Moving-load dynamic problems: A tutorial (with a brief overview)

Minimal models for disk brake squeal

Uncertainty quantification of squeal instability via surrogate modelling

Considerable effort is spent in the design and testing of disk brake systems installed in modern passenger cars. This effort can be reduced if appropriate mathematical-mechanical models are used for studying the dynamics of these brakes. In this context, the mechanism generating brake squeal in particular deserves closer attention. The present paper is devoted to the modeling of self-excited vibrations of moving continua generated by frictional forces. Special regard is given to an accurate formulation of the kinematics of the frictional contact in two and three dimensions. On the basis of a travelling Euler-Bernoulli beam and a rotating annular Kirchhoff plate with frictional point contact the essential properties of the contact kinematics leading to self-excited vibrations are worked out. A Ritz discretization is applied and the obtained approximate solution is compared to the exact one of the traveling beam. A minimal disk brake model consisting of the discretized rotating Kirchhoff plate and idealized brake pads is analyzed with respect to its stability behavior resulting in traceable design proposals for a disk brake.

Friction Induced Vibrations in Moving Continua and Their Application to Brake Squeal

Several state of the art papers and even books on brake vibration and/or noise have been presented in the literature. Many of them have analytically and sharply accounted for the impressive amount of research undertaken on this topic. This state of the art review focuses on the still-open questions that appear crucial from the perspective of a leading brake manufacturer. The paper deals with the phenomena of brake vibration and/or noise, the experimental and theoretical methods for studying such phenomena, and the actions that are identified to be necessary to definitely solve the addressed problem. Key topics are the modelling of friction, the modelling of the dynamics of the brake as a non-linear system subjected to deterministic or random (parametric) excitation, the proper modelling of the contact between the disc and the pad, and the experimental validation of the mathematical models.

Brake comfort – a review

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.

Nonlinear stability analysis of a disk brake model

Considerable effort is spent in the design and testing of disk brakes of modern passenger cars. This effort can be reduced if refined mathematical-mechanical models and new experimental techniques are used for studying the dynamics of these brakes. The present paper is devoted to the modeling and experimental investigation of a floating caliper disk brake, special regard being given to the suppression of squeal using active elements. To actively suppress brake squeal, smart pads were designed and manufactured. These pads contain piezoceramic staple actuators, which can be independently driven at both pads and within the pads. In experiments they were successfully used for the active suppression of squeal via optimal control. As the piezoceramic elements can be used both as actuators as well as sensors, the smart pads are also useful in experimental investigations such as measuring transfer functions. In this manner, design modifications proposed for conventional disk brakes can easily be tested using this method. Active control of squeal using smart pads is presently not envisaged as a technique to suppress squeal in mass produced disk brakes, but rather as a possible tool to be used in industrial laboratories to shorten the time for optimizing new brake designs, with high potential saving benefits. The development and laboratory implementation of the active squeal control goes along with a more profound understanding of brake squeal and a better modeling of the phenomena, ultimately leading to improvements in the design of disk brakes.

Active Control of Brake Squeal Via “Smart Pads”

Rotating plates are used as a main component in various applications. Their vibrations are mainly unwanted and interfere with the functioning of the complete system. The present paper investigates the coupling of disk (in-plane) and plate (out-of-plane) vibrations of a rotating annular Kirchhoff plate in the presence of a distributed frictional loading on its surface. The boundary value problem is derived from the basics of the theory of elasticity using Kirchhoff’s assumptions. This results in precise information about the coupling between the disk and the plate vibrations under the action of frictional forces. At the same time we obtain a new model, which is efficient for analytical treatment. Approximations to the stability boundaries of the system are calculated using a perturbation approach. In the last part of the paper nonlinearities are introduced leading to limit cycles due to self-excited vibrations.

Daniel Hochlenert

Papers

Minimal models for disk brake squeal

Friction Induced Vibrations in Moving Continua and Their Application to Brake Squeal

Nonlinear stability analysis of a disk brake model

Active Control of Brake Squeal Via “Smart Pads”

In- and Out-of-plane Vibrations of a Rotating Plate with Frictional Contact: Investigations on Squeal Phenomena