%0 Journal Article
%T Phase Portraits of the Autonomous Duffing Single-Degree-of-Freedom Oscillator with Coulomb Dry Friction
%A Nikola Jak£¿i£¿
%J Advances in Acoustics and Vibration
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/465489
%X The paper presents phase portraits of the autonomous Duffing single-degree-of-freedom system with Coulomb dry friction in its parameter space. The considered nonlinearities of the cubic stiffness and Coulomb dry friction are widely used throughout the literature. It has been shown that there can be more than one sticking region in the phase plane. It has also been shown that an equilibrium point occurs at the critical combinations of values of the parameters and which gives rise to zero eigenvalue of the linearised system. The unstable limit cycle may appear in the case of negative viscous damping ; . 1. Introduction It is beneficial, if not essential, to know the system behaviour prior to using it in any other application or analysis. Phase portraits of the autonomous Duffing single-degree-of-freedom oscillator with Coulomb dry friction are studied here in a complete parameter space. In order to understand the phase portraits of the nonlinear system in the next section, it is necessary to review the phase portraits of the linear system at different parameter values. Without loss of generality, the autonomous linear single-degree-of-freedom (SDOF) system¡¯s equation of motion can be written as follows: where represents viscous damping in the system and represents system¡¯s stiffness. By defining the discriminant the system eigenvalues are as follows: and . If then and the response is . If then and the response is . Finally if then the eigenvalues are complex conjugated, , and the response is . and are constants dependent on the initial conditions. The equation of motion (1) can be represented as system of first-order differential equations: where and and finally is the phase velocity. Hence, There are only two possibilities regarding the number of the equilibrium points [1]. Either there is only one equilibrium point at the origin of the phase plane or there is infinity number of equilibrium points situated at zero velocity abscissa of the phase plane. Table 1 presents all possibilities of the phase portraits of the linear system. Table 1: Equilibrium points of the linear system. Some aspects of the viscous and frictional damping acting together are presented in [2] on the linear system with the Coulomb dry friction. Negative viscous damping is normally associated with self-exciting systems. They are most commonly found in solid-fluid interactions with the fluid induced vibrations [3, 4], but certainly not exclusively [5, 6]. There are two main reasons for using the self-excited model: either the excitation itself is too difficult to model or only the
%U http://www.hindawi.com/journals/aav/2014/465489/