مدل سازی ترمزهای کاسه ای خودرو برای جیغ ممتمد و تجزیه و تحلیل حساسیت پارامتر

Many fundamental studies have been conducted to explain the occurrence of squeal in disc and drum brake systems. The elimination of brake squeal, however, still remains a challenging area of research. Here, a numerical modeling approach is developed for investigating the onset of squeal in a drum brake system. The brake system model is based on the modal information extracted from finite element models for individual brake components. The component models of drum and shoes are coupled by the shoe lining material which is modeled as springs located at the centroids of discretized drum and shoe interface elements. The developed multi degree of freedom coupled brake system model is a linear non-self-adjoint system. Its vibrational characteristics are determined by a complex eigenvalue analysis. The study shows that both the frequency separation between two system modes due to static coupling and their associated mode shapes play an important role in mode merging. Mode merging and veering are identified as two important features of modes exhibiting strong interactions, and those modes are likely candidates that lead to coupled-mode instability. Techniques are developed for a parameter sensitivity analysis with respect to lining stiffness and the stiffness of the brake actuation system. The influence of lining friction coefficient on the propensity to squeal is also discussed.

Automotive brake noise and vibration control has become increasingly important for the improvement of vehicle quietness and passenger comfort. Over the years, brake noise has been classified by frequency contents and given various names such as grunt, judder, moan, groan, squeal, squeak and so on. In a recent review on disc brake squeal, Kinkaid et al. [1] stated that there has not been a precise definition of brake squeal. Since Nishiwaki [2] showed that groan and squeal are generated by the same phenomenon of dynamic instability, both low- and high-frequency noise can be studied by using the same modeling and analysis techniques. Brake squeal here is defined as any type of elastic instability that involves elastic modes of various brake components and is within the audible range of frequencies. Systematic research on brake squeal can be traced back to the 1950s and still is an active subject for current researchers and engineers. The structure of brakes which consist of several components is complicated, and the fugitive nature of friction makes the problem more difficult. Research on brake squeal has been conducted using theoretical, experimental, and computational approaches. Many theoretical approaches have been presented to explore the squeal mechanisms. Early attempts to explain brake squeal emphasized that the negative slope of the friction coefficient with respect to the relative velocity caused the self-excited vibration. Spurr [3] proposed the sprag-slip model to introduce a new mechanism called geometry instability, without including the friction characteristic. Millner [4] also reported that brake squeal may occur even if the friction coefficient is constant. North [5] first presented a simple 2-dof model, in which the friction leads to an asymmetric stiffness coupling indicating non-conservative forces and the instability may occur. This mechanism was developed and advanced by many other investigators, and in this approach it is believed that brake squeal is mainly caused by dynamic instability of the brake system with variable friction forces [6] and [7]. In recent years the main focus on brake squeal problems has shifted from fundamental theoretical research to more practical and problem-solving oriented efforts. Instead of a simple schematic model, the brake system model tends to include more brake components, and the effects of design parameters on the stability can be investigated. Liles [8] created a linear system model based on the modal information of the disc brake components, and performed a complex eigenvalue analysis to solve the equations of motion. Guan and Jiang [9] constructed a coupled linear model including all disc brake components and identified the substructure modes which have great influence on the system stability. Chowdhary et al. [10] developed an assumed modes model for squeal prediction of a disc brake, and found that the separation between the frequencies is an important factor in determining the onset of flutter-type instability. Ouyang et al. [11] considered the effects of rotation of the disc, and the friction-induced vibration of the disc brake was treated as a moving load problem. With the improvement of numerical techniques, Hamabe et al. [12] and Nack [13] directly conducted a complex eigenvalue analysis with a finite element (FE) model of a brake system including the friction force. In their work on disc brake squeal using FE analysis, Lee et al. [14] performed a nonlinear contact analysis to determine the pressure distribution at the friction interface followed by system linearization and a complex modal analysis. Thus, in their study, the contact stiffness was dependent on local contact pressure. In the present work, a numerical approach is presented to study the drum brake squeal. FE models are first created for brake components including the drum and the shoes. The shoe lining is modeled as a layer having a distributed compliance that produces a stiffness coupling between the drum and shoes. There are two components of this coupling, one corresponding to transverse displacements and the other due to tangential forces arising as a result of friction coupling between drum and the shoes. The total degrees of freedom are reduced by transforming the physical coordinate model into a modal coordinate model. The resulting system model has symmetric as well as asymmetric stiffness matrices, and a complex eigenvalue analysis is carried out to determine the stability characteristics of the brake assembly. It is seen that eigenvalue veering and compatible mode shapes for the coupled drum brake system without friction are necessary conditions for brake squeal to occur in the presence of friction. To authors’ knowledge, this is the first instance when eigenvalue-loci veering and strong modal coupling are clearly identified as necessary conditions for coupled-mode instability to arise. Based on this model, the influences of the friction coefficient, the lining stiffness, and the hydraulic cylinder stiffness on system stability are discussed.

Brake squeal is a phenomenon of self-excited friction induced vibrations resulting from mode coupling. This paper presents an approach to develop a numerical drum brake model for squeal prediction, and analyzes the conditions that can lead to brake squeal. Based on the numerical model and results of complex eigenvalue analysis of the linear equations of motion, the following conclusions can be drawn: (1) The component modal characteristics can be extracted from either analytical analysis or FE analysis. FE analysis makes it easy to capture geometry complexities of the components and incorporate the results of contact analysis in the system model. By creating virtual mesh of contact elements, the approach does not require the FE meshes of different components at the interfaces to match, thus greatly facilitating the FE modeling of the components. (2) If the separation between the two modes due to static coupling is small enough and their mode shapes are compatible, the two modes have a good likelihood that they will merge when the friction is introduced and increased. Only compatible modes have the possibility to interact and become identical to result in instability. Though mode shapes of brake components are often measured with experimental methods, there are few investigations on the influence of mode shapes on mode merging. The understanding of the important role of mode shapes is expected to be of great help to prediction of the occurrence of squeal. (3) The statically coupled modes which tend to merge in the presence of friction always exhibit curve veering phenomenon, while the modes which simply cross do not cause instability. There is no coupling between the statically coupled modes showing curve crossing. Curve veering reflects the coupling of compatible modes, and this coupling may cause the two modes to merge as the friction increases. Eigenvector sensitivity clearly allows for differentiation between modes that undergo curve crossing and those that undergo veering. (4) The stability boundaries are sensitive to changes in parameters such as lining stiffness. Due to the correlation between the critical values of friction coefficient and the separations of the statically coupled frequencies, the changes in separations partially reveal the effects of the parameters on system stability and can provide an explanation to some squeal reduction techniques. These will be the subject of a work under preparation.