Nonlinear stability analysis of a disk brake model

It has become commonly accepted by scientists and engineers that brake squeal is generated by friction-induced self-excited vibrations of the brake system. The noise-free configuration of the brake system loses stability through a flutter-type instability and the system starts oscillating in a limit cycle. Usually, the stability analysis of disk brake models, both analytical as well as finite element based, investigates the linearized models, i.e. the eigenvalues of the linearized equations of motion. However, there are experimentally observed effects not covered by these analyses, even though the full nonlinear models include these effects in principle. The present paper describes the nonlinear stability analysis of a realistic disk brake model with 12 degrees of freedom. Using center manifold theory and artificially increasing the degree of degeneracy of the occurring bifurcation, an analytical expression for the turning points in the bifurcation diagram of the subcritical Hopf bifurcations is calculated. The parameter combination corresponding to the turning points is considered as the practical stability boundary of the system. Basic phenomena known from the operating experience of brake systems tending to squeal problems can be explained on the basis of the practical stability boundary.     

Study on the stability of drum brake non-linear low frequency vibration model

A 5-DOF non-linear model is presented to simulate the vibration of a drum brake at low frequency in the course of applying the brake. Analysis and calculation are carried out to illuminate that even when the friction coefficient is constant, the vibration and instability can occur with the combination of some specific parameters. And the stable-unstable area on the parameter plane on the condition of the combination of some specific parameters is presented to illuminate the effect of the structure parameter on the system stability.     

A polynomial chaos approach to the robust analysis of the dynamic behaviour of friction systems

This paper presents a dynamic behaviour study of non-linear friction systems subject to uncertain friction laws. The main aspects are the analysis of the stability and the associated non-linear amplitude around the steady-state equilibrium. As friction systems are highly sensitive to the dispersion of friction laws, it is necessary to take into account the uncertainty of the friction coefficient to obtain stability intervals and to estimate the extreme magnitudes of oscillations. Intrusive and non-intrusive methods based on the polynomial chaos theory are proposed to tackle these problems. The efficiency of these methods is investigated in a two degree of freedom system representing a drum brake system. The proposed methods prove to be interesting alternatives to the classic methods such as parametric studies and Monte Carlo based techniques.     

Extension of the Center Manifold Approach,Using Rational Fractional Approximants,Applied to Non-Linear Stability Analysis

A method is presented which extends the domain of the validity of non-linear systems reduced by using the centermanifold approach. This methodology applies rational fractional approximants in order to enhance the convergence ofthe series expansions of the center manifold theory.Effectively, a sequence of rational fractional approximantsmay converge even if the associated series does not; one canthen extend the domain of convergence. In this study, thedomain of validity of the solution is successfully enhanced byemploying rational fractional approximants. Here the basic ideas are outlined, an example ispresented, and some natural extensions and possibleapplications of this methodology are briefly described in theconclusions.     

Mode coupling instability in friction-induced vibrations and its dependency on system parameters including damping

Friction-induced vibrations due to coupling modes can cause severe damage and are recognized as one of the most serious problems in industry. In order to avoid these problems, engineers must find a design to reduce or to eliminate mode coupling instabilities in braking systems. Though many researchers have studied the problem of friction-induced vibrations with experimental, analytical and numerical approaches, the effects of system parameters, and more particularly damping, on changes in stable-unstable regions and limit cycle amplitudes are not yet fully understood.The goal of this study is to propose a simple non-linear two-degree-of-freedom system with friction in order to examine the effects of damping on mode coupling instability. By determining eigenvalues of the linearized system and by obtaining the analytical expressions of the Routh–Hurwitz criterion, we will study the stability of the mechanical system's static solution and the evolution of the Hopf bifurcation point as functions of the structural damping and system parameters. It will be demonstrated that the effects of damping on mode coupling instability must be taken into account to avoid design errors. The results indicate that there exists, in some cases, an optimal structural damping ratio between the stable and unstable modes which decreases the unstable region. We also compare the evolution of the limit cycle amplitudes with structural damping and demonstrate that the stable or unstable dynamic behaviour of the coupled modes are completely dependent on structural damping.     

盘式制动器摩擦噪声特性的试验分析

盘式制动器制动时产生的摩擦噪声不仅影响车辆的驾乘舒适性，还会带来严重的环境噪声污染. 摩擦噪声具有瞬时性与不确定性的特点，至今尚未形成对制动器摩擦噪声机理的统一论断. 基于盘式制动器声振试验台，研究了摩擦系数、摩擦力波动系数和等效A声级三个参数在不同制动压力和制动初速度下的变化规律，结合对摩擦片微观特性的试验分析，从摩擦学角度揭示了盘式制动器摩擦噪声的发生规律及机理. 结果表明:摩擦噪声随制动初速度的增大而减小，随制动压力的增加而先增大后减小；制动摩擦噪声由于摩擦系统不稳定而产生，其产生原因是因为摩擦副表面的沟壑与黏着引起，沟壑与黏着程度与噪声的声压与频率呈正相关关系. 研究结果对于解释制动器摩擦噪声发生机理、控制制动器噪声污染具有重要理论意义和实际价值.      

Analytical approximation for period-doubling following a hopf bifurcation

The application of center manifold techniques involves a reduction in the dimension of a dynamical system by restricting attention to an invariant subspace (the center manifold) which contains all of the essential behavior. This is the usual procedure for dealing with Hopf bifurcations, where the dimension of a system may be reduced from n to 2. In this work we have extended this procedure by noting that the n-2 dimensions that are usually discarded contain significant information concerning the attractivity, i.e., the stability, of the center manifold itself, as well as of the limit cycle which it contains. The resulting instability of the limit cycle has been seen to give rise to period-doubling.     

汽车盘式制动器摩擦-振动耦合特性试验研究

本文针对9种具有不同厚薄差和端面跳动量组合的汽车盘式制动器,利用制动器惯性试验台在四种制动压力工况下进行了制动转矩和制动压力的测量,并通过阶次分析与小波分析的方法研究了盘式制动器摩擦特性与制动压力波动之间的耦合关系.研究发现:制动盘的厚薄差和端面跳动会引起法向力的波动,进而引起摩擦系数的波动,并且摩擦系数、制动压力以及制动转矩的波动都与转速存在2阶的阶次关系;摩擦系数静态成分随着相对速度的减小和制动压力的减小而增大;摩擦系数动态部分随制动压力的增大而减小,且主要由制动盘厚薄差引起,端面跳动对其影响不明显.     

Construction of Dynamically Equivalent Time-Invariant Forms for Time-Periodic Systems

In this study dynamically equivalent time-invariant forms are obtained for linear and non-linear systems with periodically varying coefficients via Lyapunov–Floquet (L–F) transformation. These forms are equivalent in the sense that the local stability and bifurcation characteristics are identical for both systems in the entire parameter space. It is well known that the L–F transformation converts a linear periodic first order system into a time-invariant one. In the first part of this study a set of linear second order periodic equations is converted into an equivalent set of time-independent second order equations through a sequence of linear transformations. Then the transformations are applied to a time-periodic quadratic Hamiltonian to obtain its equivalent time-invariant form. In the second part, time-invariant forms of nonlinear equations are studied. The application of L–F transformation to a quasi-linear periodic equation converts the linear part to a time-invariant form and leaves the non-linear part with time-periodic coefficients. Dynamically equivalent time-invariant forms are obtained via time-periodic center manifold reduction and time-dependent normal form theory. Such forms are constructed for general hyperbolic systems and for some simple critical cases, including that of one zero eigenvalue and a purely imaginary pair. As a physical example of these techniques, a single and a double inverted pendulum subjected to periodic parametric excitation are considered. The results thus obtained are verified by numerical simulation.     

The construction of non-linear normal modes for systems with internal resonance

A numerical method, based on the invariant manifold approach, is presented for constructing non-linear normal modes for systems with internal resonances. In order to parameterize the non-linear normal modes of interest, multiple pairs of system state variables involved in the internal resonance are kept as ‘seeds’ for the construction of the multi-mode invariant manifold. All the remaining degrees of freedom are then constrained to these ‘seed’, or master, variables, resulting in a system of non-linear partial differential equations that govern the constraint relationships, and these are solved numerically. The computationally-intensive solution procedure uses a combination of finite difference schemes and Galerkin-based expansion approaches. It is illustrated using two examples, both of which focus on the construction of two-mode models. The first example is based on the analysis of a simple three-degree-of-freedom example system, and is used to demonstrate the approach. An invariant manifold that captures two non-linear normal modes is constructed, resulting in a reduced order model that accurately captures the system dynamics. The methodology is then applied to a larger order system, specifically, an 18-degree-of-freedom rotating beam model that features a three-to-one internal resonance between the first two flapping modes. The accuracy of the non-linear two-mode reduced order model is verified by comparing time-domain simulations of the two DOF model and the full system equations of motion.     

Non-linear dynamic phenomena in the behaviour of a railway wheelset model

A dicone moving on a pair of cylindrical rails can be considered as a simplified model of a railway wheelset. Taking into account the non-linear friction laws of rolling contact, the equations of motion for this non-linear mechanical system result in a set of differential-algebraic equations. Previous simulations performed with the differential-algebraic solver DASSL, [2], and experiments, [7], indicated non-linear phenomena such as limit-cycles, bifurcations as well as chaotic behaviour. In this paper the non-linear phenomena are investigated in more detail with the aid of special in-house software and the path-following algorithm PATH [10]. We apply Poincaré sections and Poincaré maps to describe the structure of periodic, quasiperiodic and chaotic motions. The analyses show that part of the chaotic behaviour of the non-linear system can be fully understood as a non-linear iterative process. The resulting stretching and folding processes are illustrated by series of Poincaré sections.     

半金属盘式刹车材料的试验研究

通过正交试验设计和蔡氏试验机测试,得到了摩擦系数受温度、载荷和速度影响较小,磨损也较小的半金属盘式刹车材料配方,其平均摩擦系数是0.444.结果表明,在4种被研究的成分中温和的研磨剂氧化铁黑对与温度有关的摩擦系数的标准偏差影响最大,原因在于其只对冷制动起增摩作用而对热制动没有增摩作用;过多使用强的研磨剂氧化铝会导致摩擦系数和比磨损率增大,出现过恢复,并且摩擦系数随载荷增加;而过多使用混合润滑剂(石景和三硫化二锑)会减小摩擦系数和比磨损率.     

Analysis of heat conduction in a disk brake system

In this paper, the governing heat equations for the disk and the pad are extracted in the form of transient heat equations with heat generation that is dependant to time and space. In the derivation of the heat equations, parameters such as the duration of braking, vehicle velocity, geometries and the dimensions of the brake components, materials of the disk brake rotor and the pad and contact pressure distribution have been taken into account. The problem is solved analytically using Green’s function approach. It is concluded that the heat generated due to friction between the disk and the pad should be ideally dissipated to the environment to avoid decreasing the friction coefficient between the disk and the pad and to avoid the temperature rise of various brake components and brake fluid vaporization due to excessive heating.     

MHD free convective flow and mass transfer over a stretching sheet with chemical reaction

The effect of chemical reaction on free convective flow and mass transfer of a viscous, incompressible and electrically conducting fluid over a stretching surface is investigated in the presence of a constant transverse magnetic field. The non-linear boundary layer equations with the boundary conditions are transferred by a similarity transformation into a system of non-linear ordinary differential equations with the appropriate boundary conditions. Furthermore, the similarity equations are solved numerically by using a fourth order Runge-Kutta scheme with the shooting method. Numerical results of the skin friction coefficient, the local Nusselt number Nu, the local Sherwood number Sh, as will as the velocity, temperature and concentration profiles are presented for gases with a Prandtl number of 0.71 for various values of chemical reaction parameter, order of reaction, magnetic parameter and Schmidt number.     

Non-linear analysis of creeping flow on the inclined permeable substrate plane subjected to an electric field

The effect of an externally applied electric field on the stability of a thin fluid film over an inclined porous plane is analyzed using linear and non-linear stability analysis in the long wave limit. The principle aim of this study is to illustrate the influence of electric field on the non-linear stability of a thin liquid layer flow down incline substrate when the plane is porous. The driving force for the instability under an electric field is an electrostatic force exerted on the free charges accumulated at the dividing interface. The coupled non-linear evolution equations for the local film thickness and the interfacial charge for two-dimensional disturbances are derived to analyze the effect of long-wave instabilities. The method of multiple scales is applied to obtain approximate solutions and analyze the stability criteria. Numerical simulations of this system of non-linear evolution equations are performed. It is found that the permeability parameter as well as the inclination of the plane plays a destabilizing role in the stability criteria, while the damping influence is observed for increasing of the electrical conductivity in both linear and non-linear behavior.     

等离子喷涂铁－镍－钴－碳化钨涂层制动摩擦特性的研究

重载制动摩擦磨损特性制约着刹车装置的刹车性能及其使用寿命．在ＭＭ－１０００试验机上，对等离子喷涂Ｆｅ－Ｎｉ－Ｃｏ－ＷＣ涂层的重载制动摩擦特性进行了试验研究．结果表明：在相同的试验条件下，等离子喷涂涂层分别与石棉摩擦材料和半金属摩擦材料对摩时的摩擦系数均比与基体３５ＣｒＭｏ钢对摩时的高，制动时间短，制动效率和耐磨性能都明显提高，制动力矩峰比小，刹车平稳，制动特性好；制动摩擦系数与制动压力、速度和转动惯量等参数密切相关，压力增大，摩擦系数减小，速度和转动惯量增大，石棉材料摩擦副的摩擦系数降低，半金属材料摩擦副的摩擦系数升高，而且半金属材料的摩擦系数的热稳定性比石棉摩擦材料的好     

A semi-analytical buckling analysis of imperfect cylindrical shells under axial compression

In the framework of the cellular bifurcation theory, we investigate the effect of distributed and/or localized imperfections on the buckling of long cylindrical shells under axial compression. Using a double scale perturbative approach including modes interaction, we establish that the evolution of amplitudes of instability patterns is governed by a non-homogeneous second order system of three non-linear complex equations. The localized imperfections are included by employing jump conditions for their amplitude and permitting discontinuous derivatives. By solving these amplitude equations, we show the influence of distributed and/or localized imperfections on the reduction of the critical load. To assess the validity of the present method, our results are compared to those given by two finite element codes.     

纤维增强合成闸片摩擦磨损特性的试验研究

采用MM-1000型摩擦磨损试验机进行了纤维增强合成闸片和HZ408合成闸片与H300制动韫材料配副的制动摩擦试验，记录了制动过程中摩擦系数、磨损量与制动盘温度的变化情况，采用三维激光共焦扫描显微镜（CLSM）分析了磨损表面形貌．结果表明：制动初速度对纤维增强合成闸片的平均摩擦系数影响小于HZ408合成闸片，在高速制动工况下，纤维增强合成闸片的平均摩擦系数大于HZ408，耐磨性略低于HZ408合成闸片；在低速制动工况下，纤维增强合成闸片和HZ408合成闸片所引起的制动盘温升大致相同；在高速制动工况下前者引起的制动盘温升略高．     

Stability and response of a self-amplified braking system under velocity-dependent actuation force

Wedge brakes, featuring self-amplification, inspire good opportunity to obtain large normal force by small actuation force. A single degree of freedom torsional model with harmonic excitation for a driveline with a wedge brake is developed to investigate the effect of velocity-dependent actuation force. The stability analysis indicates that instability can occur even with a constant friction coefficient and is greatly influenced by the slope of the actuation force. Three bifurcation points are found: one stable, one unstable, and the other one Hopf. Phase portraits, time domain responses, Poincaré maps, and frequency spectra are provided by nonlinear computation. Three motions are observed: unidirectional stick-slip, bidirectional stick-slip, and non-stick-slip. Due to the self amplification, the wedge brake leads to more stick motions and more side bands compared with the conventional brake. By varying the slope, the dynamic response of the driveline can be synchronous or irregular multi-periodic motion. The dynamics at negative slope is studied further considering three other influencing factors, i.e., the initial actuation force, excitation frequency, and wedge angle. The results are provided by the comparison with those of the driveline with a conventional brake.