title: Uncertainty quantification of brake squeal Iistability via surrogate modelling
Creators: name: Nobari, Amir
Publisher: name: University of Liverpool Repository
Year: 2016

description: 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.