A Geometrical Model Producing a Surface Equivalent To the Working Surface of the Gear Tool «Hob»

Abstract

Existing mathematical models for calculating worm gearing [34; 38] are quite complex and do not always provide an opportunity to quickly and accurately obtain the desired result [1; 3; 24–26]. A simpler way to find a suitable gearing option that satisfies the task is using computer simulation methods and computer graphics, and in particular solid modeling algorithms [4; 5; 30–33; 36; 37]. This information can be entered into the computer in order to simulate control of the movement of the cutting tool. Ultimately, this boils down to the problem of analytic description and computer representation of curves and surfaces in three-dimensional space [18–20]. Despite the diversity and good development of the calculation methods, and the analysis of the geometrical parameters of the worm gear, there is a lack of means and methods for displaying the process of forming the working surfaces of the worm gear elements [28; 29; 41]. There are no computer algorithms for obtaining the producing surfaces of a worm cutter, which are obtained by a tool with a modified producing surface. A change in the geometric shape of the tool producing surface will lead to a change in the working surfaces of the worm wheel and turns of the worm, which may lead to an improvement in their contact. This article shows the application of the developed methods and algorithms of geometric and computer modeling, which are designed to form the helical surface of the turns of the worm and the teeth of the worm wheel. Their use will speed up the process of calculating intermediate adjustments of machines used for cutting worm gears, bypassing complex mathematical calculations that, under conditions of aging of the gear-cutting machine fleet, their wear and inevitable reduction in the accuracy of their kinematic chains. This can be achieved only by applying a deliberate modification of the contacting surfaces, which reduces the sensitivity of the worm gear to the manufacturing errors of its elements, which allows to maintain the quality of the gears produced at a sufficiently high level.