La ciencia de la clasificación 3.6 Las relaciones filogenéticas

There are a number of issues associated with transient analysis of FE brake models in Abaqus. These issues are briefly discussed here. First of all, there is no extensive study on the damping of friction material in the literature. In a brake system, friction material contributes mostly to damping. Hence, the available options for modelling damping such as Rayleigh damping, is not sufficiently accurate for friction materials. Secondly, as illustrated in section ‎5.5, the nonlinear terms play a significant role in the amplitude of a limit cycle motion. What mostly behaves nonlinearly in the brake squeal problem is friction material. Unfortunately, in Abaqus it is only possible to define nonlinear isotropic material and there

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is no option to define nonlinear behaviour in different directions of a single material. As a result, modelling a nonlinear friction material is not yet feasible. Future advances may resolve this issue.

Finally, as discussed earlier, the computational workload of a dynamic transient analysis is the biggest hurdle. Abaqus/Standard uses implicit methods to find the numerical solutions of differential equations. However, for highly nonlinear problems or problems containing many contact interfaces, users have been provided with Abaqus/Explicit. To get an idea about how small the step size must be for the brake squeal problem, it is useful to review the stability limit of Abaqus/Explicit. This limit is defined based on the highest frequency of the system (Abaqus documentation). The stability limit is ⁄ when there is no damping in

the system. Therefore, it can be seen that in order to keep the contribution of high frequencies in the solution, the step size must be very small which lead to a very long simulation.

Moreover, the interval of integration (time span) is another significant issue. A transient analysis can be done in a few days if a few seconds of the response is needed. However, if the response does not tend to a limit cycle quickly, it will massively increase the computational workload. This situation is exacerbated when it is known that many attempts must be made until a limit cycle motion is found. In other words, all runs of transient analysis do not necessarily converge to a limit cycle motion.

Taking these factors into consideration, no doubt will remain that CEA is the right choice for the purpose of this study.

5.11.

Conclusions

The implementation of CEA and transient analysis are illustrated in this chapter. The pros and cons of each method are fully discussed and it is unveiled why CEA has been selected for the purpose of this study. The results of CEA on the full brake assembly are presented and validated with dynamometer test data. An example of structural modification on brakes for reducing the squeal index is also given here.

Moreover, the results of CEA of an asymmetric low-order system with non-proportional damping are presented in this chapter. This model will be used later for extending the perturbation method to be applicable on friction-induced vibration.

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6.

Non-deterministic Approaches

Recently particular attention has been given to uncertainty analysis of brake systems by car manufacturers. The deterministic approach of either CEA or transient analysis is only carried out for a set of design parameters and there is no information about the effect of variations of these parameters on the outputs. Although a brake design may remain stable at the baseline design point, the reliability and robustness of the design in terms of the likelihood of unstable vibration leading to squeal noise is still unclear. It is believed that variability and uncertainty are responsible for squeal noise in many cases and cause a significant warranty cost to car manufactures.

In this chapter, a brief review of different methods of uncertainty analysis is presented and then it is explained why surrogate modelling has been selected for the purpose of this study. This chapter may be taken as the transition from deterministic approaches to statistical approaches toward the brake squeal problem.

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6.1.

Variability and uncertainty

There are two terms which are frequently used in this study: variability and uncertainty. In the literature, structural uncertainties are classified into two groups: aleatory and epistemic (Möller and Beer, 2008). Yet it is not always possible to make a distinction between them. Aleatory uncertainty, also called irreducible uncertainty or variability, refers to the random nature in the system properties, which originates from manufacturing processes. Variations of material properties, component geometries and assemblies are the most typical examples of this type of uncertainty. Epistemic uncertainty, known as reducible uncertainty, is mainly due to the lack of information. This type of uncertainty is called reducible since future advances and/or investigations can provide new insights into the problems which are not fully understood yet.

In spite of the extensive investigations that have been conducted so far, predicting the behaviour of friction, contact and wear remains very complicated. In addition to the tribological interactions, the degree of uncertainty in a friction-induced vibration problem is increased by the thermal effects, humidity, diverse loading cases, etc. Variability also imposes a considerable level of uncertainty to the problem. The growing importance of variability in brake noise is well explained in (Day, 2014):

“One of the most interesting (and frustrating) features of brake noise is its fugitive nature; it is well known that even across different vehicles of the same make and model with the same brake installation, some exhibit brake noise, while others do not. Given the complexity of the underlying physical principles of brake noise generation that have been outlined above and the emphasis on instability, it appears that relatively small changes in one or more of the many design and operational parameters associated with friction brakes can tip a brake installation from quiet to noisy. Operational parameters such as temperature, speed, actuation pressure and humidity have all been mentioned, and many vehicle and brake manufacturers have studied the possible influence of variation arising from manufacturing tolerances. The role of the pad assembly in disc brake noise is becoming increasingly well understood, and the nature of the abutment contact is influenced by the dimensional tolerances between the pad back-plate and the calliper abutments, which may change over time as a result of wear, plastic deformation, and the build-up of debris and corrosion. Brake pads are generally closely toleranced in terms of thermo-physical properties during and immediately after manufacture, but once they enter the customer domain, variability increases. The manufacture of brake discs and drums is closely controlled but achieving and maintaining close tolerances on the internal geometry of ventilated brake discs, for example, is costly. Examination by the author of samples of ventilated brake discs within the same batch and

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across different batches has indicated a 3 variation of in resonant frequencies up to 6 kHz, with pair (doublet) modes separated by amounts ranging from 10 to 60 Hz. The close examination of components from examples of brakes that exhibit noise, and comparison with nominal dimensions and values, is thus a useful exercise.”

Ignoring the effect of variability and uncertainty in CEA causes underestimation or overestimation of the number of unstable modes. Therefore, in order to make more reliable predictions of unstable modes, it is necessary to perform an uncertainty analysis.