Bifurcation of the roots of the characteristic polynomial and the destabilization paradox in friction induced oscillations

Abstract

Paradoxical effect of small dissipative and gyroscopic forces on the stability of a linear non-conservative system, which manifests itself through the unpredictable at first sight behavior of the critical non-conservative load, is studied. By means of the analysis of bifurcation of multiple roots of the characteristic polynomial of the non-conservative system, the analytical description of this phenomenon is obtained. As mechanical examples two systems possessing friction induced oscillations are considered: a mass sliding over a conveyor belt and a model of a disc brake describing the onset of squeal during the braking of a vehicle.Proucava se paradoksalni efekt malih disipativnih i ziroskopskih sila na stabilnost linearnog nekonzervativnog sistema. Ovaj se manifestuje, na prvi pogled, nepredvidivim ponasanjem kriticne nekonzervativne sile. Analiticki opis ove pojave se dobija analizom bifurkacije visestrukih korenova karakteristicnog polinoma nekonzervativnog sistema. Dva sistema koji poseduju vibracije izazvane trenjem se posmatraju kao mehanicki primeri i to: masa koja klizi preko konvejerske trake kao i model disk kocnice koji opisuje pocetak cviljenja tokom kocenja vozila.