Geometric And Computer Modeling of Forming the Gear Wheels with Elliptical Centroid

Abstract

A fairly simple and accurate methodology of automated forming of the teeth of a wheel with an elliptical centroid is proposed. This methodology is based on computer simulation of the real process of forming and does not require the application of complex mathematical body used in most traditional methods. The key point of the methodology is the proposed solution for determining the instantaneous centers of the cutting tool and the angles of its rotation around these centers. The relative position of the cutting tooth profiles is determined by two arrays of point coordinates: one on the elliptical centroid of the non-circular wheel, the other - on its equidistant curve. Compared to the traditional methods, this methodology is a universal and much more efficient and accurate one, especially for wheels with an elliptical centroid. The proposed automated solid-state computer simulation of the forming allows to obtain not only the envelope part of the profile, but also to observe possible formation of transition curves on it and undercutting, which leads to a qualitative solution of the profiling problem. According to the developed algorithms and programs, a computer model of a non-circular wheel is obtained, on the basis of which a corresponding item used in hydraulic unit pumps is made on a CNC machine.

Designing an outer toothed gear whose wheel teeth are outlined by the logarithmic spiral arcs

“…Paper [10] proposes a geometric model for solving the problem of profiling an uncircular toothed wheel whose centroid consists of interconnected arcs. The same authors in article [11] proposed a fairly simple and accurate procedure for the automated formation of wheel teeth with an elliptical centroid. An eccentric helical toothed pair is a new type of geared pair consisting of a screw noncircular gear and an eccentric oblique curved face, which can implement the rotation of intersecting axes, as well as the movement of the output axes, proposed in [12].…”

“…1, a). Substituting the resulting angle values of α=α о , φ=0 in (11) and solving relative to r produces the following:…”

“…Thus, the value of the center-to-center distance is not set but determined depending on the number of teeth n=π/α о , n 1 =π/φ о of the drive and driven wheels and the values of ρ о and angle ε. Substituting (12) in (11) could produce the required dependence φ=φ(α). If it is substituted in (6) along with constant (12), then the parametric equations of the curve could be obtained, on which it is possible to highlight the arc А 1 В 1 (Fig.…”

“…Fig. 2, a shows the curves constructed from equations ( 1) and ( 6) taking into consideration (12) and (11), on which we highlighted with thickened lines the arcs АВ and А 1 В 1 , corresponding to the change of angle α within α=0…α о . Visually, these arcs are congruent.…”

“…Thus, in equations ( 6), instead of the polar angle φ, there should be the angle φ+π+φ о . The resulting expression of angle φ can be obtained after adding to it the previous value (11) of the angle π+φ о :…”