Natural Frequencies and Stability of a Flexible Spinning Disk- Stationary Load System With Rigid-Body Tilting

Transcription

1 Jen-San Chen Department f Mechanical Engineering, atinal Taiwan University, Taipei, Taiwan 107 D. B. Bgy Cmputer Mechanics Labratry, Department f Mechanical Engineering, University f Califrnia, Berkeley, CA atural Frequencies and Stability f a Flexible Spinning Disk- Statinary Lad System With Rigid-Bdy Tilting The effects f rigid-bdy tilting n the natural frequencies and stability f the headdisk interface f a flexible spinning disk are studied bth by analysis and numerical cmputatin. Anther prblem f a statinary flexible disk with rigid bdy tilting subject t a rtating lad system is als studied. In the cnventinal case withut rigid-bdy tilting, these tw prblems are mathematically identical except fr the additinal membrane stress terms arising frm the centrifugal frce. When the rigidbdy tilting is included, hwever, these tw prblems differ nt nly by the centrifugal effect, but als by additinal terms which arise frm the gyrdynamic effect. After prperly defining the inner prduct between tw state vectrs, the set f equatins f mtin is prven t fall int the categry f gyrspic systems. Then the derivatives f the eigenvalues f the cupled system are btained t prvide a better understanding f the numerical results. Intrductin Interest in the dynamics f a spinning disk in cntact with a statinary lad system arises frm its applicatins in such fields as cmputer disk drives and circular saws. Cnventinally, the circular disk is free at the uter radius and clamped at the inner radius by a central clamp, usually assumed t be rigid. The statinary lad system, which may cntain mass, damping, and stiffness elements, bth linear and rtary, represents the head suspensin in the cmputer disk drive, r the saw guide in the circular saw, respectively. Varius techniques have been used t predict the natural frequencies and the stability prperties f such a cupled system, fr example, the classical eigenfunctin expansin (Iwan and Meller, 1976; Shen and Mte, 1991) and the finite element methd (On et al., 1991). In all these analyses the disk, tgether with the rigid central clamp, rtate abut a fixed axis f symmetry f the disk. In a recent experiment with hard disk drives, Jeng and Bgy (1991) reprted that the disk may underg a small rigidbdy displacement as well as defrmatin when the read-write system is laded nt the spinning disk. It has als been knwn in the wd cutting industry that allwing the central clamping Cntributed by the Applied Mechanics Divisin f THE AMERICA SOCIETY OF MECHAICAL EGIEERS fr presentatin at the First Jint ASCE-EMD, ASME-AMD, SES Meeting, Charlttesville, VA, June 6-9, 199. Discussin n this paper shuld be addressed t the Technical Editr, Prfessr Lewis T. Wheeler, Department f Mechanical Engineering, University f Hustn, Hustn, TX , and will be accepted until fur mnths after final publicatin f the paper itself in the ASME JOURAL OF APPLIED ME CHAICS. Manuscript received by the ASME Applied Mechanics Divisin, Jan. 16,1992; final revisin, June 22, Assciate Technical Editr: D. J. Inman. Paper. 9-APM-2. cllar t slide transversely alng the axis f rtatin can imprve the stability f the cutting prcess. In rder t understand the effects f the rigid-bdy mtin n the natural frequencies and stability f the spinning flexible disk, it is necessary t generalize the cnventinal mdel t include the rigid-bdy mtin f the clamp r cllar. The dynamic respnse f the head-disk interface f a spinning flexible disk with axial spindle translatin has been thrughly studied (Mte, 1977; Price, 1987; Chen and Bgy, 1992c). Hwever, the effects f rigid-bdy tilting n the natural frequencies and stability f the system are still nt well understd. Arnld and Maunder (1961) derived the equatins f mtin and described the gyrdynamic behavir f an elastically supprted spinning rigid disk. Stahl and Iwan (197a) investigated the dynamic respnse f an elastically supprted statinary rigid disk subject t a rtating lad system. It was shwn that the respnse behavir f the rigid disk is directly analgus t that f a statinary elastic disk withut rigidbdy tilting and subject t a circumferentially mving lad system. Yang (1988) frmulated the equatins f mtin fr a general system with cupled rigid and flexible bdy mtin, ' and applied it t the stability analysis f a spinning flexible disk with rigid bdy translatin and tilting. Hwever, his numerical results shwed the cmbined effects f rigid-bdy translatin and tilting, frm which it is difficult t btain physical insight int the gyrdynamic effects f rigid-bdy tilting alne. In the present paper we cnsider a spinning flexible disk clamped at the inner radius and subject t a statinary lad system. The clamp is cnsidered as rigid and allwed t tilt abut a fixed pint in space, but resisted by a trsinal spring 470/Vl. 60, JUE 199 Transactins f the ASME Cpyright 199 by ASME Dwnladed 16 Oct 2008 t Redistributin subject t ASME license r cpyright; see

2 ph(w + 2Qwig + Q w ee )+DV w - (a r rw r ) r r ' ' h e w,ee + r ph(q x sin 0-9,, cs 6) 1 + 2tirph(B x cs sin d) = - 8(r - %)b{0)f z ////////// (a) disk (b) Fig. 1 (a) Glbal picture f a spinning disk with rigid-bdy tilting and the statinary lad system; (/>) parameters in the lad system and a trsinal damper. In the special case when the stiffness f the clamp spring and the value f the clamp damping are zer, we shw by numerical simulatin, as well as by an analytical methd, that the natural frequencies f all the nendal diameter mdes (as seen by an bserver mving with the tilting clamp) increase, as culd be expected due t the inertial effect. Anther prblem f a statinary flexible disk with rigid-bdy tilting and subject t a rtating lad system is als studied. It is fund that these tw types f prblems differ nt nly by the stress terms arising frm the centrifugal frce, but als by additinal terms arising frm the gyrdynamic effects. After prperly defining the inner prduct between tw state vectrs, we shw that the set f equatins fall int the categry f gyrscpic systems. Then the derivatives f the eigenvalues f the system with respect t varius lad parameters in the lad system are derived t explain several phenmena discvered in the numerical simulatin when the lad parameters are intrduced. Equatins f Mtin and umerical Results Figure 1 shws a flexible spinning disk with rigid bdy tilting. The spinning disk is clamped at the inner radius r = a by a rigid clamp, and is free at the uter radius r = b. The rigid clamp is allwed t tilt, but it is supprted by a trsinal spring k e and a trsinal damper c e. The center f the clamp is fixed at a pint O in space. The mtin f a spinning flexible disk can be described in tw crdinate frames, as shwn in Fig. 1. The frame -xyz, tgether with the rigid clamp, tilts relative t the inertial frame O-XYZ. The rientatin f O-xyz with respect t O-XYZ is btained by rtatin 9 X abut OX t O- xtfizi and then rtatin 9^, abut Oy y t O-xyz. The elastic transverse displacement w f the disk is measured with respect t the lcal nninertial O-xyz frame. The disk rtates abut the Oz-axis and is in cntact with a statinary lad system cntaining transverse inertia m z, spring k z, dashpt c z, and the analgus pitching elements 7^,, k$, and c^. By using D'Alembert's principle and by assuming that tilting angles 9 X and 9^, are infinitesimal, the equatin f mtin f the disk in terms f vc and with respect t the lcal crdinate system (/, 6) can be written as where F z =-m z (w M - Q y )-c z (w,,- 9 y )-k z (w- Q y ) and -psir-mmtfiwh (1) M 4,= ~-[I 4,( w im + 9*) + c 0 ( w, + &) + k^w + & x )] D = - Eh 12(1 A The standard subscript ntatin fr partial derivatives is used here, and the superpsed dt means material time derivative. The parameters fi, p, h, E, v, and D are the rtatin speed, density, thickness, Yung's mdulus, Pissn's rati, and flexural rigidity f the disk. 5(:) is the Dirac delta functin. The cupling psitin between the lad system and disk is assumed t be r = J and 8 = 0. The generalized plane stress cmpnents a r and a e are due t the centrifugal effect. We next cnsider a particle n the flexible spinning disk with crdinates (r cs 0, r sin 6, w) with respect t the lcal O-xyz frame. The crdinates f the psitin vectr p with respect t the inertial frame O-XYZ can be calculated by the fllwing transfrmatin cs Q y 0 sin Q y sin 9_y 0 cs Q y cs Q x - sin 9^ 0 sin 9v cs Q x where Q is the prduct matrix f the tw rtatin matrices. The abslute velcity f this particle is (r cs d) (-rqsindl p = Q ] r sin d j + Q < rfl cs The ttal angular mmentum H f the disk, with respect t the rigin O, is 2TT nb H = p/j I K p x prdrdd. The material derivative f H is { 2ir nb \ p x prdrdd. K By cnsidering the balance between H and the resultant mment f the frces exerted by the statinary lad system, the clamp spring k e, and the clamp damper c e, and by linearizing the equatins in Q x, Q y, and w, ne can btain tw additinal equatins f mtin, { 2X nb ph sin d(wilt + 2Qw yte )r 2 drd6 J«= MMr-SWO)-k e e x -c e e x (2) Jurnal f Applied Mechanics JUE 199, Vl. 60/471 Dwnladed 16 Oct 2008 t Redistributin subject t ASME license r cpyright; see

3 (/+/ c )(e^-2fie! 2TT 0 nb J a ph cs 0(w, + 2Qw iw )r drdd = V&r-MP)-kQB y -c Q B () where / = irph(b 2 -a*)/4 is the mment f inertia f the circular plate and I c is the mment f inertia f the rigid clamp. These additinal equatins f mtin accunt fr the fact that there are tw additinal degrees-f-freedm, i.e., Q x and 6^,, f the present system. Fr the cnventinal freely spinning disk with Are =,c Q = Q,F Z = 0, andal, = 0, the eigenvalues X m are all purely imaginary, i.e., A" i<jl, where w T* mn is real. The crrespnding eigenfunctins are, in general, cmplex and assume the frm w mn = Rl {r)e ±ine, n= 1,2,,... (4) in which R mn is a_real-valued functin f r. The eigenfunctin crrespnding t \ m is w nm, where the verbar means cmplex cnjugate. If we cnsider nly the psitive w%, then w m in Eq. (4) with + ino is a backward travelling wave with n ndal diameters and m ndal circles, which is als dented by (m, n) b. Similarly, w mn with -ind is a frward travelling wave (m, n)/. The critical speed Q c fr the mde (m, n) is defined as the rtatin speed at which <jp mn f the backward travelling wave (m, n)/, is zer. Fr Q greater than Q c, this mde is a frward travelling wave, and is called a "reflected wave," dented by (m, n) r. In the case f a freely spinning disk with the clamp allwed t tilt freely, i.e., k Q = 0 and c e = 0, ne can multiply Eqs. (2) and () by sin 6 and cs 9, respectively, and substitute the results int Eq. (1) t btain ph(w,t + 2Qw, e + tfw ee ) + DV*w (<vw r ) r r - P»2T P6 r. r w ee [cs 01 \ r 2 cs 6 (w M + 2Qw ite )drdd (I + I c ) J 0 J 0 ^a r 2 sin 0 (w + 2Qw,, fl )cfrtf0] = 0. (5) The nly difference between Eq. (5) and that f a cnventinal freely spinning disk with a nntilting clamp is the additinal integral terms at the end f Eq. (5), which accunt fr the inertial cupling between the elastic mdes and the rigid-bdy tilting. It has been shwn by Chen and Bgy (1992a,b) that the eigenfunctins w mn are rthgnal and frm a cmplete system f functins in r and 6 in the dmain 0 < 8 < 2TT, a < r < b. We therefre can express the slutin w f Eq. (5) as the fllwing infinite series: >= 2 2 C mn W mn (r, 6). Frm Eqs. (5) and (6) it is bvius that due t the integratin nly the ne-ndal diameter mdes will be affected by the inertial frce assciated with the rigid-bdy tilting. Using a technique presented in Chen and Bgy (1992a), the natural frequency change fr mde (m, 1) due t the rigid-bdy tilting can be estimated as X c i / 7^> / ccv' ^ y 4/ y y / *0y^, (0,l) h Rtatin Speed O, Hz Fig. 2 atural frequency versus rtatin speed fr a freely spinning flexible disk be an (w + 1, 1) mde. Therefre, AX ml in Eq. (7) can be reinterpretated by using the mde label system with respect t the inertial frame as the change f natural frequency frm an (m, 1) mde f the spinning disk withut rigid bdy tilting t an (m + 1, 1) mde f the spinning disk with rigid-bdy tilting. The advantage f using the mdel label with respect t the inertial frame is that as the clamp trsinal spring k e varies frm zer t infinity, the natural frequency f the (m, 1) mde with rigid-bdy tilting cnverges t that f the (m, 1) mde withut rigid-bdy tilting. Althugh the mde label with respect t the tilting frame is mre cnvenient fr analysis and understanding in the cntext f ur system f equatins, the mde label with respect t the inertial frame gives a mre standard ntatin. Therefre we will adpt the latter mde label with respect t the inertial frame in the fllwing discussins and all the figures, unless stated therwise explicitly. Hwever, the switch between these tw slightly different ntatins can be easily made. It is bvius that AX ml in Eq. (7) is nnnegative fr a backward travelling wave. Hwever, fr a frward travelling wave the term u m \ - 2Q casts sme uncertainties n whether AX m i is psitive r negative. Figure 2 shws the relatin between the natural frequencies f the spinning disk and the rtatin speed. This result is btained by a finite element cmputatin which is similar t the ne presented in On et al. (1991). The material prperties f the disk used in the calculatin are p = 1. x (6) 10 kg/m, I c = 0, E = 4.9 X 10 9 /m 2 0., h = mm, a = 0.0 mm, b = 65.0 mm, and %/b = 0.9. Fr simplicity, nly the first few mdes with less than three ndal diameters are presented here. The dashed lines and the slid lines are the results fr the spinning disk withut and with rigid-bdy tilting, respectively. It is nted that fr the system with rigid-bdy tilting, the mde label system with respect t iirp 2 h 2 <J s ml(ij > mi±2q) AX,,=- J/7 (r)rdr 2(/+/ c )(a&,±q) J IRli (r)\ rdr + fr backward wave - fr frward wave. (7) It is emphasized here that the displacement w in Eqs. (1)- (6) is measured relative t the nninertial tilting frame. S, the (m, 1) mde f a spinning disk with a tilting clamp in Eq. (7) is a mde with m ndal circles and ne ndal diameter as seen by an bserver mving with the tilting frame. T an bserver in the inertial frame, hwever, this mde appears t the inertial frame has been adpted. The (0, 1)/ mde (the diagnal line) and the (0, l) b mde (the hrizntal line with zer natural frequency) fr the system with rigid-bdy tilting are the rigid displacement mdes, i.e., w = 0. This can be easily checked by shwing that the mde shapes w = 0, Q x = 1, and 9 V = ± / with natural frequencies w = 2fi and 0 satisfy 472 /Vl. 60, JUE 199 Transactins f the ASME Dwnladed 16 Oct 2008 t Redistributin subject t ASME license r cpyright; see

4 0 s OH Pi Fig Rtatin Speed Q., Hz Effects f clamp damping c = 10~ 5 ms/rad Eqs. (l)-(). Because the rati f the vertical and hrizntal scales in Fig. 2 is 2, the slpe f the diagnal line is actually 2. w, in rder fr the term i n,i - 20 in Eq. (7) t be negative, the dashed curve crrespnding t the frward travelling wave wuld have t intersect the diagnal line. As we can see frm Fig. 2, the dashed curve starts with unit slpe at 0 = 0, and increases its slpe t mre than 2 due t the effect f the centrifugal frce. Cnsequently, the dashed line des nt meet the diagnal line and the term u mx - 20 is always psitive. This explains why the natural frequencies f bth the frward and backward travelling waves (as seen by an bserver mving with the tilting frame) increase fr fixed 0 when the clamp is allwed t tilt freely. The mdeshape f each ne-ndal diameter mde (except fr the rigid displacement mdes (0, 1)/ and (0, \)b) invlves rigid bdy tiltings Q x and G_, with amplitudes wph " Qyml ' ~T J/7 i(r) ±ib x + fr backward wave - fr frward wave where R m \{r) crrespnds t the new mdeshape f the disk with rigid-bdy tilting. We nw cnsider the effects f changing the value f clamp damping CQ frm 0 t 10~ 5 ms/rad n the eigenvalues f the freely spinning disk with rigid-bdy tilting. It is fund that the natural frequency-rtatin speed diagram is nt affected by the change f c e and remains the same as the slid lines in Fig. 2. Hwever, the real parts crrespnding t the ne-ndal diameter mdes becme negative fr all the rtatin speeds, as shwn in Fig.. The real parts f all ther mdes are nt affected by the change f c e. It is interesting t investigate the rle that the centrifugal frce plays in the dynamic behavir f this system. Figure 4 is the calculatin result similar t that pltted in Fig. 2, except that the membrane stresses a r and a s in Eq. (1) are set t zer. w the slpe f the dashed frequency lcus crrespnding t the frward travelling wave with ne ndal diameter is exactly ne and it meets the diagnal line at rtatin speed Q A, at which bp ml - 20,4 is equal t zer. T an bserver mving with the tilting frame, as can be predicted by Eq. (7), the natural frequency f the frward travelling wave increases (frm dashed line (0, l)/t slid line (1, l)/in Fig. 4) due t rigid bdy tilting when 0 < Q,4, and decreases when 0 > Q A. The natural frequency f the backward travelling wave increases fr all 0 except at the critical speed, at which the natural frequency remains zer. As 0 exceeds Q A, the frward travelling wave decreases and eventually merges with the reflected wave at pint B, and instability is induced. Of curse n physical phenmenn crrespnding t Fig. 4 will ccur because the centrifugal frce always exists when the disk is spinning. Hwever, this discussin shws that the centrifugal frce plays a very imprtant rle in stabilizing the present system with rigid bdy X O c <D <U 1 l-h 4» V (l,l)f 1 - % / Xf-\ \ 1 4V he" B r (0,0), (0,Db Rtatin Speed Q., Hz Fig. 4 The same as in Fig. 2 except that the membrane stresses are neglected in the calculatin tilting. In the cnventinal case withut rigid-bdy tilting, n the ther hand, neglecting the centrifugal frce in the calculatin will nt induce any instability (Chen and Bgy, 1992a). Statinary Disk With Rtating Lad System It is interesting t cmpare the results btained in the previus sectin t the case f a statinary flexible disk with rigidbdy tilting and subject t a rtating lad system. T derive the equatins f mtin fr this system, we first neglect F z and M^ and set fi t zer in Eqs. (1), (2), and () t get the equatins f mtin f a statinary disk with respect t the O-XYZ frame. Suppse the O-xyz frame is rtating with the lad system abut the OZ-axis with rtatin speed -U. The terms (W) X YZ, (Q X )XYZ, and {Q y ) X Yz bserved in the O-XYZ frame are related t the crrespnding terms bserved in the O-xyz frame thrugh the equatins (W)XYZ = (w.tdxyz + 2Q(w tie ) xyz + Sl\w fi e) xyz {Q X )XYZ = {Q x ) xyz + 2Q(.Q y ) xyz - &(e x ) xyz {Qy)xYZ = (Qy) X yz ~ 20(6 X )^, Z Q (Qy) X y Z. Equatin (8) shws the relatin between the Lagrangian and Eulerian descriptins f the functin w. Equatins (9) and (10) are the cmpnent frms f the well-knwn transfrmatin equatin f a single vectr bserved in tw different frames (Meriam, 1980), (Q)XYZ = (Q) X y Z + 20 x (9 ) xyz + 0x0x0 where vectr 9 is in the XY-plane with cmpnents (Q x, Q y, 0), and vectr 0 is in the OZ-directin with amplitude 0. As a cnsequence, the equatins f mtin fr a statinary flexible disk with a rtating lad system are, (relative t the rtating crdinate system) Jurnal f Applied Mechanics JUE 199, Vl. 60/47 Dwnladed 16 Oct 2008 t Redistributin subject t ASME license r cpyright; see (8) (9) (10)

5 C <D cr >-. *» >1 U c <D 1 l-i Rtatin Speed 2, Hz Fig. 5 atural frequency versus rtatin speed fr a statinary flexible disk with rtating bserver Rtatin Speed O, Hz Fig. 6 atural frequency versus rtatin speed fr an elastically supprted rigid disk; k g = 0.01 ms/rad ph (w M + 2Qw :, e + Q w it,) + DV 4 w + phr[q x + 2UQ y -Q 2 e x )sm 6 (0JJ 2QQ X -fi 2 G_,,)cs d]=-8(r- )5(6)F z -^2 «(r-f)[j^6(a)],, (11) bdy tilting, respectively. The dashed lines in Fig. 5 are the same as thse in Fig. 4, as explained in (Chen and Bgy, 1992a). The slid lines fr the (1,1) mdes are parallel t the dashed lines fr the (0, 1) mdes with a cnstant shifting upward. instability is induced due t the additin f the rigid-bdy tilting. f.27t (/+/ c )(0 x +2ne. 1 ph sin 0(w i + 2Ow,,fl 0 ^a + Qfw M )r drdo = M^b (r -/r e e x -c e e x (12)»2ir (/+/ c )(9 J,-2ne : -n 2 e )- \ p/? cs 9(w /( + 2fiw,, e -1)5(5)-Ar e e -c e e,. (1) After setting k e = 0, c e = 0, F, 0, M 0 = 0, and substituting Eqs. (12) and (1) int Eq. (11), we btain fr the freely tilting disk in the absence f the lad system ph(w tll + 2Qw tie + Q w i$g )+DV w rp 2 h 2 r r 2T f» -Ji777) [csd { { r cs 6(w jlt + 2Qw iie + Q, w :ee )drdd 1-K nb + sin r 1 sin 8(w! t + 2fiw,, fl +fl 2 w,^)rf/-rfe] = 0 (14) 0 J 0 It is imprtant t nte that neglecting the membrane stresses a r andct 9 in Eq. (5) des nt result in Eq. (14) exactly, which cntains the additinal w <ee terms in the integrals. The natural frequency change fr mde (m, 1) (with respect t the tilting frame) due t rigid-bdy tilting can nw be estimated as A\ iirp 2 h\u> nl ±Q),b 12 Rlx(r)rdr 2(/+/ c ) [S m i (r)frdr fr backward wave fr frward and reflected waves. A Special Case: Elastically Supprted Rigid Disk The equatins f mtin in the inertial frame f a spinning rigid disk with tilting supprted by a trsinal spring k e are btained frm Eqs. (2) and () by setting w = 0, /(9 x + 20e,) + A:ee x +c e e x = M^(r- )5(f?) (16) I(Q y - 2QQ X ) + k e B y + c e 9 y = F z b(r - )5(0). (17) On the ther hand, the equatins f mtin in the rtating frame fr a statinary rigid-disk subjected t a rtating lad system are derived frm Eqs. (12) and (1), I(e x + 2Qe y -n 2 ) + k e e x + c e Q x = M l i,5(r-z)8(d) (18) I(,e y -2Qe x -n 2 ) + k e Q y + c e e y = ZF z 8(r-Z)8(d). (19) It is nted that Stahl and Iwan (197a,b) used different Euler angles in writing the equatins f mtin fr a statinary disk with rtating lad system. As a result their equatins are mre cmplicated than Eqs. (18) and (19), althugh they are essentially the same. We find here that the prblem f a spinning rigid disk with a statinary lad system is nt exactly equivalent t the prblem f a statinary rigid disk with rtating lad system when rigid-bdy tilting is included. The differences between Eqs. (16), (17) and Eqs. (18), (19) are the terms cn- (15) It is nted that R%i(r) in Eq. (15) is independent f rtatin speed H because n centrifugal frce is invlved. Furthermre, the term i mi ± 0 is a cnstant and is equal t the natural frequency when fi = 0 (Chen and Bgy, 1992a). Cnsequently, AX ml is a cnstant and independent f the rtatin speed. Figure 5 shws the relatin between the natural frequency f the statinary flexible disk as bserved frm a rtating frame and the rtatin speed f the frame. The dashed lines and the slid lines are the results fr the cases withut and with rigidtaining fi 2. In the case withut rigid-bdy tilting, the equatins f these tw prblems are exactly the same. The dashed straight lines in Fig. 6 are the natural frequencies f the statinary rigid disk with rtating bserver. The stiffness f the clamp spring k Q is 0.01 m/rad and the value f the clamp damping Ce is 0. The slpes f the frward and backward waves are 1 and - 1, respectively. There is a critical speed, beynd which the backward wave becmes a reflected wave. The slid lines are the natural frequencies f a spinning rigid disk bserved 474 / Vl. 60, JUE 199 Transactins f the ASME Dwnladed 16 Oct 2008 t Redistributin subject t ASME license r cpyright; see

6 20 0 OH OH OS df. 1 Ss // CD 0< C cr -y^/ >> C <u PL, 20 ^ ^, (,Db, '"- "^>s.t -- " (O^)^--^^- /^^^ Rtatin Speed Q, Hz Fig. 7 Effects f transverse stiffness k z quencies and stability f a flexible disk 5 /m n the natural fre- Fig. 8 Effects f transverse stiffness k z quencies and stability f a rigid disk Rtatin Speed Q., Hz 5 /m n the natural freby a statinary bserver. The frward and backward frequency lci start with slpe 1 and - 1, respectively, at Q = 0. Hwever, the slpe f the frward wave appraches 2 as fi increases. On the ther hand, the natural frequency f the backward wave appraches zer asympttically s there is n critical speed, and the backward wave will never becme a reflected wave. These behavirs f the slid lines have been explained analytically by Arnld and Maunder (1961). The imprtance f this cmparisn between these tw types f prblems lies in the fact that instability will be induced when the rtating lad system is laded nt the statinary disk, as discussed by Stahl and Iwan (197a,b). Hwever, n instability will be induced when the statinary lad system is laded nt the spinning disk. An example will be given in the next sectin. Effects f Lad Parameters We next cnsider the effects f the statinary lad system spring with cnstant k z n the natural frequencies and stability f the spinning flexible disk with rigid-bdy tilting. The clamp spring ICQ and clamp damper c e are set t 0. The dashed and the slid lines in Fig. 7 are the results fr k z = 0 and k z = 5 /m, respectively. By cmparing the dashed lines and the slid lines, we can see that veering ccurs whenever a frward r backward wave meets either anther zer-ndal diameter mde, a frward r a backward wave. At the intersectin pint f any tw curves fr k z = 0 at least ne eigenvalue remains unchanged when k z is intrduced. When the eigenvalue f interest is well separated frm the thers the additin f k z increases the natural frequency f a zer-ndal diameter mde, a frward r backward wave, but it decreases the natural frequency f a reflected wave. Furthermre, k z causes a statinary-type instability (On et al., 1991) in a finite rtatinal speed range just beynd the critical speed. These prperties are qualitatively the same as the effects f k z in a cnventinal disk withut rigid-bdy tilting (Chen and Bgy, 1992a,b). Unlike the case f a spinning flexible disk with axial translatin (Chen and Bgy, 1992c), the additinal rigid-bdy tilting mde des nt eliminate the statinary-type instability induced by a lad system transverse spring k z. Figure 8 shws the effects f k z = 5 /m n the elastically supprted rigid disk, with the clamp spring k e = 0.01 m/ rad. Similar t thse in Fig. 6, the dashed lines are the results fr a statinary disk with rtating lad system, and the slid lines are the results fr a spinning disk with statinary lad system. By cmparing with Fig. 6 we can see that fr a statinary rigid disk k z induces a statinary-type instability, as reprted by Stahl and Iwan (197a,b). On the ther hand, fr a spinning rigid disk k z simply increases the natural frequencies f bth the frward and backward waves, and induces n instability. Figure 9 shws the effects f the lad system mass m z n the eigenvalues f a disk with rigid-bdy tilting. The dashed lines are the results fr a freely spinning disk, and the slid lines are the results fr m z = 1 g. By cmparing these results with thse presented in Chen and Bgy (1992a,b) fr the cnventinal case, we bserve that the effects f m z are qualitatively the same fr bth the cases, with and withut rigid-bdy tilting. Additinal calculatins fr the parameters c z, k$, c^, and 7^, als shw that the effects f these parameters are qualitatively the same fr bth the cases, with and withut tilting. In particular, nne f the pitching parameters have an effect n the zer ndal diameter mdes. The dashed lines in Fig. 10 are fr the freely spinning disk with rigid-bdy tilting and Jurnal f Applied Mechanics JUE 199, Vl. 60/475 Dwnladed 16 Oct 2008 t Redistributin subject t ASME license r cpyright; see

7 I-I <u T t; C* 1 CM C* >1 G <u 60 a 20 " CO - Fig. 9 - ^ - * A Sy^4 _ i ^ - ^, (0,l)b Rtatin Speed 2, Hz Effects f transverse inertia m z = 1 g - u c (U cr 1 c Z Rtatin Speed O, Hz Fig. 10 Effects f pitching stiffness k v = m/rad the slid lines are fr k$ = m/rad. It is nted that rigid-bdy tilting des nt eliminate the statinary-type instability induced by the pitching stiffness k$. Derivatives f the Eigenvalues In Chen and Bgy (1992a), we shwed that fr a freely spinning disk withut rigid-bdy tilting, the eigenfunctins are rthgnal t each ther. Als it was shwn that expressins fr the derivatives f the eigenvalues with respect t the varius parameters in the statinary lad system can be derived and used t predict the change f eigenvalues fr small values f the parameters. In Chen and Bgy (1992c), we extended this analysis t the case when the axial translatin is included, and shwed that the equatins f mtin fall int the categry f gyrscpic systems. In this sectin we further extend the analysis t the case when the rigid-bdy tilting is included. By fllwing the same prcedure as in Chen and Bgy (1992c), we can rewrite Eqs. (1), (2), and () in the matrix frm (M + M)u + (G + G)u,, + (K + K)u = 0 where the vectr u is defined as (20) The matrix peratrs M, G, and K are assciated with the freely spinning disk, while M, G, and K are assciated with the statinary lad system. It is nted that Q x and G, are functins f time nly while w is als a functin f r and 6. The inner prduct between tw vectrs u, and u 2 is defined as 476/Vl. 60, JUE 199 wi w 2 rdrdd + Q xl Q x2 + G yl O y2. In the case f a freely spinning disk, Eq. (20) reduces t Mu + Gu, + Ku = 0. (21) It can be shwn that the matrix peratrs M and K are symmetric and G is skew, i.e., <Mui, u 2 > = <U!, Mu 2 > <Ku 1; u 2 > = <ui, Ku 2 > <Gui, u 2 )= -<ui, Gu 2 > Equatin (20) can als be written in the first rder frm by defining the state vectr and the matrix peratrs As M 0 0 K, B^ -G -K K 0 (A + A)x,-(B + B)x = 0 (22)» As M 0 0 K, B = -G -K K 0 w, the inner prduct between vectrs X[ and x 2 is defined as»2ir \ (w h,w2,,+ w i w 2 )rdrdd 0 J a + *1 X2 + 9xl ;t2 + vl y2 + yl Jyl^yl Again, it can be shwn that A is symmetric and B is skew. After establishing the abve definitins and fllwing the same prcedure as that presented in Chen and Bgy (1992a), Transactins f the ASME Dwnladed 16 Oct 2008 t Redistributin subject t ASME license r cpyright; see

8 we can readily shw that the eigenfunctins x, and x M f a freely spinning disk with rigid-bdy tilting are rthgnal with respect t the peratr A, and that the derivative f eigenvalues with respect t a lad parameter p k is where m "' k ~ ~ <i Ax~~> ii(j>mnwmn\ W w mn I A 1 f\ j ^ymn J By taking int accunt the fact that Q xmn and Q ymn are zer unless n = 1, a, 4TTU simple m ( ( u m calculatin ±nq)\ Rl (Chen,(r)rdr and Bgy, 1992a) nji- ±1 leads t Cwnj *^X mn ) 4 R m (r)r z dr\ c* \J» / 4irc m ( m + nq) j R 2 mn(r)rdr- ^ - A «=±1 Fr n 7^ ± 1, the term <x,, Ax, > is psitive fr bth frward and backward travelling waves, and is negative fr the reflected wave, as explained in Chen and Bgy (1992a). At the critical speed, hwever, the term <x,, Ax m > vanishes because > mn becmes zer. Fr n = ± 1 the term <x,, Ax, > can be shwn t be nn-negative by using the Schwartz inequality ( n2ir nb \ 2 «2ir nb /»2T nb \ 1 WiW 2 rdrd6 < 1 W\W { rdrdd \ \ w 1 w 2 rdrdd J K J J K J Ja where we chse w t = R m \(r)e' 6 and w 2 = re' e. With Eq. (2) we nw can calculate the derivative f the eigenvalue with respect t the transverse stiffness parameter k z. We first derive da/dk z and db/dk z, and substitute the results int Eq. (2) t btain d\nn JUmnJRmn ( ) ~ ynm],~«\ k z pn<x m, Ax ra ) Since Q xm = 0, Q ymn = 0, and R mn (r) = R a mn(r) fr n * 1, this expressin is the same as that fr the cnventinal disk withut rigid-bdy tilting presented in Chen and Bgy (1992a) fr the mdes with ther than ne ndal diameter. This means that fr n ^ 1 the effects f k z n the eigenvalues are the same fr bth cases, with and withut rigid-bdy tilting. Fr the ne-ndal diameter mdes, n the ther hand, k z increases the natural frequencies but by different amunts fr the cases with and withut rigid-bdy tilting. By fllwing similar prcedures, we btain the derivatives f the eigenvalues with respect t c z and m z by replacing iw mn in Eq. (25) by -«m and -iw, respectively, and similar cnclusins fllw accrdingly. The derivatives with respect t the pitching parameters are the same as thse f their transverse cunterparts, except that there is a multiplicative factr n 2 / 2 in the numeratrs. This cnfirms the numerical results, which shwed that pitching parameters have n effect n the zer-ndal diameter mdes because n = 0. Cnclusins The effects f rigid-bdy tilting n the dynamics f a flexible spinning disk in cntact with a statinary lad system are examined bth numerically and analytically. The results f these analyses can be summarized as fllws: (1) T an bserver mving with the tilting frame, the natural frequencies f ne-ndal diameter mdes increase due t the inertial cupling between the rigid-bdy tilting and the elastic { ' defrmatin, while the natural frequencies f the mdes with ther than ne-ndal diameter are nt affected. (2) In the cnventinal case withut rigid-bdy tilting, neglecting the centrifugal frce in the calculatin will nt affect the stability prperty f the freely spinning disk. Hwever, in the present case with rigid-bdy tilting, the centrifugal frce is very imprtant in stabilizing the system. () Anther type f prblem in which a statinary flexible disk is subject t a rtating lad system is als cnsidered. In the cnventinal case withut rigid-bdy tilting, these tw types f prblems are mathematically identical except fr the additinal stress terms arising frm the centrifugal frce. In the case with rigid-bdy tilting, hwever, these tw types f prblem differ nt nly by the centrifugal effect, but als by the terms arising frm the gyrdynamic effect. (4) The rigid-bdy tilting des nt eliminate any instability induced by any parameter in the statinary lad system. It may be futile trying t imprve the stability prperty f a spinning disk by allwing it t tilt. (5) The system f equatins f mtin is again prven t be gyrscpic in the presence f rigid-bdy tilting. As a result, the techniques develped in previus papers (Chen and Bgy, 1992a,b,c) are extended t the current case t btain the derivatives f eigenvalues with respect t varius lad system parameters. These derivatives prvide a theretical understanding fr the numerical results btained. Acknwledgment This wrk was supprted by the Cmputer Mechanics Labratry at the University f Califrnia at Berkeley. References (24) Arnld, R.., and Maunder, L., 1961, Gyrdynamics and Its Engineering Applicatins, Academic Press, ew Yrk. Chen, J.-S., and Bgy, D. B., 1992a, "Effects f Lad Parameters n the atural Frequencies and Stability f a Flexible Spinning Disk With a Statinary Lad System," ASME JOURAL OF APPLIED MECHAICS, Vl. 59, pp. S20- S25. Chen, J.-S., and Bgy, D. B., 1992b, "Mathematical Structure f Mdal Interactins in a Spinning Disk-Statinary Lad System," ASME JOURAL OF APPLIED MECHAICS, Vl. 59, pp Chen, J.-S., and Bgy, D. B., 1992c, "atural Frequencies and Stability f a Flexible Spinning Disk-Statinary Lad System With Axial Spindle Displacement," submitted t ASME JOURAL OF APPLIED MECHAICS. Iwan, W. D. and Meller, T. L., 1976, "The Stability f a Spinning Elastic Disk With a Transverse Lad System,'' ASME JOURAL OF APPLIED MECHAICS, Vl. 4, pp Jeng, T. G., and Bgy, D. B., 1991, "Slider-Disk Cntacts During Dynamic Lad-Unlad," IEEE Transactins n Magnetics, Vl. 27, pp Meriam, J. L., 1980, Dynamics, Jhn Wiley and Sns, ew Yrk. Mte, C. D., Jr., 1977, "Mving Lad Stability f a Circular Plate n a Flating Central Cllar," Jurnal f Acustical Sciety f America, Vl. 61,. 2, pp On, K., Chen, J.-S., and Bgy, D. B., 1991, "Stability Analysis f the Head- Disk Interface in a Flexible Disk Drive," ASME JOURAL OF APPLIED ME CHAICS, Vl. 58,. 4, pp , Price, K. B., 1987, "Analysis f the Dynamics f Guided Rtating Free Center Plates," Ph.D. Dissertatin, University f Califrnia, Berkeley. Shen, I.-Y., and Mte, C. D., Jr., 1991, "On the Mechanisms f Instability f a Circular Plate Under a Rtating Spring-Mass-Dashpt System," Jurnal f Sund and Vibratin, Vl. 148,. 2, pp Stahl, K. J., and Iwan, W. D., 197a, "On the Respnse f an Elastically Supprted Rigid Disk With a Mving Massive Lad," Internatinal Jurnal f Mechanical Science, Vl. 15, pp Stahl, K. J., and Iwan, W. D., 197b, "On the Respnse f a Tw-Degreef-Freedm Rigid Disk With a Mving Massive Lad," ASME JOURAL OF APPLIED MECHAICS, pp Yang, S.-M., 1988, "Vibratin and Stability f Rtating and Translating Plates," Ph.D. Dissertatin, University f Califrnia, Berkeley. Jurnal f Applied Mechanics JUE 199, Vl. 60/477 Dwnladed 16 Oct 2008 t Redistributin subject t ASME license r cpyright; see