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An imprecise probability approach based on evidence theory is proposed for squeal instability analysis of uncertain disc brakes in this paper. First, the squeal instability of the finite element (FE) model of a disc brake is investigated and its dominant unstable eigenvalue is detected by running two typical numerical simulations, i.e., complex eigenvalue analysis (CEA) and transient dynamical analysis. Next, the uncertainty mainly caused by contact and friction is taken into account and some key parameters of the brake are described as uncertain parameters. All these uncertain parameters are usually involved with imprecise data such as incomplete information and conflict information. Finally, a squeal instability analysis model considering imprecise uncertainty is established by integrating evidence theory, Taylor expansion, subinterval analysis and surrogate model. In the proposed analysis model, the uncertain parameters with imprecise data are treated as evidence variables, and the belief measure and plausibility measure are employed to evaluate system squeal instability. The effectiveness of the proposed approach is demonstrated by numerical examples and some interesting observations and conclusions are summarized from the analyses and discussions. The proposed approach is generally limited to the squeal problems without too many investigated parameters. It can be considered as a potential method for squeal instability analysis, which will act as the first step to reduce squeal noise of uncertain brakes with imprecise information. [ABSTRACT FROM AUTHOR]

Squeal reduction of disc brake systems have been extensively investigated for both academic and industrial purposes. However, most of the existing optimization designs of squeal reduction are based on deterministic approaches which have not considered the uncertainties of material properties, loading conditions, geometric dimensions, etc. In this paper, a hybrid probabilistic and interval model is introduced to deal with the uncertainties existing in a disc brake system for squeal reduction. The uncertain parameters of the brake system with enough information are treated as probabilistic variables, while the parameters with limited information are treated as interval variables. To improve computational efficiency, the response surface methodology (RSM) is introduced to replace the time-consuming finite element (FE) simulations. By the hybrid uncertain model, an optimization design based on reliability and confidence interval is proposed to explore the optimal design for squeal reduction. In the proposed optimization, both the design objective and the design constraint are interval probabilistic functions due to the effects of hybrid uncertainties. In this case, the maximum of the upper bound of confidence interval of design objective is selected as the objective function, while the minimal value of the probabilistic constraint is selected as the constraint function. The combinational algorithm of Genetic Algorithm and Monte-Carlo method is employed to perform the optimization. The results of a numerical example demonstrate the effectiveness of the proposed optimization on reducing squeal propensity of the disc brake systems with hybrid uncertainties. [ABSTRACT FROM AUTHOR]

An uncertain optimization method for brake squeal reduction of vehicle disc brake system with interval parameters is presented in this paper. In the proposed method, the parameters of frictional coefficient, material properties and the thicknesses of wearing components are treated as uncertain parameters, which are described as interval variables. Attention is focused on the stability analysis of a brake system in squeal, and the stability of brake system is investigated via the complex eigenvalue analysis (CEA) method. The dominant unstable mode is extracted by performing CEA based on a linear finite element (FE) model, and the negative damping ratio corresponding to the dominant unstable mode is selected as the indicator of instability. The response surface method (RSM) is applied to approximate the implicit relationship between the unstable mode and the system parameters. A reliability-based optimization model for improving the stability of the vehicle disc brake system with interval parameters is constructed based on RSM, interval analysis and reliability analysis. The Genetic Algorithm is used to get the optimal values of design parameters from the optimization model. The stability analysis and optimization of a disc brake system are carried out, and the results show that brake squeal propensity can be reduced by using stiffer back plates. The proposed approach can be used to improve the stability of the vehicle disc brake system with uncertain parameters effectively. [ABSTRACT FROM AUTHOR]

The hybrid Finite Element/Statistical Energy Analysis (FE-SEA) model for mid-frequency analysis consists of two parts: the FE components and the SEA components, which are called as the master system and the subsystem respectively. The hybrid Finite Element/Statistical energy analysis approach is established on the assumption that the FE components have fully deterministic properties, while the SEA subsystems have a high degree of randomness. This method has been recently extended to allow the properties of the FE components of built-up structures to be uncertain like interval or probabilistic, and the hybrid model with parametric and non-parametric uncertainties is obtained. By dealing with the non-parametric uncertainty analytically and the parametric uncertainty with Monte Carlo Simulations, the distribution of the responses of the hybrid model can be obtained. In this paper, the interval parametric uncertainty is introduced into the hybrid FE–SEA framework for structure−acoustic systems, and the interval dynamic equilibrium equations are established, thus a hybrid model with non-parametric and interval parametric uncertainties for structure−acoustic systems is obtained; to improve the computational efficiency, the second-order interval perturbation finite element method is introduced into the hybrid FE-SEA framework and the second-order interval perturbation finite element/statistical energy analysis (SIPFEM/SEA) method is proposed. For the structure−acoustic system with interval parameters modeled by FE-SEA, the parameters of FE components are considered as interval parameters instead of deterministic ones, thus the expectations of the second order quantities such as the vibrational energy and the response cross-spectrum in the FE-SEA framework become interval variables, too. In the SIPFEM/SEA, the expectations of the second order response quantities of a structure−acoustic system are expanded with the second-order Taylor series at the nominal values of interval parameters, and for the sake of simplicity and efficiency, the non-diagonal elements of the Hessian matrices are neglected; then by searching the target positions of interval parameters that maximize or minimize the objective functions, the bounds of the vibrational energy and the response cross-spectrum can be obtained. Because of the neglect of the higher order terms of Taylor series, SIPFEM/SEA is limited to the interval analysis with narrow parameter intervals. For larger parameter intervals, the sub-interval perturbation method based on the SIPFEM/SEA is introduced. The proposed methods are illustrated by applications to two example structure−acoustic systems in which the acoustic cavities are modeled by using the interval FE method and the structures are modeled by using the SEA. Reference comparisons are made with the Monte Carlo simulations of the hybrid FE/SEA model. The numerical results verify the accuracy and efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

Considering the epistemic uncertainties within the hybrid Finite Element/Statistical Energy Analysis (FE/SEA) model when it is used for the response analysis of built-up systems in the mid-frequency range, the hybrid Evidence Theory-based Finite Element/Statistical Energy Analysis (ETFE/SEA) model is established by introducing the evidence theory. Based on the hybrid ETFE/SEA model and the sub-interval perturbation technique, the hybrid Sub-interval Perturbation and Evidence Theory-based Finite Element/Statistical Energy Analysis (SIP-ETFE/SEA) approach is proposed. In the hybrid ETFE/SEA model, the uncertainty in the SEA subsystem is modeled by a non-parametric ensemble, while the uncertainty in the FE subsystem is described by the focal element and basic probability assignment (BPA), and dealt with evidence theory. Within the hybrid SIP-ETFE/SEA approach, the mid-frequency response of interest, such as the ensemble average of the energy response and the cross-spectrum response, is calculated analytically by using the conventional hybrid FE/SEA method. Inspired by the probability theory, the intervals of the mean value, variance and cumulative distribution are used to describe the distribution characteristics of mid-frequency responses of built-up systems with epistemic uncertainties. In order to alleviate the computational burdens for the extreme value analysis, the sub-interval perturbation technique based on the first-order Taylor series expansion is used in ETFE/SEA model to acquire the lower and upper bounds of the mid-frequency responses over each focal element. Three numerical examples are given to illustrate the feasibility and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]