Ricerc@Sapienza

The study of the gears is based on the conjugated geometries according to which two curves or two surfaces in mutual movement maintain in constant contact. The geometric theory of the gears until the end of the nineteenth century was one of many branches of the applications of descriptive geometry. The study is based on knowledge of the main properties of plane curves and humps and their derivatives. The specificity of the theme is that these geometries when have to relate with their conjugated, must meet the constraints that would otherwise not have. Through the analysis of some case this essay aims to highlight the role of descriptive geometry from theory to practise, applying methods and procedures of investigation often forgotten.
Some types of gears were known since the ancient age. Erone of Alessadria for example described an “odometro”, a length counter device done with wheels, levers and pulleys. The “renaissance engineers” like Leonardo da Vinci designed several gears but without developing a real theory.
The first treaties, in which the geometric theory of the gears is developed, were published at the end of ‘600th by Philippe de la Hire and then Charles Etienne Louis Camus. Treaties whole dedicated to this topic were Théorie géometrique des engraneges by Oliver in 1842 and, immediately after, La Teoria geometrica degli Ingranaggi by Codazza in 1854.
Nowaday the possibilities of 3d digital modeling and 3d print allow us to study this topic with a renovated enthusiasm making experiment on new solutions and applications. A gear is a rotating machine part that is comprised of a set of toothed wheels, with the purpose of transmitting power from one part of a machine to another.The shape of the tooth and of the nucleus is based on the conjugated geometries. In a classic gear, done from two toothed wheels, the line of the single tooth is based on the evolving of the circumference. This curved line is defined by a point P of a straight line that roll, without slipping, along a circumference. In a bevel gear the profile of the tooth is based on a spherical epicycloid, a curved line defined by a point of a circumference that roll around another no coplanar one. 3D parametric modelling allow us to create dynamic models through which it is possible verify in real time their correctness and effectiveness.