Multidegree of Freedom Systems with Dry Friction

In engineering practice, dry friction often causes undesirable side effects such as the generation of self-sustained oscillations also called stick—slip vibrations.

Torsional stick—slip vibrations for example have been observed in drill strings, that are used in rotary drilling for oil and gas. In rotary drilling, deep wells are drilled with a rock-crushing tool, called a drill bit. The drill bit is driven by an electric motor at the surface, whose torque is transmitted by the (long and flexible) drill string. Between the drill bit and parts of the drill string on the one hand, and the rock on the other hand, dry friction occurs. When a drill string is undergoing a torsional stick—slip vibration, the top is rotating at a constant speed, whereas the speed at the drill bit varies between zero (the stick phase) and a speed, much higher than the speed at the top (the slip phase), which can lead to damage of drilling components.

In literature, different dry friction models can be found, of which Coulomb's friction law is the most familiar. Because of the discontinuous nature of dry friction, highly nonlinear differential equations are found. In this study, periodic solutions of systems, experiencing dry friction are determined, using a module STRDYN of the finite element code DIANA. This module offers several numerical algorithms for investigating multi-degree-of-freedom finite element models with local nonlinearities. To avoid numerical complications a smooth approximation of the discontinuous friction model will be used. The techniques will be applied to a 2-masses-on-belt model with dry friction, taken from literature. The friction model is approximated, using an arctan-function. Very good correspondence is found between the DIANA results and results which could also be found in literature but also some new solutions will be discussed.