Stability Analysis of a Three-Ball Automatic Balancer

Transcription

1 Chung-Jen Lu 1 Associate Pofesso cjlu@ntu.edu.tw Chia-Hsing Hung Gaduate Student Depatment of Mechanical Engineeing, National Taiwan Univesity, No. 1 Roosevelt Road Section 4, Taipei 1617, Taiwan, R.O.C. Staility Analysis of a Thee-Ball Automatic Balance Ball-type automatic alances can effectively educe the viations of optical disk dives due to the inheent imalance of the disk. Although the all-type automatic alance used in pactice consists of seveal alls moving along a cicula oit, few studies have investigated the dynamic chaacteistics of all-type alances with moe than two alls. The aim of this pape is to study the dynamic chaacteistics of a thee-all automatic alance. Emphasis is put on the effects of the nume of alls on the staility of the pefect alancing positions the equiliium positions whee the disk is pefectly alanced. A theoetical model of an optical disk dive packed with a thee-all automatic alance is constucted fist. The govening equations of the theoetical model ae deived using Lagange s equations. Closed-fom fomulas fo the equiliium positions ae pesented. The staility of the pefect alancing positions is checked with the vaiations fo a pai of design paametes. Stale egions of the pefect alancing positions in the paamete plane of a thee-all alance ae identified and compaed with those of a two-all alance. DOI: / Intoduction 1 Coesponding autho. Contiuted y the Technical Committee on Viation and Sound of ASME fo pulication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscipt eceived June 4, 27; final manuscipt eceived Apil 1, 28; pulished online August 14, 28. Assoc. Edito: Weidong Zhu. Optical disk dives have een widely used fo data stoage. The speed of the spindle has een ought up to 1, pm to incease the data tansfe ate. At such a high otational speed, the optical disk dive may suffe fom seious viations due to the eccenticity of the optical disk used. Since the imalance vaies fom disk to disk, it is desiale to have an automatic alance system ABS equipped with an optical disk dive that can eliminate the imalance associated with each disk automatically. The most popula ABS adopted y optical disk dive industy is the all-type ABS. A all-type ABS consists of seveal alls moving along a cicula oit. Unde pope woking conditions, the alls will settle at specific positions such that the viation due to the eccentic mass of the disk can e totally suppessed. These specific positions ae efeed to as the pefect alancing positions hencefoth. The pefomance of the ABS is closely elated to the staility of the pefect alancing positions. Bövik and Högfos 1 analyzed the staility of the pefect alancing position of a two-all alance using the method of multiple scales. Lee and Van Moohem 2 studied the staility of the equiliium positions of a one-all ABS using the Floquet theoy. Rajalingham et al. investigated the nonlinea system consisting of an undamped oto and a one-all ABS. The staility of the equiliium position was detemined y the coesponding lineaized system. Hwang and Chung 4 studied the dynamic chaacteistics of a two-all alance with doule aces. Chung and Ro 5 and Kang et al. 6 investigated the staility of a two-all ABS compehensively. The staility was checked with the vaiations fo a pai of design paametes. Kim and Chung 7 employed the Floquet theoy to study the effects of the stiffness and damping of ue suspensions on the staility of the equiliium positions. Huang et al. 8 investigated the pefomance of a one-all ABS on the adial viation eduction of optical dives. Lu 9 analytically studied the staility of the equiliium positions of a one-all alance. The esults indicated that at most one stale equiliium state may exist at a otating speed. Wang and Lu 1 poposed a new design of all-type alance that may incease the stale egion of the pefect alancing position. Chung and Jang 11 consideed the effects of the flexiility of the shaft on the pefomance of the ABS. The staility of the alanced equiliium state of a two-all ABS was analyzed y petuation methods. Kim et al. 12 examined the dynamic ehavio of a two-all alance ased on a thee-dimensional model. Chao et al. 1,14 pesented a nonplana model fo a spindle-disk system with an ABS. Thei esults indicated that the ABS can effectively educe the tilting angle as well as the level of adial viation. Chao et al. 15 evaluated the pefomance of a two-all ABS with consideation to the in-plane otational motions. The effects of the nonlineaity of the suspension washes on the pefomance of the automatic alance wee examined in Ref. 16. Rajalingham and Bhat 17 and Geen et al. 18 found that the system may settle to a peiodic attacto in addition to a alanced state. A nonlinea ifucation analysis of a two-all alance fo otating machines was conducted y Geen et al. 19 Although the all-type alance cuently used in pactice consists of moe than two alls, much wok has een done on the dynamic chaacteistics of one-all o two-all alances. In the pesent pape, we study the staility of the pefect alancing positions of a otating disk equipped with a thee-all ABS. The nonlinea equations of motion ae deived using Lagange s equations with espect to a cootating coodinate system. Closed-fom fomulas fo the equiliium positions ae deived. Stale egions of the pefect alancing positions in a paametic plane ae identified and compaed with those of a two-all alance. 2 Mathematical Model and Govening Equations The fictional foce due to the elative motion etween the alls and the oit plays an impotant ole in detemining the pefomance of the ABS. Thee diffeent models of fictional foce on the alls have een employed in the liteatue: 1 viscous damping foce 1,2,4,5,7,9,11 14,17 19, 2 dy fiction 2,21, and olling fiction 6,8,1,15,16,22. The majo diffeence etween these models is that 1 has no effects on the equiliium positions while 2 and can pevent the alls fom settling at the desied alancing positions. When the fictional foce is small, analytical studies show that these thee models esult in simila qualitative ehavios in the steady state. Moeove, expeimental esults conducted with optical disk dives agee well with the analytical pedictions ased on the viscous damping model 2,8,14. These esults imply that the viscous damping model not only geatly simplifies the analysis wok ut also povides useful insights fo the dynamical chaacteistics of all-type alances Jounal of Viation and Acoustics Copyight 28 y ASME OCTOBER 28, Vol. 1 / 518-1

2 y all i x whee J is the moment of inetia of the equivalent oto and M indicates the total mass of the system. The potential enegy is given y c x k x k y d O C G e t disk c y V = 1 2 Kx2 + y 2 Rayleigh s dissipation function can e epesented y R = 1 2 Cx2 + y 2 2ẋy +2xẏ + 2 x y C Bd 2 2 i The sustitution of Eqs. 2 4 into Eq. 1 yields the nonlinea equations of motion as follows: 4 Fig. 1 Schematic of the ABS-oto system and cootating efeence fame used in pactice. Theefoe the viscous damping model is employed in this wok to account fo the enegy dissipation due to the fictional foce etween the alls and the oit. Figue 1 shows the schematic of the ABS-oto system and the cootating efeence fame. The ABS is composed of a cicula disk with a goove containing alls and a damping fluid whose damping constant is C B. The alls, each of mass m, move feely along the goove, sujected to viscous damping only. The adius of the goove is d. The poduct of m and d is the amount of imalance caused y a single all. Fo simplicity, m d is efeed to as the imalance of a single all heeafte. The oto consists of an unalanced otating disk and the suspension system. The disk of mass m d otates with a constant angula velocity. The mass cente G of the disk is located at a distance e fom the disk s geometic cente C. Theefoe, the amount of imalance of the disk is m d e. The flexiility of the suspension system is chaacteized y equivalent linea spings and viscous dampes, denoted y k x,k y and c x,c y, espectively. Fo simplicity, we assume that k x =k y =K and c x =c y =C. The govening equations of an ABS with an aitay nume of alls have een deived in Refs. 2,5,17,19 and expessed in diffeent dimensionless foms. Fo the sake of completeness, the deivation of the equations of motion is descied iefly elow. The xy-efeence fame otates aout a fixed point O with the same speed as the disk. When the disk is at est, the geometic cente C of the disk coincides with O at which the suppoting spings ae undefomed. The position of C is defined y the coodinates x,y with espect to the cootating fame. The position of the ith all is given y the angle i elative to the mass cente G. The eason fo the choice of this cootating fame is that the equations of motion expessed in this fame ae autonomous. The equations of motion ae deived fom Lagange s equations given y d dt T q k T + V + R = 1 q k q k q k whee T is the kinetic enegy, V the potential enegy, R Rayleigh s dissipation function, and q k the genealized coodinates. The kinetic enegy can e expessed as T = 1 2 J Mẋ2 + ẏ 2 +2xẏ 2ẋy + 2 x y m d2eẏ +2 2 ex + 2 e m d 2 i + 2 2d i + ẋ ysin i +2d i + ẏ + xcos i 2 Mẍ 2Mẏ + Cẋ + K M 2 x Cy m d i sin i + i + 2 cos i = m d e 2 Mÿ +2Mẋ + Cẏ + K M 2 y + Cx + m d i cos i i + 2 sin i = m d 2 i + C B d 2 i + m dÿ cos i +2ẏ sin i 2 y cos i m dẍ sin i 2ẋ cos i 2 x sin i =, i = 1,2, 7 In ode to simplify the susequent analysis, we intoduce the following dimensionless vaiales: x* = x/e, y* = y/e, = d/e, t* = t n whee n = K/M indicates the undamped natual fequency of the system when the disk is not otating. The equations of motion in tems of the dimensionless vaiales defined aove can e witten in matix fom as Mq + C + Gq + g + f = whee q=x*,y*, 1, 2, T, indicates diffeentiation with espect to t*, and 1 sin 1 sin 2 sin 1 cos 1 cos 2 cos sin 2 cos 2 sin cos = M sin 1 cos 1 2 C =2 2 B 2 B 2 B cos 1 cos 2 cos 1 sin 1 sin 2 sin G cos =2 1 sin 1 cos 2 sin 2 cos sin / Vol. 1, OCTOBER 28 Tansactions of the ASME

3 cos i 2 2x* y* 2 sin i 2 x*sin 1 y*cos 1 2 x*sin 2 y*cos 2 2 x*sin y*cos i g = 2 cos i i 2 sin i 2y* 2 f =1 2x* in which = m d m d e, = C, =, n 2M n = m d M, B = C 2m n 9 It is woth noting that is the atio of the imalance of a single all to that of the disk. Equiliium Positions The fist step in analyzing a nonlinea system is to detemine the equiliium positions. Setting q, q, and g in Eq. 8 to zeo, the location of the equiliium position q =x,ỹ, 1, 2, T is given y fq =. This set of equations can e expessed explicitly as 1 2 x 2ỹ 2 cos 1 + cos 2 + cos = 2 1 2x ỹ 2 sin 1 + sin 2 + sin = 11 x sin i ỹ cos i =, i = 1,2, 12 To solve Eqs. 1 12, we intoduce the pola coodinates x = cos and ỹ= sin and ewite Eq. 12 as sin i =, i = 1,2, 1 The aove equations equie that sin i = o =. The equiliium positions associated with these two conditions ae efeed to as the unalanced equiliium positions sin i = and pefect alancing positions =. If the alls ae indistinguishale, the unalanced equiliium positions can e futhe divided into fou cases: i 1= 2= =, ii 1= 2= = +, iii 1 = +, 2= =, and iv 1=, 2= = +. Including the solution v =, thee ae totally five diffeent cases of solutions to Eq. 1. Cases i and ii ae equivalent to a single all with mass m located at = and = +, espectively. Similaly, cases iii and iv ae equivalent to a single all with mass m located at = and = +, espectively. Consequently, the equiliium positions of cases i iv can e detemined fom the fomulas fo the equiliium positions of a one-all alance y sustituting appopiate values to the imalance. Since the equiliium positions fo a one-all alance have een analyzed compehensively, the detailed algeaic pocess fo detemining the equiliium positions fo cases i iv is omitted. Only the esults ae listed elow. i 1= 2= =. The two solutions fo ae efeed to as 11 and 12 and expessed as = K1 2 D 14 whee and K = D = The coesponding is detemined y = tan ii 1= 2= = +. The two solutions fo ae denoted y 21 and 22 and expessed as = K 2 D 18 whee K and D ae defined y Eqs. 15 and 16, espectively. The coesponding is detemined y = tan iii 1= +, 2= =. The two solutions fo ae 2 1 = K1 2 D whee K is defined y Eq. 15 and D = The coesponding is detemined y = tan iv 1=, 2= = +. The two solutions fo ae = K 1 2 D whee K and D ae defined y Eqs. 15 and 21, espectively. The coesponding is detemined y = tan In the aove fou cases, the effects of the thee alls ae equivalent to a single all with pope mass located at eq. Heeafte, the value of eq associated with ij is denoted y ij and is used to identify the equiliium positions of the thee alls. Fo example, when = j, the system is equivalent to a single all with mass m located at j=, whee is detemined y sustituting j into Eq. 22. In this case, moe specifically, two alls stick togethe at = j while the othe all is located at = j+. Similaly, when = 4j, one all is located at = while the othe two alls stick togethe at = 4j= +, whee is detemined Jounal of Viation and Acoustics OCTOBER 28, Vol. 1 / 518-

4 y sustituting 4j into Eq. 24. Then we study the equiliium positions associated with case v, =, in detail. v =. In this case the system is pefectly alanced and has no esidual viation. Sustituting x =ỹ= into Eqs. 1 and 11 yields cos 1 + cos 2 + cos =/ sin 1 + sin 2 + sin = The positions of the thee alls cannot e detemined uniquely fom these two equations. Howeve, if the position of one of the alls is given, say,, the positions of the othe two alls, 1 and 2, can e expessed in tems of. When the position of the thid all is specified, the system can e teated as a new disk packed with a two-all alance, whee the new disk is composed of the oiginal disk and the thid all. Since the pefect alancing positions of a disk packed with a two-all alance have een studied thooughly, all we need to do is to detemine the imalance of the new disk and then sustitute appopiate values of the imalance atio into the fomulas fo the equiliium positions of a two-all alance. Let G and B indicate the position vectos of the mass cente of the oiginal disk and the thid all, espectively. With espect to the cootating efeence fame, G =eî and B =dcos î+sin ĵ. The position vecto G of the cente of mass of the new disk can e expessed as whee and G = 1 m d + m m B + m d G = ecos î + sin ĵ e = m de m + m d 2 +2 cos +1 = tan sin cos +1 Thus the imalance atio of the new disk is given y m d = m + m d e It is well known that fo a two-all alance, pefect alancing is possile only if the imalance atio is no less than.5. Consequently, fo the new disk to e pefectly alanced, m d = m + m d e By sustituting Eq. 28 into Eq. 1 and eaanging tems, we otain the condition fo pefect alancing fo the thee-all alance as cos A = Based on this esult, we can detemine the estiction on fo pefect alancing and the coesponding ange of. By noticing that A is stictly inceasing fo and using the facts A1/=, A1/ =, and A1=1, we have the following esults. 1 Fo 1/, A. In this case, the total imalance of the thee alls is less than that of the disk. Hence pefect alancing is impossile. 2 Fo 1/1, A1. Fom Eq. 2, cos A. Hence, must lie in the ange eal pat of eigenvalue Fig. 2 Vaiation of the eal pats of eigenvalues with the otational speed fo a pefect alancing position of a thee-all alance with =.4 *,2 *, whee * = cos A Since the alls ae indistinguishale, all thee alls must lie in this ange. Fo 1, A1. As a esult, thee is no estiction on the position of the alls. The aove discussion indicates that fo 1/, pefect alancing is possile. In this case, it is not difficult to show that the positions of the othe two alls can e expessed in tems of as 1 = cos 2 +2 cos tan sin cos +1 4 cos +1 2 = + cos 2 +2 cos tan sin In summay, the system is pefectly alanced if 1 and 2 ae given y Eq. 4 and lies in the ange *,2 * 5 whee * = cos 2 1, 1 2 6, 1 4 Staility Analysis The staility of an equiliium position can e detemined y the eigenvalues of the associated lineaized system if all the eigenvalues have negative eal pats o if at least one of the eigenvalues has a positive eal pat. The lineaization fails when the lineaized system has some eigenvalues with zeo eal pats and no eigenvalues with positive eal pats. Fo unalanced equiliium positions, i.e.,, none of the eigenvalues have zeo eal pat. In this case, the staility of the equiliium position can e detemined using the Routh citeion. Fo pefect alancing positions, y contast, thee may exist eigenvalues with zeo eal pats. Figue 2 is a typical esult of a thee-all alance showing the vaiation of the eal pats of the eigenvalues of a pefect alancing position with the otational speed. As can e seen fom the figue, when the otational speed is lowe than aout 1.51, one eigenvalue / Vol. 1, OCTOBER 28 Tansactions of the ASME

5 has a positive eal pat. Hence, the pefect alancing position is unstale fo On the othe hand, when the otational speed is highe than 1.51, one eigenvalue is zeo while the othe eigenvalues have negative eal pats. Since none of the eigenvalues in this speed ange has a positive eal pat, the staility of the pefect alancing position cannot e detemined fom the lineaized system. Theefoe, in the pesent study, the staility of a pefect alancing position is detemined fom its steady state ehavios sujected to distuances. We conside distuances in each coodinate diection of the state space in sequence, one diection at a time. The steady state ehavio is otained y integating numeically the nonlinea govening equation Eq. 8 fo a long time until the tansient viation dies away. A pefect alancing position is stale if the steady state solutions unde all distuances appoach the initial position; othewise, it is unstale. Thee is anothe issue egading the staility of the pefect alancing positions needed to e addessed. Fo a two-all alance, each all should stay at a specific position when the system is pefectly alanced. In othe wods, thee is a unique pefect alancing position. By contast, fo a thee-all alance, thee ae infinitely many pefect alancing positions. The system is pefectly alanced if all the alls stay in the angula ange defined y Eq. 5 and thei positions satisfy Eq. 4. Fom the point of view of pactical application, the exact locations of the alls ae not impotant as long as the disk is pefectly alanced. Hence, when studying the staility of the pefect alancing positions, we divide evenly the possile angula ange of the alls, as defined y Eq. 5, into seveal small intevals. By setting to the cental value of each small inteval and detemining 1 and 2 fom Eq. 4, we get a set of equiliium positions. The staility of each equiliium position in the set is examined y the method discussed in the peceding paagaph. In this way, the stale egion of each equiliium position in a paamete plane can e identified. The stale egion of = is taken as the union of the stale egions of the set of equiliium positions examined. Pevious investigations on the all-type automatic alance indicate that in addition to the imalance atio, the otational speed, damping atio of the suspension, and damping atio etween the alls and the oit ae impotant paametes influencing the pefomance of the automatic alance. Theefoe, we study the staility of the pefect alancing positions in the - plane point y point fo diffeent values of and. Fo a specified value of, stale egions fo = in the - plane fo vaious values of ae identified, and the esults ae compaed with those of a two-all alance =.2, =.1, =.1, = Results and Discussion Figue shows the vaiation of the equiliium positions with the otational speed fo =.2. As discussed efoe, fo 1/, pefect alancing can not e achieved. Hence, fo =.2, all the equiliium adii ae nonzeo. In addition, the equiliium adii i2, 4, ae negative. Since negative adii ae unealistic, all the negative equiliium adii i2 and the associate s ae not shown in Fig.. Fo simplicity, fom now on, we use the equiliium adius to identify the coesponding equiliium configuation. Fo example, 11 indicates the equiliium configuation in which the thee alls stick togethe at 11; 21 indicates the equiliium configuation in which the thee alls stick togethe at 21. On the othe hand, 1 indicates the equiliium configuation in which one all is at 1+ and the othe two alls ae at 1; 41 indicates the equiliium configuation in which one all is at 41 and the othe two alls ae at 41. The meaning of ij is explained in the paagaph elow Eq. 24. The cicles in Fig. denote the steady state esponses otained fom numeical integation. The esults imply that among all the equiliium positions, 11 may e the only stale equiliium position at the specific values of paametes. Then we check the staility of 11 with the vaiations of,, and. The staility of 11 is detemined y the eigenvalues of the coesponding lineaized system. The stale egions of 11 in the - plane at =.2 fo diffeent values of ae shown in Fig. 4. Each ell-shaped cuve indicates the ounday of the unstale egion of 11 fo the coesponding value of. Specifically, inside the aea enclosed y the ellshaped cuve and the hoizontal axis, 11 is unstale. Fo example, =.2, =.1, =.1, =. Fig. Vaiation of the equiliium positions with the otational speed fo = A. Unstale Fig. 4 Stale egion of 11 at =.2 fo vaious values of Jounal of Viation and Acoustics OCTOBER 28, Vol. 1 / 518-5

6 =.4, =.1, =.1, = IC_1 IC_ t 2. =.4, =.1, =.1, = i 1 IC_1 IC_ Fig. 5 Vaiation of the equiliium positions with the otational speed fo = t Fig. 6 Responses stating fom two diffeent sets of initial conditions fo =.4 and Ω=2 at point A, 11 is stale fo =. ut unstale fo =.1. The esults indicate that the unstale egion fo 11 deceases with inceasing. Figue 5 shows the vaiation of the equiliium positions with the otational speed fo =.4; diffeent lines indicate diffeent equiliium configuations. Fo the meanings of ij and ij, efe to the fist paagaph of this section. The cicles and tiangles denote steady state esults otained y numeical integation fom diffeent initial conditions. The esults of numeical integation indicate that all equiliium positions, except 11 and =, ae unstale. Futhemoe, 11 is stale when the otational speed is less than aout.9, while = is stale when the otational speed is highe than aout As can e seen fom the ottom of Fig. 5, when =, the steady-state positions of the alls depend on the initial conditions. To clealy illustate the dependence of the steady-state positions on the initial conditions, the esponses of the system stating fom two diffeent sets of initial conditions ae plotted in Fig. 6. The two sets of initial conditions ae 1 x=y =, 1 =.1, 2 =.2, =., and ẋ=ẏ= i= and 2 x=y=, 1 =., 2 =.2, =.4, and ẋ=ẏ= i=. The esults indicate that in the steady state, the system is pefectly alanced ut the alls occupy two diffeent sets of positions. Then we poceed to compae the stale egions of = fo a thee-all alance to those fo a two-all alance. To make the compaison of stale egions meaningful, the total imalance atio the atio of the imalance of all the alls to that of the disk should e the same. Recall that indicates the atio of the imalance of a single all to that of the disk. The total imalance atio total can e expessed as total =2 two = thee whee two and thee ae the imalance atios of a single all fo the two-all and thee-all alances, espectively. Figue 7 compaes the stale egions of = fo the thee-all and two-all alances in the - plane at total =1.2. The solid and dashed lines indicate the oundaies of the stale egions fo the thee-all and two-all alances, espectively. Inside the aea to the ight and elow the solid/dashed line, = is stale fo the thee =.4 two = A Stale B.15.1 Fig. 7 Compaison of the stale egions of = fo the theeall solid and two-all dashed alances at total = / Vol. 1, OCTOBER 28 Tansactions of the ASME

7 .2.1 Two-Ball Thee-Ball t Fig. 8 Radial viations of the two-all and thee-all alancesatpointbinfig.7fo =.1 associated value of. Fo example, when =.1, = is unstale at point A fo oth the two-all and thee-all alances. On the othe hand, at point B, = is stale fo the thee-all alance ut unstale fo the two-all alance. Time esponses fo the two-all and thee-all alances associated with point B fo =.1 ae shown in Fig. 8. The esults veify that = is unstale fo the two-all alance ut stale fo the thee-all alance. Figue 9 compaes the stale egions of the thee-all and two-all alances in the - plane fo total =2.4. Figues 7 and 9 show that the thee-all alance has lage stale egions than the twoall alance unde the conditions examined. This, in tun, implies that the oustness of an ABS can e enhanced y employing moe alls if the othe paametes ae fixed. 6 Conclusion Ball-type automatic alances ae employed widely y the optical disk dive industy to suppess the viations induced y the eccenticity of the optical disk. Pope conditions unde which the imalance viations can e suppessed pefectly ae closely elated to the staility of the pefect alancing positions. Although much wok has een done on the analysis of the staility of alltype automatic alances, the effects of the nume of alls on the pefomance of the alance have not een clealy examined. This pape investigates the staility of the pefect alancing positions of a thee-all automatic alance. Closed-fom fomulas fo the ζ η thee =.8 η two = Ω Fig. 9 Compaison of the stale egions of = fo the two-all and thee-all alances at total =2.4 equiliium positions ae deived, and the conditions fo pefect alancing ae examined. The esults show that, to achieve pefect alancing, the imalance atio of a single all should e no less than one-thid. Moeove, the alls should all stay in an angula ange elative to the mass cente of the disk. The staility of the pefect alancing positions is studied numeically and the stale egions in a paamete plane ae identified. The esults indicate that the thee-all alance has lage stale egions compaed with the two-all alance in the paamete space examined. Theefoe inceasing the nume of alls may enhance the oustness of the all-type automatic alance. Acknowledgment This wok was suppoted y the National Science Council of R.O.C. unde Gant No. NSC E-2-. Refeences 1 Bövik, P., and Högfos, C., 1986, Autoalancing of Rotos, J. Sound Vi., 111, pp Lee, J., and Van Moohem, W. K., 1996, Analytical and Expeimental Analysis of a Self-Compensating Dynamic Balance in a Rotating Mechanism, ASME J. Dyn. Syst., Meas., Contol, 118, pp Rajalingham, C., Bhat, R. B., and Rakheja, S., 1998, Automatic Balancing of Flexile Vetical Rotos Using a Guided Ball, Int. J. Mech. Sci., 49, pp Hwang, C.-H., and Chung, J., 1999, Dynamic Analysis of an Automatic Ball Balance With Doule Races, JSME Int. J., Se. C, 422, pp Chung, J., and Ro, D. S., 1999, Dynamic Analysis of an Automatic Dynamic Balance fo Rotating Mechanisms, J. Sound Vi., 2285, pp Kang, J. R., Chao, C. P., Huang, C. L., and Sung, C. K., 21, The Dynamics of a Ball-Type Balance System Equipped With a Pai of Fee-Moving Balancing Masses, ASME J. Vi. Acoust., 124, pp Kim, W., and Chung, J., 22, Pefomance of Automatic Ball Balances on Optical Disc Dives, Poc. Inst. Mech. Eng., Pat C: J. Mech. Eng. Sci., 21611, pp Huang, W. Y., Chao, C. P., Kang, J. R., and Sung, C. K., 22, The Application of Ball-Type Balances fo Radial Viation Reduction of High-Speed Optic Disk Dives, J. Sound Vi., 25, pp Lu, C. J., 26, Staility Analysis of a Single-Ball Automatic Balance, ASME J. Vi. Acoust., 1281, pp Wang, M.-C., and Lu, C.-J., 27, Dynamic Chaacteistics of a One-Unit Ball-Rod-Sping Balance, Tans. ASME, J. Vi. Acoust., 1294, pp Chung, J., and Jang, I., 2, Dynamic Response and Staility Analysis of an Automatic Ball Balance fo a Flexile Roto, J. Sound Vi., 2591, pp Kim, W., Lee, D.-J., and Chung, J., 25, Thee-Dimensional Modelling and Dynamic Analysis of an Automatic Ball Balance in an Optical Disk Dive, J. Sound Vi., 285, pp Chao, P. C. P., Huang, Y.-D., and Sung, C.-K., 2, Non-Plana Dynamic Modeling fo the Optical Disk Dive Spindles Equipped With an Automatic Balance, Mech. Mach. Theoy, 811, pp Chao, P. C. P., Sung, C. K., Wu, S. T., and Huang, J. S., 27, Nonplana Modeling and Expeimental Validation of a Spindle-Disk System Equipped With an Automatic Balance System in Optical Disk Dives, Micosyst. Technol., 18 1, pp Chao, P. C. P., Sung, C.-K., and Wang, C.-C., 25, Dynamic Analysis of the Optical Disk Dives Equipped With an Automatic Ball Balance With Consideation of Tosional Motions, ASME J. Appl. Mech., 726, pp Chao, P. C. P., Yo, B. C., Sung, C. K., and Chiug, C. W., 26, Nonlinea Dynamic Effects of Damping Washes on the Pefomance of Automatic Ball Balances in Optical Disc Dives, Int. J. Nonlinea Sci. Nume. Simul., 7, pp Rajalingham, C., and Bhat, R. B., 26, Complete Balancing of a Disk Mounted on a Vetical Cantileve Shaft Using a Two Ball Automatic Balance, J. Sound Vi., 291 2, pp Geen, K., Champneys, A. R., and Fiswell, M. I., 26, Analysis of the Tansient Response of an Automatic Dynamic Balance fo Eccentic Rotos, Int. J. Mech. Sci., 48, pp Geen, K., Champneys, A. R., and Lieven, N. J., 26, Bifucation Analysis of an Automatic Dynamic Balancing Mechanism fo Eccentic Rotos, J. Sound Vi., 291 5, pp Van de Wouw, N., Van den Heuvel, M. N., Nijmeije, H., and Van Rooij, J. A., 25, Pefomance of an Automatic Ball Balance With Dy Fiction, Int. J. Bifucation Chaos Appl. Sci. Eng., 151, pp Yang, Q., Ong, E.-H., Sun, J., Guo, G., and Lim, S.-P., 25, Study on the Influence of Fiction in an Automatic Ball Balancing System, J. Sound Vi., , pp Chao, P. C. P., Sung, C.-K., and Leu, H.-C., 25, Effects of Rolling Fiction of the Balancing Balls on the Automatic Ball Balance fo Optical Disk Dives, ASME J. Tiol., 1274, pp Jounal of Viation and Acoustics OCTOBER 28, Vol. 1 / 518-7