EXTERNAL ROLLING OF A POLYGON ON CLOSED CURVILINEAR PROFILE

Zynovii Ruzhylo,Serhii Pylypaka,Oleksandra Trokhaniak,Ivan Rogovskii,Tetiana Kresan

doi:10.14311/ap.2020.60.0313

Zynovii Ruzhylo, Serhii Pylypaka + Show 3 more

Open Access

https://doi.org/10.14311/ap.2020.60.0313

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Abstract

The rolling of a flat figure in the form of an equilateral polygon on a curvilinear profile is considered. The profile is periodic. It is formed by a series connection of an arc of a symmetrical curve. The ends of the arc rely on a circle of a given radius. The equation of the curve, from which the curvilinear profile is constructed, is found. This is done provided that the centre of the polygon, when it rolls in profile, must also move in a circle. Rolling occurs in the absence of sliding. Therefore, the length of the arc of the curve is equal to the length of the side of the polygon. To find the equations of the curve of the profile, a first-order differential equation is constructed. Its analytical solution is obtained. The parametric equations of the curve are obtained in the polar coordinate system. The limits of the change of an angular parameter for the construction of a profile element are found. It is a part of the arc of the curve. According to the obtained equations, curvilinear profiles with different numbers of their elements are constructed.

Highlights

Some flat figures, including polygons that can be rolled on a curvilinear periodic profile without sliding, are considered in [1]
The profile is formed by equal symmetrical curvilinear elements of their serial connection so that the ends of the elements abut on a straight line
Constructing a closed profile in which curvilinear elements touch a circle is important for the design of centroids of non-circular wheels

Summary

Introduction

Some flat figures, including polygons that can be rolled on a curvilinear periodic profile without sliding, are considered in [1]. When rolling a polygon on such profile, its centre moves in a straight line. Constructing a closed profile in which curvilinear elements touch a circle is important for the design of centroids of non-circular wheels. In [3], information on the rolling of second-order curves along a straight line is given. The basics of designing non-circular wheels for gears are given in [5]. The purpose of the article is to develop an analytical description of a curvilinear closed profile, in which an equilateral polygon will be rolled without sliding and its centre will move in a circle of a given radius

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