Accurate design and modification method for tooth profile of gear

Abstract

The invention belongs to the technical field of mechanical transmission, and particularly relates to a gear tooth profile accurate design and modification method. A gear tooth profile is divided into a tooth top section, a working section, a transition section and a tooth root section, a continuous and smooth tooth profile model is constructed by combining a quadratic B-spline curve based on a segmentation design principle, and a tooth profile is optimized by adopting a dynamic control point adjustment strategy. Through accurate modeling, real-time curvature verification and a multi-tooth profile collaborative modification algorithm, the problems of tooth root stress concentration, scanning stretching tooth thickness deficiency, discontinuous tooth profile connection and the like are solved. The gear design precision and machining feasibility are remarkably improved, and the method is suitable for the complex design requirements of straight gears, bevel gears and modified gears.

Description

Technical Field

[0001] The present invention relates to the field of gear transmission technology, and in particular to a precise modeling method for gear tooth profiles and a segmented curve modification strategy, which is particularly suitable for the optimization design of modeling and modification of involute cylindrical gears, helical gears and displacement gears. Background Art

[0002] Traditional gear design methods generally use standard tooth profile templates or software default module modeling methods, which are difficult to meet the needs of personalized tooth profile design. In particular, in non-standard components such as helical gears and displacement gears, there are typical problems such as uneven tooth profile transition, sudden changes in root curvature, and missing tooth thickness in scanning modeling.

[0003] In existing technologies, common methods include constructing gear tooth profiles based on standard involute theory, batch-generating tooth profile models using unified formulas, or performing direct extrusion modeling based on CAD platforms. Although these methods can achieve modeling operations with a certain degree of accuracy, they generally suffer from the following problems:

[0004] 1. There is a lack of a segmented design mechanism tailored to the characteristics of different tooth profile areas, making it impossible to flexibly adjust the local characteristics of each part of the tooth profile;

[0005] 2. Lack of dynamic optimization methods for tooth profile control points, resulting in sudden changes in curvature, discontinuous connections, and local stress concentration in the modeling curve;

[0006] 3. For the problem of missing tooth thickness modeling in the middle of complex tooth shapes (such as helical gears), existing methods lack effective parameter correction and scanning path control strategies;

[0007] 4. It is impossible to achieve the linkage closed-loop adjustment of the tooth profile modeling and modification process, which seriously affects the modeling efficiency and accuracy. Summary of the Invention

[0008] To solve the above problems, the purpose of the present invention is to provide a method for precise design and shaping of gear tooth profiles, which realizes precise reconstruction of gear tooth profiles through segmented modeling and control point adjustment, combines modeling accuracy and structural optimization, and improves its transmission performance under complex working conditions through parametric shaping methods, thereby improving gear design efficiency and processing adaptability.

[0009] A method for accurately designing and modifying a gear tooth profile comprises the following steps:

[0010] S1, division structure;

[0011] S2, 3D modeling;

[0012] S3. Design control points within the segmented area;

[0013] S4, connecting tooth profile curve;

[0014] S5, scan path control;

[0015] S6. Design the tooth profile modification control points.

[0016] Furthermore, step S1 is specifically as follows:

[0017] S1.1. Divide the entire gear tooth profile into individual tooth shapes. The individual gear tooth profile is bilaterally symmetrical and adopts a single-sided tooth profile design.

[0018] S1.2. The entire tooth profile is designed in sections. Key point A is the midpoint of the tooth top arc, key point B is the intersection of the involute and the tooth top arc, key point C is the starting point of the involute on the base circle, key point D is the intersection of the tooth root transition curve and the tooth root arc, and key point E is the midpoint of the tooth root arc. Key points A, B, C, D, and E are divided.

[0019] S1.3. The tooth profile curve between points A and B is defined as the tooth top segment S1, the tooth profile curve between points B and C is defined as the working segment S2, the curve between points C and D is defined as the transition segment S3, and the curve between points D and E is defined as the tooth root segment S4.

[0020] Furthermore, step S2 is specifically as follows:

[0021] S2.1. Using the gear module, number of teeth, pressure angle and other parameters, calculate the pitch circle radius, base circle radius, addendum circle radius and root circle radius;

[0022] S2.2, let the coordinates of the gear rotation center O be (0,0), relative to point O, according to Figure 4 As shown, through the angle and the tooth tip radius r a , the coordinates of the key point A can be obtained through formula (1),

[0023]

[0024] S2.3, relative to point O, according to formula (2), through the base circle radius r b , combined with the key point's spread angle θ k , calculate the exact coordinates of key point B;

[0025]

[0026] S2.4, the parametric equation of the tooth root transition curve in the rectangular coordinate system is shown in formula (3). According to the gear pitch circle radius r, the angle With the angle ψ, relative to point O, the coordinates of the key point C are obtained by formula (3);

[0027]

[0028] S2.5. Similarly, calculate the coordinates of the intersection key point D according to formula (3);

[0029] S2.6, passing angle Tooth root circle r f , combined with formula (4), the coordinates of the key point E can be calculated;

[0030]

[0031] Furthermore, step S3 is specifically as follows:

[0032] S3.1, tooth top segment S1 is the tooth top arc, and the precise arc is fitted by multiple control point coordinates;

[0033] S3.2, working section S2 tooth profile curve is positioned by multiple control points. The coordinates of these control points are determined by the tooth thickness calculation formula and the coordinate function derived from the pressure angle. The number of control points is determined by the quantity equation. The control points are connected in sequence to draw the involute curve. The coordinate function is:

[0034]

[0035] Where (x k-1 ,y k-1 ) is the coordinate of the previous point, and the equation for the number of control points is:

[0036]

[0037] Where: m is the module, k1 is the demand constant, k2 is the module influence coefficient, and k3 is the gear number adjustment coefficient, which accurately controls the spacing between control points in different areas.

[0038] S3.3. To ensure that the transition curve does not interfere with meshing and does not cause root undercutting, the transition segment S3 is obtained by machining the trajectory curve of a standard involute cylindrical gear using the Fancheng method using a rack tool. The number of transition segments is determined by a quantity equation. During machining, the rack tool moves from left to right at a constant speed v1 (mm / s), while the gear wheel rotates around its axis at a constant speed v2. The relative speed between the wheel wheel rotation speed and the tool feed is:

[0039]

[0040] S3.4 and tooth root segment S4 are tooth root arcs. The coordinates of the control points on the tooth root arc are calculated by formula (4), and the number of control points is determined according to formula (6);

[0041] Furthermore, step S4 is specifically as follows:

[0042] Use a quadratic B-spline curve to connect all the control points of the tooth top segment S1 in sequence, and then connect all the control points of the working segment S2, transition curve segment S3 and tooth root segment S4 at once;

[0043] S4.1. A parameterized curve based on a quadratic B-spline basis function is used. The quadratic B-spline curve is used to construct an involute cylindrical gear tooth profile curve using control points and node vectors. It is usually expressed as:

[0044]

[0045] Where, d i is the control point, N i,2 (u) is the quadratic B-spline basis function, u is the parameter, and m is the number of control points;

[0046] S4.2. Quadratic B-spline basis function N i,2 The calculation formula of (u) is given by the De Boor algorithm, and the quadratic basis function is defined as:

[0047]

[0048] Furthermore, step S5 is specifically as follows:

[0049] S5.1. The scanning path is usually defined by a parameterized curve that describes the motion trajectory of the tooth profile in three-dimensional space. To ensure the integrity of the tooth profile, the scanning path function must meet the following conditions:

[0050] a) Continuity: The scanning path must be continuous between the starting and ending points of the tooth profile to avoid breaks or mutations.

[0051] b) Smoothness: The smoothness of the scanning path directly affects the smoothness of the tooth profile, and the curvature change of the path should be as smooth as possible.

[0052] c) Matching with the tooth profile curve: The scanning path must accurately match the geometric shape of the tooth profile curve to ensure that the tooth profile is not missing or deformed during the scanning process.

[0053] S5.2. The scanning path needs to accurately describe the geometric relationship of the helix angle. The original scanning path function is defined as:

[0054]

[0055] Among them, when the helix angle β is 0, it indicates a spur gear, and when it is not 0, it indicates a helical gear, b is the tooth width, d is the pitch circle diameter, and t is a parameter;

[0056] S5.3, axial displacement increment:

[0057] φ(t)=b·t(t∈[0,1]) (11)

[0058] Among them, t is a parameter;

[0059] S5.4. By accurately mapping the helix angle, the helix angle β is combined with the tooth profile involute equation to modify the angle parameterization formula:

[0060]

[0061] Where z is the number of teeth, β is the helix angle, r is the pitch circle radius, and Z(t) is the time series function. By introducing the tooth number correction term, the geometric matching of the helix angle and the tooth profile curve is ensured;

[0062] S5.5. Use a segmented parameterization strategy to increase the parameter density in the middle area of the scan stretch:

[0063] t n =t+k·sin(2πt)(t∈[0,1],k∈[0,0.1]) (13)

[0064] Furthermore, step S6 is specifically as follows:

[0065] S6.1. By adjusting the position of the adjustment point, combined with the direction coefficient and the unit vector, the adjusted tooth profile control point can be expressed as:

[0066]

[0067] In the formula, P is the original control point, ΔP is the adjustment amount, k is the adjustment coefficient, is a unit direction vector. By reasonably selecting the adjustment coefficient k and the unit direction vector

[0068] S6.2. By adjusting the control points and optimizing the curvature, the tooth profile curve can be ensured to have a smooth transition throughout the entire working range. The expression for the curvature change of the tooth profile curve at u is:

[0069]

[0070] Where P′(u) and P″(u) represent the first-order and second-order derivatives of the tooth profile curve in the direction of parameter u, respectively. BRIEF DESCRIPTION OF THE DRAWINGS

[0071] Figure 1 Schematic diagram of the tooth profile precision design and modification method of the present invention;

[0072] Figure 2 It is a schematic diagram of the tooth profile structure decomposition of the present invention;

[0073] Figure 3 Generate schematic diagrams for involutes;

[0074] Figure 4 is a schematic diagram of the gear tool angle;

[0075] Figure 5 Create schematic diagrams for precise gear tooth profiles;

[0076] Figure 6 Schematic diagram of gear end face tooth profile modification;

[0077] Figure 7 This is the gear drum modification diagram;

[0078] Figure 8 This is the gear pair meshing side clearance diagram after gear drum modification;

[0079] Figure 9 This is the curvature comb diagram of the tooth profile after gear drum shaping;

[0080] Figure 10 Create a result plot for the gear 3D solid model. DETAILED DESCRIPTION

[0081] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0082] A method for precise design and modification of gear tooth profile, such as Figure 1 As shown, the following steps are included:

[0083] S1. Division structure:

[0084] S1.1、 Figure 2 As shown, the entire gear tooth profile is divided into individual tooth shapes. The individual gear tooth profile is bilaterally symmetrical and adopts a single-sided tooth profile design.

[0085] S1.2. The entire tooth profile is designed in sections. Key point A is the midpoint of the tooth top arc, key point B is the intersection of the involute and the tooth top arc, key point C is the starting point of the involute on the base circle, key point D is the intersection of the tooth root transition curve and the tooth root arc, and key point E is the midpoint of the tooth root arc. Key points A, B, C, D, and E are divided.

[0086] S1.3. The tooth profile curve between points A and B is defined as the tooth top segment S1, the tooth profile curve between points B and C is defined as the working segment S2, the curve between points C and D is defined as the transition segment S3, and the curve between points D and E is defined as the tooth root segment S4.

[0087] S2. 3D modeling:

[0088] S2.1. Using a standard involute cylindrical gear with a module of 2 mm, 20 teeth, and a pressure angle of 20°, calculate the pitch circle radius r to be 20.0000 mm and the base circle radius r to be 20.0000 mm. b The radius of the tooth top circle is 18.7938 mm.a 22.0000mm, tooth root radius r f 17.5000mm;

[0089] S2.2, such as Figure 4 As shown, let the coordinates of the gear rotation center O be (0,0). Relative to point O, through the angle and the tooth tip radius r a , combined with the following formula, we can get the coordinates of the key point A,

[0090]

[0091] is 1.6rad, the radius of the tooth top circle r a is 22.0000 mm, and the coordinates of the key point A are (-0.4926, 21.9945) mm according to formula (1);

[0092] S2.3, relative to point O, according to formula (2), through the base circle radius r b , combined with the key point's spread angle θ k ,like Figure 3 As shown, substitute the exact coordinates of the key point B into the calculation,

[0093]

[0094] Base circle radius r b is 18.7938 mm, and the angle θ at the key point k is 1.35rad, and substituting it into formula (2) yields: the exact coordinates of the key point B are (0.1978, 21.9991)mm;

[0095] S2.4. The parametric equation of the tooth root transition curve in the rectangular coordinate system is shown in formula (3). Figure 4 As shown, according to the gear pitch circle radius r, the angle With the angle ψ, relative to point O, the coordinates of the key point C are obtained by formula (3);

[0096]

[0097] Among them, the pitch circle radius r is 20.0000mm, is 11.2mm, a is 9.1mm, α is 0.23rad, is 1.64 rad, ψ is 0.122 rad, and the coordinates of the key point C are obtained as (1.3272, 18.7300) mm when substituted into formula (3);

[0098] S2.5. Similarly, calculate the coordinates of the intersection key point D according to formula (3);

[0099] Among them, the pitch circle radius r is 20.0000mm, is 13.4mm, a is 7.44mm, α is 0.26rad, is 1.53 rad, ψ is 0.132 rad, and the coordinates of the key point D are obtained by substituting them into formula (3): (1.7888, 17.4083) mm;

[0100] S2.6, passing angle Combining the formula, the coordinates of the key point E can be calculated;

[0101]

[0102] Among them, the angle Substituting into formula (4), the coordinates of the key point E are (2.3771, 17.3378) mm;

[0103] S3. Design control points within the segmented area:

[0104] S3.1, tooth top segment S1 is the tooth top arc, and the precise arc is fitted by multiple control point coordinates;

[0105] S3.2, working section S2 tooth profile curve is positioned by multiple control points. The coordinates of these control points are determined by the tooth thickness calculation formula and the coordinate function derived from the spread angle. The number of control points is determined by the quantity equation. The control points are connected in sequence to draw the involute curve. The coordinate function is:

[0106]

[0107] The equation for the number of control points is:

[0108]

[0109] Where: m is the module, k1 is the demand constant, k2 is the module influence coefficient, and k3 is the gear number adjustment coefficient, which accurately controls the spacing between control points in different areas.

[0110] S3.3. To ensure that the transition curve does not interfere with meshing and does not cause root undercutting, the transition segment S3 is obtained by machining the trajectory curve of a standard involute cylindrical gear using the Fancheng method using a rack tool. The number of transition segments is determined by a quantity equation. During machining, the rack tool moves from left to right at a constant speed v1 (mm / s), while the gear wheel rotates around its axis at a constant speed v2. The relative speed between the wheel wheel rotation speed and the tool feed is:

[0111]

[0112] S3.4 and tooth root segment S4 are tooth root arcs. The coordinates of the control points on the tooth root arc are calculated by formula (4), and the number of control points is determined according to formula (6);

[0113] like Figure 5 As shown, the tooth top segment S1 is calculated as 8 control points by formula (6), and the following are obtained by formula (1):

[0114] <![CDATA[S11=(-0.4926,21.9945)mm]]> <![CDATA[S12=(-0.4271,21.9959)mm]]> <![CDATA[S13=(-0.3247,21.9976)mm]]> <![CDATA[S14=(-0.2244,21.9989)mm]]> <![CDATA[S15=(-0.1121,21.9997)mm]]> <![CDATA[S16=(-0.0168,22.0000)mm]]> <![CDATA[S17=(0.0834,21.9998)mm]]> <![CDATA[S18=(0.1978,21.9991)mm]]>

[0115] The working section S2 is calculated by formula (6) as 36 control points, which are obtained by formula (5):

[0116]

[0117]

[0118] The transition section S3 is calculated as 26 control points by formula (6). The rack tool speed is set by formula (7) to simulate the processing of the gear blank, and the tool trajectory coordinates are obtained as follows:

[0119] <![CDATA[S31=(1.3272,18.7300)mm]]> <![CDATA[S32=(1.3240,18.6697)mm]]> <![CDATA[S33=(1.3196,18.6088)mm]]> <![CDATA[S34=(1.3162,18.5406)mm]]> <![CDATA[S35=(1.3143,18.4593)mm]]> <![CDATA[S36=(1.3144,18.4032)mm]]> <![CDATA[S37=(1.3160,18.3356)mm]]> <![CDATA[S38=(1.3195,18.2602)mm]]> <![CDATA[S39=(1.3259,18.1852)mm]]> <![CDATA[S3 10 =(1.3324,18.1252)mm]]> <![CDATA[S3 11 =(1.3432,18.0486)mm]]> <![CDATA[S3 12 =(1.3534,17.9948)mm]]> <![CDATA[S3 13 =(1.3670,17.9338)mm]]> <![CDATA[S3 14 =(1.3809,17.8787)mm]]> <![CDATA[S3 15 =(1.3989,17.8184)mm]]> <![CDATA[S3 16 =(1.4177,17.7667)mm]]> <![CDATA[S3 17 =(1.4469,17.6985)mm]]> <![CDATA[S3 18 =(1.4695,17.6534)mm]]> <![CDATA[S3 19 =(1.4941,17.6111)mm]]> <![CDATA[S3 20 =(1.5238,17.5685)mm <!-- 7 -->]]> <![CDATA[S3 21 =(1.5552,17.5301)mm]]> <![CDATA[S3 22 =(1.5866,17.4982)mm]]> <![CDATA[S3 23 =(1.6298,17.4641)mm]]> <![CDATA[S3 24 =(1.6660,17.4435)mm]]> <![CDATA[S3 25 =(1.7202,17.4224)mm]]> <![CDATA[S3 26 =(1.7888,17.4083)mm]]>

[0120] The tooth root segment S4 is calculated as 8 control points by formula (6), and is obtained by formula (4):

[0121] <![CDATA[S41=(1.7888,17.4083)mm]]> <![CDATA[S42=(1.9057,17.3908)mm]]> <![CDATA[S43=(1.9879,17.3863)mm]]> <![CDATA[S44=(2.0779,17.3765)mm]]> <![CDATA[S45=(2.1654,17.3656)mm]]> <![CDATA[S46=(2.2371,17.3564)mm]]> <![CDATA[S47=(2.2948,17.3489)mm]]> <![CDATA[S48=(2.3771,17.3378)mm]]>

[0122] S4. Connecting tooth profile curve:

[0123] Use a quadratic B-spline curve to connect the eight control points of the tooth top segment S1 in sequence, and then connect all the control points of the working segment S2, transition curve segment S3 and tooth root segment S4 at once;

[0124] S4.1. A parameterized curve based on a quadratic B-spline basis function is used. The quadratic B-spline curve is used to construct an involute cylindrical gear tooth profile curve using control points and node vectors. It is usually expressed as:

[0125]

[0126] Where, d i is the control point, N i,2 (u) is the quadratic B-spline basis function, u is the parameter, and m is the number of control points;

[0127] S4.2. Quadratic B-spline basis function N i,2 The calculation formula of (u) is given by the De Boor algorithm, and the quadratic basis function is defined as:

[0128]

[0129] S5, Scan path control:

[0130] S5.1. The scanning path is usually defined by a parameterized curve that describes the motion trajectory of the tooth profile in three-dimensional space. To ensure the integrity of the tooth profile, the scanning path function must meet the following conditions:

[0131] a) Continuity: The scanning path must be continuous between the starting and ending points of the tooth profile to avoid breaks or mutations.

[0132] b) Smoothness: The smoothness of the scanning path directly affects the smoothness of the tooth profile, and the curvature change of the path should be as smooth as possible.

[0133] c) Matching with the tooth profile curve: The scanning path must accurately match the geometric shape of the tooth profile curve to ensure that the tooth profile is not missing or deformed during the scanning process.

[0134] S5.2. The scanning path needs to accurately describe the geometric relationship of the helix angle. The original scanning path function is defined as:

[0135]

[0136] Among them, when the helix angle β is 0, it indicates a spur gear, and when it is not 0, it indicates a helical gear, b is the tooth width, d is the pitch circle diameter, and t is a parameter;

[0137] S5.3, axial displacement increment:

[0138] φ(t)=b·t(t∈[0,1]) (11)

[0139] S5.4. By accurately mapping the helix angle, the helix angle β is combined with the tooth profile involute equation to modify the angle parameterization formula:

[0140]

[0141] Where z is the number of teeth, β is the helix angle, r is the pitch circle radius, and Z(t) is the time series function. By introducing the tooth number correction term, the geometric matching of the helix angle and the tooth profile curve is ensured;

[0142] S5.5. Use a segmented parameterization strategy to increase the parameter density in the middle area of the scan stretch:

[0143] t n =t+k·sin(2πt)(t∈[0,1],k∈[0,0.1]) (13)

[0144] Among them, in the parameter setting, the helix angle β = 0 (spur gear), the scanning increment Δz = 0.5mm; the path correction process adopts the sinusoidal modulation function Δu = sin(πt)·Δz, and the control point density of the middle end face is increased to 48 points;

[0145] S6. Design tooth profile modification control points:

[0146] S6.1, such as Figure 6 As shown, by adjusting the position of the adjustment point, combining the direction coefficient and the unit vector, the adjusted tooth profile control point can be expressed as:

[0147]

[0148] In the formula, P is the original control point, ΔP is the adjustment amount, k is the adjustment coefficient, is a unit direction vector. By reasonably selecting the adjustment coefficient k and the unit direction vector

[0149] S6.2. By adjusting the control points and optimizing the curvature, the tooth profile curve can be ensured to have a smooth transition throughout the entire working range. The expression for the curvature change of the tooth profile curve at u is:

[0150]

[0151] Where P′(u) and P″(u) represent the first-order and second-order derivatives of the tooth profile curve in the direction of parameter u, respectively.

[0152] like Figure 7 As shown in the figure, the middle end face (numbered L5) generated by scanning and stretching after initial precise modeling has local tooth thickness loss, and the curvature distribution of the transition section S3 is uneven. By setting the shaping parameters: scanning increment Δz = 0.5mm, target tooth profile segment S3, tooth profile line L5, shaping direction is the positive direction of the x-axis, and shaping amount ΔP = 0.001mm. Based on the segmented parameterization strategy, the control point density of the S3 segment is increased to 36 points, and the sinusoidal modulation function is used to optimize the curvature continuity; at the same time, the distribution of the L5 end face control points is dynamically adjusted through the axial curvature compensation strategy to reconstruct the scanning path. After shaping, the middle tooth thickness deviation is reduced to 0.001mm, and the sudden change value of the curvature of the transition section is optimized to 0.03mm. The shaping effect is as follows Figure 8 、 9 and 10.

[0153] The foregoing is merely a preferred embodiment of the present invention. It should be noted that those skilled in the art may make various improvements and modifications without departing from the principles of the present invention, and such improvements and modifications should be considered within the scope of protection of the present invention. Any components not specified in this embodiment may be implemented using existing technologies.

Claims

1. A method for precise design and modification of gear tooth profile, characterized in that: The following steps are involved: S1, division structure; S2, 3D modeling; S3. Design control points within the segmented area; S4, connecting tooth profile curve; S5, scan path control; S6. Design the tooth profile modification control points.

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