An analytical algorithm for profile grinding of helical gear tooth surfaces

Abstract

The present invention relates to the field of gear forming technology, and discloses an analytical algorithm for forming and grinding tooth surfaces of helical gears, comprising: S1, listing Q(t); S2, calculating Q n (t); S3, list S f (θ, t); S4, calculate n f (θ, t) = S f (θ, t)‑Q n (t); S5, get S w (θ, t) and n w (θ, t); S6, get v w (θ, t); S7, S f (θ, t) is simplified to S f (t); S8, #imgabs0# The analytical algorithm for forming and grinding the tooth surface of helical gears disclosed in the present invention improves the machining accuracy of the gears.

Description

Technical Field

[0001] The present invention relates to the technical field of gear forming, and in particular to an analytical algorithm for forming and grinding tooth surfaces of helical gears. Background Art

[0002] The machining accuracy requirements of gears determine the operating conditions of the gears, such as power consumption, gear life, and noise level. Grinding, the last machining step in the gear machining process, is often an important process that affects the gear tooth surface accuracy. Among them, profile grinding is widely used due to its advantages such as high machining efficiency and high machining accuracy.

[0003] During the profile grinding process, determining the gear tooth surface based on the grinding wheel's information is crucial for profile grinding simulation, error detection during the grinding process, and grinding path planning. The process of profile grinding with a profile grinding wheel is manifested through the contact line between the grinding wheel and the gear. This contact line forms the tooth surface along the grinding wheel's path. Therefore, calculating the contact line between the tooth surface and the grinding wheel is crucial for determining the gear tooth surface. Existing techniques typically employ numerical calculations to generate analytical formulas for the contact line. However, these methods are time-consuming and difficult to select initial values, resulting in gears with machining accuracies that fail to meet user requirements. Summary of the Invention

[0004] Based on the above, the purpose of the present invention is to provide an analytical algorithm for the profile grinding of helical gear tooth surfaces, which solves the problem of low gear machining accuracy caused by errors in the analytical formula obtained by the numerical calculation method.

[0005] To achieve the above object, the present invention adopts the following technical solutions:

[0006] An analytical algorithm for the profile grinding of helical gear tooth surfaces, including:

[0007] S1. Establish a grinding wheel coordinate system and a gear coordinate system, and list the coordinates Q(t) of any point on the curve of the axial section of the grinding wheel, where t is the B-spline parameter;

[0008] S2. Calculate the normal vector of any point Q(t) on the curve of the axial section of the grinding wheel at the z-axis of the grinding wheel. c Component Q on the axis n (t), z c The axis is the axis where the central axis of the grinding wheel is located;

[0009] S3. Rotate the axial section of the grinding wheel 360° around the central axis of the grinding wheel to form a grinding wheel, and list the coordinates S of any point on the rotating surface of the grinding wheel. f (θ, t), where θ is the angle of rotation;

[0010] S4. Calculate the normal vector n of any point on the surface of the grinding wheel f (θ, t), n f (θ, t) = S f (θ, t)-Q n (t);

[0011] S5, S f (θ, t) is transformed into the gear coordinate system through the coordinate transformation matrix to obtain the coordinate S of any point on the rotating surface. w (θ, t), and get n f The coordinate n of (θ, t) in the grinding wheel coordinate system w (θ, t);

[0012] S6. The grinding wheel rotates around the axis of the gear and moves along the axis of the gear. The S at any point on the surface of the grinding wheel is obtained. w The velocity vector v of (θ, t) w (θ, t);

[0013] S7, by n w (θ, t) and v w (θ, t) is perpendicular and can eliminate θ, point S on the contact line between the gear and the grinding wheel f (θ, t) is simplified to S f (t);

[0014] S8, move the contact line along the gear spiral line to form the tooth surface, the point on the tooth surface The calculation formula is as follows:

[0015]

[0016] Where, is the helix rotation angle, ∈ is the grinding wheel installation angle, a is the normal distance between the center axis of the grinding wheel and the center axis of the gear, and P is the lead parameter of the tooth surface.

[0017] As a preferred solution for the analytical algorithm of helical gear profile grinding, the calculation formula of Q(t) in S1 is as follows:

[0018]

[0019] Where Q i is the control point of B-spline, F i，3 (t) is the basis function of B-spline, Q y (t) and Q z (t) is the grinding wheel section curve at y c axis and z c Component of the axis, y c The axis is a vector axis along the radial direction of the grinding wheel and pointing to the center of the gear.

[0020] As a preferred solution for the analytical algorithm of helical gear profile grinding, Q n The calculation formula of (t) is as follows:

[0021]

[0022]

[0023] Where k is Q z (t) and Q y (t) component in z c Components on the axis.

[0024] As a preferred solution for the analytical algorithm of helical gear profile grinding, S f The calculation formula of (θ, t) is as follows:

[0025]

[0026] Wherein, the range of θ is (0, 2π].

[0027] As a preferred solution for the analytical algorithm of helical gear profile grinding, the normal vector n in S4 f The calculation formula of (θ, t) is as follows:

[0028]

[0029] Where q x =-Q y (t)sinθ,q y =Q y (t)cosθ,

[0030] As a preferred solution for the analytical algorithm of helical gear profile grinding, S5 w The calculation formula of (θ, t) is as follows:

[0031]

[0032] As a preferred solution for the analytical algorithm of helical gear profile grinding, n w The calculation formula of (θ, t) is as follows:

[0033]

[0034] As a preferred solution for the analytical algorithm of helical gear profile grinding, the velocity vector v in S6 w The calculation formula of (θ, t):

[0035]

[0036] Where, ω w is the angular velocity of the grinding wheel rotating around the gear axis, v is the speed of the grinding wheel moving along the axis of the gear, v = m n zω w / 2sin(β), where m n is the normal module, z is the number of teeth, and β is the helix angle.

[0037] As a preferred solution for the analytical algorithm of helical gear profile grinding, the n w (θ, t) and v w (θ, t) vertically, v w (θ, t)·n w (θ, t) = 0, and then we get the following formula:

[0038] Dsinθ+Ecosθ=F

[0039] Where D = -Q y (t)vsin∈-aω w Q y (t)cos∈, E=-kω w Q y (t)sin∈,F=(Q z (t)-k)(aω w sin∈-vcos∈), and after extracting sinθ and cosθ, we get the following system of equations:

[0040]

[0041] Where D, E, and F are parameters.

[0042] As a preferred solution for the analytical algorithm of helical gear profile grinding, cosθ and sinθ are substituted into S f (θ, t), we get S f The calculation formula of (t):

[0043]

[0044] Where S f (t) is a point on the contact line.

[0045] The beneficial effects of the present invention are as follows: the analytical algorithm for the helical gear forming grinding tooth surface disclosed in the present invention first understands the curved surface of the grinding wheel as a rotating surface formed by the axial cross-section of the grinding wheel rotating around the central axis of the grinding wheel. Then, according to the properties of the rotating surface, it is obtained that the normal vector of any point on the rotating surface intersects with the central axis of the grinding wheel at the same point. Then, using the principle of gear meshing, it is obtained that the point on the grinding wheel surface is the point on the gear tooth surface, and finally the analytical formula of the contact line is obtained, which shortens the time spent in the calculation process, greatly improves the calculation speed, does not need to select initial values, increases the accuracy of the analytical formula of the contact line, and ensures the processing accuracy of the gear finally processed. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following briefly introduces the drawings required for use in describing the embodiments of the present invention. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on the contents of the embodiments of the present invention and these drawings without any creative work.

[0047] Figure 1 is a flow chart of an analytical algorithm for profile grinding of helical gear tooth surfaces provided by a specific embodiment of the present invention;

[0048] Figure 2 Schematic diagram of the grinding wheel and gear during grinding according to the analytical algorithm for the profile grinding of the tooth surface of the helical gear provided by a specific embodiment of the present invention;

[0049] Figure 3 The normal vector Q(t) of any point on the cross section of the grinding wheel of the analytical algorithm for the helical gear profile grinding tooth surface provided by the specific embodiment of the present invention is at the z of the grinding wheel. c Component plots on axes;

[0050] Figure 4 This is a coordinate diagram of a circular arc curve obtained by using an analytical algorithm for the profile grinding of helical gear tooth surfaces provided by a specific embodiment of the present invention;

[0051] Figure 5 The gear tooth surface is obtained by fitting the analytical algorithm for the helical gear profile grinding tooth surface provided by the specific embodiment of the present invention.

[0052] In the picture:

[0053] 100. Grinding wheel; 200. Gear. DETAILED DESCRIPTION

[0054] To make the technical problems solved, the technical solutions adopted, and the technical effects achieved by the present invention more clearly understood, the technical solutions of the embodiments of the present invention will be further described in detail below with reference to the accompanying drawings. It is apparent that the described embodiments are only some of the embodiments of the present invention, not all of them. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort shall fall within the scope of protection of the present invention.

[0055] In the description of the present invention, unless otherwise expressly specified or limited, the terms "connected," "connected," and "fixed" should be understood in a broad sense. For example, they may refer to fixed connections, detachable connections, or integration; mechanical connections or electrical connections; direct connections or indirect connections through an intermediate medium; and internal communication between two components or interaction between two components. Those skilled in the art will understand the specific meanings of the above terms in the present invention in specific circumstances.

[0056] In the present invention, unless otherwise expressly specified or limited, a first feature being "above" or "below" a second feature may include the first and second features being in direct contact, or may include the first and second features being in contact not directly but through another feature between them. Furthermore, a first feature being "above," "above," and "above" a second feature may include the first feature being directly above or obliquely above the second feature, or may simply mean that the first feature is higher in level than the second feature. A first feature being "below," "below," and "below" a second feature may include the first feature being directly below or obliquely below the second feature, or may simply mean that the first feature is lower in level than the second feature.

[0057] In the description of this embodiment, terms such as "upper," "lower," and "right" are used to refer to positions or locations based on the positions or locations shown in the accompanying drawings. These terms are intended solely to facilitate description and simplify operation, and are not intended to indicate or imply that the devices or components referred to must have, be constructed, or operate in a specific orientation. Therefore, they should not be construed as limitations on the present invention. Furthermore, the terms "first" and "second" are used solely for descriptive purposes and have no special meaning.

[0058] This embodiment provides an analytical algorithm for the profile grinding of helical gear tooth surfaces. Figure 1 Shown, including:

[0059] S1. Establish a grinding wheel coordinate system and a gear coordinate system, and list the coordinates Q(t) of any point on the curve of the axial section of the grinding wheel 100. The calculation formula of Q(t) is as follows:

[0060]

[0061] Where Q iis the control point of B-spline, F i，3 (t) is the basis function of B-spline, t is the B-spline parameter, Q y (t) and Q z (t) is the truncation curve of grinding wheel 100 at y c axis and z c The component of the axis, z c Axis is the central axis of the grinding wheel 100, y c The z axis is a vector axis along the radial direction of the grinding wheel 100 and pointing to the center of the gear 200. c axis and y c Axis Figure 2 As shown, Figure 2 Also drawn in the x c Axis, ω c 、ω w 、x w axis, y w Axis and z w Axis. It should be noted that the B-spline parameters belong to the existing technology, F i，3 (t) can also be listed based on the existing technology and will not be repeated here.

[0062] S2. Calculate the normal vector of any point Q(t) on the curve of the axial section of the grinding wheel 100 at the z-axis of the grinding wheel 100. c Component Q on the axis n (t), Q n The calculation formula of (t) is as follows:

[0063]

[0064]

[0065] Where z c The axis is the axis where the central axis of the grinding wheel 100 is located. Figure 2 As shown, k is Q z (t) and Q y (t) component in z c Component on the axis, dQ y (t) / dt is Q y (t) Take the derivative of t, dQ z (t) / dt is Q z (t) Take the derivative with respect to t.

[0066] Specifically, if Figure 3 As shown, the normal vector of any point on the axial section of the grinding wheel 100 is c The coordinates of the intersection point on the axis are Q n (t).

[0067] S3. Rotate the axial section of the grinding wheel 100 360° around the central axis of the grinding wheel 100 to form the grinding wheel 100, and list the coordinates S of any point on the rotating curved surface of the grinding wheel 100. f (θ, t), S f The calculation formula of (θ, t) is as follows:

[0068]

[0069] Where θ is the angle of rotation, and the range of θ is (0, 2π].

[0070] Since the grinding wheel 100 is obtained by rotating the axial section of the grinding wheel 100, the coordinate S on the rotating surface of the grinding wheel 100 can be obtained through the coordinate change matrix. f (θ, t).

[0071] S4. Calculate the normal vector n of any point on the surface of the grinding wheel 100 f (θ, t), n f (θ, t) = S f (θ, t)-Q n (t), specifically, the normal vector n f The calculation formula of (θ, t) is as follows:

[0072]

[0073] Where q x =-Q y (t)sinθ,q y =Q y (t)cosθ,

[0074] Specifically, according to the characteristics of the rotating surface, the normal vector of any point on the curved surface of the grinding wheel 100 must intersect with its rotation axis (i.e., the central axis of the grinding wheel 100). At the same time, the normal vectors of all points on the same arc in the rotating surface intersect with the central axis of the grinding wheel 100 at the same point, so there is n f (θ, t) = S f (θ, t)-Q n (t).

[0075] S5, S f (θ, t) is transformed into the gear coordinate system through the coordinate transformation matrix to obtain the coordinate S of any point on the rotating surface. w (θ, t), and get n f The coordinate n of (θ, t) in the grinding wheel coordinate system w (θ, t)

[0076] Specifically, S w The calculation formula of (θ, t) is as follows:

[0077]

[0078] n w The calculation formula of (θ, t) is as follows:

[0079]

[0080] Where q x =-Q y (t)sinθ,q y =Q y (t)cosθ,

[0081] S6, the grinding wheel 100 rotates around the axis of the gear 200, and at the same time the grinding wheel 100 moves along the axis of the gear 200, and any point S on the curved surface of the grinding wheel 100 is obtained. w The velocity vector v of (θ, t) w (θ, t), velocity vector v w The calculation formula of (θ, t) is as follows:

[0082]

[0083] Where, ω w is the angular velocity of the grinding wheel 100 rotating around the axis of the gear 200, such as Figure 2 As shown, v is the speed of the grinding wheel 100 moving along the axis of the gear 200, v = m n zω w / 2sin(β), where m n is the normal module, z is the number of teeth, and β is the helix angle.

[0084] Specifically, assuming that the gear 200 remains stationary during the grinding process, the grinding wheel 100 is rotated at a speed ω w Around the axis z of the gear 200 w The grinding wheel 100 rotates at a speed v along the central axis z of the gear 200. w Sports, z w like Figure 2 As shown, for the right-handed gear 200, ω w is greater than 0, so v w (θ, t) can be calculated using the above formula.

[0085] S7, by n w (θ, t) and v w (θ, t) is perpendicular and can eliminate θ, point S on the contact line between the gear 200 and the grinding wheel 100 f (θ, t) is simplified to S f (t). Specifically, n in S7 w(θ, t) and v w (θ, t) vertically, v w (θ, t)·n w (θ, t) = 0, and then we get the following formula:

[0086] Dsinθ+Ecosθ=F

[0087] Where D = -Q y (t)vsin∈-aω w Q y (t)cos∈, E=-kω w Q y (t)sin∈,F=(Qz(t)-k)(aω w sin∈-vcos∈), and after extracting sinθ and cosθ, we get the following system of equations:

[0088]

[0089] Where D, E, and F are parameters.

[0090] Then, substituting cosθ and sinθ into Sf(θ, t), we can obtain the formula for Sf(t):

[0091]

[0092] Where S f (t) is a point on the contact line.

[0093] Specifically, since the gear 200 is meshed with the grinding wheel 100, the velocity vector and the normal vector at the meshing point are perpendicular, so v w (θ, t)·n w (θ, t) = 0. and Substituting this into the above equation and rearranging it yields Dsinθ+Ecosθ=F.

[0094] S8, move the contact line along the helical line of gear 200 to form the tooth surface, and the points on the tooth surface The calculation formula is as follows:

[0095]

[0096] Where, is the helical rotation angle, ∈ is the installation angle of the grinding wheel 100, a is the normal distance between the central axis of the grinding wheel 100 and the central axis of the gear 200, and P is the lead parameter of the tooth surface.

[0097] Specifically, for an involute helicoid, P = r b tanγ b , rb =m n zcosα n / cosβ,r b is the base circle radius, γ b is the base cylindrical helix angle, α n is the normal pressure angle.

[0098] The analytical algorithm for the profiled grinding tooth surface of helical gears provided in this embodiment first understands the curved surface of the grinding wheel 100 as a rotating surface formed by rotating the axial cross-section of the grinding wheel 100 around the central axis of the grinding wheel 100. Then, based on the properties of the rotating surface, it is obtained that the normal vector of any point on the rotating surface intersects the central axis of the grinding wheel 100 at the same point. Then, using the principle of gear meshing, it is obtained that the point on the surface of the grinding wheel 100 is the point on the tooth surface of the gear 200, and finally the analytical formula for the contact line is obtained, which shortens the time spent in the calculation process, greatly improves the calculation speed, and does not require the selection of initial values, thereby increasing the accuracy of the analytical formula for the contact line and ensuring the processing accuracy of the gear 200 obtained by the final processing.

[0099] Furthermore, the normal modulus m of this embodiment is n =17, number of teeth z=36, helix angle β=7°, normal pressure angle α n =20°, the installation angle of the grinding wheel 100 ∈=83°, and the common normal distance a between the central axis of the grinding wheel 100 and the central axis of the gear 200 =695 mm. The truncation of the grinding wheel 100 is an arc. The specific arc curve equation is:

[0100] Q=[0 112+112cosδ 434+112sinδ] T

[0101] Where δ is between -0.25 and -0.58.

[0102] The computational efficiency of the method proposed in this embodiment is about 15 times that of the traditional numerical algorithm. The cross-sectional shape is used to obtain the arc curve according to the analytical algorithm proposed in this embodiment, as shown in FIG. Figure 4 As shown, the tooth surface of the gear 200 is fitted using this curve group, as shown in Figure 5 shown.

[0103] It should be noted that, in other embodiments of the present invention, the parameter normal modulus m n , number of teeth z, helix angle β, normal pressure angle α n , the installation angle ∈ of the grinding wheel 100, and the common normal distance a between the center axis of the grinding wheel 100 and the center axis of the gear 200 are not limited to the limitations of this embodiment, and can also be other values, which are specifically determined according to the actual gear 200 to be processed.

[0104] The analytical algorithm for the helical gear forming and grinding tooth surface provided in this embodiment increases the accuracy of the analytical formula for the contact line because it does not require the selection of initial values. It has the characteristics of short calculation time and fast calculation speed, so that the final processed gear 200 has higher accuracy.

[0105] Note that the above are only preferred embodiments of the present invention and the technical principles employed. Those skilled in the art will understand that the present invention is not limited to the specific embodiments described herein, and that various obvious changes, readjustments, and substitutions can be made by those skilled in the art without departing from the scope of protection of the present invention. Therefore, although the present invention has been described in detail through the above embodiments, the present invention is not limited to the above embodiments and may include many other equivalent embodiments without departing from the concept of the present invention. The scope of the present invention is determined by the scope of the appended claims.

Claims

1. An analytical algorithm for the profile grinding of helical gear tooth surfaces, characterized in that: include: S1. Establish the grinding wheel coordinate system and the gear coordinate system, and list the coordinates of any point on the curve of the axial section of the grinding wheel , where t is the B-spline parameter; S2. Calculate any point on the curve of the axial section of the grinding wheel The normal vector of the grinding wheel is at z c Components on the axis , z c The axis is the axis where the center axis of the grinding wheel is located; S3. Rotate the axial section of the grinding wheel 360° around the central axis of the grinding wheel to form a grinding wheel, and list the coordinates of any point on the rotating surface of the grinding wheel. ,in, is the angle of rotation; S4. Calculate the normal vector of any point on the surface of the grinding wheel , ; S5. The coordinates of any point on the rotating surface are obtained by transforming the coordinate matrix into the gear coordinate system. , and get Coordinates in the grinding wheel coordinate system ; S6. The grinding wheel rotates around the axis of the gear and moves along the axis of the gear at the same time. Velocity vector ; S7, by and vertical elimination , point on the contact line between the gear and the grinding wheel Simplified to ; S8, move the contact line along the gear spiral line to form the tooth surface, the point on the tooth surface The calculation formula is as follows: Where, is the helix rotation angle, is the grinding wheel installation angle, a is the normal distance between the center axis of the grinding wheel and the center axis of the gear, is the lead parameter of the tooth surface; S1 Where, are the control points of the B-spline, is the basis function of B-spline, and The tectonic curve of the grinding wheel is in y c axis and z c Component of the axis, y c The axis is the vector axis along the radial direction of the grinding wheel and pointing to the center of the gear; S2 The calculation formula is as follows: Where, for and Component in z c Components on the axis; S3 The calculation formula is as follows: Where, The range is (0,2π]; Normal vector in S4 The calculation formula is as follows: Where, , , ; S5 The calculation formula is as follows: ； S5 The calculation formula is as follows: ； Velocity vector in S6 The calculation formula is: = Where, is the angular velocity of the grinding wheel rotating around the gear axis, is the speed of the grinding wheel moving along the axis of the gear, , where is the normal module, z is the number of teeth, is the helix angle; By S7 and Vertical knowledge, , and then we get the following formula: Where, , , ,Will and After extraction, the following system of equations is obtained: In the formula, D, E, and F are parameters; Will and Substitution ,get The calculation formula is: Where, is a point on the contact line.

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