Abstract: In engineering design，the supporting force of the orbit to the object is often taken into consideration， and the non-smooth orbit is closest to the physical reality. This paper studies the constraint force problem of the small ring constrained on the non-smooth orbit. By using the differential equation of particle motion，the analytical solution of the constraint reaction of a small ring at any position on the parabolic orbit is obtained. At the same time，the effect of friction on the motion process of small ring is studied. And the numerical calculations and analysis of the complete process of forward and return movement is performed with the help of Mathematica software. The example of solving this kind of problem is given. The solution technique of the non-smooth orbit problem can be applied to other forms of orbits.

Key words: friction, non - smooth orbit, Mathematica software, constraint reaction force, numerical calculation and analysis