The anisotropy of a friction force is proved to be an important factor in various contact problems. We study the dynamical behavior of thin plates with respect to symmetric and asymmetric orthotropic friction. The terminal motion of plates with circular and elliptic contact areas is analyzed. The evaluation of friction forces for both symmetric and asymmetric orthotropic cases is shown using an analytic approach. Regular pressure distribution is considered. Differential equations are formulated and solved numerically for a number of initial conditions. The method used gives more accurate results compared to the previous study. Examples show the significant influence of friction force asymmetry on motion.