A MATHEMATICAL METHOD FOR EVALUATING THE GEOMETRIC CONFORMITY OF TOOTH SURFACES IN BEVEL AND HYPOID GEAR TRANSMISSIONS

ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-12 1493 A MATHEMATICAL METHOD FOR EVALUATING THE GEOMETRIC CONFORMITY OF TOOTH SURFACES IN BEVEL AND HYPOID GEAR TRANSMISSIONS Abdurakhmanov Azamat Erkinovich Senior Lecturer, Ishmuratov Hikmat Kakharovich PhD., professor, Hamroev Ramzjon Komiljon ugli PhD., Associate Professor Email: [email protected] Tashkent State Technical University Abstract. This paper proposes a method for evaluating the geometric conformity of tooth surfaces in bevel and hypoid gear transmissions. The study determines the mutual geometric deviations between theoretical and actual (conjugate) tooth surfaces based on their spatial mathematical models. The deviations between the tooth surfaces are calculated in the normal direction for discretely selected grid points along the axial section, and the obtained values are approximated using a second-order surface. The conformity coefficients are determined using the least squares method, enabling a quantitative characterization of the spatial mismatch of the tooth surfaces. Keywords: bevel gear; hypoid gear; tooth surface; geometric conformity; spatial deviation; discretization; least squares method. МАТЕМАТИЧЕСКИЙ МЕТОД ОЦЕНКИ ГЕОМЕТРИЧЕСКОЙ СОГЛАСОВАННОСТИ ПОВЕРХНОСТЕЙ ЗУБЬЕВ В КОНИЧЕСКИХ И ГИПОИДНЫХ ЗУБЧАТЫХ ПЕРЕДАЧАХ Абдурахманов Азамат Эркинович старший преподаватель, Ишмуратов Хикмат Кахарович PhD., профессор, Хамрoев Рамзжон Комилжон угли PhD., доцент Эмаил: [email protected], Ташкентский государственный технический университет Аннотация. В статье предложен метод оценки геометрической согласованности поверхностей зубьев в конических и гипоидных зубчатых передачах. В рамках исследования на основе пространственных математических моделей определены взаимные геометрические отклонения между теоретической и фактической (сопрягаемой) поверхностями зубьев. Отклонения между поверхностями зубьев рассчитываются в нормальном направлении для дискретно выбранных узловых точек по аксиальному сечению, а полученные значения аппроксимируются поверхностью второго порядка. Коэффициенты согласованности определяются методом наименьших квадратов, что позволяет количественно охарактеризовать пространственную несогласованность поверхностей зубьев. Ключевые слова: коническая зубчатая передача; гипоидная зубчатая передача; поверхность зуба; геометрическая согласованность; пространственное отклонение; дискретизация; метод наименьших квадратов. ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-12 1494 Introduction. The performance quality and reliability of bevel and hypoid gear transmissions are directly dependent on the geometric accuracy and mutual conformity of tooth surfaces. Geometric mismatches between tooth surfaces lead to non-uniform formation of contact zones during the meshing process, localized load concentration, and an increase in wear and noise levels. Therefore, the quantitative evaluation of tooth surface geometric conformity represents an important scientific and engineering task. Despite the use of high-precision grinding technologies in practice, kinematic and geometric factors inherent in the machining process cause certain deviations in tooth surfaces. These deviations appear as differences between the theoretical and actual spatial shapes of the tooth surfaces and adversely affect the operational performance of the transmission. In this paper, a method for evaluating the geometric conformity of tooth surfaces in bevel and hypoid gear transmissions is proposed. The method enables the identification and quantitative assessment of spatial geometric deviations between theoretical and actual tooth surfaces and can be applied to analyze the meshing process and to substantiate technological solutions aimed at reducing wear and noise. Research Methods. In this study, spatial models of the theoretical and actual tooth surfaces of bevel and hypoid gear transmissions were developed. The geometric deviations between the tooth surfaces were calculated in the normal direction for discretely selected grid points along the axial section. The obtained deviations were approximated by a second-order surface, and the conformity coefficients were determined using the least squares method [1, 2]. Conjugate Tooth Surface of the Driving Bevel Gear. Figure 1 presents a mathematical model of the meshing process of a bevel gear pair. In this model, Sb(Xb, Yb, Zb) and Sd(Xd, Yd, Zd) are auxiliary coordinate systems, S₂(X₂, Y₂, Z₂) is the coordinate system of the driven bevel gear rotating about its axis, and S₁₀(X₁₀, Y₁₀, Z₁₀) is the coordinate system of the driving bevel gear rotating about its axis. The instantaneous rotation angles of the driven and driving gears are denoted by φ₂ and φ₁, respectively. Points O₂ and O₁ represent the intersection points of the gear axes, and E denotes the axial offset of the driving gear. Figure 1. Mathematical model of the meshing process of a bevel gear pair. ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-12 1495 When the spatial equation of the driven gear tooth surface r₂(u₂, θ₂) and its unit normal vector n₂(u₂, θ₂) are transformed into the coordinate system S₁₀ of the driving gear, the following equations are obtained: (0) 1 10 2 (2 2 2 2 2 2 2 2 2 2 2 2 0) 1 10 2 , , ( , , ) , ( , () ( ) ) d db b d db b r u M M M r u n u L L uL n       =  =   (1) Here, M denotes the transformation matrices between the corresponding coordinate systems, and L represents their 3×3 submatrices [3, 4]. If the tooth surface of the driving gear is fully conjugate to the tooth surface of the driven gear, the ratio of their angular velocities satisfies the following condition: where z₁ is the number of teeth of the driving gear and z₂ is the number of teeth of the driven gear. During the meshing process, the following meshing equation must be satisfied: (0) (0) 11 (0) (0) (21) 1 1(0) ( 22 2 2 2 2 2 2 2 0) 11 2 ( ) 0,,f rr ur un rr u              =       = = =  (2) where v denotes the relative velocity between the driving and driven gears, determined in the coordinate system S₁₀. The tooth surface determined by equation (2) represents the fully conjugate tooth surface of the driving gear, as it ensures continuous and smooth contact between the gear pair. Shape of Tooth Surface Mismatch. To determine the shape of the mismatch between tooth surfaces, the surface obtained based on equation (1) is regarded as the theoretical tooth surface of the driving bevel gear and is denoted by r₁. Figure 2. Geometric relationship between the theoretical and actual tooth surfaces of the driving gear. The surface generated according to equation (1) is considered the actual (conjugate) tooth surface of the driving gear and is denoted by r′₁. The spatial geometric differences arising between the theoretical and actual tooth surfaces are treated as deviation values between the tooth surfaces, which characterize the degree of their mutual mismatch [5]. ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-12 1496 Figure 2 illustrates the spatial relationship between the theoretical and actual tooth surfaces of the driving gear. Point M represents the midpoint of the tooth surface along the grid. If the vector equation of an arbitrary point P₀ on the actual tooth surface is given as: 1 2 (0) 1 (0) (0) 2 11 (0 22 2 2 2 2 2 ) 2 22 , , ( , , , , ( , , ) ( ) ( ) ) xu r u y u zu         =   The unit normal vector of this point is expressed as follows: 2 2 2 2 2 2 2 2 2 2 (0) 1 (0) (0) 2 11 (0) 1 2 ,() ( ) ( ) () , , , , , ,, x y z nu n u n u nu         =   P₀ nuqtadan chizilgan normal chiziq nazariy tish sirtini P₁ nuqtada kesib o‘tadi. Ushbu nuqtaning koordinatalari shaklida belgilanadi. Shunda, P₀ va P₁ nuqtalar orasidagi masofa: The normal line drawn from point P₀ intersects the theoretical tooth surface at point P₁. The coordinates of this point are expressed in the form. Accordingly, the distance between points P₀ and P₁ is given by: 01 PP  = which represents the magnitude of the deviation between the tooth surfaces (i.e., the degree of their mutual mismatch). According to the vector relationship shown in Figure 2, the following equation can be written: (0) (0) 1 1 1 rr n  =+ By expanding this expression into its components and combining it with equation (2), the following system of equations is obtained: 2 2 2 2 2 2 1 2 2 2 2 2 2 (1 2 2 2 (0) (0 ) 2 2 2 2 2 ) 1 2 2 1 2 2 2 1 (0 0) 11 (0) 1(0) (0) 11 , , , , , , , , ,, ,, ( ) ( ) () (,, ,, () () ) ( ) ( ) 0 p p x z p y x u n u y u n u r x y z u f z u n u u               += +=  =+=  =  (3) By substituting the coordinates of all grid points of the theoretical tooth surface into equation (3) and solving it sequentially, the deviation values Δδᵢ (i=1,2,3,…,m×n) at each point are determined. These values characterize the degree of geometric mismatch between the theoretical and actual tooth surfaces of the driving gear. The set of deviation values at all grid points constitutes the shape of the mutual mismatch of the tooth surfaces. This shape indirectly represents the level of geometric accuracy between the tooth surfaces and has a direct influence on contact accuracy, load distribution, and noise level. ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-12 1497 Results and Discussion. The deviation values between the tooth surfaces are essentially composed of the deviations at all grid points between the theoretical and actual tooth surfaces, and their combined effect is considered as the overall deviation of the tooth surface. Figure 3. Deviations of the tooth surface approximated by a second-order surface. In general, the deviations of the tooth surface can be approximated by a second-order surface (a paraboloid-shaped surface), as illustrated in Figure 3. In this case, the deviation magnitude is expressed as follows: 22 LH a X a Y a X a Y a XY     = + + + + (4) Here, (X, Y) are the coordinates of the grid points on the tooth surface (X denotes the coordinate along the tooth length direction, and Y denotes the coordinate along the tooth height direction), with the surface center taken as the origin of the coordinate system; a is the spiral angle error coefficient; b is the pressure angle error coefficient; c is the crowning coefficient along the tooth length; d is the crowning coefficient along the tooth profile; and e is the coefficient characterizing the twist distortion of the tooth surface. If the deviation values at all grid points on the tooth surface are known, equation (4) can be written in matrix form as follows: 22 11 1 1 1 1 1 22 22 2 2 2 2 2 22 3 3 3 3 3 3 3 22 ... ... ... ... ... ... m n m n L H m n m n m n m n mn a X Y X Y X Y a X Y X Y X Y X Y X Y X Y a a X Y X Y X Y a                          =            (5) The coordinates of each grid point on the tooth surface, (XLij, RLij), are determined in the coordinate system S(XL, YL). Based on this, the deviation values Δδᵢ (i=1,2,3,…,m×n) are calculated for all points along the tooth length and tooth height directions. To solve equation (5), the least squares method is applied. As a result, the coefficients , , , , LH a a a a a    which characterize the deviations between the tooth surfaces, are determined. Conclusion. In this study, spatial vector equations r₁(u, θ) and r′₁(u, θ) were obtained for the theoretical and actual (conjugate) tooth surfaces of bevel and hypoid gear transmissions. For discretely selected grid points along the axial section, the normal-direction deviation values Δδᵢ (i = ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-12 1498 1, 2, …, m×n) between the two tooth surfaces were calculated. During the study, spatial vector equations were derived for the theoretical and actual (conjugate) tooth surfaces of bevel and hypoid gear transmissions. The geometric deviations between the tooth surfaces were calculated using discretely selected grid points along the axial section, and the normal-direction deviation values were obtained for each point. 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