fade, since a reduction in pad friction material height or thickness also reduces pad force. In contrast, for a brake without self-assist, such as a conventional disc brake, a loss of pad friction material does not change the pad force, so there is no necessary loss in the brake torque reaction for a given amount of input force. The self-assist mechanism affects the water pump and the amount of fade. For example, the Ausco Lambert and Murphy brakes have self-assist roughly proportional to pad friction, so total braking is reduced as roughly the square of the loss in friction. Many other self-assist designs, such as band brakes and many common drum brakes, have exponential self-assist, described by , where is the natural logarithm base, is the coefficient of friction between shoes and drum, and is the angle of engagement between shoes and drum. A small change in friction causes an exponential change in self assist. In many common brakes, a slight increase in friction can lead to wheel lockup with even light application. For example, on damp mornings, drum brakes can lock on first application, skidding to a stop even after the brake pedal has been released. Conversely, a slight decrease in friction can lead to severe brake fade. Factors contributing to fade Brake fade failures can cascade. For example, a typical 5-axle truck/trailer combination has 10 brakes. If one brake fades, brake load is transferred to the remaining 9 brakes, causing them to work harder, get hotter, and thus fade more. Where fade is non-uniform, fade may cause a vehicle to swerve. Because of this, heavy vehicles often use disproportionately weak brakes on steered wheels, which hurts the stopping distance and causes brakes on non-steered wheels to work harder, worsening fade. An advantage of low-fade brakes such as disc brakes is steered wheels can do more braking without causing brake steer. Brake fade typically occurs during heavy or sustained braking. Many high-speed vehicles use disc brakes, and many European heavy vehicles use disc brakes. Many U.S. and third-world heavy vehicles use drum brakes due to their lower purchase price. On heavy vehicles, air drag is often small compared to the weight, so the brakes dissipate proportionally more energy than on a typical car or motorcycle. Thus, heavy vehicles often need to use engine compression braking, and slow down so braking energy is dissipated over a longer interval. Recent studies