Instability of systems with a frictional point contact. Part 2: model extensions

In a companion paper, a theory was presented which allows the study of the linear stability of a class of systems consisting of two subsystems coupled through a frictional contact point. A stability criterion in terms of transfer functions was derived and used to simulate the behaviour of generic systems. In the present paper, this approach was pursued and generalized by relaxing in turn certain of the assumptions made earlier. By doing this, it is possible to catalogue systematically all the routes to instability conceivable within the scope of linearity for the class of systems considered. The additional routes to instability identified are as follows. First, the contact point was made compliant by adding a linear contact spring at the interface between the two subsystems. This feature proved to have a significant influence on stability when the contact spring stiffness takes values of the same order of magnitude or lower than that of the average structural stiffness of the system. Second, a route to instability is possible if the system structural damping possesses a slight non-proportional component. The last and most elaborate extension consisted in allowing the coefficient of friction to vary linearly with the sliding speed. Simulation results suggest that a coefficient falling with increasing sliding speed can destabilize an otherwise stable system or can make it even more unstable. In accordance with previous results, a coefficient of friction rising with the sliding speed tends to make a system more stable, although this is not systematic. The theory presented here allows these possible routes to instability to be combined, so that data from vibration measurements or modelling and from frictional measurements can be used directly to predict the region of instability in parameter space.