Parameter Determination And Stability Analysis Of A Minimal Model For Brake Squeal

In the research into the mechanism of brake squeal,the finite element model and the minimal model with two degrees of freedom are two widely used numerical modeling methods.The finite element model can simulate the complex shape of disk brake parts because of its less simplification and a large number of degrees of freedom(DoFs).The complex eigenvalue analysis(CEA)of the finite element model can obtain the multiple unstable modes of the actual brake,but the CEA cannot calculate the vibration response,and the finite element transient analysis(TA)is extremely time-consuming.Compared with the finite element method,the minimal model with fewer DoFs is more concise and its CEA and TA are both efficient,making it easier to analyze the effect of parameters.A variety of minimal models have been proposed in the literature,however,how to accurately determine its kinetic parameters is rarely reported.Therefore,this paper aims to explore a more accurate method for parameter determination of the 2-DoF minimal model,so as to fill in a gap in the literature.Firstly,a model test is carried out for the brake disc,and an accurate finite element disc brake model is established based on the test results.And the CEA is carried out to obtain unstable modes of the brake.By changing the friction coefficient between the contact surfaces of the disk and the pad,the evolution of the real part,the natural frequency and the Hopf bifurcation point versus the friction coefficient of each unstable mode is observed.The results show that the unstable mode with seven nodal diameters is representative and can be used as a reference target for further establishing the minimal model.Then,a 2-DoFs minimal model is established to make up the shortcomings of the existing brake squeal minimal models.The model also considers the torsional mode of the disc and the normal mode of pad.The dynamic equation of the model is derived,and the theory of CEA for determining the stability of the model is expounded.The results show that when the real part of the eigenvalue is greater than zero,the model is unstable and has a higher squeal propensity.Otherwise,it is stable and the squeal propensity is small.Next,in order that the natural frequency,Hopf bifurcation point and real parts of eigenvalues of the minimal model coincide with that of the unstable mode with seven nodal diameters,the response surface method(RSM)is applied to determine the kinetic parameters of the minimal model.Finally,the parameter-optimized minimal model is achieved.The results show that the response surface fitting has high accuracy and acceptable error.The coupling characteristics of the parameter-optimized minimal model is consistent with that of the target mode,which can provide more reliable numerical results for subsequent squeal stability analysis.Finally,the stability analysis is performed for the minimal model.On the one hand,based on the mode coupling mechanism,the CEA is used to investigate the influence of mass,stiffness and damping on the coupling characteristics.The results show that whether the stiffness and mass ratio of the two modes is close and whether the two modes is equally-damped have a significant effect on modal coupling.Furthermore,the negative slope of friction-velocity characteristic is introduced into the model,and transient analysis(TA)is used to calculate the dynamic response,and study the effect of the negative slope,braking velocity,and brake pressure on the friction coefficient and stability of the brake system.The results show that a relative smaller slope,lower speed,and greater braking pressure corresponds to greater amplitude of the vibration response.That is,the stability of the model becomes worse and the squeal propensity of brake becomes higher.