Tribology International

Tribology International ] (]]]]) ]]] ]]] Contents lists available at SciVerse ScienceDirect Tribology International journal homepage: www.elsevier.com/locate/triboint A generalized Reynolds equation and its perturbation equations for fluid dynamic bearings with curved surfaces Hakwoon Kim, Gunhee Jang n, Heonjeong Ha PREM, Department of Mechanical Engineering, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, Republic of Korea article info Article history: Received 2 April 2011 Received in revised form 26 December 2011 Accepted 29 December 2011 Keywords: Reynolds equation Fluid dynamic bearings Curved surface Dynamic coefficient abstract We derived a generalized Reynolds equation and its perturbation equations using meridian and angular coordinates to calculate the characteristics of fluid dynamic bearings (FDBs) with various shapes, namely journal, thrust, conical, and spherical bearings. They were transformed to the finite element equation to calculate the pressure, load, stiffness, and damping coefficients of FDBs. We verified our proposed method by comparing the measured flying height of the coupled conical and journal bearings in the spindle motor of a hard disk drive with the simulated height. Additionally, we investigated their characteristics according to flying height and conical angle. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction Fluid dynamic bearings (FDBs) have been applied to the spindle motor of a computer hard disk drive (HDD) to support the rotating disk spindle system because of their outstanding low noise and vibration characteristics. FDBs in a HDD are required to support both axial and radial loads so that they have coupled structures with journal, thrust, or conical bearings. A rotating shaft-type HDD spindle motor with FDBs composed of two journal bearings, one thrust bearing, and a plain journal and thrust bearing is shown in Fig. 1(a). In this design, the fluid lubricant completely fills the clearance of the coupled journal and thrust bearings so that an air oil interface exists at the upper end of the upper journal bearing. Curved-shape FBDs such as conical and spherical bearings may also be used in HDD spindle motors; these types of bearings can support axial and radial loads at the same time. A fixed shaft-type HDD spindle motor with two conical and two journal bearings is shown in Fig. 1(b). This fixed shaft design has air oil interfaces at both the upper and lower ends of the upper and lower conical bearings, respectively, so there is a greater possibility of oil leakage than in the rotating shaft-design. However, the fixed shaft-type HDD spindle motor is stiffer than the rotating shaft-design HDD spindle motor because the former is tied with the cover of a HDD. The static and dynamic characteristics of coupled journal and thrust bearings have been investigated using several methods. Zang and Hatch [1] analyzed coupled journal and thrust bearings n Corresponding author. Tel.: þ82 2 2220 0431; fax: þ82 2 2292 3406. E-mail address: ghjang@hanyang.ac.kr (G. Jang). using the finite volume method, while Rahman and Leuthold [2] analyzed the static characteristics of coupled journal and thrust bearings using the finite element method (FEM). These latter authors used the Half-Sommerfeld boundary condition that does not describe the cavitation phenomenon. Jang et al. [3] proposed a finite element method with a Reynolds boundary condition to calculate the static characteristics of coupled journal and thrust bearings, and they verified the accuracy of their method by comparing the simulated flying height of the rotor with the measured flying height. Jang and Lee [4] proposed a mathematical perturbation method for coupled journal and thrust bearings. However, none of the methods described above include a Reynolds equation for a general curved surface; these methods can therefore not be applied to analyze FDBs with various curved surfaces. Researchers have proposed several methods to analyze the characteristics of conical or spherical bearings. Bootsma [5] analyzed the characteristics and stability of spherical or conical bearings with spiral grooves; however, he assumed that the FDBs were fully grooved. Yuan and Di-Gong [6] extended Bootsma s research to analyze the performance of partially grooved spherical or conical bearings. Hannon et al. [7] developed a generalized universal Reynolds equation for variable fluid-film lubrication and variable geometry for a self-acting bearing by introducing a spherical coordinate. Choi et al. [8] calculated the characteristics of a FDB with a curved surface using a finite difference method and a physical perturbation method. Hong et al. [9] also calculated the dynamic coefficients and stability of a conical bearing using the FEM and a physical perturbation method, and they verified their simulation results by experiment. However, prior researchers did not investigate the characteristics of coupled FDBs 0301-679X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2011.12.019

2 H. Kim et al. / Tribology International ] (]]]]) ]]] ]]] Nomenclature d distance from the tilting center to the tangent line of the bearing surface (m) h film thickness (m) h 0 film thickness in quasi-equilibrium (m) l distance from the tilting center to the bearing surface (m) P pressure (N/m 2 ) P 0 pressure in quasi-equilibrium (N/m 2 ) P x pressure change in x direction q 0 y,q0 s flow rate per unit length (m 2 /s) S meridian coordinate z 0 axial coordinate of the center of mass (m) [C] damping matrix F x, F y, F z reaction force in x, y, and z direction (N) [K] stiffness matrix M x, M y reaction moment in x and y direction (N m) R radius of journal (m) S distance from the bearing surface to the line that starts from the tilting center and parallel to the radius of curvature (m) T f friction torque (N m) V s, V y velocities in s and y direction (m/s) a index for the direction of axial load Df variation in the tilting angle y angular coordinate _y rotational speed of a rotor (rad/s) m viscosity coefficient (Pa s) t ky shear stress (N/m 2 ) c angle between the axis of rotation and the bearing surface (rad) z angle between the axis of rotation and the line from the tilting center to the bearing surface (rad) boundary of cavitated area G C Subscript x perturbed direction ð¼x, y, z, y x, y y, _x, _y, _z, _ y x, _ y y Þ composed of several bearings with connected groove or plain areas. In these systems, fluid lubricant can circulate through each bearing section, so that the pressure and flow of one section can affect the other sections of the FDBs. Furthermore, previous studies have used the physical perturbation method to calculate dynamic coefficients; this method determines dynamic coefficients by numerically differentiating the bearing forces with respect to finite displacements and finite velocities of the bearing center. However, the accuracy of this method is significantly dependent on the perturbation amplitude [4]. In this paper, we derive a Reynolds equation and its perturbation equations for a generalized curved surface to enable various characteristics of journal, thrust, conical, and spherical FDBs to be determined. We then transform these equations using the finite element equation to calculate the pressure, load, stiffness, and damping coefficients of FDBs. We also incorporate the Reynolds boundary condition in our calculations to simulate the cavitation phenomenon. We verify our proposed method by comparing the measured flying height of coupled conical and journal bearings in the spindle motor of a HDD with the simulated flying height. Additionally, we compare the dynamic coefficients calculated using our proposed mathematical perturbation method with those calculated using the conventional physical perturbation method. Finally, we use our method to investigate the characteristics of coupled conical and journal bearings according to flying height and conical angle. 2. Method of analysis 2.1. Governing equation Fig. 2 shows the generalized coordinates of FDBs with a curved surface. The bearing surface is determined by the angular Fig. 1. Mechanical structure of a HDD spindle motor with FDBs. (a) FDB system composed of coupled journal and thrust bearing and (b) FDB system composed of coupled journal and conical bearing. Fig. 2. Generalized coordinates of FDBs with a curved surface.

H. Kim et al. / Tribology International ] (]]]]) ]]] ]]] 3 Fig. 4. Parameters used to calculate the film thickness of FDBs with a curved surface. Table 1 Major design parameters of the FDBs. Journal bearing Conical bearing Fig. 3. Various types of FDBs. (a) Thrust bearing, (b) journal bearing, (c) conical bearing and (d) spherical bearing. coordinate y and the meridian coordinate s, and the film thickness is determined by k [5]. The radius from the axis of rotation to the bearing surface r(s) determines the shape of the FDB as described in the equation below. The various FDB shapes according to r(s) are shown in Fig. 3. 8 s : thrust bearing >< R : journal bearing rs ðþ¼ ssinc : conical bearing >: Rsin s R : spherical bearing ð1þ where c is the angle between the axis of rotation and the bearing surface in a conical bearing. The equations of flow rate and the continuity equation in the y s plane can be written as follows: q 0 y ¼ 1 1 @p 12m h3 þ V y 2 h ð2þ q 0 s ¼ 1 @p 12m @s h3 þ V s 2 h ð3þ 1 rs ðþ 1 ðþrq0 s þ rs ðþ @ @s rs @ @ @y rq0 y þ @t rh ¼ 0 ð4þ where q 0 y, q0 s, V s, and V y are the flow rates and velocities in the y and s directions, respectively, and h, P, and m are the film thickness, pressure, and viscosity coefficients, respectively. The Reynolds equation in the y s plane can be derived by substituting the flow rate equation in the continuity equation as follows: 1 @! rs ðþ h3 @p 12m @s þ 1 rs ðþ @ @y! h 3 1 @p 12m ¼ V y 2 @h þ @h @t Fig. 4 shows the film thickness and curvature of an FDB with a curved surface. Here O, r, and z 0 are the center of curvature, the radius of curvature, and the tilting center of the rotor, respectively, d is the distance from the tilting center to the tangent line of the bearing surface, and S is the distance from the bearing surface of interest to the line that starts from the tilting center ð5þ Bearing width (mm) 1.15 Upper: 1.25 (at c¼301) Lower: 1.25 (at c¼301) Clearance (lm) 2.0 10.0 Number of grooves 15 Upper: 20 Lower: 15 Groove depth (lm) 5.87 6.38 Groove angle (deg.) 30 13 Area (mm 2 ) 12.62 Upper: 24.26 Lower: 18.30 Groove pattern Spiral Viscosity (Pa s) 0.018 Rotating speed (rpm) 15000 and runs parallel to the radius of curvature. Because it is not necessary for the center of curvature to be located on the axis of rotation or at the tilting center, the bearing surface can be defined by various curved shapes. When the bearing has a motion with five degrees of freedom (x, y, z, y x, and y y ), the film thickness of FDBs with curved surfaces can be calculated as follows: h ¼ h 0 þxcosycoscþysinycoscþa S siny x sinyþsiny y cosy þd 2 cosy x cosy y þzsinc ð6þ where h 0 is the film thickness in quasi-equilibrium and a is the index to determine the direction of the reaction force of the FDBs. If the axial force of the FDBs is positive, a is þ1. Otherwise, a is 1. The first three and the last terms of Eq. (6) indicate the change in film thickness due to translational motion while the fourth and fifth terms indicate the change in film thickness due to rotational motion. Perturbation equations were derived by substituting a firstorder expansion of the film thickness and pressure with respect to small displacements and velocities in the Reynolds equation. The film thickness, its time derivative, and pressure were expanded as follows: h ¼ h 0 þ X @h @x Dx x¼ x, y, z, y x, y y @h @t ¼ @h 0 @t þ X @h @x D x _ þ X @ @h Dx @t @x p ¼ p 0 þ X @p @x Dxþ X @p @ x _ D x _ where P 0 is the pressure in quasi-equilibrium. ð7þ ð8þ ð9þ

4 H. Kim et al. / Tribology International ] (]]]]) ]]] ]]] By substituting Eqs. (7) (9) into Eq. (5), retaining only the first order terms, and separating variables with respect to each perturbed displacement or velocity, the following perturbation equations were obtained for FDBs with a curved surface [4]: applied to the cavitated area: p x ¼ 0 inside G C ð16þ rs ðþ h3 @p 0 x 12m @s þ 1 @ h 3 0 12m 1 @p x rs ðþ @s 8 V y 1 @h 0 2 : x ¼ 0 ð10-1þ 1 @ rs ðþ h2 0 4m @p 0 cosy @s 1 @ h 2 0 4m 1 @p 0 cosy þ V y 1 @ cos y 2 rs ðþ @y cosc : x ¼ x ð10-2þ 1 @ rs ðþ h2 0 4m @p 0 siny @s 1 @ h 2 0 4m 1 @p 0 siny þ V y 1 @ sin y 2 rs ðþ @y cosc : x ¼ y ð10-3þ a 1 @ rs ðþ h2 0 4m @p 0 @s 1 @ h 2 0 4m 1 @p 0 sinc : x ¼ z ð10-4þ >< a 1 @ rs ðþ h2 0 a ¼ 4m @p 0 @s 1 @ h 2 0 a 4m 1 @p 0 rs ðþ @s þ V y 1 @a 2 : x ¼ y x ð10-5þ a 1 @ rs ðþ h2 0 b 4m @p 0 @s 1 @ h 2 0 b 4m 1 @p 0 rs ðþ @s þ V y 1 @b 2 : x ¼ y y ð10-6þ 1 @ cosycosc : x ¼ _x ð10-7þ sinycosc : x ¼ _y ð10-8þ asinc : x ¼ _z ð10-9þ aa : x ¼ y _ x ð10-10þ >: ab : x ¼ y _ y ð10-11þ The terms a and b are defined as follows: a ¼ dsiny x Scosy x siny ð11þ b ¼ dsiny y þscosy y cosy ð12þ FEM was used to solve the general Reynolds equation in Eq. (5) as well as the general perturbation equations in Eq. (10). The global matrix equation of the finite element equation of the general Reynolds equations can be used to calculate the pressure of FDBs with a curved surface in quasi-equilibrium. Once the pressure in the fluid film is determined, the global matrix equation of the finite element equation corresponding to each perturbation equation can be determined. 2.2. Boundary conditions External and internal boundary conditions were applied to calculate the pressure distribution in FDBs using the finite element method. The external boundary conditions are related to the geometry of the FDBs or to the pressure supplied from outside the FDBs. In our study, we applied two external boundary conditions. One was that the pressure and pressure change at the air fluid interface were assumed to be zero, and the other was that the pressure and the pressure change were both continuous along the circumferential direction: p x ¼ 0 ong a ð13þ p x 9 y ¼ 0 ¼ p x 9 y ¼ 2p ð14þ where G a is the geometric boundary exposed to air. The Reynolds boundary condition was applied as the internal boundary condition. The continuity of pressure and the pressure gradient across the cavitated area was guaranteed using an iterative method [10,11]: p 0 ¼ @p 0 @n ¼ 0 ong C ð15þ where G C is the boundary of the cavitated area and n is the coordinate normal to G C. We assumed that a pressure change did not occur inside the cavitated area. As such, the following boundary condition was 2.3. Calculation of static characteristics and dynamic coefficients The reaction force and moment were obtained by integrating the pressure, p, along the fluid film as follows: F X ¼ pcosycoscdo ð17 1Þ F Y ¼ psinycoscdo F Z ¼ psincdo M X ¼ SpsinydO M Y ¼ SpcosydO ð17 2Þ ð17 3Þ ð17 4Þ ð17 5Þ The shear stress, t ky, was determined as in Eq. (18), and the friction torque was obtained by integrating along the fluid film as follows: t ky 9 k ¼ h ¼ h @p 2 T f ¼ rs ðþt ky 9 k ¼ h do þm rs ðþ_ y h ð18þ ð19þ We calculated dynamic coefficients using our mathematical perturbation method that numerically integrates pressure changes across the fluid film as follows: 8 9 cosycosc ½KŠ¼ >< >: sinycosc asinc assiny ascosy >= n p x p y p z p yx p yy >; o do

H. Kim et al. / Tribology International ] (]]]]) ]]] ]]] 5 2 3 K xx K xy K xz K xyx K xyy K yx K yy K yz K yyx K yyy ¼ K zx K zy K zz K zyx K zyy 6 K yxx K yxy K yxz K yxyx K 7 4 yxyy 5 K yyx K yyy K yyz K yyyx K yyyy 8 9 cosycosc >< sinycosc >= n ½CŠ¼ asinc p _x p _y p _z p yx _ assiny >: >; ascosy 2 3 C xx C xy C xz C xyx C xyy C yx C yy C yz C yyx C yyy ¼ C zx C zy C zz C zyx C zyy 6 4 C yxx C yxy C yxz C yxyx C 7 yxyy 5 p _ yy o do ð20þ ð21þ of the rotor at standstill. The rotor weight of the HDD spindle system with a dummy disk was 138.5 g. Once the HDD spindle system started to rotate, the flying height was measured using a capacitance sensor with 5 nm resolution. The flying height at steady state was determined at 25 s after the HDD spindle system started to rotate. The measured flying heights, which were 8.82 mm, 9.10 mm, 9.27 mm, and 9.39 mm at 8000, 12000, 15000, C yyx C yyy C yyz C yyyx C yyyy 3. Simulated model and verification We applied our proposed method to coupled conical and journal bearings of the spindle motor of the HDD shown in Fig. 1(b). The major design specifications are listed in Table 1. The model consisted of separate upper and lower bearings composed of two spiral grooved conical bearings and a spiral grooved journal bearing connected by plain conical and journal bearings. The finite element model and the calculated pressure distribution are shown in Fig. 5. The fluid film was discretized by four-node isoparametric bilinear elements. The total numbers of elements and degrees of freedom is 5280 and 6120, respectively. The accuracy of the developed finite element model was verified by comparing the convergence of pressure, load capacity, and friction torque with the increase of the number of elements. When the HDD spindle system starts to rotate, the rotating part of the HDD spindle system moves upward to the equilibrium position until the weight of the rotating parts is equal to the axial load generated by the FDBs. Therefore, the flying height can be estimated if the weight of the rotating part of the HDD spindle system and the variation of the axial load capacity of the FDBs due to the flying height are known. We verified our proposed method by comparing the analytical results of the flying height at various rotating speeds with experimental results. A dummy disk with additional mass was used to measure the flying height at various rotating speeds as well as to determine the lowest axial position Fig. 6. Flying heights of the spindle at various rotating speeds. (a) Analytical result and (b) experimental result. Fig. 5. Finite element model and calculated pressure distribution (c¼301). (a) Finite element model and (b) calculated pressure distribution.

6 H. Kim et al. / Tribology International ] (]]]]) ]]] ]]] and 18000 rpm, respectively, are shown in Fig. 6. The calculated axial load capacity according to flying heights of 9000, 12000, 15000, and 18000 rpm is shown in Fig. 6(a); the dotted line in this figure indicates the weight of the rotor. In Fig. 6(b), the intersection between the weight line and the calculated load capacity represents the simulated flying heights of the rotor, which were 8.73 mm, 8.88 mm, 9.16 mm, and 9.25 mm for rotating speeds of 9000, 12000, 15000, and 18000 rpm, respectively. The measured and simulated flying heights at various rotating speeds are presented in Table 2; it is clear that the simulated flying heights match well with the measured ones. Thus, our proposed method can accurately predict the static characteristics of curved FDBs. The stiffness and damping coefficients of the coupled conical and journal bearings according to flying height as calculated by our mathematical perturbation method and by the physical perturbation method are shown in Tables 3 and 4, respectively. Using the physical perturbation method, the perturbed displacement was calculated as 1.0 10 7 m, which is 1/100th of the clearance of the conical bearing, and the velocity was calculated as 1.0 10 7 m/s. The perturbed angle was 5.0 10 6 rad, which Table 2 Comparison of the measured flying heights with the calculated flying heights at various rotating speeds. Rotating speed (rpm) Measured result (lm) Calculated result (lm) Error (%) 9000 8.82 8.73 0.99 12000 9.10 8.88 2.47 15000 9.27 9.16 1.16 18000 9.39 9.25 1.50 is 1/100th of the possible tilting angle with respect to the shaft center. The angular velocity was 5.0 10 6 rad/s. The dynamic coefficients were determined by dividing the difference between the two bearing forces by the difference in the initial and final displacements or velocities. Tables 3 and 4 show that the dynamic coefficients estimated using our mathematical perturbation method match well with those calculated using the physical perturbation method. 4. Results and discussion In this paper, we investigated the characteristics of coupled conical and journal FDBs according to changes in conical angle and flying height. The bearing area of the conical bearing remained constant while the width of the conical bearing changed to keep the volume of the fluid lubricant in the FDBs constant. The finite element models for conical angles of 01, 301, 601, and 901 are shown in Fig. 7. The eccentricity ratio and tilting angle were assumed to be 0.1 and 01, respectively. The radial load of the coupled conical and journal FDBs according to changes in the conical angle and flying height is shown in Fig. 8(a). The conical bearing can support both radial and axial loads at the same time. The clearance of the journal bearing in this model was smaller than that of the conical bearing; thus the pressure change of the conical bearing due to the change in flying height barely affected the journal bearing. Furthermore, the pressure change of the journal bearing was almost independent of flying height because the clearance of the journal bearing remained constant as the flying height changed. Therefore, the radial load of the coupled conical and journal FDBs decreased as the conical angle increased. The axial loads of the Table 3 Comparison of the stiffness coefficients for various flying heights calculated using the mathematical perturbation method with those calculated using the physical perturbation method (c¼301). Flying height (lm) 7 8 9 10 11 12 13 K xx ( 10 7 N/m) Mathematical perturbation 2.483 2.452 2.438 2.435 2.438 2.452 2.483 Physical perturbation 2.476 2.444 2.430 2.426 2.430 2.444 2.476 Error (%) 0.28 0.33 0.33 0.37 0.33 0.33 0.28 K zz ( 10 5 N/m) Mathematical perturbation 20.011 13.254 10.122 9.203 10.122 13.245 20.011 Physical perturbation 19.777 13.141 10.076 9.205 10.170 13.371 20.250 Error (%) 1.17 0.85 0.45 0.02 0.47 0.95 1.19 K hxhx ( 10 2 Nm/rad) Mathematical perturbation 2.052 1.811 1.699 1.666 1.699 1.811 2.052 Physical perturbation 2.040 1.798 1.688 1.655 1.689 1.799 2.042 Error (%) 0.59 0.72 0.65 0.66 0.59 0.66 0.49 Table 4 Comparison of the damping coefficients for various flying heights calculated using the mathematical perturbation method with those calculated using the physical perturbation method (c¼301). Flying height (lm) 7 8 9 10 11 12 13 C xx ( 10 4 N s/m) Mathematical perturbation 3.663 3.389 3.247 3.203 3.247 3.389 3.663 Physical perturbation 3.664 3.39 3.247 3.203 3.247 3.39 3.664 Error (%) 0.03 0.03 0.00 0.00 0.00 0.03 0.03 C zz ( 10 3 N s/m) Mathematical perturbation 14.187 11.148 9.615 9.146 9.615 11.148 14.187 Physical perturbation 14.187 11.148 9.616 9.147 9.616 11.148 14.187 Error (%) 0.00 0.00 0.01 0.01 0.01 0.00 0.00 C hxhx ( 10 1 Nm s/rad) Mathematical perturbation 5.372 4.612 4.217 4.094 4.217 4.612 5.372 Physical perturbation 5.369 4.608 4.213 4.09 4.213 4.608 5.369 Error (%) 0.06 0.09 0.10 0.10 0.10 0.09 0.06

H. Kim et al. / Tribology International ] (]]]]) ]]] ]]] 7 Fig. 7. Finite element models of the coupled conical and journal bearings according to the conical angle. (a) c¼01, (b) c¼301, (c) c¼601 and (d) c¼901. Fig. 8. Radial load, axial load, and friction torque according to flying height and conical angle. (a) Radial load capacity, (b) axial load capacity and (c) friction torque. coupled conical and journal FDBs according to different conical angles and flying heights are shown in Fig. 8(b). Only the conical bearing of the coupled conical and journal bearing generated axial load to support the rotor, and the axial load increased as the conical angle increased. Furthermore, the conical bearing of the coupled conical and journal bearing generated zero axial load

8 H. Kim et al. / Tribology International ] (]]]]) ]]] ]]] when the conical angle was equal to zero, because the conical bearing turned to become the journal bearing. The upper and lower bearings were identically symmetrical, so that the axial load was symmetric with respect to zero axial load, corresponding to a flying height of 10 mm. The friction torque of the coupled conical and journal FDBs due to changes in flying height is shown in Fig. 8(c). Friction torque was calculated using the product of the shear stress of the fluid lubricant and the distance from the axis of rotation to the bearing surface, as shown in Eq. (19). The shear stress was proportional to the pressure gradient and the rotating linear velocity, but it was mainly dependent on the rotating linear speed. Thus, the friction torque increased as the conical angle increased because the latter increased the rotating linear velocity at the outer area of conical bearing. The direct stiffness and damping coefficients in the radial direction due to changes in the conical angle and flying height are shown in Fig. 9. The eccentricity ratio and tilting angle were assumed to be 0.1 and 01, respectively. The variation in flying height was affected not by the film thickness of the journal bearing, but by the conical bearing. Fig. 9(a) shows that the radial direct stiffness coefficient of this model was maximal when the conical angle was equal to 301. However, the variation in radial direct stiffness due to the conical angle was less than 2.3%. The clearance of the conical bearing was 5-fold larger than that of the journal bearing; therefore the journal bearing generated most of the radial load. Fig. 9(b) shows that the radial direct damping coefficient of this model was maximal for a medium flying height where the conical angle was equal to 01. This can be explained by the fact that the perturbation equations with respect to radial perturbed velocity as shown in (10-8) and (10-9) are independent of the rotational velocity and are proportional to cos c. When the flying height was higher than 12 mm or lower than 8 mm, the radial damping coefficient when the conical angle was 301 was slightly larger than that for a conical angle of 01. Fig. 10 shows the direct stiffness and damping coefficients in the axial direction according to changes in the conical angle and flying height. Fig. 10(a) shows that the axial direct stiffness coefficient increased as the conical angle increased. The axial direct stiffness was zero when the conical angle was equal to 01, because the conical bearing with a conical angle of 01 turned to become a journal bearing. The axial direct stiffness was maximal when the conical angle was equal to 901, because the conical bearing with a conical angle of 901 turned to become the thrust bearing. Fig. 10(b) shows that the axial direct damping coefficient of this model had a maximum value when the conical angle was equal to 901. This can be explained by the fact that the perturbation equations with respect to axial perturbed velocity in (10-10) are independent of the rotational velocity and are proportional to sin c. Fig. 11 shows the direct stiffness and damping coefficients in the tilting direction according to changes in the conical angle and flying height. The tilting direct stiffness and damping coefficients of this model had maximum values when the conical angle was equal to 301. This can be explained by the change and the rate change in film thickness according to tilting angle, as shown in Fig. 12. The change in film thickness can be defined as follows: Dh ¼ ldjcosðz cþ ð22þ Fig. 9. Radial direct stiffness and damping coefficients according to flying height and conical angle. (a) K xx and (b) C xx. Fig. 10. Axial direct stiffness and damping coefficients according to flying height and conical angle. (a) K zz and (b) C zz.

H. Kim et al. / Tribology International ] (]]]]) ]]] ]]] 9 Fig. 11. Tilting direct stiffness and damping coefficients according to flying height and conical angle. (a) K yx y x and (b) C yx y x. Fig. 12. Change of film thickness according to the conical angle and tilting center. where l, z, and Df are the distance from the tilting center to the surface of the FDBs, the angle between the axis of rotation and the line from the tilting center to the surface of the FDBs, and the variation in the tilting angle, respectively. The decrease in film thickness on the left side of the FDBs increased the film thickness on the right side, which increased the restoring moment of the FDBs. According to Eq. (22), the change in film thickness is maximal when c is equal to z. The value of z in this model was 29.271; a conical bearing with a conical angle of 301 has maximum tilting stiffness and damping coefficients. 5. Conclusions We derived the Reynolds equation and its perturbation equations to calculate the characteristics of various curved FDBs such as journal, thrust, conical, and spherical bearings. They were then transformed to the finite element equation to calculate the pressure, load, stiffness, and damping coefficients of FDBs. We also included the Reynolds boundary condition in our model to simulate the cavitation phenomenon. We verified our proposed method by comparing the measured flying height of the coupled conical and journal bearings in the spindle motor of a HDD with the simulated flying heights. Furthermore, we verified our model by comparing the dynamic coefficients from our mathematical perturbation method with those derived using a conventional physical perturbation method. We applied our method to investigate the characteristics of coupled conical and journal bearings according to flying height and conical angle. We anticipate that this research will contribute to the design and development of robust FDBs of various shapes. Acknowledgment This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (2010-0021919). References [1] Zang Y., Hatch M.R. Analysis of coupled journal and thrust hydrodynamic bearing using finite volume method. Proceedings of the ISPS/ASME, vol. 1. San Jose, CA; 1995. p. 71 80. [2] Rahman M., Leuthold J. Computer simulation of a coupled journal and thrust hydrodynamic bearing using a finite-element method. Proceedings of the 25th Annual IMCSD Symposium. San Jose, CA; 1996. p. 103 12. [3] Jang GH, Lee SH, Kim HW. Finite element analysis of the coupled journal and thrust bearing in a computer hard disk drive. Journal of Tribology: Transactions of the ASME 2006;128:335 40.

10 H. Kim et al. / Tribology International ] (]]]]) ]]] ]]] [4] Jang GH, Lee SH. Determination of the dynamic coefficients of the coupled journal and thrust bearings by the perturbation method. Tribology Letters 2006;22(3):239 46. [5] Bootsma J. Liquid-lubricated spiral-groove bearings. Phillips Research Reports Supplements 1975;7. [6] Yuan H, Di-Gong C. Effects of partial-grooving on the performance of spiral groove bearings: analysis using a perturbation method. Tribology International 1996;29(4):281 90. [7] Hannon W, Braun M, Hariharan S. Generalized universal Reynolds equation for variable properties fluid-film lubrication and variable geometry selfacting bearings. Tribology Transactions 2004;27(2):171 81. [8] Choi WC, Shin YH, Choi JH. Numerical analysis of stiffness of self-acting air bearings of various curvatures. JSME International Journal 2001;44(2): 470 5. [9] Hong G, Xinmin L, Shaoqi C. Theoretical and experimental study on dynamic coefficients and stability for a hydrostatic/hydrodynamic conical bearing. Journal of Tribology: Transactions of the ASME 2009;131(4):041701. [10] Zheng T, Hasebe N. Calculation of equilibrium position and dynamic coefficients of a journal bearing using free boundary theory. Journal of Tribology: Transactions of the ASME 2000;122(3):616 21. [11] Hamrock BJ. Fundamentals of fluid film lubrication. New York: McGraw-Hill; 1994.