ON THE POSSIBILITY OF APPROXIMATION OF THE CYLINDERS INTERSECTION LINE PROJECTION WITH A CIRCULAR ARC

Abstract

The work is devoted to the study of the possibility of approximating the projection of the intersection line of two cylinders intersecting at a right angle by an arc of a circle with a diameter equal to the diameter of the largest of the intersecting cylinders, in a certain range of ratios of their diameters. A study was carried out, within the framework of which a graphical solution of the problem under consideration was made for various ratios of the diameters d and D of the cylinders. The deviations Δ of the approximating arcs from the exact projections of the intersection lines are measured. Based on the measurement results, a plot of the dependence of the maximum deviation of the approximating arc Δ on the ratio of the diameters of the intersecting cylinders d/D was plotted. It is shown that in the range of ratios d ≤ 0.6D, the approximating arc deviates from the exact projection of the intersection line within 0.5% of D. A variant of the practical application of the results of the work to optimize the process of connecting pipeline elements is proposed. It is shown that when using the proposed approximation method to determine the contours of the cutout on the butt sections of the pipe surfaces, the scheme for forming cutouts is greatly simplified, and the process of obtaining them can be performed using one end mill, that is, without changing the tool. Schemes for calculating the contours of cutouts on joined pipes according to the proposed method are given.

Subject of study: the case of the intersection of two cylinders at a right angle with a varying ratio of their diameters.

Materials and methods: a geometric algorithm for modeling the line of mutual intersection of two cylinders intersecting at a right angle - a curve of the 2nd order - using the method of flat mediators and computer simulation; approximation method in the context of the problem under consideration.

Results: it is shown that the projection of the line of mutual intersection of two cylinders intersecting at a right angle on a plane parallel to their plane of symmetry, for a certain range of cylinder diameter ratios, can be approximated by a circular arc with a diameter equal to the diameter of the largest of the intersecting cylinders. The dependence of the approximation accuracy on the ratio of the diameters of intersecting cylinders is established. The values of the maximum deviations of the approximating arc from the exact line for various ratios of cylinder diameters are determined, and a graph of this dependence is plotted based on the data obtained. A variant of the practical application of the proposed approximation method for optimizing the technological process of connecting pipeline elements is proposed. A scheme for calculating cutouts on joined pipes is given.

Conclusions: the projection of the line of intersection of cylinders intersecting at right angles onto a plane parallel to their plane of symmetry, in a certain range of ratios of their diameters, can be approximated with sufficient accuracy by an arc of a circle, the diameter of which is equal to the diameter of the largest of the cylinders. The accuracy of the approximation depends on the ratio of the diameters of the intersecting cylinders. The possibility of using the proposed approximation method and the boundaries of the confidence range of diameter ratios should be chosen taking into account the acceptable accuracy of constructions in a specific engineering problem. This approximation method can be used, in particular, in the processes of connecting pipeline elements for calculating the contours of cutouts on joined pipes, as a result of which the scheme and the process of forming cutouts are greatly simplified.