Stability of Water-Lubricated, Hydrostatic, Conical Bearings With Spiral Grooves for High-Speed Spindles

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1 S. Yoshimoto Professor Science University of Tokyo, Department of Mechanical Engineering, 1-3 Kagurazaka Shinjuku-ku, Tokyo Japan S. Oshima Graduate Student Science University of Tokyo, Department of Mechanical Engineering, 1-3 Kagurazaka Shinjuku-ku, Tokyo Japan S. Danbara Research Associate Science University of Tokyo, Department of Mechanical Engineering, 1-3 Kagurazaka Shinjuku-ku, Tokyo Japan T. Shitara Union Tool Co. Ltd Aza-Sotokawa, Settaya-Cho Nagaoka-City, Niigata Prefecture Japan Stability of Water-Lubricated, Hydrostatic, Conical Bearings With Spiral Grooves for High-Speed Spindles In this paper, the stability of water-lubricated, hydrostatic, conical bearings with spiral grooves for high-speed spindles is investigated theoretically and experimentally. In these bearing types, pressurized water is first fed to the inside of the rotating shaft and then introduced into spiral grooves through feeding holes located at one end of each spiral groove. Therefore, water pressure is increased due to the effect of the centrifugal force at the outlets of the feeding holes, which results from shaft rotation. In addition, water pressure is also increased by the viscous pumping effect of the spiral grooves. The stability of the proposed bearing is theoretically predicted using the perturbation method, and calculated results are compared with experimental results. It was consequently found that the proposed bearing is very stable at high speeds and theoretical predictions show good agreement with experimental data. DOI: / Introduction Contributed by the Tribology Division of THE AMERICAN SOCIETY OF ME- CHANICAL ENGINEERS for presentation at the STLE/ASME Tribology Conference, San Francisco, CA, October 4, 001. Manuscript received by the Tribology Division February 7, 001; revised manuscript received February 31, 001. Associate Editor: C.-P.R. Ku. Recently, the size of many electric and opto-electric devices has been reduced and printed-circuit boards used in these devices have also had to be made smaller. Accordingly, the diameters of the small holes for fixing electric parts to the printed-circuit board have had to be reduced. Water-lubricated, hydrostatic, conical bearings with spiral grooves were designed for a high-speed spindle to drill these small holes in printed-circuit boards. Compared with oil-based hydrostatic bearings, water-lubricated hydrostatic bearings have lower power consumption at high speeds and do not cause any environmental problems. Moreover, this bearing has a larger load capacity than aerostatic bearings, as a higher supply pressure can be used due to the incompressibility of water. Hydrostatic, conical bearings have the advantage of being able to carry both axial and radial loads, and many studies of this type of bearing have been published. However, there are only a few reports that treat hydrostatic, conical bearings with grooves on the bearing surface. Rodkiewicz and Kalita 1 investigated the effect of radial and inclined grooves formed on the bearing surface on the bearing load capacity. They demonstrated that the load capacity of the conical bearing with inclined grooves varied with the rotational direction of a rotor. Yoshimoto et al. studied waterlubricated, conical, hydrostatic bearings with spiral grooves at high speeds. It was shown that spiral grooves formed on the shaft surface acted as a viscous pump and were very effective for increasing the water pressure in the bearing clearance and the load carrying capacity. It was also experimentally shown that a rotor supported by this bearing could rotate stably up to a speed of 10,000 rpm. Yoshimoto et al. 3 also reported the static characteristics of this type of conical bearing with a compliant bearing surface. As mentioned previously, the demand for higher rotational speed in drilling machines is increasing strongly as smaller diameter of holes for printed-circuit boards become necessary. Therefore, it is very important to accurately predict the threshold speed of instability of bearings used in high-speed drilling machines. The objectives of this paper are to investigate theoretically and experimentally the threshold speed of instability of waterlubricated, hydrostatic, conical bearings with spiral grooves, and to clarify the usefulness of the proposed conical bearings for highspeed spindles. Structure and Operating Principle of the Bearing Figure 1 shows the water-lubricated, hydrostatic, conical bearing with spiral grooves proposed in this paper. Spiral grooves are formed on the cone surface of the shaft, and feeding holes are drilled at one end of each spiral groove. Pressurized water is fed to the inside of the shaft and introduced into the bearing clearance through feeding holes. Therefore, water pressure is increased at the outlet of the feeding holes due to the centrifugal force generated as a result of rotation, and water in the bearing clearance is pumped up by the viscous effect of the spiral grooves. This hydrodynamic effect raises the water pressure in the bearing clearance, which results in a bearing with a large load carrying capacity and excellent stability at high speeds. Figure shows a detailed drawing of the conical bearing treated in this paper. 3 Numerical Calculation Method 3.1 Governing Equations. In numerical calculations, water flow in the bearing clearance is assumed to be laminar, viscous, and isothermal. In calculating the pressure distribution for the groove-ridge region, narrow-groove theory by Vohr and Chow 4 is adopted. Therefore, the mass flow rates for a unit width in the s and directions are given as follows: p q s k 1 s k p r k 4 cos p q k s k p 3 r k 4 sin h mr, (1) 398 Õ Vol. 14, APRIL 00 Copyright 00 by ASME Transactions of the ASME Downloaded From: on 01/7/016 Terms of Use:

2 Fig. 1 Water-lubricated hydrostatic conical bearing with spiral grooves where k 0 1h g 3 h r 3 k 1 h 3 g h 3 r 1h 3 g h 3 r sin /1k 0 k 1h 3 g h 3 r cos sin /1k 0 k 3 h 3 g h 3 r 1h 3 g h 3 r cos /1k 0 k 4 r1h g h r h 3 g h 3 r sin /k 0 h m h g 1h r, rs sin. () For the smooth region located in the upper and lower parts of a conical bearing, the following equations for the mass flow rate are given for the s and directions: q s h r 3 p 1 s q h r 3 p 1 r h rr. (3) Fig. 3 Mass flow continuity in the resistance network method In order to solve the pressure distribution, the resistance network RN method, which was presented by Kogure and Kaneko 5 is used. In the RN method, the mass flow continuity in a small control volume surrounded by dashed lines is considered, as shown in Fig. 3a, and the following equation is obtained: Fig. Detailed drawing of the conical bearing Q s1 Q s Q 1 Q h i rs, (4) t where Q si q si r, Q i q i s, h i is h m for the grooveridge region, and h r for the smooth region, respectively. The water pressure, p i, at the outlet of a feeding hole, including the effect of the centrifugal force generated by shaft rotation, is given as follows: p i p s r 1 r l 1 cos /. (5) Therefore, the total mass flow rate through n feeding holes, including the orifice effect at the outlet of a feeding hole, is given as follows: q in C D nd 3 h r h g p sp i, (6) where n is the number of feeding holes, and p i is the pressure at the outlet of a feeding hole. Consequently, we may obtain the following equation of mass continuity for a control volume including a feeding hole as shown in Fig. 3b, by making use of the line source assumption: Journal of Tribology APRIL 00, Vol. 14 Õ 399 Downloaded From: on 01/7/016 Terms of Use:

3 K 0 1H 3 3 g H r K 1 H 3 g H 3 r 1H 3 g H 3 r sin /K 0 K 1H 3 g H 3 r cos sin /K 0 K 3 H 3 g H 3 r 1H 3 g H 3 r cos /K 0 K 4 R1H g H r H 3 g H 3 r sin /K 0 H m H g 1H r (11) pp a P, ss 0 S, rs 0 R, h r h r0 H r, h g h r0 H g k 1 h r0 3 1 K 1, k h 3 r0 1 K, k 3 h 3 r0 1 K 3, k 4 s 0 h r0 K 4 6s 0 h, r0 p a. (1) From Eqs. 4 and 10, the continuity of mass flow in a control volume can be expressed in a dimensionless form as follows: Fig. 4 Fixed and moving coordinate systems Q s1 Q s Q 1 Q Q 3 Q 4 q in r /r h m r t s h r t r s. (7) It is considered under the line source assumption that water enters the bearing clearance through a circumferentially continuous restrictor called the line source, of which mass flow rate equals the sum of those through n feeding holes. Therefore, the mass flow rate through a width of r of the line source is expressed as mass flow rate for unit width r (q in /r )r. 3. Calculation Method for the Threshold Speed of Instability. In order to obtain the threshold speed of instability of the shaft supported by two identical conical bearings, it is assumed that the coordinate system is fixed on the shaft and rotates at an angular whirl velocity, as reported by Fleming et al. 6. Therefore, the circumferential co-ordinate * in the moving system shown in Fig. 4 is expressed by in the fixed system as follows: *t. (8) Thus, the derivatives with regard to time t and circumferential coordinate can be written as follows, using * in the moving system: * t * * t *. (9) Substituting Eqs. 8 and 9 into Eq. 1, the following dimensionless equations are obtained by introducing the dimensionless variables shown below: P Q sk 1 S K P R* K 4 cos P Q K S K P 3 R* K 4 sin H m R, (10) where Q s1 Q s R*Q 1 Q S H m * R*S. (13) Similarly, for the smooth region, Q s1 Q s R*Q 1 Q S H r * R*S. (14) In the region including the line source, the following equation is derived from Eq. 7: Q s1 Q s R*Q 1 Q S Q 3 Q 4 S H m S R* * 1 * h 3 r0 p a q in R* S. H r * (15) To obtain the threshold speed of instability, the perturbation method was used. In this method, it is assumed that the shaft is whirling with a small amplitude, e, under the concentric condition. Therefore, it can be considered that the bearing clearance and the pressure consist of two terms. One term is related to the equilibrium condition and the other term is related to the change due to the whirl of the shaft with a small amplitude, e. Accordingly, we can express the bearing clearance and the pressure, respectively, as follows: H r H r0 H r1 1 cos *, H g H g0 H g1 H d 1 cos * PP 0 P 1 cos *P sin *, (16) where e/h r0. Moreover, K 1, K, K 3, and K 4 shown in Eq. 11 are functions of the bearing clearance, and they are also perturbed in the following forms: K 1 K 01 Kˆ 1 cos * K K 0 Kˆ cos * K 3 K 03 Kˆ 3 cos * K 4 K 04 Kˆ 4 cos *, (17) 400 Õ Vol. 14, APRIL 00 Transactions of the ASME Downloaded From: on 01/7/016 Terms of Use:

4 where, K 01, K 0, K 03, and K 04 are obtained by replacing H r and H g in Eq. 11 by H r0 and H g0. K 01H 3 3 g0 H r0 Kˆ 031H g0 H r0 /K 0 Kˆ 1K 01 Kˆ 03H g0 H r0 H g0 H r0 61H g0 H r0 H 3 g0 H 3 r0 sin /K 0 Kˆ K 0 Kˆ 061H g0 H 3 r0 H g0 H 3 r0 cos sin /K 0 Kˆ 3K 03 Kˆ 03H g0 H r0 H g0 H r0 61H g0 H r0 H 3 g0 H 3 r0 cos /K 0 Kˆ 4K 04 Kˆ 03R1H g0 H r0 H g0 H r0 sin /K 0 (18) Substituting Eqs. 16 and 17 into Eqs. 10 and 13, and then arranging Eq. 13 with regard to 0, cos *, and sin *, three equations are derived. 0 is not explicitly written in Eqs. 16 and 17 because 0 equals unity. Terms without are ones related to 0. This perturbation method is also applied to Eq. 14 for the smooth region and to Eq. 15 for the outlet of a feeding hole. By numerically solving the equation related to 0, the pressure P 0 for the equilibrium condition can be obtained. From the equations related to cos * and sin *, pressures, P 1 and P are determined. By integrating P 1 and P, the radial and tangential forces, f r and f t, exerted by the fluid film on the shaft are obtained: P 1 cos RdS m ch r0 m ch r0 p a s 0 p a s 0 M c, () where M c is the dimensionless critical mass. 4 Numerical Results 4.1 Static Characteristics. Before the stability of the proposed bearing is discussed, the static characteristics of this bearing are shown. By solving the equation related to 0, the pressure, P 0, for the equilibrium condition can be obtained. Figure 5 shows the static pressure distribution in the s direction at 100,000 rpm with various bearing clearances. Principal dimensions of the bearing and the shaft used for this calculation are shown in Table 1 and water temperature is 30 C. It can be seen from Fig. 5 that pressure at the feeding holes is increased to about 0.6 MPa by the effect of the centrifugal force resulting from the shaft rotation, even when supply pressure to the shaft is only 0.1 MPa. In addition, it can be seen that the pressure at the edge of the ridgegroove region for the bearing clearance of 10 m is considerably increased due to the pumping effect of the spiral grooves. The effect of the centrifugal force and the spiral grooves on the axial load carrying capacity is clearly seen in Fig. 6. The load carrying capacity of the proposed bearing is increased as the bearing clearance becomes small and the rotational speed of the shaft is increased. In Fig. 6, experimental results by Yoshimoto et al. 3 are also shown for comparison and it can be seen that the experimental and theoretical results are generally in good agreement. However, some discrepancies are seen in large bearing clearances at high speeds of 100,000 and 10,000 rpm, and in f r s e 0 p h r0 a P 1 cos RdS (19) f t s e 0 p h r0 a P cos RdS. (0) 3.3 Procedure for the Numerical Calculations. At the threshold speed of instability, the tangential force, f t, which acts as a damping force, equals zero and the radial force, f r, equals the centrifugal force, m c e, according to the work of Pan 7. Therefore, the numerical calculation to obtain the threshold speed of instability proceeds as follows: 1 Numerically calculate the pressure P 0 at a certain rotational speed, using the equation related to 0. Assume that the initial value, 1 / equals 0.5 in the equations related to cos * and sin *. 3 Calculate pressures, P 1 and P, and obtain the value of p P cos RdS. 4If p 0, change the value, 1 to the new value, by using the bisection method so as to satisfy the equation of P cos RdS0. 5 Calculate ( n1 n )/ n. If 110 5, then return to step 3. If 110 5, the calculation is finished. As the radial force, f r, equals the centrifugal force at the threshold speed of instability, the following equation is obtained, where the critical mass is denoted by m c : Fig. 5 Pressure distribution in the s direction Table 1 Principal dimensions of the conical bearing and the shaft used for obtaining the static characteristics f r s e 0 p h r0 a P 1 cos RdSm c e. (1) In addition, the following dimensionless relations can be obtained by normalizing Eq. 1 with h r0 /(p a s 0 ): Journal of Tribology APRIL 00, Vol. 14 Õ 401 Downloaded From: on 01/7/016 Terms of Use:

5 Table Principal dimensions of the conical bearing and the shaft used for obtaining the dynamic characteristics Fig. 6 Relationship between bearing clearance and load carrying capacity small clearances at speed of 40,000 rpm. The causes for the discrepancy are not clear at this moment, but it is considered that geometrical errors such as semi-cone angle, groove angle and the depth of spiral grooves may be a major cause for the discrepancy. In addition, at speed of 40,000 rpm, both the pumping effect and the outlet pressure of feeding holes are small and the bearing stiffness is not so large. Therefore, if an axial load imposed by an air cylinder has any radial component, the shaft easily tilts by this external force, which may affect the relationship between the bearing clearance and the load capacity of the conical bearing. 4. Threshold Speed of Instability. Figure 7 shows the relationship between the critical mass and the bearing clearance. In the calculations for the dynamic characteristics mentioned below, the conical bearing with the principal dimensions shown in Table is used. Water temperature of 0 C and supply pressure of 0.1MP are used. When the values shown in Table are changed, such values are indicated in each figure. As the bearing clearance increases, the value of critical mass decreases because of the reduction of the pumping effect of the spiral grooves. Even for a bearing clearance of 0 m, the proposed conical bearing can stably support a shaft with a mass of 50 g at 00,000 rpm, i.e., the proposed bearing has an excellent stability at high speeds. Figure 8 shows the whirl ratio as a function of the bearing clearance. The same parameters were used as in Fig. 7. We observed that the whirl ratio is insensitive to changes in the rotational speed and there exists a maximum value of the whirl ratio at a bearing clearance of around 5 m, irrespective of the rotational speeds. Beyond a bearing clearance about 15 m where the pumping effect becomes small, the whirl ratio shows an almost constant value of 0.5. In Figs. 9 1, the influence of various design parameters, such as groove width ratio, groove angle, and groove depth ratio h d /h r0, on the stability of the proposed bearing is discussed. Figure 9 shows the relationship between the groove width ratio,, and the dimensionless critical mass, M c. As mentioned previously, M c is defined as m c h r0 /(p a s 0 ). Therefore, a knowledge of M c allows the critical mass, m c, to be calculated at a certain rotating speed,. As the value of approaches to either 0 or 1, the pumping effect of spiral grooves decreases because the bearing surface becomes smooth at 0 and 1. Consequently, the value of M c rapidly decreases near 0 or 1. As mentioned previously, spiral grooves contribute much to the stability of the rotating shaft. The optimum value of, which corresponds to the maximum value of M c, exists at bearing clearances between 15 and 0 m. However, at the bearing clearance of 10 m, M c is not sensitive to changes of in the range from 0.4 to 0.9. Figures 10 and 11 show the effect of groove angle on M c. There also exists an optimum value of, which corresponds to the maximum value of M c, and the optimum value of becomes small as the bearing clearance decreases as seen in Fig. 10. At a Fig. 7 Relationship between bearing clearance and critical mass Fig. 8 Relationship between bearing clearance and whirl ratio 40 Õ Vol. 14, APRIL 00 Transactions of the ASME Downloaded From: on 01/7/016 Terms of Use:

6 Fig. 9 Effect of groove width ratio on M c bearing clearance of 10 m, the optimum value of is about 0 degrees. In addition, theoretical results corresponding to speeds of 80,000, 100,000, and 10,000 rpm are shown in Fig. 11 for a bearing clearance of 10 m. It can be seen that the optimum groove angle does not change with the rotating speed. Fig. 1 Effect of groove depth ratio on M c Figure 1 shows the effect of groove depth on M c. We found that the influence of groove depth on M c increases as the bearing clearance decreases because the pumping effect of the spiral grooves increases at smaller bearing clearances. The optimum groove depth for the maximum M c does not change with the bearing clearance and the optimum h d /h r0 is around Comparison With Experimental Results In order to validate the theoretical predictions, experiment was performed. Figure 13 shows the experimental apparatus. The shaft was set vertically, and two identical, conical bearings were located at the upper and lower part of the shaft. Pressurized water was fed into the rotating shaft through a non-contact seal. The bearing clearance was adjusted using the screw attached to the top of the shaft. The spiral grooves with a depth of 0 m and a width of 0.8 mm were manufactured on the bearing surface using a milling Fig. 10 Effect of groove angle on M c with various h r0 Fig. 11 Effect of groove angle on M c with various N Fig. 13 Experimental apparatus Journal of Tribology APRIL 00, Vol. 14 Õ 403 Downloaded From: on 01/7/016 Terms of Use:

7 machine. The groove width is constant in the s direction and the groove width ratio changes from 0.48 to 0.63 due to the cone. As shown in Fig. 9, the influence of is not significant in this region. Therefore, was set 0.5 in the calculations. In order to experimentally determine the threshold speed of instability, the radial motion of the shaft was monitored by two non-contact displacement probes. The shaft was driven by a high frequency motor with a power of 500 W. Figure 14 shows the measured displacement of the shaft under the unstable condition. The displacement was measured at the upper and lower parts of the shaft. Dotted curves in this figure are drawn by tracing the peak values of the shaft displacement during one revolution and present the whirlmotion of the shaft. It can be seen from Figure 14 that the whirl frequency of the shaft, indicated by dotted curves, is almost half of the rotating frequency of the shaft. In addition, we observed that the whirl motion of the shaft measured by the two displacement probes is in phase, indicating that the parallel whirl mode occurred in this experiment. In Figure 15, we compare the experimental results and the theoretical predictions of the threshold speed of instability, for a shaft with a mass of 190 g, as a function of the supply pressure. In the theoretical calculation, the threshold speed of instability is numerically solved by changing the rotational speed,, and making the critical mass equal to the shaft mass. In the experiment, the water temperature remained constant at about 0 C. It can be seen from Fig. 15 that the measured threshold speed of instability increases remarkably as the bearing clearance decreases. At a bearing clearance around 3 m, the threshold speed of instability Fig. 14 Shaft displacements for an unstable condition Fig. 15 Relationship between bearing clearance and threshold speed of instability reached 10,000 rpm. It was not possible to measure the threshold speed of instability at bearing clearances less than 3 m because of the 10,000 rpm speed limitation of the motor. The arrows attached to experimental results in Fig. 15 mean that the threshold speed of instability is higher than 10,000 rpm. From the experiment, the good stability of the proposed conical bearing was also confirmed. 6 Conclusions The stability of water-lubricated, hydrostatic, conical bearings with spiral grooves was investigated theoretically and experimentally. As a result, the following conclusions were drawn: 1 The proposed bearings with spiral grooves can stably support a shaft at high speeds, and spiral grooves make a large contribution to the stability of the rotating shaft. Design parameters, such as groove width ratio, groove angle, and groove depth ratio h d /h r0, have a large influence on the stability of the shaft and there exist optimum values of these parameters corresponding to the maximum critical mass. 3 Theoretical and experimental results show good agreement and, therefore, the theoretical calculation method presented in this paper can predict the threshold speed of instability of the proposed bearings. Nomenclature C D flow coefficient in this paper, C D is assumed to be 0.8 d 1 diameter of the shaft d diameter of the shaft at the edge of the cone d 3 diameter of the feeding hole e amplitude of whirl f r, f t radial and tangential components of the film force h bearing clearance h d groove depth h m (1)h r h g average bearing clearance h r,h g h r h d bearing clearances in the ridge and groove region, respectively h r0 bearing clearance in the ridge region under the concentric condition in the equilibrium condition Hh/h r0 dimensionless bearing clearance H r0,h g0 dimensionless bearing clearances in the ridge and groove region under equilibrium condition H r1,h g1 variations of dimensionless bearing clearances in the ridge and groove region pertaining to small whirl motion of the shaft l 1 length of a feeding hole m c critical mass M c m c h r0 /p a s 0 dimensionless critical mass n number of feeding holes n c threshold speed of instability p pressure p a ambient pressure p i outlet pressure of a feeding hole p s supply pressure P(pp a )/p a dimensionless pressure q mass flow rate for unit width Q mass flow rate r 1 radius of the inner hole of the shaft r radius at the shaft at the outlet of a feeding hole s,z, conical coordinates in the fixed system S s/s Õ Vol. 14, APRIL 00 Transactions of the ASME Downloaded From: on 01/7/016 Terms of Use:

8 s 0,s 1,s,s 3 slant length of the cone t time T water temperature ratio of groove width and the width sum of groove and ridge angle of spiral groove semi-cone angle e/h r0 dimensionless amplitude density of water 6s 0 /(p a h r0 ) / whirl ratio viscosity of water * circumferential coordinate in the moving system angular velocity of the shaft rotation angular velocity of the whirling of the shaft Subscripts s slant direction circumferential direction References 1 Rodkiewicz, C. M., and Kalita, W., 1995, Experimental Investigation Regarding the Effects of Grooves on Conical Bearing Performance, Tribol. Trans., 38, No. 1, pp Yoshimoto, S., Anno, Y., Tamura, M., Kakiuchi, Y., and Kimura, K., 1996, Axial Load Capacity of Water-Lubricated Hydrostatic Conical Bearings With Spiral Grooves On The Case of Rigid Surface Bearings, ASME J. Tribol., 118, No. 4, pp Yoshimoto, S., Kume, T., and Shitara, T., 1998, Axial Load Capacity of Water-Lubricated Hydrostatic Conical Bearings With Spiral Grooves for High Speed Spindles On the Case of a Compliant Surface Bearing, Tribol. Int., 31, No. 6, pp Vohr, J. H., and Chow, C. Y., 1965, Characteristics of Herringbone-Grooved Gas-Lubricated Journal Bearings, ASME J. Basic Eng., 87, No. 3, pp Kogure, K., Kaneko, R., and Ohtani, K., 198, A Study on Characteristics of Surface-Restriction Compensated Gas Bearing With T-Shaped Grooves, BULL. JSME, 5, No. 10, p Fleming, D. P., Cunningham, R. E., and Anderson, W. J., 1970, Zero-Load Stability of Rotating Externally Pressurized Gas-Lubricated Journal Bearings, ASME J. Lubr. Technol., 9, No., pp Pan, C. H. T., Spectral Analysis of Gas Bearing Systems for Stability Studies, in Dynamics and Fluid Mechanics, Vol. 3, Part of Developments in Mechanics, Wiley, New York, pp Journal of Tribology APRIL 00, Vol. 14 Õ 405 Downloaded From: on 01/7/016 Terms of Use: