Derivation of Friction Torque-Speed Relation Due to Viscoelastic Contact in a Two-Disk Brake

Using a nonlinear model of a two disk brake system, coupled equations of motion are found for their frictional interaction. The mathematical formulation relates the tribological events at micron-scale and the macroscopic scale vibration response of a two-disk brake system. This is accomplished by a visco-elastic account of interaction at the micron scale, its statistical quantification through the approximate analytical representation of the statistical expectation of contact force and the introduction of the contact force into the macro-scale dynamics of the two-disk system. Steady-state analysis of the system establishes the relation between friction torque and speed and supports observed behavior of many mechanical systems with friction. It is shown that, as a result of coupling of the macro-system’s dynamics and contact, there are combinations of parameters at the micro-and macro-scale that yield negative slope in friction torque/sliding speed relation, a well known source of dynamic instability. This results in an effective negative damping that tends to decrease with decrease in the normal load and/or increase in structural damping of the system.