Method for calculating actual measurement tooth surface geodesic curvature of spiral bevel gear

Abstract

The invention relates to a spiral bevel gear actually-measured tooth surface geodesic curvature calculation method, which comprises the following steps: acquiring tooth surface actually-measured coordinates and a tooth thickness angle of a gear pair by means of gear measurement equipment, and determining a relative position relation of two actually-measured tooth surfaces through rotation transformation. The method comprises the following steps of: forming a pairing dot matrix by actual measurement point coordinates of a large gear and a small gear according to a geometric position system when gear teeth work, calculating a meshing rotation angle of the large gear and the small gear and a spatial distance between pairing points under the condition that normal vectors are equal, and calculating a tooth surface contact path by taking a tooth surface parameter and the meshing rotation angle corresponding to a point pair with the minimum spatial distance as initial values of actual measurement tooth surface contact analysis. And determining the position of a design reference point through a transmission ratio condition or an iteration method, and projecting all contact points to a tangent plane of the design reference point. The projection points are fitted into a plane curve, the geodesic curvature at the design reference point is calculated, the gear meshing performance is judged through the geodesic curvature, the solving process has high precision, and complexity brought by direct use of a space curve NURBS curve for fitting is avoided.

Description

Technical Field

[0001] The invention belongs to the technical field of gear transmission, and in particular relates to a method for calculating the geodesic curvature of a measured tooth surface of a spiral bevel gear. Background Art

[0002] Spiral bevel gears are key transmission components in automobile drive axles. Their performance directly affects the comfort, load-bearing capacity and service life of automobiles. Due to thermal deformation, machine tool errors, cutting force changes and other unpredictable factors, the actual gear tooth surface will deviate from the theoretical tooth surface, resulting in tooth surface errors. Tooth surface errors will cause adverse factors such as gear contact area movement and transmission errors, which will further lead to problems such as high stress concentration and premature failure caused by edge contact during operation. It is also one of the main factors for vibration and noise in automobile rear axle transmission. The ideal gear transmission is conjugate, but the fully conjugated gear pair has no adjustability. Manufacturing and installation errors and load deformation will cause load concentration and damage the gear teeth. The quasi-conjugate characteristics of the point contact tooth surface overcome the above shortcomings. The instantaneous transmission ratio changes during the entire meshing process. Only the instantaneous transmission ratio at the design reference point is equal to the nominal transmission ratio of the gear pair. The meshing performance of the tooth surface is controlled by the first-order and second-order contact parameters of the design reference point. On the one hand, the contact path can no longer be calculated using the traditional tooth surface contact analysis method for the measured tooth surface; on the other hand, the design reference point position of the measured tooth surface and its second-order contact characteristics are of great significance to the gear meshing performance. Therefore, it is impossible to accurately estimate the meshing performance of the measured tooth surface, and it is necessary to use methods such as rolling inspection and VH inspection to achieve it, which will increase the trial production time and cost of the product. The geodesic curvature of the geodesic (also known as the short-range line) is the second-order contact parameter of the tooth surface, especially the geodesic curvature at the design reference point has an important influence on the direction of the contact path. The geodesic refers to the shortest curve connecting two points on the surface. The common geodesic feature is that the principal normal of the curve at any point on the surface coincides with the normal of the surface, or the occurrence line on the surface is a straight line. Therefore, the geodesic curvature of the geodesic is equal to zero. When the geodesic curvature of the contact point on the contact path is equal to zero, it can ensure that the contact path is not bent and obtain better meshing performance. In 1981, Professor Litvin of the United States proposed using the local synthesis method to ensure that the contact path is a geodesic near the design reference point, that is, the geodesic curvature of the design reference point is equal to zero, thereby avoiding the bending of the contact path, reducing the sensitivity of the contact area to installation errors, and improving the lubrication characteristics of the gear pair. However, this method is limited to theoretical tooth surfaces, and there are few literature reports on the geodesic characteristics of the contact points of the measured tooth surfaces at home and abroad. Therefore, it is of great theoretical significance and engineering value to use the geodesic curvature calculation of the design reference point to evaluate the meshing performance of the measured tooth surface of the gear pair.

[0003] CN104330065A discloses a method for testing the curvature interference of spiral bevel gear tooth surfaces. The method can directly determine whether the two tooth surfaces have curvature interference according to the position vector and normal vector of the tooth surface point detected by the gear measurement center. If interference occurs, the interference position can be determined. However, the above patent cannot solve the problem of accurately estimating the meshing performance of the measured tooth surface of spiral bevel gears. Summary of the invention

[0004] In order to solve the problem of accurately predicting the meshing performance of the measured tooth surface of spiral bevel gears, the present invention provides an accurate calculation method of the geodesic curvature at the design reference point based on the measured tooth surface data obtained by a three-dimensional coordinate measuring machine, and the meshing transmission performance of the gear pair is judged by its value.

[0005] The technical solution adopted by the present invention to solve the technical problem includes the following steps:

[0006] A method for calculating the geodesic curvature of a spiral bevel gear tooth surface, comprising the following steps:

[0007] Step (1): Based on the tooth surface mesh division, the measured tooth surface coordinates of the concave and convex surfaces of the large and small wheels are measured; the measured tooth surfaces are rotated according to the tooth pitch or detection dimension data to obtain the accurate positional relationship between the two measured tooth surfaces;

[0008] Step (2): After the measured tooth surface coordinates are processed in step (1), the concave and convex surfaces of the large and small wheels are respectively fitted with high precision using bicubic non-uniform rational B-spline surfaces to obtain the mathematical expressions of the digital tooth surface position vector and unit normal vector of the measured tooth surface; in the meshing coordinate system, during the meshing process of the digital paired tooth surfaces of the large and small wheels, a contact analysis model of the measured tooth surfaces of the spiral bevel gear pair is established according to the continuous tangency condition at the contact point;

[0009] Step (3): Take the grid points of the paired tooth surface in order to form a paired point matrix, calculate the meshing angle of the large and small wheels and the spatial distance of the paired points according to the normal vector equality condition; traverse the paired point matrix of the entire tooth surface to find a pair of points with the smallest spatial distance, and use the parameters and the rotation angle of the point as the initial values ​​for solving the measured tooth surface contact analysis; solve the nonlinear equation group of the tooth surface contact analysis with the help of numerical methods, obtain the tooth surface contact points, and form a contact path from a series of contact points;

[0010] Step (4): by the constraint condition that the instantaneous transmission ratio of the contact point is equal to the theoretical transmission ratio, combined with the measured tooth surface contact analysis model, the position of the design reference point is obtained, or the position of the design reference point that meets the accuracy requirements is obtained through an iterative method; the contact point on the contact path is projected onto the tangent plane of the design reference point, and then the NURBS curve is used to fit the plane curve, and the curvature of the design reference point is calculated according to the curvature formula of the parameterized curve, that is, the geodesic curvature of the design reference point along the tangent direction of the contact path, and the gear meshing performance is judged by the geodesic curvature.

[0011] The step (1) is specifically as follows: after the spiral bevel gear is heat treated and ground, the tooth surface measurement grid is divided according to the requirements of AGMA2009-B01, the tooth length is taken as MW rows and the tooth height is taken as MI columns, and MI×MW grid points are obtained. After the measurement path is planned, the actual tooth surface coordinates of the concave and convex surfaces of the large and small wheels are respectively measured on a three-coordinate measuring machine or a gear measurement center; according to the relative positions between the concave and convex surfaces of the large and small wheels, an accurate gear tooth model is obtained.

[0012] In step (1), if the double-sided method is used, the convex surface of the small wheel and the concave surface of the large wheel are used as reference surfaces, and the concave surface of the small wheel and the convex surface of the large wheel are rotated around their respective axes by a pitch angle θ=2π / z, where z is the number of teeth of the gear being measured; if the single-sided method is used, the relative position between the concave surface and the convex surface of the large and small wheels is determined according to the tooth thickness angle θ in the measured tooth surface data file to obtain an accurate gear tooth model.

[0013] Step (2) comprises the following steps:

[0014] Step (2.1): Taking the measured tooth surface as the shape value point, first calculate the control vertices of each NURBS curve along the u or v direction according to the shape value point, and then use the obtained control vertices as the shape value point to calculate the control polygon mesh vertices along the u or v direction, that is, the control vertices of the NURBS surface and the corresponding weight factors. Therefore, the position vector of the bicubic NURBS surface with second-order continuity is

[0015]

[0016] The unit normal vector is expressed as

[0017]

[0018] Where: m is the number of control vertices in the u direction; n is the number of control vertices in the v direction; V i,j is the control vertex of the surface; W i,j V i,j The weight factor of B i,3 is the cubic B-spline basis function along the u direction; B j,3 is the cubic B-spline basis function along the v direction; P u (u,v) and Pv (u,v) are the first-order derivatives of the position vector P(u,v) along u and v respectively;

[0019] Step (2.2): The tooth surfaces of the large and small wheels are matched, that is, the driving surface is the concave surface of the small wheel and the convex surface of the large wheel, and the non-driving surface is the convex surface of the small wheel and the concave surface of the large wheel. The position vectors of the matched digital tooth surfaces are r1=r1(u1,v1) and r2=r2(u2,v2), where u1,v1 are the tooth surface parameters of the small wheel and 0≤u1≤1,0≤v1≤1; u2,v2 are the tooth surface parameters of the large wheel and 0≤u2≤1,0≤v2≤1; the unit normal vectors of the matched gears are n1(u1,v1) and n2(u2,v2); the digital tooth surfaces Σ1 and Σ2 are in the fixed coordinate system S h The position vector and normal vector of the tooth surface should be equal; the coordinate systems S1 and S2 are the digital tooth surface dynamic coordinate systems of the small wheel and the large wheel respectively. h is a fixed coordinate system; the small wheel and the large wheel rotate around their respective axes X1 and X2, the angular velocity of the large wheel is ω2, and the angular velocity of the small wheel is ω1. The position vector and unit normal vector of the digitized tooth surface ∑1 and ∑2 are expressed in the coordinate system S h middle

[0020]

[0021] In the formula, and are the meshing angles of the small wheel digital tooth surface ∑1 and the large wheel digital tooth surface ∑2 respectively; M h1 From coordinate system S1 to coordinate system S h The homogeneous coordinate transformation matrix, M h2 =M hd M d2 From coordinate system S2 to coordinate system S h The homogeneous coordinate transformation matrix, L h1 and L h2 M h1 and M h2 The matrix without the last row and column; the digital tooth surfaces ∑1 and ∑2 of the small wheel and the large wheel are in the fixed coordinate system S h The continuous contact tangency in the tooth surface is obtained to obtain the real tooth contact analysis model (RTCA: RealTooth Contact Analysis), and its basic equation is:

[0022]

[0023] Step (3) is specifically as follows: the measurement sequence of the actual tooth surface is as follows: the tooth length direction is from the outer end to the inner end, the tooth shape direction is from the tooth root to the tooth top, according to the corresponding relationship between the geometric positions of the large and small wheels during meshing transmission, that is, the condition that the small wheel tooth root contacts the large wheel tooth top or the small wheel tooth top contacts the large wheel tooth root, take the same tooth length grid number, and the tooth shape direction should be paired with the tooth root and the tooth top, that is, the large wheel tooth shape grid number 1 point and the small wheel tooth shape grid number MI point form a point pair, the large wheel tooth shape grid number MI point and the small wheel tooth shape grid number 1 point form a point pair, and the remaining points are similar, forming a total of MI×MW paired point matrices; for any paired point, the small wheel meshing angle is

[0024]

[0025] in,

[0026]

[0027] The large wheel meshing angle is

[0028]

[0029] in,

[0030]

[0031] Where, the wheel rotation axis vector p1 = [1 0 0] T , large wheel rotation axis vector p2 = [cosΣ 0 -sinΣ] T , ∑ is the axis intersection angle; after substituting the obtained large and small wheel rotation angles into the first equation of equations (1) and (2), according to the distance formula between two points in space

[0032]

[0033] Traverse all MI×MW pairing point arrays, calculate their spatial distance, and take the small wheel digital tooth surface parameter u of the pairing point with the smallest distance 10 ,v 10 、Digital tooth surface parameters of large wheel u 20 ,v 20 And the big and small wheel turning angle As the initial value for solving the measured tooth surface contact analysis

[0034] Small wheel turning angle As input, that is, fixed The value of u1,v1,u2,v2, is a nonlinear equation set with unknown quantities, thus obtaining a contact point between two meshing digital tooth surfaces; then changing The value of is calculated and the solution is continued until the contact point is beyond the effective boundary of the tooth surface. These instantaneous contact points of the tooth surface constitute the tooth surface contact path.

[0035] Step (4) specifically includes the following steps:

[0036] Step (4.1): The meshing tooth surface of the spiral bevel gear is a point contact conjugate surface, which is given by the formula Calculation shows that the instantaneous transmission ratio at the design reference point is equal to the theoretical transmission ratio, that is, the transmission ratio condition is met.

[0037] i 21 =z2 / z1(4)where z1 and z2 are the number of teeth of the large and small wheels respectively; On the one hand, the position vector r of the design reference point M is obtained by using the quasi-Newton method m 、Normal vector n m and tangent vector t m On the other hand, the position of the design reference point that meets certain accuracy requirements is obtained through the iteration method, that is, given the error ε, the small wheel rotates a small amount The given transmission ratio accuracy requirements can be met through continuous iteration;

[0038] Step (4.2): Establish a tangent plane coordinate system S with the design reference point M as the origin t , whose three coordinate axes are respectively tangent vector t m For X t Axis, normal vector n m Z t Axis, vector t m ×n m Y t Axis, in coordinate system S i The subscript i=1 indicates a small wheel, i=2 indicates a large wheel, and X i axisi i =[1 0 0] T ,Y i Axis j i =[0 1 0] T ,z axis i =[0 0 1] T , from the coordinate system S i To the tangent plane coordinate system S t The transformation matrix is

[0039]

[0040] Discrete points P on the tooth surface on the contact path i , i=1,2,…,n c , after the above transformation, converted to S t The corresponding point after is Q i , i=1,2,…,nc , using the NURBS curve formula to fit the projection points, the k-order NURBS curve can be expressed as a rational polynomial vector function

[0041]

[0042] Among them, d i ,i=0,1,…,n c is called the control vertex, w i ,i=0,1,…,n c is the weight factor attached to the corresponding control vertex, N i,k (u) is the i-th k-order B-spline basis function, and the curvature at the design reference point M is p′(u) and p″(u) are the first-order derivative and second-order derivative of p(u) respectively. The curvature is the geodesic curvature of the contact path. If the geodesic curvature value of the design reference point is larger, the contact path is more curved. Conversely, the closer the geodesic curvature value is to zero, the closer the contact path is to the geodesic line.

[0043] The present invention adopting the above technical solution has the following technical effects:

[0044] (1) First fix one side of the tooth surface, and then determine the relative position relationship between the concave and convex sides of the measured tooth surface through the rotation transformation of the pitch angle or tooth thickness angle, so as to provide an accurate model for the subsequent reverse modeling and analysis of the measured tooth surface.

[0045] (2) Based on the measured grid points, a pairing point matrix is ​​formed, and the contact point tooth surface parameters and meshing angle are preliminarily determined by the same normal vector condition, which effectively solves the initial value problem in the measured tooth surface contact analysis and enables fast and accurate convergence when solving the measured tooth surface contact points.

[0046] (3) The position of the design reference point is accurately calculated through the transmission ratio condition or iterative method, thereby determining the center position and deviation of the contact mark. If the design reference point is not within the effective range of the tooth surface or is near the edge of the gear tooth, it can be determined that the actual meshing performance of the gear is poor.

[0047] (4) The discrete points on the contact path are projected onto the tangent plane of the design reference point, and the geodesic curvature of the design reference point is calculated using the NURBS plane curve fitting. The geodesic curvature value is used to determine whether the area near the design reference point is a geodesic line, thereby estimating the sensitivity of the contact mark to installation errors and the lubrication performance. In addition, the solution process of the geodesic curvature of the design reference point is highly accurate, avoiding the complexity brought by directly fitting the spatial curve NURBS curve. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 It is the measured tooth surface of the small wheel before the rotation treatment of the present invention;

[0049] Figure 2 It is the measured tooth surface of the small wheel after the rotation treatment of the present invention;

[0050] Figure 3 It is the digital tooth surface and control vertex of the concave surface of the small wheel;

[0051] Figure 4 It is the digitized tooth surface and control vertex of the large wheel convex surface;

[0052] Figure 5 is the meshing coordinate system of the paired tooth surfaces of the present invention;

[0053] Figure 6 It is the measured grid point pairing lattice of the present invention;

[0054] Figure 7 It is the design reference point tangent plane coordinate system of the present invention;

[0055] Figure 8 It is the measured tooth contact path and design reference point;

[0056] Fig. 9 It is the overall flow chart of the present invention. DETAILED DESCRIPTION

[0057] like Fig. 9 As shown, a method for calculating the geodesic curvature of the measured tooth surface of a spiral bevel gear comprises the following steps:

[0058] Step (1): Based on the tooth surface mesh division, the measured tooth surface coordinates of the concave and convex surfaces of the large and small wheels are measured by means of a three-dimensional measuring machine or a gear measurement center. The measured tooth surface is rotated according to the pitch or detection dimension data to obtain the accurate positional relationship between the two measured tooth surfaces.

[0059] Specifically, after the spiral bevel gears are heat treated and ground, they are divided into tooth surface measurement grids according to the requirements of AGMA2009-B01. The tooth length is taken in MW rows and the tooth height is taken in MI columns. MI×MW grid points are obtained. After the measurement path is planned, the actual tooth surface coordinates of the concave and convex surfaces of the large and small wheels are measured on a three-coordinate measuring machine or a gear measurement center, such as Figure 1 As shown, Figure 1 The tooth surface of the small wheel is measured before the rotation process, where 1 is the concave surface of the small wheel and 2 is the convex surface of the small wheel.

[0060] The double-sided method is used for machining. The convex surface of the small wheel and the concave surface of the large wheel are used as reference surfaces. The concave surface of the small wheel and the convex surface of the large wheel are rotated around their respective axes by a pitch angle θ=2π / z, where z is the number of teeth of the gear being measured. If the single-sided method is used for machining, the relative position between the concave and convex surfaces of the large and small wheels is determined according to the tooth thickness angle θ in the measured tooth surface data file to obtain an accurate gear tooth model, such as Figure 2 shown.

[0061] Step (2): After the measured tooth surface coordinates are processed in step (1), the concave and convex surfaces of the large and small wheels are fitted with high precision using bicubic non-uniform rational B-spline surfaces to obtain the mathematical expressions of the digital tooth surface position vector and unit normal vector of the measured tooth surface. In the meshing coordinate system, during the meshing process of the digital paired tooth surfaces of the large and small wheels, a contact analysis model of the measured tooth surfaces of the spiral bevel gear pair is established based on the continuous tangency condition at the contact point.

[0062] The specific steps include:

[0063] Step (2.1): Taking the measured tooth surface as the form value point, first calculate the control vertices of each NURBS curve (non-uniform rational B-spline) along the u (or v) direction according to the form value point, and then use the obtained control vertices as the form value point to calculate the control polygon mesh vertices along the u (or v) direction, that is, the control vertices of the NURBS surface and the corresponding weight factors. Therefore, the position vector of the bicubic NURBS surface with second-order continuity is

[0064]

[0065] The unit normal vector is expressed as

[0066]

[0067] Where: m is the number of control vertices in the u direction; n is the number of control vertices in the v direction; V i,j is the control vertex of the surface; W i,j V i,j The weight factor of B i,3 is the cubic B-spline basis function along the u direction; B j,3 is the cubic B-spline basis function along the v direction; P u (u,v) and P v (u,v) are the first-order derivatives of the position vector P(u,v) along u and v respectively.

[0068] Step (2.2): The tooth surfaces of the large and small wheels are matched, that is, the driving surface is the concave surface of the small wheel and the convex surface of the large wheel, and the non-driving surface is the convex surface of the small wheel and the concave surface of the large wheel. The position vectors of the matched digital tooth surfaces are r1=r1(u1,v1) and r2=r2(u2,v2). Among them, u1,v1 are the parameters of the tooth surface of the small wheel and 0≤u1≤1,0≤v1≤1; u2,v2 are the parameters of the tooth surface of the large wheel and 0≤u2≤1,0≤v2≤1. The unit normal vectors of the matched gears are n1(u1,v1) and n2(u2,v2). The digital tooth surfaces ∑1 and ∑2 are in the fixed coordinate system S h In continuous contact tangency, the position vector and normal vector of the tooth surface should be equal. Figure 3 As shown in the figure, coordinate systems S1 and S2 are the digital tooth surface dynamic coordinate systems of the small wheel and the large wheel respectively.h is a fixed coordinate system. The small wheel and the large wheel rotate around their respective axes X1 and X2, with the angular velocity of the large wheel being ω2 and the angular velocity of the small wheel being ω1. The position vector and unit normal vector of the digitized tooth surfaces Σ1 and Σ2 are expressed in the coordinate system S h middle

[0069]

[0070] In the formula, and are the meshing angles of the small wheel digital tooth surface ∑1 and the large wheel digital tooth surface ∑2 respectively; M h1 From coordinate system S1 to coordinate system S h The homogeneous coordinate transformation matrix, M h2 =M hd M d2 From coordinate system S2 to coordinate system S h The homogeneous coordinate transformation matrix, L h1 and L h2 M h1 and M h2 The matrix without the last row and column. The digital tooth surfaces ∑1 and ∑2 of the small and large wheels are in the fixed coordinate system S h The continuous contact tangency in the tooth surface is obtained to obtain the real tooth contact analysis model (RTCA: RealTooth Contact Analysis), and its basic equation is:

[0071]

[0072] Formula (3) is the basic equation for RTCA analysis. The first vector equation contains three scalar equations, which are the conditions satisfied by the two tooth surfaces at point M. The second vector equation is the condition satisfied by the tangency at point M. Only two equations are independent, so there are five independent scalar equations in total in equation (3), but the system of equations contains six unknowns u1, v1, u2,v2,

[0073] Step (3): Take the grid points of the paired tooth surface in a certain order to form a paired point matrix, and calculate the meshing angle of the large and small wheels and the spatial distance of the paired points based on the normal vector equality condition. Traverse the paired point matrix of the entire tooth surface to find a pair of points with the smallest spatial distance, and use the parameters and angle of this point as the initial values ​​for solving the measured tooth surface contact analysis. Use numerical methods to solve the nonlinear equations of the tooth surface contact analysis, obtain the tooth surface contact points, and form a contact path from a series of contact points.

[0074] Specifically, the measurement order of the actual tooth surface is as follows: the tooth length direction is from the outer end to the inner end, and the tooth shape direction is from the tooth root to the tooth top. According to the corresponding relationship between the geometric positions of the large and small wheels during meshing transmission, that is, the small wheel tooth root contacts the large wheel tooth top or the small wheel tooth top contacts the large wheel tooth root, the same tooth length grid number is selected, and the tooth shape direction should be paired with the tooth root and the tooth top, that is, the large wheel tooth shape grid number 1 point and the small wheel tooth shape grid number MI point form a point pair, the large wheel tooth shape grid number MI point and the small wheel tooth shape grid number 1 point form a point pair, and the remaining points are similar, forming a total of MI×MW paired point arrays. For any paired point, the small wheel meshing angle is

[0075]

[0076] in,

[0077]

[0078] The large wheel meshing angle is

[0079]

[0080] in,

[0081]

[0082] Where, the wheel rotation axis vector p1 = [1 0 0] T , large wheel rotation axis vector p2 = [cos∑ 0 -sin∑] T , ∑ is the axis intersection angle. Substituting the obtained large and small wheel rotation angles into the first equation of equations (1) and (2), according to the distance formula between two points in space:

[0083]

[0084] Traverse all MI×MW pairing point arrays, calculate their spatial distance, and take the small wheel digital tooth surface parameter u of the pairing point with the smallest distance 10 ,v 10 、Digital tooth surface parameters of large wheel u 20 ,v 20 And the big and small wheel turning angle As the initial value for solving the measured tooth surface contact analysis

[0085] Small wheel turning angle As input, that is, fixed The value of u1,v1,u2,v2, is a nonlinear equation set with unknown quantities, thus obtaining a contact point between two meshing digital tooth surfaces. Then change The value of is calculated and the solution is continued until the contact point is beyond the effective boundary of the tooth surface. These instantaneous contact points of the tooth surface constitute the tooth surface contact path.

[0086] Step (4): By combining the constraint condition that the instantaneous transmission ratio of the contact point is equal to the theoretical transmission ratio and the measured tooth surface contact analysis model, the position of the design reference point is obtained, or the position of the design reference point that meets certain accuracy requirements is obtained through an iterative method. The contact point on the contact path is projected onto the tangent plane of the design reference point, and then the NURBS curve is used to fit the plane curve. The curvature of the design reference point is calculated according to the curvature formula of the parameterized curve, which is the geodesic curvature of the design reference point along the tangent direction of the contact path. The gear meshing performance is judged by the geodesic curvature.

[0087] The specific steps include:

[0088] Step (4.1): The meshing tooth surface of the spiral bevel gear is a point contact conjugate surface, and its transmission ratio is not constant. It can be expressed by Calculation shows that the instantaneous transmission ratio at the design reference point is equal to the theoretical transmission ratio, that is, the transmission ratio condition is met.

[0089] i 21 =z2 / z1(4) Where z1 and z2 are the number of teeth of the large and small wheels respectively. The contact path obtained by (5) above generally does not pass through the design reference point. Here, a nonlinear equation system containing 6 unknowns and 6 equations is established by combining equations (3) and (4). On the one hand, the position vector r of the design reference point M is obtained by solving it with the help of the quasi-Newton method m 、Normal vector n m and tangent vector t m ,like Figure 5 On the other hand, the position of the design reference point that meets certain accuracy requirements is obtained by iteration, that is, given an error ε, the small wheel rotates a small amount The given transmission ratio accuracy requirements can be met through continuous iteration.

[0090] Step (4.2): Establish a tangent plane coordinate system S with the design reference point M as the origin t , whose three coordinate axes are respectively tangent vector t m For X t Axis, normal vector n m Z t Axis, vector t m ×n m Y t Axis. In the coordinate system S i The subscript i=1 indicates a small wheel, i=2 indicates a large wheel, and X i axisi i =[1 0 0] T ,Y i Axis ji =[0 1 0] T ,z axis i =[0 0 1] T , from the coordinate system S i To the tangent plane coordinate system S t The transformation matrix is

[0091]

[0092] Discrete points P on the tooth surface on the contact path i (i=1,2,…,n c ) After the above transformation, it is converted to S t The corresponding point after is Q i (i=1,2,…,n c ),like Figure 5 As shown, the projection points are fitted using the NURBS curve formula. Then the k-order NURBS curve can be expressed as a rational polynomial vector function

[0093]

[0094] Among them, d i (i=0,1,…,n c ) is called the control vertex, w i (i=0,1,…,n c ) is the weight factor attached to the corresponding control vertex, N i,k (u) is the i-th k-order B-spline basis function, and the curvature at the design reference point M is p′(u) and p″(u) are the first-order derivative and second-order derivative of p(u) respectively. The curvature is the geodesic curvature of the contact path. If the geodesic curvature value of the design reference point is larger, the contact path is more curved. Conversely, when the geodesic curvature value is closer to zero, the contact path is closer to the geodesic line. At this time, the sensitivity of the meshing performance to the installation error can be reduced, the lubrication conditions of the gear pair can be improved, and the quality of the product can be improved.

[0095]

Design Example

[0096] Taking a pair of rear axle spiral bevel gears as an example, the basic parameters of the gear pair are: the number of small gear teeth z1 = 11, the number of large gear teeth z2 = 41, the axis angle ∑ = 90°, and the offset distance E = 35mm. Taking the driving surface (small gear concave surface and large gear convex surface) as an example, the tooth surface measurement grid is 11×11. Figure 1 The tooth surface coordinate data obtained from the three-dimensional coordinate measuring machine, the small gear tooth thickness angle is 17.9005°, the large gear tooth thickness angle is 3.8968°, the initial two sides of the tooth surface cross, after the rotation processing of the measured tooth surface as shown Figure 2 shown. Figure 3 and Figure 4They are the concave and convex digitized tooth surfaces of the small wheel after bicubic NURBS fitting. The black dots are control vertices, where the number of control vertices in the u direction is m=13, and the number of control vertices in the v direction is n=13. The weight coefficient W is taken to simplify the calculation. i,j =1.0, and the remaining tooth surfaces are digitized using the same method. Figure 6 The paired point matrix of the measured tooth surface of the driving surface has a total of 11×11 pairs of paired points. When the initial value of the tooth surface contact analysis is determined, the paired points formed by the 8th point of the tooth height and the 6th point of the tooth length in the small wheel tooth surface mesh and the 3rd point of the tooth height and the 6th point of the tooth length in the large wheel tooth surface mesh have the minimum spatial distance d=0.4170mm. At this time, the initial value X0=[0.7143 0.5232 0.6968 0.5171 0.1828 0.0649] T The value of the small wheel angle step is related to the length of the contact path. If the step is larger, there will be fewer contact points, which will have a certain impact on the plane curve fitting accuracy and geodesic curvature calculation results; if the step is smaller, there will be more contact points, the calculation accuracy is high, and the calculation efficiency is low. Therefore, the calculation accuracy and efficiency should be considered comprehensively to determine the step. Here, the small wheel step is 1.3636°, and the driving surface has a total of 40 contact points. The theoretical transmission ratio of the gear pair is 3.72727. When using the iteration method to calculate the design reference point, the small wheel angle is taken as Accuracy ε=1.0×10 -4 , the instantaneous transmission ratio obtained by iteration is 3.72717, which meets the accuracy requirements. At this time, the position vector, unit normal vector and unit tangent vector of the small wheel and the large wheel at the design reference point are r m1 =[165.14524.5794 -53.6113] T 、n m1 =[-0.7348 -0.6468 -0.2039] T ,t m1 =[0.2704 -0.5551 0.7865] T and r m2 =[53.7741 -2.5196 -168.4172] T 、n m2 =[0.1709 0.7968 0.57949] T ,t m2 =[-0.86690.4011 -0.2958] T Finally, the geodesic curvature of the wheel at the design reference point is calculated to be 3.4375×10 -3 The geodesic curvature of the large wheel at the design reference point is 4.3896×10 -3,From this it can be seen that the geodesic curvature of the measured tooth surface at the design reference point is small, and it has better meshing transmission performance.

Claims

1. A method for calculating the geodesic curvature of the measured tooth surface of a spiral bevel gear, characterized in that: The following steps are involved: Step (1): Based on the tooth surface mesh division, the measured tooth surface coordinates of the concave and convex surfaces of the large and small wheels are measured; the measured tooth surfaces are rotated according to the tooth pitch or detection dimension data to obtain the accurate positional relationship between the two measured tooth surfaces; Step (2): After the measured tooth surface coordinates are processed in step (1), the concave and convex surfaces of the large and small wheels are respectively fitted with high precision using bicubic non-uniform rational B-spline surfaces to obtain the mathematical expressions of the digital tooth surface position vector and unit normal vector of the measured tooth surface; in the meshing coordinate system, during the meshing process of the digital paired tooth surfaces of the large and small wheels, a contact analysis model of the measured tooth surfaces of the spiral bevel gear pair is established according to the continuous tangency condition at the contact point; Step (3): Take the grid points of the paired tooth surface in order to form a paired point matrix, and calculate the meshing angle of the large and small wheels and the spatial distance of the paired points according to the normal vector equality condition; traverse the paired point matrix of the entire tooth surface to find a pair of points with the smallest spatial distance, and use the parameters and angle of the point as the initial values ​​for solving the measured tooth surface contact analysis; The nonlinear equations of tooth surface contact analysis are solved by numerical methods to obtain the tooth surface contact points, and the contact path is formed by a series of contact points; Step (4): by the constraint condition that the instantaneous transmission ratio of the contact point is equal to the theoretical transmission ratio, combined with the measured tooth surface contact analysis model, the position of the design reference point is obtained, or the position of the design reference point that meets the accuracy requirements is obtained through an iterative method; the contact point on the contact path is projected onto the tangent plane of the design reference point, and then the NURBS curve is used to fit the plane curve, and the curvature of the design reference point is calculated according to the curvature formula of the parameterized curve, that is, the geodesic curvature of the design reference point along the tangent direction of the contact path, and the gear meshing performance is judged by the geodesic curvature.

2. The method for calculating the geodesic curvature of the measured tooth surface of a spiral bevel gear according to claim 1, characterized in that: The step (1) is specifically as follows: after the spiral bevel gear is heat treated and ground, the tooth surface measurement grid is divided according to the requirements of AGMA2009-B01, the tooth length is taken as MW rows and the tooth height is taken as MI columns, and MI×MW grid points are obtained. After the measurement path is planned, the actual tooth surface coordinates of the concave and convex surfaces of the large and small wheels are respectively measured on a three-coordinate measuring machine or a gear measurement center; according to the relative positions between the concave and convex surfaces of the large and small wheels, an accurate gear tooth model is obtained.

3. The method for calculating the geodesic curvature of the measured tooth surface of a spiral bevel gear according to claim 2, characterized in that: In step (1), if the double-sided method is used, the convex surface of the small wheel and the concave surface of the large wheel are used as reference surfaces, and the concave surface of the small wheel and the convex surface of the large wheel are rotated around their respective axes by a pitch angle θ=2π / z, where z is the number of teeth of the gear being measured; if the single-sided method is used, the relative position between the concave surface and the convex surface of the large and small wheels is determined according to the tooth thickness angle θ in the measured tooth surface data file to obtain an accurate gear tooth model.

4. The method for calculating the geodesic curvature of the measured tooth surface of a spiral bevel gear according to claim 1, characterized in that: Step (2) comprises the following steps: Step (2.1): Taking the measured tooth surface as the shape value point, first calculate the control vertices of each NURBS curve along the u or v direction according to the shape value point, and then use the obtained control vertices as the shape value point to calculate the control polygon mesh vertices along the u or v direction, that is, the control vertices of the NURBS surface and the corresponding weight factors. Therefore, the position vector of the bicubic NURBS surface with second-order continuity is The unit normal vector is expressed as Where: m is the number of control vertices in the u direction; n is the number of control vertices in the v direction; V i,j is the control vertex of the surface; W i,j V i,j The weight factor of B i,3 is the cubic B-spline basis function along the u direction; B j,3 is the cubic B-spline basis function along the v direction; P u (u,v) and P v (u,v) are the first-order derivatives of the position vector P(u,v) along u and v respectively; Step (2.2): The tooth surfaces of the large and small wheels are matched, that is, the driving surface is the concave surface of the small wheel and the convex surface of the large wheel, and the non-driving surface is the convex surface of the small wheel and the concave surface of the large wheel. The position vectors of the matched digital tooth surfaces are r1=r1(u1,v1) and r2=r2(u2,v2), where u1,v1 are the tooth surface parameters of the small wheel and 0≤u1≤1,0≤v1≤1; u2,v2 are the tooth surface parameters of the large wheel and 0≤u2≤1,0≤v2≤1; the unit normal vectors of the matched gears are n1(u1,v1) and n2(u2,v2); the digital tooth surfaces Σ1 and Σ2 are in the fixed coordinate system S h The position vector and normal vector of the tooth surface should be equal; the coordinate systems S1 and S2 are the digital tooth surface dynamic coordinate systems of the small wheel and the large wheel respectively. h is a fixed coordinate system; the small wheel and the large wheel rotate around their respective axes X1 and X2, the angular velocity of the large wheel is ω2, and the angular velocity of the small wheel is ω1. The position vector and unit normal vector of the digitized tooth surface Σ1 and Σ2 are expressed in the coordinate system S h middle In the formula, and are the meshing angles of the small wheel digital tooth surface Σ1 and the large wheel digital tooth surface Σ2 respectively; M h1 From coordinate system S1 to coordinate system S h The homogeneous coordinate transformation matrix, M h2 =M hd M d2 From coordinate system S2 to coordinate system S h The homogeneous coordinate transformation matrix, L h1 and L h2 M h1 and M h2 The matrix without the last row and column; the digital tooth surfaces ∑1 and ∑2 of the small wheel and the large wheel are in the fixed coordinate system S h The continuous contact tangency in the tooth surface is obtained, and the actual tooth surface contact analysis model (RTCA: Real Tooth Contact Analysis) is obtained. Its basic equation is 5. The method for calculating the geodesic curvature of the measured tooth surface of a spiral bevel gear according to claim 1, characterized in that: Step (3) is specifically as follows: the measurement order of the actual tooth surface is: the tooth length direction is from the outer end to the inner end, the tooth shape direction is from the tooth root to the tooth top, according to the corresponding relationship between the geometric positions of the large and small wheels during meshing transmission, that is, the condition that the small wheel tooth root contacts the large wheel tooth top or the small wheel tooth top contacts the large wheel tooth root, take the same tooth length grid number, and the tooth shape direction should be paired with the tooth root and the tooth top, that is, the large wheel tooth shape grid number 1 point and the small wheel tooth shape grid number MI point form a point pair, the large wheel tooth shape grid number MI point and the small wheel tooth shape grid number 1 point form a point pair, and the remaining points are similar, forming a total of MI×MW paired point arrays; For any matching point, the small wheel meshing angle is in, The large wheel meshing angle is in, Where, the wheel rotation axis vector p1 = [1 0 0] T , large wheel rotation axis vector p2 = [cos∑ 0 -sin∑] T , ∑ is the axis intersection angle; after substituting the obtained large and small wheel rotation angles into the first equation of equations (1) and (2), according to the distance formula between two points in space Traverse all MI×MW pairing point arrays, calculate their spatial distance, and take the small wheel digital tooth surface parameter u of the pairing point with the smallest distance 10 ,v 10 、Digital tooth surface parameters of large wheel u 20 ,v 20 And the big and small wheel turning angle As the initial value for solving the measured tooth surface contact analysis Small wheel turning angle As input, that is, fixed The value of is a nonlinear equation set with unknown quantities, thus obtaining a contact point between two meshing digital tooth surfaces; then changing The value of is calculated and the solution is continued until the contact point is beyond the effective boundary of the tooth surface. These instantaneous contact points of the tooth surface constitute the tooth surface contact path.

6. The method for calculating the geodesic curvature of the measured tooth surface of a spiral bevel gear according to claim 1, characterized in that: Step (4) specifically includes the following steps: Step (4.1): The meshing tooth surface of the spiral bevel gear is a point contact conjugate surface, which is given by the formula Calculation shows that the instantaneous transmission ratio at the design reference point is equal to the theoretical transmission ratio, that is, the transmission ratio condition is met. i 21 =z2 / z1(4)where z1 and z2 are the number of teeth of the large and small wheels respectively; On the one hand, the position vector r of the design reference point M is obtained by using the quasi-Newton method m 、Normal vector n m and tangent vector t m On the other hand, the position of the design reference point that meets certain accuracy requirements is obtained through the iteration method, that is, given the error ε, the small wheel rotates a small amount The given transmission ratio accuracy requirements can be met through continuous iteration; Step (4.2): Establish a tangent plane coordinate system S with the design reference point M as the origin t , whose three coordinate axes are respectively tangent vector t m For X t Axis, normal vector n m Z t Axis, vector t m ×n m Y t Axis, in coordinate system S i The subscript i=1 indicates a small wheel, i=2 indicates a large wheel, and X i axisi i =[1 0 0] T ,Y i Axis j i =[0 1 0] T ,z axis i =[0 0 1] T , from the coordinate system S i To the tangent plane coordinate system S t The transformation matrix is The discrete points P on the tooth surface on the contact path i ,i=1,2,…,n c , after the above transformation, converted to S t The corresponding point after is Q i ,i=1,2,…,n c , using the NURBS curve formula to fit the projection points, the k-order NURBS curve can be expressed as a rational polynomial vector function Among them, d i ,i=0,1,…,n c is called the control vertex, w i ,i=0,1,…,n c is the weight factor attached to the corresponding control vertex, N i,k (u) is the i-th k-order B-spline basis function, and the curvature at the design reference point M is p′(u) and p″(u) are the first-order derivative and second-order derivative of p(u) respectively. The curvature is the geodesic curvature of the contact path. If the geodesic curvature value of the design reference point is larger, the contact path is more curved. Conversely, the closer the geodesic curvature value is to zero, the closer the contact path is to the geodesic line.

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