Instability of systems with a frictional point contact. I - Basic modelling

The linear stability is investigated of systems which contain a sliding frictional contact at a single point. A condition for instability is found, in terms of the transfer functions of the two systems at the point of contact. This condition is explored for generic systems, to establish the circumstances under which instabilities might be expected. A major conclusion is that if the coefficient of friction is assumed to be constant, then at least one mode of one or other of the contacting systems must have a displacement at the contact with a particular pattern of signs. If such a mode exists then instability is possible, depending on the value of the coefficient of friction and on the frequencies and mode shapes of the other modes of the system. Stability boundaries are shown to be extremely sensitive to distribution of damping in the system, suggesting that damping might be one of the causes of the typical 'capricious' behaviour of friction-instability experiments. Systems consisting of three modes are studied in detail. This is shown to be an important case since much of the behaviour of a system consisting of many modes can be understood by breaking it down into clusters of three. In a subsequent paper, some of the assumptions made here will be relaxed so as to catalogue systematically all the possible routes to instability within linear theory.