PSI - Issue 16

Andrzej Gębura et al. / Procedia Structural Integrity 16 (2019) 184 – 191 Andrzej Gębura, Sylwester Kłysz, Tomasz Tokarski / Structural Integrity Procedia 00 (2019) 000 – 000

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1. Introduction

The first transmissions were made of wood, and there were initially friction gears (mentioned already by Aristotle in ‘Mechanics’), but soon afterwards (as far back as in IV B.C.), they transformed to gear wheels. II B.C. is a period, during which gear wheels found a broad scope of applications in the whole Mediterranean. There were still wooden wheels, where one of them was loaded with pegs, and the other resembled a basket. Simon Stevin (1548÷1620) – the Flemish engineer, economic activist, mathematician and scientist, worked on increasing the efficiency of such transmission mechanism combined with the windmill for pumping water from polders, which was typical for the Netherlands. Thanks to the mathematical analysis of the whole power unit (windmill, transmission and power transmission to pumps), having considered the radial and longitudinal movements of transmission wheels, he modified the system in such a way that it became three times more efficient. Wooden transmission wheels survived in windmills until the beginning of the 20th century. Gear transmissions with metal gear wheels were already known in Ancient times in precise mechanisms with low power made of bronze to reflect the movement of planets and in the Middle Age (from the VIII century) in clock mechanisms. The first steel gear wheels transmitting substantial power were applied in England already from the XVII century in some machines based on water drive and in some agricultural machinery. However, in general use, transmissions with steel gear wheels appeared only in the XIX century along with the invention of the steam engine. Nowadays, the treatment of gear wheels attained perfection – the quality of mechanical and chemical treatment of these wheels guarantees its silent operation, while carburization and nitration ensure a considerably increased abrasion resistance. It is, however, worth noting that after the long operation, especially under fast-changing loads, it may result in the inequality of the wear of wheels and inequality of output angular speed of transmission. Due to the variable forces acting on the tooth of such a wheel, cases of its breakage are sometimes reported, developed by Augustyn and Gębura ( 2012), Dudziński et al. (2011), Gębura and Stefaniuk (2017). Evolvent shape of teeth guarantees the cooperation of gear teeth during power transmission between a driving wheel and a drive wheel without the displacement of the point of contact (contact line) of the teeth’ surface. Since there is no displacement in such a situation, there is also no friction. If there is no friction, there is also no abrasive wear. However, it is only a theory. In reality, gear wheels have their own manufacturing defects advancing throughout the wear process. These defects include:  shape defect,  scale defect,  eccentricity defect. Moreover, rotation axes are not stable due to the wear of bearing supports, by which gear wheels make complex movements:  wheels move further away and then come closer due to the eccentricity of suspension,  frequent axis skew is observed, which results in the surface skew of the contact surface,  shafts of gear wheels are characterized by frequent longitudinal clearances, which results in post-axial movements of the cooperating gear wheels,  angular speed of input shaft of gear transmission may be substantially modulated (most often caused by the power unit), which results in accumulating mechanical stresses at the teeth’ surface during acceleration and deceleration of angular speed. The complex character of the movement of gear wheels in real conditions in a power unit (windmill transmission-water pump) was already addressed by Simon Stevin in XVII century. The mathematical description, which he utilized, enabled to improve the efficiency of pumping water from polders in the Netherlands. 2. Theory of meshing

3. Teeth breakage process in the transmission – reasons and effects

There are many publications concerning the computer simulation of wear of gear wheels i.a. due to fatigue cracks devoleped by: Bukowski and Kłysz (1993), Jakielaszek and Nowakowski (2014), Kłysz and Lisiecki (2009),