Peg-on-hole: mathematical investigation of motion of a peg and of forces of its interaction with a vertically fixed hole during their alignment with a three-point contact

The task of the peg-in-hole assembly is a very common task in industry. Despite its intuitive comprehensibility and everyday experience, the industrial assembly faces specific problems. The naive approach of inserting with bare hands an industrial peg inside into an industrial hole can very certainly bring to the jamming and damaging because of a very small gap between them. Therefore, the initial alignment of the details before insertion, so-called peg-on-hole phase, or simply peg-on-hole, is of the special importance in industry to avoid jamming. This paper develops the description and analysis of this phase based on the methods of analytical mechanics. As a common approach, the details in this study are considered as cylindrical, and such that peg is supported at the edge of vertically fixed hole, and could freely move keeping a three-point contact with three degrees of freedom, namely (i) nutation angle or align describing planar motion of aligning parts, (ii) precession angle or slide describing rotational motion around hole axis, and (iii) self-rotation angle or slip describing rotational motion about its axis. The analytical approach implies the system of Lagrange Equations for this particular case of motion, which is called Dynamic Differential Equations (DDE). DDE describes interconnections between motion in three degrees of freedom on one side, reaction forces on another side, and the external forces and torques on the third side. DDE includes geometrical properties of the details, and full descriptions of velocities, normal reactions, and friction forces at the contact points. It is the most general description of the peg-on-hole case. As the example of application, the normal reactions were found for relative large alignment angles far from reduction into a two-point contact, and for a small alignment angles, when system transits to the two-point contact. It was shown that for both cases, reaction forces become larger during alignment because they have to balance the force of gravity. For the larger angles, slip reduces the reaction forces, whereas for small alignment angles, slip and slide increases them.