Journal of Advanced Mechanical Design, Systems, and Manufacturing JSME, Vol

Journal of Advanced Mechanical Design, Vol. 4, No. 1, 2010

Systems, and Manufacturing

Optimum Design of Thrust Air Bearing for Hard Disk Drive Spindle Motor*

Mohd Danial IBRAHIM**, Tadashi NAMBA**, Masayuki OCHIAI*** and Hiromu HASHIMOTO*** **Graduate School of Science and Engineering, Tokai University 1117, Kitakaname, Hiratsuka City, Kanagawa Prefecture 259-1292, Japan E-mail: [email protected] *** Depatment of Mechanical Engineering, Tokai University 1117, Kitakaname, Hiratsuka City, Kanagawa Prefecture 259-1292, Japan

Abstract This paper describes the application of geometry optimization method proposed by Hashimoto to the design of air lubricated thrust bearings used for HDD spindle motors. The optimization is carried out to maximize the dynamic stiffness of air films because the low stiffness is a serious problem of thrust air bearings in the actual application to HDD spindle motor. The optimized dynamic stiffnesses are obtained by changing the allowable film thickness, which is corresponding to the tolerance of bearings. The results obtained show that the optimized thrust air bearings have the comparable stiffness to the oil lubricated thrust bearings and it is verified theoretically that this type of thrust air bearing can be used for HDD spindle motors.

Key words: Hard Disk Drive, Spindle Motor, Thrust Air Bearing, Geometrical Optimum Design, Bearing Stiffness

1. Introduction Hard disk drive (HDD) has been playing a big role in the area of informative multimedia as the main storage media for electronics devices. The back bone of these media storage technology actually lies in the revolutionary part of the spindle motor supported mainly by the oil lubricated thrust and journal bearings. Among various types of thrust bearings used in HDD spindle motors, spiral and herringbone types of grooved bearings are widely used. Although oil lubricated thrust bearing has high dynamic stiffness compared to air lubricated thrust bearing, it is not an exaggeration to say that the current commercialized oil lubricated thrust bearings are coming to its limit where high rotational speed leads to high friction losses, wear, seal problems due to oil leakage, oil contaminations and degradations, as explained by Landsdown(1). The air lubricated thrust bearing which has lower friction losses has a higher probability for reducing the total power consumption, as it poses low environmental load. Furthermore, the air lubricated thrust bearing does not have sealing problems, and its oil-less design makes it a maintenance free equipment. Therefore, high speed air lubricated thrust bearing with high dynamic stiffness and low friction torque is strongly expected to be the next most applicable thrust bearing. Several investigators(2)~(7) tried to analyzed the thrust bearing characteristics for HDD spindle. Most of these papers treated the oil lubricated thrust bearings. Regarding to air lubricated thrust bearings, Xue and Stolarski(8) analyzed the performance of air lubricated spiral groove thrust bearings using the control volume method. Zhang et. al(9) proposed a spindle motor for 3.5" HDD with a combination *Received 3 Aug., 2009 (No. 09-0424) [DOI: 10.1299/jamdsm.4.70] of herringbone type grooved thrust and journal air bearings, in which spindle designs were Copyright © 2010 by JSME considered with a very severe tolerance of 0.5µm to improve bearing stiffness. Despite that

70 Journal of Advanced Mechanical Design, Vol. 4, No. 1, 2010 Systems, and Manufacturing there were only few researches regarding air lubricated thrust bearings in HDD, most of the researches only focus on the available conventional groove geometries such as spiral or herringbone groove as the subject to improve the bearing stiffness. Hashimoto et. al(10),(11) originally developed the optimization method to maximize the dynamic stiffness of air lubricated thrust bearings, but there is no application example of this model for HDD spindle motor. In this paper, a new application of Hashimoto’s method to design an optimal groove geometry for thrust air bearing used for HDD spindle motor is described, and some numerical examples of optimized results were presented. 2. Nomenclature

b1 : groove width [m]

b2 : land width [m] c : damping coefficient of air film [N·s/m] f (X) : objective function

gi(X) : constraint functions; (i =1~16)

ha : allowable film thickness [m]

hg : groove depth [m]

hr : flying height (gap between shaft and lower groove surface) [m]

hu : gap between shaft and upper groove surface [m] K : dynamic stiffness of fluid film [N/m] k : spring coefficient of fluid film [N/m] N : groove number

ns : rotational speed of spindle [rpm]

Pa : atmospheric pressure [Pa]

p0 : static component of fluid film pressure [Pa]

pt : dynamic component of fluid film pressure [Pa]

ri(i =1 ~ 4) : coordinate of radius direction [m]

Rout : bearing outer radius [m]

Rin : bearing inner radius [m]

Rs : seal radius [m]

rs : seal ratio; ( = Rs /R out ) Δr : divided divisions of radius [m]

Tr : frictional torque on the bearing surface [N·m] W : load capacity [N] X : vector of design variables

α : groove width ratio; (=b1 /(b1+ b2)) β : air inflow angle [deg] δ : gap between shaft and upper and lower groove surface [m] μ : viscosity of lubricant [Pa·s] θ : circumference direction of coordinate [deg] ξ : radial coordinate of groove geometry η : angular coordinate of groove geometry Φ : phase angle [deg] th φi : groove geometry design parameter; (displacement for i point of shift from the initial groove geometry to a new point ) [deg]

ωd : natural damped frequency [rad/s]

ωf : squeeze frequency of shaft revolution [rad/s]

ωs : shaft angle velocity [rad/s] Subscripts max : maximum value of stated variables min : minimum value of stated variables

71 Journal of Advanced Mechanical Design, Vol. 4, No. 1, 2010 Systems, and Manufacturing 3. Formulation of Optimum Design and Vibration Analysis 3.1 Modification Method of Groove Geometry In this paper, the initial groove geometry of thrust air bearing is flexibly modified using the cubic spline interpolation function. The cubic spline function is a cubic polynomial equation in each section of (ri, ri+1) (i=1,2,…,n). The condition required for the cubic spline function is a continuity of a second order derivative of the function at each nodal point. The cubic spline interpolation function is expressed as the following equation.

θ" (r ) 3 θ" (r ) 3 θ()r = i ()r − r + i+1 ()r − r 6Δr i+1 6Δr i (1) ⎛ θ" ()r Δr 2 ⎞ r − r ⎛ θ" ()r Δr 2 ⎞ r − r + ⎜θ r − i ⎟ i+1 + ⎜θ r − i+1 ⎟ i ⎜ ()i ⎟ ⎜ ()i+1 ⎟ ⎝ 6 ⎠ Δr ⎝ 6 ⎠ Δr

Considering that θ˝(ri) = 0 for i = 0 and i = n+1, θ˝(ri) (i=1~n) are given as follows;

⎡ 2 1 2 ⎤ ⎡θ" ()r1 ⎤ ⎢ ⎥ ⎡d1 ⎤ ⎢ ⎥ 1 2 2 1 2 ⎢ ⎥ θ" ()r ⎢ ⎥ d θ" = ⎢ 2 ⎥ = A−1d , A = ⎢ % % % ⎥ , d = ⎢ 2 ⎥ (2) ⎢ # ⎥ ⎢ ⎥ ⎢ # ⎥ ⎢ ⎥ 1 2 2 1 2 ⎢ ⎥ θ" r ⎢ ⎥ d ⎣ ()n ⎦ ⎢ ⎥ ⎣ n ⎦ ⎣ 1 2 2 ⎦

where d in Eq. (2) is expressed as the following.

1 d = 2θ" ()r + θ" ()r (3) 1 1 2 2

3()θ(r )−2θ(r )+θ(r ) d = i−1 i i+1 ()i = 2,3,",n −1 (4) i Δr 2

1 d = θ" ()r + 2θ" ()r (5) n 2 n−1 n

An arbitrary groove geometry can be represented by finding the spline function in all sections of the target groove geometry by using Eqs.(1) and (2). 3.2 Governing Equations for Optimization Figure 1 shows an overview of a HDD spindle motor, which is supported by a combination of thrust and journal bearings. Figure 1(a) illustrates a typical HDD structure being considered in this paper which is made of a combination of clamped upper and lower surfaces of thrust bearings shown in Fig. 1(b). Figure 2 shows the initial groove geometry and feasible region for optimum design of the thrust air bearing treated in this paper. In the optimization, the initial geometry is selected as a spiral groove as shown in Fig. 2(a), and the feasible region is divided into four regions in the radial direction as shown in Fig. 2(b), because there is insignificant effect on the optimized results for the number of regions over four. In formulating the optimum design problems on groove geometry, the design variable vector X, consisting of the angle variation φi (i=1~4) in the circumferential direction from the initial geometry to k-th step geometry as shown in Fig. 2(b) and the other bearing dimension parameters rs, hg , and α, are defined as follows.

72 Journal of Advanced Mechanical Design, Vol. 4, No. 1, 2010 Systems, and Manufacturing

X = (rs ,hg ,α,ϕ1 ,ϕ2 ,ϕ3 ,ϕ4 ) (6)

As can be seen from Fig. 2(b), in the optimization process, the most inner point of the curve is fixed, the groove geometry is gradually evolved to optimize the later mentioned objective function from initial stage to the final stage, and the groove geometry is revised step by step using the cubic spline interpolation function. In order to design the thrust air bearing and apply it to the current commercialized HDD, the bearing design parameters have to be constrained with some restrictions based on the total specification of a HDD characteristic. The constraint conditions are shown in the following;

gi ()X ≤ 0 (i =1~ 16) (7)

where the constraint functions in Eq.(7) are defined as follows.

g1 = rs min − rs , g 2 = rs − rs max , ⎫ ⎪ g3 = hg min − hg , g 4 = hg − hg max , ⎪ ⎪ g5 = α min − α , g 6 = α − α max , g 7 = ϕ 1min − ϕ1 , ⎪ ⎬ (8) g8 = ϕ1 − ϕ1max , g9 = ϕ 2min − ϕ 2 ,g10 = ϕ 2 − ϕ 2max , ⎪ g = ϕ − ϕ , g = ϕ − ϕ , g = ϕ − ϕ ,⎪ 11 3min 3 12 3 3max 13 4min 4 ⎪ ⎪ g14 = ϕ 4 − ϕ 4 max , g15 = ha − hr , g16 = −c ⎭

Thrust bearing Shaft Journal bearing

(a) Overview of a spindle motor (b) Initial spiral grooved thrust bearing Fig. 1 Bearings in 2.5" HDD spindle motor

Shaft W Upper groove surface ωs

h Δr Δr Δr Δr u O hg hr r’4 Lower groove surface r’2 r’1 r’3 Rin ϕ4 ϕ1 r1 ϕ2 ϕ3 R k-th step s r 2 r r4 3 θ

R Groove Initial r b out 1 b2 Land (a) Initial geometry (b) Feasible region Fig. 2 Variables of bearing dimensions and geometries

73 Journal of Advanced Mechanical Design, Vol. 4, No. 1, 2010 Systems, and Manufacturing In Eq. (8), besides the groove dimensions and geometries design variables of g1 through g14, the constraint functions also cover the allowable flying height, or tolerance of the bearing in spindle motor to avoid contact and seizures shown as g15. The final constraint function shown as g16 represents the constraint variable to make sure that the damping coefficient of spindle is kept positive at all times to avoid self-induced vibrations. The performance improvement of the thrust air bearing lies in the dynamic stiffness improvements. Therefore, in this paper, the objective function f(X) is defined as follows;

2 2 f ()X = K = k + (ω f c) (9)

where k and c are the spring and damping coefficients of air film, which are calculated based on the equations as shown in Appendix. The optimization problem for HDD thrust bearing is, then, formulated as follows.

max f (Χ ) (10) subject to gi ()Χ ≤ 0 (i =1~16 )

In this paper, the optimization is conducted by Hybrid Method combining direct search method and SQP method(11). The values of the prescribed design parameters based on 2.5” commercialized HDD specifications are shown in Table 1. The optimization was being conducted under the fixed rotational speed of spindle from 10,000 rpm to 30,000 rpm. The prescribed values for the constraint conditions, which are previously mentioned in Eq. (8), are shown in Table 2. These values are mainly being set based on the previous papers by Hashimoto et al.(10)-(13). 3.3 Vibration Analysis The vibration analysis was also being conducted through the simulation under the assumption that the thrust lubricated air bearing supporting the spindle motor is treated as a viscously damped one degree free vibration model. The spring and damping coefficients obtained by the optimum design are then being taken into consideration for the response of the vibration analysis. The equation of motion is given by;

mx + cx + kx= f (t) (11)

Table 1 Prescribed design parameters Table 2 Prescribed constraint values Parameters Values Constraint Values

Rout, mm 4.3 hg min, μm 10.0 Rin, mm 1.0 h , μm 30.0 W, N 0.15 g max

ns, rpm 10,000 ~ 30,000 rs min 0.1 N 16 rs max 0.9 rs 0.58 αmin 0.1 hg, μm 11.0

α 0.636 αmax 0.9 β, deg 18 φi min (i=1~4) -π δ, μm 30.0 -5 φi max (i=1~4) π μair at 60°C, Pa·s 2.002×10 -3 μoil at 60°C, Pa·s 6.090×10 ha, μm 1.0, 2.0 ,3.0

74 Journal of Advanced Mechanical Design, Vol. 4, No. 1, 2010 Systems, and Manufacturing where m is the mass of the disk platters, c and k are the damping and spring coefficients obtained by the above mentioned optimum design, respectively, and f(t) is the impulsive external force being applied to the disk platters. The external impulsive force, which is being applied to the HDD disk platters, is assumed as follows;

⎛ −ζω t ⎞ mω x()t I = ⎜e n ⎟sinω t (12) n ⎜ 2 ⎟ d ⎝ 1− ζ ⎠

where the natural frequency is expressed as ωn = k m ; the damped natural frequency is 2 expressed as ωd = ωn 1−ζ ; the damping ratio and impulse are, respectively, expressed

as ζ = c 2 mk and I = f 0Δt . The nondimensional expressions for the gain G and phase angle Φ are, then, calculated using the following equations.

1 G = (13) 2 2 2 2 {}1− ()ω f ωn + 4ζ ()ω f ωn

−1 2ζ (ω f ωn ) Φ = − tan 2 (14) 1− ()ω f ωn

4. Results and Discussions 4.1 Geometry and Dimension of Optimized Bearings for HDD Spindle Figure 3 shows the bearing geometries and pressure distributions for (a) initial spiral groove bearing and optimized groove bearings at 30,000 rpm with an allowable film thickness of (b) 3.0μm, (c) 2.0μm and (d) 1.0μm, respectively, for the reason that the tolerance factor with respect to air film thickness plays an important role in designing high performance precision machineries. In this figure, the left hand side shows the groove geometries, while the middle and right hand side show the static pressure distributions in the r-θ coordinates and in the ξ coordinate for the fixed value of η = 0, respectively. The optimized bearings as shown in Fig.3 (b)~(d) have a geometry structure basically with steep inclination of a spiral geometry in the inner vicinity; with an opposite spiral geometry in the outer vicinity. The inner division which creates a positive pressure is generated as the fluid flows from the outer vicinity towards the inner vicinity and collides with the seal, hence causing the bearing flying height to rise. This concept is similar with the initial spiral groove pressure generation as shown in Fig. 3(a). On the other hand, as can be seen from the middle and right hand side figures, the outer vicinity creates a "negative pressure", by means of a pressure smaller than the atmospheric pressure is generated. This combination of inner spiral geometry and outer vicinity bends, optimally balances the pressure distribution which will eventually improves the objective function of dynamic stiffness, K. In order to improve the dynamic stiffness of spindle, the clearance has to be severely small to gain performance improvements. In this paper, such types of bearings that have an outer vicinity bends are called modified spiral groove bearings (MSGBs). The details of dimensions of initial spiral groove bearing and MSGBs shown in Fig. 3 are presented in Table 3. From this table, it is understood that all of these optimized values generally pose a geometry which has bends in the outer vicinity, but with different angles of φ1 through φ4. The seal ratio, rs, and groove width ratio, α, also show a slightly different values depending on the prescribed bearing tolerances, which are the allowable film thickness constraints. The optimization is first carried out step by step for the number of grooves from N = 6 to N = 24, which means the feasible region for the number of grooves. After that, the optimum number, which gives the largest dynamic stiffness, is selected from the candidate obtained at the first step.

75 Journal of Advanced Mechanical Design, Vol. 4, No. 1, 2010 Systems, and Manufacturing

p0max = 0.106 MPa ) 0.12

ξ MPa p = 0.106 MPa η ( 0max 0.11 0 p 0.10

ressure Pa p 0.09 p0min = 0.099 MPa η = 0

Static Static 0.08 p0min = 0.099 MPa 0.2 0.4 0.6 0.8 1.0 ξ (a) Initial air spiral groove at 30,000 rpm

p0max = 0.109 MPa ) 0.12

ξ p0max = 0.109 MPa η MPa (

0.11 0 p 0.10

ressure Pa p 0.09 p = 0.096 MPa η = 0 0min

Static Static 0.08 p0min = 0.096 MPa 0.2 0.4 0.6 0.8 1.0 ξ (b) Optimized air at 30,000 rpm; ha=3µm

p0max = 0.114 MPa ) 0.12 η ξ p0max = 0.114 MPa MPa (

0.11 0 p 0.10

ressure Pa

p 0.09 η = 0 p0min = 0.092 MPa

p0min = 0.092 MPa Static 0.08 0.2 0.4 0.6 0.8 1.0 ξ (c) Optimized air at 30,000 rpm; ha=2µm

p = 0.120 MPa 0max ) 0.12 η p0max = 0.120 MPa

ξ MPa (

0.11 0 p 0.10

ressure Pa p 0.09 p0min = 0.085 MPa η = 0

p0min = 0.085 MPa Static 0.08 0.2 0.4 0.6 0.8 1.0 ξ (d) Optimized air at 30,000 rpm; ha=1µm Fig. 3 Groove geometry and pressure distribution of bearings

These changes of groove dimensions and geometries furthermore proved that in order to improve the dynamic stiffness of thrust air bearing in HDD spindle motors, groove geometry should basically be a MSGB, with slightly different values of groove geometries and dimensions for each and every different rotational speed. This unique different characteristics obtained from different groove geometries and dimensions for different rotational speed is indeed a merit for the spindle motor allocated in HDD as the rotational speed for HDD is generally fixed constant upon maneuvers depending on the prescribed bearing tolerances. 4.2 Static and Dynamic Characteristics of Optimized Bearings of HDD Spindle Figure 4(a), (b) and (c) show the relations between the rotational speed and the optimized values of dynamic stiffness K, friction torque Tr and film thickness hr, respectively, in which the green lines show the initial air lubricated spiral groove bearing,

76 Journal of Advanced Mechanical Design, Vol. 4, No. 1, 2010 Systems, and Manufacturing

Table 3 Design variables of initial and optimized bearings at 30,000 rpm Initial spiral Optimized with Optimized with Optimized with Design variables groove ha=3µm ha=2µm ha=1µm

Angle, φ1, deg 0.000 0.031 2.373 4.846

Angle, φ2, deg 0.000 -0.336 -5.371 -10.743

Angle, φ3, deg 0.000 113.808 118.674 122.756

Angle, φ4, deg 0.000 91.763 80.572 69.471 Groove number, N 16 18 17 17

Seal ratio, rs 0.58 0.66 0.55 0.52

Groove depth, hg, μm 11.0 10.8 10.9 10.9 Groove width ratio, α 0.636 0.702 0.653 0.669

the lines with plot of cyan, blue and violet, represent the optimized air bearings for allowable film thicknesses of ha=1μm, ha=2μm, and ha=3μm, respectively. The thick red lines show the results for the oil lubricated spiral groove bearing and the dashed red curves show the results for the oil lubricated bearings used at 4,200 rpm, which is selected as a reference of conventional bearing. As can be seen from Fig. 4(a), the dynamic stiffness characteristics of the optimized MSGB for three types of allowable film thicknesses are improved compared to the initial air spiral groove bearing. The most significant improvement can be seen at allowable film thickness of ha=1μm. However the film thickness is relatively severe, suggesting that an appropriate selection of bearing tolerance is very important for optimum bearing designs. On the other hand, the dynamic stiffness of air lubricated bearings are comparable to those of oil lubricated bearings hence it is found that MSGB is suitable for practical use. Figure 4(b) shows the friction torque between bearing surfaces. The friction torque of optimized MSGB has drastically smaller values than that of conventional oil lubricated spiral groove bearing in spite of very high rotational speed. Even though the dynamic stiffness of MSGB has the same level of oil lubricated bearing, the friction torque is very low. Therefore it can be said that the bearing performance of MSGB is drastically improved compared to that of conventional oil lubricated bearing. Figure 4(c) demonstrates the film thickness from 10,000 rpm to 30,000 rpm. As expressed by the figure, even though the MSGBs have relatively low film thickness compared to oil lubricated spiral groove bearings, all of the values of film thickness for optimized MSGBs are still above the allowable film thickness prescribed as the constraint values, holding up the fact that the MSGBs are functioning without contact with the bearing surfaces avoiding seizures and bearing malfunctions. The figures also demonstrate that by optimizing the groove geometry, despite the high value of flying height, the performance improvement can be obtained well enough depending on the maneuver conditions. These optimized bearing grooves pose convincing characteristics, furthermore supporting the fact that the applicable of the novel geometry of MSGB onto high precision machinery generally, and HDD spindle motors particularly, is promising and highly expected. 4.3 Vibration Analyses Results Figure 5 shows the vibration characteristics of optimized MSGBs optimized at 20,000 rpm with an allowable flying height of 2μm compared with conventional air and oil lubricated spiral groove bearings. As impulsive excitation being applied to the bearings platter disks, air spiral groove bearing shows a continuous damping sinusoidal wave response. On the other hand, the optimized MSGB’s displacement response becomes steady and converges faster compared to that of oil lubricated spiral groove bearing. Therefore it is numerically verified that the impulse response of MSGB is drastically improved compared to that of conventional spiral groove bearings. Although the displacement response of oil lubricated spiral groove bearing is relatively smaller than that of MSGB, oil bearing has other problems such as high friction torque and oil leakage as mentioned before.

77 Journal of Advanced Mechanical Design, Vol. 4, No. 1, 2010 Systems, and Manufacturing

[×105] 8 Optimized; ha=1µm Air spiral Optimized; ha=2µm Oil spiral Optimized; h =3µm Oil spiral(4,200 rpm) 6 a (N/m) K

2 Dynamic stiffness 0 10000 20000 30000 Rotational speed ns (rpm)

(a) Dynamic stiffness K vs. rotational speed ns 10-2 Optimized; ha=1µm Air spiral Optimized; ha=2µm Oil spiral Optimized; h =3µm Oil spiral(4,200 rpm) 10-3 a (N·m) r T 10-4

10-5 Friction torque 10-6 10000 20000 30000 Rotational speed ns (rpm)

(b) Friction torque Tr vs. rotational speed ns

15 m) μ ( r

h 10 Optimized; ha=1µm Air spiral Optimized; ha=2µm Oil spiral Optimized; ha=3µm Oil spiral(4,200 rpm)

5 Film thickness

0 10000 20000 30000 Rotational speed ns (rpm)

Fig. 4 Bearing characteristics vs. rotational speed ns

Figures 5(b) and (c) show the results for gain G and phase angle Φ, respectively. As can be seen from the figures, the gain of MSGB shows a comparable trend compared to oil lubricated bearing, where the response of MSGB decreases slowly with the increase of frequency. The spiral groove air lubricated bearing shows a peak of gain as shown in Fig. 5(b), which means a relatively higher displacement of output is initiated with a relatively small input of external vibrations. On the other hand, it can be seen that MSGBs have superior vibration characteristics than the conventional spiral groove air lubricated bearing towards

78 Journal of Advanced Mechanical Design, Vol. 4, No. 1, 2010 Systems, and Manufacturing

Optimized air (20,000rpm)

0.5 Air spiral (20,000rpm) z/I n ω m 0

Oil spiral (4,200 rpm) -0.5 Oil spiral (20,000rpm)

Displacement Temperature: 65(℃) 0 0.001 0.002 0.003 0.004 0.005 Time t (s) (a) Displacement by impulsive force

0 0 -10 (deg)

Φ G -20

Gain Optimized -100 Optimized Air spiral (20,000 rpm) -30 Air spiral (20,000 rpm) Oil spiral (20,000 rpm) Oil spiral (20,000 rpm) Oil spiral (4,200 rpm) Phase angle Oil spiral (4,200 rpm) -40 100 101 102 103 104 100 101 102 103 104 Frequency f (Hz) Frequency f (Hz) (b) Gain G vs. frequency f (c) Phase angle Φ vs. frequency f

Fig. 5 Vibration characteristics of bearings

frequency response; hence promising the probability of application to HDD compared to conventional grooves geometries. 5. Conclusions In this paper, the applicability of optimum design method of air thrust bearing oriented for small size HDD spindle was theoretically examined. In order to improve the bearing characteristics, the dynamic stiffness of thrust air bearing was set as an objective function and optimum design was conducted. The results obtained are summarized as follows;

i. The bearing tolerance of air lubricated bearings plays an important role in determining the performance of bearings. It has been proven that depending on the prescribed allowable film thicknesses, different values of geometries and dimensions are obtained without jeopardizing the maximization of the objective function of dynamic stiffness. ii. The optimized groove designs obtained through the combination of direct search and SQP method show that the dynamic stiffness of MSGBs is improved compared to the conventional spiral groove air lubricated bearings. In all of the cases shown, the prescribed bearing tolerance values are cleared and comprehended above the prescribed constraints. iii. Despite the dynamic stiffness improvements of MSGBs, the values for friction losses are drastically decreased compared to the conventional oil lubricated spiral groove bearings. iv. It is verified from the vibration analysis that the MSGBs pose better performance compared to conventional spiral groove bearing which gives promising results for future application of MSGBs onto HDD spindle motors towards high precision machinery improvements.

79 Journal of Advanced Mechanical Design, Vol. 4, No. 1, 2010 Systems, and Manufacturing References

[1] Lansdown, A. R., “Lubrication and Lubricant Selection: A Practical Guide,” 3rd Edition, ASME Press, New York, (2004), pp. 37. [2] Jang, G. H., Lee, S. H., and Kim, H. W., “Finite Element Analysis of the Coupled Journal and Thrust Bearing in a Computer Hard Disk Drive,” Journal of Tribology ASME, Vol.128, No. 2(2006), pp. 335-340. [3] Kobayashi, T., and Yabe, H., “Numerical Analysis of a Coupled Porous Journal and Thrust Bearing System,” Journal of Tribology ASME, Vol. 127, No. 1(2005), pp. 120-129. [4] Jang, H. G., and Kim, Y. J., “Calculation of Dynamic Coefficients in a Hydrodynamic Bearing Considering Five Degrees of Freedom for a General Rotor-Bearing System,” Journal of Tribology ASME, Vol.121, No. 3(1999), pp. 499-505. [5] Zhu, J. and Ono, K., “A Comparison Study on the Performance of Four Types of Oil Lubricated Hydrodynamic Thrust Bearings for Hard Disk Spindles,” Journal of Tribology ASME, Vol.121, No. 1(1999), pp. 114-120. [6] Zhang, Q. D., Chen, S. X., and Liu, J., “Design of A Hybrid Fluid Bearing System for HDD Spindles,” IEEE Trans. on Magnetics, Vol. 35, No. 2(1999), pp. 821-826. [7] Roger Ku, C. P., “Effect of Compliance of Hydrodynamic Thrust Bearing in Hard Disk Drive on Disk Vibration,” IEEE Trans. on Magnetics, Vol. 33, No. 5(1997), pp. 2641-2643. [8] Xue, Y., and Stolarski, T.A., “Numerical Prediction of the Performance of Gas-lubricated Spiral Groove Thrust Bearings,” IMechE Proceedings, Professional Engineering Publishing, Vol. 211, No.2, (1997), pp. 117-128. [9] Zhang, Q. D., Guo, G. X. and Chao, B., “Air Bearing Spindle Motor for Hard Disk Drives,” STLE Tribology Transactions, 48 (2005), pp. 468-473. [10] Hashimoto, H., and Ochiai, M., “Optimization of Groove Geometry for Thrust Air Bearing to Maximize Bearing Stiffness (Optimization of Groove Geometry and Its Experimental Verification),” Journal of Tribology ASME, Vol. 130, Issue 3, (2008), 031101, 11 pages. [11] Hashimoto, H., Ochiai, M. and Namba, T., “Theoretical Analysis and Optimum Design of High Speed Air Film Thrust Bearings (Application to Optimum Design Problem),” Journal of Advanced Mechanical Design, Systems, and Manufacturing JSME, Vol. 1, No. 3, (2007), pp. 306-318. [12] Hashimoto, H., and Ochiai, M., “Theoretical Analysis and Optimum Design of High Speed Gas Film Thrust Bearings (Static and Dynamic Characteristic Analysis with Experimental Verifications),” Journal of Advanced Mechanical Design, Systems, and Manufacturing JSME, Vol. 1, No. 1, (2007), pp. 102-112. [13] Hashimoto, H., “Optimum Design of Fluid Film Bearing Elements,” Journal of Japanese Society of Tribologists, Vol. 54, Issue 3, (2009), pp. 20-26.

Appendix The equations governed for theoretical modeling of thrust air bearing is summarized as follows. The control volume of fluid flowing between the bearing surface and the base comes from the effect of rotational and squeeze motion of bearing. The volume control equation of fluid flow is represented by the following equations;

η2 ⎛⎞∂∂pp QpABDEdξ =−+++η (A-1) ∫η ⎜⎟ 1 ⎝⎠∂∂ξη

ξ2 ⎛⎞∂∂pp QpBCFGdη =−++ξ (A-2) ∫ξ ⎜⎟ 1 ⎝⎠∂∂ξη

ξη22∂ ( ph) QJddV = ηξ (A-3) ∫∫ξη 11 ∂t where A until G values are represented by the following equations.

80 Journal of Advanced Mechanical Design, Vol. 4, No. 1, 2010 Systems, and Manufacturing h3 h3 h3 rωh A = a , B = b , C = c , D = − ()r , 12μ J 12μ J 12μJ 2 η

ρrω2h3 rωh ρrω2h3 E = ()rθ , F = ()r , G = − ()rθ , 40μ η 2 ξ 40μ ξ (A-4)

2 2 2 2 a = (rθ ) + (r ) , b ={(rθ )(rθ )+ (r )(r )}, c = (rθ ) + (r ) , η η ξ η ξ η ξ ξ

∂r ∂θ ∂r ∂θ J = r − r ∂ξ ∂η ∂η ∂ξ Here, from the control volume of the law of conservation of mass, the equivalent equation of Reynolds equation is expressed as follows.

ξ ξ ξ ξ η η η η V Q2Ⅰ + Q1Ⅲ - Q2Ⅱ - Q1Ⅳ + Q2Ⅰ + Q1Ⅱ - Q2Ⅲ - Q1Ⅳ = Qi , j (A-5)

The clearance between bearing surface and base is assumed to be supported by the film thickness or flying height of h, while pressure distribution of p is expressed, respectively, as follows;

jω f t h = h0 + εe (A-6)

jω f t p = p0 + εpt e (A-7)

where ε represents the amplitude of an infinitesimal vibration of bearing caused by film thickness, while p0 and pt, respectively, represent the static and dynamic components of pressure distribution. Substituting Eqs. (A-6) and (A-7) into Eq. (A-5), and neglecting small quantities, yields the following two equations.

ξ ξ ξ ξ η η η η F ()p0 = Q2Ⅰ0 + Q1Ⅲ 0 − Q2Ⅱ 0 − Q1Ⅳ 0 + Q2Ⅰ0 + Q1Ⅱ 0 − Q2Ⅲ 0 − Q1Ⅳ 0 (A-8)

G()p = Qξ + Qξ − Qξ − Qξ + Qη + Qη − Qη − Qη − QV (A-9) t 2Ⅰt 1Ⅲt 2Ⅱt 1Ⅳt 2Ⅰt 1Ⅱt 2Ⅲt 1Ⅳt t

Solving Eqs. (A-8) and (A-9) in turns by the Newton-Raphson iteration, the static and dynamic components, p0 and pt, are obtained. The spindle motor’s load capacity and friction torque are given respectively by the following integrations.

r 2π WpPrddr=−2 ()θ (A-10) ∫∫r 0 0 a 1

3 r2 2π ⎛⎞ μωrrhps 00∂ (A-11) Tdrdr =−θ ∫∫r 0 ⎜⎟ 1 ⎝⎠h0 2 ∂θ

Similarly, the spring and damping coefficient of air film are given, respectively, by the following integrations.

r 2π kprddr=−2 Re ()θ (A-12) ∫∫r 0 t 1

1 r2 2π cprddr=−Im ()θ (A-13) ∫∫r 0 t ω f 1