The elastic analysis of load distribution in wide-faced helical gears

Abstract

For a gear designer, the meshing gear tooth root bending stresses, and contact stresses are of major importance. To be able to obtain accurate values of these stresses, it is essential to determine the actual load distribution along the contact lines of the meshing gear tooth pairs. this load distribution. The objective of this work is to predict In the current gear design standards such as AGMA 2001¹, B5436², DIN3990³ and 150-0156336⁴the contact line load distribution is estimated by using a two-dimensional "thin slice" model of the meshing gear teeth. Clearly, this cannot account accurately for maldistribution of loading across the tooth face width, which is essentially a three-dimensional phenomenon. As a result, the effects of tooth lead, profile and pitch deviations are inadequately modelled. In this work, the elastic compliance of wide-faced helical gears of standard tooth form, zero addendum modification, and between 10 and 100 teeth, was determined using a 3-D finite element elastic model of the whole gear. These results were incorporated into a micro-computer program which calculates the load distribution across the meshing tooth pair faces. The effects of a number of parameters such as U, Z, b, and β* on the load distribution and contact stresses of an error-free gear were also investigated using the micro-computer program and the results were compared with other published data and those obtained from the standards²,³,⁴ Vedmar⁵ and Simon⁴³. The load peaks near the start and end of contact, attributed by some⁶,⁷ to the resistance of the unloaded portion of the tooth beyond the shorter contact lines in those regions, is very clearly demonstrated by Vedmar⁵, others⁶,⁷ and this work, but certainly not by the standards (this effect is usually referred to as the "buttressing" effect). The thin slice model largely over estimates the tooth mesh stiffness cγ since the convective effects of loading are completely ignored. The effects of lead deviations such as helix angle error and face crowning (barrelling), profile deviations such as profile angle error, profile crowning and tip relief, and pitch deviations such as adjacent base pitch error, were also studied. Their effect on the load distribution factors KH(3' KHO! and the overall load distribution factor KH, were obtained from the compared with the results from the standards2,3,4. micro-computer program and As expected, the standards considerably overestimate these factors due to their overestimation of mesh stiffness. Nevertheless, the pattern of variation in the load distribution factors was similar. The theoretical predictions were compared with experimental results measured on wide-faced test gears (specifications given in Table 5.1) with known (measured) mounting and tooth form errors. Measurement of tooth root strains to determine the load distribution along the simultaneous contact lines showed that the experimental and theoretical results agreed on the average to within 3.5% (end of tooth results not included). Also the total applied load upon comparison with theory agreed to within 6%. Experimental absolute values of transmission error "ft" were not available, however, the pattern of variation of 11ft" during meshing showed excellent consistency with the theoretical results (variations were very small anyway and within the error band). A separate test however, which gave the approximate absolute transmission error (tooth misalignments and form errors not included) agreed to within 1 % with theory.