International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 12 Issue: 09 | Sep 2025
p-ISSN: 2395-0072
www.irjet.net
A BEVEL GEAR ROOT BENDING STRESS MODEL 1Edward E. Osakue & 1Rasoul Saneifard 1Department of Engineering, Texas Southern University, Houston, Texas, USA
--------------------------------------------------------------------------***---------------------------------------------------------------------------straight bevel gears. The pressure angles in use for bevel Abstract A reversed Lewis root bending stress capacity model for bevel gears is presented. The model is developed from first principle based on the Tredgold’s approximation of a bevel gear. The approximation conceptually transforms a bevel gear into a spur gear. The load distribution of the bevel gear facewidth is assumed to be triangular so that the stress at the gear is constant over the facewidth. Six design cases of root bending stress computations from different references are done using the new and AGMA models. The stress estimates from the new and AGMA models were compared. The stress deviations associated with the pinions in the gearsets for the design problem examples vary between -14 to -4%, approximately. Based on the values of the deviations for the pinion stresses, the new root bending stress model verification may be said to be favorable when compared with AGMA stress values. However, the root bending stress predictions are relatively less conservative in values. Keywords: Bending, fatigue failure, nominal helix angle, base helix angle, virtual plane, Lewis stress factor
1.0 Introduction
A bevel gear has teeth cut on a conical pitch surface, making the tooth to vary in thickness and height from the front end or toe to the back end or heel. A bevel gearset consists of a pinion (smaller gear) and a ring gear (larger gear). The common types of bevel gears are straight and helical gears. The teeth of straight bevel gears are straight on the conical disk surface. Helical bevel gears have curved teeth on conical disk surfaces and include skew, spiral, zerol, and hypoid bevel gears [1, 2]. Skew bevel gears have a constant nominal helix angle like helical cylindrical gears, while spiral and zerol gears have variable helix angle. The nominal helix angle of spiral bevel is taken as the mean spiral angle defined at the mid-facewidth of the gear. The most popular helix angle for spiral bevel gears is 35o [3], but nominal helix angles of 20o to 45o may be used with the same tooth proportions as for 35o. Though spiral bevel gears resemble helical gears, they however, do not have a true helical spiral [4, 5]. Zerol bevel gears have nominal helix angle of zero degree. Hypoid gears are spiral bevel gears with nonintersecting axes. The offset allows for more compact design and efficient power transmission at greater angles than
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gears can vary from 14.5o to 25o. Pressure angles of 20o and higher are often used to avoid interference [1]. However, 22.5o and 25o pressure angles are used mainly in heavy-duty drives [6]. In North America, the most popular pressure angle is 20o. The International Standardization Organization (ISO) has adopted 20o pressure angle as the standard pressure angle, which makes it more attractive for international collaboration and commerce. A gear tooth in a mesh during operation acts like a loaded cantilever beam as it resists the forces transmitted. In 1892, Wilfred Lewis developed a gear root bending stress model [7]. He modeled a gear tooth as a short cantilever beam on a rigid base with the transmitted load applied near the tip of the gear tooth. The maximum tensile stress occurs at the root radius on the loaded side of the gear tooth. At the gear root, the involute profile merges with a root fillet [8] which creates some stress concentration. Due to the repeated loading of a gear tooth, this region becomes the preferential site for initiation of fatigue crack. The model of Lewis is the basis of national and international gear beam strength standards. These include American Gear Manufacturers Association (AGMA), International Standardization Organization (ISO), Deutsches Institut Normung (DIN), and Japanese Industrial Standard (JIS), which have introduced modification factors into the original Lewis model. Bending fatigue failure is one of the two most prominent modes of gear failure. Factors that can influence gear failures include transmission error, design error, manufacturing error and assembly error. Generally, severe surface wear-out in gears, which can accelerates bending fatigue failure, may be caused by thin lubrication film between two mating surfaces and the presence of debris in the lubricant. Bending fatigue failure of straight bevel pinions is the main failure mode in engineering machinery [9]. A review of bevel gear failures by Joshi and Kothari [10] concluded that bevel gears commonly fail by tooth breakage or fatigue. Such failures are largely caused by overloading, shaft misalignment, material and metallurgical deficiencies. Mohan Raj and Jayaraj [11] performed studies using distributed and concentrated loads applied at the pitch point on a bevel gear. They found that the stresses at the toe of the tooth were higher than those at the heel. Irsel [12] showed
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