Design method of gear tooth to drum-shaped modification curve

Abstract

The present application belongs to the field of gear design, and particularly relates to a design method of a drum-shaped modification curve of gear tooth direction, comprising: 1) constructing a whole C 2 smoothly continuous double circular-arc spline drum-shaped modification curve; 2) obtaining the coordinates of the common tangent point F i of the i-th double circular-arc segment of the double circular-arc spline drum-shaped modification curve; 3) constructing a multi-element nonlinear equation set; 4) obtaining an initial solution by solving the linear term of the equation set, and iteratively calculating the accurate solution of the multi-element nonlinear equation set; and 5) obtaining the drum-shaped modification curve of the gear tooth direction according to the accurate value of the intermediate variable chord tangent angle of the i-th double circular-arc segment. The drum-shaped modification curve of the gear tooth direction obtained by the present application adopts segmented low-order interpolation, can avoid the "Runge" effect of high-order interpolation, and has good properties of the first and second order derivatives of the drum-shaped curve. The curve has high fitting accuracy, is continuous and smooth, reduces the impact and vibration in the gear transmission process, improves the tooth surface contact state, and improves the gear transmission performance.

Description

Technical Field

[0001] This invention belongs to the field of gear design, specifically relating to a design method for a gear tooth profile drum-shaped modification curve. Background Technology

[0002] Gear modification is increasingly required in engineering applications, such as tooth tip trimming, tooth profile shaping, and tooth profile bulging. Gear modification can improve tooth surface contact and enhance transmission performance. However, there is still no complete mathematical description method for tooth profile bulging. The amount of bulging along the tooth width is generally estimated empirically, lacking a scientific and reasonable calculation method, resulting in low accuracy in tooth profile bulging modification. Double circular arc spline curves can achieve C... 2 The overall shape is smooth and continuous, belonging to piecewise low-order interpolation, which avoids the "Runge" effect of high-order interpolation. Both its first and second derivatives exhibit good behavior. Furthermore, the double circular arc spline CNC programming is easy to implement and applicable to general-purpose CNC machine tools, making it very convenient to achieve tooth-direction drum-shaped shaping. The drum-shaped curve obtained based on the double circular arc spline design is continuous and smooth, which can reduce impact and vibration during gear transmission and improve the overall performance of the gear. Summary of the Invention

[0003] In order to solve the above-mentioned technical problems in the background art, the present invention provides a design method for a gear tooth profile drum-shaped modification curve that can improve gear transmission performance.

[0004] To achieve the above objectives, the present invention adopts the following technical solution:

[0005] A method for designing a gear tooth profile drum-shaped modification curve, characterized in that the method includes the following steps:

[0006] 1) Analyze the geometric characteristics of the tooth-shaped bulge, establish a mathematical model based on the bulge amount, and construct the overall C. 2 A smooth and continuous double-circular-arc spline drum-shaped shaping curve;

[0007] 2) Based on the double circular arc spline drum-shaped shaping curve constructed in step 1), and using the chord length equal division method, the common tangent point F of the i-th segment of the double circular arc spline drum-shaped shaping curve is obtained. i The coordinates;

[0008] 3) To solve for the intermediate variable chord-tangent angle of the i-th double circular arc in the double circular arc spline drum-shaped modification curve, a multivariate nonlinear equation system is constructed based on the relationship of equal curvature at the nodes;

[0009] 4) Solve the linear terms of the multivariate nonlinear equation system constructed in step 3) to obtain the initial solution. Use the Newton-Raphson algorithm to calculate the exact solution of the multivariate nonlinear equation system through multiple iterations. The exact solution of the multivariate nonlinear equation system is the exact value of the chord-tangent angle of the intermediate variable of the i-th double circular arc.

[0010] 5) The gear tooth profile curve is obtained based on the precise value of the intermediate variable tangential angle of the i-th double circular arc.

[0011] Preferably, in step 1) of this invention, the angle formed between the common tangent point F of the double circular arc spline drum-shaped shaping curve and the adjacent node is... The following angular relationships exist:

[0012]

[0013] in:

[0014] It is the angle between the point of tangency F and the line connecting the two adjacent nodes;

[0015] It is node k i The tangent angle of the right-hand arc;

[0016] It is node k i+1 The tangent angle of the left arc;

[0017] when and Once determined, γ is a constant, and the locus of the common tangent point F passes through point k. i With k i+1 The arc has infinitely many common tangent points F.

[0018] Preferably, in step 2) of the present invention, the common tangent point F of the i-th segment of the double circular arcs is... i The method for solving the coordinates is:

[0019] node k i k i+1 The tangent line passing through the node forms a triangle, and the incenter G of the triangle also lies on the arc locus of the common tangent point F; then the incenter G and k i、 k i+1 The arc determined by the three points is the locus of the common tangent point F; the intersection of this arc with the perpendicular bisector of the line connecting the adjacent nodes is defined as the common tangent point F. i Then the common tangent point F of the i-th double circular arc segment i The specific expression for the coordinates is:

[0020]

[0021] in:

[0022] The distance between adjacent nodes;

[0023] It is node k i-1 The tangent angle of the right-hand arc;

[0024] It is node k i The tangent angle of the left arc.

[0025] Preferably, the expression for the multivariate nonlinear equation system in step 3) of this invention is as follows:

[0026]

[0027] in:

[0028] P is a linear term, and also the principal term;

[0029] for The higher-order infinitesimals are nonlinear terms, i.e., correction terms;

[0030] The expression for the coefficient matrix A is as follows:

[0031]

[0032] .

[0033] As a preferred embodiment, step 4) of the present invention is specifically implemented as follows:

[0034] Solving for linear terms The initial solution is obtained. The chord-tangent angle was obtained by iterating multiple times using the Newton-Raphson algorithm. The exact value of is obtained by the iterative formula:

[0035]

[0036] in The Jacobi iteration matrix is ​​expressed as:

[0037] .

[0038] As a preferred embodiment, the specific implementation of step 5) in this invention is as follows:

[0039] 5.1) Calculate the center coordinates and radii of the two arcs adjacent to the common tangent point F based on the precise value of the intermediate variable chord-tangent angle of the i-th double arc obtained in step 4).

[0040] 5.2) Perform radial coordinate transformation on the center coordinates and radii obtained in step 5.1), rotate and translate the center of each arc to the global coordinate system, and draw the gear tooth direction drum-shaped modification curve.

[0041] Preferably, the specific implementation of step 5.1) in this invention is as follows:

[0042] Let node k i The center of the left arc is , radius is Then the coordinates of the center and the radius of the left arc are expressed as follows:

[0043]

[0044]

[0045] Let node k i-1 The center of the right arc is , radius is Then the coordinates of the center and the radius of the right arc are expressed as follows:

[0046]

[0047] .

[0048] Preferably, the specific implementation method of radial coordinate transformation in step 5.2) of the present invention is as follows:

[0049]

[0050] in:

[0051] The coordinates of the interpolation nodes in the global coordinate system;

[0052] These are the coordinates of the center of the circle in the local coordinate system.

[0053] These are the coordinates of the center of the circle, translated to the global coordinate system.

[0054] This represents the rotation angle of the local coordinate system relative to the global coordinate system.

[0055] Preferably, the design method for the gear tooth profiled curve used in this invention further includes, after step 5):

[0056] 6) Ensure the global convexity of the gear tooth profile curve.

[0057] As a preferred embodiment, step 6) of the present invention is specifically implemented as follows:

[0058] Construct a mathematical model for global convexity, wherein the mathematical model is:

[0059] The tangential angle α0(+) at the starting point k0 of the drum-shaped curve is the angle between the feed direction and the vector. The included angle; let The slope is The slope of the feed direction is node k i The coordinates are The formula for calculating the tangent-chord angle α0(+) is:

[0060]

[0061] To ensure the global convexity of the drum-shaped curve, the feed direction should meet the following conditions:

[0062]

[0063] Then the range of values ​​for α0(+) is:

[0064]

[0065] The endpoint k of the drum-shaped curve n The tangent angle α at the point n (-) represents the retraction direction and vector. The included angle; let The slope is The slope of the retraction direction is tangent-chord angle α n The formula for calculating (-) is:

[0066]

[0067] To ensure the global convexity of the drum-shaped curve, the retraction direction should meet the following conditions:

[0068]

[0069] but The range of values ​​for is:

[0070] .

[0071] The advantages of this invention are:

[0072] The tooth profile curve obtained by this invention employs piecewise low-order interpolation, which avoids the "Runge" effect of high-order interpolation. Both the first and second derivatives of the curve exhibit good behavior. This curve has high fitting accuracy, is continuous and smooth, reduces impact and vibration during gear transmission, improves tooth surface contact, and enhances gear transmission performance. Attached Figure Description

[0073] Figure 1 This is a schematic diagram of a double circular arc spline curve;

[0074] Figure 2 It is a mathematical model of a double circular arc curve;

[0075] Figure 3 This is a schematic diagram for solving the problem of finding the center and radius of a circle;

[0076] Figure 4 This is a schematic diagram of radial coordinate transformation;

[0077] Figure 5 It is the tangent angle. The solution results;

[0078] Figure 6 It is the tangent angle. The solution results;

[0079] Figure 7 The radius of the left arc of node ki The solution results;

[0080] Figure 8 The radius of the right arc of node ki-1 The solution results;

[0081] Figure 9 This is the result diagram of the double circular arc spline drum-shaped curve;

[0082] Figure 10 This is the result after adjusting the node coordinates (xi, yi);

[0083] Figure 11 Is it to increase? The result;

[0084] Figure 12 Is it to increase? The result;

[0085] Figure 13 This is a schematic diagram illustrating the derivation of the convexity preservation condition;

[0086] Figure 14 The results of gear tooth profile modification curves obtained based on the method provided in this invention are compared. Detailed Implementation

[0087] To enable those skilled in the art to better understand the present invention, the specific embodiments of the technical solutions of the present invention will be clearly and completely described below. Obviously, the following description is only a specific embodiment and does not limit the scope of protection of the present invention.

[0088] The design concept of this invention is as follows:

[0089] A mathematical model is established, a double circular arc spline curve is constructed, and the calculation formulas for the coordinates of the tangent point and center of the arc are derived. A system of multivariate nonlinear equations is constructed, and based on the Newton-Raphson algorithm, the exact solution for the intermediate variable, the chord-tangent angle, is obtained. n+1 interpolation nodes k are input. i (i=0,...,n) Position coordinates and feed direction at the first node k0 and the last node k n The retraction direction at the point yields a double-circular-arc spline drum-shaped shaping curve.

[0090] For example, the present invention can generally adopt the following technical solution:

[0091] (1) Select any two interpolation nodes, establish a rectangular coordinate system, construct two tangent circular arcs, and derive the solution for the tangency point F. i The calculation method is used to obtain the formula for calculating the center of the arc.

[0092] (2) In order to solve the intermediate variable chord-tangent angle, a multivariate nonlinear equation system is constructed based on the curvature relationship. The initial solution is obtained by solving the linear terms of the equation system. The exact solution of the equation system is obtained by iterating multiple times using the Newton-Raphson algorithm.

[0093] (3) Select n+1 interpolation nodes, calculate the center and radius of the circle from the exact solution, and translate each arc segment to the main coordinate system based on the radial coordinate transformation formula to obtain the double arc spline drum-shaped modification curve.

[0094] Based on the above ideas and technical solutions, this invention provides a design method for a gear tooth profile drum-shaped modification curve, which includes the following steps:

[0095] 1) Analyze the geometric characteristics of the tooth-shaped bulge, establish a mathematical model based on the bulge amount, and construct the overall C. 2 A smooth, continuous double-circular-arc spline drum-shaped modified curve, wherein the angle formed between the common tangent point F of the double-circular-arc spline drum-shaped modified curve and the adjacent node is . The following angular relationships exist:

[0096]

[0097] in:

[0098] It is the angle between the point of tangency F and the line connecting the two adjacent nodes;

[0099] It is node k i The tangent angle of the right-hand arc;

[0100] It is node k i+1 The tangent angle of the left arc;

[0101] when and Once determined, γ is a constant, and the locus of the common tangent point F passes through point k. i With k i+1 The arc has infinitely many common tangent points F.

[0102] 2) Based on the double circular arc spline drum-shaped shaping curve constructed in step 1), and using the chord length equal division method, the common tangent point F of the i-th segment of the double circular arc spline drum-shaped shaping curve is obtained. i The coordinates of the i-th segment of the double circular arc are given by F. i The method for solving the coordinates is:

[0103] node k i k i+1 The tangent line passing through the node forms a triangle, and the incenter G of the triangle also lies on the arc locus of the common tangent point F; then the incenter G and k i、 k i+1 The arc determined by the three points is the locus of the common tangent point F; the intersection of this arc with the perpendicular bisector of the line connecting the adjacent nodes is defined as the common tangent point F. i Then the common tangent point F of the i-th double circular arc segment i The specific expression for the coordinates is:

[0104]

[0105] in:

[0106] The distance between adjacent nodes;

[0107] It is node k i-1 The tangent angle of the right-hand arc;

[0108] It is node k i The tangent angle of the left arc.

[0109] 3) To solve for the intermediate variable chord-tangent angle of the i-th segment of the double circular arc in the double circular arc spline drum-shaped shaping curve, a system of multivariate nonlinear equations is constructed based on the equality of nodal curvature. The expression of the system of multivariate nonlinear equations is as follows:

[0110]

[0111] in:

[0112] P is a linear term, and also the principal term;

[0113] for The higher-order infinitesimals are nonlinear terms, i.e., correction terms;

[0114] The expression for the coefficient matrix A is as follows:

[0115]

[0116] .

[0117] 4) Solve the linear terms of the multivariate nonlinear equation system constructed in step 3) to obtain the initial solution. Use the Newton-Raphson algorithm to iteratively calculate the exact solution of the multivariate nonlinear equation system. The exact solution of the multivariate nonlinear equation system is the exact value of the intermediate variable chord-tangent angle of the i-th double circular arc. The specific implementation of this process is: solve the linear terms. The initial solution is obtained. The chord-tangent angle was obtained by iterating multiple times using the Newton-Raphson algorithm. The exact value of is obtained by the iterative formula:

[0118]

[0119] in The Jacobi iteration matrix is ​​expressed as:

[0120] .

[0121] 5) The gear tooth profile shaping curve is obtained based on the precise value of the intermediate variable tangential angle of the i-th double circular arc. The specific implementation method is as follows:

[0122] 5.1) Based on the precise value of the intermediate variable chord-tangent angle of the i-th double arc obtained in step 4), calculate the center coordinates and radii of the two arcs adjacent to the common tangent point F. Specifically, let node k i The center of the left arc is , radius is Then the coordinates of the center and the radius of the left arc are expressed as follows:

[0123]

[0124]

[0125] Let node ki-1 The center of the right arc is , radius is Then the coordinates of the center and the radius of the right arc are expressed as follows:

[0126]

[0127] .

[0128] 5.2) Perform radial coordinate transformation on the center coordinates and radii obtained in step 5.1), rotate and translate the centers of each arc to the global coordinate system, and draw the gear tooth profile curve. The specific implementation of the radial coordinate transformation is as follows:

[0129]

[0130] in:

[0131] The coordinates of the interpolation nodes in the global coordinate system;

[0132] These are the coordinates of the center of the circle in the local coordinate system.

[0133] These are the coordinates of the center of the circle, translated to the global coordinate system.

[0134] This represents the rotation angle of the local coordinate system relative to the global coordinate system.

[0135] 6) Ensure the global convexity of the gear tooth profile curve. The specific implementation method for this step is as follows:

[0136] Construct a mathematical model for global convexity, wherein the mathematical model is:

[0137] The tangential angle α0(+) at the starting point k0 of the drum-shaped curve is the angle between the feed direction and the vector. The included angle; let The slope is The slope of the feed direction is node k i The coordinates are The formula for calculating the tangent-chord angle α0(+) is:

[0138]

[0139] To ensure the global convexity of the drum-shaped curve, the feed direction should meet the following conditions:

[0140]

[0141] Then the range of values ​​for α0(+) is:

[0142]

[0143] The endpoint k of the drum-shaped curve n The tangent angle α at the point n (-) represents the retraction direction and vector. The included angle; let The slope is The slope of the retraction direction is tangent-chord angle α n The formula for calculating (-) is:

[0144]

[0145] To ensure the global convexity of the drum-shaped curve, the retraction direction should meet the following conditions:

[0146]

[0147] but The range of values ​​for is:

[0148] .

[0149] The meanings of the character parameters mentioned in this article are shown in Table 1.

[0150] Table 1. Meaning of Character Parameters

[0151]

[0152] I. Construction of Double Circular Spline Curves

[0153] Assume there are n+1 nodes k in the plane. i A double circular arc spline is a spline curve constructed by using a pair of tangent circular arcs between two adjacent nodes, with the included angle between the lines connecting the two adjacent nodes set as θ. ,like Figure 1 As shown.

[0154] Establishing mathematical models, such as Figure 2 As shown, any node k is defined i The line connecting the tangent of the arc at point k to the adjacent node i k i-1 The included angle is the chord-tangent angle α. i (-), and the line k i k i+1 The included angle is the chord-tangent angle α. i (+), let Figure 2 Chinese K i and k i+1 The tangent and chord angles at the points are respectively and The intersection point of the two tangents to the chord is C.

[0155] Passing point k i Do k i The extension of C, take k i Let A have length a, and in k... i+1 C cut Connect AB, and draw the perpendicular bisector ED of AB, intersecting K. i Connect point C and point E, and link BE through point K. i Draw a perpendicular line to AC, intersecting ED at point O. i Over O i Draw the perpendicular line O to BE. i F, intersects BE at point F, and passes through point k. i+1 Do k i+1 The perpendicular line from C intersects O. i F extension line at O i+1 .

[0156] The following geometric relationship is clearly present.

[0157] (1)

[0158] but

[0159] (2)

[0160] Then the arc With arc They are tangent at point F. Since a is less than k... i+1 For any real number C, there are infinitely many points of tangency F. Let angle... The following geometric relationship exists:

[0161] (3)

[0162] when and Once confirmed, Let F be a constant, meaning the trajectory of the tangent point F is an arc. Let k be the line connecting this arc and its adjacent node. i k i+1 The intersection of the perpendicular bisectors is the common tangent point F. i Then the point of tangency F i The coordinates are:

[0163] (4)

[0164] In the formula This represents the distance between adjacent interpolation nodes.

[0165] With k i-1 k i The x-axis passes through point k. i-1 Do k i-1 ki Draw a perpendicular line to the y-axis, and establish a coordinate system as follows: Figure 3 As shown, analyze node k i The following geometric relationship holds true for the arc on the left.

[0166] (5)

[0167] Let node k i The center of the left arc is , radius is Using the geometric relationship of formula (5), we obtain the following formula:

[0168] (6)

[0169] (7)

[0170] Let node k i-1 The center of the right arc is , radius is Similarly, we can obtain:

[0171] (8)

[0172] (9)

[0173] II. Construction of Multivariate Nonlinear Equation Systems

[0174] In the above formulas for solving the radius and center of the arc, the chord-tangent angle is unknown. Constructing the equation for solving the chord-tangent angle is the core of solving the double-circular-arc spline drum-shaped curve.

[0175] If two circular arcs passing through the same node have the same curvature, then the following equation holds:

[0176] (10)

[0177] in (11)

[0178] To facilitate the solution of the nonlinear equation system, the following identity is constructed:

[0179] (12)

[0180] The given conditions are:

[0181] (13)

[0182] Subtracting equation (10) from equation (12) and substituting the condition of equation (13), we obtain the following system of equations after simplification.

[0183] (14)

[0184] in

[0185] (15)

[0186] (16)

[0187] Boundary conditions are

[0188] (17)

[0189] The matrix form of the above system of equations is:

[0190] (18)

[0191] Where P is the principal term. for The higher-order infinitesimal terms, i.e., the correction terms, have the following coefficient matrix A:

[0192] (19)

[0193] III. Solving and Plotting Curves for Multivariate Nonlinear Equations

[0194] Let the coordinates of the interpolation nodes be... If the number of interpolation nodes is n, then in the interpolation coefficient matrix A, a i b i The calculation formula is:

[0195] (20)

[0196] (twenty one)

[0197] Angle between adjacent nodes The calculation formula is

[0198] (twenty two)

[0199] Calculate the system of equations using formulas (20), (21), and (22). Given the coefficient matrix A and the constant vector P, use MATLAB to calculate... Let the solution be... .

[0200] (twenty three)

[0201] make A system of multivariate nonlinear equations The initial solution is used to obtain the exact solution of the system of equations through multiple iterations using the Newton-Raphson algorithm. The iterative formula is as follows:

[0202] (twenty four)

[0203] In the formula The Jacobi iteration matrix is ​​expressed as follows:

[0204] (25)

[0205] Let the exact solution obtained by iterative calculation be

[0206] (26)

[0207] and The sum of ,but The calculation formula is:

[0208] (27)

[0209] Will and Substituting into formulas (6), (7), (8), and (9), the coordinates of the center and the radius of the circle can be determined. For example... Figure 4 As shown, by using formula (28) to perform radial coordinate transformation, the centers of each arc are rotated and translated into the global coordinate system, thus drawing the double-circular-arc spline drum-shaped curve.

[0210] (28)

[0211] in These are the interpolation node coordinates in the global coordinate system. The coordinates of the center of the circle in the local coordinate system. To translate to the center coordinates of the circle in the global coordinate system, This represents the rotation angle of the local coordinate system relative to the global coordinate system.

[0212] To achieve better design results, the design method provided by this invention also includes global convexity processing of the shaping curve, the specific implementation of which is as follows: See Figure 13 The tangential angle α0(+) at the starting point k0 of the drum-shaped curve is the angle between the feed direction and the vector. The included angle; let The slope is The slope of the feed direction is node k i The coordinates are The formula for calculating the tangent-chord angle α0(+) is:

[0213]

[0214] To ensure the global convexity of the drum-shaped curve, the feed direction should meet the following conditions:

[0215]

[0216] Then the range of values ​​for α0(+) is:

[0217]

[0218] The endpoint k of the drum-shaped curve n The tangent angle α at the point n (-) represents the retraction direction and vector. The included angle; let The slope is The slope of the retraction direction is tangent-chord angle α n The formula for calculating (-) is:

[0219]

[0220] To ensure the global convexity of the drum-shaped curve, the retraction direction should meet the following conditions:

[0221]

[0222] but The range of values ​​for is:

[0223]

[0224] The double-circular arc spline tooth-direction drum-shaped profile curve obtained by this invention adopts segmented low-order interpolation, which is simple to calculate, has high interpolation accuracy, avoids the "Runge" phenomenon of high-order interpolation, and has good drum-shaped curve properties, continuous and smooth, reducing the flexible and rigid impacts in the gear transmission process, improving the tooth surface contact state, and improving the overall performance of the gear.

[0225] Example

[0226] (1) Determine the input conditions for the tooth-shaped drum curve.

[0227] The gear tooth width is 54 mm, and the maximum camber is 110 μm. The camber curve is divided into 54 equal parts along the tooth width direction, and the camber at each node is shown in Table 2. The feed direction, i.e. The direction of retraction, i.e. .

[0228] Table 2. Cadence Scale (Unit: μm)

[0229]

[0230] (2) Calculate the tangent-chord angle , .

[0231] Substitute the coordinates of the 54 interpolation nodes into formulas (20), (21), and (22), calculate the coefficient matrix A and the constant vector P of the system of equations, and solve the system of equations. Obtain the initial solution ,Will Substituting into formula (24), the exact solution is obtained through iterative calculation. ,like Figure 5 As shown. Substituting into formula (27), we get ,like Figure 6 As shown.

[0232] (3) Calculate the radius and center of the circle.

[0233] Will and Substituting these equations into formulas (7) and (9), we can obtain node k. i left arc radius With node k i-1 Right arc radius The calculation results are as follows Figure 7 and Figure 8 As shown,

[0234] (4) Draw the double circular arc spline drum-shaped curve.

[0235] Using formula (28), a radial coordinate transformation is performed to translate the centers of each arc in the local coordinate system to the global coordinate system. Then, in MATLAB software, the gear tooth profile curve is plotted, as shown below. Figure 9 As shown.

[0236] (5) Adjustment of the curve drum shape.

[0237] The drum-shaped curve can be formed by node k. i , , Adjustments are made. Changing the interpolation nodes has the most significant impact on the curve, and can alter the overall shape of the curve.

[0238] To increase the maximum bulge of the curve by 20 μm, i.e. Only the node coordinates need to be adjusted as follows:

[0239] (29)

[0240] Where y i The node's ordinate before adjustment. The values ​​represent the adjusted ordinates of the nodes. After adjustment, the maximum bulge of the curve becomes 130 μm, and the bulge at other nodes also increases accordingly, such as... Figure 10 As shown.

[0241] Adjust the feed and retraction directions, i.e. and It has a relatively small impact on the curve, mainly changing the local bulge shape of the curve, and and The larger the value, the larger the bulge at the endpoint of the curve. To ensure the bulge curve has global convexity, by the convexity-preserving condition, we obtain... and Adjustment range:

[0242] (30)

[0243] (31)

[0244] when When the value is 0~0.00058 rad, The larger the value, the greater the bulge shape, and the bulge curve exhibits global convexity. If it continues to increase... , such as When the value is increased from 0.00058 rad to 0.00068 rad, the bulge near curve k0 increases from 109.5 μm to 111.1 μm, but local concave arcs appear, which does not meet the requirements of actual engineering. Figure 11 As shown.

[0245] when When the value is 0~0.0015 rad, The larger the value, the greater the bulge shape, and the bulge curve exhibits global convexity. If it continues to increase... , such as When the value increases from 0.0015 rad to 0.0055 rad, the curve k... n The bulge in the vicinity increased from 1.1 μm to 2.2 μm, and a local concave arc appeared, which also does not meet the requirements of actual engineering. Figure 12 As shown.

[0246] When adjusting the drum-shaped curve, adjustments should be made within the allowable range of convexity preservation conditions; otherwise, local concave arcs will occur, significantly affecting the curve's behavior and hindering the design of the gear tooth-direction drum-shaped shaping curve. According to the method provided by this invention, the final drum-shaped shaping curve is as follows: Figure 14 As shown.

[0247] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for designing a gear tooth profiled curve, characterized in that: The design method for the gear tooth profile drum-shaped modification curve includes the following steps: 1) Analyze the geometric characteristics of the tooth-shaped bulge, establish a mathematical model based on the bulge amount, and construct the overall structure. C 2 A smooth and continuous double-circular-arc spline drum-shaped shaping curve; 2) Based on the double circular arc spline drum-shaped shaping curve obtained in step 1), and using the chord length equal division method, the first... i Common tangent point of the double circular arc F i The coordinates; 3) To solve the first circular arc spline drum-shaped shaping curve... i The intermediate variable chord-tangent angle of a double circular arc is used to construct a system of multivariate nonlinear equations based on the relationship of equal curvature at the nodes; 4) Solve the linear terms of the multivariate nonlinear equation system constructed in step 3) to obtain the initial solution, and then use... Newton- Raphson The algorithm iterates multiple times to calculate the exact solution of a system of multivariate nonlinear equations, where the exact solution is the first... i The precise value of the intermediate variable chord-tangent angle of a double circular arc segment; 5) According to the first i The precise value of the intermediate variable chord-tangent angle of the double circular arc segment yields the gear tooth profile shaping curve, specifically: 5.1) Based on the result obtained in step 4), the first... i Calculate the precise value of the intermediate variable chord-tangent angle of a double circular arc segment, and the coordinates and radii of the centers of the two circular arc segments to the left and right of the common tangent point F; 5.2) Perform radial coordinate transformation on the center coordinates and radii obtained in step 5.1), rotate and translate the center of each arc to the global coordinate system, and draw the gear tooth direction drum-shaped modification curve; 6) Ensure the global convexity of the gear tooth profile curve, specifically: Construct a mathematical model for global convexity, wherein the mathematical model is: The tangential angle α0(+) at the starting point k0 of the drum-shaped curve is the angle between the feed direction and the vector. The included angle; let The slope is The slope of the feed direction is node k i The coordinates are The formula for calculating the tangent-chord angle α0(+) is: To ensure the global convexity of the drum-shaped curve, the feed direction should meet the following conditions: Then the range of values ​​for α0(+) is: End point of the drum-shaped curve k n tangential angle at the point α n (-) represents the retraction direction and vector. The included angle; let The slope is The slope of the retraction direction is tangent angle α n The formula for calculating (-) is: To ensure the global convexity of the drum-shaped curve, the retraction direction should meet the following conditions: ,but The range of values ​​for is: 。 2. The design method for the gear tooth profile drum-shaped modification curve according to claim 1, characterized in that: The common tangent point of the double circular arc spline drum-shaped shaping curve in step 1) F The angle formed with the adjacent node is The following angular relationships exist: ,in: It is the tangent point F The angle between the line connecting the two adjacent nodes; It is a node k i The tangent angle of the right-hand arc; It is a node k i+1 The tangent angle of the left arc; when and Once determined, γ is a constant, and the common tangent point is... F The trajectory is passing through the point k i and k i+1 The arc, common tangent point F There are countless.

3. The design method for the gear tooth profile drum-shaped modification curve according to claim 2, characterized in that: In step 2), the first i Common tangent point of the double circular arc F i The method for solving the coordinates is: node k i , k i+1 The triangle is formed by the tangent line passing through the node, and the incenter of the triangle is... G Also located at the common tangent point F On the arc trajectory; then the inner... G and k i , k i+1 The arc determined by the three points is the common tangent point. F The trajectory; the intersection of the arc and the perpendicular bisector of the line connecting the adjacent nodes is defined as the common tangent point. Then the first i Common tangent point of the double circular arc F i The specific expression for the coordinates is: ,in: The distance between adjacent nodes; It is node k i-1 The tangent angle of the right-hand arc; It is a node k i The tangent angle of the left arc.

4. The design method for the gear tooth profile drum-shaped modification curve according to claim 3, characterized in that: The expression for the system of multivariate nonlinear equations in step 3) is: ,in: P This is a linear term, and also the principal term; for The higher-order infinitesimals are nonlinear terms, i.e., correction terms; Where A is the coefficient matrix, its expression is as follows: ， 。 5. The design method for the gear tooth profile drum-shaped modification curve according to claim 4, characterized in that: The specific implementation method of step 4) is as follows: Solving for linear terms The initial solution is obtained. ,use Newton-Raphson The algorithm iterates multiple times to obtain the chord-tangent angle. The exact value of is obtained by the iterative formula: ,in for jacobi The iteration matrix is ​​expressed as: 。 6. The design method for the gear tooth profile drum-shaped modification curve according to claim 5, characterized in that: The specific implementation method of step 5.1) is as follows: Set nodes k i The center of the left arc is , radius is Then the coordinates of the center and the radius of the left arc are expressed as follows: , Set nodes k i-1 The center of the right arc is , radius is Then the coordinates of the center and the radius of the right arc are expressed as follows: ， 。 7. The design method for the gear tooth profile drum-shaped modification curve according to claim 6, characterized in that: The specific implementation method for the radial coordinate transformation in step 5.2) is as follows: ,in: The coordinates of the interpolation nodes in the global coordinate system; These are the coordinates of the center of the circle in the local coordinate system. These are the coordinates of the center of the circle, translated to the global coordinate system. This represents the rotation angle of the local coordinate system relative to the global coordinate system.

Images