Acta Mathematica Vietnamica

This paper deals with the influence of nonlinear frictions to the parametric oscillations of dynamical systems described by the equation with the cubic term at the modulation depth \begin{equation} x+\omega^2x+\varepsilon(cx+dx^3)\cos \gamma t+\varepsilon\alpha x^3+\varepsilon R(x,\dot{x}),\tag{0.1} \end{equation} where $\omega, c, d,\alpha$ are constants, $\varepsilon$ is a small positive parameter, $R(x,\dot{x})$ is a nonlinear function of $x,\dot{x}$ characterized the frictions considered. Three forms of nonlinear frictions will be investigated here [1, 2]: the Coulomb friction, the turbulent one and their combination. As will be seen later in the analysis, the sign and value of parameter $d$ sharply change the motion picture and the stable regions. It must be emphasized that the equation (0.1) describes the real physical systems more precisely than the one in which $d=0$ [4, 5, 6]. The system of type (0.1) with linear friction was studied qualitatively by Minorsky [3] but no attempt has yet been made to investigate it with the Coulomb friction, turbulent one and their combination.

PDF