Robust Optimum Design of Thrust Hydrodynamic Bearings for Hard Disk Drives

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1 Appled Mathematcs, 0, 3, Publshed Onlne October 0 ( Robust Optmum Desgn of Thrust Hydrodynamc Bearngs for Hard Dsk Drves Hromu Hashmoto, Yuta Sunam Department of Mechancal Engneerng, Toka Unversty, Hratsuka, Japan Emal: hromu@keyak.cc.u-toka.ac.jp, 0btad005@mal.toka-u.jp Receved July 4, 0; revsed August 4, 0; accepted August, 0 ABSTRACT Ths paper descrbes the robust optmum desgn whch combnes the geometrcal optmzaton method proposed by Hashmoto and statstcal method. Recently,.5 hard dsk drves (HDDs) are wdely used for moble devces such as laptops, vdeo cameras and car navgaton systems. In moble applcatons, hgh durablty towards external vbratons and shocks are essentals to the bearngs of HDD spndle motor. In addton, the bearng characterstcs are nfluenced by manufacturng error because of small sze of the bearngs of HDD. In ths paper, the geometrcal optmzaton s carred out to maxmze the bearng stffness usng sequental quadratc programmng to mprove vbraton characterstcs. Addtonally, the bearng stffness s analyzed consderng dmensonal tolerance of the bearng usng statstcal method. The dmensonal tolerance s assumed to dstrbute accordng to the Gaussan dstrbuton, and then the bearng stffness s estmated by combnng the expectaton and standard devaton. As a result, n the robust optmum desgn, new groove geometry of bearng can be obtaned n whch the bearng stffness s four tmes hgher than the stffness of conventonal spral groove bearng. Moreover, the bearng has lower varablty compared wth the result of optmum desgn neglectng dmensonal tolerance. Keywords: Robust Optmum Desgn; Hard Dsk Drve; Hydrodynamc Bearng; Tolerance; Statstcal Method. Introducton Recently, demand of hard dsk drves (HDDs) has been contnued to expand because of development of nformaton technology n ndustres. Especally,.5 HDDs are wdely used for many nformaton devces such as laptops, dgtal vdeo cameras and car navgaton systems. Consequently, convenence and performance of nformaton devces have been mprovng year by year. In the meantme, usage envronments n whch HDDs are used under the condtons occurrng vbratons have been beng severe than before. Therefore, mprovement of vbraton characterstcs of HDDs has been strongly demanded. Hydrodynamc bearngs, whch are used n spndle motor of HDDs, are one of key machne elements to mprove vbraton characterstcs of HDDs. The hydrodynamc bearngs whch manly have grooves called the spral or herrng-bone groove are tradtonally used for HDDs spndle motor. There are several researchers [-7] who attempted to nvestgate the research related to the mprovement of vbraton characterstcs of the hydrodynamc bearngs. However, they are stll nsuffcent for a sgnfcant mprovement of bearng characterstcs. One reason s because the groove geometry s fxed on the spral or herrng-bone grooves. On the other hand, Hash- moto and Ocha [8] dealt wth the geometrcal optmzaton amed at dscoverng the optmum groove geometry whch have never found before and mprovng dramatcally bearng stffness of thrust ar bearngs. In addton, the hgher performance of the bearng havng optmum groove geometry s expermentally verfed. Moreover, Ibrahm et al. [9] appled the same method of the geometrcal optmzaton to thrust ar bearngs on HDD spndle motor and the effectveness was theoretcally dscussed. In the process of manufacturng thrust hydrodynamc bearngs for.5 HDDs, t s compulsory to mantan hgh dmensonal accuracy due to hgh nfluence by manufacturng errors towards the bearng characterstcs. Therefore, t s necessary to severely desgn consderng the dmensonal tolerance of desgn varables of the bearngs. In that case, a new desgnng method whch treats the tolerance numercally n advance s reasonable compared wth the conventonal method whch determnes the tolerance by tral-and-error method. In the conventonal desgnng method of the bearngs, a determnstc method neglectng the dmensonal tolerance of desgn varables s beng manly used. For that reason, there are several researchers [0-4] who attempted to nvestgate the nfluence of manufacturng errors on the bearng Copyrght 0 ScRes.

2 H. HASHIMOTO, Y. SUNI 369 characterstcs through senstvty analyss. However, as far as the authors know, there s no research whch had carred out the optmum desgn wth the consderaton of manufacturng errors for the hydrodynamc bearngs of HDD spndle motors. In ths paper, the nfluence of dmensonal tolerance on bearng characterstcs s conducted usng the statstcal method. And then, the robust optmum desgn based on the geometrcal optmzaton combned wth the statstcal method [5] s appled to thrust hydrodynamc bearngs of.5 HDD spndle motor. The results obtaned are compared wth the result by the conventonal optmum desgn neglectng tolerance to verfy the effectveness of the proposed method.. Geometrcal Optmzaton and Analyss of Bearng Characterstcs.. Geometrcal Optmzaton In ths paper, the geometrcal optmzaton proposed by Hashmoto and Ocha [8] s appled to thrust hydrodynamc bearngs for.5 HDD spndle motor to drastcally mprove the bearng stffness. In the process of optmzng the groove geometry, the ntal geometry s establshed and a groove shape, whch s lkely to provde a maxmal bearng stffness, s determned by usng the method of successve evoluton from the ntal geometry as shown n Fgure. Therefore, spral groove bearngs are consdered as the ntal geometry to rase calculaton effcency because the spral groove bearngs have relatvely hgh bearng stffness compared wth other types of bearngs. The ntal spral groove geometres are flexbly modfed usng the cubc splne nterpolaton functon as shown n Appendx I. The whole groove s parttoned nto n parts n the r drecton, and then each P r ns provded to the ntersecton wth the cubc splne nterpolaton functon. At the tme when a groove geometry s gradually evolved nodal pont, + r r - r + - r Orgnal geometry (Spral grooved) r r P (r, ) P + (r +, ) + P (r, ) - k th step geometry P (r, + ) (k+) th step geometry P + (r +, + ) P (r, + ) Fgure. Method of changng groove boundary geometry. from the prevous stage (k step) to the present stage (k + step) shown by the broken lne n Fgure, the r coordnate of each nodal pont P r, s fxed, and the θ coordnate s changed when the angle δ moves n the postve drecton or negatve drecton, resultng to a new nodal pont P r, v. Then, the groove geometry s revsed, usng the new coordnate value that can be obtaned usng ths way, and gradually evolved untl the value of the bearng stffness becomes maxmum... Analyss of Bearng Characterstcs In ths paper, the calculaton method of bearng characterstcs s derved by applyng the boundary-ftted coordnate system to adjust the geometrcal optmzaton method. Moreover, n the process of analyzng the statc and dynamc characterstcs of the bearngs, the perturbaton method s appled to the Reynolds equvalent equaton. The Reynolds equvalent equaton can be solved by usng the Newton-Raphson teraton method, and the statc component p 0 and dynamc component p t of pressure are obtaned. These detaled calculaton method s shown n Appendx II. The load-carryng capacty W s obtaned by the followng ntegraton: π 0 r0 a dd () W p h p r r 0 where p a ndcates the atmospherc pressure. The mnmum ol lubrcatng flm thckness h rmn s smultaneously determned from the equlbrum condton between the axal load actng on a thrust bearng and the bearng load-carryng capacty. The mnmum ol lubrcatng flm thckness h rmn s obtaned by solvng the followng force balance equaton: W h r mn mg 0 () The sprng coeffcent k and dampng coeffcent c can be obtaned, respectvely, by ntegratng the real and magnary parts of the dynamc pressure components, p t, as follows: π t (3) k Re p r d r d 0 π t (4) c Im p r d r d 0 Fnally, the bearng stffness s gven by the followng equaton. K k f c (5) 3. Estmaton of Varablty Usng Statstcal Method Desgnng of the bearngs for HDD spndle motor s re- Copyrght 0 ScRes.

3 370 H. HASHIMOTO, Y. SUNI qured the hgh bearng performance and qualty. Therefore, n ths case, the optmum desgn s appled to the bearngs for obtanng hgh bearng performance. On the other hand, bearng performance s hghly nfluenced by manufacturng error because of small sze of the bearngs of HDD. However, the conventonal method of desgnng the bearngs was conducted usng determnstc method, whch neglected the dmensonal tolerance. Hence, consderng the nfluence of dmensonal tolerance under product desgn s very mportant n terms of the qualty and productvty of HDD spndle motors. Fgure shows a conceptual dagram between desgn varables and varablty of bearng performance. The vertcal axs shows the typcal bearng performance and the horzontal axs shows the typcal desgn varables, for example, groove depth, groove wdth rato and so on. As can be seen n the fgure, varablty of bearng performance shows non-unform under the same range of dmensonal tolerance. In the conventonal optmum desgn neglectng tolerance, soluton A can be obtaned as optmzed soluton. Soluton A s maxmum value of bearng performance, but the varablty becomes large when desgn varable s changed due to manufacturng error. On the other hand, varablty of bearng performance of soluton B s lower than that of soluton A, although the bearng performance of soluton A s better than soluton B. That means, soluton B has larger robustness of bearng performance compared wth soluton A. Therefore, t s necessary to fnd the desgn varable whch can be obtaned low varablty wth hgh bearng performance lke soluton B. Fgure 3 shows the relatonshp between bearng performance and varablty. The fgure shows the trade-off correlaton between an ncrease of the varablty and the bearng performance. In addton, t s possble that the spatal dstrbuton of bearng performance s a multmodal dstrbuton because the hydrodynamc bearngs have Fgure. Varablty of desgn varable and bearng performance. Fgure 3. Trade-off correlaton between bearng performance and varablty of bearng performance. relatvely a lot of dmensons. In ths case, t s mportant to treat mult-objectve problem to obtan hgh bearng performance and robustness. Moreover, optmum desgn s needed to consder a statstcal factor for desgnng of commercal HDDs. In ths paper, the robust optmum desgn wth hgh robustness s newly ntroduced to thrust hydrodynamc bearngs. The evaluaton method for the varablty of the bearng performance usng the statstccal method s descrbed as follows. The varablty of desgn varables s expressed by the probablty densty functon, and then the robustness s estmated based on the expectaton and standard devaton. In the robust optmum desgn of thrust hydrodynamc bearngs, the desgn varables such as groove depth and groove wdth rato are gven by the followng expresson as the random varable vector s. s s, s, sn (6) Fgure 4 shows a conceptual dagram consderng random varables s and s. In the fgure, a target value μ ( = ~ n, where n represents the number of random varable) s the central value of the dstrbuton of desgn varable, n whch the desgn varable s assumed to dstrbute accordng to the Gaussan dstrbuton wthn the range of ±Δs from the central value μ. Consequently, the margnal probablty densty functon for the component s of random varable vector s can be expressed as follows. s f ps e 0 ~n (7) π In ths paper, the tolerance of the bearng dmenson s consdered as ±3σ. Copyrght 0 ScRes.

4 H. HASHIMOTO, Y. SUNI 37 Fgure 4. Probablty densty functon. When the margnal probablty densty functons of s are ndependent each other, the conjunctve probablty densty functon s expressed as follows. f s f s f s f s ~n (8) p p p p As can be seen n Fgure 4, when the varablty s gven to the desgn varables by the Gaussan dstrbuton, the bearng performance wll be dstrbuted at the same tme. Therefore, n estmatng the bearng performance, t s necessary to use the expectaton and standard devaton. The expectaton and standard devaton of bearng performance are obtaned, respectvely, as follows, E Δs Δsn q s n f p s nd n d s s Δ Δsn s Δs Δsn q s d d s s n E sn fp sn sn s Δ Δ s s (9) n where q(s ) ndcates the bearng characterstcs. 4. Robust Optmum Desgn of HDDs Consderng Dmensonal Tolerance (0) In ths paper, the optmum problem s defned as maxmzng the bearng stffness of thrust hydrodynamc bearngs to mprove vbraton characterstc of HDDs. Fgure 5 shows a schematc dagram of a.5 HDD spndle motor wth hydrodynamc bearngs. A rotor whch conssts of a shaft, a hub, and two dsks s supported by two thrust hydrodynamc bearngs n the axal drecton. In desgnng the bearngs, we fxed the followng desgn varables, n whch outsde radus of bearng r s.55 mm, nsde radus of bearng r s.5 mm, rotatonal speed of shaft n s s 400 rpm, seal rato R r r s s s 0.636, nflow angle β s 5 deg., rotor mass m s 8.6 g and ol lubrcant vscosty μ s Pa s. The values are defned by referrng specfcatons of an actual.5 HDDs. Fgure 5. Schematc dagram of spndle motor of.5 HDD and hydrodynamc thrust bearng, () Overall vew of spndle motor, () Spral grooved thrust bearng; (a) Poston of thrust bearngs and (b) Groove geometry. In the present study, realzng an optmal desgn s frst done by examnng the magntude of the bearng stffness by changng the number of parttons of the cubc splne nterpolaton functon n the r drecton from two to sx to process the optmal number of parttons. As a result, because the maxmum value of the bearng stffness s found when there are four parttons, the number of parttons n the r drecton s fxed to four. In addton, groove depth h g, groove wdth rato α and number of grooves N are gven as desgn varables. Consequently, The desgn varable vector X consstng of the bearng dmensons s defned as follows:,,,, h,, N () 3 4 where 4 are extents of angle change from the ntal geometry. In the robust optmum desgn, the dmensonal tolerance of desgn varables s consdered. In ths case, t s essental to consder the tolerance for all desgn varables. However, there s a possblty that computatonal tme wll be enormously large because of relatvely large number of desgn varables. Therefore, the bearng dmensons wth hgh senstvty on the bearng performance are checked through the senstvty analyss. As a result of the senstvty analyss, groove depth h g and groove wdth rato α have hgh senstvty to the bearng characterstcs, whch are the bearng stffness K and ol lubrcatng flm thckness h r. Consequently, the dmensonal tolerances of groove depth h g and groove wdth rato α should be consdered. In ths paper, we determned expermentally the tolerance ranges of groove depth of Δh g = ±0.5, ±.0, ±.5, ±.0 μm and groove wdth rato of Δα g Copyrght 0 ScRes.

5 37 H. HASHIMOTO, Y. SUNI = ±0.03, ±0.05. On the other hand, as for the constrant functons, the upper and lower lmts of the desgn varables n Equaton () are consdered. In addton, the allowable ol lubrcatng flm thckness h a and non-negatve dampng coeffcents c wthn varablty are also consdered as the constrants to guarantee the safety operaton of the bearng. Moreover, the mnmum groove wdth L mn of grooved part as shown n Fgure 6 s consdered because there are some cases where the groove geometry wll be n rregular shapes. The mnmum groove wdth L mn should be larger than the dameter of the ndustral tool d a. Then, n ths paper, the dameter of ndustral tool of d a = 0.0 mm s determned wth reference to prove dameter of ndustral tool for an actual.5 HDD spndle motor. The values of constrant condtons are shown n Table. In the robust optmum desgn, t s necessary to reduce the varablty of bearng characterstc value. Therefore, n ths paper, the value of the standard devaton of bearng characterstc value obtaned by Equaton (0) ncludng 3σ (F(X)) has to be less than 0% of the expectaton value obtaned by Equaton (9). Then, the constrant equaton s defned as follows, F 0.E F X 3 X 0 () All of the constrant condtons are summarzed as the followng nequalty expresson: g 0 ~ X 8 (3) where the constrant functons n Equaton (3) are defned as follows. Fgure 6. Relatonshp between groove wdth and dameter of ndustral tool. Table. Constrant condtons. Parameters Values Mnmum extents of angle change δ mn (= ~ 4) (deg.) 80 Maxmum extents of angle change δ max (= ~ 4) (deg.) 80 Mnmum groove depth h gmn (μm) 5 Maxmum groove depth h gmax (μm) 5 Mnmum groove wdth rato α mn 0. Maxmum groove wdth rato α max 0.9 Mnmum number of grooves N mn 6 Maxmum number of grooves N max Allowable flm thckness h a (μm) 5.0 g mn, g max, g3 mn, g4 max, g5 3mn 3, g6 3 3max, g7 4mn 4, g8 4 4max, g9 hgmn hg, (4) g0 hg hgmax, g mn, g max, g3 Nmn N, g4 N Nmax, g5 ha hrmn, g6 c, g7 da Lmn, g8 5 KEK The objectve functon s expressed as follows: f X E K (5) The optmum desgn problem of thrust hydrodynamc bearngs s formulated from the above equaton as follows: X X ~ Maxmze : f subjected to g 0 8 (6) Ths optmum desgn problem s a typcal nonlnear optmum desgn problem because the objectve functon and constrant functons are nonlnear ncludng seven desgn varables. Therefore, the objectve functon has complcated multdmensonal dstrbuton. Consequently, at frst several solutons are nvestgated through parametrc study to fnd some local optmum solutons. Then, only the global optmum soluton wll be calculated usng sequental quadratc programmng (SQP) [8]. To clarfy the valdty of present robust optmum desgn, the results of the robust optmum desgn are compared wth the results neglectng tolerance. Fgures 7 and 8 show the flowcharts of the present robust optmum desgn and the conventonal optmum desgn neglectng tolerance, respectvely. As shown n these fgures, the process of obtanng the values of the expectaton and standard devaton neglectng tolerance are dfferent from that of the robust optmum desgn. The calculaton method of these values s descrbed as follows. In the optmum desgn neglectng tolerance, the same desgn varables and prescrbed values used for the robust optmum desgn are gven. However, the constrant condton of varablty n Equaton () and mnmum groove wdth L mn are excluded. Moreover, the objectve functon neglectng tolerance s dfferent from the functon of the robust optmum desgn because the dmensonal tolerance s neglected. The objectve functon s defned as follows. f X K (8) The flow of the optmum desgn neglectng tolerance s shown n the range enclosed wth sngle dotted lne n Fgure 8. The values of the expectaton and standard devaton are calculated usng the same tolerances of groove depth Δh g and groove wdth rato Δα by Equatons (9) and (0). Copyrght 0 ScRes.

6 H. HASHIMOTO, Y. SUNI 373 Fgure 7. Flowchart of present robust optmum desgn. 5. Example of Optmzed Results and Dscussons Fgure 9 shows the results of groove geometres and statc pressure dstrbutons, respectvely. In the fgures, the groups () and () show respectvely the results obtaned by the robust optmum desgn (tolerance range, Δh g = ±.0 μm, Δα = ±0.05) and by the optmum desgn neglectng tolerance and the group (3) shows the result of spral groove bearng. The optmzed groove bearngs by the robust optmum desgn and by the optmum desgn neglectng tolerance as shown n Fgures 9() and () have new groove geometres wth smlar geometry of spral groove bearng n the nner vcnty; wth an opposte spral geometry n the outer vcnty. In ths paper, such types of bearngs that have an outer vcnty bends are called modfed spral groove bearng. Table shows the example of optmzed results. As can be seen n Table, the values of bearng stffness of modfed spral groove bearngs s more than four tmes the value of spral groove bearng. The bearng stffness s ncreased wth decreasng the ol lubrcatng flm thckness because the relatonshp between the ol flm Fgure 8. Flowchart of calculaton of objectve functon by conventonal optmum desgn neglectng tolerance. thckness and the bearng stffness s trade-off. The pressure of outer bearng boundary s decreased by the nverse step effect as shown n Fgures 9() and () by accompanyng the ol lubrcatng flm thckness s decreased. However, mnmum ol lubrcatng flm thcknesses h rmn are the same value as the allowable flm thckness (h a = 5.0 μm). Consequently, there s a low rsk of contact between the bearng and housng. On the other hand, the value of mnmum groove wdth L mn by the robust optmum desgn s exceeded the constrant value of 0.0 mm, whle the value of the optmum desgn neglectng tolerance s mm. As can be seen n the Table, the number of grooves N obtaned by the robust optmum desgn s reduced compared wth the number of optmum desgn neglectng tolerance. Addtonally, the extents of angle change are dfferent from the changes of the optmum desgn neglectng tolerance. As a result, the optmzed values of the number of grooves N and extents of angle change 4 are newly found to secure the mnmum groove wdth. Therefore, n the robust optmum desgn, t s possble to determne the dameter of ndustral tool under product desgn by provdng the constrant condton for groove wdth and to reduce the Copyrght 0 ScRes.

7 374 H. HASHIMOTO, Y. SUNI Fgure 9. Bearng geometry and statc pressure dstrbuton of each optmum desgn and spral grooved bearng (ntal groove), () Robust optmum desgn consderng tolerance (tolerance range, h g = ±.0 m, = ±0.05 ), () Optmum desgn neglectng tolerance and (3) Spral grooved bearng; (a) Groove geometry and (b) Statc pressure dstrbuton. Parameters Table. Optmzed results Robust opt. consderng tolerance (tolerance range, Δh g = ±.0 μm, Δα = ±0.05 ) Opt. neglectng tolerance Spral groove Extent of angle change δ (deg.) Extent of angle change δ (deg.) Extent of angle change δ 3 (deg.) Extent of angle change δ 4 (deg.) Groove depth h g (μm) Number of grooves N 6 Groove wdth rato α Mnmum flm thckness h rmn (μm) Mnmum groove wdth L mn (mm) Bearng stffness K (N/m) manufacturng costs due to easy manufacturng of bearng grooves. Fgure 0 shows a comparson of the results of robust optmum desgn and optmum desgn neglectng tolerance. In the fgures, (a), (b) and (c) show the mnmum ol lubrcatng flm thckness wthn varablty, the varablty of bearng stffness and the expectaton of bearng stffness, respectvely. As can be seen n Fgure 0(a), the values of mnmum flm thckness obtaned by the optmum desgn neglectng tolerance are less than the allowable flm thckness. On the other hand, the values of the robust optmum desgn are exceeded the allowable flm thckness for all tolerances. Ths means that there s a low rsk of contact between the bearng and housng when the bearngs are manufactured wthn the settng ranges of tolerance. In Fgure 0(b), the vertcal axs shows the rato of expectaton value and standard devaton of bearng stffness. Ths means that when the rato becomes smaller, the expectaton becomes larger and standard devaton becomes smaller. The values of rato obtaned by the robust optmum desgn are less than the rato by the optmum desgn neglectng tolerance. In addton, the tendency of rato becomes flat and the values have not ncreased for all tolerances of groove depth. As a result, t s confrmed that the robust optmum desgn s effectve of suppressng the varablty of bearng stffness. As can be seen n Fgure 0(c), the expectaton of bearng stffness by the robust optmum desgn s equvalent to the value neglectng tolerance when the tolerance of groove depth s set wthn Δh g = ±.5 μm. On the other hand, the value s decreased from Δh g = ±.5 μm to Δh g = ±.0 μm. Because the relatonshp between the ol flm thckness and the bearng stffness s trade-off, the expectaton of bearng stffness s decreased wth ncreasng the mnmum ol flm thckness n the tolerance range as Copyrght 0 ScRes.

8 H. HASHIMOTO, Y. SUNI 375 press the varablty wth hgh bearng stffness. (a) 6. Conclusons Ths paper descrbed the methodology and sample of robust optmum desgn consderng dmensonal tolerance based on the statstcal method combned wth the geometrcal optmzaton for thrust hydrodynamc bearngs of a.5 HDD spndle motor. The conclusons are brefly summarzed as follows. ) The optmzed groove geometres obtaned by the robust optmum desgn and by the conventonal optmum desgn neglectng tolerance are the modfed spral groove wth bends n the vcnty of the outer crcumference of the bearng. The bearng stffness of the modfed spral groove bearng becomes more than four tmes compared wth the stffness of spral groove bearng. ) It s possble to determne the dameter of ndustral tool under product desgn by provdng the constrant condton for groove wdth. 3) The expectaton of bearng stffness by the robust optmum desgn s equvalent to the value neglectng tolerance, and the standard devaton can be suppressed compared wth the standard devaton neglectng tolerance when the bearngs are manufactured wthn the tolerance of groove depth of Δh g = ±.5 μm. (b) (c) Fgure 0. Comparson of results of robust optmum desgn consderng tolerance and optmum desgn neglectng tolerance: (a) Mnmum ol lubrcatng flm thckness; (b) Varablty of bearng stffness; and (c) Expectaton of bearng stffness. shown n Fgure 0(a). The reason why such result obtaned s because of suppressng the varablty of bearng stffness by Equaton (). Consequently, the tolerance of groove depth of Δh g = ±.5 μm s recommended to sup- REFERENCES [] K. Ono and J. Zhu, A Comparson Study on Characterstcs of Fve Types of Hydrodynamc Ol Bearng for Applcaton to Hard Dsk Drve Spndles, Transactons of the Japan Socety of Mechancal Engneers, Seres (C), Vol. 64, No. 6, 998, pp [] J. Zhu and K. Ono, A Comparson Study on the Performance of Four Types of Ol Lubrcated Hydrodynamc Thrust Bearngs for Hard Dsk Spndles, Transactons of the ASME, Journal of Trbology, Vol., No., 999, pp. 4-. do:0.5/ [3] K. Matsuoka, S. Obata, H. Kta and F. Toujou, Development of FDB Spndle Motors for HDD Use, IEEE Transactons on Magnetcs, Vol. 37, No., 00, pp do:0.09/ [4] G. H. Jang, S. H. Lee, H. W. Km and C. S. Km, Dynamc Analyss of a HDD Spndle Wth FDBs Due to the Bearng Wdth and Asymmetrc Grooves of Journal Bearng, Mcrosystem Technologes, Vol., No. 7, 005, pp do:0.007/s [5] T. Hrayama, S. Saka, N. Yamaguch, N. Hshda and H. Yabe, A Proposal of Optmum Groove Desgn Concepts of Spral-Grooved Journal Bearngs Appled to Spndles of Precson Equpments (st Report, A Gude for Reducton of Ampltude of Repeatable Run-Out n Incompressble Flud Flm Bearng), Transactons of the Japan Socety of Mechancal Engneers, Seres (C), Vol. 7, No. 73, 006, pp do:0.99/kkac.7.4 [6] H. Yabe, T. Hrayama, S. Saka, N. Yamaguch and N. Copyrght 0 ScRes.

9 376 H. HASHIMOTO, Y. SUNI Hshda, A Proposal of Optmum Groove Desgn Concepts of Spral-Grooved Journal Bearngs Appled to Spndles of Precson Equpments (nd Report, A Gude for Reducton of Ampltude of Indcal Response), Transactons of the Japan Socety of Mechancal Engneers, Seres (C), Vol. 7, No. 79, 006, pp [7] S. Tohma, T. Isoga, T. Hrayama, T. Matsuoka, K. Yokozuka and S. Mor, Frequency Analyss of Hard Dsk Drve Spndle System Supported by Hydrodynamc Bearngs, Journal of Advanced Mechancal Desgn, Systems, and Manufacturng, Vol., No. 5, 007, pp do:0.99/jamdsm..77 [8] H. Hashmoto and M. Ocha, Optmzaton of Groove Geometry for Thrust Ar Bearng to Maxmze Bearng Stffness, Transactons of the ASME, Journal of Trbology, Vol. 30, No. 3, 008, pp. -. do:0.5/ [9] M. D. Ibrahm, T. Namba, M. Ocha and H. Hashmoto, Optmum Desgn of Thrust Ar Bearng for Hard Dsk Drve Spndle Motor, Journal of Advanced Mechancal Desgn, Systems, and Manufacturng, Vol. 4, No., 009, pp do:0.99/jamdsm.4.70 [0] K. Y. Koak, G. H. Jang and H. W. Km, Whrlng Tltng and Axal Motons of a HDD Spndle System Due to the Manufacturng Errors of FDBs, Mcrosystem Technologes, Vol. 5, No. 0-, 009, pp do:0.007/s [] M. Fllon, W. Dmochowsk and A. Dadouche, Numercal Study of the Senstvty Tltng-Pad Journal Bearng Performance Characterstcs to Manufacturng Tolerances: Steady-State Analyss, STLE Trbology Transactons, Vol. 50, No. 3, 007, pp do:0.080/ [] W. Dmochowsk and M. Fllon, Numercal Study of the Senstvty Tltng-Pad Journal Bearng Performance Characterstcs to Manufacturng Tolerances: Dynamc Analyss, STLE Trbology Transactons, Vol. 5, No. 5, 008, pp do:0.080/ [3] W. Xu, P. J. Ogrodnk, M. J. Goodwn and G. A. Bancroft, The Stablty Analyss of Hydrodynamc Journal Bearngs Allowng for Manufacturng Tolerances. Part: Effect Analyss of Manufacturng Tolerances by Taguch Method, Proceedngs of Measurng Technology and Mechatroncs Automaton 009, Vol., Zhangjaje, - Aprl 009, pp [4] P. J. Ogrodnk, M. J. Goodwn, G. A. Bancroft and W. Xu, The Stablty Analyss of Hydrodynamc Journal Bearngs Allowng for Manufacturng Tolerances. Part: Stablty Analyss Model wth Consderaton of Tolerances, Proceedngs of Measurng Technology and Mechatroncs Automaton 009, Vol., Zhangjaje, Hunan, - Aprl 009, pp [5] G. J. Park, T. H. Lee, H. L. Kwon and K. H. Hwang, Robust Desgn: An Overvew, Amercan Insttute of Aeronautcs and Astronautcs, Vol. 44, No., 006, pp [6] H. Hashmoto and M. Ocha, Theoretcal Analyss and Optmum Desgn of Hgh Speed Gas Flm Thrust Bearngs (Statc and Dynamc Characterstc Analyss wth Expermental Verfcatons), Journal of Advanced Mechancal Desgn, Systems, and Manufacturng, Vol., No., 007, pp. 0-. do:0.99/jamdsm..0 [7] N. Kawabata, Study on Generalzaton of Calculatons of Lubrcant Flow Usng Boundary Ftted Coordnate System (Part, Basc Equatons of DF Method and Case of Incompressble Fluds), Transactons of the Japan Socety of Mechancal Engneers, Seres (C), Vol. 53, No. 494, 987, pp Copyrght 0 ScRes.

10 H. HASHIMOTO, Y. SUNI 377 Nomenclature b b c d a E(F(X)) f(x) f p (X) : Wdth of groove (m) : Wdth of land (m) : Dampng coeffcent of ol lubrcatng flm (N s/m) : Dameter of ndustral tool (m) : Expectaton of bearng characterstcs : Objectve functon (N/m) : Probablty densty functon g : Acceleraton of gravty (m/s ) g (X) ( = ~ n) : Constrant functon h g h r k K N n s P ( = ~n) : Depth of groove (m) : Ol lubrcatng flm thckness (m) : Sprng coeffcent of ol lubrcatng flm (N/m) : Bearng stffness (N/m) : Number of grooves : Rotatonal speed of shaft (rpm) : Nodal ponts parttonng cubc splne nterpolaton functon n the r drecton p 0 : Statc component of ol lubrcatng flm pressure (absolute pressure) (Pa) p a : Atmospherc pressure (Pa) p t : Dynamc component of ol lubrcatng flm pressure (Pa/m) r : Coordnate of radal drecton (m) r : Outsde radus of bearng (m) r : Insde radus of bearng (m) r s : Seal dameter (m) R s : Seal dameter rato (=r s r ) T r : Frcton torque of bearng surface (nm) W : Load-carryng capacty of bearng (n) X : Vector of varables used n calculatons : Groove wdth rato (=b (b + b )) : Inflow angle (deg.) r : Equpartton space of r (m) : Coordnate of crcumferental drecton (rad) : Angle of ntal geometry (spral curvature) at the th nodal pont (rad) (F(X)) δ : Standard devaton of the bearng characterstcs : Extent of angle change from ntal geometry (spral curvature) at the th nodal pont (rad) : Extent of angle change durng evoluton at the th nodal pont (rad) : Vscosty of ol lubrcatng flm (Pa s) : Densty of ol lubrcatng flm (kgm 3 ) : Coordnates of change based on boundary-ftted coordnate system (m) : Coordnates of change based on boundary-ftted coordnate system (rad) f : Squeeze frequency of the rotatng shaft (rad/s) Subscrpts Max : Maxmum value of state varables Mn : Mnmum value of state varables Appendx I The cubc splne functon s a cubc polynomal equaton n each secton of r, r,,, n. The condton requred for the cubc splne functon s a contnuty of the second order dervatve of the functon at each nodal pont. The cubc splne nterpolaton functon s expressed as the followng equaton. r 3 r 3 r r r rr 6r 6r r r r r r (I-) 6 r r r r r r 6 r Consderng that r 0 r where for 0 and n, n are gven as follows: A d (I-) r r, A, rn d d d dn where d n Equaton (I-3) s expressed as the followng. (I-3) d r r (I-4) r r r 3 d,3,, n(i-5) r d n r n r (I-6) An arbtrary groove geometry can be represented by fndng the splne functon n all sectons of the target groove geometry usng Equatons (I-) and (I-). Appendx II When optmzng the groove geometry, t s necessary to perform a sequental analyss of characterstcs of a bearng wth a groove geometry modfed n successon by the fnte dfference method, but because of the r-θ polar coordnate system, drect processng s rather dfferent to n Copyrght 0 ScRes.

11 378 H. HASHIMOTO, Y. SUNI realze. Therefore, an analyss of the bearng stffness s performed frst by usng a boundary-ftted coordnate system as shown n Fgure [6,7] and transformng a complex groove geometry nto a smple fanlke geometry. A boundary transformaton functon used for the transformaton s gven as r (II-) r r r r r r 6r 6r r r r r r 6 r 3 3 r r r r r 6 r (II-) The followng Reynolds equvalent equaton can be obtaned from the equlbrum between the mass flow rates of ol nflowng nto and outflowng from the control volume due to the shaft rotaton and the squeezng moton: Q Q Q Q Q Q Q Q Q (II-3) I III II IV I II III IV where Q, Q ndcate the mass flow rates, across the boundary of = const. and across the boundary of = const. as shown n Fgure. On the other hand, Q ndcates the mass flow rate due to squeezng moton nsde the control volume. Q, Q and Q are expressed, respectvely, as follows: In that case, p p d (II-4a) Q A B DE p p Q B C F G Q d (II-4b) h Jdd (II-4c) t h h h r sh A a, B b, C c, D J J J 3 3 r sh rsh r sh E r, F r, G r, 40 40, a r r b r r r r, c r r, J r r r r, r r (II-4c) r, r, r r, r r Then, n Equaton (II-3), subscrpts l and and I-IV r, (a) Orgnal bearng geometry (b) Transformed bearng geometry Fgure. Bearng geometry transformaton based on the boundary-ftted coordnate system. Fgure. Defnton of control volume. ndcate the domans n the control volume. Assumng that varatons of the bearng clearance are mcroscopc, the mnmum ol lubrcatng flm thckness h and pressure p can be expressed by the followng equaton: j f t f h h0 e p p0 pe j t t (II-6) In that case, ε ndcates the ampltude of small varatons of the ol lubrcatng flm thckness and p 0 and p t express a statc component and a dynamc component, respectvely. The substtuton of Equaton (II-6) nto Equaton (II-3) and the neglgence of terms of small magntude of ε of above second order allow the ntroducton of two equatons for terms ε of orders 0 and l as follows: F p Q Q Q Q Q 0 0 I 0 III 0 II 0 IV 0 I 0 Q Q Q II 0 III 0 IV 0 0 F p, p Q Q Q Q Q t t 0 It IIIt IIt IVt It Q Q Q Q 0 IIt IIIt IVt t (II-7a) (II-7b) Copyrght 0 ScRes.

12 H. HASHIMOTO, Y. SUNI 379 In that case, subscrpt 0 ndcates a statc component of the mass flow rate determned from Equaton (II-4) and subscrpt t smlarly ndcates a dynamc component. Solvng Equatons (II-7a) and (II-7b) n turns by the Newton-Raphson teraton method, the statc and dynamc components, p 0 and p t, are obtaned. Copyrght 0 ScRes.