Computer Simulation of the Impact of Optimization of Width in the Helical Cylindrical Gear on Bearing and Durability Part 1 . Height Correction of the Gear Profile

Based on the elaborated calculation method he authors method for determining the wear and durability of gears was employed to measure the maximum contact pressures, linear wear of teeth and durability of the gear with height correction of the profile. The optimal condition width in involute helical gears is indicated ensuring constant length of the line of contact between the meshing gears. As a result, it was possible to determine variations in the parameters for the optimized gear describing the meshing gears at different values of the profile correction coefficients.


INTRODUCTION
In helical gears, the width of gear wheels ranges from 0.2 to 1.4 of the pitch diameter of a rack and it depends on the position of the gear wheels relative to supports and the hardness of the gear teeth.An increase in the width of the gears leads to an increase in the length of the contact line and the total tooth contact ratio, which results in a higher load-carrying capacity and durability of the gear.The overlap ratio ε β depends on the width of the gear wheel and tooth inclination angle.To ensure constant load on the teeth during gear operation, in practice the constant load on the teeth during gear operation, in practice the width of the gear wheel must be selected such that the minimum length of the contact line can be maintained constant.The literature [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] of the subject offers hardly any studies on this problem, particularly when it comes to gears with profile correction.

FORMULATION OF THE PROBLEM AND ITS SOLUTION
The total tooth contact ratio of a helical gear is ε γ = ε α + ε β , where: Where ε α is the end-face tooth contact ratio; ε β is the overlap ratio; + is the addendum radius of the gear; β is the inclination angle of the teeth; b W is the width of the pinion; m is the engagement modulus; z 1 , z 2 are the numbers of gear teeth; α t is the end-face pressure angle; α w is the pressure angle of the corrected profile.
Hence, the length of the contact line during teeth engagement will be constant if ε β = const (1 or 2).Accordingly, using the above formula of ε β , the width of gear teeth is calculated as: The minimum length of contact between a pair of teeth is calculated using the formula [18]: where are the fractional parts of the coef- , , where: tact ratio;   is the overlap ratio; e formula of   , the width of gear teeth is calculated as contact between a pair of teeth is calculated using the formula [18]:  F ) to single and then, again, to double ulated in the following way: , , on of engagement, the addendum radii: (1 ) , ( 1) fficients of displacement (correction) of the profile.  1 , In the case of triple-double-triple engagement After the height correction of engagement, the addendum radii: (1 ) , ( 1) where 1 2 x x = − are the coefficients of displacement (correction) of the profile.
Other parameters of the gear are the same as those in the gear without profile correction.
The changes in the initial maximum contact pressures p fmax during one cycle of tooth engagement are determined using the method for determining the wear and durability of toothed gears [16,17], considering the profile correction and the type of engagement [18].This is done using the Hertz formula: max 0.418 / is the force acting in tooth engagement; P n is the rated torque on the drive shaft; P is the power on the drive shaft; n 1 is the number of revolutions of the drive shaft; K g is the dynamic coefficient; E is Young's modulus of steel teeth; l min is the minimum length of a line of contact; j = 0, 1, 2, …, s are the points of contact on the tooth profile; w is the number of engagement pairs which transmit power simultaneously; ρ j is the reduced radius of curvature of the tooth profile in normal section; ρ 1j , ρ 2j are the radii of curvature of the side profiles of the teeth of the pinion and gear, respectively where ( )  is the number of revolutions of the gear; kj h′ is the linear wear of the teeth at j-th point of their profile during single engagement; the minimum durability min t of the gear will be observed at the point where the profile reaches the highest wear.
According to [16,18]: where v j = v is the sliding velocity at j-th points of the tooth profiles; j t′ = 2 bj /v 0 is the time of mesh- ing during the displacement of j-th point of tooth contact along their profile per the width of tooth contact area; v 0 = ω 1 r 1 sin α t the velocity of shift of the contact point along the tooth profile; ω 1 is the angular velocity of the pinion; f is the sliding friction factor; R m is the immediate tensile strength of the material; C k , m k are the factors of frictional wear resistance of gear materials at limit friction determined in compliance with the methodology presented in [16] based on the results of experimental tribological tests; 2 3.044 / is the width of tooth contact area.The sliding velocity is determined in the following way: compliance with the methodology presented in [16] based is the width of tooth contact area The sliding velocity is determined in the following way: ( ) . (10) where Changes in the profile curvature of the teeth due to their determined in the following way: The length of the chord replacing the involute between th way: 2ρ sin ε const Where The linear wear kjn h of the teeth at every j-th point of t block in the time jh t of their engagement.Accordingly, ( ( 0.35 where where compliance with the methodology presented in [16] based on the 2 3.044 / is the width of tooth contact area.
The sliding velocity is determined in the following way: ( ) . (10) where Due to the wear of the gear teeth, the curvature radii jh Changes in the profile curvature of the teeth due to their wear du determined in the following way: The length of the chord replacing the involute between the points way: 2ρ sin ε const Where The linear wear kjn h of the teeth at every j-th point of their pro block in the time jh t of their engagement.Accordingly, ( ) is the change in reduced radius of tooth profile curvature due to tooth wear.
Changes in the radii of tooth profile curvature can be measured after every revolution of the gear.This, however, leads to the extending of computational time.To avoid this, we applied a autors block method ([19] -Fig.1), which consists in measuring changes in the process parameters (h 1j , h 2j , ρ 1jh , ρ 2jh , ρ jh , p jhmax , 2b jh , jh t′ ) after a certain number of gear revolutions (engagement block B).Accordingly, a change in the radii of the curvature ρ kjh is determined in the following way [16]: ) , ( 3 Changes in the profile curvature of the teeth due to their wear during every block of their engagement is determined in the following way: The length of the chord replacing the involute between the points j -1, j + 1 is calculated in the following way: Changes in the profile curvature of the teeth due to their wear during every block of their engagement is determined in the following way: The length of the chord replacing the involute between the points j -1, j + 1 is calculated in the following way: 2ρ sin ε const Where arctan tan 1 arccos / cos t j w j w r r  .The linear wear kjn h of the teeth at every j-th point of their profile is calculated in this case after every block in the time jh t of their engagement.Accordingly, ( ) where 1 Changes in the profile curvature of the teeth due to their wear during every block of their engagement is determined in the following way: The length of the chord replacing the involute between the points j -1, j + 1 is calculated in the following way: 2ρ sin ε const Where arctan tan 1 arccos / cos t j w j w r r  .The linear wear kjn h of the teeth at every j-th point of their profile is calculated in this case after every block in the time jh t of their engagement.Accordingly, ( ) The linear wear kjn h′ of the teeth at every j-th point of their profile is calculated in this case after every block in the time jh t′ of their engagement.
The total wear 1 jn h and 2 jn h of the gear teeth at the j-points j of their profiles for a selected number of pinion rotations n 1s or gear rotations n 2s is determined by the following formulas: / 60 / 60
The results of the numerical solution are given in the figures below.Figure 1a shows the maximum initial contact pressures The gear teeth are the first to reach the maximum allowable wear at different characteristic points of tooth contact depending on the coefficient of profile correction at the start of triple engagement or at the end of the double engagement.Similar observations with respect to the points marking maximum wear can be made about the pinion teeth.Figure 4 illustrates the relationship between minimum gear durability and profile correction.
The maximum durability is exhibited by the gear with profile correction when 1 2 x x = − = 0.2 -its durability is higher by 1.55 times than that of the gear without profile correction.
To determine the effect of gear wheel width on the type of engagement and the above contact and tribological parameters, two types of helical gear were tested: one described by the gear wheel width W b = 30 mm and double-single- double engagement, and the other described by W b = 54.275mm and triple-double-triple engage- ment.Accordingly, Figure 5 shows the results of the maximum contact pressure max j p for the two tested width of gear wheels.
As a result, increasing the gear width by 1.81 times leads to a nearly proportionate de-

CONCLUSIONS
1. We determined the optimal gear width in helical cylindrical gears which ensures that the length of the contact line is maintained constant.
2. Using a new method for the determination of wear and durability of helical gears, depending on the profile correction and the type of engagement, a numerical solution block method was proposed to the problem of determining the maximum contact pressures, linear wear of the teeth and durability of the gear.3. The study was performed on a helical gear with optimized gear width ensuring a constant meshing force and on a helical gear with decreased wheel width.
4. It has been found that the increase in the gear width in the range between 30 and 54.275 mm results in an almost proportionate decrease in the maximum contact pressures.
5. The increase in the gear width by 1.81 times leads to a significant increase in the minimum gear durability -by 10.24÷8 times, depending on the applied coefficients of profile correction.
at the start of engagement; r w1 , r w2 are the radii of the pitch circles of the pinion and gear, respectively; r 1s = r a1 -r, r 20 = r a2 -r; r = 0.2 m is the rounding radius of the top land of a gear tooth; 1 1 cos b t r r = α is the radius of the base circle in the pinion; radius of the base circle in the gear; 1 1 / 2 cos r mz = β is the radius of the pitch circle in the pinion; Advances in Science and Technology Research Journal Vol.13(1

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follows is the analysis of two helical cylindrical gears with profile correction described by two different widths of gear wheels: l min = const when ε β = 1 and b W = 54.275mm (according to (2)); l min ≠ min l const when ε β <1 and b W = 30 mm.The gear with b W = 54.275mm is described by tripledouble-triple engagement while that with b W = 30 mm by double-single-double engagement.The angles of transition from double engagement ( ∆ϕ F ) to single and then, again, to double engagement ( ∆ϕ F ) are calculated in the follow-

is the length of tooth contact at the 2 w
are the radii of the pitch circles of the pinion and gear, respectively; m 2 , 0 = is the rounding radius of the top land of a gear tooth; radius of the base circle in the gear; us of the pitch circle in the pinion; 2 2 / 2 cos r mz =  is the radius of the pitch m is the addendum radius of the pinion; 2 2 a r r m = + is the addendum radius of on angle of the teeth; W b is the width of the pinion; m is the engagement ers of gear teeth; t  is the end-face pressure angle; w  is the pressure angle of ntact line during teeth engagement will be constant if   = const (1 or 2).
al parts of the coefficients   ,   .lysis of two helical cylindrical gears with profile correction described by two els: min l = const when   = 1 and W b = 54.275mm (according to (2));  min l 30 mm.The gear with W b = 54.275mm is described by triple-double-triple W b = 30 mm by double-single-double engagement.n from double engagement ( 2 1

;
the tooth pitch; α = 20 O is the pressure angle.
is the linear wear of teeth at selected j -th points of their profiles during one hour of operation of the gear; k = 1 -pinion, k =

1 
wear of the gear teeth, the curvature radii jh the change in reduced radius o Changes in the radii of tooth profile curvature ca This, however, leads to the extending of computational method ([19] -Fig.1), which consists in measuring cha 2 ρ jh , ρ jh , max jh p , 2 jh b , jh t ) after a certain numbe Accordingly, a change in the radii of the curvature ρ kjh is

2 ρ
in reduced radius of tooth p Changes in the radii of tooth profile curvature can be mea This, however, leads to the extending of computational time.T method ([19] -Fig.1), which consists in measuring changes in jh , ρ jh , max jh p , 2 jh b , jh t ) after a certain number of ge Accordingly, a change in the radii of the curvature ρ kjh is determi of block can be proportionate to the == 1 revolution (accurate solution), of block can be proportionate to the number of revolutions of the pinion -B = 1 revolution (accurate solution), in the type of engagement and in the initial maximum contact pressures max j p due to tooth wear, the gear durability min •10 6 , m 2 = 2.5; E =2.1•10 5 МPа.Lubrication involved the use of an oil described by the kinematic viscosity 50 Figure 1b illustrates the variations in theirmax jh p caused by the tooth wear * 2 h = 0.5 mm.The contact pressures

Fig. 1 .Fig. 2 .
Fig. 1.Variations in the initial contact pressures j p max during the meshing of teeth due to wear