Accurate modeling of logarithmic spiral bevel gear based on the tooth flank formation and Boolean addition operation

The logarithmic spiral bevel gear is a new type of spiral bevel gear and has received great attention in the industry for its excellent engineering characteristics. The objective of this study is to develop an accurate geometric model for the logarithmic spiral bevel gear. Based on the gear tooth flank formation mechanism, a conical logarithmic spiral curve with a constant spiral angle was constructed as the tooth trace curve. The profiles for the exterior and interior transverse of the tooth were built with an accurate involutes curve, with transition being circular arcs and straight lines. The first tooth model was established by sweeping the tooth profile accurately along the tooth trace curve, and the rest of the teeth were created by an array operation. The accurate three-dimensional model of logarithmic spiral bevel gear was finally obtained by means of a Boolean addition operation among all of the gear teeth and the root cone. An experimental study was carried out for such a gear whose number of teeth was 37, with modules being 4.5mm, normal pressure angle being 20 degrees and spiral angle being 35 degrees. A DMU 40 monoBLOCK five-axis computer numerical control milling machine tool was used to produce the prototype, and a Zeiss CONTURA G3 three-dimensional coordinate instrument was employed to measure the exterior transverse addendum diameter. The linear error between the theoretical model value and the measured average value was 0.0027mm, indicating the effectiveness and practicability of this modeling method, which provides an accurate geometric model for design and for subsequent tasks like computer-aided engineering analysis and manufacturing.