On The Brake Squeal Complex Eigenvalue Analysis Model Concerning Disc Rotation

Brake squeal remains one of the most elusive dynamic problems in automobile industry since it is affected not only by component structures and material properties,but also by environments and braking conditions.Disc rotates during braking,however,it is difficult to build a rotating disc finite element(FE)model,thus the rotation effect has always been neglected when squeal issues are studied through FE and modal analysis techniques.Based on modal parameters extracted from the non-rotating disc FE model,this paper concerns the variations of dynamic characters of the disc under moving load condition and innovatively constructed a brake squeal complex eigenvalue analysis model which contains disc rotation effect.Influence of rotation effect on system stability is also investigated.Modelling process is divided into substructure level and system level.At substructure level,modal shapes of the stationery disc,which are extracted from FE modal analysis are expressed in mathematic expressions through curve fitting.The transfer function of the rotating disc is derived through coordinates and time-frequency transformations.Then equivalent modal parameters are extracted,which are speeddependent and can be used to describe dynamic character of the disc under moving loads.The orthogonality of equivalent modal parameters in state space are validated.For other stationery substructures,dynamic equations are expanded to state space and treatment of rigid modes is discussed.At system level,non-symmetric coupling stiffness and negativeslope friction matrices are derived,dynamic equation of the whole system expressed in the form of substructures modal parameters and coupling parameters are formulated through modal synthesis technique.Thus the system stability problem is converted into a general complex eigenvalue problem under modal coordinates.Squeal modes under different braking conditions are calculated through complex eigenvalue analysis.Influences of rotation,negative-slope friction character and friction coupling on system stability are analyzed.Results show that disc doublet modes are split and travelling waves are generated after rotation being introduced to the model.Under constant friction assumption,rotation has negative influence on system stability since it generates new unstable modes related to disc/pad in-plane vibration and the squeal tendency increases with growing speed.Under negative-slope friction assumption,negative damping effect that intensifies the vibration is generated and also leads to new unstable modes.However,brake system becomes more stable with growing speed due to the descending friction coefficient in high speed range,which means that the vibration divergence resulted from negative damping is neutralized by the convergence effect resulted from the descending friction coupling at high speed level.Thus it is believed that the major mechanism of squeal is the mode-merging and self-induced vibration due to friction coupling.In comparison,doublet modes splitting and negative damping effects originated from rotation and negative-slope friction respectively only have limited influences on system stability.Finally,the modelling process and conclusions are validated through squeal test.This paper completed the process of modelling a disc brake complex eigenvalue analysis model for squeal analysis,the predicted squeal modes are in agreement with the test result,at the same time comprehensively revealed the influences of disc rotation,negative-slope friction character and friction coupling on system stability.It has both theoretical and practical value in predicting and analyzing brake squeal in product design phase.