From Jerry Czarnecki,
23 March 1998
Mathematics lies at the heart of the Sciences and Engineering. Indeed is is a language with which Physicists, Engineers, Economists and many others communicate ideas and solve problems. The fact that it is a universal requirement in the education of such professionals though many times does not adequately illustrate its real importance to these professions. It is my intention here to briefly describe some interestin aspects of the Gear industry, in which I am currently employed and which relies heavily on what may be termed higher Mathematics.
Thoughout the Gearing industry and especially at the Gleason works here in Rochester, NY, a central problem that is faced is to characterize the shape that a gear tooth assumes as it is produced, that is, as it is cut and ground in the machine. Suffice it to state here that the particular shape of the tooth is important to achieve desirable between the gear tooth in question and the teeth of the mating or contacting gear as they run together in their application. Good contract between gear teeth will in turn minimize vibrations, hazardous to the gear housing as well as the frictional losses that result between teeth that have not assumed the correct shape or profile and that occur due to resulting rubbing motion as opposed to rolling motion. These frictional losses not only lead to operating inefficiencies of the gear set but also to gear tooth wear and ultimate failure of the gear.
Describing the shape of a gear tooth involves a good knowledge of Differential Geometry and Kinematics. One must be able to describe spatial motions of the gear cutting blade in the machine relative to the gear being cut, which itself moves during the cutting process. The result is a mathematical description of the gear tooth surface from which valuable information about the surface such as its principle curvatures, important for subsequent stress calculations can be extracted. This description isn't exact, however due to factors beyond human control during the cutting process like randon machine motion and inexact machine calibration, and these models only offer an approximation to the real surface, albeit a good one.
Certain characteristics of the running gears, like vibration frequencies can be analyzed to offer insight into the possivle reasons for the inaccuracies in the aforementioned modeling process and to provide a basis for corrective actions during production. Today these types of projects are ongoing. They promise to provide more accurate analytical models upon which later, tooth shape dependent analysis of subjects ranging from gear stress evaluations and tribollgy(study of lubrication and wear) can be themselves advanced.
While this exposition into the Gearing profession is indeed short, it may well explain the need for individuals skilled in and comfortable with Mathematics beyond the elementary Calculus and Linear Algebra training standard throughout the sciences. Mathematics is certainly alive within the Gear industry and even so throughout other manufacturing industries.