THE NUMERICAL EVALUATION OF THE LEVITATION FORCE IN A HYDROSTATIC BEARING WITH ALTERNATING POLES

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1 THE NUMERICAL EVALUATION OF THE LEVITATION FORCE IN A HYDROSTATIC BEARING WITH ALTERNATING POLES MARIAN GRECONICI Key words: Magnetic iquid, Magnetic fied, 3D-FEM, Levitation, Force, Bearing. The magnetic fied and the evitation force produced by a rotor with aternating magnetic poes in a magnetic fuid hydrostatic bearing is numericay evauated. For the magnetic fied computation has used a 3D-FEM program, MagNet 5.0 and for the numerica evauation of the evitation force the MatLab program has been used. 1. INTRODUCTION The cyindrica bearing with magnetic iquid represents one of the appications of the second order magnetic evitation. The bearing with magnetic fuid, aternating magnetic poes on the stator, and nonmagnetic rotor has been presented in [1] and anayticay investigated in [, 3]. A numerica computation of the force that acts on the rotor, using a D-FEM program under the hypothesis of a magnetic fied with pane meridian symmetry, can be found in [4]. The bearing with magnetic fuid, aternating magnetic poes in rotor and nonmagnetic stator has been anayzed anayticay in [5 7]. In the present paper, the magnetic fied and the evitation force are evauated numericay in a bearing with magnetic fuid, aternating magnetic poes in rotor and nonmagnetic stator. It has been used Magnet 5.0, a 3D program of Infoytica, based on the finite eement method (FEM), and the MatLab medium. The sketch of a bearing with magnetic fuid and aternating magnetic poes paced in rotor is presented in Fig.1, where we have denoted: 1 the rotor (the shaft) of r 1 radius and µ 0 permeabiity; the magnetic poes with h p height, supposed to have a radia permanent magnetization M p of zero divergence and µ m = µ 0 µ rm permeabiity; 3 magnetic iquid that fis the space between the rotor and stator, considered as a inear medium of permeabiity µ = µ 0 µ r ; 4 the nonmagnetic stator of µ 0 permeabiity. Poitehnica University of Timisoara, e-mai: marian@et.utt.ro Rev. Roum. Sci. Techn. Éectrotechn. et Énerg., 5,, p , Bucarest, 007

2 00 Marian Greconici Fig. 1 The sketch of a bearing with aternating poes in rotor. The dispacement between the rotor axes and stator axes wi be denoted by. Because of the dispacement, the magnetic fied has not a radia symmetry, and on the rotor acts a magnetic force in the direction to bring the rotor in equiibrium (when it is centered in the bearing). The evitation force depends on the dispacement, the magnetic properties of the magnetic iquid, the permanent magnetization of the magnetic poes and on the geometrica design of the bearing. To evauate numericay the evitation force, we need to know the magnetic fied distribution produced by the magnetic poes.. MAGNETIC FIELD EQUATIONS AND FINITE ELEMENT FORMULATION Fig. The bearing heaving a ong extension on z axes.

3 3 Numerica evauation of evitation force in a hydrostatic bearing 01 The bearing that wi be anaysed is considered to have a ong extension aong the z axis (Fig. ). The ength of the 3D mode has been denoted λ/ = p + 0, a haf of waveength, [6]. It represents the ength between two orthogona panes on the z axes that pass through the middes of two consecutive poes (Fig. 3). The radia extension of the 3D mode, a cyinder of r ext radius, has been adopted so that the magnetic fied is considered to vanish outside the mode. The Dirichet condition A = 0 has been considered for the boundary of the 3D mode. In a domains of the 3D mode, Fig. 3, the magnetic potentia vector A satisfies a Lapace equation: A =0, (1) as divb = 0, curh = 0, divm p = 0 and, B=µ H+µ 0M p in the magnetic poes, B=µ H in the magnetic iquid and B=µ 0 Hin the shaft, stator and outside the bearing. Fig. 3 The geometrica mode used in 3D-FEM anayses. The 3D-FEM program MagNet 5.0 of the Infoytica was used to sove numericay the fied probem. The version 5.0 of the program doesn t aow defining a radia permanent magnetization. To simuate the radia magnetization of the permanent magnets has been used the mode in Fig. 4. The toroidas magnets has been divided into 0 domains, ba, bb,, bt, each of them having a constant permanent magnetization, as shown in Fig. 4.

4 0 Marian Greconici 4 Fig. 4 The mode adopted for simuating a radia magnetization. The geometry and the finite eements in a orthogona pane on the z axis of the mode, generated by the MagNet is presented in Fig. 5. Fig. 5 The finite eements in a pane orthogona on the z axis. To generate the 3D mode, the extrusion principe has been used. The first order tetrahedrons were used in a the regions of the mode except the region of the iquid where second order tetrahedrons have been used. In the anaysed cases there has been used a number of nodes and tetrahedrons.

5 5 Numerica evauation of evitation force in a hydrostatic bearing THE NUMERICAL FORCE EVALUATION The force that acts on the rotor is [8, 6]: H F= d d + ( ) d, Σ B H S 0 H B n S () Σ where Σ is the cyindrica interface surface between the magnetic iquid and rotor and ds= dsn is the surface unit vector oriented outwards the rotor (Fig. 6). Fig. 6 The cyindrica system of coordinates. Due to the symmetry of the magnetic fied, the integras on the two bases of the cyinder has been negected. Considering the magnetic iquid as a inear medium of µ =µ 0µ r permeabiity, we have: H H 1 B dh =µ d = µ 0 H H H, 0 and the expression () becomes: F 1 = µ H d S +µ H H n d S, (3) S S ( ) where S is the surface of the cyinder of r radius and λ/ height taken in the points of the magnetic iquid. Using the cyindrica coordinates (r, θ, z), we have successivey (Fig. 6): H= Hrur + Hθuθ + Hzu z, d = ds = ds r S n u, and ( ) ( ) r θ z H = H + H + H, H H n = H u + H u + H u H = H u + H H u + H H u. r r θ θ z z r r r θ r θ z r The force expression (3) becomes: z

6 04 Marian Greconici 6 H F=µ Hr ds ur +µ HrHθdS uθ +µ HrHzd S u z. S S S (4) As the goba evitation force that acts on the rotor is different of zero ony in the direction of the maximum dispacement, the evitation force is: H Fm = r sin θ+ Fθcosθ=µ H r sin θ+ H rh θcosθ d S. S (5) The intersection between the 3D mesh of tetrahedron and the cyindrica sice of r radius and λ/ height generates a D mash of trianges as finite eements. Fig. 7 How the cyinder unwraps for a D computation. For an incision made at θ = 0, MagNet unwrapped the cyinder of r radius, peeing it open from the incision ine in the manner iustrated in Fig. 7. The resuting rectanguar surface has about eements (trianges). As the eements are very sma, the magnetic fied and its components has been considered constant on each eement, and (5) becomes: H i Fm =µ Hr sin θ cos. i i + Hr H i θ θ i i Si i (6) The evitation force that acts on the rotor for the anaysed mode has been computed numericay, using (6) and MatLab medium. The evitation force that acts on an unit ength of rotor, is: * F F λ = F λ. (7) m = m m

7 7 Numerica evauation of evitation force in a hydrostatic bearing 05 The nodes coordinates and the magnetic fied components generated by MagNet (as text fies) has been used in numerica computation of the evitation force. 4. NUMERICAL RESULTS The vaues of the geometrica quantities used in the numerica computation are: r 1 = 10 mm, r = mm, δ = 1 mm, r 3 = mm, λ = 8.8 mm, p =.93 mm, 0 = mm and r ext = 100 mm, where δ represents the maximum dispacement. The resuts of the numerica simuation of the magnetic fied inside the bearing are presented in a graphica form. They have been generated by the MagNet 3D-FEM program. Fig. 8 presents the magnetic fux density distribution (in Tesa) in the rotor with magnetic poes and the magnetic iquid. There has been used a Sa-Co permanent magnet with M p = 754 ka/m, µ rm = 1 and a iquid with µ r = 1. for a dispacement = 0.8 mm. The higher vaues of the fux density in the iquid takes pace where the iquid ayer is the thinnest (the bottom part of the iquid). Fig. 8 The magnetic fux density distribution in the rotor and magnetic iquid. The magnetic fux density distribution in the magnetic iquid and in the magnetic poes, for the same quantities, is presented in Fig. 9. The highest vaues of the fux density are paced in the magnetic poes and immediatey around them.

8 06 Marian Greconici 8 In Fig. 10 the evitation force that acts on the unit ength of rotor has been represented versus the dispacement for a Sa-Co permanent magnet with M p = 754 ka/m, µ rm = 1 and a iquid with µ r = 1.. Fig. 9 The magnetic fux density distribution in the magnetic iquid and poes. Fig. 10 The evitation force versus the dispacement.

9 9 Numerica evauation of evitation force in a hydrostatic bearing 07 The curve denoted by (1) has been obtained using a anaytica approximate expression for the evitation force, [5, 6], and the curve () used the numerica 3D- FEM computation. The two curves match quite we. 5. CONCLUSIONS In [6, 7] there has been proposed an anaytica expression for the evitation force that acts on the rotor with magnetic poes of a hydrostatic bearing with magnetic iquid. The magnetic iquid has been considered as a inear medium and for the magnetic fied has been adopted a pan-parae mode. The reation for the second order evitation has been used for the force evauation. The numerica computation deveoped in present paper has adopted a 3D mode for the magnetic fied and the evitation force has been cacuated using an expression derived from the Maxwe tensor. The resuts for the evitation force cacuated in present paper match quite we with the corresponding resuts obtained by using the anaytica expression. As a consequence, the anaytica expression of the evitation force coud be used to anayse and design cyindrica bearings with aternating poes paced in rotor and magnetic iquid. The numerica computation deveoped in present paper is appied aso in noninear cases (considering the magnetic iquid as a noninear medium), where the anaytica soution is no more possibe. Received on 9 November 005 REFERENCES 1. R.E. Rosensweig, Thermomecanics of magnetic fuids, Ed. B.Berkovskii, Hemisphere, Washington, B.N. Berkovskii, A.N. Visovich, Desiatoe Rijkoe Sovesk po Magn. Ghidrodin., vo. III, Riga, pp , I. De Sabata, L. Vekas, On the restoring force of magnetic fuid bearings, Rev. Roum. Sci. Techn. Méc. App., 34, 1, pp. 13 0, (1989). 4. C. Baj, Modeization of the magnetic evitation appied on the design of magnetic bearings with aternating poes (in Romanian), PhDThesis, Poitehnica University of Timişoara, I. De Sabata, M. Greconici, A. De Sabata, Restoring force acting on a rotor with aternating poes in a magnetic fuid, Rev. Roum. Sci. Techn. Éectrotechn. et Énerg., 46, 1, pp , (001). 6. M. Greconici, Research on the cyindrica bearings with magnetic iquid (in Romanian), PhD Thesis, Poitehnica University of Timisoara, I. De Sabata, A. De Sabata, Magnetic fuid bearings with poes on the shaft, Rev. Roum. Sci. Techn. Éectrotechn. et Énerg., 48, 3, pp (001). 8. I. De Sabata, Forţa exercitată de câmpu magnetic a unui corp având magnetizaţie permanentă şi temporaă, imersat într-un ichid magnetic şi în câmp magnetic, Anae Univ. Oradea, vo. II, Fasc. Eectrotehnica, 6 14, Oradea, 1991.