Dry-Friction Oscillator with an Elastic Stop

Abstract—The work considers an oscillator with two degrees of freedom excited by dry friction. The system consists of two masses connected by linear springs. The second mass is in contact with a driving belt moving at a constant velocity and can collide with an elastic stop. Coulomb friction forces act between the mass and the belt. Several periodic motions including stick phases, slip phases and collision with the stop are discovered.

APPENDIX

$$\begin{gathered} {{L}_{i}}(t) = \Sigma {{A}_{i}}(t){{\Sigma }^{{ - 1}}}\quad (i = 1,2,3), \\ {{A}_{2}}(t) = \left( {\begin{array}{*{20}{c}} {{{S}_{1}}{\text{/}}{{\Omega }_{1}}}&0 \\ 0&{{{S}_{2}}{\text{/}}{{\Omega }_{2}}} \end{array}} \right),\quad {{S}_{j}} = \sin ({{\Omega }_{j}}t),\quad (j = 1,2). \\ \end{gathered} $$

$$\begin{gathered} {{A}_{1}}(t) = A_{2}^{'}(t),\quad {{A}_{3}}(t) = A_{2}^{{''}}(t), \\ \Sigma = \left( {\begin{array}{*{20}{c}} 1&1 \\ {{{\Sigma }_{1}}}&{{{\Sigma }_{2}}} \end{array}} \right). \\ \end{gathered} $$

The natural frequencies $({{\Omega }_{1}},{{\Omega }_{2}})$ are the roots of the characteristic equation

$$\begin{gathered} D({{s}^{2}}) \equiv det(K - I{{s}^{2}}) = 0, \\ K = \left( {\begin{array}{*{20}{c}} 1&{ - \chi } \\ { - \eta \chi }&{\eta (\chi + k)} \end{array}} \right),\quad I = \left( {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right). \\ \end{gathered} $$

The eigenvectors ${{\Phi }_{j}} = \left( {\begin{array}{*{20}{c}} 1 \\ {{{\Sigma }_{j}}} \end{array}} \right)$ $(j = 1,2)$ are defined as $(K - I\Omega _{j}^{2})$ ${{\Phi }_{j}} = 0$.