On the effect of non-smooth Coulomb damping on flutter-type self-excitation in a non-gyroscopic circulatory 2-DoF-system

This article deals with self-excited vibrations, attractivity of stationary solutions, and the corresponding bifurcation behavior of two-dimensional differential inclusions of the type $mathbf{M}mathbf{q}'' + mathbf{D}mathbf{q}' + (mathbf{K} + bar{mu}mathbf{N})mathbf{q} in-mathbf{R}operatorname{Sign}(mathbf{q}')$ . For the smooth case R=0, the equilibrium may become unstable due to non-conservative positional forces stemming from the circulatory matrix N. This type of instability is usually referred to as flutter instability and the loss of stability is related to a Hopf bifurcation of the steady state, which occurs for a critical parameter $bar{mu}= bar{mu}_{mathrm{crit}}$ . For R≠0, the steady state is a set of equilibria, which turns out to be attractive for all values of the bifurcation parameter $bar{mu}$ . Depending on $bar{mu}$ , the basin of attraction of the equilibrium set can be infinite or finite. The transition from an infinite to a finite basin of attraction occurs at the stability threshold $bar{mu}_{mathrm{crit}}$ of the underlying smooth problem. For the finite basin of attraction, its size is proportional to the Coulomb friction and inverse-proportional to $(bar{mu}- bar{mu}_{mathrm{crit}})$ . By adding Coulomb damping the notion of steady state stability for the smooth problem is replaced by the question whether the basin of attraction of the steady state is infinite or finite. Simultaneously, the local Hopf-bifurcation is replaced by a global bifurcation. This implies that in the presence of Coulomb damping the occurrence of self-excited vibrations can only be investigated with regard to the perturbation level.