Tooth profile modification method of involute tooth surface based on the principle of gradual expansion and contraction of generating line

Abstract

The present invention provides a method for tooth profile modification of an involute tooth surface based on the principle of gradual expansion and contraction of a generating line. The tooth profile modification of the involute tooth surface of a gear is achieved by gradual expansion and contraction of the generating line of the involute. The present invention adopts a method for modifying the tooth surface by adjusting the length (expansion) of the generating line of the involute, thereby solving the difficult problems of tooth profile modification of involute cylindrical gears and tooth surface modeling of worm transmission media. The specific parameters can be adjusted according to the working conditions, and the method has better applicability.

Description

Technical Field

[0001] The present invention relates to the field of mechanical parts processing, and in particular to a method for modifying the tooth profile of an involute tooth surface based on the principle of gradual expansion and contraction of a generating line. Background Art

[0002] Gears are commonly used components in mechanical transmissions. In ordinary cylindrical gear transmissions, in order to moderate the change in meshing stiffness, reduce the impact of engagement and engagement caused by pitch error and load deformation, improve the lubrication of the tooth surface and prevent bonding, the standard involute tooth profile is partially trimmed at the tooth top or near the root fillet to form a modified tooth surface gear. At the same time, in worm transmissions, in order to achieve point contact transmission with low error sensitivity, a medium tooth surface is often introduced to achieve local conjugate point contact engagement in new worm transmissions (including new enveloping toroidal worms and new enveloping drum worms).

[0003] Prior art research on medium tooth surfaces has primarily focused on planar medium tooth surfaces and medium tooth surfaces with small tooth difference. These types of surfaces are relatively limited, and it is difficult to adjust medium tooth surface parameters to suit specific application conditions. Therefore, a new method for modifying involute tooth surfaces with adjustable parameters and wider applicability is urgently needed. Therefore, based on this background, the present invention proposes a new involute tooth surface modification method to address the aforementioned challenges in modifying the tooth profile of involute cylindrical gears and modeling medium tooth surfaces in worm gear transmissions. Summary of the Invention

[0004] In view of this, the present invention provides a method for modifying the tooth profile of an involute tooth surface based on the principle of gradual expansion and contraction of the generating line, which solves the difficult problems of tooth profile modification of involute cylindrical gears and tooth surface modeling of worm transmission media, can adjust specific parameters according to the operating conditions, and has better applicability.

[0005] The present invention provides an involute tooth surface profile modification method based on the principle of gradual expansion and contraction of a generating line. The tooth profile modification of the involute tooth surface of a gear is achieved by gradual expansion and contraction of the generating line of the involute.

[0006] Further, the following steps are included:

[0007] a. Determine the shaping amount function:

[0008]

[0009] Where: θ K is the angle of any point K on the generating line of the involute;

[0010] And the function f(θ K ) should be continuous and differentiable, symmetrical with respect to a given target modification point, monotonically increasing on both sides of the given target modification point, and the modification amount should have a maximum and a minimum value;

[0011] b. Obtain the curve equation after the involute line shape is modified:

[0012]

[0013] Where: r K is the radius of point K on the involute line; r b is the base circle radius; α K is the pressure angle at point K on the involute;

[0014] c. Based on the curve equation in step b, obtain the tooth profile modification tooth surface equation:

[0015] Left modified tooth surface equation for:

[0016]

[0017] Right modified tooth surface equation for:

[0018]

[0019] Where: f(τ) is the function f(θ K ) is converted into a function of the involute tooth surface parameter τ, and τ=θ K +α K ; r b II is the base circle radius of the modified involute surface; τ and θ∈[θ1,θ2] are the parameters of the involute tooth surface; δ is the base circle half angle; p is the tooth surface helical parameter; α t is the pressure angle of the pitch circle end face, and where α n is the normal pressure angle of the pitch circle; β is the helix angle of the tooth surface; (i, j, k) are the unit vectors of the coordinate axes in the coordinate system O-xyz.

[0020] Furthermore, the curve equation of the involute after the shape modification in step b is obtained by substituting the shape modification function in step a into the standard involute polar coordinate equation. The standard involute polar coordinate equation is:

[0021]

[0022] Furthermore, it is characterized in that:

[0023] The tooth surface equation of the tooth profile modification in step c is obtained by substituting the function f(τ) into the tooth surface equation of the standard involute cylindrical gear. The tooth surface equation of the standard involute cylindrical gear is:

[0024] Left tooth surface equation for:

[0025]

[0026] Right tooth surface equation for:

[0027]

[0028] Where: is the base circle radius of the standard involute surface.

[0029] Beneficial effects of the present invention: The involute tooth profile modification method based on the principle of gradual expansion and contraction of the generating line of the present invention adopts the method of adjusting the length (expansion) of the generating line of the involute to modify the tooth surface, which solves the difficult problems of involute cylindrical gear tooth profile modification and worm transmission medium tooth surface modeling, can adjust specific parameters according to the working conditions, and has better applicability. BRIEF DESCRIPTION OF THE DRAWINGS

[0030] The present invention will be further described below in conjunction with the accompanying drawings and embodiments:

[0031] Figure 1 Form a schematic diagram for a standard involute;

[0032] Figure 2 A schematic diagram is formed for the involute curve of the gradual expansion and contraction of the line;

[0033] Figure 3 This is the schematic diagram of the involute cylindrical gear;

[0034] Figure 4 This is the principle diagram of the end face of the involute cylindrical gear;

[0035] Figure 5 Schematic diagram of the tooth surface after the tooth profile is modified by gradual shrinkage of the generation line;

[0036] Figure 6 It is a schematic diagram of the tooth surface of a standard involute cylindrical gear;

[0037] Figure 7 Schematic diagram of the tooth surface after the tooth profile is modified by gradual elongation of the generation line;

[0038] Figure 8 It is a schematic diagram of the plane tooth surface after the tooth profile is modified by the gradual elongation of the limit line;

[0039] Figure 9 The diagram shows the relative position relationship of the tooth surfaces before and after tooth profile modification;

[0040] Figure 10 Schematic diagram of local gradual shrinkage shaping of dual-target shaping points on the generation line. DETAILED DESCRIPTION

[0041] Items and names marked by reference numerals in the table below:

[0042]

[0043]

[0044] The tooth profile modification method of the involute tooth surface based on the principle of gradual expansion and contraction of the generating line of the present invention realizes the tooth profile modification of the involute tooth surface of the gear by gradual expansion and contraction of the generating line of the involute; the shape of the involute is changed by gradual change of the length of the generating line, thereby realizing the tooth profile modification.

[0045] like Figure 1 As shown in the diagram of the formation principle of the standard involute, when the line When pure rolling is performed along the base circle 12, the trajectory AK of any point K on the line is the involute 11 of the base circle, where the center of the base circle 12 is point O and the radius of the base circle is r. b The radius of point K on the involute line is r K , the angle of point K on the involute line is θ K , the pressure angle at point K on the involute is α K .

[0046] According to the geometric relationship of the right triangle, from ΔBOK, we can know that the pressure angle α K With base circle radius r b and radial r K The relationship between them is:

[0047]

[0048] at the same time,

[0049]

[0050] Therefore, according to the above relationship, the spread angle θ can be obtained K About the pressure angle α K The involute function invα K for:

[0051] invα K =θ K =tanα K -α K (10)

[0052] Then the polar coordinate equation of the involute 11 can be obtained as:

[0053]

[0054] Based on the above standard involute formation theory, the length of the generating line is gradually extended and retracted to achieve the linear shape modification of the involute, such as Figure 2 As shown, the specific shaping method includes the following steps:

[0055] a. Determine the shaping amount function:

[0056]

[0057] Where: θ K is the angle of any point K on the involute line, that is, the modification function is about the angle θ K function, and the function f(θ K ) should be continuous and differentiable, symmetrical with respect to the given target modification point, monotonically increasing on both sides of the given target modification point, and the modification amount has a maximum and a minimum value, that is, the function f(θ K ) should meet the following conditions: ① Gradual differentiability criterion: the modified line shape still follows the continuous differentiability at any point, that is, the stretching amount of the modified line is a continuously differentiable function in a given interval, without breakpoints and non-differentiable inflection points; ② Bilateral symmetry criterion: the stretching amount of the modified line is a symmetric function about the given target modification point, that is, the modification amount is a symmetric function about the expansion angle θ at any point K on the involute K ③ Monotonically increasing criterion: the farther the two sides are from the target modification point K, the greater the modification amount, that is, the modification amount is proportional to the angle θ at any point K on the involute line. K The increase of is first monotonically decreasing to zero, and then symmetrically monotonically increasing; ④ Limit boundary criterion: the maximum limit of the modification amount should not exceed the tangent value at the target modification point, that is, the maximum limit modification line is a straight line, and the minimum limit should be defined according to the specific target tooth shape, that is, the minimum limit modification should not modify the root and addendum widths to zero;

[0058] At the same time, the direction of the modification amount is defined as follows: the elongation of the generating line is the positive direction, and the contraction of the generating line is the negative direction;

[0059] b. According to the defined modification function and the standard involute polar coordinate equation, we get:

[0060]

[0061] Therefore, after modification, the angle θ K About the pressure angle α K The involute function invα K for:

[0062]

[0063] That is, the modification function is substituted into the standard involute polar coordinate equation to obtain the curve equation of the involute after modification:

[0064]

[0065] Where: r K is the radius of point K on the involute line; r bis the base circle radius; α K is the pressure angle at point K on the involute;

[0066] Solve the value of the modified line according to formula (2) to obtain a series of polar coordinate points, and then draw a curve graph after modification based on the obtained polar coordinate points;

[0067] c. Figure 2 As shown, the corresponding trimming line can be obtained according to the above steps. Figure 2 In the figure, 25 is the base circle, and the curve It is a standard involute 22 without modification. It is the negative shaping line 21 after the gradual shrinkage shaping of the generating line. It is the positive shaping line 23 after the gradual elongation and shaping of the generating line, the straight line It is the maximum modification line 24 after the generation line is gradually stretched and modified; point K o is the target shaping point, and the corresponding standard involute parameters at this point are as follows: the expansion angle is θ Ko , the pressure angle is α Ko , the radius is r Ko ,straight line is the corresponding generating line 0 state; when the expansion angle is less than the expansion angle value corresponding to the target modification point, the corresponding arbitrary point on the standard involute is K o1 , at this time the corresponding occurrence line 1 state is a straight line When the angle of extension is greater than the angle of extension corresponding to the target modification point, the corresponding arbitrary point on the standard involute is K o2 , at this time the corresponding occurrence line 2 state is a straight line

[0068] like Figure 2 As shown, because point K o is the target shaping point, the spread angle θ Ko The corresponding generation line stretch modification amount is zero, so here the negative modification line 21, the standard involute 22, the positive modification line 23 and the maximum modification line 24 coincide, that is, point K n , K o , K m and K g When the extension angle is less than the extension angle value corresponding to the target modification point, the corresponding arbitrary point on the negative modification line 21 is K n1 , the corresponding shrinkage of the line modification is The corresponding arbitrary point on the positive modification line 23 is K m1 , the corresponding elongation of the occurrence line modification is The corresponding arbitrary point on the maximum modification line 24 is K g1 , the corresponding elongation of the occurrence line modification is When the extension angle is greater than the extension angle value corresponding to the target modification point, the corresponding arbitrary point on the negative modification line 21 is K n2 , the corresponding shrinkage of the line modification is The corresponding arbitrary point on the positive modification line 23 is K m2 , the corresponding elongation of the occurrence line modification is The corresponding arbitrary point on the maximum modification line 24 is K g2 , the corresponding elongation of the occurrence line modification is

[0069] The involute is converted to the xOy plane in the spatial rectangular coordinate system O-xyz and spirally moved around the z axis to form the involute surface 31, that is, the involute cylindrical gear tooth surface, as shown in FIG. Figure 3 As shown; where the xOy plane is the middle section in the tooth width direction, the tooth surface is equivalent to the pure rolling of multiple generating lines 33 on the base cylinder 32, and the base cylinder radius is r b , the helix angle is β;

[0070] Will Figure 3 Projecting the involute surface in the figure onto the xOy plane, we get Figure 4 The principle diagram of the involute cylindrical gear end face is shown, and the left tooth surface equation of the standard involute cylindrical gear can be obtained. L I for:

[0071]

[0072] Similarly, the right tooth surface equation of the standard involute cylindrical gear can be obtained for:

[0073]

[0074] Where: is the standard involute base circle radius; τ and θ∈[θ1,θ2] are the involute tooth surface parameters; δ is the base circle (tooth thickness) half angle; p is the tooth surface helical parameter; α t is the pressure angle of the pitch circle end face, and where α n is the normal pressure angle of the pitch circle; β is the helix angle of the tooth surface; (i, j, k) are the unit vectors of the coordinate axes in the coordinate system O-xyz.

[0075] Of course, if the helix angles of the left and right tooth surfaces are not equal, that is, β L ≠β R , then the above parameters are adjusted to β (the left tooth surface is β L , the right tooth surface is β R ), α t (The left tooth surface is α tL, the right tooth surface is α tR ), α t (The left tooth surface is α tL , the right tooth surface is α tR ), (The left tooth surface is The right tooth surface is ) and p (the left tooth surface is p L , the right tooth surface is p R ).

[0076] The above is the standard involute cylindrical gear tooth surface equation. The standard involute generating line gradual expansion and contraction modification theory defined above is introduced and converted to Figure 4 On the xOy plane in the rectangular coordinate system O-xyz shown in the figure, the previously defined modification amount is about the spread angle θ K The function f(θ K ) is converted into a function f(τ) about the involute tooth surface parameter τ, and τ=θ K +α K ; Therefore, the tooth surface equation of the involute cylindrical gear tooth profile modification based on the principle of gradual expansion and contraction of the generating line can be obtained; that is, based on the curve equation in step b and the standard involute cylindrical gear tooth surface equation, the function f(τ) is substituted into the standard involute cylindrical gear tooth surface equation to obtain the tooth surface equation of the tooth profile modification:

[0077] Left modified tooth surface equation for:

[0078]

[0079] Right modified tooth surface equation for:

[0080]

[0081] Where: f(τ) is the function f(θ K ) is converted into a function of the involute tooth surface parameter τ, and τ=θ K +α K ; is the base circle radius of the modified involute surface; similarly, if the helix angles of the tooth surfaces on the left and right sides are not equal, that is, β L ≠β R , then the above parameters are adjusted to β (the left tooth surface is β L , the right tooth surface is β R ), α t (The left tooth surface is α tL , the right tooth surface is α tR ), (The left tooth surface is The right tooth surface is ) and p (the left tooth surface is p L , the right tooth surface is p R ). It should also be noted that when the shaping is positive, the “±” in the formula is taken as “+”, and when the shaping is negative, it is taken as “-”.

[0082] The present invention will be further described below with examples:

[0083] First, take the modification function as Then the equation of the modified tooth surface on the left is:

[0084]

[0085] The equation of the modified tooth surface on the right is:

[0086]

[0087] Where: w is the tooth surface modification factor, which determines the size of the modification amount. At the same time, it can be seen that with τ as the x-axis and f(τ) as the y-axis, the modification amount function On the xOy plane, it is about τ = tanα t Symmetric cosine function, and f(τ)∈[0,w]. At this time, the target training point is on the involute pitch circle of the tooth surface. From this, it can be determined that the function meets the four major principles followed by the gradual expansion and contraction of the involute line shape modification.

[0088] In the present invention, the number of teeth Z is selected to be 60, and the normal pressure angle of the pitch circle is α n is 20°, tooth width B is 50mm, normal module m n is 3mm, and the tooth surface modification factor w is 0.2; the above standard parameters are substituted into the above standard involute cylindrical gear tooth surface equation and the modified involute cylindrical gear tooth surface equation, solved by matlab programming, and the tooth surface diagram is drawn by UG, respectively. Figure 5 The tooth surface of the gear 5 after the tooth profile is modified by gradual contraction of the generating line, Figure 6 The standard involute cylindrical gear with 6 tooth surfaces and Figure 7 The tooth surface of the gear 7 after the tooth profile modification is gradually elongated. Figure 8 The schematic diagram of the plane tooth surface after the gear 8 is modified by the gradual elongation of the tooth profile; placing the above four tooth surfaces in the same coordinate system, the relative position relationship between them can be obtained as follows Figure 9 A comparison of the four tooth surfaces shown in the gear 9, namely the tooth surface 91 after the linear gradual contraction tooth profile modification, the standard involute cylindrical gear tooth surface 92, the tooth surface 93 after the linear gradual extension tooth profile modification and the flat tooth surface 94 after the linear gradual extreme extension tooth profile modification; in the example of the present invention, the above four tooth surfaces are tangent to each other at the pitch circle at the same time, and the tangent contact line is the intersection line of the pitch cylindrical surface and the tooth surface.

[0089] When the tooth surface of cylindrical gear transmission is modified using the modification method proposed above, only the generation line gradual shrinkage modification method can be used, and the entire tooth surface in the tooth height direction cannot be modified. In this case, the local gradual shrinkage modification method of the generation line double target modification point can be used, such as Figure 10 As shown, at the target shaping point K o1 and K o2 The standard involute 101 is subjected to local linear gradual shrinkage modification. The modification mathematical model is the same as the aforementioned full tooth surface modification model. Only the number and position of the target modification points are changed. The modified curve is the local shrinkage modification segment a102 and the local shrinkage modification segment b103. The non-modified segment is Among them, the shaping range angle θ of the local shrinkage shaping section a is K 0≤θ K ≤θ Ko1 , the shaping range of the local shrinkage shaping section b is angle θ K is θ Ko2 ≤θ K , so the non-modified segment angle θ K is θ Ko1 ≤θ K ≤θ Ko2 .

[0090] Of course, first of all, for the present invention, the shaping function is not limited to the cosine function, but can also be other curve functions that meet the above four conditions, such as: quadratic curve function, sine function, perfect circle function, elliptical function and other similar functions. Under the guidance of the method of the present invention, the purpose of the present invention can be achieved.

[0091] Secondly, in the present invention, the trained tooth surface is not only used for a pair of ordinary cylindrical gear transmissions, but can also be applied to medium tooth surface enveloping worm transmissions (enveloping toroidal worm transmissions and enveloping drum worm transmissions) to provide a forming modeling method for medium tooth surfaces.

[0092] Third, in the present invention, for the tooth surface modification of a pair of involute cylindrical gears, only the local gradual shrinkage modification method of the generating line double target modification points can be used, and the modification function, modification amount and modification range need to be selected according to the specific situation.

[0093] In the present invention, when the modified tooth surface is applied to the enveloping worm drive (annular envelope and drum envelope), the modification type (elongation and contraction) can be selected according to the specific situation and matched with the corresponding worm tooth surface.

[0094] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not limiting. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solutions of the present invention may be modified or replaced by equivalents without departing from the purpose and scope of the technical solutions of the present invention, which should all be included in the scope of the claims of the present invention.

Claims

1. A method for modifying the tooth profile of an involute tooth surface based on the principle of gradual expansion and contraction of the generating line, characterized by: The tooth profile modification of the involute tooth surface of the gear is achieved by gradually expanding and contracting the involute generating line; The following steps are involved: a. Determine the shaping amount function: Where: θ K is the angle of any point K on the generating line of the involute; And the function f(θ K ) should be continuous and differentiable, symmetrical with respect to a given target modification point, monotonically increasing on both sides of the given target modification point, and the modification amount should have a maximum and a minimum value; b. Obtain the curve equation after the involute line shape is modified: Where: r K is the radius of point K on the involute line; r b is the base circle radius; α K is the pressure angle at point K on the involute; c. Based on the curve equation in step b, obtain the tooth profile modification tooth surface equation: Left modified tooth surface equation for: Right modified tooth surface equation for: Where: f(τ) is the function f(θ K ) is converted into a function of the involute tooth surface parameter τ, and τ=θ K +α K ; is the base circle radius of the modified involute surface; τ and θ∈[θ1,θ2] are the parameters of the involute tooth surface; δ is the base circle half angle; p is the tooth surface helical parameter; α t is the pressure angle of the pitch circle end face, and where α n is the normal pressure angle of the pitch circle; β is the helix angle of the tooth surface; (i, j, k) are the unit vectors of the coordinate axes in the coordinate system O-xyz.

2. The involute tooth profile modification method based on the principle of gradual expansion and contraction of the generating line according to claim 1, characterized in that: The curve equation of the involute after the shape modification in step b is obtained by substituting the shape modification function in step a into the standard involute polar coordinate equation. The standard involute polar coordinate equation is:

3. The involute tooth profile modification method based on the principle of gradual expansion and contraction of the generating line according to claim 1, characterized in that: The tooth surface equation of the tooth profile modification in step c is obtained by substituting the function f(τ) into the tooth surface equation of the standard involute cylindrical gear. The tooth surface equation of the standard involute cylindrical gear is: Left tooth surface equation for: Right tooth surface equation for: Where: is the base circle radius of the standard involute surface.

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