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Automotive powertrain mounting systems (PMSs) may exhibit hybrid interval-random (HIR) uncertainties in engineering practice. In HIR case, the uncertain parameters of PMSs are characterized as the random variables with interval distributions. A comprehensive methodology is proposed for the design optimization of the PMSs involving HIR uncertainties in this study. The hybrid interval-random perturbation central difference method (HIRP-CDM) is firstly derived to calculate the uncertain responses of PMSs, in which HIR uncertainties are addressed by twice perturbation analyses and twice central difference analyses. Meanwhile, the hybrid interval-random Monte Carlo method is presented as a reference method. Then, reliability assessment models are constructed to evaluate the probability intervals of system inherent characteristics satisfying design requirements. Next, a design optimization model is formulated to explore the optimal design of PMSs. The expectation intervals and variance intervals of system inherent characteristics are taken to create optimization objective, and the minimal probabilities of system inherent characteristics meeting requirements are utilized to establish reliability constraints. Both robustness and reliability are considered in the optimization, and the optimization can be simplified with the aid of HIRP-CDM. The effectiveness of the proposed methodology is finally demonstrated by the numerical example. [ABSTRACT FROM AUTHOR]

In the traditional uncertainty-based analyses and optimizations of the automotive powertrain mounting system (PMS), uncertain parameters are usually treated as either random variables or interval variables. In this article, an efficient analysis and optimization method is proposed for PMS designs with hybrid random and interval uncertainties. An efficient analysis method, named the hybrid perturbation–finite difference method (HPFDM), is first derived to calculate the uncertain responses of the natural frequencies (NFs) and the decoupling ratios (DRs) of the PMS. Then, an optimization model of the PMS with hybrid uncertainties is established based on the HPFDM, where the uncertain responses of the relevant NFs and DRs are taken to create optimization constraints and optimization objectives. With the aid of the HPFDM, the nested optimization model can be solved efficiently. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Uncertainty widely exists in the powertrain mounting system of a vehicle. The traditional analysis and optimization methods for powertrain mounting system design are often based on deterministic or random models. In this study, an efficient analysis and optimization method is developed for powertrain mounting system design involving interval uncertainty. In the proposed method, the uncertain parameters of powertrain mounting system are treated as interval variables, and an efficient method called as Chebyshev-Vertex method is developed to fast calculate the lower and upper bounds of the natural frequencies and decoupling ratios of powertrain mounting system. Monte-Carlo method is taken as a reference method to verify the calculation. Then, an optimization model is established based on Chebyshev-Vertex method, in which the interval responses of decoupling ratios are used to build up optimization objective while the interval responses of natural frequencies and decoupling ratios are taken to create optimization constraints. The optimization model of the powertrain mounting system with interval parameters is generally a double-loop nested problem, and it is rather time-consuming on calculation. However, based on Chebyshev-Vertex method, the optimization model can be approximately simplified into a single-loop one and the calculation efficiency is greatly improved. A numerical example is provided to demonstrate the effectiveness of the proposed method on the analysis and optimization design of the powertrain mounting system involving interval uncertainty. [ABSTRACT FROM AUTHOR]

Abstract Most of the existing methods of brake squeal instability analysis are merely available to handle single type of uncertain case. In this study, an improved unified method is developed for uncertainty quantification, which is capable of handling two types of fuzzy–interval cases. In the first fuzzy–interval case, uncertain parameters of engineering structures are assumed as either fuzzy variables or interval variables, which exist in structures simultaneously and independently. In the second fuzzy–interval case, all uncertain parameters are represented by interval variables, but their lower and upper bounds just can be expressed as fuzzy variables instead of deterministic values. In the proposed method, fuzzy–boundary interval variables are introduced to handle fuzzy–interval uncertainties, and based on which an improved response analysis model is established. In the improved model, the fuzzy–boundary interval variables are firstly converted into interval-boundary variables by level-cut technique. Then by temporarily neglecting boundary uncertainties, the initial interval responses can be approximated via conducting once Taylor series expansion and subinterval analysis. Next, Taylor series expansion and central difference method are combined twice to deal with boundary uncertainties, and the interval responses of the structures with interval-boundary variables are yielded. Finally, the fuzzy–interval responses of the structures are derived on the basis of interval union operation and fuzzy decomposition theorem. The improved method is subsequently extended to quantify the uncertainties in brake squeal instability analysis involving two types of fuzzy–interval uncertainties. The effectiveness of the proposed method on tackling fuzzy–interval problems is demonstrated by numerical examples. [ABSTRACT FROM AUTHOR]

A new hybrid uncertainty analysis method with random and interval variables is proposed in this paper. In the proposed method, the uncertain parameters with sufficient information are treated as random variables, while the uncertain parameters with limited information are modeled as interval variables. Both random variables and interval variables can be viewed as special evidence variables. From this special point of view, both random variables and interval variables are represented by equivalent evidence variables in this study, and a unified framework for hybrid uncertainty analysis is developed based on evidence theory and subinterval perturbation technique. The effects of the mixture of random and interval uncertainties on uncertain output are assessed by belief measure and plausibility measure. On the base of the proposed method, the squeal analysis model of automotive disc brakes involving both random and interval uncertainties can be developed. A numerical example of brake squeal analysis is provided to illustrate the effectiveness of the proposed method. The analysis results show that the equivalent evidence variables can be explored to represent random variables and interval variables reasonably and the proposed method has good accuracy and efficiency in the hybrid uncertainty analysis of squeal instability The proposed method gives a unified framework to tackle several types of uncertain cases, and it is quite general and not only limited to the uncertainty analysis of brake squeal. [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]

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]

• The control problem of a soft robotic snake is for mulated as constraint following. • An adaptive robust control is designed for the soft robotic snake. • A smooth zone approach is proposed to construct the adaptation law. • Advantages of smooth zone approach over dead zone approach are demonstrated. This paper targets the motion control problem of a soft robotic snake with uncertainty. The problem is formulated as constraint-following. From practical point of view, the uncertainty is assumed to be (possibly fast) time-varying and bounded. The bound is unknown. To render constraint-following for the soft robotic snake, a new adaptive robust control is designed based on a novel design of the adaptation law. A smooth-zone approach is proposed to construct the adaptation law. Compared with the past discontinuous (hence non-smooth) dead-zone approach, the proposed approach ensures the adaptation law to be, besides saving control effort, continuous so that the adaptive parameters can vary smoothly, which in turn assures a smooth robot operation. We demonstrate that, even in the presence of uncertainty, the proposed control is capable of rendering approximate constraint-following by guaranteeing uniform boundedness and uniform ultimate boundedness. Simulation results on the soft robotic snake demonstrate the effectiveness and advantages of the proposed control. [ABSTRACT FROM AUTHOR]