Squeal reduction of a disc brake system with fuzzy uncertainties

Automotive brake squeal has become one of the major concerns in the automotive industry. The researches on suppressing brake squeal are of great practical significances. However, most of the existing researches have not taken into account the parameters uncertainties, although it is well known that uncertain factors widely exist in the brake systems. To reduce disc brake squeal more effectively, a practical approach for the stability analysis and improvement of an automotive disc brake system with fuzzy uncertainties is proposed. In the proposed approach, the uncertainties associated with friction coefficient, material properties and loading properties are taken into consideration, and the uncertain parameters of the brake system are represented by fuzzy numbers. The brake system stability is investigated by performing complex eigenvalue analysis (CEA), and response surface methodology (RSM) is employed to approximate the implicit relationship between the dominant unstable mode and system parameters. Then, the stability analysis model of the brake is constructed based on RSM, CEA and fuzzy analysis. As a numerical example, the stability analysis of a commercial disc brake system with fuzzy uncertainties is carried out, and the influences of different uncertain parameters on system stability are investigated. The analysis results show that the stability of the fuzzy brake can be improved effectively by increasing the specific modulus of back plate. The proposed approach can be considered as a potential method for squeal reduction of automotive disc brake systems under fuzzy case

Uncertainty quantification of brake squeal Iistability via surrogate modelling

Noise, vibration and Harshness (NVH) of automotive disc brakes have been an active research topic for several decades. The environmental concerns, on one hand, and the rising customer expectations of their car quality, on the other hand, have made NVH of brakes an important issue for car manufacturers. Of different types of noise and vibration that a brake system may generate, squeal is the main focus of the current study. Brake squeal is an irritating high-frequency noise causing a significant warranty cost to car manufacturers. There are a number of reasons leading to squeal noise either at the end of production or during usage and services. Of these reasons, it is believed that manufacturing variability, several sources of uncertainty (such as friction and contact) and diverse loading cases have the most contribution in this problem. Car manufacturers are then recently encouraged to look into the uncertainty analysis of the brake systems in order to cover the influence of these variations on brake designs. The biggest hurdle in the uncertainty analysis of brakes is the computational time, cost and data storage. In general, stochastic studies are done on the premise of deterministic analyses of a system. As the deterministic analyses of brake squeal instability essentially involve a great deal of computational workload, their stochastic (non-deterministic) analyses will be consequently very expensive. To overcome this issue, the method of surrogate modelling is proposed in this study. Briefly speaking, surrogate modelling replaces an expensive simulation code with a cheap-to-evaluate mathematical predictor. As a result, instead of using the actual finite element model of a brake for statistical analyses, its replacement model will be used alternatively. There are three main advantages in surrogate modelling of brakes. First of all, it paves the way of structural modification of brakes, which are conventionally done for reducing squeal propensity. Secondly, structural uncertainties of a brake design can cost-effectively be propagated onto the results of the stability analysis. Thereafter, instead of making a single design point stable, a scatter of points should meet the stability criteria. Finally, the reliability and robustness of a brake design can be quantified efficiently. These two measures indicate the probability of unstable vibration leading to squeal noise for a brake design. Accordingly, car manufacturers will be able to estimate the cost of warranty claims which may be filed due to this particular issue. If the probability of failure which is calculated for squeal propensity is significant, surrogate modelling helps come up with a solution during the design stage, before cars go into production. In brief, two major steps must be taken toward constructing a surrogate model: making a uniform sampling plan and fitting a mathematical predictor to the observed data. Of different sampling techniques, Latin hypercube sampling (LHS) is used in this study in order to reduce the amount of computational workload. It is worth mentioning that the original LHS does not enforce the uniformity condition when making samples. However, some modifications can be applied to LHS in order to improve the uniformity of samples. Note that the uniformity of samples plays a crucial role in the accuracy of a surrogate model. A surrogate model, in fact, is built on the premise of the observations which are made over a design space. Depending on the nonlinearity of the outputs versus the input variables and also depending on the dimensions of a design space, different mathematical functions may be used for a surrogate predictor. The results of this study show that Kriging function brings about a very accurate surrogate model for the brake squeal instability. In order to validate the accuracy of surrogate models, a number of methods are reviewed and implemented in the current study. Finally, the validated surrogate models are used in place of the actual FE model for uncertainty quantification of squeal instability. Apart from surrogate modeling, a stochastic study is conducted on friction-induced vibration. Statistics of complex eigenvalues of a simplified brake models are studied under the influence of variability and uncertainty. For this purpose, the 2nd order perturbation method is extended to be applicable on an asymmetric system with non-proportional damping. The main advantage of this approach is that the statistics of complex eigenvalues can be calculated in just one run, which is massively more efficient than the conventional techniques of uncertainty propagation that use a large number of simulations to determine the results

Interval Fuzzy Segments

In this paper, we bring together two concepts related to uncertainty and vagueness: fuzzy numbers and intervals. With them, we build a new structure whose elements we call interval fuzzy segments. We have undertaken this based on the conviction that the fuzzy numbers are a correct representation of the real numbers under situations of indeterminacy. We also believe that if it makes sense to consider the set of real numbers between two real bounds, then it also makes sense to consider the set of all the fuzzy numbers between two fuzzy number bounds. In this way, we extend the concept of real interval to the concept of interval fuzzy segment defined by two fuzzy bounds and a transition mapping that leads from the lower fuzzy bound to the upper fuzzy bound and this transition mapping generates the set of all the fuzzy numbers comprised between those fuzzy bounds. At the same time, this transition mapping brings the concept of interval fuzzy segment closer to the concept of line segment

A Possibilistic Approach for the Prediction of the Risk of Interference between Power and Signal Lines Onboard Satellites

This work presents a hybrid random/fuzzy approach for uncertainty quantification in electromagnetic modelling, which combines probability and possibility theory in order to properly account for both aleatory and epistemic uncertainty, respectively. In particular, a typical intrasystem electromagnetic-compatibility problem in aerospace applications is considered, where some parameters are affected by fabrication tolerances or other kinds of randomness (aleatory uncertainty) and others are inherently deterministic but unknown due to human's lack of knowledge (epistemic uncertainty). Namely, a differential-signal line in a satellite is subject to crosstalk due to a nearby dc power line carrying conducted emissions generated by a dc-dc converter in a wide frequency range (up to 100 MHz). The nonideal features of the signal line (e.g., weak unbalance of terminal loads) are treated as random variables (RVs), whereas the mutual position of signal and power line is characterized by possibility theory through suitable fuzzy variables. Such a hybrid approach allows deriving a general and exhaustive description of uncertainty of the target variable of interest, that is, the differential noise voltage induced in the signal line. The obtained results are compared versus a conventional Monte Carlo simulation where all parameters are treated as RVs, and the advantages of the proposed approach (in terms of completeness and richness of information gained about sensitivity of results) are highlighted

Brake-clutch squeal prediction and supression

This work studies the high frequency noise in brake–clutch systems known as squeal. The work is divided into two parts, the first is focused on the development of both a theoretical and experimental simplified model of a brake–clutch and the second is centred on squeal modelling in the real system. For the simple model, on the theoretical side, a FE model was developed including anisotropic material properties, pressure and speed dependent friction coefficient and friction damping. The pertinent characterisation tests were performed as needed. On the experimental side, squeal tests were performed in the test bench in order to check the ability of the system for squeal prediction. Once the model was thought as accurate enough, a methodology to decide over point structural modifications for squeal suppression based on the receptance function was designed. Using this process squeal was successfully eliminated from the simplified model both theoretically and experimentally. In the second part the modelling of squeal in a real brake–clutch system was tackled. With this objective a FE model of the whole system was developed and its validity was checked first by EMA and after, comparing the experimental squeal frequencies with the ones predicted by simulation. To finish, the methodology for structural modifications previously designed was applied to the system and several theoretical modifications were proposed and studied.Lan honetan balazta–enbrage unitate konbinatuetan agertzen den squeal izeneko frekuentzia altuko zarata aztertzen da. Lana bi zatitan banatuta dago: alde batetik, balazta–enbragearen eredu sinplifikatu bat garatu da, bai teorikoa zein esperimentala; bestetik, squeala sistema errealean modelizatu da. Eredu sinplean, alde teorikoari dagokionez, elementu finitutako eredu bat garatu da. Eredu honek marruskadura materialaren propietate anisotropoak, presio eta abiaduraren menpeko marruskadura koefizientea eta marruskadurak eragindako moteltzea kontutan hartzen ditu. Propietate hauek saiakuntza independentetan neurtu dira. Alde esperimentalean, squeal saiakuntzak burutu dira ereduak squeala aurreikusteko duen gaitasuna balioztatzeko asmoz. Behin eredua nahiko zehatza izanik, zarata kentzeko aldaketa estrukturalak proposatu eta baloratzeko metodologia bat diseinatu da. Proposatutako metodoa errezeptantzia funtzioan oinarrituta dago. Metodo honi esker squeala eredu teorikoan lehenengo eta ondoren saiakuntza bankuan kentzea posible izan da. Bigarren atalean, balazta–enbragearen modelizazioari ekin zaio squealari dagokionez. Helburu honekin, sistema osoaren eredu bat garatu da eta lehengo AME bidez eta gero squeal frekuentzia esperimentalak ereduak aurreikusitakoekin konparatuz balioztatu da. Azkenik, aldaketa estrukturalak proposatzeko metodologia sistema errealari aplikatu zaio eta aldaketa teoriko batzuk planteatu eta aztertu dira.El presente trabajo estudia el ruido de alta frecuencia que se da en los freno-embragues conocido como squeal. El trabajo se divide en dos partes, la primera enfocada en el desarrollo de un modelo simple tanto teórico como experimental del freno–embrague y la segunda centrada en la modelización del squeal en el sistema real. En lo referente al modelo simple, en la parte teórica se desarrolló un modelo de elementos finitos que incluía propiedades anisótropas para el material de fricción, un coeficiente de fricción dependiente de la presión y la velocidad y amortiguamiento por fricción. Estas propiedades se caraterizaron en los correspondientes ensayos independientes cuando fue necesario. En la parte experimental, se llevaron a cabo ensayos de squeal en banco con la idea de verificar la capacidad del modelo para predecir el squeal. Una vez que el modelo se consideró lo suficientemente exacto, se desarrolló una metodología para proponer y valorar modificaciones estructurales para la supresión de squeal. El método propuesto está basado en la función de receptancia. Gracias a este proceso fue posible eliminar el squeal del modelo primero en la teoría y a continuación en el banco de ensayos. En la segunda parte se aborda la modelización de squeal en un freno–embrague real. Con esto en mente se desarrolló un modelo de elementos finitos del sistema completo y se validó en primer lugar mediante AME y a continuación comparando las frecuencias experimentales de squeal con las predichas por la simulación. Por último, la metodología desarrollada para modificaciones estructurales se aplicó al sistema real y se propusieron y analizaron varias modificaciones teóricas

Non-smooth Dynamic Behaviour of Friction-induced Self-excited Vibration

Friction is everywhere and important in our daily life as well as in industry. In general, dry friction acts as a resistance to the relative motion and dissipates energy of a system; however, under certain conditions, it can cause self-excited vibration of a system, which is known as friction-induced vibration. Friction-induced vibration can potentially cause problems like wear, fatigue failure, and noise, among which brake squeal is a typical engineering problem. As a comprehensive understanding of friction-induced vibration has not been achieved, friction-induced vibration is still a challenging research topic. The aim of this research is to carry out a theoretical study of the dynamic behaviour of nonlinear/non-smooth friction-induced vibration of phenomenological mechanical models. Discrete and continuous mechanical models with dry friction that involve nonlinear contact stiffness, stick-slip motion, or separation and reattachment at the contact interface are proposed, and numerical simulations of the transient dynamic behaviour of the non-smooth frictional systems are implemented. Their complex dynamic behaviour and the influences by various system parameters are predicted. In addition, a reduction strategy for the complicated frictional systems is presented and validated via theoretical and experimental results, which is a preliminary step in analysing complicated systems (real structures) with nonlinear/non-smooth friction behaviour in future research. The main objectives of the research work reported in this thesis are: 1. To carry out the transient dynamic analysis of non-smooth friction-induced vibration. A varying time-step numerical algorithm, which combines Runge-Kutta method that is specifically for the second-order differential equation of motion and the bisection method, is proposed (Chapter 3). This algorithm ensures the accuracy of the results of the non-smooth vibration in which the motion states keep switching among distinct motion states of separation and reattachment, or stick and slip, which is testified by providing the same results of a classic non-smooth stick-slip vibration of a single-degree-of-freedom model. 2. In consideration of improving previous research on theoretical mechanical models with the assumptions that the contact stiffness is linear or separation is ignored, a nonlinear 2-degree-of-freedom slider-on-moving-belt model developed from Hoffmann’s model and the theoretical formulations are proposed (Chapter 4), in which a cubic contact spring is included; loss of contact (separation) at the slider-belt interface is allowed and importantly reattachment of the slider to the belt after separation is also considered. The stability and dynamic behaviour of the system are investigated. Complex eigenvalue analysis (CEA) indicates that the roles of the preload and the nonlinear stiffness on the stability of the nonlinear system are not monotonous. Transient dynamic analysis (TDA) shows that separation and reattachment could happen. Ignoring separation between bodies in sliding frictional contact in vibration is unsafe as this may underestimate the vibration amplitude, and predicts incorrect effects of the key parameters on the vibration, thus considering separation is very important. Moreover, frequency domain results show the necessity of implementing both CEA and TDA in the study of nonlinear friction-induced vibration and the importance of considering separation from the frequency domain point of view. Finally, non-smooth Coulomb’s law of friction is introduced in the nonlinear 2-degree-of-freedom (2-DoF) slider-belt model. Numerical results show that diverse dynamic behaviour of this 2-DoF system with nonlinearity/non-smoothness can be generated when both of the stick-slip and mode-coupling instability are involved. 3. Separation, reattachment and impact are considered in the study of friction-induced vibration of a system having an elastic disc, excited by the in-plane stick–slip vibration of a moving mass-damper-spring slider attached to a rigid wall that is dragged around on the disc surface at a constant rotating speed (Chapter 5). Theoretical formulations and the numerical procedure for the current non-smooth system are developed. Numerical results show that separation and reattachment could occur in a low speed range well below the critical disc speed in the context of a constant rotating load. Poincare maps of the system of the two distinct cases (considering separation and ignoring separation) are plotted which exhibit the diversity of nonlinear dynamic behaviour of the system and the importance of considering separation. Furthermore, the roles of the key system parameters on the vibration are investigated. Time-frequency analysis reveals the time-varying properties of this system and the contributions of separation and in-plane stick-slip vibration to the system frequencies. One major finding is that ignoring separation, as is usually done, often leads to very different dynamic behaviour and possibly misleading results. 4. Based on the idea of mode synthesis method, a reduction strategy for complicated frictional systems is put forward, in which the natural contact interfaces and the tangential friction force are involved, and its applications and experimental validation are presented (Chapter 6). Firstly, its application to a theoretical multi-degree-of-freedom model with linear contact verifies the accuracy and feasibility of the strategy. The influence of the system parameter, and the mode number that is used in the reduced model on the stability of the reduced model are investigated. The reduced model is capable of preserving the key features (bifurcation of the eigenvalue and unstable frequencies) of the original model. Furthermore, a specific reduction strategy, for a real pad-on-disc test rig and its corresponding finite element model which involve direct contact of the interface, is proposed. The results of the reduced model with a small number of modes of the substructures correlate fairly well with theoretical results of the full model and the test results in terms of predicting mode-coupling instability and unstable frequencies, which validates this promising method for future work on friction-induced vibration of complicated frictional systems or real structures