Prediction of the Service Life of Metal-Polymer Gears Made of Glass and Carbon Fibre-Reinforced Polyamide, Considering the Impact of Height Correction

Calculation methods for predicting the service life of metal-polymer gears with height correction are proposed. Two methods for gear service life estimation are presented: a simplified method that assumes constant tooth interaction conditions and an improved method that takes into account wear-induced changes in tooth interaction conditions. Spur cylindrical gears with a steel gear and a pinion made of polyamide composite reinforced with carbon and glass fibres are studied. The effect of height correction on the service life of the gears is examined, considering the type of tooth engagement. It is established that with the improved method the service life of the gear is by up to 1.3 times longer than that calculated by the simplified method at optimal height correction coefficients. Optimal service life of the gear is achieved when the correction coefficients are x1 = – x2 = 0.1. The effect of the coefficient of friction on gear service life is investigated. It is found that the service life of the gear with a carbon fibre-reinforced pinion is by 8.1...9.3 times longer than that of the gear with a glass fibre-reinforced pinion.


INTRODUCTION
Gear teeth are modified to increase their load bearing capacity by reducing contact pressures. This, in turn, results in reduced wear of gear teeth and longer service life of the gear. Nevertheless, the literature of the subject offer no methods for determining the effect of addendum modification on the service life of metal and metal-polymer gears, considering both tooth engagement and tooth wear. It must be stressed that in comparison to steel, the mechanical properties of metal-polymer (MoP) gears are 10 times lower while their Young's modulus is lower from 40 to 50 times, which has a significant effect on their deformation and thus service life and wear. The calculation methods [1,10,11,13,14,18,19,22,23] widely use Archard's law of adhesive/abrasive wear for metal toothed gears. However, this type of wear practically does not occur either in metal gears in maximum lubrication conditions or in metal-polymer gears in dry friction conditions.
To perform numerical calculations, one requires quantitative data regarding wear characteristics of polymeric composite materials used in MoP gears. In literature there are only few experimental studies on quantitative wear resistance of polymer reinforced composites, the results of which could be used to predict the service life and wear of MoP gears using particular calculation methods. The studies [15,16] experimentally investigated the wear volume of teeth in model metal-polymer gears made of different polyamide composite materials (РA6-Mg, РA6-Na, РA66+GF-30, polyoxymethylene РОМ-С). These studies were conducted in dry friction and abrasive wear conditions. Detailed results of a wide group of polyamide composites for miniature slide bearings with different fillers are given in [20,21] (without lubrication: PA6, PA66,  PA66+GF30, PA66+GF50, PA66+GS28, PA610,  PA11, PA12, POM-C, POM-H; with grease: PA6+GF25+graphite, PA6+GF25+MoS 2 ). However, the experimental studies for the above-mentioned composite materials were conducted at low contact pressures and sliding velocities that are inappropriate for MoP gears. What is more, these studies did not investigate the mass wear of the composites, but identified the radial wear of journal bearings as wear characteristics. Also in [25] the author gives the results obtained for different types of polymers and polymer matrix composites (polyamides (PA), polyacetals (POM), polytetrafluoroethylenes (PTFE), polyethylenes (PE-HD, PE-UHMW), polyetheretherketones (PEEK), polyimides (PI, PEI, PAI) etc.), including tribological. Nevertheless, the linear wear rate of the above materials were determined only for single values of contact pressure and sliding velocity. Similar information can be found in brochures provided by companies manufacturing polymer composites and friction pair components. Still, the results of the above and other studies [3,12,17] are difficult to use as data when analysing MoP gears by the above-mentioned calculation methods, particularly the author's method. This is one of the basic reasons for not using calculation methods to predict the service life of not only metal-polymer gears but also of gears made of different polymeric compositions.
This study estimates the predicted service life of corrected MoP gears according to a modified method based on the principles of the author's calculation method [3][4][5][6][7] for toothed gears with metal gears. The proposed calculation method is based on the well-known phenomenological mathematical model of wear [3] at sliding friction due to friction fatigue.

METHOD FOR CALCULATING GEAR SERVICE LIFE
The predicted service life of cylindrical gears ( Fig. 1) was calculated with the use of two author's methods: the simplified method [4,5], which assumes constant contact pressures until reaching the acceptable tooth wear of one of the gears, and the improved method [6,7], which takes account of the effect of gear tooth wear on the reduction in contact pressures.
According to the simplified method [4,5], where p jmax = const at selected j-th point of the tooth profile, the predicted gear service life t min at which the gear teeth reach the acceptable wear h k* is determined as follows: The hourly rate of linear wear of the gears at contact points is here determined in an obvious way: A single (relating to a single interaction of the gear teeth) linear wear h' kj is determined according to the tribokinetic model of wear at sliding friction [3,5,9] in the following way: where: Maximum contact pressures p jmax formed during meshing at j-th contact points between the gear teeth and the contact area width 2b j are determined according to the Hertz equations In spur cylindrical gears the initial curvature radii ρ 1j , ρ 2j of the pinion teeth and gear teeth profiles are determined according to equations (5) where: r b1 = r 1 cosα ,  2 20 20 cos α cos α r r  , 20 20 cos α cos α r r  , According to the improved method [6,7], the predicted minimum gear life t Bmin for a specific number of gear revolutions n 1s and n 2s is calculated in the following way: The number of maximum revolutions n max1s and n max2s is determined from the condition of reaching the acceptable gear wear h k* at one of the contact points.
For any number of the maximum pinion revolution n 1s and gear revolution n 2s , the total linear wear h 1jn and h 2jn at j-th point of contact will be expressed as where: Sliding velocity is calculated with the equation: where: 1 The improved method considers the effect of teeth wear on increasing the curvature radius of their profile. As a result, the initial contact pressures p jmax decrease while the contact area width A method for determining changes in the tooth curvature radius due to wear is described in [3,4]. The change in the tooth curvature radius ρ kjh is determined with the formula The proposed method uses a block-based calculation procedure. This procedure assumes that changes in the tooth curvature radii ρ 1jh , ρ 2jh , the reduced tooth curvature radius ρ kh , the maximum tribocontact pressures p jh max and the contact area width 2b jh will only be considered after a certain number of revolutions (interaction block В), rather than after every single revolution (meshing cycle). In every single block, the degree of wear is calculated at constant contact The length of a chord replacing the involute between points j -1, j + 1 is determined according to the following formula: Spur cylindrical gears are characterized by double-single-double-pair meshing. The angles describing the shift from double to single and again double-pair meshing were determined according to [3,7,9].
The method for determining the geometrical parameters of a gear with height correction (P -O) is given in [4][5][6][7]. The application of this type of modification changes the addendum circle radii of the gear and pinion teeth 1 1 Other gear parameters were exactly the same as those of the unmodified gear.
Numerical results of the predicted minimal service life t min (simplified method) and t Bmin (block method) of both analysed gears when f = 0.3 are given in Figs. 2 and 3, for the gear with a carbon-fibre reinforced pinion and the gear with a carbon fibre-reinforced pinion, respectively.
The above results demonstrate that compared to the unmodified gear, the P -O correction has a considerable impact on the service life t Bmin of the analysed gears. With increasing the height correction coefficients, the minimal service life of the gear reaches the highest value when x 1 = -x 2 = 0.1. However, a further increase in the correction coefficients has a negative effect on the gear life. A similar relationship between increasing the height correction coefficients and t min is demonstrated by the simplified method to.
At the optimal value of correction coefficients, i.e., x 1 = -x 2 = 0.1, the minimal gear life determined by the simplified method is much shorter than that calculated with the improved (block) method considering a decrease in the maximum contact pressures due to pinion teeth wear.
The coefficient of friction has a significant effect on the service life of gears, as shown in Figure 4.
An analysis of the figure reveals that increasing the coefficient of friction leads to a nonlinear decrease in the service life of the investigated gears -by 3.73 times in the case of the gear with a glass fibre-reinforced pinion and by 4.92 times in the case of the gear with a carbon fibre-reinforced pinion.
The type of fibre has a significant effect on the service life of gears. The predicted service life of the gear with carbon fibre-reinforced pinion is approx. 9 times higher than that of the gear with a glass fibre-reinforced pinion over the full range of addendum modification coefficients. Such a considerable difference in the service ls from the fact that carbon fibres have significantly higher anti-friction properties than glass fibres, the latter being characterized by higher abrasive properties.

CONCLUSIONS
By considering tooth wear in the improved method, it is possible to increase the service life of the gear by approx. 1.3 times in a certain rage of the height correction coefficients. It has been found that the predicted service life of the gear with a carbon fibre-reinforced pinion is higher by 8.1 … 9.3 times than that of the gear with a glass fibre-reinforced pinion. At the optimal value of correction coefficients, i.e., x 1 = -x 2 0.1, the minimal gear life of both analysed gear types as estimated by the employed methods is the highest. When the correction coefficients are A two-fold decrease in the coefficient of friction leads to increasing the service life of the gears by 3.73 times (glass fibre-reinforced pinion) and 4.92 times (carbon fibre-reinforced pinion). The proposed method for predicting service life of metal-polymer gears allows optimization of height correction coefficients with respect to gear service life. The literature of the subject offers no such justified approach to optimized addendum modification.