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TECHNICAL PAPER A method of reducing the windage power loss in a laser scanner motor using spiral-groove aerodynamic thrust bearings functioning as a viscous vacuum pump Shigeka Yoshimoto Masaaki Miyatake Tomoatsu Iwasa Akiyoshi Takahashi Received: 29 June 2006/Accepted: 2 November 2006/Published online: 1 December 2006 ?Springer-Verlag 2006 AbstractWe propose a spiral-groove aerodynamic thrust bearing functioning as a viscous vacuum pump in a laser scanner motor to reduce the windage power loss of a polygon mirror. The proposed bearing pumps out the air in the scanner housing using the pumping effect of the spiral-groove thrust bearing, reducing the inner pressure of the housing. The pumping performances and the static characteristics of the spiral-groove thrust bearings were investigated numerically and experi- mentally. Two numerical calculation methods were used to study the pumping characteristics of the spiral- groove thrust bearing. It was found that a bearing with 15 spiral grooves reduced the inner pressure of the housing to 0.01 in the bearing clearance, and that the slip fl ow in the bearing clearance infl uenced the pumping characteristics of the viscous vacuum pump. As mentioned above, we proposed using a spiral- groove aerodynamic thrust as a viscous vacuum pump, which would enable us to reduce the windage power loss of the polygon mirror in a laser scanner motor, and reduce the size of the laser scanner motor. In addition, the pumping and load-carrying characteristics of our proposed bearing were investigated theoretically and experimentally. Two numerical calculation methods considering the slip fl ow were used. The fi rst method employed was the narrow-groove theory reported by Whipple (1958), Vohr and Chow (1965) and Malanoski and Pan (1965), and the second method used was the divergenceformulation(DF)methodusingthe boundary-fi tted coordinate system proposed by Ka- wabata (1986). Therefore, the objective of this work was to confi rm the usefulness of our proposed bearing for a laser scanner motor, both theoretically and experimentally. 2 The proposed scanner motor using a spiral-groove aerodynamic thrust bearing as a viscous vacuum pump Figure 3 shows the structure of a laser scanner motor using spiral-groove aerodynamic thrust bearings func- tioning as a viscous vacuum pump. The shaft is fi xed to the housing, and the rotor and mirror rotate. Two identicalaerodynamicthrustbearingswithspiral grooves are located at the upper and lower parts of the rotor. In the inner circle of the thrust bearing, several small holes are drilled so that the air inside the housing can be pumped out through these holes. The housing is sealed tightly to maintain a high vacuum in the hous- ing. Figure 4 shows the geometrical confi guration of the thrust bearing with spiral grooves, along with the symbols used in this paper. A number of spiral grooves with groove tilt angle, b, are formed at even intervals in the circumferential direction on the bearing surface. In addition,alandregion,withlength,lw(where lw= r1 r2), is created, allowing the bearing to both pump air and support the weight of the rotor and mirror. 3 Numerical calculation methods Two calculation methods were used to obtain the pumping characteristics of the proposed thrust bearing numerically. The fi rst method used was narrow-groove theory, and the second method was the DF method with a boundary-fi tted coordinate system. Further- more, the fi rst-order slip fl ow proposed by Burgdorfer (1959) was considered in the bearing clearance in both calculations. 3.1 Narrow-groove theory In narrow-groove theory presented by Malanoski and Pan (1965), it is assumed that the number of spiral grooves is infi nite, and that the width is infi nitesimal. Since there is no pressure gradient in the circumfer- ential direction, the mass fl ow rate in the spiral-groove region in the r-direction is q ? p RT k1 dp dr ? k2rcosb ? rdh;1 where k0 1 ? aAg aAr k1AgAr a1 ? aAg? Ar2sin2b no. 12lk0 k2 xa1 ? ahg? hrAg? Arsinb?2k0 and Ag h3 g 1 6k hg ? ;Ar h3 r 1 6k hr ? :2 For the land region in the inner circle of the thrust bearing, the mass fl ow rate is q ? p 12lRT h3 r 1 6k hr ? dp dr rdh:3 To obtain the distribution in pressure in numerical calculations, the equation of continuity is assumed in a small element in the bearing clearance using Eqs. (1) and (3). The derived equation is solved numerically assuming the following boundary conditions, at Microsyst Technol (2007) 13:112311301125 123 r r2; p pa; and at r r0; p pu;4 where pu is defi ned as the ultimate pressure where no fl ow rate exists in the bearing clearance. Therefore, the ultimate pressure, pu, can be obtained by solving Eq. (1) under the boundary condition where q = 0. Furthermore, when a load is imposed on the rotor, the continuity of mass fl ow rate is assumed between the upper and lower bearing clearances, and therefore the inner housing pressure can be obtained. 3.2 DF method with boundary-fi tted coordinate system In narrow-groove theory, the number of grooves is assumed to be infi nite and hence, the effect of the number of grooves on the pumping characteristics is not applicable using this theory. Therefore, we used the DF method with a boundary-fi tted coordinate system to clarify the effect of the number of grooves. The mass fl ow rates in the boundary-fi tted coordinate system, n g coordinates, as shown in Fig. 5, were derived by transformation from the polar r h coor- dinates as presented by Kawabata (1986). For the mass fl ow rate in the n-direction qn p RT ffi ffiffi a p?A p n B p g D ? :5 For the mass fl ow rate in the g-direction qg p RT ffi ffiffi c pB p n ? C p g E ? 6 where A h3 12l 1 6k h ? a J B h3 12l 1 6k h ? b J C h3 12l 1 6k h ? c J D ? rx 2 h r g E rx 2 h r n a r g ?2 rh g ?2 b r n r g rh n rh g c r n ?2 rh n ?2 J r n rh g ? r g rh n : Figure 5 shows a control volume in the bearing clearance where a continuity of the mass fl ow rate was assumed. Therefore, the mass fl ow rate in the normal direction to the constant n or g boundary is given by Qn Zg2 g1 p RT ?A p n B p g D ? dg Qg Zn2 n1 p RT B p n ? C p g E ? dn: 7 The following equation is derived from the continuity of mass fl ow rate Rotor Pump out Sealed housing Polygon mirror Motor Pump out Exhaust holes Thrust bearing with spiral grooves Journal bearing Fig. 3 A laser scanner motor using spiral-groove aerodynamic thrust bearings as a viscous vacuum pump Groove Ridge Shaft Rotor Pump in Pump in Groove e r0 r1 r2 hd Groove Ridge Land Thrust bearing Exhaust holes hr hd Pump out r hr0 region 1- a Fig. 4 The confi guration of the spiral grooves and symbols 1126Microsyst Technol (2007) 13:11231130 123 Qn 2A1 Q n 1A3? Q n 2A2? Q n 1A4 Q g 2A1 Q g 1A2 ? Qg 2A3? Q g 1A4 0:8 3.3 Calculation of the pressure in a sealed housing with time The pressure in the sealed housing was calculated using Eq. (1) after the rotor had begun to rotate. Assuming that no axial load was imposed on the rotor, and that the upper and lower bearing clearances had the same value, then, in the narrow-groove theory, the mass fl ow rate in unit time is given by Qt 2 Z2p 0 qjrr2r2dh:9 Next, assuming that the mass of air in the housing with a volume, Vi,was G(t) at time t, then, under isothermal conditions, G(t + Dt) is expressed as G t Dt G t ? Q t Dt:10 Furthermore, using pressure, pi, and density, qi, in the housing, the mass of air included in the housing at time t, is given by G t qit Vi pit Vi=RT*qit pit =RT: 11 SubstitutingEq. (11)intoEq. (10),thehousing pressure at time, t + Dt, is given as follows. pit Dt pit ? Q t RTDt=Vi:12 In our calculations, Dt = 0.0167 s, and at t = 0, the initial value of the housing pressure is assumed to be atmospheric pressure. The housing pressure obtained by Eq. (12) is used as the boundary condition of Eq. (1) for the next time step. 4 Results of the calculations As mentioned above, narrow-groove theory cannot be used to estimate the effect of the number of grooves on the pumping characteristics of the spiral-groove bear- ing, although this theory is very simple, and is very useful for designing aerodynamic bearings with her- ringbone or spiral grooves. Therefore, the suitability of narrow-groove theory to predict the pumping charac- teristics of the spiral-groove thrust bearing treated in this paper was confi rmed by comparing data obtained using the DF method. The principal dimensions used in our calculations are shown in Table 1. Figure 6 shows the pressure distribution in the bearing clearances obtained using the DF method with a boundary-fi tted coordinate system for n = 8 and 15. The DF method can be used to calculate the pressure distribution by considering the groove shape, as shown in Fig. 6. Accordingly, it can be seen that the pressure distribution varied between the grooves in the cir- cumferential direction. By increasing the number of grooves, the pressure variation between the grooves decreased, and it can be predicted that the pressure distribution would approach that obtained using nar- row-groove theory. Figure 7 shows a comparison of the theoretical ultimate pressure obtained using narrow-groove theory and the DF method considering the slip fl ow in the bearing clearance. The number of grooves was changed from 6 to 15 using the DF method. In Fig. 7, a theo- retical result without considering the slip-fl ow effect using narrow-groove theory is also shown for refer- ence. As can be seen in Fig. 7, the ultimate pressure for n = 15 using the DF method had almost the same value as that using narrow-groove theory, and the ultimate pressure increased with decreasing the number of grooves. However, the effect of the number of grooves was not large under the conditions shown in Fig. 7. From these observations, the theoretical calculations discussed below were carried out using narrow-groove theory because the spiral-groove bearing with n = 15 was used in the experiment. A1 A3 A4 A2 i i i j j j Q2A1 Q1A2 Q2A3 Q1A4 Q2A1 Q1A3 Q2A2 Q1A4 rq x x x x x x h h h h h h Fig. 5 The n and g coordinates and mass fl ow continuity in a control volume Table 1 Principal dimensions of the spiral-groove thrust bearing r0(mm)r1(mm)r2(mm)ab (?)hd(lm)n 14.09.08.50.515.012.015 Microsyst Technol (2007) 13:112311301127 123 Figure 8 shows the theoretical pressure distribution with and without the inner land region for a dimen- sionless axial displacement of 0.4 to clarify the effect of the inner land region on the load capacity. When there was no inner land region, the difference between the pressure distributions in the upper and lower bearing clearances was very small, and in this case even a negative load capacity was generated. In contrast, the difference became large in the bearings with an inner land region. In addition, the ultimate pressure, Pu, was not sensitive to the presence of an inner land region. Figure 9showstherelationshipbetweenthe dimensionless axial displacement and the dimension- less load capacity at 20,000 rpm. In Fig. 9, the load capacity of the proposed bearing is compared to that of a conventional bearing that was operated at atmo- spheric pressure. The load capacity of the proposed Fig. 6 Pressure distribution obtained using the DF method and spiral-groove shapes 0100002000030000 0 0.2 0.4 0.6 0.8 1 Rotational Speed : N rpm Ultimate Pressure : Pu a=0.5b = 15deg. hr0=2.5 m DF Method (Slip Flow) n= 6 8 15 Narrow Groove Theory No Slip Flow Slip Flow Fig. 7 Effect of the number of spiral grooves on the ultimate pressure in the housing 0.20.40.60.81 0.5 1 1.5 0 Dimensionless Pressure P R Coordinate Land Region Groove Region : With Land Region lw=0.5mm : No Land Region hr0=4m : 30000rpm :=0.4 Lower Bearing Pressure Upper Bearing Pressure Fig. 8 Effect of the inner land region on the pressure distribu- tion 1128Microsyst Technol (2007) 13:11231130 123 bearing showed relatively smaller values compared to those of the conventional bearing, but it was large enough to support the scanner rotor. The dimension- less load capacity of 0.1 corresponds to a load of about 4N in the proposed bearing. 5 Comparison with the experimental results To verify the calculated results, we conducted a series of experiments. Figure 10 shows the experimental apparatus used to measure the pumping characteristics and the load capacity of the proposed bearing. The polygon mirror was attached to one end of the rotor, which was supported by three aerostatic journal bear- ings. The polygon mirror was located between two identicalaerodynamicthrustbearingswithspiral grooves. The mirror and the thrust bearings were made of ceramic. The outer space of the polygon mirror was connected by a tube to a tightly sealed housing with a volume of 65 cm3. The pressure inside the housing was measured using a vacuum pressure gauge. The bearing clearance of the thrust bearing could be changed using spacers with different thicknesses, denoted by Plate A in Fig. 10. The rotor was operated using a DC motor. Figure 11 shows the relationship between the inner pressure of the housing and the elapsed time after the rotor had begun rotating. The inner pressure decreased rapidly immediately after the rotor had begun rotating, and leveled out within a period of 5 min. 0. 20. 40. 60.8 0.2 0.4 0.6 0.8 0 Dimensionless Axial Displacement Dimensionless load-carrying capacity W 20000rpm Present Conv. hr0=3.0m : hr0=4.0m : hr0=5.0m : Fig. 9 The relationship between the axial displacement and the load capacity Tube Digital Tachometer Polygon Mirror Plate A Motor Housing Aerostatic Journal Bearings Shaft Spiral-Grooved Thrust Bearing Pressure Gauge Volume Air out Air out Air in Fig. 10 The experimental apparatus 1020 0.2 0.4 0.6 0.8 1 0 Cal. Exp. : hr0=2.5 m : hr0=3.5 m : hr0=4.5 m Elapsed time : t min Dimensionless Pressure in the Chamber Pi lw=0.5mm : 20000rpm Pu Fig. 11 Variation of the housing pressure with elapsed time 100002000030000 0.2 0.4 0.6 0.8 1 0 Rotational speed N rpm Ultimae Pressure Pu Exp. Cal. : hr0=2.5m : hr0=3.5m : hr0=4.5m Fig. 12 Effect of the rotational speed and bearing clearance on the ultimate pressure Microsyst Technol (2007) 13:112311301129 123 Figure 12 shows the ultimate pressure when the rotational speed and the bearing clearance were changed. As can be seen in Fig. 12, the proposed thrust bearing with spiral grooves decreased the inner pres- sure of the housing to 0.01 MPa at hr0= 2.5 lm at a speed of 20,000 rpm. Figure 13 shows the variation in the ultimate pres- sure in the housing when a load was imposed on the bearings. In this experiment, a rotor mass = 180 g was used to impose a load on the bearings. The test rig was set on a magnetic chuck, which could be tilted from the horizontal to a vertical angle. As can be seen in Fig. 13, the ultimate pressure was not greatly infl uenced by the imposed load, and the proposed bearing could support the rotor without any deterioration in the pumping characteristics. 6 Conclusions We have developed a spiral-groove aerodynamic thrust bearing functioning as a vacuum pump to reduce the windage power loss of the polygon mirror and the size of a laser scanner motor. The pumping characteristics of the proposed bearing were investigated theoretically and experimentally. As a result, the following conclu- sions were derived: 1.The proposed bearing with spiral grooves reduced the pressure in the sealed housing to 0.01 MPa at hr0= 2.5 lm at 20,000 rpm. 2.Though the load capacity of the proposed bearing was relatively small compared with that of a bearing operating at atmospheric pressure, it was large enough to support the rotor mass in this type of device. 3.Calculation method presented in this paper can predict well the pumping characteristics of the proposed bearing. References Burgdorfer A (1959) The infl uence of the molecular mean free path on the performance of hydrodynamic gas lubricated bearings. Trans ASME J Basic Eng 81(1):94100 James DD, Potter AF (1967) Numerical analysis of the g