外文资料--Squeal analysis of gyroscopic disc brake system based on finite element method.pdf

Squeal analysis of gyroscopic disc brake system basedon finite element methodJaeyoung KangDivision of Mechanical and Automotive Engineering, College of Engineering, Kongju National University, Cheonan-Si, Republic of Koreaa r t i c l e i n f oArticle history:Received 9 October 2008Received in revised form30 January 2009Accepted 12 February 2009Available online 9 March 2009Keywords:GyroscopicDisc brakeBrake squealMode-couplinga b s t r a c tIn this paper, the dynamic instability of a car brake system with a rotating disc in contact with twostationary pads is studied. For actual geometric approximation, the disc is modeled as a hat-disc shapestructure by the finite element method. From a coordinate transformation between the referenceand moving coordinate systems, the contact kinematics between the disc and pads is described. Thecorresponding gyroscopic matrix of the disc is constructed by introducing the uniform planar-meshmethod. The dynamic instability of a gyroscopic non-conservative brake system is numericallypredicted with respect to system parameters. The results show that the squeal propensity for rotationspeed depends on the vibration modes participating in squeal modes. Moreover, it is highlighted thatthe negative slope of friction coefficient takes an important role in generating squeal in the in-planetorsion mode of the disc.& 2009 Elsevier Ltd. All rights reserved.1. IntroductionDisc brake squeal has been investigated by many researchersforseveraldecades.Muchvaluableinformationonsquealmechanisms has been accumulated throughout the research.Kinkaid et al. [1] presented the overview on the various discbrake squeal studies. Ouyang et al. [2] published the review articlefocused on the numerical analysis of automotive disc brakesqueal. They have shown that one major approach on brakesqueal study is the linear stability analysis. From the linearizedequations of motion, the real parts of eigenvalues have beencalculated for determining the equilibrium stability. In theliterature, there are two major directions on the linear squealanalysis: the complex eigenvalue analysis of the static steady-sliding equilibrium [3–8] and the stability analysis of rotatingbrake system [9–12,14].The stability analysis at the static steady-sliding equilibriumof the stationary disc and pads provides the squeal mechanism asmode-merging character in the friction–frequency domain. Parti-cularly, Huang et al. [6] used the eigenvalue perturbation methodto develop the necessary condition for mode-merging without thedirect eigensolutions. Kang et al. [7] derived the closed-formsolution for mode-merging between disc doublet mode pair. Dueto the stationary disc assumption, the finite element (FE) methodhas been easily implemented as referred to the review article [2].Alternately, Cao et al. [13] studied the moving load effect from aFE disc brake model with moving pads, where the disc wasstationary, and therefore, the gyroscopic effects were neglected.Giannini et al. [15,16] validated the mode-merging behavior assqueal onset by using the experimental squeal frequencies.On the other hand, the stability of a rotating disc brake hasbeen investigated in the analytical manner. The rotating discbrake system has been modeled as a ring [10] and an annular plate[12] in point contact with two pads, and an annular plate subjectto distributed frictional traction [9]. With inclusion of gyroscopiceffect, the real parts of eigenvalues have been solved with respectto system parameters. Due to the complexity of the rotating discmodeling, however, a rotating FE disc brake model has not beendeveloped yet.Recently, Kang et al. [14] developed a theoretical disc brakemodel in the comprehensive manner. The disc brake modelconsists of a rotating annular plate in contact with two stationaryannular sector plates. The comprehensive analysis explained thestability character influenced by mode-coupling and gyroscopiceffect, and provided the physical background on the approxima-tions and mechanisms used in the previous squeal literature.However, it still contains limitations on examining brake squealmechanisms since the annular plate approximation does notrepresent all of modal behaviors existing on the physical discbrake, for example, the in-plane mode and hat mode of the disc.In this paper, the methodology of constructing a rotating FEdisc brake model is developed. Consequently, it enables usto examine the squeal mechanisms in the physical FE brakemodel subject to rotation effects. The global contact model [10]describing the contact kinematics under the undeformed config-uration is utilized to develop contact modeling between therotating disc and two stationary pads. From the assumed modeARTICLE IN PRESSContents lists available at ScienceDirectjournal homepage: Journal of Mechanical Sciences0020-7403/$-see front matter & 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijmecsci.2009.02.003E-mail address: jkang@kongju.ac.krInternational Journal of Mechanical Sciences 51 (2009) 284–294method, the equations of motion of the friction-engaged brakesystem are derived. The numerical results demonstrate severalsqueal modes and explain the corresponding squeal mechanisms.2. Derivation of equations of motionThe disc part of a brake system is modeled as a hat-disc shapestructure as shown in Fig.1. The hat-disc is subject to the clampedboundary condition at the inner rotating shaft and the freeboundary condition at the outer radius. Owing to the complexityof the geometry, the finite element method is utilized for modalanalysis. The disc rotation with constant speed (O) generatesfriction stresses over the contact with two stationary pads loadedby pre-normal load (N0). The friction material of the pad ismodeled as the uniform contact stiffness (kc), where contactstresses are defined on the global contact model. Centrifugal forceis neglected due to the slow rotation in the brake squeal problem.In order to describe the contact kinematics, the displacementvectors of the disc and top pad are expressed in the referencecoordinates (Fig. 2), respectively, such thatuðr;y;z;tÞ ¼ uðr;y;z;tÞerþ vðr;y;z;tÞehþ wðr;y;z;tÞez(1)up1ðr;y;z;tÞ ¼ up1ðr;y;z;tÞerþ vp1ðr;y;z;tÞehþ wp1ðr;y;z;tÞez(2)where the superscripts, p1 and p2 denote the top and bottompads, respectively, and the disc displacement is also defined in thelocal coordinates (Fig. 2):~uðr;c;z;tÞ ¼~uðr;c;z;tÞerþ~vðr;c;z;tÞehþ~wðr;c;z;tÞez(3)As shown in Fig. 3, the contact point P0of friction material of thetop pad is assumed to be in contact with P of the disc and laterallyfixed with R of the top pad, which results inup1P0ðr;y;tÞ ¼ up1Rðr;y;tÞerþ vp1Rðr;y;tÞehþ wPðr;y;tÞez(4)The velocity vectors of the disc and top pad are obtained fromthe following time-derivatives. First, the position vectors of thedisc are expressed in the local coordinates as~r ¼ ðr þ~uÞerþ~vehþ ðz þ~wÞez(5)~rP¼~rjz¼h=2(6)For describing the direction vector of friction force, the contactvelocity vector of the disc is derived by taking the time-derivativeARTICLE IN PRESSConnected with contact stiffness oNoNConnected with contact stiffness ΩClamped at an inner Zrotating shaft Fig. 1. Hat-disc brake system.tΩ⋅ψXθeZθreckΩNeutral surface Rotor part Fig. 2. Coordinate system of the rotating disc, reference (y) and local (c)coordinates.1ppo+kc (wP–wR)ZSegment of undeformed top surface of the disc Disc 1N−ze1Nzeck1−F1FBottom surface of the top Pad: contact area (Ac ) Pw1pRw1pRv1pRuPuPvZPRPR'PFig. 3. Contact kinematics at a contact point P (or P0) in the global contact model: (a) contact displacements; (b) contact forces. P0of friction material of the top pad is incontact with P of the disc.J. Kang / International Journal of Mechanical Sciences 51 (2009) 284–294285in Eq. (6) in the reference coordinates:VP¼D~rPDt(7)where the coordinate transformation is given by the differentia-tion in the local coordinates such thatD~uðr;c;z;tÞDt¼@uðr;y;z;tÞ@tþO@uðr;y;z;tÞ@y(8)D~vðr;c;z;tÞDt¼@vðr;y;z;tÞ@tþO@vðr;y;z;tÞ@y(9)D~wðr;c;z;tÞDt¼@wðr;y;z;tÞ@tþO@wðr;y;z;tÞ@y(10)Since the brake pad is stationary, the contact velocity vector at P0of the top pad is simply the partial time-derivative of Eq. (4):Vp1P0¼@up1R@terþ@vp1R@tehþ@wP@tez(11)From Coulomb’s law of friction, contact friction force isexpressed asF1¼ ?m1? N1VreljVrelj(12)where the normal load is the sum of pre-stress (p0¼ N0/Ac) andthe normal load variation:N1¼ p0þ kcðwP? wp1RÞ(13)and the relative velocity at top contact is given byVrel¼ VP? Vp1P0(14)In order to capture the negative slope effect, the continuousfriction curve [14] is used such thatm1ðtÞ ¼ fmkþ ðms?mkÞe?ajVreljgr¼rctr(15)wherems,mkandaare the control parameters determiningthe magnitude and the slope of the friction coefficient, and thefriction coefficient is assumed to be uniform and calculated at thecentroid of the contact area (rctr).The transverse vibrations of the disc and pad components areexpressedinthemodalexpansionformofN ¼ ðNdþ 2NpÞtruncated modes using the assumed mode method:wp1ðx;tÞ ffiXNpn¼1jp1z;nðxÞqp1nðtÞ(16)wðx;tÞ ffiXNdn¼1jz;nðxÞqnðtÞ(17)wp2ðx;tÞ ffiXNpn¼1jp2z;nðxÞqp2nðtÞ(18)where Ndand Npare the numbers of the truncated modes of thedisc and the pad, respectively, and whereqp1¼ fqp11qp12... qp1Npg(19)q ¼ fq1q2... qNdg(20)qp2¼ fqp21qp22... qp2Npg(21)jp1z;nðxÞ,jz;nðxÞ andjp2z;nðxÞ are the nth transverse mode shapefunctions obtained from the eigenfunctions of the top pad, discandbottompadcomponents,respectively.Theradialandtangential vibrations, ðup1;u;up2Þ and ðvp1;v;vp2Þ can be writtenin the modal expansion form associated with the correspondingmode shape functions fjp1r;nðxÞ;jr;nðxÞ;jp2r;nðxÞg, fjp1y;nðxÞ;jy;nðxÞ;jp2y;nðxÞg as well. The modal coordinates are rearranged in thevector form for the following discretization:fag ¼qp1p28><>:9>=>;¼ fa1a2... aNgT(22)FromthediscretizationofLagrangeequationbymodalcoordinates, the friction-coupled equations of motion are given byddt@L@_am???@L@am¼XNn¼1QmnðanÞ;m ¼ 1;...;N; n ¼ 1;...;N(23)L ¼ T ? ðU þ UcÞ(24)dW ?XNm¼1XNn¼1QmnðanÞdam(25)where U is the total strain energy of the uncoupled componentdisc and two pads, andT ¼ Tp1þ Tdþ Tp2(26)Td¼rZVdD~rDt?D~rDt??dV(27)Tp1¼rpZVp@up1@t?@up1@t??dV(28)Tp2¼rpZVp@up2@t?@up2@t??dV(29)Uc¼kc2ZAcðwP? wp1RÞ2dA þ Uc;bottom(30)dW ¼ZAcfð?N1? F1Þ ?dup1P0þ ðN1þ F1Þ ?duPgdA þdWbottom(31)Here Vdand Vpare the volumes of the disc and pad, respectively. Inthe similar manner of obtaining the virtual work and contactstrain energy at the top contact,dWbottomand Uc,bottomon thebottom contact can be derived as well.The direction vector of friction force at the top contact islinearized by Taylor expansion at the steady sliding equilibriumsuch thatVreljVrelj¼1rOð@uP=@t ? @up1R=@tÞ þ1rð@uP=@y? vPÞ??erþ ehþ1r@wP@yezþ h:o:t.(32)where h:o:t denotes the higher order terms. Here @wP=@yisassociated with frictional follower force as explained in [11] andneglected in the subsequent analysis due to the insignificance ofthe frictional follower force as referred to [5], [10,11] and [14].Using the finite element method, the transverse mode shapefunctions are discretized in the matrix formup1z??¼up1z;1up1z;2???up1z;Nphi¼jp1z;jðxiÞhi(33)½uz? ¼ ½uz;1uz;2...uz;Nd? ¼ ½jz;jðxiÞ?(34)up2z??¼up2z;1up2z;2???up2z;Nphi¼jp2z;jðxiÞhi(35)where the lengths of their columns correspond to the numbersof nodes in the component FE model. The radial and tangentialARTICLE IN PRESSJ. Kang / International Journal of Mechanical Sciences 51 (2009) 284–294286mode functions are also denoted as f½up1r?;½ur?;½up2r?g and f½up1y?;½uy?;½up2y?g.From the mass-normalization and the linearization at thesteady-sliding equilibrium of Eq. (23), the homogeneous part ofthe linearized equations of motion takes the (N?N) matrix formsuch that€a þ ð½G? þ ½C? þ ½Rd? þ ½Ns?Þ_a þ ð½o2? þ ½A? þ ½B? þ ½F?Þa ¼ 0(36)where the system matrices are described in Eq. (37) andEqs. (A.1)–(A.7) of Appendix A. Substituting aðtÞ ¼ aoeltintoEq. (36) and solving Re(l) and Im(l) of the characteristic equationresult in the determination of the modal stability and frequency.Here the physical meaning of each system matrix of Eq. (36) isprovided in the following. ½G?ð¼ ?½G?TÞ is the gyroscopic matrix tobe described in Eq. (37), [C] is the structural modal dampingmatrix, and ½Ns?ð¼ ½Ns?TÞ is the negative slope matrix. The negativefriction-slope effect can be referred to [17]. ½Rd?ð¼ ½Rd?TÞ is theradial dissipative matrix stemming from 1=rOð@uP=@t ? @up1R=@tÞin Eq. (32). Also, [o2] is the natural frequency matrix of thedisc and pad components, ½A?ð¼ ½A?TÞ is the contact stiffnessmatrix. Of the non-symmetric stiffness matrix, ½B?ða½B?TÞ isthenon-symmetricnon-conservativeworkmatrixproducedbyfriction-couple.½F?ða½F?TÞisderivedfromthein-planefrictional followerforceassociatedwith1=rð@uP=@y? vPÞinEq. (32), but neglected in the subsequent analysis due toits insignificance [14] as well. The local contact model [10]incorporated with the frictional follower forces can be referred toKang et al. [14], where the frictional follower force effects wereshown to be marginal due to the dominant role of [B] in thenumerical and analytical manners.In the finite element approach, several technical difficultiesare encountered in calculating Eqs. (27)–(31) numerically andsummarized as:?The mesh of the contact area between the disc and pad shouldbe identical in order to connect the finite contact forceelements on the same contact positions of the mating parts.?Tdrequires the numericaly-derivatives of modal vectors.Particularly, the gyroscopic matrix is given by½G? ¼½0?½0?½0?½0?½Gd?½0?½0?½0?½0?264375(37)where½Gd? ¼OrZVd½ur?T@ur@y??? ½uy????@ur@y??? ½uy???T½ur?(þ ½uy?T@uy@y??þ ½ur????@uy@y??þ ½ur???T½uy?þ ½uz?T@uz@y???@uz@y??T½uz?)dV(38)In order to resolve the above, the hat-disc and each pad should beuniformly meshed in the cylindrical coordinates by ANSYS (or anyother pre-processing FE software). In general, this task is trickyand not recommended for the practical purpose. Alternately, theuniform discretization will be achieved by interpolating the modalvectors of irregular meshes onto those of uniform meshes. Theonly pre-requisite for this task is to discretize the disc geometryin the axial direction (as Fig. 1) generating the planar mesh oneach layer perpendicular to the axis, where the planar meshes arenot yet uniform. Then, the modal vectors assigned to the planarmesh of each layer are interpolated onto those of the uniformmesh in polar coordinates by MATLAB, which will be referred tothe uniform planar-mesh method. Fig. 4 illustrates how the modalvector on the irregular planar mesh is interpolated onto that of theuniform planar mesh. For the mode shape shown on the irregularmesh (Fig. 5a), the interpolated modal vector on the uniformplanar meshes assigned to the top surfaces of the rotor and hatparts is demonstrated as in Figs. 5b and c.From the uniform planar mesh in the cylindrical coordinates,the numericaly-derivatives of the nth mode vector can becalculated at ðri;yj;zkÞ, for example,@jz;nðri;yj;zkÞ@y¼jz;nðri;yjþ1;zkÞ ?jz;nðri;yj;zkÞyjþ1?yj(39)where i ¼ 1;...;Mr, j ¼ 1;...;My, k ¼ 1;...;Mz, and Mr, My, Mzarethe numbers of the nodes of the hat-disc, respectively, in thecylindrical coordinates (r,y,z). Fig. 6 demonstrates the severaly-derivative modal vectors of the hat-disc at a given zk.In order to assign the finite contact force element to each finiteelement of the disc and pad contacts at the same location,the planar mesh taken in the disc contact surface is defined on thepad contact surface as well. Moreover, the modal vectors on thepad contact are interpolated onto those of the defined planarmesh. Connecting the finite contact force element between thedisc and pad is referred to Fig. 7 and [18]. As a result, theARTICLE IN PRESS( ,,)ijkrzθ,( ,,)z nijkrzϕθ(),( , ,)z nkx y zϕx,( ,,)z nlmkx yzϕ( ,,)lmkx yzFig. 4. Transverse modal vector at z ¼ zkinterpolated by the uniform planar-mesh method: (a) modal vector on the irregular mesh; (b) modal vector interpolated on theuniform planar mesh in the polar coordinates.J. Kang / International Journal of Mechanical Sciences 51 (2009) 284–294287numerical volume and area integrations are available in such away thatZAcfðr;yÞdA ffiXMrci¼1XMycj¼1fðri;yjÞri?Dri?Dyj(40)ZVdgdðr;y;zÞdV ffiXMri¼1XMyj¼1XMzk¼1gdðri;yj;zkÞri?Dri?Dyj?Dzk(41)ZVpgpðr;y;zÞdV ffiXMrpi¼1XMypj¼1XMzpk¼1gpðri;yj;zkÞri?Dri?Dyj?Dzk(42)where Mc, Mpdenote the number of the nodes, respectively, of thecontact area and pad in the cylindrical coordinates, and f, gd, gparethe quantities associated with modal vectors interpolated on theuniform planar mesh.As previously mentioned, one of the major differences betweenthe current model and the previous gyroscopic annular platemodels is the vibration modes obtained from the different discgeometry. In Kang et al. [14], the only vibration modes of the discare the transverse modes of the annular plate. In automotiveapplications, however, the transverse mode approximation cannotcapture the general modal behavior, for example, of the in-planemode, hat-mode, and so forth. Fig. 8 illustrates several vibrationmodes of the FE hat-disc model that the annular plate approx-imation is not likely to capture.ARTICLE IN PRESSFig. 5. The uniform planar-mesh method: (a) mode shape on the irregular mesh; (b) modal vector of A on the uniform planar mesh; (c) modal vector of B on the uniformplanar mesh. A: top rotor surface; B: top hat surface.Fig. 6. Derivatives of a transverse modal vector (nth mode, n ¼ 19) at zk¼ 4mm. (a)jz,n(r,y), (b) qjr,n(r,y)/qy, (c) qjy,n(r,y)/qy, and (d) qjz,n(r,y)/qy.J. Kang / International Journal of Mechanical Sciences 51 (2009) 284–2942883. Numerical resultsThe numerical simulation is conducted for the system para-meters given in Tables 1 and 2. The following numerical stabilityanalysis will be divided into two sections: constant frictioncoefficient and velocity-dependent friction coefficient. Fig. 9illustrates the friction–velocity curves used in the subsequentnumerical analysis. The velocity-dependent friction coefficientis linearized in the form of negative slope matrix [Ns], whereasthe negative slope effect disappears under the assumption of theconstant friction coefficient. The natural frequencies obtainedthrough modal analysis at kc¼ 0 can be found from those atARTICLE IN PRESS−−F1(ri,?j)F1(ri,?j)N1(ri,?j)ezN1(ri,?j)ezContact surface of the top pad Contact surface of the discFig. 7. Scheme of the contact mesh: (a) irregular mesh; (b) the contact forces assigned to ðri;yjÞ on the identical uniform mesh of the contact surface.(727Hz) (7528Hz) (4920Hz) (4820Hz) (1959Hz) (1896Hz) Fig. 8. Several vibration modes of the hat-disc.Table 1Nominal values of disc parameters.ParameterSymbolValueYoung’s modulusE88.9GPaDensity of discr7150kgm?3Poisson ration0.285Thickness of rotorh16.0mmPre-loadN02000NTable 2Nominal values of pad parameters.ParameterSymbolValueOuter radius of contactro142mmInner radius of contactri100mmContact angleyc621Young’s modulusE207GPaDensity of padrp7820kgm?3Poisson rationp0.29Thickness of padhp11.0mmNominal contact stiffnessknom0.35?1011Nm?3Fig. 9. Constant friction coefficient (broken line:m¼ 0.42) and velocity-dependentfriction coefficient (solid line:ms¼ 0.5,mk¼ 0.32,a¼ 1.0).J. Kang / International Journal of Mechanical Sciences 51 (2009) 284–294289K ¼ 0[%] in Fig. 10. Here, 3431 and 5720Hz correspond to the firstbending and torsion mode, respectively, of the pad, and 1365Hzcorresponds to the in-plane torsion mode of the disc.3.1. The constant friction coefficient assumptionUnder the assumption of the constant friction coefficient,the squeal mechanisms associated with the flutter modes areinvestigated by the eigenvalue sensitivity analysis. The contactstiffness and the friction coefficient are chosen for the parametersof the following sensitivity analysis. Fig. 10 demonstrates thefrequency loci of the friction-coupled system (kca0,m¼ 0.42)with respect to the contact stiffness variation atO¼ 5rads?1. Inthe stiffness–frequency domain, the unstable frequency loci areidentified by marking on the frequency loci corresponding to theeigenvalues having positive real parts. The mode shapes asso-ciated with the unstable frequency loci are used for examiningthe squeal character. The pad rigid modes and the third disctransverse doublet mode pair are found to participate in thesqueal modes as shown in Figs. 11a and 12a. It is relevant to trackthe frequency and real part loci with respect to friction coefficientfor demonstrating the mode-coupling between two closely spacedmodes in the following.Fig. 11 illustrates the binary flutter modes stemming from tworigid modes of the pad. From the mode-merging character shownin the eigenvalue loci without rotation effects, it is found that themode-coupling of the binary mode is engaged enough to causesthe modal instability by splitting the branches of Re(l). Therotation effects influence on equilibrium stability by alteringthe eigenvalue loci of the rotation-free disc approximation [14](thediscmodelwithoutrotationeffects).Particularly,themodification of Re(l) loci is attributed to the modal dampingseparation. In them–Re(l) domain, the radial dissipative effectrotates the loci of Re(l) clockwise around the pivot point (m¼ 0)increasing the criticalm, whereas the separation between the twomodal radial dissipative (viscous damping) terms destabilizes thesteady-sliding equilibrium by strengthening the splitting of Re(l),which is called ‘‘viscous damping instability [19]’’. This eigenvalueperturbationduetorotationeffectshasbeenanalyticallyinvestigated in [14].ARTICLE IN PRESSFig. 10. Stability map in the stiffness–frequency domain for constant frictioncoefficient; the mark ‘3’ denotes Re(ln)40 for the nth mode of the friction-coupledsystem (kca0,m¼ 0.42, N ¼ 92),xn¼ 0.002, K½%? ¼ 100 ? kc=knom.Fig. 11. Eigenvalue loci of modes 25,26 at K ¼ 100[%] in Fig. 10 with rotation effects (solid line) and w/o rotation effects (broken line): (a) mode shapes; (b) frequency loci;(c) real part loci.J. Kang / International Journal of Mechanical Sciences 51 (2009) 284–294290Fig. 12 demonstrates that the modes 13,14 of Fig. 10 areassociated with the 3rd disc transverse doublet pair and theirmode-coupling generates the modal destabilization. The rotationeffects on the mode-coupled eigenvalues are also seen in Figs.12band c. For the disc doublet mode pair, the gyroscopic destabilizingeffect is further involved due to disc rotation. The gyroscopiceffect has been shown to strengthen the splitting of Re(l) aswell [14].The increase of rotation speedOis shown to change the modalstability map of Fig. 10 by comparison with that of Fig. 13. Dueto the gyroscopic destabilization on the nonconservative brakesystem, the additional squeal modes appear at the higher speed.Fig. 14 illustrates the mode shapes of the squeal modes newlyappearing atO¼ 20rads?1, but being stable atO¼ 5rads?1. Byusing the comprehensive analytical model, Kang et al. [14] hasexplained that the gyroscopic destabilization stems from thegyroscopic frictional mode-coupling. Since the gyroscopic fric-tional mode-coupling is proportional toO, it can be said that thenewly appearing squeal modes in Fig. 13 have the sufficientamount of friction-coupling due to the increase ofO. Therefore,the rotation-dependent Re(l) of the rotating disc brake systemmay be the subject in predicting squeal occurrence with accuracy.3.2. The velocity-dependent friction coefficientThe constant friction coefficient has been one conventionalassumption in the brake squeal analysis. However, this assump-tion cannot capture the squeal character stemming from therotation-dependent friction curve. Since the friction material of anautomotive brake pad normally generates negative slope-typefriction curve [20] with respect to sliding speed, the negativeslope effect should be considered the potential mechanism of thefriction-induced vibration. Moreover, the magnitude variationof the friction coefficient is expected to directly influence on thesqueal propensity for disc rotation speed. Therefore, the velocity-dependent friction coefficient shown in Fig. 9 is used in thefollowing stability analysis.For demonstration of rotation-dependent squeal propensity atsteady-sliding equilibrium, Relð Þ is traced with respect to discrotation speed. In Fig. 15, the squeal propensity of the pad rigidmodes and the disc in-plane torsion mode is shown to increasewith the decrease of rotation speed, whereas the third and fourthdisc doublet modes become stable below certain low speeds. Thedivergence instability is also found to arise due to the negativeslope effect.ARTICLE IN PRESSFig.12. Eigenvalue loci of modes 13,14 at K ¼ 100[%] in Fig.10 with rotation effects (solid line) and w/o rotation effects (broken line), (a) mode shapes, (b) frequency loci, (c)real part loci.Fig. 13. Stability map in the stiffness–frequency domain for constant frictioncoefficient (m¼ 0.42) atO¼ 20rads?1.J. Kang / International Journal of Mechanical Sciences 51 (2009) 284–294291One interesting squeal mode (the in-plane torsion mode of thedisc as shown in Fig. 16) appears when negative slope-typefriction curve is introduced. The mode-coupling by the in-planemode cannot be incorporated into [B], but [F] because the in-planemode does not have the out-of-plane displacement ([jz] ¼ 0 inEq. A.6), but produce friction follower force due to rotation. It hasbeen known that the influence of [F] on the flutter instability isnegligible. Therefore, it is notable that the in-plane disc squealmode is not related to mode-coupling effect, but purely caused bythe negative slope effect.From the speed-dependent friction curve and the correspond-ing stability figure, it is concluded that the squeal propensity ofARTICLE IN PRESSFig. 14. Additional squeal modes at K ¼ 100[%] andm¼ 0.42 due to the increase ofOin Fig. 13, (a) modes 19,20, (b) modes 39,40.Fig. 15. Real parts of eigenvalues versus disc rotation speed [rads?1] at K ¼ 100[%].J. Kang / International Journal of Mechanical Sciences 51 (2009) 284–294292the disc brake system for the rotation speed is dependent of thevibration modes associated with squeal modes.4. Conclusions and discussionThe finite element model of a rotating hat-disc in contact withtwo stationary pads has been constructed. The uniform planar-mesh method enables us to calculate the numerical derivativesand integrations resulting in system matrices. The linearizedequations of motion determine the stability at the steady-slidingequilibrium of the gyroscopic non-conservative disc brake system.In the analysis, two types of squeal mechanism have beenfound: mode-coupling type and negative slope type. The mode-coupling effect can be demonstrated by mode-merging characterof the rotation-free approximation. Rotation effects on the mode-coupled binary mode arise from gyroscopic effect, radial compo-nent of friction force, the negative slope of friction coefficient, andthe variation of friction coefficient with respect to sliding speed.Particularly, the negative slope effect takes an important role ongenerating squeal of non-mode-coupled mode such as the in-plane torsion mode of the disc. Each rotation effect contributes tosqueal propensity depending on disc rotation speed.For the determination of squeal propensity for rotation speed,the speed-dependent friction curve should be introduced in thesqueal analysis. From the numerical calculation, it is found thatthe squeal propensity for rotation speed depends on the vibrationmode participating in squeal mode.AcknowledgmentsThe author is grateful to Professors Charles Krousgrill andFarshid Sadeghi at Purdue University for the advice on brakesqueal study.Appendix AThe components of the system matrices of Eq. (36) are given by½C? ¼ diagð2xnonÞ(A.1)½o2? ¼ diagðo2nÞ(A.2)½Rd? ¼pom?O?ZAc1r½up1r?T½up1r?jP?½up1r?T½ur?jP½0??½ur?T½up1r?jP½½ur?T½ur?jPþ ½ur?T½ur?jP???½ur?T½up2r?jP?½0??½up2r?T½ur?jP?½up2r?T½up2r?jP?2666437775dA(A.3)½Ns? ¼ p0m?_v?ZAc½up1y?T½up1y?jP?½up1y?T½uy?jP½0??½uy?T½up1y?jP½½uy?T½uy?jPþ ½uy?T½uy?jP???½uy?T½up2y?jP?½0??½up2y?T½uy?jP?½up2y?T½up2y?jP?266664377775dA(A.4)½A? ¼ kc?ZAc½up1z?T½up1z?jP?½up1z?T½uz?jP½0??½uz?T½up1z?jP½½uz?T½uz?jPþ ½uz?T½uz?jP???½uz?T½up2z?jP?½0??½up2z?T½uz?jP?½up2z?T½up2z?jP?2666437775dA(A.5)½B? ¼m?kc?ZAc½up1y?T½up1z?jP?½up1y?T½uz?jP½0??½uy?T½up1z?jP½½uy?T½uz?jP? ½uy?T½uz?jP??½uy?T½up2z?P?½0?½up2y?T½uz?jP??½up2y?T½up2z?jP?2666437775dA(A.6)½F? ¼ p0m?ZAc1r½0??½up1r?T@ur@y??????Pþ ½up1r?T½uy?jP??½0?½0?½up1r?T@ur@y??????P? ½up1r?T½uy?jPþ½ur?T@ur@y??????P?? ½ur?T½uy?jP?2666437775½0?½0??½up2r?T@ur@y??????P?þ ½up2r?T½uy?P?"#½0?26666666666666643777777777777775dA(A.7)where the subscripts P and P?denotes the top and bottom contactlocations at z ¼ h/2 and z ¼ ?h/2, respectively.xnandonare thedamping coefficients and the circular natural frequencies of thedisc and pad components, andm?¼mkþ ðms?mkÞe?a?O?rctr(A.8)m?_v¼ ?aðms?mkÞe?a?O?rctr(A.9)m?_vrepresents the slope of the friction coefficient with respect tosliding speed.References[1] Kinkaid NM, O’Reilly OM, Papadopoulos P. 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