Tooth root strength of bevel and hypoid gears : development of a practical design method for rear axle gears in commercial vehicles

These investigations are focussed on the tooth root strength of rear axle gears for trucks. Four different standards for calculating the tooth root stress of bevel gears have been compared and analysed. Although the expressions that are used in these standards appear to be quite different, they have been rewritten in such a way that the individual load, geometry and material factors can be directly compared. This shows that in fact these standards are all built up in a comparable way. A calculation example has been performed on two hypothetical gear sets, representative for truck applications. The largest differences are attained on the Allowable Stress, the Tooth Form Factor and the Face Load Distribution Factor. Smaller differences may be observed on the Load Sharing, the Helical and the Size Factor. The calculated stresses and the allowable material stresses are to be closely linked to each other within one and the same calculation standard. The Face Load Distribution Factor was calculated to be 1.30 instead of the normally recommended value of 1.50 according to DIN, by considering a reviewed stress distribution over the gear facewidth. Three normally applied methods for calulating tooth root stress of hypoid gears have been compared. The differences between these methods are mainly determined by the geometry of the virtual bevel gears. The influence of hypoid offset on tooth root stress, calculated according to these three methods showed very large differences. The tendencies of these calculations have been compared with results of other investigations. This has shown that the influence of hypoid, calculated according to Winter, showed the best correlation with results of external investigations. Several endurance driveline tests have been performed on four different types of rear axle hypoid gears, assembled in a driving head. The tests were run at a constant amplitude load and speed, until fatigue breakage of the pinion teeth occurred. The testresults were statistically evaluated and have been described by a lognormal and a two parameter Weibull failure distribution. A damage analysis on several of the failed pinions showed consistent failure types that partly correspond to assumptions in stress calculations. The DIN 3991 method was used for calculating the tooth root stress, by using the Winter method for determining the geometry of the virtual bevel gears. These were then fitted with the test results, by which a synthetic SN curve was established. The established fatigue limit is in line with standard values. The slope of the SN curve and the ratio of static to endurance strength were however different from the values, used for helical gears of the same material. This is believed to be caused by the non linearity of the stress-torque relationship, being a consequence of the growth of the contact pattern with increasing torque. In all standards however, the stress is calculated as being linear with torque. Therefore, a Load Factor is introduced to account for the influence of the contact pattern. This assures a non linear relation between torque and tooth root stress for bevel and hypoid gears. When this factor is applied, comparable slope values for the SN curve of helical gears in identical materials may be used. With this mathematical adaptation, the difference between calculated and registered endurance lifes for three of four axle types became far leSS than piUS-minUS 1 0°/o fOr a failure probability of 1 0°/o. Vehicle Driveline Loading Spectra have been measured and calculated. A comparison shows good agreement. On this basis, several loading spectra have been simulated for typical vehicle routes. Two basic types of loading spectra have been determined here, for which analytical expressions have been developed as well as equations for the equivalent torque. A limited number of endurance tests at variable amplitude loading have been performed on one axle type. It was found that the fatigue damage accumulation theory according to CortenDolan was the best suitable for variable amplitude loading when the non linear stress-torque relation was used, although still a reduction in endurance strength may be noticed. At variable amplitude loading mostly a mix of failures may be expected, where the early occurence of surface damage may influence the endurance limit for tooth root fatigue. A simple relation has been derived for the gear outer diameter of bevel and hypoid gears, based on the maximum output torque. For preliminary dimensioning, this torque can be considered to be mainly based on the vehicle weight. With this expression, it is possible to give a practical first order estimate on the gear outer diameter for a given vehicle weight.