In this paper, we analyze the problem on the verti� cal stability and instability of equilibrium locations of a wheel carriage. The consideration is restricted to the case of the uniform and rectilinear motion of the car� riage in the presence of rollingfriction forces. The coefficients of rolling friction as functions of the veloc� ity of motion are supposed to be known. The necessary and sufficient conditions for system parameters corre� sponding to stable equilibrium locations are found. 1. DESCRIPTION OF THE MODEL We consider the model of a carriage (see figure) con� sisting of three solid bodies, namely, two pairs of wheels 1 and 2 of masses m1 and m2 and of radii r1 and r2, respec� tively, with the centers of mass at the points O1 and O2. In addition, the carriage body O1O2C is of mass m, and its central radius of inertia is ρ, whereas the center of mass corresponds to the point C. The system moves translationally along the positive direction of the Ox axis. The carriage motion is studied in the vertical plane and in the immovable coordinate system Oxz, with the Ox axis coinciding with the plane of support (the road). The Oz axis is directed vertically and oppo� sitely to the force of gravity. The carriage body is con� nected to the wheels 1 and 2 at the points O1 and O2 by means of perfect cylindrical hinges. The carriage body is equipped with an engine that generates the moment M1 applied clockwise to the drive wheel 1 (correspondingly, the reactive moment M1 = -M1 directed counterclockwise acts on the car� riage body. It is assumed to be an absolutely rigid body, whereas the wheels are supposed to be deformable (with the energy dissipation) in the vertical direction. This is modeled by suspending (above the road) the centers of mass O1 and O2 of the wheels by springs and dampers having the stiffness and damping coefficients c 1 , γ 1 and c 2 , γ 2 , respectively. The assumption on the deformable wheels (with the energy dissipation) natu� rally results in the appearance of resistance moments Mk1 and Mk2 of friction to their rolling. These moments equal to f1N1 and f2N2 are directed counter� clockwise, i.e., oppositely to rolling. Here, N1 and N2 are normal reactions at points of wheel contacts with the road, and f1 and f2 are coefficients of rolling friction for the wheels, respectively. These coefficients are dependent on different reasons, the main reason being the velocity of the wheel motion. In the spirit of (1-3), we assume that