Hindawi Publishing Corporation International Journal of Rotating Machinery Volume 2013, Article ID 896148, 11 pages http://dx.doi.org/10.1155/2013/896148

Research Article Optimum Design of Oil Lubricated Thrust for Hard Disk Drive with High Speed Spindle Motor

Yuta Sunami,1 Mohd Danial Ibrahim,2 and Hiromu Hashimoto1

1 Department of Mechanical Engineering, Tokai University, 4-1-1 Kitakaname, Hiratsuka-shi, Kanagawa-ken 259-1292, Japan 2 Department of Mechanical and Manufacturing Engineering, Faculty of Engineering, Universiti Malaysia Sarawak, 94300 Kota Samarahan, Sarawak, Malaysia

Correspondence should be addressed to Yuta Sunami; [email protected]

Received 29 July 2013; Accepted 15 October 2013

Academic Editor: Masaru Ishizuka

Copyright © 2013 Yuta Sunami et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper presents the application of optimization method developed by Hashimoto to design oil lubricated thrust bearings for 2.5 inch form factor hard disk drives (HDD). The designing involves optimization of groove geometry and dimensions. Calculations are carried out to maximize the dynamic stiffness of the thrust bearing spindle motor. Static and dynamic characteristics of the modeled thrust bearing are calculated using the divergence formulation method. Results show that, by using the proposed optimization method, dynamic stiffness values can be well improved with the bearing geometries not being fixed to conventional grooves.

1. Introduction spindlebyintroducingpermanentmagneticthrustplatesinto thebearingspindlestructure[6]orintroducingmagnetic HDD has been used as the main storage multimedia for fluid as lubricants [7]. However, there are very few attempts at electronics devices. Currently, HDD widely depends on oil finding an optimum design with a novel geometry to replace lubricated hydrodynamic bearings. Hydrodynamic bearing the conventional herringbone or spiral grooves for HDD. In is mainly supported by thrust and journal bearings. A the attempts at improving HDD performance, Arakawa et al. schematicviewofbearingsina2.5inchHDDisshownin [8] proposed a nonuniform spiral groove for journal bearings Figures 1 and 2. to expand the critical bearing number for higher revolution Hydrodynamic bearing gives better performance char- speed of HDD spindle. The approach indicated that the novel acteristics compared to conventional ball bearings as it nonuniform spiral grooves hydrodynamic bearings manage has high dynamic stiffness with additional much higher to increase the stability of rotation in high speeds. damping effects. These damping effect characteristics provide This suggests that there is a probability of a further smaller nonrepeatable run-out (NRRO). NRRO is the major improvement if the groove geometry is not being fixed to any contributor to the track misregistration in HDD read-write conventional grooves, either spiral or herringbone grooves. mechanism. Even though the repeatable run-out (RRO) of However, as far as authors know, there are no attempts at oil lubricated hydrodynamic bearing is higher than the ball drastically improving the bearing characteristics changing bearing spindle motor, the RRO can be corrected by a read- the geometry and dimensions for oil lubricated 2.5 inch write servo. Therefore, to reduce NRRO and increase spindle HDD spindle motor. Recently, HDD is being demanded to performance is to improve the bearing performance. be thinner. If the characteristic of thrust bearing can be Currently, groove geometries that are being widely used drastically improved, it is possible for HDD to be thinner. in HDD thrust bearings are mainly a spiral or a herringbone Therefore, in this paper, by adapting the optimization method grooved geometries. Several investigators have conducted initiated by Hashimoto, new optimum groove geometry and numerical analysis predictions of these grooves for HDD dimension to replace conventional grooves and increase the bearing performance [1–5]. There were also some attempts bearing performance of an oil lubricated 2.5 inch HDD had at improving the dynamic stiffness and damping of HDD been calculated. The groove geometry and dimension of the 2 International Journal of Rotating Machinery

Thrust bearing Journal bearing Table 1: Initial 2.5 inch bearing parameters.

Parameters Values 𝑅 2.55 × 10−3 Outer radius out (m) 𝑅 1.0 × 10−3 Inner radius in (m) Weight 𝑊𝑑 (N) 0.185

Revolution speed 𝑛𝑠 (rpm) 5000∼20,000 𝑁 Figure 1: Bearings in spindle motor. Groove number 16 Seal ratio 𝑟𝑠 0.58 −5 Groove depth ℎ𝑔 (m) 1.0 × 10

Groove width ratio 𝛼=(𝑏1/𝑏1 +𝑏2) 0.5

Geometry parameter 𝜑1 (deg) 0.0

Geometry parameter 𝜑2 (deg) 0.0

Geometry parameter 𝜑3 (deg) 0.0

Geometry parameter 𝜑4 (deg) 0.0 −3 Lubricant viscosity 𝜇 (Pa⋅s) 6.523 × 10

Shaft W 𝜔s

hr (a) hg

rs =Rs/Rout

R 𝛼=b1/(b1 +b2) in

Rs b1

b2 Land R out Groove

Figure 3: Design parameters for thrust bearing.

(b) method to obtain the optimum solutions for nonlinear Figure 2: Initial spiral thrust bearing groove geometry and pressure optimization problems. However, it sometimes can not find distribution; (a) groove geometry, (b) pressure distribution. the global optimum solution when the objective function has the multipeaks. Therefore, the direct search method is used first to roughly search for several local optimum solutions; thrust bearing for spindle motor are calculated using the then, SQP is used to find the local optimum solutions hybrid method [9, 10] with the improvement of dynamic 𝐾 when we set the initial values obtained by the direct search stiffness , being set as the objective function. method. The global optimum solution is finally obtained by The optimization results showed that the dynamic stiff- choosing the maximum or minimum value. Furthermore, the ness of oil lubricated thrust bearing can be improved by confirmation of the optimum solution is needed to clarify the introducing a new optimum geometry into the system. validity of the optimum solution. The validity of the optimum Vibration analyses were also presented to numerically verify solution is confirmed by comparing the value of the objective the applicability of the improved bearings. function calculated from the optimum design variables with those from randomly selected design variables. 2. Geometrical Optimization and Figure 3 shows the design parameters used in this opti- Analysis of Bearing Characteristics mization. The initial prescribed design parameters for thrust bearing in this paper, based on a commercialized 2.5 inch 2.1. Modification of Groove Geometry. The geometrical opti- form factor HDD design, are shown in Table 1.Takingthe mization process is carried out using the hybrid method spiral groove geometry as a threshold groove geometry, the [9, 10], a combination of direct search method and successive groove geometry parameters of angle variation 𝜑1 through 𝜑4 quadratic programming (SQP) method. SQP is a powerful are initially set to zero. International Journal of Rotating Machinery 3

From there, the groove geometry design parameters Δr Δr Δr Δr are changed arbitrarily using the cubic spline interpolation function [9]. The cubic spline interpolation function is a cubic O polynomial equation where each section connected to nodal 󳰀 points is a continuity of a second order derivative of the r4 function. Figure 4 shows the spiral groove geometry represented 󳰀 r as initial values in the feasible region for optimum design of O󳰀󳰀 2 󳰀 r3 thrust bearing treated in this paper. Referring to the same 󳰀 figure,theinitiallinerepresentingthespiralgroovegeometry, r1 k th step is shown with the label initial groove geometry while the 𝜑1 𝜑4 𝜑 𝜑3 new line of geometry after the spline interpolation function is r1 2 calculated step by step as shown with the label 𝑘th step. The 𝜑 𝜑 groove geometry parameter of 1 through 4 is set starting r2 r4 from inner region towards the outer region. The feasible r3 region for the groove geometry is divided equally into four regions in the radial direction as there were no significant effects on the optimization with regions over the divisions of 𝜃 Initial groove four. geometry With the most inner point of the initial spiral groove 󸀠󸀠 curve fixed at point 𝑂 , the angle variation of groove geometry is then evolved gradually from initial stage to the r final stage. Figure 4: Modifications of groove geometry.

2.2. Analysis of Bearing Characteristics. In this paper, apply- The spring coefficient 𝑘 and damping coefficient 𝑐 can be ing the boundary-fitted coordinate system to adjust the obtained, respectively, by integrating the real and imaginary geometrical optimization method derives the calculation partsofthedynamicpressurecomponents,𝑝𝑡, as follows: method of bearing characteristics. Moreover, in the process 𝑟 2𝜋 of analyzing the static and dynamic characteristics of the 2 𝑘=∫ ∫ Re (−𝑝𝑡)𝑟𝑑𝜃𝑑𝑟 bearings, the perturbation method is applied to the Reynolds 𝑟1 0 equivalent equation. The Reynolds equivalent equation can 𝑟 2𝜋 (4) 1 2 be solved using the Newton-Raphson iteration method, and 𝑐= ∫ ∫ (−𝑝 )𝑟𝑑𝜃𝑑𝑟, 𝑝 𝑝 𝜔 Im 𝑡 the static component 0 and dynamic component 𝑡 of 𝑓 𝑟1 0 pressure are obtained. This detailed calculation method is 𝜔 shown in the appendix. where 𝑓 isthesqueezefrequencyoftheshaft,whichis The load-carrying capacity 𝑊 is obtained by the following angular frequency of harmonic vibration when the bearing integration: is small vibrated in the axial direction under equilibrium condition, in the HDD obtained through the perturbation 2𝜋 𝑟 method the and modified Reynolds equation [9]. 2 𝐾 𝑊=∫ ∫ {𝑝0 (ℎ𝑟0)−𝑝𝑎}𝑟𝑑𝑟𝑑𝜃, (1) Finally, the bearing stiffness, ,isgivenbythefollowing 0 𝑟1 equation:

2 𝑝 √ 2 where 𝑎 indicates the atmospheric pressure. 𝐾= 𝑘 +(𝜔𝑓 ⋅𝑐). (5) The minimum oil lubricating film thickness ℎ𝑟 is simulta- neously determined from the equilibrium condition between In this paper, the oil leak is not considered because the the axial load acting on a thrust bearing and the bearing load- surround of thrust bearing of the spindle motor for 2.5 inch carrying capacity. The minimum oil lubricating film thickness HDD, as shown in Figure 1,issealed. ℎ𝑟 is obtained by solving the following force balance equation: 3. Formulation of Optimum Design and 𝑊 (ℎ ) −𝑚𝑔=0. 𝑟 (2) Vibration Analysis

The spindle motor’s torque on bearing surface is 3.1.FormulationofOptimumDesign.In formulating the given by the following integration: optimum design problems for groove geometry, the design variable of vector X consisting of groove geometry design 𝑟 2𝜋 3 variables of angle variations 𝜑1 through 𝜑4 and groove 2 𝜇𝑟 𝜔 𝑟ℎ 𝜕𝑝 𝑠 0 0 𝑟 ℎ 𝛼 𝑇𝑟 = ∫ ∫ ( − )𝑑𝑟𝑑𝜃, (3) dimension design variables of 𝑠, 𝑔,and as shown in 𝑟 0 ℎ 2 𝜕𝜃 1 0 Figures 3 and 4 is defined as follows:

X =(𝑟𝑠,ℎ𝑔,𝛼,𝜑1,𝜑2,𝜑3,𝜑4), (6) where 𝜔𝑠 is the angular velocity of the rotating shaft. 4 International Journal of Rotating Machinery

ℎ ,𝑟 =(𝑅/𝑅 ) 𝛼=(𝑏/(𝑏 +𝑏)) where 𝑔 𝑠 𝑠 out ,and 1 1 2 are the Table 2: Prescribed constraint values. groove depth, the seal ratio, and the ratio of groove width, Parameters Values respectively. The seal ratio means ratio of seal part, and the 𝑁 6,7,8,...,16 ratio of groove width means ratio of groove part of the total Groove number 𝑟 surface area. Minimum seal radius ratio 𝑠 min 0.4 𝑟 Another design variable considered in this optimization Maximum seal radius ratio 𝑠 max 0.8 ℎ 5.0 × 10−6 is the number of grooves. However, since design variables of Minimum groove depth 𝑔 min (m) −5 groove numbers are of discrete values, the optimization is Maximum groove depth ℎ𝑔 max (m) 1.5 × 10 preliminarily calculated step by step for the groove number Minimum groove width ratio 𝛼min 0.4 𝑁=6 𝑁=16 from to . Then, the groove numbers, which Maximum groove width ratio 𝛼max 0.9 gives the highest dynamic stiffness, is then selected as the Minimum geometry parameter 𝜑1∼4 min (deg) −180 optimumgroovenumber.Wehadconfirmedthatthebearing Maximum geometry parameter 𝜑1∼4 max (deg) 180 characteristic, which is especially the dynamic stiffness, is −6 2.0 × 10 , decreased if the groove number is larger than 𝑁=16and −6 Allowable film thickness ℎ𝑎 (m) 3.0 × 10 , less than 𝑁=6in both cases. In addition, it is difficult 4.0 × 10−6 to manufacture the grooves on the bearing surface because the width of grooves becomes narrow due to increasing the groove numbers. Therefore, we chose the range of groove 3.2. Vibration Analysis. Once the optimization is being car- numbers from 𝑁=6to 𝑁=16. This is also to save the ried out, the vibration analysis is calculated to numerically computational time for obtaining the optimum solution. verify the improvements. The vibration is conducted assum- Furthermore, the constraint conditions based on a ing that the oil film supporting the bearing is a viscously 2.5 inch HDD thrust bearing are expressed as damped one degree free vibration model. The values of the spring and damping coefficient obtained by the optimum 𝑔𝑖 (X) ≤0, (𝑖=1∼16) , (7) design are applied for the vibration response. where the details of the constraint functions are defined as in The equation of motion is given by (8). 𝑚𝑥+𝑐̈ 𝑥+𝑘𝑥=𝑞̇ (𝑡) , In (8), besides the constraint function of groove geome- (11) 𝑔 tries and dimensions from g1 through 14,theminimumoil 𝑚 𝑐 𝑘 ℎ where is the mass of the disk platters, and are lubricating film thickness 𝑟 obtained by (2)mustbelarger the damping and spring coefficients obtained by the above than the allowable film thickness to avoid bearing contact 𝑞(𝑡) 𝑔 mentioned optimum design, respectively, and is the and seizures shown as 15. Meanwhile, the final constraint impulsive external force being applied to the disk platters. function represents the constraint variable of the damping The external impulsive force being applied to the HDD coefficient being kept positive at all times to avoid self- disk platters is assumed as follows: induced vibrations. These values of prescribed constraint −𝜁𝜔𝑛𝑡 values are shown in Table 2: 𝑚𝜔𝑛𝑥 (𝑡) 𝑒 =( ) sin 𝜔𝑑𝑡, (12) 𝑔1 =𝑟𝑠 min −𝑟𝑠,𝑔2 =𝑟𝑠 −𝑟𝑠 max,𝑔3 =ℎ𝑔 min −ℎ𝑔, 𝐼 √1−𝜁2

𝑔4 =ℎ𝑔 −ℎ𝑔 max,𝑔5 =𝛼min −𝛼,6 𝑔 =𝛼−𝛼max, where the natural angular frequency is expressed as 𝜔𝑛 = √𝑘/𝑚; the damped natural angular frequency is expressed as 𝑔7 =𝜑1 min −𝜑1,𝑔8 =𝜑1 −𝜑1 max,𝑔9 =𝜑2 min −𝜑2, 2 𝜔𝑑 =𝜔𝑛√1−𝜁 which is angular frequency of oscillatory 𝑔10 =𝜑2 −𝜑2 max,𝑔11 =𝜑3 min −𝜑3, wave form with viscous damping; the damping ratio and impulse are, respectively, expressed as 𝜁=𝑐/2√𝑚𝑘 and 𝐼= 𝑔12 =𝜑3 −𝜑3 max,𝑔13 =𝜑4 min −𝜑4, 𝑓0Δ𝑡. 𝐺 𝑔14 =𝜑4 −𝜑4 max,𝑔15 =ℎ𝑎 −ℎ𝑟,𝑔16 =−𝑐. The nondimensional expressions for the gain and phase angle Φ are, then, calculated using the following equations: (8) 1 The objective function being set to improve the perfor- 𝐺= 2 2 2 mance characteristics of the spindle in hydrodynamic bearing √ 2 {1 − (𝜔𝑓/𝜔𝑛) } +4𝜁 (𝜔𝑓/𝜔𝑛) is the dynamic stiffness because thrust bearing plays an (13) important role in supporting the total axial direction load 2𝜁 (𝜔 /𝜔 ) 𝐾 −1 𝑓 𝑛 of the HDD spindle. The dynamic stiffness, ,ofthethrust Φ=−tan . bearing can be obtained through the following equation: 2 1−(𝜔𝑓/𝜔𝑛) 𝑓 (X) =𝐾. (9) TheoptimizationproblemforHDDthrustbearingisthen 4. Characteristics of Optimized Grooves formulated as follows: 4.1. Groove Geometry and Dimensions of Optimized Bearings. 𝑓 (Χ) max (𝑖=1∼16) . Figures 5(a)–5(c) show the groove geometry and pressure dis- 𝑔 (Χ) ≤0, (10) subject to 𝑖 tribution of optimized oil lubricated bearings at 10,000 rpm International Journal of Rotating Machinery 5

MPa p0max = 0.192 theoillubricantflowsawayfromtheouterbendstowards the most outer radius of the bearing surface. Hence, as canbeseenfromthepressuredistributionfigures,there are no pressure peaks generated in the vicinity of the most outer regions of the bearing surface. This is because the pressure in the outer periphery of the bearing is neutralized to the atmospheric pressure. The pressure equivalent to the atmospheric pressure caused by the outer geometry bends willactasaforcetopulldownthebearingandlowerthe p0 = 0.101 MPa min film thickness. With the combination of spiral geometry in (a) the inner periphery and the opposite spiral bends in the outer p = 0.154 MPa periphery, the film thickness decreased, thus maximizing the 0max dynamic stiffness of the spindle. Even though all groove geometries basically possess the modified spiral geometry, Figure 5(a) shows that the region of spiral groove of optimized with ℎ𝑎 =2.0𝜇missmaller than those of optimized with ℎ𝑎 =3.0𝜇mandℎ𝑎 = 4.0 𝜇m. The geometry is concentrated in the outer regions only, since the seal radius ratio turned to the maximum constraint value being set in this paper, which is 0.8. The oil p = 0.101 MPa 0min pressure distribution on the right hand side in the same figure (b) shows pointed sharp pressure distribution. These peaks are the pressure where the groove part and seal part meet, a result p = 0.194 MPa 0max of collisions between oil lubricant flow and seal. For the groove geometries with allowable film thickness of ℎ𝑎 = 3.0 𝜇mandℎ𝑎 = 4.0 𝜇m shown in Figures 5(b) and 5(c), the inner part of the bearing possesses a larger portion of spiral groove geometries compared to that optimized with allowable film thickness of ℎ𝑎 = 2.0 𝜇m. Even though the inner division creates comparatively smaller pressure peaks, this time the pressure is well distributed throughout the p = 0.101 MPa 0min bearing surface. (c) If we focus on the inner periphery of the total geometry closely, that is, the end part of inner spiral just before the seal, Figure 5: Groove geometry and pressure distribution of optimized wecanseethatanotherbendwiththeoppositedirectionof oil lubricated bearings. (a) Optimized at 10,000 rpm, ℎ𝑎 =2𝜇m, (b) ℎ =3𝜇 the outer bends started to appear. These inner bends however optimized at 10,000 rpm, 𝑎 m, (c) optimized at 10,000 rpm, are smaller than the outer ones. This can be explained by ℎ𝑎 =4𝜇m. the fact that in order to maintain the prescribed allowable film thickness, it is essential to increase the pump in effect with allowable film thickness of 2.0 𝜇m, 3.0 𝜇m, and 4.0 𝜇m, of the inner spiral. It is insufficient with only the inner spiral respectively. geometry and outer bends. Therefore, another bend in the As can be seen from the figures, all geometries of inner periphery, just beside the seal, appears to increase the the optimized bearings changed from initial spiral groove filmthicknessofthebearing.Theinnerbendscanbeseen geometry to a new geometry, a spiral groove with bends in clearly for the case of allowable film thickness of ℎ𝑎 = 4.0 𝜇m. the outer periphery. In this paper, such geometry is called the The combination of inner bends, outer bends, and inner modified spiral geometry. spiral geometry helps to optimally balance the pressure and Generally, it is widely known that spiral groove geometry improve the dynamic stiffness. has the ability to elevate the film thickness with its pump in The details of groove geometry and dimension of the effects as the revolution increases due to pressure generation. optimized bearings from Figure 5 are shown in Table 3.As This explanation is applicable for the inner periphery of can be seen from the table, the values of geometry parameters the optimized groove geometry. As the bearing rotates, the changed from initial to a new groove geometry with slightly oil lubricants flows inwards, starting from the outer bends different angles of 𝜑1 through 𝜑4 depending on the allowable periphery, and flows along the grooves and collides with film thickness. However, in general all the groove geometry the seal, hence generating pressure. From all of the pressure changes from spiral groove geometry to spiral with bends in distribution figures, the point where the groove and seal meet the outer periphery. has the highest point of pressure generated. The other design parameters of groove dimensions such However, the phenomenon is different from the outer as seal radius ratio, groove depth, and groove width ratio periphery for opposite spiral groove geometries. Instead of alsochange.Thesealradiusratioshowsthatthelowerthe generating the pressure and increasing the film thickness, allowable film thickness is being set, the more the groove 6 International Journal of Rotating Machinery

×10−6 12 (m) r h 10 (N/m)

K 8 106

6

4 Dynamic stiffness 105 2 0 5000 10000 15000 20000 film thickness lubricating oil Minimum 5000 10000 15000 20000

Rotational speed ns (rpm) Rotational speed ns (rpm) (a) (b)

×10−4

2 (Nm) r T

1 Friction torque Friction

0 5000 10000 15000 20000

Rotational speed ns (rpm)

Optimized; ha =2𝜇m Optimized; ha =4𝜇m Optimized; ha =3𝜇m Conventional spiral (c)

Figure 6: Bearing characteristics versus rotational speed; (a) dynamic stiffness 𝐾, (b) minimum oil lubricating film thickness ℎ𝑟,(c)friction torque 𝑇𝑟. geometry surface will be occupied with the seal area. This Table 3: Design variables of optimized bearings at 10,000 rpm. eventually made the groove geometry to be concentrated at Optimized Optimized Optimized the outer region of the total bearing surface. On the other Design variables ℎ𝑎 = 2.0 𝜇m ℎ𝑎 = 3.0 𝜇m ℎ𝑎 = 4.0 𝜇m hand, for the groove depth and groove width ratio values, Angle 𝜑1 (deg) 7.94 4.46 2.26 results show that the lower the allowable film thickness is Angle 𝜑2 (deg) −20.7 −11.2 −4.70 being set, the smaller those values turned into. Angle 𝜑3 (deg) 79.9 68.1 61.3 Angle 𝜑4 (deg) 17.5 26.4 32.3 𝑁 4.2. Static and Dynamic Characteristics of Optimized Bearings. Groove number 61113 𝑟 Figures 6(a), 6(b),and6(c) show the relations between Seal ratio 𝑠 0.80 0.69 0.63 ℎ 5.0 × 10−6 7.5 × 10−6 8.8 × 10−6 therotationalspeedandtheoptimizedvaluesofdynamic Groove depth 𝑔 (m) Groove width ratio 𝛼 0.4 0.45 0.48 stiffness, 𝐾, minimum oil lubricating film thickness, ℎ𝑟,and friction torque, 𝑇𝑟, respectively. The solid lines, alternate long and short dashed lines, and short dashed lines with dots The objective function for this study is the improvement represent the optimized bearings for ℎ𝑎 = 2.0 𝜇m, ℎ𝑎 = of the HDD spindle dynamic stiffness. As can be seen 3.0 𝜇m, and ℎ𝑎 = 4.0 𝜇m,respectively.Thedotsrepresent from Figure 6(a), with the proposed optimization method, the characteristics values for each optimized bearing at that the dynamic stiffness values are improved compared to the specific rotational speed. The short dashed lines without dots values of conventional spiral bearings. The oil lubricated represent the initial spiral bearings. bearing for conventional spiral showed a trend of decreasing International Journal of Rotating Machinery 7

1 0

x(t)/I 0.5 −20 n G m𝜔

−40 Gain 0

Displacement −60

−0.5 −80 0 0.01 0.02 0.03 0.04 0.05 0.06 100 101 102 103 104 Time t (s) Frequency f (Hz) (a) (b)

(deg) 0 Φ Phase angle

−100

100 101 102 103 104 Frequency f (Hz)

Optimized; ha =2𝜇m Optimized; ha =4𝜇m Optimized; ha =3𝜇m Conventional spiral (c)

Figure 7: Vibration characteristics of optimized at 10,000 rpm bearings and conventional spiral bearing; (a) displacement with impulsive force, (b) gain versus frequency, and (c) phase angle versus frequency. with the increase of rotational speed. However, optimized On the other hand, the dynamic stiffness is increased with bearings still show an improvement where different rotational adecreaseinthefilmthicknessasshowninFigure6(a). speed gives different stiffness characteristics but with all Therefore, in this optimization, the minimum oil lubricating of them being improved and maintained compared to the film thickness comes close to the allowable film thickness to conventional ones. drastically improve the dynamic stiffness changing the groove Figure 6(b) shows the relation of minimum oil lubricating numbers of 𝑁=6, 11, 13 under the allowable film thickness film thickness for all optimized results with rotational speed. of ℎ𝑎 = 2.0 𝜇m, 3.0 𝜇m, 4.0 𝜇m. The minimum oil lubricating film thickness of conventional For the friction torque values shown in Figure 6(c),all spiral shows an increase with rotational speed. However, optimizedbearingsshowslightlylargerfrictiontorquevalues compared to the initial spiral groove, the geometry of opti- compared to initial spiral groove. However, the increase of mized bearings has a much lower film thickness. This means friction torque is small if we were to compare the dynamic that the outer bends of optimized bearings have the ability stiffness improvement obtained using the proposed hybrid to pull down the bearings towards bearing base. This is what method. contributes to the increase of the dynamic stiffnesses of the To confirm the effectiveness of the optimum design spindle. Moreover, the results obtained are attributable to presented here, the values of design variables and objective the groove numbers as shown in Figure 5. Generally, the function for the optimized bearing with ℎ𝑎 = 3.0 𝜇mare load carrying capacity 𝑊 is increased with an increase in the compared to those for randomly choosen design variables groove numbers. Therefore, the oil lubricating film thickness rotating at 10,000 rpm in Table 4.Asshowninthetable, isincreasedwithanincreaseinthegroovenumbersbecause the objective function of the optimized bearing shows max- we set constant load 𝑊𝑑 = 0.185 (N) in this optimization. imum value compared with other values. Furthermore, we 8 International Journal of Rotating Machinery

Spindle motor for Eddy-current proximity probe 2.5 inch HDD

Acceleration pick-up Motor fixing jig Spindle motor for Eddy-current proximity probe 2.5 inch HDD Motor fixing jig

See saw Acceleration pick-up Vibration exciter

Vibration isolation Pole table Vibration isolation table

(a) Vibration test rig (b) Impulse test rig

Figure 8: Experimental test rigs for verification of calculated data.

Groove Groove

Coordinate Land Land transformation

𝜃 𝜂

𝜃land r 𝜃land 𝜉 𝜃groove 𝜃groove

(a) (b)

Figure 9: Bearing geometry transformation based on the boundary-fitted coordinate system; (a) Original bearing geometry, (b) Transformed bearing geometry. confirmed the values of the optimized bearing in the cases order system of an overdamped wave form can be seen. This of changing allowable film thickness ℎ𝑎 = 2.0 𝜇m, 4.0 𝜇mto confirms that the optimized bearings have higher stability be the largest using the same method. than the conventional spiral ones. Figures 7(b) and 7(c) show the frequency response for 4.3. Results of Vibration Analysis. Figures 7(a)–7(c) show the bearings. The optimized bearings have a smoother line 𝜇 the comparison of vibration characteristics for optimized of gain and phase angle. Gain shows that optimized 2.0 m 𝜇 𝜇 bearings with ℎ𝑎 = 2.0 𝜇m, ℎ𝑎 = 3.0 𝜇m, ℎ𝑎 = 4.0 𝜇m, and has the highest damping followed by 3.0 m, 4.0 m, and con- conventional spiral bearing rotating at 10,000 rpm. ventional spiral, respectively. This proves that the optimized Figure 7(a) shows the response of bearings towards bearings have better system responses. impulsive force. The magnitude of the amplitude for impul- sive force shows the highest value for conventional spiral 5. Conclusions groove bearings where a second order system wave response of an underdamped oscillatory response can be seen. On the In this paper, the optimum design method for 2.5 inch HDD other hand, the responses towards impulsive force for the rest was carried out to improve the dynamic stiffness of a thrust of the bearings show smaller displacement values where a first bearing spindle. The results can be concluded as follows. International Journal of Rotating Machinery 9

1 j+ 2 1 𝜉 i+ D Q1IV 2 j 𝜂 Q1IV

+ + 𝜉 IV (i , j ) Q1II i 1 𝜂 j− Q2III (i, j) 2 II(i+, j−) III (i−, j+) B 1 i− 2 C 1 i+ − − 𝜂 2 I(i , j ) Q1II 1 j+ Q𝜉 i 2 2III j 𝜂 𝜉 Q𝜉 𝜂 2I Q2I A 1 1 j− i− 2 2

Figure 10: Definition of control volume.

−6 Table 4: Comparison of the optimized solution and random solutions under allowable film thickness of ℎ𝑎 = 3.0 × 10 (m).

Case Obj. f (X)(N/m) 𝜑1 (deg) 𝜑2 (deg) 𝜑3 (deg) 𝜑4 (deg) 𝑁𝑟𝑠 ℎ𝑔 (m) 𝛼 6 −6 Optimum solution 1.89 × 10 4.46 −11.2 68.1 26.4 11 0.69 7.50 × 10 0.45 5 −6 Random 1 2.91 × 10 −60.0 −105.7 −31.4 82.8 6 0.79 1.47 × 10 0.87 5 −6 Random 2 2.04 × 10 −54.3 −37.1 159.9 162.8 9 0.44 7.90 × 10 0.79 5 −5 Random 3 4.05 × 10 −131.4 −42.8 −125.7 57.1 11 0.4 1.20 × 10 0.66 5 −5 Random 4 8.90 × 10 17.1 −5.7 −128.5 88.5 7 0.65 1.08 × 10 0.49 6 −5 Random 5 1.02 × 10 −168.5 142.8 −97.1 −25.7 10 0.44 1.09 × 10 0.73 5 −5 Random 6 4.25 × 10 20.0 17.1 −119.9 −105.7 11 0.68 1.09 × 10 0.5 6 −6 Random 7 1.44 × 10 40.0 −60.0 74.2 −91.4 6 0.58 8.10 × 10 0.58 5 −5 Random 8 4.85 × 10 77.1 62.8 −117.1 −14.3 12 0.64 1.25 × 10 0.72 5 −5 Random 9 3.46 × 10 102.8 −42.8 −165.6 −122.8 6 0.71 1.21 × 10 0.41 5 −5 Random 10 3.54 × 10 131.4 142.8 −142.8 −157.1 8 0.45 1.30 × 10 0.85

(1) Bearing characteristics using the proposed hybrid 6. Future Work method showed that, compared with conventional spiral groove bearings, improvements of dynamic In this paper, we confirmed that dynamic stiffness of thrust stiffnesses can be obtained when a new geometry of bearing for 2.5 inch HDD was drastically improved numer- modified spiral groove geometry is introduced. ically using the proposed hybrid method. However, it is necessary to verify the calculated data by experiments. We (2) To maximize the dynamic stiffness is to set the have prepared the experiments to verify the calculated data maximum seal radius ratio, that is, by concentrating by using original experimental test rigs as shown in Fig- the geometry at the most outer periphery of the ure 8.Figures8(a) and 8(b) showtheimpulseandvibration bearing. experimental test rigs, respectively. The response of bearings (3) Vibration analysis of optimized bearings showed towards impulse force, which can be applied using see-saw better characteristics than the initial spiral bear- mechanism as shown in Figure 8(a),canbemeasured.On ings. Therefore, optimized bearings are expected to the other hand, it is very difficult to measure the spring improve vibration characteristic of HDD. and damping coefficients of spindle motor for 2.5 inch HDD 10 International Journal of Rotating Machinery because the displacement of the bearing is very small. We where, in that case, proposedtheidentificationmethod[11]ofoilfilmcoefficients of bearings for HDD spindle using the response waveform ℎ3 ℎ3 ℎ3 obtained by vibration experiment using the test rig as shown 𝐴=𝑎 ,𝐵=𝑏,𝐶=𝑐, in Figure 8(b). As mentioned above, therefore, we plan to 12𝜇𝐽 12𝜇𝐽 12𝜇𝐽 submit the further study including these contents. 𝑟𝜔 ℎ 𝜌𝑟𝜔2ℎ3 𝑟𝜔 ℎ 𝐷=− 𝑠 𝑟 ,𝐸=𝑠 𝑟𝜃 ,𝐹=𝑠 𝑟 , 2 𝜂 40𝜇 𝜂 2 𝜉 Appendix 2 3 𝜌𝑟𝜔 ℎ 2 When optimizing the groove geometry, it is necessary to 𝐺= 𝑠 𝑟𝜃 ,𝑎=(𝑟𝜃) +𝑟2, 40𝜇 𝜉 𝜉 𝜂 perform a sequential analysis of characteristics of a bearing with a groove geometry modified in succession by the finite 2 𝑏=(𝑟𝜃)(𝑟𝜃 )+𝑟𝑟 ,𝑐=(𝑟𝜃) +𝑟2, difference method, but because of the 𝑟-𝜃 polar coordinate 𝜉 𝜂 𝜉 2 𝜉 𝜉 system, direct processing is rather different to realize. There- 𝜕𝑟 𝜕𝑟 fore,ananalysisofthebearingstiffnessisperformedfirst 𝐽=𝑟 (𝑟𝜃 )−𝑟 (𝑟𝜃 ), 𝑟 = ,𝑟= , 𝜉 𝜂 𝜂 𝜉 𝜉 𝜕𝜉 𝜂 𝜕𝜂 by using a boundary-fitted coordinate system as shown in Figure 9 [12, 13] and transforming a complex groove geometry 𝜕𝜃 𝜕𝜃 𝑟𝜃 =𝑟 ,𝑟𝜃=𝑟 . into a simple fanlike geometry. A boundary transformation 𝜉 𝜕𝜉 𝜂 𝜕𝜂 function used for the transformation is given as (A.4)

𝜉=𝑟 Then, inA.2 ( ), subscripts l and 2 and I–IV indicate the 󸀠󸀠 󸀠󸀠 domains in the control volume. 𝜃 (𝑟𝑖) 3 𝜃 (𝑟𝑖+1) 3 𝜂=𝜃−[ (𝑟𝑖+1 −𝑟) + (𝑟 −𝑖 𝑟 ) Assuming that variations of the bearing clearance are 6Δ𝑟 6Δ𝑟 microscopic, the minimum oil lubricating film thickness ℎ and pressure 𝑝 can be expressed by the following equation: 𝜃󸀠󸀠 (𝑟 )Δ𝑟2 𝑟 −𝑟 (A.1) +(𝜃(𝑟)− 𝑖 ) 𝑖+1 𝑖 6 Δ𝑟 𝑗𝜔 𝑡 ℎ=ℎ +𝜀𝑒 𝑓 𝜃󸀠󸀠 (𝑟 )Δ𝑟2 𝑟−𝑟 0 +(𝜃(𝑟 )− 𝑖+1 ) 𝑖 ]. (A.5) 𝑖+1 𝑗𝜔𝑓𝑡 6 Δ𝑟 𝑝=𝑝0 +𝜀𝑝𝑡𝑒 .

The following Reynolds equivalent equation can be In the equations stated above, 𝜀 indicates the amplitude obtained from the equilibrium between the mass flow rates of small variations of the oil lubricating film thickness and 𝑝0 of oil inflowing into and outflowing from the control volume and 𝑝𝑡 express a static component and a dynamic component, duetotheshaftrotationandthesqueezingmotion: respectively. The substitution ofA.5 ( )into(A.2) and the negligence of 𝜀 𝑄𝜉 +𝑄𝜉 −𝑄𝜉 −𝑄𝜉 +𝑄𝜂 +𝑄𝜂 −𝑄𝜂 −𝑄𝜂 =𝑄Γ, terms of small magnitude of of above second order allow for 2I 1III 2II 1IV 2I 1II 2III 1IV the introduction of two equations for terms 𝜀 of orders 0 and (A.2) l as follows:

𝑄𝜉 𝑄𝜂 where , indicate the mass flow rates, across the bound- 𝐹 (𝑝 )=𝑄𝜉 +𝑄𝜉 −𝑄𝜉 −𝑄𝜉 ary of 𝜉 = const. and across the boundary of 𝜂 =const.as 0 0 2I0 1III0 2II0 1IV0 Γ (A.6a) showninFigure10. On the other hand, 𝑄 indicates the mass 𝜂 𝜂 𝜂 𝜂 +𝑄2 0 +𝑄1 0 −𝑄2 0 −𝑄1 0 =0, flow rate due to squeezing motion inside the control volume. I II III IV 𝜉 𝜂 Γ 𝑄 , 𝑄 ,and𝑄 are expressed, respectively, as follows: 𝐹 (𝑝 ,𝑝 )=𝑄𝜉 +𝑄𝜉 −𝑄𝜉 −𝑄𝜉 +𝑄𝜂 𝑡 𝑡 0 2I𝑡 1III𝑡 2II𝑡 1IV𝑡 2I𝑡 (A.6b) 𝜂 𝜂 𝜂 Γ 𝜂 +𝑄 −𝑄 −𝑄 −𝑄 =0, 2 𝜕𝑝 𝜕𝑝 1II𝑡 2III𝑡 1IV𝑡 𝑡 𝑄𝜉 = ∫ 𝜌(−𝐴 +𝐵 +𝐷+𝐸)𝑑𝜂 𝜕𝜉 𝜕𝜂 (A.3a) 𝜂1

𝜉 where subscript 0 indicates a static component of the mass 𝜂 2 𝜕𝑝 𝜕𝑝 𝑄 =∫ 𝜌 (𝐵 −𝐶 +𝐹+𝐺) 𝑑𝜉 (A.3b) flow rate determined from (A.3a), (A.3b), and (A.3c), and 𝜉 𝜕𝜉 𝜕𝜂 1 subscript 𝑡 similarly indicates a dynamic component. 𝜉 𝜂 Solving Equations (A.6a)and(A.6b)inturnbythe Γ 2 2 𝜕(𝜌ℎ) 𝑄 = ∫ ∫ |𝐽| 𝑑𝜂 𝑑𝜉, (A.3c) Newton-Raphson iteration method, the static and dynamic 𝜉 𝜂 𝜕𝑡 1 1 components, 𝑝0 and 𝑝𝑡,areobtained. International Journal of Rotating Machinery 11

Nomenclature References

𝑏1: Groove width (m) [1] G. H. Jang, S. H. Lee, and H. W.Kim, “Finite element analysis of 𝑏2: Land width (m) the coupled journal and thrust bearing in a computer hard disk 𝑐: Damping coefficient of oil lubricated film drive,” Journal of ,vol.128,no.2,pp.335–340,2006. (N⋅s/m) [2] K. Y. Park and G. H. Jang, “Dynamics of a hard disk drive 𝑓(X): Objective function spindle system due to its structural design variables and the 𝑔𝑖(X): Constraint functions; (𝑖=1∼16) design variables of fluid dynamic bearings,” IEEE Transactions 𝑞(𝑡): Impulse external force (N) on Magnetics,vol.45,no.11,pp.5135–5140,2009. 𝐺:Gain [3] T. Kobayashi and H. Yabe, “Numerical analysis of a coupled ℎ𝑎: Allowable film thickness (m) porous journal and thrust bearing system,” Journal of Tribology, ℎ𝑔: Groove depth (m) vol. 127, no. 1, pp. 120–129, 2005. ℎ𝑟: Minimum oil lubricating film thickness (gap [4]J.ZhuandK.Ono,“Acomparisonstudyontheperformance between shaft and lower groove surface) (m) of four types of oil lubricated hydrodynamic thrust bearings for 𝑘: Spring coefficients of oil lubricated film (N/m) hard disk spindles,” Journal of Tribology,vol.121,no.1,pp.114– 𝐾: Dynamic stiffness of oil lubricated film (N/m) 120, 1999. 𝑚: Mass of the rotor (kg) [5]T.Asada,H.Saitou,andD.Itou,“Designofhydrodynamic 𝑁:Groovenumber bearing for miniature hard disk drives,” IEEE Transactions on Magnetics,vol.43,no.9,pp.3721–3726,2007. 𝑛𝑠: Rotational speed of spindle (rpm) 𝑝: Oil lubricated film pressure (Pa) [6] G. H. Jang and C. I. Lee, “Development of an HDD spindle motor with increased stiffness and damping coefficients by 𝑝𝑎: Atmospheric pressure (Pa) 𝑝 utilizing a stationary permanent magnet,” IEEE Transactions on 0: Static component of oil lubricated film pressure Magnetics,vol.43,no.6,pp.2570–2572,2007. (Pa) 𝑝 [7]Q.Zhang,S.Chen,S.H.Winoto,andE.-H.Ong,“Design 𝑡: Dynamic component of oil lubricated film of high-speed magnetic fluid bearing spindle motor,” IEEE pressure (Pa/m) Transactions on Magnetics,vol.37,no.4,pp.2647–2650,2001. 𝑟: Coordinate of radius direction (m) 𝑅 [8] Y.Arakawa,S.Ikeda,T.Hirayama,T.Matsuoka,andN.Hishida, out:Bearingouterradius(m) “Hydrodynamic bearing with non-uniform spiral grooves for 𝑅 in:Bearinginnerradius(m) high-speed HDD spindle,” in Proceedings of the IEEE 13th 𝑅𝑠:Sealradius(m) International Symposium on Consumer Electronics (ISCE ’09), 𝑟 =(𝑅/𝑅 ) 𝑠:Sealratio; 𝑠 out pp.511–512,Kyoto,Japan,May2009. 𝑡: Times (s) [9] H. Hashimoto and M. Ochiai, “Optimization of groove geome- 𝑇𝑟: Friction torque on bearing surface (N⋅m) try for thrust air bearing to maximize bearing stiffness,” Journal 𝑊: Load-carrying capacity (N) of Tribology,vol.130,no.3,ArticleID031101,2008. X: Vector of design variables [10] M. D. Ibrahim, T. Namba, M. Ochiai, and H. Hashimoto, Δ𝑟: Equipartition space of 𝑟 (m). “Optimum design of thrust air bearing for hard disk drive spindle motor,” Journal of Advanced Mechanical Design, Systems Greek Symbols and Manufacturing,vol.4,no.1,pp.70–81,2010. [11] M. Ochiai, Y. Sunami, and H. Hashimoto, “Experimental study on dynamic characteristics of fluid film bearing for HDD spin- 𝛼:Ratioofgroovewidth;=(𝑏1/(𝑏1 +𝑏2)) dle motor,”in Proceedings of the 2nd International Conference on 𝜇: Viscosity of lubricant (Pa⋅s) Design Engineering and Science, pp. 217–222, 2010. 𝜃: Coordinate of circumferential direction (rad) [12] H. Hashimoto and M. Ochiai, “Theoretical analysis and opti- 𝜉: Coordinate of change based on boundary-fitted mum design of high speed gas film thrust bearings (static coordinate system (m) and dynamic characteristic analysis with experimental verifi- 𝜂: Coordinate of change based on boundary-fitted cations),” JournalofAdvancedMechanicalDesign,Systems,and coordinate system (rad) Manufacturing JSME,vol.1,no.1,pp.102–112,2007. Φ: Phase angle (deg) [13] N. Kawabata, “Study on generalization of calculations of lubri- 𝜑𝑖: Groove geometry design parameter; cant flow using boundary fitted coordinate system (part 1, basic (displacement for 𝑖thpointofshiftfromthe equations of DF method and case of incompressible fluids),” initialgroovegeometrytoanewpoint)(deg) Transactions of the Japan Society of Mechanical Engineers C,vol. 53,no.494,pp.2155–2160,1987. 𝑤𝑑: Natural damped frequency (rad/s) 𝜔𝑓: Squeeze frequency of the shaft revolution (rad/s) 𝜔𝑠: Angular velocity of the shaft (rad/s).

Subscripts max: Maximum value of state variables min: Minimum value of state variables. International Journal of

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