Friction Hysteresis in Simulink - Dahl's Model

A Simulink block diagram for Dahl’s model of friction hysteresis in the pre-sliding regime.

Dahl’s model is widely used to represent friction hysteresis at very small displacements. Hysteretic friction refers to the position or path- dependence of friction in the pre-sliding regime which arises due to microscopic sliping of the surface asperities [1]. This type of friction can be visualized as the stretching of 'soft springs' between surfaces in relative motion; the springs are linearly elastic at small micro-displacements and yield plastically at larger displacements [2].

The Dahl model was originally applied to the simulation of ball bearing friction which demands precise position control. The model features a friction slope function with ‘rest stiffness’ and ‘shape/exponential’ parameters to represent material-specific force-deflection properties [2]. Representative figures are provided below to demonstrate the model performance. Note - the provided Simulink block diagram represents equation #2 from Dahl's original paper [2].

Figure 1: The Effect of the Rest Stiffness Parameter on Dahl Friction for a sinusoidal input velocity with a frequency and amplitude of 1. The shape/exponential factor was held constant at i=1 for ductile materials. The Coulombic friction was held constant at Fc=0.9. Simulation time t=10 s.

Figure 2: The Effect of the Shape/Exponential Parameter on Dahl Friction for a sinusoidal input velocity with a frequency and amplitude of 1. The rest stiffness parameter was held constant at σ=1. The Coulombic friction was held constant at Fc=0.9. Simulation time t=10 s.

References:

[1] Al-Bender, F. Fundamentals of Friction Modeling. 2010. ASPE- The American Society of Precision Engineering.

[2] Dahl, P.R., Solid Friction Damping of Mechanical Vibrations.AIAA Journal, 1976. 14(12): p. 1675-1682.