On a simple nonlinear system with circulatory forces

Abstract

AbstractSelf‐excited vibrations in mechanical system have been always gathering attention since being problematic for various applications, including brake squeal, aerodynamic flutter, machining chatter, and galloping transmission lines, or in some cases beneficial as in some musical instruments (ex. violin) or resonant MEMS. These vibrations can have different origins, such as ‘negative damping’ (such as in the classical Van der Pol oscillator) and/or circulatory forces (i.e. ‘follower forces’, non‐conservative positional forces). Both the negative damping and the circulatory forces can have different physical origins. It is well known that frictional forces generated between bodies in sliding contact can generate circulatory terms as well as damping and other types of forces. In particular, if one of two bodies in frictional contact moves with a constant speed or angular velocity, the other one being capable to oscillate, this implies an energy source for the oscillating part of the system. This has been identified as the cause for many self‐excited vibrations. In this presentation a very simple nonlinear two‐degree‐of‐freedom system of this general type is examined in some detail. It has recently been proposed in the literature as a paradigm for frictionally generated circulatory forces, for which the equations of motion can be derived from first principles. Other related systems have been presented in the literature before, but this newer system seems to be much simpler than the earlier ones and presents a great wealth of dynamic behavior. The nonlinear equations of motion of this 2‐DoF system always have the trivial solution and, depending on the system's parameters, also non‐trivial stationary solutions. The stability of the different stationary solutions is discussed in some detail. It was interesting to note that the increase of the friction parameter drives the system towards instability. This does however not happen, if the stiffness of both springs are identical, which practically can't be guaranteed.