The Simulation of Electromagnetic Suspension System Based on the Finite Element Analysis

Transcription

1 308 JOURNAL OF COMPUTERS, VOL. 8, NO., FEBRUARY 03 The Smulaton of Suspenson System Based on the Fnte Element Analyss Zhengfeng Mng Shool of Eletron & Mahanal Engneerng, Xdan Unversty, X an, Chna Emal: mngzf@xdan.edu.n Tao Wen and Tuo Chen Natonal Key Laboratory of Antennas and Mrowave Tehnology, Xdan Unversty, X an, Chna Emal: wt00537@6.om, kvty@63.om Abstrat In ths artle, the large ar gap eletromagnet suspenson system model s aurately dvded nto the mesh by the method of the fnte element analyss, then the eletromagnet fore of the suspended matter n the system s alulated. Fnally, the fttng urves of the expermental data of a new knd of dstrbuted eletromagnet array system an be obtaned through the least squares method and the fttng expressons of dfferent model system n the large ar gap are aheved. Index Terms eletromagnet suspenson system; fnte element analyss; large ar gap I. INTRODUCTION The fnte element analyss method s a numeral method solvng the numeral and physal boundary value approxmately. Its development experene s onssted of Node Fnte Element Method (N-FEM) and Edge-Based Fnte Element Method (EB-FEM). The soalled Node Fnte Element Method uses salar nterpolaton funton as a base funton, but the great nonvenene of dealng wth ondutor and medum edge horn s the bggest weakness of ths method. Thus the vetor fnte element method s proposed, whh gves the freedom to the edge nstead of node. The ore dea of fnte element s the dsretzaton of the struture. Frstly, a real struture s magnarly dsretzed nto a fnte number of nerrat ombnaton unts. Seondly, the physal parameters of the atual struture an be alulated by the analyss of the dsrete body. Fnally, the approxmate result that meets the engneerng auray s obtaned and replaes the analyss of the atual struture. As one of the popular advaned tehnology n reent years, the tehnology researh s reevng more and more attenton [] [] [3]. In ths paper, several knds of mathematal models of the large ar gap eletromagnet levtaton system have been gven. On the bass of the theory of fnte element analyss and the Maxwell's equatons, the spae magnet feld ntensty s analyzed and alulated. The fnte element analyss s appled to the eletro-magnet suspenson tehnology, and we get the spae magnet feld ntensty and alulate several hange trends of eletromagnet fore under the large ar gap. Fnally, the method of least squares s used to get the relevant fttng mathematal expressons. II. THE FINITE ELEMENT ANALYSIS THEORY AND ELECTROMAGNETIC SUSPENSION SYSTEM A. The Fnte Element Analyss Theory The fnte element analyss method s a numeral method based on varaton prnple approxmate nterpolaton and dsretzaton. Frstly, the boundary value problem to be solved s onverted nto the orrespondng varaton problem and the varaton problem s onverted nto the extremal problem of general multvarate funton. Fnally, an algebra equaton s obtaned, the soluton of whh s a numeral soluton. The most mportant step of the fnte element analyss s the regonal dsretzaton, whh wll nfluene the omputng tme, the requrements of omputer memory and the numeral results preson. For the three dmensonal (3D) problem, the most ommon unt s the tetrahedron whh s the smplest and most sutable for dsrete arbtrarly volume area, as shown n Fg.. After dsretzaton of the volume area, a sutable bass funton s needed to approxmately express the unknown funtons n eah unt whh manly nludes Node Fnte Element Method (N-FEM) and Edge-Based Fnte Element Method (EB-FEM) (whh s ommonly used now). For the vetor funton w, j, we have w = λ λ λ λ (), j j j 03 ACADEMY PUBLISHER do:0.4304/jp

2 JOURNAL OF COMPUTERS, VOL. 8, NO., FEBRUARY where, j are the vertexes. Sne a tetrahedron has four vertexes, the volume oordnates of the tetrahedron an be desrbed as λ = a + bx + y+ dz () where b d a x x x3 x4 b d a y y y3 y 4 (3) = b3 3 d3 a 3 z z z3 z 4 b4 4 d4 a4 TABLE I. EDGE DEFINITION OF TETRAHEDRON UNIT Edge Start node End node Fgure. Tetrahedron unt of the fnte element dvson The followng result an be obtaned w = 0, w = λ λ (4) Assume that e represents a unt vetor from node to node, λ s the lnear funton that hanges from (node ) to 0 (node ) and λ s the lnear funton that hanges from (node ) to 0 (node ), therefore e λ =, e λ = (5) l l where l represents the length of the edge onnetng nodes and, so λ+ λ e w = = (6) l l It means that w has a onstant tangental omponent along the edge (, ). As a vetor base funton of an edge feld related to the edge (, ), w has all the neessary features. If the edge s defned as edge, we have N = wl = ( λ λ λ λ) l (7) Smlarly, the vetor base funton of the edge s desrbed as below N = w l ( ) = λ λ λ λ l (8) where the defntons of edge number, node and node are shown n Fg. and Table I. So the vetor feld of the unt an be dvded nto 6 e e e E = NE (9) = e where E represents the feld of the unt, N e represents the vetor base funton of the unt and E e represents the tangental felds along the edge. The fnal results of fnte element formula s a group of lnear equatons, whh an be desrbed as ax + ax + a3x a nxn = b ax + ax + a3x anxn = b (0) a3x + a3x + a33x a3nxn = b3... an x+ anx+ an3x annxn = bn whh an be redued as Ax= b () where A s a oeffent matrx of sze n n, x s an unknown quantty and b s a known vetor. For 3D eletromagnet feld problem, the dmenson of the oeffent matrx A nreases rapdly wth the nrease of solvng regon and A s a sparse matrx. B. The Bas Prnple of Suspenson System The edge method s used to solve the stat magnet feld, whh s usng the freedom of dvson on the unt. The bas Maxwell's equatons of 3D stat magnet feld are shown as follow H ( x, y, z) = J( x, y, z) () Bxyz (,, ) = 0 B = μh where B( xyz,, ) s the magnet flux densty, H( x, y, z ) s the magnet feld ntensty and J ( x, y, z ) s the urrent densty. The thrd equaton desrbes the onsttutve relatonshp between magnet feld ntensty and the magnet flux densty, where μ s the deletr magnet permeablty. The ylndral oordnate system and the magnet vetor potental A are brought n for the axsymmetr problems [4] [5]. The magnet flux densty of the permanent magnet s desrbed as B = μμ 0 H + μ0m (3) P 7 where μ = 4π 0 H/m s the magnet permeablty n 0 absolute vauum, μ s the relatve magnet permeablty r and H P (Mp) s the ntensty of polarzaton of the permanent magnet materal. The number of the degrees of freedom to be solved on the tetrahedral partton adopted s up to 0, four are the magnet salar potental on the four vertes of 03 ACADEMY PUBLISHER

3 30 JOURNAL OF COMPUTERS, VOL. 8, NO., FEBRUARY 03 tetrahedron, the other sx are the magnet flux denstes of the sx sdes of tetrahedron. The method of quadrat nterpolaton s used to approxmate the feld quantty n a sngle dvson. The eletromagnet fore equaton s Φ (, δ ) (, ) 0.5 ( ) ( ) ( ) ( ) B δ = = μ0cφ N() t S δ S δ dδ S δ δ (4) where CΦ s the magnet flux leakage oeffent, whh has relatons wth the leakage magnet flux Φ l ; S ( δ ) s the magnet flux equvalent area funton, whh has relatons wth δ () t. Suspenson suffered eletromagnet attraton s B A μ0 AN () t Fm (,) z = = * (5) μ0 4 z ( t) where F s the levtaton fore (N) of eah eletromagnet; A s the area of permanent magnet (m ); B s the magnet feld strength (T). C. Model suspenson system makes use of the repulson between the eletromagnet ols and the permanent magnets n the spae, whh an provde enough eletromagnet levtaton fore for offsettng the gravty of the suspended. III. THE ANALYSIS OF THE SINGLE ELECTROMAGNETIC SUSPENSION SYSTEM Aordng to the analyss above, the model of the eletromagnet levtaton system s establshed. As one of the most mportant fators for the eletromagnet levtaton fore, the ar gap s analyzed and studed, model spef parameters are shown n the Table II. Fgure. Fnte mesh dsseton map of sngle eletromagnet suspenson model Fgure 3. Spae magnet feld dstrbuton map of sngle eletromagnet mode TABLE II. SINGLE ELECTROMAGNETIC MODEL GEOMETRY PARAMETERS magnet radus magnet thkness Current Varable R /mm T /mm I/A Value ore radus ore thkness Varable R /mm T /mm Value The fnte mesh dsseton of the sngle eletromagnet suspenson model an be made through the fnte element method, and the results are shown n Fg.. Other ondtons reman unhanged, the workng envronment s ar and the ar gap sze hanges from 0m to 4 m. Fg. 3 shows the magnet feld dstrbuton of the eletromagnet suspenson system and the date of eletromagnet fore s shown n Fg. 4. Fgure 4. Relaton between eletromagnet fore and heght In order to explore the rules of the eletromagnet levtaton system under the large ar gap, the least square method s used n data fttng. Compared to other data fttng method suh as nterpolaton method and approahng funton method, the least square method works better [6] [7]. The base funtons that an be seleted n the least square method are varous, suh as the ommon polynomal, Chebyshev polynomal, Bemsten polynomal, et. Aordng to the eletromagnet feld theory formula, the least square method s used n data fttng based on the ratonal fttng funton, The fttng formula s f( x) = 0.59x + 653x (6) Fg. 4 shows the fttng graphs. From the fttng formula we an see that the seond tem oeffent of the 03 ACADEMY PUBLISHER

4 JOURNAL OF COMPUTERS, VOL. 8, NO., FEBRUARY 03 3 formula s small whle the frst tem oeffent s large, they are dfferent from the theoretal formula. Beause durng -4 meters, the eletromagnet magnet felds ntensty produed by eletromagnet n spae s abate, the haratersts of magnet feld has hanged. IV. THE ANALYSIS OF THE DISTRIBUTED ELECTROMAGNETIC SUSPENSION SYSTEM In order to be more lose to requrements of real experment, we nrease the number of the researh objet. Takng four-dstrbuton array as an example, the orgnal sngle eletromagnet s dvded nto four small sngle eletromagnets. Model geometry parameters are shown n the Table III. the fttng urve of the axal eletromagnet fores that hanged wth the horzontal dsplaement. The fttng formula s f( x) =4.05x 6.44x (7) Fg.8 shows the fttng urve of the lateral eletromagnet fores whh responses wth the horzontal dsplaement. The fttng formula s f( x) =4.5x 7.39x (8) TABLE III. DISTRIBUTED ELECTROMAGNETIC MODEL GEOMETRY PARAMETERS magnet radus magnet thkness Current Varable R /mm T /mm I/A Value ore radus ore thkness Varable R /mm T /mm Fgure 5. Spae magnet feld dstrbuton map of dstrbuted eletromagnet model. Value The fnte mesh dsseton of the dstrbuted eletromagnet suspenson model an be made and the result s shown n Fg.5. The magnet feld dstrbuton of the eletromagnet suspenson system s shown n Fg.6. The workng envronment s ar and the suspenson s m n heght, movng the suspended matter along horzontal dreton, we an observe the hangng law of the eletromagnet fore, the result s shown as below. Fgure 7. Relaton between axal fore and horzontal dsplaement. Fgure 8. Fnte mesh dsseton map of dstrbuted eletromagnet model. Aordng to the eletromagnet feld theory formulas, the data s ftted by the least square method. Fg. 7 shows Fgure 6. Relaton between lateral fore and horzontal dsplaement. In the dstrbuted system model, the axal fore hange formula manly presents the delne of the parabol tendeny, the seond tem oeffent s far bgger than 03 ACADEMY PUBLISHER

5 3 JOURNAL OF COMPUTERS, VOL. 8, NO., FEBRUARY 03 the frst one; On the ontrary, the seond tem oeffent of the lateral fore hange formula s less than the frst one, whh s loser to lnear haratersts. The rule that eletromagnet fore hanges wth the dstane s nversely proportonal to the dstane, and the x here s just a part of the dret dstane. V. CONCLUSION Ths paper analyzes the eletromagnet feld of mxed suspenson eletromagnet wth fnte element analyss method. We model and analyze the suspenson eletromagnet to study the relatonshp between the eletromagnet fore, urrent, and suspenson gap. Then, the hange law of the eletromagnet fore wth horzontal offset eletromagnet feld s also studed. Then the alulated dates are ftted by the least square method. Fnally, we ompare the fttng formula wth the theory one. These results lay the foundaton for the further expermental study. REFERENCES [] M. Hano, Fnte-element soluton of three-dmensonal resonator problems: Novel retangular parallelepped elements, Eletron. Commun. Jpn. Pt., vol. 7, no. 7, pp.7-34, 998. [] G. Mur and A. T. de Hoop, A fnte-element method for omputng three- dmensonal eletromagnet fleds n nhomogeneous meda, IEEE Trans. Magn. vol. MAG-, pp. 88-9, Nov [3] Mmpe Morshta, Teruo Azukzawa, Shuj Kanda. A New Maglev System for Magnetally Levtated Carrer System, IEEE Transatons on Vehular tehnology. November 989, vo. 38, No 4. [4] Coulomb. J, A methodology for the determnaton of global eletromehanal quanttes from a fnte element analyss and ts applaton to the evaluaton of magnet fores, torques and stffness, IEEE Trans. on Mag, 983, 9(6): [5] Golob M, Tovornk B. Modelng and ontrol of the magnet suspenson system [J]. ISA Transatons, 003, 4 (): [6] Kas F, An nterpolatve fuzzy nferene usng least square prnple by means of β-funton and hgh order polynomals, IEEE Internatonal Conferene on Mehatons & Automaton, Vol., pp , 005. [7] Royhowdhury S, Fuzzy urve fttng usng least square prnples, 998 IEEE Internatonal Conferene on Systems, Man and Cybemets, 4: Zhengfeng Mng (M 08) was born n X an County, Shaanx Provne, Chna. He reeved the B.S. degree n automat ontrol from X an Unversty of Tehnology, X an, Chna, n 988, the M.S. degree n the ndustral eletral automaton from X an Sene Unversty, X an, n 993, and the Ph.D. degree n eletral engneerng from X an Unversty of Tehnology, n 00. From Marh 003 to Aprl 005, he was a Postdotoral Researher of eletral engneerng n Zhejang Unversty, Hangzhou, Chna. He joned Department of Eletral Engneerng, Xdan Unversty, thereafter and s urrently an Assoate Professor and Char of the Department. He was a vstng Assoate Professor n the Department of Eletral and Computer Engneerng at New Jersey Insttute of Tehnology (NJIT), Newark, n 009. He has authored more than 30 publshed tehnal onferene and journal papers n the area of power eletrons. Hs researh nterests nlude dgtal ontrol for power eletrons applatons, hgh-power onverters, three-phase nverters and retfers, motor drves, soft-swthng tehnques, and ntegrated power eletrons systems. 03 ACADEMY PUBLISHER