NEW APPROACH BASED ON ANALYTIC-NUMERICAL METHOD FOR CALCULATING THE SLIDING DISTANCE OF TEETH PROFILE POINTS DURING GEARS MESHING

N. Agbetossou 1 , D. Koffi 2 , K. Attipou 1 , S. Tiem 3 and A. Afio 1 . 1. Département de Génie Mécanique, Ecole Nationale Supérieure d’Ingénieurs, Université de Lomé, B.P. 1515 Lomé, Togo. 2. Centre de recherche en Matériaux Ligno Cellulosiques, Département de génie mécanique, Ecole d’ingénierie, Université du Québec à Trois-Rivières, CP 500, Trois-Rivières, Québec, G9A 5H7, Canada. 3. Professeur Titulaire, Département de Génie Mécanique, Ecole Nationale Supérieure d’Ingénieurs, Université de Lomé, B.P. 1515. ...................................................................................................................... Manuscript Info Abstract ......................... ........................................................................ Manuscript History Received: 03 October 2019 Final Accepted: 05 November 2019 Published: December 2019

:-Below presents the characteristic data of the meshing modeling.  Figure 1a summarizes the principle for a meshing of spur gear. Here the driving gear rotates clockwise.     Figure 1b and Figure 2 show the relationships between the normalized positions and the corresponding points on the profiles. They also make it possible to calculate the geometrical data of the meshing. S 2 ''/P n S 2 */P n S 1 */P n S 1 ''/P n F(S/P n )

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In the normalized reference, the position of the theoretical contact start point is , the real contact start point is , the position of the end of the theoretical contact is , and the end of the real contact is . For the metal-to-metal meshing, the theoretical beginning and the real beginning are practically confounded, so the theoretical end and the real end of contact are also practically confounded. The formulas for calculating its different positions will be established later.

Calculation of geometrical data:-
Referring to the characteristic data and figures 1b and 2, we have:-(1); (2); (3); (4); (5); (6); (7); (8); √ (9); √ (10); √ (11) ; √ (12) ; The contact ratio is given by: ; As defined by the normalized reference, we have: (15); For gears in thermoplastic materials there is an extension of the contact before and after the beginning and the end of the theoretical contact [3]. We have indeed: The relative velocities of the point M i come from the rotation of the profiles around the instantaneous points N 1 and N 2 . The rolling speeds of the profiles with respect to the points N 1 and N 2 have the same intensity as the relative speeds, but are of opposite directions [7].
The sliding velocities V g1i and V g2i are such that: (26); (27); -The absolute velocities V 1 and V 2 are such that: (28); -The slip ratios which are the ratios of the sliding speeds over the rolling speeds are such that:   This being so, for any type of meshing (metal/metal, plastic/plastic or plastic/metal), it is known to determine for each meshing position on the line of action, the load supported by each tooth.
For the case of plastic/plastic meshing which interests us more particularly in this study, we have: Theoretical analysis of the calculation of the sliding distance of each point of the tooth profiles in meshing:-For this study, we limit ourselves between the beginnings of theoretical contact A and the end of theoretical contact B on the action line. This is the reality of metal gears. For gears made of polymeric materials, a similar study will be made for contact before point A and contact after point B.
The references and parameters calculation are defined in Figures 9, 10, 11 and 12.

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The reference (I, x, y) consists of the axis (I, y) which is the straight line linking the centers O 1 O 2 of the two gears, and the axis (I, x) perpendicular to the axis (I , y). The angle between the action line (A, B) and the axis (I, x) is the pressure angle α. Figure 9 shows the contact at point A and the contact at point B.
If we consider that a pair of teeth come into contact at point A, then immediately after the contact, because of the sudden Hertz deformation, the points of profile 1 of the driving tooth, from A towards its top over the half -Hertz contact length b, come all into contact at the same time with the profile 2 of the driven tooth.
Similarly, the points of the profile 2 of the driven tooth, from its top R a2 towards the base, on the half-contact length of Hertz b, come all into contact at the same time with the profile 1 of the driving tooth.   Apart from the edge cases at A and B, at each meshing position M between A and B, a pair of points (R 1ME , R 2ME ) comes into contact and another pair of points (R 1MS , R 2MS ) leaves the contact ( Figure 10, 11, 12). We obtain the following formulas: In approach between A and I, at the point of contact M, we obtain in the reference (I, x, y): In recess, we make the mesh in N 2 following positions: (59); (60); (the position of the pitch point I).
We set, the mesh step in recess, ; , as well as their relative velocities and corresponding input and output sliding velocities. By identifying the input assembly and the output assembly, the sliding distance of several representative positions is then calculated according to the desired accuracy and the sliding distance curve is constructed in the normalized reference. The flowchart of the Matlab simulation program is shown in Figure 13.
Below we do the simulation with materials commonly used in the field of plastic gears such as nylon, acetal and UHMWPE as well as our new composite material studied HDPE40B. In order to compare the appearance of the results with the results of the experimental wear tests available in the literature, we use the same meshing characteristics as those of the tests.  With the new approach, a non-zero sliding distance is obtained at the pitch point.

Results and Applications:-
The points of the profile of the driven gear in the vicinity of the outside radius vertex including the point of the vertex, have short sliding time because their contact takes place at the same time at the moment of the theoretical beginning at point A. Just after, they are the first to come out of contact starting with the point of the vertex. The phenomenon is the same for the points in the vicinity of the contact start point of the driver gear profile. The similar phenomenon occurs at the sudden mesh exit points in the vicinity of the driver gear tooth vertex and in the vicinity of the contact ending point of the driven gear profile. This explains a steep climb from the beginning of the curve and a steep descent also from the end of the curve. The big differences between the two methods are found in these areas. The relative error is 100% at the pitch point since the old method gives zero sliding distance at this point.
When we see the difference between the two methods from the curves and error evaluation in tables 4 to 6, we estimate that the impact of the improvement to the neighborhoods of the pitch point and the points of the head and foot that would bring the new approach in real applications will be notorious. The relative error evaluation between the two methods for different meshing conditions is shown in tables 4 to 6. Tableau 4-Sliding distance old approach dso1, new approach dsn1 and relative error Er between the two methods according to the normalized positions for acetal/acetal and T=10N.m driver gear.