Structural optimization of an asymmetric automotive brake disc with cooling channels to avoid squeal

Brake squeal is still a major issue in the automotive industry due to comfort complaints of passengers and resulting high warranty costs. Many measures to avoid squeal have been discussed in the engineering community reaching from purely passive measures like the increase of damping, e.g. by the application of shims, to the active or semiactive suppression of squeal. While active measures can be effective but are elaborate and therefore more expensive, passive measure are less complex in most cases. This leads to the necessity to develop passive, economic and robust measures to avoid squeal. Asymmetry of the brake rotor has been proposed to achieve this goal and the resulting split of all double eigenfrequencies of the brake rotor has lately been shown to stabilize the system.     

The method of feed-in energy on disc brake squeal

Brake squeal is studied in this paper by feed-in energy analysis. Based on the brake closed-loop coupling model, a calculation method of feed-in energy for squeal mode is derived. Result of the feed-in energy indicates squeal tendency of the brake system, while formula for calculating it discloses the relation among brake squeal phenomenon and structural parameters, such as frictional coefficient, geometric shape of brake pads, elastic modulus of frictional material, substructure modal shape, etc. The method also helps to analyze the effectiveness of various structural modification schemes attempted to eliminate the squeal noise. Finally, this method is illustrated by application to a typical squealing disc brake.     

Analysis of disc brake squeal using the complex eigenvalue method

A new functionality of ABAQUS/Standard, which allows for a nonlinear analysis prior to a complex eigenvalue extraction in order to study the stability of brake systems, is used to analyse disc brake squeal. An attempt is made to investigate the effects of system parameters, such as the hydraulic pressure, the rotational velocity of the disc, the friction coefficient of the contact interactions between the pads and the disc, the stiffness of the disc, and the stiffness of the back plates of the pads, on the disc squeal. The simulation results show that significant pad bending vibration may be responsible for the disc brake squeal. The squeal can be reduced by decreasing the friction coefficient, increasing the stiffness of the disc, using damping material on the back plates of the pads, and modifying the shape of the brake pads.     

Finite element modelling for the investigation of in-plane modes and damping shims in disc brake squeal

Squeal propensity of the in-plane modes and the constrained-layer type damping shims for disc brake system is investigated by using the finite element method. Theoretical formulation is derived for a rotating disc in contact with two stationary vibrating pads attached to the damping shim components. By the conversion from the theoretical to FE brake model, the full equations of motion for the actual disc brake system describes the disc rotation, the in-plane friction characteristics and damping shims in association with squeal vibration. It is concluded from the results that the in-plane torsion modes can be generated by the negative friction slope, but they cannot be controlled by the damping shims. The in-plane radial mode is also investigated and found to be very insensitive in squeal generation.     

Suppression of brake squeal noise applying finite element brake and pad model enhanced by spectral-based assurance criteria

An enhanced dynamic finite element (FE) model with friction coupling is applied to analyze the design of disc brake pad structure for squeal noise reduction. The FE model is built-up from the individual brake component representations. Its interfacial structural connections and boundary conditions are determined by correlating to a set of measured frequency response functions using a spectral-based assurance criterion. The proposed friction coupling formulation produces an asymmetric system stiffness matrix that yields a set of complex conjugate eigenvalues. The analysis shows that eigenvalues possessing positive real parts tend to produce unstable modes with the propensity towards the generation of squeal noise. Using a proposed lumped parameter model and eigenvalue sensitivity study, beneficial pad design changes can be identified and implemented in the detailed FE model to determine the potential improvements in the dynamic stability of the system. Also, a selected set of parametric studies is performed to evaluate numerous design concepts using the proposed dynamic FE model. The best pad design attained, which produces the least amount of squeal response, is finally validated by comparison to a set of actual vehicle test results.     

Design optimization for dynamic response of vibration mechanical system with uncertain parameters using convex model

The concept of uncertainty plays an important role in the design of practical mechanical system. The most common methods for solving uncertainty problems are to model the parameters as a random vector. A natural way to handle the randomness is to admit that a given probability density function represents the uncertainty distribution. However, the drawback of this approach is that the probability distribution is difficult to obtain. In this paper, we use the non-probabilistic convex model to deal with the uncertain parameters in which there is no need for probability density functions. Using the convex model theory, a new method to optimize the dynamic response of mechanical system with uncertain parameters is derived. Because the uncertain parameters can be selected as the optimization parameters, the present method can provide more information about the optimization results than those obtained by the deterministic optimization. The present method is implemented for a torsional vibration system. The numerical results show that the method is effective.     

具有不确定参数的Liu混沌系统的同步

基于控制Lyapunov函数,对具有不确定参数的Liu混沌系统提出一种同步控制方法,控制律只依赖参数变化范围而不必知道参数的具体数值. 与已有结果相比,该设计比较简单,有较强的鲁棒性,能克服较大参数误差对同步的影响. 仿真结果说明了所提方法的有效性. 关键词： 参数系统 同步 鲁棒控制 控制Lyapunov函数     

A new treatment for predicting the self-excited vibrations of nonlinear systems with frictional interfaces: The Constrained Harmonic Balance Method,with application to disc brake squeal

Brake squeal noise is still an issue since it generates high warranty costs for the automotive industry and irritation for customers. Key parameters must be known in order to reduce it. Stability analysis is a common method of studying nonlinear phenomena and has been widely used by the scientific and the engineering communities for solving disc brake squeal problems. This type of analysis provides areas of stability versus instability for driven parameters, thereby making it possible to define design criteria. Nevertheless, this technique does not permit obtaining the vibrating state of the brake system and nonlinear methods have to be employed. Temporal integration is a well-known method for computing the dynamic solution but as it is time consuming, nonlinear methods such as the Harmonic Balance Method (HBM) are preferred. This paper presents a novel nonlinear method called the Constrained Harmonic Balance Method (CHBM) that works for nonlinear systems subject to flutter instability. An additional constraint-based condition is proposed that omits the static equilibrium point (i.e. the trivial static solution of the nonlinear problem that would be obtained by applying the classical HBM) and therefore focuses on predicting both the Fourier coefficients and the fundamental frequency of the stationary nonlinear system.The effectiveness of the proposed nonlinear approach is illustrated by an analysis of disc brake squeal. The brake system under consideration is a reduced finite element model of a pad and a disc. Both stability and nonlinear analyses are performed and the results are compared with a classical variable order solver integration algorithm.Therefore, the objectives of the following paper are to present not only an extension of the HBM (CHBM) but also to demonstrate an application to the specific problem of disc brake squeal with extensively parametric studies that investigate the effects of the friction coefficient, piston pressure, nonlinear stiffness and structural damping.     

不确定声场分析的二阶区间摄动有限元法

针对一阶区间摄动有限元法在声场参数不确定程度增大时误差过大的缺陷,在二阶Taylor展开的基础上推导了声学二阶区间摄动有限元法,并将其应用于区间不确定声场的声压响应分析。该方法先对声学区间有限元方程的声压响应向量进行二阶Taylor展开,获取声压响应的二阶近似响应向量;再根据二次函数极值定理获得声压响应向量的上下界。二维管道声场与轿车声腔模型的数值分析算例表明,与一阶区间摄动有限元法相比,二阶区间摄动有限元法有效提高了计算精度。因此二阶区间摄动有限元适合不确定度更大的区间不确定声场声压响应分析,具有良好的工程应用前景。     

不确定声场分析的二阶区间摄动有限元法

针对一阶区间摄动有限元法在声场参数不确定程度增大时误差过大的缺陷,在二阶Taylor展开的基础上推导了声学二阶区间摄动有限元法,并将其应用于区间不确定声场的声压响应分析。该方法先对声学区间有限元方程的声压响应向量进行二阶Taylor展开,获取声压响应的二阶近似响应向量;再根据二次函数极值定理获得声压响应向量的上下界。二维管道声场与轿车声腔模型的数值分析算例表明,与一阶区间摄动有限元法相比,二阶区间摄动有限元法有效提高了计算精度。因此二阶区间摄动有限元适合不确定度更大的区间不确定声场声压响应分析,具有良好的工程应用前景。     

不确定混沌系统的反同步与参数辨识

针对一类混沌系统,提出了一种系统的混沌反同步设计方案,基于该方案,设计一种自适应控制方法,实现了参数不确定系统的混沌反同步和未知参数的辨识.数值模拟结果表明了提出方法的有效性.     

具有完全不确定参数的五项双曲型混沌系统的投影同步

提出了一个新的简单的双曲型三维自治混沌系统,该三维混沌系统只含有五项, 并且其非线性特征主要依赖于一个非线性二次双曲正弦项和一个非线性二次交叉项. 较已有的三维混沌系统而言, 不仅系统的项要少一些, 而且在参数变化时, 呈现混沌的参数范围也很大. 对系统的一些基本动力学特性进行了数值模拟和理论分析. 同时, 还研究了具有完全不确定参数的该五项双曲型混沌系统的投影同步. 基于Lyapunov指数稳定性理论和Barbalat引理, 设计了一个新的具有参数自适应律的自适应同步控制器, 利用该控制器分别实现了两个结构相同和相异混沌系统的渐进性和全局性投影同步. 数值模拟验证了该方法的有效性和可行性.     

参数不确定永磁同步电机混沌系统的有限时间稳定控制

针对含有参数不确定性的永磁同步电机混沌系统, 结合主动控制与有限时间稳定控制理论, 提出一种改进的主动有限时间稳定控制器. 该控制器不仅具有较强的鲁棒性, 而且可以通过调整终端吸引子比例系数, 有效地缩短系统的响应时间, 以提高系统的响应能力. 通过仿真实验, 验证了该控制器比传统的控制器的具有更强的鲁棒性和响应能力. 关键词： 永磁同步电机 混沌控制 有限时间稳定 主动控制     

Adaptive function project synchronization of Rössler hyperchaotic system with uncertain parameters

This Letter addresses the function project synchronization problem of two Rössler hyperchaotic in the presence of unknown system parameters. Based on Lyapunov stability theory an adaptive control law is proposed to make the states of two identical Rössler hyperchaotic systems asymptotically synchronized. Numerical simulations are presented to show the effectiveness of the proposed schemes.     

Adaptive modified function projective synchronization between hyperchaotic Lorenz system and hyperchaotic Lu system with uncertain parameters

This Letter investigates modified function projective synchronization between hyperchaotic Lorenz system and hyperchaotic Lu system using adaptive method. By Lyapunov stability theory, the adaptive control law and the parameter update law are derived to make the state of two hyperchaotic systems modified function projective synchronized. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive controllers.     

Pragmatical generalized synchronization of chaotic systems with uncertain parameters by adaptive control

A new kind of generalized synchronization of two chaotic systems with uncertain parameters is proposed. Based on a pragmatical asymptotical stability theorem and an assumption of equal probability for ergodic initial conditions, an adaptive control law is derived so that it can be proved strictly that the common null solution of error dynamics and of parameter dynamics is actually asymptotically stable, i.e. these two identical systems are in generalized synchronization and the estimated parameters approach the uncertain values. It is called pragmatical generalized synchronization. Finally, two numerical examples are studied for two Quantum-CNN oscillator chaotic systems to show the effectiveness of the proposed generalized synchronization strategy with a double Duffing chaotic system as a goal system.     

参数不确定量子细胞神经网络与Lorenz超混沌系统的函数投影同步研究

研究了具有不确定参数的量子细胞神经网络系统与Lorenz超混沌系统的函数投影同步.设计了自适应追踪控制器来实现两个不同系统渐进同步到已知的比例函数,并运用Lyapunov稳定性方法进行了证明.理论分析了量子细胞神经网络系统不确定参数的系数向量线性无关性,实现了不确定参数的识别和估计.数值仿真结果验证了函数投影同步和参数估计的有效性.     

Complete synchronization of double-delayed RSssler systems with uncertain parameters

In this paper, we investigate complete synchronization of double-delayed RSssler systems with uncertain parameters as the master system is in chaotic synchronization. The uncertain parameters can be nonlinearly expressed in the system. The analysis and proof are given by means of the Lyapunov stability theorem. Based on theoretical analysis, some sufficient conditions of complete synchronization are proved. In order to validate the proposed scheme, numerical simulations are performed and the numerical results show that our scheme is very effective.     

Combined synchronization of time-delayed chaotic systems with uncertain parameters

The article presents combined synchronization among time delayed chaotic systems in the presence of uncertain parameters using nonlinear control method. Control functions are designed to achieve combined synchronization using Lyapunov-Krasovskii function for stability analysis. The synchronization among three and four time delayed chaotic systems have been shown as examples of combined synchronization. Double delay Rossler system, the advanced Lorenz system, time delay Liu and Chen systems have been taken to show the combined synchronization. Numerical simulation and graphical results are carried out using Runge–Kutta method for delay-differential equations, which show that the designing of control functions are very effective and reliable and can be applied for combined synchronization among time-delayed chaotic systems.