Example Based Mini-Course on Mathematical Modelling: Brake Squeal

Driven by the desire for safety and automobile regulations, there has been a movement from the drum brake system, where braking force is not as predictable, to the disc brake system that still continues to this day. This shift is perhaps most noticeable in the heavy vehicle industry. However, despite the improvements in stopping distance, the open nature of the disc brake system makes it more prone to contamination and brake squeal; caused by unstable resonance of the brake and its components. On account of this, many attempts have been made to understand the dynamics of the brake and the thermoelasticity that underpins this. We shall focus on and attempt to model an experiment by J. R. Barber [1]. In an initial experiment, it was apparent that a large part of the vibration was due to only small areas of the surface being in contact with the pad at any one time. This was partly because of the initial rough nature of the surface at a microscopic level and partly because of the accentuation of the roughness of the surface due to thermal expansion. Thus, in order to model this more simply and to understand the effects of this non-uniform distribution of load and even the subsequent transfer of load, Barber [1] devised a simpler experiment. In this experiment, three cast iron pins of diameter 843 mm were attached to the loading arm to represent roughness of a surface in its most manageable form. The loading arm was then able to press the pins against a wheel that was moving at speeds of between 5 m/s and 35 m/s to simulate braking. The arm itself was