Analytical Description of Meshing of Constant Pressure Angle Teeth Profiles on a Variable Radius Gear and its Applications

Abstract

This paper presents a method for determining the profile of the gear teeth on a variable radius wheel characterized by a constant pressure angle. The method can generate special gears using numerically controlled milling machines. As will be shown, the method, applied to a constant radius gear, generates an involute profile. The method is based on the integration of a differential equation describing the mesh between gears of variable radius, where the mesh point position is computed during rotation starting from the point, freely selected, where the tooth crosses the pitch line. The individual point is subsequently rotated in the opposite direction by an angle equal to the angle of rotation from the initial pitch line point, thereby generating the tooth profile. The method, applied to a wheel of variable radius, defined analytically or numerically, can compute teeth profiles on pairs of pitch lines of any shape. In particular, the motion of a slotted rotating link mechanism has been reproduced, but for the sign. Teeth profiles of other variable radius wheels have also been obtained. The results are more than satisfactory and are presented below. A numerically controlled milling machine has been programmed to actually build the antirotating slotted link equivalent gear. The present method, however, has much broader application, such as assigning the speed law to consequentially determine the gear form, as can be done with cams. Furthermore, a special planetary gear train makes it also possible to obtain reciprocating motion driven solely by gears. This has been built and its picture and scheme are presented in the paper. However, due to the low efficiency of the said mechanism, the best way to utilize this new technology seems to be to couple a crank and a rod to the pair of variable radius gears, as has been done at Hanover University. Some possible applications are presented. The special feature of these gears is the programmability of the shape of the pitch lines during the design phase, and thus of the velocity and acceleration profiles. In this way velocity profiles that could formerly only be obtained electro-pneumatically can be produced from purely mechanical components, with the added advantage of being able to control the level of inertia forces during the design phase. [S1050-0472(00)01601-9]