Analytical Assessment of the Sliding Friction Coefficient Influence on Durability, Wear and Contact Pressure in Spur Gears

The research of influence of sliding friction coefficient on durability is carried out for gear train containing steel wheels and metal-polymer gear trains containing polyamide gears reinforced with carbon and glass dispersion fibers. The teeth engagement conditions (two teeth pairs – single tooth pair – two teeth pairs) and the change of teeth tribocontact interaction conditions due to wear are also taken into account. The gear train containing the carbon-filled composite gear has the highest durability in comparison with other types in all ranges of change of sliding friction coefficient. The highest maximum contact pressures will be at the point of entry into the one-pair engagement. The change of initial maximum contact pressures in gear train due to tooth wear was also investigated and its regularities were established. The kinetics of the tooth profile wear was studied. It is established that the maximum wear will be at the point of entry into the one-pair engagement. Close to it will be the wear at the entrance into the two-pair engagement. The course of wear at different points of engagement for the studied gear trains containing steel toothed wheels and gear trains containing steel and composite gears is almost the same except for the point of the teeth exit from the engagement.


INTRODUCTION
In order to effectively reduce the friction coefficient in tribomechanical systems of various kinds, lubricants are widely used, including the use of antifriction and antiwear additives, resulting in reduced wear and increased durability. In particular, this applies to gears. One of the fundamental factors determining the conditions of friction in the tribological system is the sliding friction coefficient f. Therefore, the study of the influence of this parameter on the gearing service life at the boundary, mixed and dry friction is of great practical importance. In addition, in metal-polymer gears without lubrication in the case of dry friction, the modification of polymers with dispersed particles of micron size or short fibers is used to adjust the value of the friction coefficient. In the literature of the subject there are no studies of the friction coefficient influence on the change of durability of gears made of metal (ММ) and nonmetallic materials using known computational methods [1][2][3][4][5][6][7][8][9][10]. Only according to the author's calculation method in [11] the influence of the coefficient of sliding friction on the durability of spur MM gears of an electric locomotive under changing contact conditions in the corrected gearing due to teeth wear was inves-tigated. Regarding metal-polymer (MP) gears, there are no calculation methods for their study, except for the author's [12,13].
In the literature on the subject there are only some quantitative results of experimental studies of wear of polymer composite materials for MP gears [14 -16], which can be used to predict their wear using existing calculation methods. In [14] the experimental and numerical study a loaded cylindrical PA66 gear was given. In [15], the results of experimental studies of the influence of the sliding speed on the friction force for NaPA6/steel S355J2 pairs with twin-disc setup are presented. In [16], quite extensive results of experimental studies in air and abrasive mediums of the coefficient of friction and relative volumetric wear of model MP gears for several types of polyamide composites (PA6-Mg, PA6-Na, PA66 + 30GF, POM-C) and steel S355 are presented. The results of these further studies are given in [17].
Using the method [18][19][20][21][22] of the estimated assessment of gearing wear and durability, the influence of the friction coefficient on the durability of metal and metal-polymer gearings under the same operating conditions will be investigated. In contrast to the above methods of gearing calculation based on the Archard law of abrasive wear, which is virtually absent in gearings, the author's method, based on the known phenomenological calculation model of friction-fatigue wear at sliding friction [23], takes into account the influence of teeth wear on durability and conditions of their engagement.

Method of calculating linear wear
To determine the linear wear of the teeth h kjn at any point j of the working surface for the cycle of interaction, the ratio [19][20][21] is used: where t jh =2b jh / v 0 = var is the time of tribocontact interaction (wear) of the teeth during the movement of the j -th point of their contact along the contour of the tooth to the contact area variable width due to wear 2b jh ; j = 1,2,3,… are points of interaction on the teeth profiles from the entrance to the out of contact; v 0 =ω 1 r 1 sinα is the velocity of movement of the contact point along the tooth profile; ω 1 is the pinion angular velocity; r 1 is the pinion pitch circle radius; v j is the sliding velocity; f is the sliding friction coefficient; p jhmax is the maximum tribocontact pressure at the j-th point of interaction arising at the teeth wear; C k , m k are the wear resistance characteristics toothed wheels materials for selected conditions [18]; τ s =0.35R m is the shear strength of metallic materials of toothed wheels; R m is the tensile strength of materials; τ s =0.5R m is the shear strength of polymer composite materials of toothed wheels.
Due to teeth wear there is an increase in the radii of curvature of their working profiles. Accordingly, there will be a decrease in the initial maximum contact pressures p j max and an increase in the width of the contact areas 2b j at each j -th point of the teeth contact. Therefore, taking into account the teeth wear, the current values are calculated by modified Hertz formulas  (2) where N =N/bw; N = 9550P/r 1 n 1 cosα is the force acting in engagement; P is the power on the driving shaft; b is the width of the pinion; w is the number of pairs of meshing teeth; ( ) ( ) the variable at wear the reduced radius of curvature of teeth profiles in normal section; ρ 1jh , ρ 2jh are respectively, the variable radii of curvature of the teeth profiles of pinion and gear.
In the process of gearing operation, due to teeth wear, the initial radii of curvatures ρ 1j , ρ 2j of their working profiles and, accordingly, the reduced radius of curvature ρ j will increase. Accordingly, they are calculated as follows [18]: where r 1 , r 2 are the pinion and gear pitch circle radii; r b1 , r b2 are the pinion and gear base circle radii; r a1 , r a2 are the pinion and gear addendum circle radii; r is rounding radius of addendums; u is the gear ratio; а is the centre distance; Δφ is the angle of rotation of the pinion tooth from the point of initial contact (p.0) to point 1, etc.; α 10 is the angle corresponding to the 1-st contact point of pinion tooth on the line of engagement; α 1s is the angle determining the position of the last contact point of pinion tooth on the line of engagement; α 20 , α 2s are the angles of the 1-st and last contact point of gear tooth on the line of engagement.

Method of calculating the change in the teeth radii of curvature
The effect of wear on the change of the initial radii of tooth curvature was studied in [18,19]. Accordingly, the change in the radii of curvature is taken into account as follows: 1 n kjh kj jk kj where n = n k = 1, 2, 3, ... are numbers of toothed wheel revolution; k is the toothed wheel numbering (1 -pinion, 2 -gear); are the dimensionless constants at each contact point j depending on the teeth wear in the general case. Change in the curvature of tooth profiles due to wear during each interaction To reduce the duration of calculations, the calculation block diagram has been developed. Here, the change of tooth profile radii of curvature, their reduced radius of curvature, maximum contact pressures, width of the contact area are not considered after each revolution (interaction cycle), but after a certain number of revolutions (interaction block). In the block, the calculation is carried out by the linear method of accumulation, i.e. under constant initial conditions. In the next block of calculations the accumulated changes are taken into account by Equations (6), (7) and according to the new current data the calculations of the above parameters continue. The calculations time decreases in proportion to the size of the block. In this case the teeth radii of curvature are calculated by the formula [19] Because the teeth wear during gearing operation causes a change in the initial radii of curvature, the values h kjn are calculated at each subsequent revolution for time t jh =2b jh /v 0 , and the variable width of the contact area 2b jh at (n k -1)-th revolution or at (B-1)-th block is calculated according to Equation (2).
The chord length of the circle replacing the involute between points j -1, j + 1, is calculated as follows: is the involute length between the points j and j + 1; α j , α j+1 are the pressure angles for selected involute points j, j + 1 (see above); ( ) z 2 are the numbers of teeth of the pinion and gear. As a result of teeth wear, after each interaction or block of interactions, all calculated parameters will change, in particular The ratio is used to calculate the sliding speed Units should be typewritten vertically, as for example: where r b1 = r 1 cosα.

Calculation of total wear
For the selected arbitrary number of revolutions of the pinion and gear n 1s and the corresponding number of blocks, the total wear h 1jn and h 2jn of teeth at j-th point of contact is calculated as follows: where n 2s = n 1s /u; kjB kj h h′ = ∑ is the wear of teeth in each block; u is the gear ratio.

Calculation of durability
The durability of gearing for a given number of revolutions of toothed wheels n 1s or n 2s is calculated as follows [21,22]: Upon reaching the accepted allowable wear of the teeth h k* in one of the points of any toothed wheel is automatically calculated the corresponding maximum number of revolutions n max1s and n max2s , which allows determining the ultimate minimum gearing life according to Equation (10).
Provided that the maximum initial contact pressures p j max remain unchanged, the gearing life can be calculated according to the simplified method [13,18] as where 60 kj k kj h n h′ = is the linear wear of teeth for one hour; h k* is the accepted allowable wear of teeth. Then the dependence in Equation (1) for h kj at p j max =const will take the form [13,[18][19][20] ( ) is the time of teeth tribocontact interaction. Hertz's formulas in Equation (2) also acquire a classical form, because jh j ρ ρ const ≡ = .

Method of determining the pressure angles
In straight spur gears, two-one-two-pair engagement of teeth is realized. The angles of transition from two-pair (Δφ 1F2 ) to one-pair and again to two-pair (Δφ 1F1 ) engagement and the teeth exit angle from the engagement Δφ 1E are calculated as follows [22]:
The results of calculations are presented in Fig. 1-7. In particular, Fig. 1 shows the minimum durability t B min of gears calculated by the block method (solid curves), when allowable wear is achieved at one of the points of teeth. Also here are the results of calculating the durability t min of gears by the simplified method (dashed curves), when the conditions of teeth contact interaction are assumed to be constant during the operation until the allowable wear is achieved, i.e. without taking into account the effect of tooth wear on the change of initial maximum contact pressures.
According to the simplified method of calculations, the durability of the metal (ММ) gear train will be 1.13… 1.3 times less than according to the specified method, which takes into account the real conditions of tribocontact interaction. However, in the metal-polymer (MP) gear train the durabilities t B min , t min will be almost the same. Fig. 1 shows that the gear train containing a gear made of carbon-filled composite has the highest durability in the entire range of changes in the sliding friction coefficient. The gear train containing steel toothed wheels has a slightly lower durability (1.46… 1.88 times lower at f = 0.05… 0.2). In contrast, the metal-polymer gear train containing glass-composite gear has significantly lower durability than the other two types of gear trains: 8.9… 3.12 times lower at f = 0.05… 0.2 relative to the gear train containing steel toothed wheels; 12.9… 3.12 times lower at f = 0.05… 0.6 relative to the metal-polymer gear train containing carbon composite gear. In closed gear trains with metal wheels lubricated with oils the boundary friction is realized f = 0.05… 0.1. However, in case of insufficient lubrication or oil destruction during the operation, the friction coefficient will increase. In some cases, the gear train can operate in the mode of semidry or even dry friction, in which the friction coefficient will reach 0.2… 0.4. Metal-polymer gear trains made of the studied filled polyamide composites work quite reliably without lubrication at dry friction conditions with the friction coefficient f = 0.3… 0.6. However, by changing the percentage and composition of the fillers or by applying lubrication, adding antifriction additives to oils, the friction coefficient can be reduced even to 0.05 and consequently it increases the durability of the gear train.
Analysis of the fig. 1 shows that the gear train with a carbon composite wheel at f = 0.05… 0.6 will have the greatest durability. At the boundary friction (f = 0.05… 0.1) MM gear train will have slightly less durability (about 1.5 times), and at higher values of the friction coefficient this difference will increase to 2. Regarding the relative durability t of MM and MP gear trains, shown in Fig.2, the increase in the friction coefficient f leads to its decrease in the case of gear trains (Steel-PA6 + 30CF)/(Steel-Steel). Therefore, the carbon-composite metalpolymer gear train will be significantly more durable in a wide range of changes in the sliding friction coefficient of the studied types of materials. An important parameter for gear trains is the load carrying capacity, which is characterized by the level of maximum contact pressures p j max in the engagement.  The friction coefficient has a significant effect on the rate of p jhmax in the zone of two-pairs engagement and has a much smaller effect in the zone of single-pair engagement.
Accordingly, Fig. 4 shows the change in the initial pressures max j p in the MP gear train during the cycle of two-one-two-pairs engagement. The nature of the change of the maximum initial pressures p j max in MP gear trains is the same as in MM gear trains. However, with the same transmitted power and geometric parameters of the toothed wheels, there is a very significant difference between the value p max in MM gear train and MP gear trains. In particular, in the metal-polymer carbon-composite gear train it is 5.07 times smaller, and in the metal-polymer glass-composite gear train it is 5.85 times smaller. The analysis of the received data testifies almost identical change of tribocontact pressures in the engagement. However, in the MP gear train Steel -PA6 + 30CF the maximum pressures are slightly higher than in the MP gear train Steel -PA6 + 30GF. As a result of calculations it is established that the friction coefficient does not show influence on p jhmax in the specified MP gear trains. Obviously, this is due to the fact that the steel teeth of the pinion are practically not worn, because it is three orders of magnitude less than the wear of the teeth of composite wheels (compared to PA6 + 30CF 1340 times, and compared to PA6 + 30GF 1610 times). Fig. 6 shows the linear wear of the working teeth profile at different points of engagement.
The maximum (allowable) wear of the teeth will occur at the point of entry into the single-pair engagement. Wear at the entrance to the two-pair engagement will also be close to it. It is established that the values of linear wear of the teeth profile at individual points in the studied MM gear train and both types of MP gear trains will be approximately the same in magnitude and nature of change. The change in sliding speed during engagement is shown in Fig.7 At the entrance of the teeth into the engagement and at the exit from it, the sliding speed will have the opposite direction and approximately the same value for all types of gear trains. According to the calculated data of the gears, the reduction of the initial thickness S 0 near the root of the composite teeth is calculated. Accordingly, the initial thickness S 0 is 7.39 mm. According to fig. 6 the teeth wear of the gears is: PA6 + 30CF -h s ≈0.21 mm, PA6 + 30GF -h s ≈0.26 mm. Due to wear, the teeth thickness S h is: PA6 + 30CF -7.18 mm (<2.84%), PA6 + 30GF -7.13 mm (<3.52%). According to the results of experimental studies of gears at different loads [25], it was found that when wearing up to 10%, the bending stresses of a worn and unworn tooth in the zone of tension and compression will be almost the same. Tooth wear by 20% will already increase these stresses in the zone of tension by 6-8%, and in the zone of compression -up to 15%. That is, for the studied MP gears with the accepted calculation data, in particular h * = 0.5 mm, the teeth wear will not affect the bending stress. This effect will be negligible even when the allowable wear is twice as high.
It should also be noted that the teeth of composite gears are more deformable and undergo greater bending than the teeth of steel gears, which leads to an increase in the engagement period, i.e. to an increase in the overlap coefficient. As a result, it is known that the performance efficiency of the gear train will be improved, because there will be some reduction of the maximum contact pressures at the entrance of the teeth into the engagement, which will increase the durability of the MP gears.

CONCLUSION
1. The author's computational method of gear trains, in particular metal-polymer, provides at the design stage the possibility of a predictive assessment of their durability, load capacity and wear of the composite gear. 2. Irrespective of materials of toothed wheels the assessment of tribotechnical and contact parameters can be realized by the developed computational method. 3. The results of the estimated assessment of the durability of gear trains (Fig. 1) indicate that in the case of calculation of MM gear trains by the specified method, the durability is higher 1.13 ... 1.3 times than calculated by the simplified method. On the other hand, in both types of MP gear trains this difference will be insignificant (up to 3%). 4. As established, the sliding friction coefficient significantly affects the change of the initial contact pressures in the MM gear train (Fig. 3). This effect is not observed in MP gear trains (Fig. 5). 5. The developed computational method also provides an opportunity to assess the teeth wear at selected points of two-one-two-pair engagement in spur gears. In helical gears, which are not studied here, such an assessment by this method will also not be difficult. 6. As a result of the conducted researches it is established that the trend of teeth wear in MM and MP gear trains is practically the same ( fig. 6). 7. The influence of tooth wear on the maximum bending stresses was also studied. 8. The presented research method, previously developed for the calculation of gear trains with metal wheels, and subsequently transformed for the calculation of metalpolymer gear trains, provides at the design stage effective assessment of durability and optimization of gear trains by choosing materials of toothed wheel, fillers of polymer composites and possible use of the appropriate type of lubricants.