity profile is significantly jagged and overshotappears to manifest near the end of motion (especiallynoticeable during free-wheelingwith 1 kg addedmass).Obviously, those phenomena should not be present dur-ing free-wheeling with no closed-loop control. Instead,

123


they are the result of the servo drive’smeasurement sys-tem specifics and the fact that there exists no method ofaccurately measuring the linear motor’s velocity. Theresults shown in Fig. 9 are actually numerically calcu-lated derivatives of motor’s displacement (Fig. 10), assuch they carry substantial noise and cannot be used inidentification or validation processes. They were nev-ertheless included for the sake of completion.

As observed on the graphs, the behaviour of the lab-oratory stand during free-wheeling is typical of sys-tems with high dry friction component. The experi-ments proved that the guideway was homogeneous andtherefore that the resistive function R was independentform displacement y. Following [21] the LuGre dryfriction model in the form of

R (ẏ.ẇ, w) = σ0w + σ1ẇ + σ2 ẏ (40)was incorporated as a resistive force model [22]. Hereσ0, σ1, σ2 represent, respectively, the stiffness, micro-damping and viscous aspect of the resistive force andwis the internal friction state [the symbol w was choseninstead of more widely used z to avoid confusion withparameters in Eqs. (33)–(39)]. The value of w can beobtained by solving a differential equation in the formof

ẇ = ẏ − σ0 |ẏ|g(ẏ)

, (41)

where g(ẏ) is a function of velocity used to incorporatetheStribeck effect andCoulomb friction into themodel.It is defined as

g(v) = FC + (FS − FC )e|v/vs |α . (42)The values of FS and FC represent the values of stic-

tion force and Coulomb friction force, while parametervs determines how quickly does the value of functionswitch from FS to FC . According to references [23–25], the values of α should be in the range between 0.5and 2. In this paper, α = 2 was chosen and used forboth identification and validation.

The parameters in Eq. (43) were identified using thefollowing free-wheelingmodel for the laboratory stand

ẇ = ẏ − σ0 |y|FC + (FS − FC )e|ẏ/vs |2

,

ÿ = −σ0w + σ1ẇ + σ2 ẏm + mA ,

(43)

where m is the forcer’s mass and mA is the knownvalue of the mass added during free-wheeling. As withall other models presented in this paper, the parameters

Table 3 Parameters obtained elliptic theta model identification


Force constant K f 80.63 [N]

Guideway density factor z0 1.047 [1/mm]

Winding length factor zL −1.376 [-]Windings displacement factor z p 1.513 [-]

Shift parameter zS 0.7680 [-]

Theta function nome q0 0.493 [-]







were found using genetic algorithms. Table 1 shows thevalues of those parameters, and Fig. 11 shows samplevalidation of the free-wheeling model.

5.2 Sine model

By incorporating the obtained resistive force functionin Eq. (31) and adding the shift parameter in the formof αS , the sine model is obtained. It is governed byequations

ẇ = ẏ − σ0 |y|FC + (FS − FC )e|ẏ/vs |2

,

ÿ = −K fm

3∑j=1

sin (ωg y + αpj + αS)I j (t)

− σ0w + σ1ẇ + σ2 ẏm

.

(44)

During identification, the excitation functions I1(t),I2(t) and I3(t) were constructed by setting the ampli-fier work mode to “Current sine function generator”and measuring actual values of current (as shown inFig. 12) along with values of displacement and veloc-ity. These values were then converted to a continuousfunction using the spline interpolation [26]. Figure 13shows all three excitation functions. The model wasthen identified in similar manner as with the resistiveforce function by using a genetic algorithm. The com-parison between best acquired simulation result andexperimental data during identification is displayed inFig. 15. Table 2 presents identified parameters value,

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Fig. 17 Validation of identified model against experimental data

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and Fig. 14 shows examples of comparison betweensimulation and experimental data for four model casesof motor operation. It has to be noted that parametersof resistive force [Eq. (40)] were identified separatelyfor the sine model and their values slightly differ fromthose listed in Table 1.

As can be seen in Figs. 14 and 15, the model withidentified parameters fits closely to experimental datafor most cases of closed-loop working conditions andwith slowly changing sine current functions. However,during sudden and rapid changes of current value, aswas the case with square wave current excitation func-tion, the model exhibited strong oscillations of velocityand displacement not present in the physical counter-part.

The identified force constant parameter has nega-tive value, which means that the phase order in math-ematical model was inverted in comparison with theactualmotor. It should also be noted that despite a slightchange of resistive function parameters the obtaineddisplacement and velocity functions for free-wheelingoperations still very closely resemble the experiment(Fig. 16).

5.3 Elliptic theta model

The second tested model was identified in a very sim-ilar pattern. Firstly, a shift parameter zs and resistiveforce were incorporated into model’s ODE (Eq. (39))resulting in the following final model’s equations

ẇ = ẏ − σ0 |y|FC + (FS − FC )e|ẏ/vs |2

,

ÿ = K′f

m

3∑j=1

Θ0(z0y + z pj + zs, q0

)I j (t)

− σ0w + σ1ẇ + σ2 ẏm

.

(45)

During identification, the current functions I j (t)were chosen and defined in the same manner as inSect. 5.2 and are represented in Figs. 12 and 13. As inprevious cases, genetic algorithm was used during theprocess. Figure 14 presents the obtained results, whilevalues of model parameters are shown in Table 3. Sameas with sine model, Fig. 17 presents examples of com-parison between four experiment and simulation datafor four cases of motor operation (experiment data andexcitation functions are exactly the same as with sinemodel presented in Fig. 15).

The simulation results provide close resemblance toexperiments for all studied cases of closed and open-loopworking conditions. In case of square current exci-tation, both the physical motor and simulation exhibitsimilar square wave oscillations; however, the experi-ment shows a slight drift of displacement values in neg-ative direction, while the simulation continues oscil-lates around a single point. One possible explanation ofthis is that the stand tends to exhibit slightly lower fric-tion for high negative values of velocity. In case of sinewave position excitation, the model exhibited an unex-pected drop and peak in velocity during first momentsof operation; it was, however, brief and did not greatlyaffect the desired position function shape and values.As with sine model, the dry friction parameters werechanged compared to previous model; however, thisdid not adversely affect themodel during free-wheelingoperation. The negative value of zL demonstrates that,as with sine model, the phases for motor were choseninversely.

6 Conclusions

The proposed model of coreless linear three-phasemotor was constructed with respect to Lorentz forceand electromagnetic induction. The functions of mag-netic induction were assumed as a sum of infinitelymany single magnets, while the magnets individualfields distributionwas calculated usingAmperesmodelof permanentmagnets andBiot–Savart law.Thenumer-ically calculated distribution between two bar mag-nets was then superseded with a single Gaussian func-tion. Two methods of calculating the sum from manymagnets were proposed. First of them uses the sineapproximation, and second provides exact values andincorporates Jacobi elliptic theta functions. For certaincases, with appropriate relationship between magnetssize, distances between magnets and number of mag-nets (as shownby the red line inFig. 6) the two solutionsprovide nearly identical values of field distribution. Inaccordance with these two methods, two mathematicalmodels of themotorwere created. Preceding their iden-tification and validation, a resistive force function waschosen as a dry friction force and model with LuGremethod. Both models were correctly identified and val-idated. In most studied cases, the results of simulationsmatched closely to experiment for both models; how-ever, during operationswith sudden and abrupt changes

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to current value the sine model produced significantlyworse results. On the other hand, while the elliptic thetafunction model provided a more accurate results, thesine model’s results were smoother and resembled theexperiment better during sine wave position excitation.The value of theta function nome q0 calculated dur-ing identification suggests that motor was indeed con-structed in a manner in which sine approximation netssimilar results as actual magnetic field values.

A huge differences between models were observedin both computation time and ease of finding the neces-sary parameters. On average, a single simulation of themodel with elliptic theta field distribution took ten totwenty times longer than sine model. At the same time,finding satisfactory convergent solution took over tenthousand passes of genetic algorithm (so-called gen-erations) for sine model, while elliptic theta modelrequired only two thousand.

Both models can be used depending on requiredtask. The elliptic theta model requires more compu-tation time, but provides more accurate solution forall cases. The sine model works well for usual motorapplications. It provides results faster and requires lessparameters to identify, but falls short during atypicaloperations.

Open Access This article is distributed under the terms of theCreative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/),whichpermits unrestric-ted use, distribution, and reproduction in any medium, providedyou give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons license, andindicate if changes were made.

References

1. Shukor, A.Z., Fujimoto, Y.: Permanentmagnet linearmotorsused as variablemechanical dampers for vehicle suspension.IEEE Trans. Industr. Electron. 61, 1063–1071 (2014)

2. Hellinger, R., Mnich, P.: Linear motor-powered transporta-tion: History, present status, and future outlook. Proc. IEEE97, 1892–1900 (2009)

3. Musolino, A., Rizzo, R., Tripodi, E.: The double-sided tubu-lar linear induction motor and its possible use in the electro-magnetic aircraft launch system. IEEE Trans. Plasma Sci.41, 1193–1200 (2013)

4. Slamani, M., Nubiola, A., Bonev, I.A.: Effect of servo sys-tems on the contouring errors in industrial robots. Trans.Can. Soc. Mech. Eng. 36(1), 83–96 (2012)

5. Kwak, H.-J., Park, G.-T.: Study on the mobility of servicerobots. Int. J. Eng. Technol. Innov. 2(1), 97–112 (2012)

6. Karnopp, D.: Permanent magnet linear motors used as vari-ablemechanical dampers for vehicle suspension. Int. J.Vehi-cle Mech. Mobil. 18(4), 187–200 (1989)

7. Compter, J.C., Frissen, P.C.M.: Electro-dynamic planarmotor. In Proceedings of Mechatronics, pp. 81–85. Univer-sity of Twente (2002)

8. Holmes, A.S., Hong, G., Pullen, K.R.: Axial-flux permanentmagnet machines for micropower generation. J. Microelec-tromech. Syst. 14(1), 54–62 (2005)

9. Lemarquand, G., Lemarquand, V.: Calculation method ofpermanent magnet pickups for electric guitars. IEEE Trans.Magn. 43(9), 3573–3578 (2007)

10. Yatchev, I., Ritchie, E.: Simulation of dynamics of perma-nent magnet linear actuator. World Academy of Science,Engineering and Technology, 42:5 (2010)

11. Hamzehbahmani, H.: Modeling and simulating of singleside short stator linear induction motor with the end effect.J. Electr. Eng. 62(5), 302–308 (2011)

12. Jeong, J., Kim, M.-H., Park, J., Gweon, D.: Mathematicalmodeling and design considerations for designing voice coilmotor. In: InternationalConference onMechanics,Materialsand Structural Engineering (ICMMSE 2016), pp. 97–101.Atlantis Press (2016)

13. Yao, N., Yan, L., Wang, T., Wang, S.: Magnetic flux dis-tribution of linear machines with novel threee-dimensionalhybrid magnet arrays. Sensors 17, 2662 (2017)

14. Bijanzad, A., Lazoglu, I.: Dynamic analysis of a novel mov-ing magnet linear actuator. IEEE Trans. Industr. Electron.64(5), 3758–3766 (2017)

15. Gajek,J.: Construction of a laboratory stand based on thepvc glazing beads saw feeder design. Master’s thesis, LodzUniversity of Technology (2012)

16. Awrejcewicz, J., Lewandowski, D., Olejnik, P.: Dynamicsof Mechatronics Systems: Modeling, Simulation, Control,Optimization and Experimental Investigations. World Sci-entific Publishing Co (2016)

17. Chari, M.V.K.: Finite Elements in Electrical and MagneticField Problems. Wiley, Somerdset (1980)

18. Derby, N., Olbert, S.: Cylindrical magnets and idealsolenoids. Am. J. Phys. 78(3), 229–235 (2010)

19. Potter, M.H., Morrey, ChB: Intermediate Calculus, 2nd edn.Springer, New York (1985)

20. Whittaker, E.T., Watson, G.N.: A Course of Modern Analy-sis, 4th edn. Cambridge University Press, Cambridge (1996)

21. Felix, J.L.P., Balthazar, J.M., Brasil, R.M.L.R.F., Pontes Jr.,B.R.: On lugre friction model to mitigate nonideal vibra-tions. J. Comput. Nonlinear Dyn. 4(3), 034503 (2009)

22. Åström, K.J., Canudas de Wit, C.: Revisiting the lugre fric-tion mode. IEEE Control Syst. Mag. 28(6), 101–114 (2008)

23. Tustin, A.: The effects of backlash and of speed-dependentfriction on the stability of closed-cycle control systems. J.Inst. Electr. Eng. 1 Gen. 94, 143–151 (1947)

24. Bo, L.C., Pavelescu, D.: The frictionspeed relation and itsinfluence on the critical velocity of the stick-slip motion.Wear 82(3), 277–289 (1982)

25. Armstrong-H Élouvry, B.: Control of Machines with Fric-tion. Kluwer Academic Publishers, Moston (1991)

26. Nievergelt, Y.: Splines in single and multivariable calculus.In: Campbell, P. J. (ed.) UMAP modules: tools for teaching1992, pp. 39–101. COMAP (1993)

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Mathematical models and nonlinear dynamics of a linear electromagnetic motorAbstract1 Introduction2 Validation platform3 Motor's physical interactions3.1 Base model3.2 Lorentz force3.3 Electromagnetic induction3.4 Magnetic field distribution

4 Stand's model4.1 Motor with sinusoidal field distribution4.2 Motor with elliptic theta function field distribution

5 Model identification and validation5.1 Resistive force5.2 Sine model5.3 Elliptic theta model

6 ConclusionsReferences

mathematical models and nonlinear dynamics of a linear ......3 motor’s physical interactions 3.1...

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