[机械模具数控自动化专业毕业设计外文文献及翻译]【期刊】用一种复杂的特征值分析方法来分析盘式制动器的尖啸情况-外文文献

Analysis of disc brake squeal usingAbstract*Corresponding author. Tel.: +65 64191218; fax: +65 64191280.E-mail address: .sg (P. Liu).Applied Acoustics 68 (2007) 603615/locate/apacoust0003-682X/$ - see front matter C211 2006 Elsevier Ltd. All rights reserved.A new functionality of ABAQUS/Standard, which allows for a nonlinear analysis prior to a com-plex eigenvalue extraction in order to study the stability of brake systems, is used to analyse discbrake squeal. An attempt is made to investigate the eects of system parameters, such as the hydrau-lic pressure, the rotational velocity of the disc, the friction coecient of the contact interactionsbetween the pads and the disc, the stiness of the disc, and the stiness of the back plates of the pads,on the disc squeal. The simulation results show that significant pad bending vibration may beresponsible for the disc brake squeal. The squeal can be reduced by decreasing the friction coecient,increasing the stiness of the disc, using damping material on the back plates of the pads, andmodifying the shape of the brake pads.C211 2006 Elsevier Ltd. All rights reserved.Keywords: Disc brake squeal; Complex eigenvalue extraction; Friction coecient; Stiness; Damping ratiothe complex eigenvalue methodP. Liua,*, H. Zhenga, C. Caia, Y.Y. Wanga,C.Lua,K.H. Angb, G.R. LiucaInstitute of High Performance Computing, 1 Science Park Road, #01-01 The Capricorn SingaporeScience Park II, Singapore 117528, SingaporebSunstar Logistic Singapore Pte Ltd., 10 Science Park Road, #04-16/17 The Alpha Singapore Science ParkII, Singapore 117684, SingaporecNational University of Singapore, 9 Engineering Drive 1, Singapore 117576, SingaporeReceived 25 April 2005; received in revised form 30 March 2006; accepted 30 March 2006Available online 5 June 2006doi:10.1016/j.apacoust.2006.03.012604 P. Liu et al. / Applied Acoustics 68 (2007) 6036151. IntroductionBrake squeal, which usually occurs in the frequency range between 1 and 16 kHz, hasbeen one of the most dicult concerns associated with vehicle brake systems. It causes cus-tomer dissatisfaction and increases warranty costs. Although substantial research has beenconducted into predicting and eliminating brake squeal, it is still dicult to predict itsoccurrence due to the complexity of the mechanisms that cause brake squeal 1.Several theories have been formulated to explain the mechanisms of brake squeal, andnumerousstudieshavetriedwithvariedsuccesstoapplythemtothedynamicsofdiscbrakes2.Therearemanymodelsforanalysingdiscbrakesqueal.Forexample,theeectofsurfacetopographyofthepad/discassemblyonsquealgenerationwasreported3andadistributed-parametermodel ofadisc brakehasbeendeveloped tosimulatefriction-induced vibrationsin the form of high-frequency squeal 4. A two-degree-of-freedom model has been used toinvestigate the basic mechanisms of instability of the disc brake system and demonstratestheconditionsnecessaryforpreventingtheinstability5.Brakesquealhasalsobeenstudiedfrom an energy perspective using feed-in energy analysis and results indicate a squeal ten-dencyofthebrakesystem6.Theuseofviscoelasticmaterial(dampingmaterial)onthebackofthebackplatesofthepadscanbeeectiveinreducingsquealwhenthereissignificantpadbending vibration 7 and another reported eective method is to modify the shape of thebrake pads to change the coupling between the pads and the disc 8.Brakenoiseismainlycausedbyfriction-induceddynamicinstability.Therearetwomaincategories of numerical methods that are used to study this problem: (1) transient dynamicanalysis and (2) complex eigenvalue analysis. Currently the complex eigenvalue method ispreferred and widely used in predicting the squeal propensity of the brake system includingdamping and contact 912,since thetransient dynamicanalysisiscomputationally expen-sive. The main idea of the complex eigenvalue method involves symmetry arguments of thestinessmatrixandtheformulationofafrictioncoupling.Thismethodismoreecientandprovides more insight to the friction-induced dynamic instability of the disc brake system.In the present study, an investigation of disc brake squeal is performed by using the newcomplex eigenvalue capability of the finite element (FE) software ABAQUS version 6.413. This FE method uses nonlinear static analysis to calculate the friction coupling priorto the complex eigenvalue extraction, as opposed to the direct matrix input approach thatrequires the user to tailor the friction coupling to stiness matrix, Thus, the eect of non-uniform contact and other nonlinear eects are incorporated. A systematic analysis is doneto investigate the eects of system parameters, such as the hydraulic pressure, the rota-tional velocity of the disc, the friction coecient of the contact interactions between thepads and the disc, the stiness of the disc, and the stiness of the back plates of the pads,on the disc squeal. The simulations performed in this work present a guideline to reducethe squeal noise of the disc brake system.2. Methodology and numerical model2.1. Complex eigenvalue extractionFor brake squeal analysis, the most important source of nonlinearity is the frictionalsliding contact between the disc and the pads. ABAQUS allows for a convenient, but gen-eral definition of contact interfaces by specifying the contact surface and the properties ofP. Liu et al. / Applied Acoustics 68 (2007) 603615 605the interfaces. ABAQUS version 6.4 has developed a new approach of complex eigenvalueanalysis to simulate the disc brake squeal. Starting from preloading the brake, rotating thedisc, and then extracting natural frequencies and complex eigenvalues, this new approachcombines all steps in one seamless run 13. The complex eigenproblem is solved using thesubspace projection method, thus a natural frequency extraction must be performed firstin order to determine the projection subspace. The governing equation of the system isMxC _xKx 0; 1whereMis the mass matrix,Cis the damping matrix, which includes friction-induced con-tributions, and K is the stiness matrix, which is unsymmetric due to friction. The govern-ing equation can be rewritten asl2M lC KU 0; 2where l is the eigenvalue and U is the corresponding eigenvector. Both eigenvalues andeigenvectors may be complex. In order to solve the complex eigenproblem, this systemis symmetrized by ignoring the damping matrix C and the unsymmetric contributions tothe stiness matrix K. Then this symmetric eigenvalue problem is solved to find the pro-jection subspace. The N eigenvectors obtained from the symmetric eigenvalue problemare expressed in a matrix as /1,.,/N. Next, the original matrices are projected ontothe subspace of N eigenvectorsMC3/1; .;/NC138TM/1; .;/NC138; 3aCC3/1; .;/NC138TC/1; .;/NC1383bandKC3/1; .;/NC138TK/1; .;/NC138: 3cThen the projected complex eigenproblem becomesl2MC3lCC3KC3UC3 0: 4Finally, the complex eigenvectors of the original system can be obtained byU /1; .;/NC138TUC3: 5A more detailed description of the algorithm may be found in 13. The complex eigen-value l, can be expressed as l = a ix where a is the real part of l, Re(l), indicatingthe stability of the system, and x is the imaginary part of l, Im(l), indicating the modefrequency. The generalized displacement of the disc system, x, can then be expressed asx Aelt eatA1cosxt A2sinxt: 6Thisanalysis determines thestability ofthesystem. When thesystem isunstable,abecomespositiveandsquealnoiseoccurs.Anextraterm,dampingratio,isdefinedasC0a/(p|x|).Ifthedampingratioisnegative,thesystembecomesunstable,andviceversa.Themainaimofthisanalysis is to reduce the damping ratio of the dominant unstable modes.2.2. Finite element modelA disc brake system consists of a disc that rotates about the axis of a wheel, a calliperpiston assembly where the piston slides inside the calliper, that is mounted to the vehiclesuspension system, and a pair of brake pads. When hydraulic pressure is applied, the pis-ton is pushed forward to press the inner pad against the disc and simultaneously the outerpad is pressed by the calliper against the disc. The brake model used in this study is a sim-plified version of a disc brake system which consists of a disc and a pair of brake pads. Thedisc has a diameter of 292 mm and a thickness with typical value of 5.08 mm and is madeof cast iron. The pair of brake pads, which consist of contact plates and back plates, arepressed against the disc in order to generate a friction torque to slow the disc rotation.Fig. 1. Geometry and finite element mesh of the simplified disc brake system.606 P. Liu et al. / Applied Acoustics 68 (2007) 603615Fig. 2. Constraints and loadings of the disc brake system.The contact plates are made of an organic friction material and the back plates are madeof steel. The FE mesh is generated using three-dimensional continuum elements for thedisc and pads as shown in Fig. 1, where a fine mesh is used in the contact regions. Thefriction contact interactions are defined between both sides of the disc and the contactplates of the pads. A constant friction coecient and a constant angular velocity of thedisc are used for simulation purposes. Figs. 2(a)(c) present the constraints and loadingsfor the pads and disc assembly. The disc is completely fixed at the five counter-bolt holes asshown in Fig. 2(a) and the ears of the pads are constrained to allow only axial directionalmovements as shown in Figs. 2(b) and (c). The calliperpiston assembly is not defined inthe simplified model of the disc brake system, hence the hydraulic pressure, which has atypical value of 0.5 MPa, is directly applied to the back plates at the contact regionsbetween the inner pad and the piston and between the outer pad and the calliper as shownin Figs. 2(b) and (c), and it is assumed that an equal magnitude of force acts on each pad.The analysis procedure contains the following four steps: (1) nonlinear static analysis forthe application of brake pressure; (2) nonlinear static analysis to impose a rotational veloc-ity on the disc; (3) normal mode analysis to extract the natural frequency to find the pro-jection subspace; and (4) complex eigenvalue analysis to incorporate the eect of frictioncoupling.P. Liu et al. / Applied Acoustics 68 (2007) 603615 607Fig. 3. (a) Variation of the damping ratio with frequency for dierent friction coecients; (b) variation of thedamping ratio with friction coecient at frequency 12 kHz.3. Results and discussionThe eects of the system parameters, such as the hydraulic pressure P, the rotationalvelocity of the disc W, the friction coecient of the contact interactions between the padsand the disc u, the stiness of the disc, and the stiness of the back plates of the pads, onthe disc squeal are investigated by the simulation model. The eect of the stiness of thedisc can be changed by varying Youngs modulus EDand the disc thickness TDof the discwhile the eect of the stiness of the back plates of the pads can be changed by varyingYoungs modulus EPof the back plates of the pads. The complex eigenvalue analysis isperformed up to 13 kHz which is the range of squeal occurrence for the present disc model.As mentioned previously, if the damping ratio is negative, the system becomes unstable,and vice versa; when the disc system is unstable, the squeal propensity increases with anincreased value of the damping ratio (absolute values are used). For clarity, only negativevalues of the damping ratio are plotted. The typical values for the system parameters usedin the simulation are: P = 0.5 MPa, W = 1.5 rad/s, u = 0.653, ED= 219.669 GPa,TD= 5.08 mm, and EP= 210 GPa. Analysis is carried out by changing the values of eachparameter while retaining the respective typical values for the others.608 P. Liu et al. / Applied Acoustics 68 (2007) 603615Fig. 4. (a) Variation of the damping ratio with frequency for dierent hydraulic pressures; (b) variation of thedamping ratio with hydraulic pressure at frequency 12 kHz.3.1. Eect of friction coecientDisc squeal is believed to be caused mainly by friction-induced dynamic instability. Thissection presents the eect of the friction coecient of the contact interactions between thepads and the disc on the disc squeal, in which the friction coecient u varies from 0.2 to0.8. Fig. 3(a) shows results in the form of the damping ratio as a function of frequency fordierent friction coecients. It can be seen that the major squeal frequency is approxi-mately 12 kHz. The value of the damping ratio is decreased significantly with a decreaseof the friction coecient as shown in Fig. 3(b) at a frequency of 12 kHz. It is understand-able that with an increase in the friction coecient, there is an accompanying increase inthe instability of the system, thus an increase in the damping ratios. This means that themost fundamental method of eliminating brake squeal is to reduce the friction between thepads and the disc. However, this obviously reduces braking performance and is not a pref-erable method to employ.3.2. Eect of hydraulic pressureThe eect of the hydraulic pressure P on the squeal propensity is studied by varying Pfrom 0.5 MPa to 2.0 MPa. Fig. 4(a) shows the change of the damping ratio with frequencyP. Liu et al. / Applied Acoustics 68 (2007) 603615 609Fig. 5. (a) Variation of the damping ratio with frequency for dierent rotational velocities of the disc; (b)variation of the damping ratio with rotational velocity of the disc at frequency 12 kHz.for dierent hydraulic pressures. The major squeal frequency is approximately 12 kHz. Itcan be seen from Fig. 4(b) that with an increase in P, the value of the damping ratio isincreased, so the squeal propensity is increased. This is due to a larger hydraulic pressureinducing more friction between the pads and the disc. However, the simulation results alsoshow that the eect of the hydraulic pressure on the disc brake squeal is not significantbecause the value of the damping ratio only changes from 0.17 to 0.193 when P increasesfrom 0.5 MPa to 2.0 MPa.3.3. Eect of rotational velocity of the discFig. 5(a) presents the variation of the damping ratio with the frequency for dierent discangular velocities W (0.78.0 rad/s). The dominant squeal frequency is approximately12 kHz. As the angular velocity increases, the value of the damping ratio graduallydecreases. However, as with the previous case, when changing the hydraulic pressure,the eect of changing the angular velocity on the squeal propensity is also not obvious:this can be seen from Fig. 5(b) which shows the value of the damping ratio varies withan increase in the rotational velocity of the disc.610 P. Liu et al. / Applied Acoustics 68 (2007) 603615Fig. 6. (a) Variation of the damping ratio with frequency for dierent Youngs moduli of the disc; (b) variation ofthe damping ratio with Youngs moduli of the disc at frequency 12 kHz.3.4. Eect of stiness of the discThe eect of the stiness of the disc on the disc brake squeal is studied by changingYoungs modulus EDand the thickness TDof the disc. Fig. 6(a) shows results of the damp-ing ratio versus frequency for dierent Youngs modulus ED, i.e. ED= 0.8ED0, 0.9ED0,1.0ED0, 1.1ED0and 1.2ED0, where ED0is the typical value of Youngs modulus of the disc,which is 219.669 GPa. It can be seen that the major squeal frequency does not change fordierent disc Youngs moduli. The value of the major squeal frequency is approximately12 kHz. As Youngs modulus EDis increased and hence as the stiness of the disc isincreased, the value of the damping ratio decreases greatly. Fig. 6(b) presents the dampingratio versus Youngs modulus of the disc at a frequency of 12 kHz. It is found that a largerdisc stiness can reduce the squeal propensity of the disc system. It is believed that a sti-ening of the disc can reduce the disc vibration magnitude, as a result, the squeal propensityof the disc system can be reduced. The stiness of the disc is also changed by varying itsthickness TD. Four cases were studied, i.e. TD= 0.9TD0, 1.0TD0, 1.1TD0and 1.2TD0,where TD0= 5.08 mm is the typical value for disc thickness. Fig. 7(a) shows results ofthe damping ratio plotted against frequency for dierent disc thicknesses and Fig. 7(b) pre-sents the damping ratio versus disc thickness at a frequency of 12 kHz. The thicker theP. Liu et al. / Applied Acoustics 68 (2007) 603615 611Fig. 7. (a) Variation of the damping ratio with frequency for dierent disc thicknesses; (b) variation of thedamping ratio with thickness of the disc at frequency 12 kHz.612 P. Liu et al. / Applied Acoustics 68 (2007) 603615disc, the higher its stiness, the smaller the damping ratio, and thus the lower the squealpropensity.3.5. Eect of stiness of the back plates of the padBrake pads consist of contact plates which are made of a friction material and backplates. In this study, the eect of Youngs modulus EPof the back plates of the pads onthe disc squeal is investigated, in which EP= 0.8EP0, 0.9EP0, 1.0EP0, 1.1EP0and 1.2EP0,where EP0= 210 GPa, is the typical value of Youngs modulus for the back plates of pads.Fig. 8(a) shows results of the damping ratio versus frequency for dierent Youngs moduliEP. It can be seen that the dominant squeal occurs at a frequency of approximately12 kHz. As Youngs modulus EP, is increased, corresponding to an increase in stinessof the back plates of the pads, the value of the damping ratio increases significantly asshown in Fig. 8(b); here the variation of the damping ratio with Youngs modulus ofthe back plates at a frequency of 12 kHz is shown. This important observation implies thatthe stier back plates of pads cause a higher squeal propensity. This is so since the frictionmaterial connected to the back plates is very soft compared with the back plate material.Hence the higher the stiness of