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Purpose: The purpose of this paper is to propose a unified optimization design method and apply it to handle the brake squeal instability involving various uncertainties in a unified framework. Design/methodology/approach: Fuzzy random variables are taken as equivalent variables of conventional uncertain variables, and a unified response analysis method is first derived based on level-cut technique, Taylor expansion and central difference scheme. Next, a unified reliability analysis method is developed by integrating the unified response analysis and fuzzy possibility theory. Finally, based on the unified reliability analysis method, a unified reliability-based optimization model is established, which is capable of optimizing uncertain responses in a unified way for different uncertainty cases. Findings: The proposed method is extended to perform squeal instability analysis and optimization involving various uncertainties. Numerical examples under eight uncertainty cases are provided and the results demonstrate the effectiveness of the proposed method. Originality/value: Most of the existing methods of uncertainty analysis and optimization are merely effective in tackling one uncertainty case. The proposed method is able to handle the uncertain problems involving various types of uncertainties in a unified way. [ABSTRACT FROM AUTHOR]

In some special engineering circumstances, it is likely that all parameters of an uncertain automotive structure can only be treated as interval variables due to limited knowledge, but meanwhile their lower and upper bounds can just be modeled as fuzzy variables rather than as deterministic values due to ambiguous information. To handle this dual uncertainties case, a reliability-based optimization method with fuzzy-boundary interval variables is developed in this study, and it is further extended to carry out squeal instability analysis and reduction of brake involving both limited and vague information. In the proposed method, fuzzy-boundary interval variables are utilized to cope with the above dual uncertainties of structure parameters and help to build up the structure response analysis model. First, the structure responses are derived on the basis of α -cut strategy, Taylor series expansion, subinterval analysis, and central difference method. Then, with the aid of fuzzy possibility theory, a reliability analysis model of structure response is developed, which can make use of extra reliability information and thus quantify the reliability more accurately. Next, a reliability-based optimization model involving fuzzy-boundary interval variables is established by integrating the uncertain response analysis model and the reliability analysis model. Finally, the proposed method is extended to carry out automotive brake squeal instability analysis and optimization. The numerical investigations demonstrate the applicability and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Automotive brake systems are always subjected to various types of uncertainties and two types of random-fuzzy uncertainties may exist in the brakes. In this paper, a unified approach is proposed for squeal instability analysis of disc brakes with two types of random-fuzzy uncertainties. In the proposed approach, two uncertainty analysis models with mixed variables are introduced to model the random-fuzzy uncertainties. The first one is the random and fuzzy model, in which random variables and fuzzy variables exist simultaneously and independently. The second one is the fuzzy random model, in which uncertain parameters are all treated as random variables while their distribution parameters are expressed as fuzzy numbers. Firstly, the fuzziness is discretized by using α -cut technique and the two uncertainty analysis models are simplified into random-interval models. Afterwards, by temporarily neglecting interval uncertainties, the random-interval models are degraded into random models, in which the expectations, variances, reliability indexes and reliability probabilities of system stability functions are calculated. And then, by reconsidering the interval uncertainties, the bounds of the expectations, variances, reliability indexes and reliability probabilities are computed based on Taylor series expansion. Finally, by recomposing the analysis results at each α -cut level, the fuzzy reliability indexes and probabilities can be obtained, by which the brake squeal instability can be evaluated. The proposed approach gives a general framework to deal with both types of random-fuzzy uncertainties that may exist in the brakes and its effectiveness is demonstrated by numerical examples. It will be a valuable supplement to the systematic study of brake squeal considering uncertainty. [ABSTRACT FROM AUTHOR]

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]

A hybrid uncertain model is introduced to deal with the uncertainties existing in a disc brake system in this paper. By the hybrid uncertain model, the uncertain parameters of the brake with enough sampling data are treated as probabilistic variables, while the uncertain parameters with limited data are treated as interval probabilistic variables whose distribution parameters are expressed as interval variables. Based on the hybrid uncertain model, the reliability-based design optimization (RBDO) of a disc brake with hybrid uncertainties is proposed to explore the optimal design for squeal reduction. In the optimization, the surrogate model of the real part of domain unstable eigenvalue of the brake system is established, and the upper bound of its expectation is adopted as the optimization objective. The lower bounds of the functions related to system stability, the mass, and the stiffness of design component are adopted as the optimization constraints. 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 improving system stability and reducing squeal propensity of a disc brake under hybrid uncertainties. [ABSTRACT FROM AUTHOR]

To suppress the brake noises of an automotive disc brake system with uncertain parameters, a stability analysis method of automotive brake systems is presented in this paper. The essential idea of this method is to combine the RSM (Response Surface Method) and complex eigenvalue analysis techniques with random parameters introduced to describe such uncertainties in the brake system as friction coefficients, brake pressure, disc thickness, pad thickness and backplate thickness. In order to improve the analysis efficiency, RSM is used to build up the approximation models to express the functional relationships between the unstable eigenvalues and the system parameters, which makes it available to study the stability of the brake system according to the random characteristics of the unstable eigenvalues. The stability analysis of a float caliper disc brake system is carried out by the proposed method with the random parameters assumed to follow normal distributions. The statistical characteristics and the sensitivities of unstable eigenvalue are calculated by Monte Carlo method. The results show that increasing the thickness of the backplate can improve the stability and reduce the brake noises of the brake system effectively. [ABSTRACT FROM AUTHOR]