Nonlinear Dynamics 20: 221–246, 1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands. Dynamics of Oscillators with Strongly Nonlinear Asymmetric Damping S. NATSIAVAS and G. VERROS Department of Mechanical Engineering, Aristotle University, 54006 Thessaloniki, Greece (Received: 31 December 1998; accepted: 14 May 1999) Abstract. Dynamics of a class of strongly nonlinear single degree of freedom oscillators is investigated. Their common characteristic is that they possess piecewise linear damping properties, which can be expressed in a general asymmetric form. More specifically, the damping coefficient and a constant parameter appearing in the equation of motion are functions of the velocity direction. This class of oscillators is quite general and includes other important categories of mechanical systems as special cases, like systems with Coulomb friction. First, an analysis is presented for locating directly exact periodic responses of these oscillators to harmonic excitation. Due to the presence of dry friction, these responses may involve intervals where the oscillator is stuck temporarily. Then, an appropriate stability analysis is also presented together with some quite general bifurcation results. In the second part of the work, this analysis is applied to several example systems with piecewise linear damping, in order to reveal the most important aspects of their dynamics. Initially, systems with symmetric characteristics are examined, for which the periodic response is found to be symmetric or asymmetric. Then, dynamical systems with asymmetric damping characteristics are also examined. In all cases, emphasis is placed on investigating the low forcing frequency ranges, where interesting dynamics is noticed. The analytical predictions are complemented with results obtained by proper integration of the equation of motion, which among other responses reveal the existence of quasiperiodic, chaotic and unbounded motions. Keywords: Coulomb friction, asymmetric damping, exact periodic motions, stability. 1. Introduction The present study investigates the dynamics of a class of strongly nonlinear mechanical systems. These systems are harmonically excited single degree of freedom oscillators with piecewise linear damping properties and dry friction. They model the dynamic response of many mechanical components, including oscillation of turbine blades, rotating members of ro- bots, vibration dampers, braking systems, clutches and belts of automotive engines as well as of metal cutting processes (see [1–10] and references therein). In particular, systems involving classical Coulomb damping fall within the category of the systems examined. Moreover, one of the most important categories of systems with variable asymmetric viscous damping are those encountered in car suspension mechanisms [11, 12]. The prediction of the steady-state response of these mechanical systems is essential in a large number of engineering areas, as evidenced by the previous research work on the subject. However, the great majority of these studies investigated response of oscillators with symmetric characteristics and provided useful information on several typical phenomena associated with their response, like stick-slip sequences. These phenomena are due to the presence of the dry friction terms in the equation of motion, which introduce discontinuities in the acceleration and the friction force. The main objective of the present study is to develop and apply an analysis that will im- prove the understanding on the dynamics of harmonically excited oscillators with asymmetric