Hakwoon Kim Gunhee Jang Sanghoon Lee. 1 Introduction

Transcription

1 Microsyst Technol (2011) 17: DOI /s TECHNICAL PAPER Complete determination of the dynamic coefficients of coupled journal and thrust bearings considering five degrees of freedom for a general rotor-bearing system Hakwoon Kim Gunhee Jang Sanghoon Lee Received: 29 July 2010 / Accepted: 16 December 2010 / Published online: 11 January 2011 Ó Springer-Verlag 2011 Abstract A complete method is presented for calculating the stiffness and damping coefficients of coupled journal and thrust bearings of a general rotor-bearing system considering five degrees of freedom. The Reynolds equations and their perturbation equations were derived by linearization of the bearing reaction with respect to the general five degrees of freedom, i.e., the tilting displacements and angular velocities as well as the translational displacements and velocities. The Reynolds equations and their perturbation equations were transformed into finite element equations by considering the continuity of pressure and flow at the interface between the journal and the thrust bearings. The Reynolds boundary condition was included in the numerical analysis so as to simulate the phenomenon of cavitation. The stiffness and damping coefficients of the proposed method were compared with those found from a numerical differentiation of the loads with respect to the finite displacements and velocities of the bearing center. It was shown that the proposed method may be used to calculate the dynamic coefficients of coupled journal and thrust bearings more accurately and efficiently than the differentiation method. The tilting motion was also been found to play an important role in the determination of force and moment coefficients. H. Kim G. Jang (&) S. Lee Department of Mechanical Engineering, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul , Republic of Korea ghjang@hanyang.ac.kr 1 Introduction Bearings support the rotor in such a way to allow only rotational motion of a rotor so that reactions are produced to restrict the remaining five degrees of freedom in the bearing. The stiffness and damping coefficients of a bearing are determined in five degrees of freedom, i.e., translation in the x, y, and z directions and rotation in the x and y directions. These coefficients are especially important design considerations in the fluid dynamic bearings (FDBs) of spindle motors in a hard disk drive (HDD) because they determine not only the dynamics of the spinning diskspindle system but also the memory capacity of a disk. FDBs in a HDD have a unique structure comprised of coupled journal and thrust bearings that work to support the radial load as well as the axial load of a rotating diskspindle. Figure 1 shows a rotating shaft-type HDD spindle motor with FDBs. The FDBs are composed of two journal bearings, one thrust bearing, and plain journal and thrust bearings. Because the fluid lubricant is filled in the clearance of the bearing, the pressure generated in the journal part affect the pressure in the thrust part and vice versa. However, the stiffness and damping coefficients in FBDs have thus far not been considered in all five degrees of freedom, even though all components play an important role in determining the dynamic characteristics of a rotor system. In the case of a short shaft, such as a spindle motor driving disk, dynamic characteristics are determined from the moment coefficients as well as from the force coefficients of the bearing. Two methods have been used to calculate the dynamic coefficients of FDBs: physical perturbation and mathematical perturbation of the Reynolds equation. The physical perturbation method determines the dynamic coefficients by numerically differentiating the bearing forces with respect

2 750 Microsyst Technol (2011) 17: Fig. 1 Structure of a 2.5 inch HDD spindle motor with FDBs to finite displacement and finite velocity of the bearing center. However, the accuracy of this method is significantly dependent on the perturbation amplitude. In addition, long computation time may be encountered because each coefficient is determined by dividing the difference of two bearing forces by the difference of the initial and final displacements or velocities. The mathematical perturbation method was proposed by Lund and Thomsen (1978) to overcome the aforementioned disadvantages of the physical perturbation method. The mathematical method determines the dynamic coefficients of a journal bearing using the Taylor expansion of the Reynolds equation and the finite element method (FEM). Jang and Kim (1999) have extended the applicability of this method in order to calculate the dynamic coefficients of journal and thrust bearings for a general rotor-bearing system considering five degrees of freedom. Zheng and Hasebe (2000) applied the free boundary condition to solve the perturbation equations of a journal bearing experiencing cavitation. In their study, separate journal bearings or thrust bearings were considered. Many researchers have been interested in coupled journal and thrust bearings. As such, several methods have been proposed to analyze their static and dynamic characteristics. Zang and Hatch (1995) and Rahman and Leuthold (1996) analyzed the static characteristics of coupled journal and thrust bearings using the finite volume method (FVM) and the FEM, respectively. They also calculated the dynamic coefficients with respect to translational motion using mathematical perturbation. The Half- Sommerfeld boundary condition, which does not include the phenomenon of cavitation, was used in the calculations. Oh and Rhim (2001) analyzed the static characteristics and the dynamic coefficients of coupled journal and thrust bearings using the Reynolds boundary condition. Jang et al. (2006) proposed a finite element method with the Reynolds boundary condition to calculate the static characteristics of coupled journal and thrust bearings. They verified the accuracy of their method by comparing the simulated flying height of the rotor with that of measured results. Jang and Lee (2006) also proposed a mathematical perturbation method for coupled journal and thrust bearings. However, only the force coefficients and not the moment coefficients were included in the analysis. This paper proposes a complete method for calculating the stiffness and damping coefficients of coupled journal and thrust bearings for a general rotor-bearing system with respect to the five degrees of freedom, i.e. the moment coefficients as well as the force coefficients. The perturbation equations of the Reynolds equations were derived for both translational and tilting motions. They were transformed into finite element equations by considering the continuity of pressure and flow at the interface between the journal and the thrust bearings. The Reynolds boundary condition was also included in the numerical analysis so as to simulate cavitation. The stiffness and damping coefficients of the proposed method were compared with those of the physical perturbation of the bearing center. The effect of tilting motion on the moment and force coefficients was also investigated. 2 Methods of analyses 2.1 Governing equations This study extended the method of Jang and Lee by including rotational degrees of freedom in the perturbed equations to investigate the moment coefficients and tilting effect (Jang et al. 2006). Figure 2 shows the coordinate system of the coupled journal and thrust bearings. The governing equations for the journal bearing and the thrust bearing were obtained by transforming the Reynolds equation into the hz and rh planes, respectively. o Roh h 3 op þ o h 3 op 12l Roh oz 12l oz ¼ R_ h oh 2 oh þ oh ot ð1þ Fig. 2 Coordinate system of the coupled journal and thrust bearings

3 Microsyst Technol (2011) 17: Fig. 3 Perturbations applied to the rotor o ror op 12l or r h3 þ o roh h 3 op 12l roh ¼ r _ h 2 oh roh þ oh ot ð2þ where R is the radius of the journal, h _ is the rotational speed of the shaft, h is the film thickness, p is the pressure, and l is the viscosity coefficient. Figure 3 shows the perturbations applied to the rotor at a quasi-equilibrium position. Perturbation equations were derived by substituting into the Reynolds equation a first-order expansion of the film thickness and the pressure with respect to small displacement and velocity. The film thickness, its time derivative, and the pressure may be expanded into the following forms: h ¼ h 0 þ X oh on Dn n ¼ x; y; z; h x; h y oh ot ¼ oh 0 ot þ X oh on D_ n þ X o ot oh on Dn ð3þ ð4þ p ¼ p 0 þ X op on Dn þ X op o n _ D_ n ð5þ where h 0 and p 0 are the film thickness and the pressure in quasi-equilibrium, respectively. By substituting (3), (4), and (5) into (1) and (2), retaining only the first order terms, and separating variables with respect to each perturbed displacement or velocity, the following perturbation equations are obtained for the journal and the thrust bearings, respectively.

4 752 Microsyst Technol (2011) 17: Table 1 Major design parameters of the FDBs Journal bearing Thrust bearing Bearing width (mm) Upper journal: 1.85 Inner diameter: 2.50 Lower journal: 1.45 Radial clearance (lm) 2.0 Axial total clearance (lm) 9.0 Groove pattern Herringbone Spiral Number of grooves 8 20 Groove depth (lm) 5 10 Groove angle (deg) Viscosity (Pas) Rotating speed (rpm) 5400 Outer diameter: 3.05 Fig. 5 Coordinate system for the tilting motion of the rotating part u J ¼ z ð z 0 Þcos h x sin h ð8þ w J ¼ ðz z 0 Þcos h y cos h ð9þ u T ¼ ðz 0 zþsin h x r cos h x sin h ð10þ w T ¼ ðz 0 zþsin h y þ r cos h y cos h ð11þ If the force of the thrust bearing is positive, a is?1. Otherwise, it is -1. The terms u J, w J, u T, and w T in (6.5), (6.6), (6.10), (6.11), (7.5), (7.6), (7.10), and (7.11) are defined as follows: where z 0 is the axial coordinate of the center of mass. The FEM was used to solve the Reynolds equations in (1) and(2), as well as the perturbation equations in (6.1) to (6.11) and (7.1) to (7.11). The global matrix equation of the finite element equation of the Reynolds equations was obtained so as to calculate the pressure of the coupled journal and thrust bearings in quasiequilibrium. Ap 0 ¼ b 0 ð12þ Once the pressure in the fluid film was determined, the global matrix equation of the finite element equation Fig. 4 The finite element model and pressure distribution of the coupled journal and thrust bearings

5 Microsyst Technol (2011) 17: Fig. 6 Variation in the stiffness coefficients of the FDBs due to the eccentricity ratio corresponding to each perturbation equation was determined. This allowed for a calculation of the perturbation pressure of the coupled journal and thrust bearing as follows: Ap n ¼ b n ðn ¼ x; y; z; h x ; h y ; _x; _y; _z; _ h x ; _ h y Þ ð13þ In (13), the matrix is identical to that previously determined in the solution stage of the Reynolds equation, as shown in (12). Equations (8), (9), (10) and (11) show that the tilting displacements and velocities Fig. 7 Variation in the damping coefficients of the FDBs due to the eccentricity ratio affect the perturbed displacements and velocities of both the journal and thrust bearings. 2.2 Boundary conditions External and internal boundary conditions were applied to solve the global matrix equation of the coupled journal and thrust bearings. The external boundary condition is related to the geometry of the FDBs or to the pressure supplied from outside the FDBs. In the present research, two

6 754 Microsyst Technol (2011) 17: external boundary conditions were applied in such a way that the pressure and the pressure change at the air-fluid interface were assumed to be zero and that they were both continuous along the circumferential direction. p n ¼ 0 on C a ð14þ p n j h¼0 ¼ p n j h¼2p ð15þ where C a is the geometric boundary exposed to air and n is 0; x; y; z; h x ; h y ; _x; _y; _z; h _ x,or h _ y. The Reynolds boundary condition was applied as the internal boundary condition. It guarantees the continuity of pressure and the pressure gradient across the cavitated area using an iterative method. p 0 ¼ op 0 on ¼ 0 on C C ð16þ where C C is the boundary of the cavitated area and n is the coordinate normal to C C. It was assumed that a pressure change does not occur inside the cavitated area. As such, the following boundary condition was applied to the cavitated area: p n ¼ 0 inside C C ð17þ where n ¼ x; y; z; h x ; h y ; _x; _y; _z; _ h x ; _ h y. 2.3 Calculation of the load capacity and dynamic coefficients Once the pressure in the fluid film was determined, the bearing forces of the FDBs were calculated by integrating the pressure across the fluid film. The dynamic coefficients of the FDBs can be obtained in two ways. One is the so-called physical perturbation method, which uses the classical definition of dynamic coefficients without solving the perturbation equations. The other method is known as the mathematical perturbation method and employs the solution of the perturbation equation (Jang et al. 2006). The dynamic coefficients of the coupled journal and thrust bearings can be calculated by integrating the pressure change across the fluid film as follows: 8 cos h 0 9 ZZ >< sin h 0 >= K J ¼ 0 p x p y p z p hx p hy dx J ðz z 0 Þsin h 0 >: >; ðz z 0 Þcos h K xx K xy K xz K xhx K xhy K yx K yy K yz K yhx K yhy ¼ ð18þ 6 K hx x K hx y K hx z K hx h x K 7 4 hx h y 5 K hy x K hy y K hy z K hy h x K hy h y 8 9 cos h ZZ >< sin h >= n o C J ¼ 0 p _x p _y p _z p _hx p _hy dx J ðz z 0 Þsin h >: >; ðz z 0 Þcos h 2 3 C xx C xy C xz C xhx C xhy C yx C yy C yz C yhx C yhy ¼ ð19þ 6 C hx x C hx y C hx z C hx h x C 7 4 hx h y 5 C hy x C hy y C hy z C hy h x C hy h y ZZ >< 0 >= K T ¼ a p x p y p z p hx T ra sin h 0 >: >; ra cos h p hy dx ¼ K zx K zy K zz K zhx K zhy 6 K hx x K hx y K hx z K hx h x K 7 4 hx h y 5 ð20þ K hy x K hy y K hy z K hy h x K hy h y ZZ >< 0 >= n o C T ¼ a p _x p _y p _z p _hx p _hy dx T ra sin h 0 >: >; ra cos h ¼ C zx C zy C zz C zhx C zhy 6 C hx x C hx y C hx z C hx h x C 7 4 hx h y 5 ð21þ C hy x C hy y C hy z C hy h x C hy h y where J and T are the bearing areas of the journal bearing and thrust bearing, respectively. The total stiffness or damping coefficients of the coupled journal and thrust bearings are a summation of each component of the stiffness or damping coefficients of the journal or thrust bearing. The zero components of the third row of K J and C J, as well as the zero components of the first and second row of K T and C T indicate that the journal bearing does not support axial loading and that the thrust bearing does not support radial loading (Jang et al. 2006). However, the cross-coupled dynamic coefficients with the subscripts xz, yz, h x z, and h y z were not zero because the axial perturbation of the thrust bearing changed the pressure of the journal bearing as well as the thrust bearing. Similarly, the cross-coupled dynamic coefficients with the subscripts zx, zy, zh x, and zh y were not zero because the

7 Microsyst Technol (2011) 17: Table 2 Variation in K hxh x and C hxh x due to the eccentricity ratio Eccentricity ratio K hxh x (Nm/rad) C hxh x (910-2 Nm s/rad) Physical Mathematical Physical Mathematical Min (%) Max (%) Min (%) Max (%) (-6.37) (0.13) (-0.01) (0.42) (-11.72) (0.11) (-1.40) (46.72) (-3.90) (0.04) (-0.01) (13.41) (-0.28) (34.33) (-1.37) (1.24) (-0.28) (-0.11) (-0.23) (0.28) (-0.52) (5.85) (0.00) (1.53) (-13.45) (12.78) (0.00) (41.66) radial perturbation of the journal bearing and the tilting perturbation of the journal and thrust bearing change the pressure of the thrust bearing as well as the thrust bearing. 3 Results and discussion Table 1 shows the major design specifications of the FDBs used in this research, which were typical design specifications of the FDBs in a 2.5 inch HDD. Figure 4 shows the finite element model and pressure distribution of the coupled journal and thrust bearings. This model consists of two grooved journal bearings, four plain journal bearings, one grooved thrust bearing, and one plain thrust bearing. The fluid film was discretized by 7,000 four-node isoparametric bilinear elements. The accuracy of the developed program was verified by comparing the calculated flying height of the coupled journal and thrust bearings in equilibrium (where the axial load generated by the FDBs was equal to the weight of a rotor) with the measured flying height at various rotating speeds (Jang et al. 2006). Figure 5 shows the coordinate system for the tilting motion of the rotating part. The tilting ratio is defined as follows: coefficients using the physical perturbation method, in which eight values between and were chosen as the perturbed displacement (m, rad) and velocity (m/s, rad/s). Table 2 shows a comparison of the angular dynamic coefficients of the FDBs for the physical and mathematical perturbation methods. As shown in Figs. 6 and 7, and Table 2, the results of the physical perturbation method depend on the applied perturbation amplitudes, especially for the tilting motion. It was also shown that the results of the mathematical perturbation were within the upper and lower limits of the results of the physical perturbation method. In the coupled journal and thrust bearings, cross-coupled dynamic coefficients existed in the radial-axial direction. g ¼ h l 1 2 L ð22þ where L and h l are the length of the journal bearing and the distance between the bearing span center, R 0, and the tilting center, R 1, respectively. If the tilting center was?z from the bearing center, the tilting ratio had a positive sign. Otherwise, it had a negative sign. Figures 6 and 7 show the variation in the stiffness and damping coefficients of the FDBs due to a change in the eccentricity ratio. In the figures, the dotted lines show the stiffness and damping coefficients of the FDBs using the mathematical perturbation method. The vertical bars show the upper and lower limits of the dynamic Fig. 8 Cavitation area in the journal bearing due to the eccentricity ratio

8 756 Microsyst Technol (2011) 17: The clearance of the journal bearing was much smaller than that of the thrust bearing in this model. This resulted in a high pressure for the journal bearing and a low pressure for the thrust bearing. The pressure change of the journal bearing due to axial displacement had little effect on the pressure of the thrust bearing. Therefore, K xz and K yz were almost negligible. However, the pressure change of the journal bearing due to radial displacement changed the pressure of the thrust bearing so that K zx and K zy reached significant values. Figure 8 shows the cavitated area of the journal bearing with eccentricity ratios of 0.4 and 0.5. As shown in Fig. 8b, where the eccentricity ratio is 0.5, the cavitation area is generated in the diverging section of the fluid lubricant of the upper journal bearing. This decreased the axial force of the thrust bearing. Therefore, in Fig. 6b, the values of K zx and K zy at an eccentricity ratio of 0.4 are larger than those of the 0.5 eccentricity ratio. Figures 9 and 10 show the variation in the dynamic coefficients of the FDBs due to a change in the flying height. The eccentricity ratio was assumed to be 0.1. These figures show that the results of the mathematical perturbation are within the upper and lower limits of the results Fig. 9 Variation in the stiffness coefficients of the FDBs due to the flying height Fig. 10 Variation in the damping coefficients of the FDBs due to the flying height

9 Microsyst Technol (2011) 17: of the physical perturbation method. The pressure change of the thrust bearing had little effect on the pressure of the journal bearing because the clearance of the journal bearing was much smaller than that of thrust bearing. Therefore, the dynamic coefficients, with the exception of K zz and C zz, remained almost independent of the flying height. Figures 11 and 12 show the variation of the dynamic coefficients of the FDBs due to a change in the tilting angle. The eccentricity ratio was assumed to be 0.0 so as to exclude the influence of the clearance change due to the eccentricity of the journal. The origin of the tilting angle was the bearing span center, at which the tilting ratio was zero. The range of the tilting angle in this work was between -0.4 and 0.4 (mrad), which is about 45% of the maximum possible tilting angle. If tilting motion was generated at the span center of the journal bearing and the eccentricity ratio was zero, the upper part of the journal bearing moved in a direction opposite to the lower part of the journal bearing. If the areas of the grooved upper and lower journal bearings were the same, the tilting effect of the journal bearing was canceled. However, a different area between the upper and lower grooved journal bearing was used for the model in this research. Therefore, the tilting Fig. 11 Variation in the stiffness coefficients of the FDBs due to the tilting angle Fig. 12 Variation in the damping coefficients of the FDBs due to the tilting angle

10 758 Microsyst Technol (2011) 17: effect in the journal bearing was not perfectly negated. The clearance of the thrust bearing was also changed by the tilting angle. The average clearance of the thrust bearing however did not change because the clearance in one side of the thrust decreases while the clearance in the other side of the thrust increases. Thus, the axial stiffness coefficient, K zz, had almost same due to the variation of tilting angle. The variations in K zx and K zy were caused by the change in the pressure of the journal bearing, especially in the upper journal bearing which was connected directly to the thrust bearing. With an increase or decrease in the tilting angle, the eccentricity of the upper part of the upper journal bearing increased. The absolute value of K zx and K zy then increased. Figures 13 and 14 show the variation in the dynamic coefficients of the FDBs due to changes in the tilting ratio. The eccentricity ratio and the tilting angle were assumed to be 0.1 and 0.0, respectively. As shown in the figures, the translational dynamic coefficients were independent of the tilting ratio. However, the angular dynamic coefficients were strongly affected by the tilting ratio. An increase of the absolute value of the tilting ratio means that the tilting center is located far away from the bearing span center. A Fig. 13 Variation in the stiffness coefficients of the FDBs due to the tilting ratio Fig. 14 Variation in the damping coefficients of the FDBs due to the tilting ratio

11 Microsyst Technol (2011) 17: more distant tilting center resulted in the application of a larger moment of the rotating part. Consequently, as the tilting ratio increased, the absolute values of the angular stiffness and damping coefficients increased. In other words, the FDBs produced larger dynamic coefficients, with respect to the tilting motion, when the tilting ratio increased. 4 Conclusion This paper proposed a method for calculating the stiffness and damping coefficients of coupled journal and thrust bearings when tilting motion is considered. It has been shown that the mathematical perturbation method is more numerically stable and computationally efficient when compared to the physical perturbation method. The stiffness and damping coefficients were investigated with respect to the eccentricity ratio, flying height, tilting angle, and tilting ratio by using both the physical perturbation method and the mathematical perturbation method. The tilting motion was shown to be one of the most important design considerations due to the fact that the tilting motion plays an integral role in the determination of both the force and the moment coefficients. References Jang GH, Kim YJ (1999) Calculation of dynamic coefficients in a hydrodynamic bearing considering five degrees of freedom for a general rotor-bearing system. J TRIBOL T ASME 121(3): Jang GH, Lee SH (2006) Determination of the dynamic coefficients of the coupled journal and thrust bearings by the perturbation method. Tribol Lett 22(3): Jang GH, Lee SH, Kim HW (2006) Finite element analysis of the coupled journal and thrust bearing in a computer hard disk drive. J TRIBOL-T ASME 128: Lund JW, Thomsen KK (1978) A calculation method and data for the dynamic coefficients of oil-lubricated journal bearings. Topics in fluid journal bearing and rotor bearing system. ASME, New York, pp 1 28 Oh SM, Rhim YC (2001) The numerical analysis of spindle motor bearing composed of herringbone groove journal and spiral groove thrust bearing. KSTLE Int J 2(2): Rahman M, Leuthold J (1996) Computer simulation of a coupled journal and thrust hydrodynamic bearing using a finite-element method. In: Proceedings of 25th Annual IMCSD Symposium, The Society, San Jose, pp Zang Y, Hatch MR (1995) Analysis of coupled journal and thrust hydrodynamic bearing using a finite-volume method. ASME AISPS 1:71 79 Zheng T, Hasebe N (2000) Calculation of equilibrium position and dynamic coefficients of a journal bearing using free boundary theory. J TRIBOL T ASME 122(3):