Bifurcations in autonomous mechanical systems under the influence of joint damping

Abstract

This contribution deals with the impact of joint damping on two classes of stability problems which are often found in engineering problems. In a first part, the principle structure of the equations of motion is derived when joints are modeled using Masing-, Prandtl- and Coulomb-elements. For these general formulations, some fundamental statements concerning stability and attractiveness of steady-state solutions may be given for large amplitudes and configurations which are not too close to the linear stability threshold. The second part focuses on analyzing the behavior at small amplitudes and in the vicinity of the linear stability threshold in more detail: to this end, a static stability problem (buckling) and two oscillatory self-excitation mechanisms (negative damping, non-conservative coupling) are discussed. For all considered problems, adding joint damping transforms the equilibrium points into sets of equilibria and bifurcations of the non-smooth problems occur near the linear stability threshold. Concerning the buckling problem adding joint damping does not alter the behavior fundamentally: still a local bifurcation occurs and attractiveness or instability of equilibrium solutions is preserved. In contrast, the oscillatory instability examples are strongly influenced by joint damping: here, global discontinuous bifurcations may occur. Besides the joint friction also the joint-stiffness may play a crucial role, since it determines whether attractive solutions in or about the equilibrium set exist. It is found that only in some cases a linear stability analysis of the corresponding system without joints may give correct indications on the behavior: consequently, neglecting joint-damping in stability analyses may lead to wrong results concerning self-excitation.