Motion of a Chain of Three Point Masses on a Rough Plane Under Kinematical Constraints

A series of papers have analyzed the rectilinear motion on a rough plane of bodies (mass points) connected by viscoelastic elements (springs and dampers) in the case when the force of normal pressure is not changed. The system is moved by forces that changed harmonically and acting between the bodies. The asymmetry of the friction force,required for a motion in a given direction,is provided by the dependence of the friction coefficient on the sign of the velocity of the bodies which make up the system. This effect can be achieved if the contact surfaces of the bodies are equipped with a special form of scales (needle-shaped plate with a required orientation of scales (needles)).

In [1–5],the dynamics of a system of two bodies joined by an elastic element with a linear characteristic were considered. The motion is excited by a harmonic force acting between the bodies. In [3],a magnetizable polymer was employed as an elastic element and the motion was excited by a magnetic field. In the case of small friction,the analytical expression for the average velocity of steady motion of the whole system was found and it is shown,that the motion with this velocity is stable. A similar investigation for a system of two bodies joined by a spring with a nonlinear (cubic) characteristic was shown in [5]. Algebraic equations were obtained for average velocities of the steady motion. It was shown that there exist up to three different motion modes,one or two of them are stable. The limiting case of asymmetric friction is the kinematic condition that admits the motion only in one direction. This condition was considered in [6] in connection with a computer model of an earthworm.