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.     

Automotive disc brake squeal

Disc brake squeal remains an elusive problem in the automotive industry. Since the early 20th century, many investigators have examined the problem with experimental, analytical, and computational techniques, but there is as yet no method to completely suppress disc brake squeal. This paper provides a comprehensive review and bibliography of works on disc brake squeal. In an effort to make this review accessible to a large audience, background sections on vibrations, contact and disc brake systems are also included.     

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.     

Wave pattern motion and stick-slip limit cycle oscillation of a disc brake

This paper examines the dynamic response of a rotating squealing disc brake subject to distributed nonlinear contact stresses where two brake pads are assumed to be stationary and rigid. The friction stresses produce high-frequency vibrations that exhibit standing or traveling waves on the disc surface. The wave pattern resulting from the binary flutter mechanism of one transverse doublet mode pair is studied here. The results show that the wave pattern is associated with mode-coupling character. For a steady-squealing mode, the stick zone of the contact area is determined by a smooth friction-velocity curve having both negative and positive slopes.     

Brake squeal reduction of vehicle disc brake system with interval parameters by uncertain optimization

An uncertain optimization method for brake squeal reduction of vehicle disc brake system with interval parameters is presented in this paper. In the proposed method, the parameters of frictional coefficient, material properties and the thicknesses of wearing components are treated as uncertain parameters, which are described as interval variables. Attention is focused on the stability analysis of a brake system in squeal, and the stability of brake system is investigated via the complex eigenvalue analysis (CEA) method. The dominant unstable mode is extracted by performing CEA based on a linear finite element (FE) model, and the negative damping ratio corresponding to the dominant unstable mode is selected as the indicator of instability. The response surface method (RSM) is applied to approximate the implicit relationship between the unstable mode and the system parameters. A reliability-based optimization model for improving the stability of the vehicle disc brake system with interval parameters is constructed based on RSM, interval analysis and reliability analysis. The Genetic Algorithm is used to get the optimal values of design parameters from the optimization model. The stability analysis and optimization of a disc brake system are carried out, and the results show that brake squeal propensity can be reduced by using stiffer back plates. The proposed approach can be used to improve the stability of the vehicle disc brake system with uncertain parameters effectively.     

Modeling of automotive drum brakes for squeal and parameter sensitivity analysis

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.     

Structural optimization for the avoidance of self-excited vibrations based on analytical models

Self-excited vibrations are a severe problem in many technical applications. In many cases they are caused by friction as for example in disk and drum brakes, clutches, saws and paper calenders. The goal to suppress self-excited vibrations can be reached by active and passive techniques, the latter ones being preferable due to the lower costs. Among design engineers it is known that breaking the symmetries of structures is sometimes helpful to avoid self-excited vibrations. This has been verified from an analytical point of view in a recent paper. The goal of the present paper is to use this analytical insight for a systematic structural optimization of rotors in frictional contact. The first system investigated is a simple discrete model of a rotor in frictional contact. As a continuous example a rotating beam in frictional contact is considered and optimized with respect to its bending stiffness. Finally a brake disk is optimized giving some attention to the feasibility of the modifications for the production process.     

Squeal and chatter phenomena generated in a mountain bike disc brake

This paper examines squeal and chatter phenomena generated experimentally in mountain bike disc brakes. There are two kinds of frictional self-excited vibrations in the bike disc brakes, called squeal with frequency of 1 kHz and chatter with frequency of 500 Hz. In order to reproduce the squeal and chatter, a bench test apparatus using an actual bike was set up to determine the associated frequency characteristics experimentally. The results show the frequencies to be independent of pad temperature and disc rotating speed. Squeal is shown to be in-plane vibration in the direction of the disc surface which is caused by the frictional characteristics having negative slope with respect to the relative velocity in the vibrating system, which includes brake unit, spokes and hub. Chatter is generated within a limited high temperature region. Again, it is frictional vibration in which the squeal and out-of-plane vibration of the disc due to Coulomb friction combine through the internal resonance relation between in-plane and out-of-plane nonlinear vibration caused by the temperature increase of the disc during braking.     

ANALYSIS OF DISC BRAKE NOISE USING A TWO-DEGREE-OF-FREEDOM MODEL

Although a brake pad and disc have many modes of vibration, when it is unstable and hence noisy then frequently only a single mode of the complete system contributes to the vibration. In this condition, only a few modes are required to model the system. In this paper, a two-degree-of-freedom model is adopted where the disc and the pad are modelled as single modes connected by a sliding friction interface. Using this model, the interaction between the pad and the disc is investigated. Stability analysis is performed to show under what parametric conditions the system becomes unstable, assuming that the existence of a limit cycle represents the noisy state of the disc brake system. The results of this analysis show that the damping of the disc is as important as that of the pad. Non-linear analysis is also performed to demonstrate various limit cycles in the phase space. The results show that the addition of damping to either the disc or the pad alone may make the system more unstable, and hence noisy.     

Instabilities in the sliding of continua with surface inertias: An initiation mechanism for brake noise

This paper presents a model for friction-induced vibration in brake systems, which takes a homogeneous tribological layer on the brake pad into account. The derived model consists of two flat elastic bodies sliding against each other with a constant coefficient of friction. In brake tribology, like in most tribological processes, a surface structure is observed, which can be modeled as an additional film of homogenized mass distribution bonded on the moving continua. The developed mechanical model and its analytical solution show an excitation mechanism that bases on the interaction of normal and frictional shear force and on the elastic coupling of spatial directions. The derived solution allows to study stability and eigenforms of the sliding process: on the frictional plane traveling surface waves are generated, with stability properties depending on parameters of the tribological layer. A parameter study analyzes the frictional couple of brake disk and pad and the related surface state. It is found that increasing inertias of surface structures on the pad strengthen instabilities of the sliding system. A comparison with experiments suggests a similar dependency between surface state and stability as observed by the model under discussion.     

Eigenvalue optimization against brake squeal: Symmetry,mathematical background and experiments

Brake squeal is still a challenge for design engineers and scientists. Due to cost reasons for the avoidance of brake noise only passive measures are meaningful for a broad industrial range. Many countermeasures against squeal are based on the introduction of damping, for example by using shims. In the literature on the modeling of brake squeal, the structural properties of the brake disc are most often not considered. It has however been shown analytically and experimentally that the stiffness properties of the disc are important and that splitting of double modes of the disc has a stabilizing effect. This knowledge can be used for structural optimization of brake rotors. The goal of this paper is to exploit the potential and to discuss some mathematical difficulties. Furthermore, experimental evidence for the relation of rotor asymmetry and squeal is given.     

Double pulsed holography used to investigate noisy brakes

The vibrational characteristics of a noisy passenger car disc brake have been studied using the double pulsed holographic technique which has been developed to allow three orthogonal visual images of a vibrating brake system to be recorded simultaneously. These images show the disc to be vibrating in a bending mode whereas the pad is seen to be excited in a variety of modes such as bending, torsion, and often a combination of both. The development of the technique includes alternative ways of triggering the laser and typical results from the application of these differing methods are also included along with mechanical signals which confirm the visual interpretations. Final results, using a laser trigger delay technique, show that the disc mode waveform rotates about the disc at a rate equivalent to the frequency of vibration divided by the diametral mode order. Early work on a passenger car drum brake is also introduced, this complementing commercial ‘noise fix’ solutions and a proposed theoretical model.     

Numerical study of friction-induced instability and acoustic radiation – Effect of ramp loading on the squeal propensity for a simplified brake model

This paper presents a numerical study of the influence of loading conditions on the vibrational and acoustic responses of a disc brake system subjected to squeal. A simplified model composed of a circular disc and a pad is proposed. Nonlinear effects of contact and friction over the frictional interface are modelled with a cubic law and a classical Coulomb?s law with a constant friction coefficient. The stability analysis of this system shows the presence of two instabilities with one and two unstable modes that lead to friction-induced nonlinear vibrations and squeal noise. Nonlinear time analysis by temporal integration is conducted for two cases of loadings and initial conditions: a static load near the associated sliding equilibrium and a slow and a fast ramp loading. The analysis of the time responses shows that a sufficiently fast ramp loading can destabilize a stable configuration and generate nonlinear vibrations. Moreover, the fast ramp loading applied for the two unstable cases generates higher amplitudes of velocity than for the static load cases. The frequency analysis shows that the fast ramp loading generates a more complex spectrum than for the static load with the appearance of new resonance peaks. The acoustic responses for these cases are estimated by applying the multi-frequency acoustic calculation method based on the Fourier series decomposition of the velocity and the Boundary Element Method. Squeal noise emissions for the fast ramp loading present lower or higher levels than for the static load due to the different amplitudes of velocities. Moreover, the directivity is more complex for the fast ramp loading due to the appearance of new harmonic components in the velocity spectrum. Finally, the sound pressure convergence study shows that only the first harmonic components are sufficient to well describe the acoustic response.     

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.     

Improvement in the predictivity of squeal simulations: Uncertainty and robustness

The objective of this paper is to improve the predictivity of squeal simulations by introducing uncertainty and robustness concepts during simulations. Complex eigenvalue analysis is a traditional way to detect numerically the unstable modes that can be associated with extensive vibration and noise pollution. This simulation, for which associated computational times are compatible with the design phase, is known to be insufficiently predictive. We first propose a complete strategy that relies on the integration of random fields into the contact interface, complex eigenvalue calculations, probabilistic analysis and a robustness criterion. Next, this strategy is applied to study the instabilities of a complete industrial brake system. Experimental comparisons highlight the efficiency of the improved squeal detection methodology.     

Optical analogy to electronic quantum corrals

We describe full multiple-scattering calculations of localized surface photonic states set up by lithographically designed nanostructures made of a finite number of dielectric pads deposited on a planar surface. The method is based on a numerical solution of the dyadic Dyson's equation. When the pads are arranged to form a closed circle, we find field patterns that look like the electronic charge density recently observed above quantum corrals. We propose two experimental techniques that could be used to observe these electromagnetic modes in direct space.