Calculation Method for the Distribution of Circumferential Backlash Deviation of Helical Gears Based on the Jacobian Spinor Model

Abstract

The present invention discloses a calculation method for the circumferential backlash deviation distribution of helical gears based on the Jacobian screw model, which specifically includes the following steps: S1: According to the assembly model, define the planes or cylindrical surfaces involved in assembly contact and tolerance marking in each part as characteristic surfaces, and then obtain the characteristic information, tolerance information, and assembly information of the characteristic surfaces; S2: Use the Jacobian screw model to calculate the deviation screw in the six degrees of freedom directions of the tooth axis; S3: Combine the deviation screw in the six degrees of freedom directions of the tooth axis in S2 and use the improved circumferential backlash formula of helical gears to calculate the circumferential backlash deviation distribution. By building a bridge between the Jacobian screw model and the circumferential backlash empirical formula, the present invention realizes the calculation of the distribution range of the circumferential backlash of helical gears under the tolerances of each part of a given assembly, realizes the simulation calculation of the assembly accuracy, helps to pre-estimate the circumferential backlash before trial assembly, guides gear processing, and shortens the assembly and grinding time.

Description

Technical Field

[0001] The present invention relates to the technical field of gearboxes, and particularly to a method for calculating the circumferential backlash deviation distribution of helical gears based on the unified Jacobian-Torsor model. Background Art

[0002] During the assembly process of large planetary gearboxes, the deviations of each part and each assembly surface will accumulate through assembly contacts, affecting the deviations of key functional parts such as the gear meshing position, and further affecting the vibration and noise performance of the gear system. The circumferential backlash is a key indicator affecting the performance of the gear system. An appropriate circumferential backlash can allow for lubricating oil and gear thermal expansion, while an excessive circumferential backlash will cause meshing impact in the gear system, resulting in vibration.

[0003] Currently, the control method for circumferential backlash is to install a pair of gears before grinding on a fixed-axis test fixture, and then insert lead wires, or plug gauges, or fix one tooth and rotate the other tooth to measure the maximum rotation angle, and then grind the tooth surface according to the test results to adjust the tooth thickness. However, the deviations of other parts outside the gears in the gear train during the assembly of the gearbox will also affect the circumferential backlash. Therefore, the calculation of the circumferential backlash tolerance needs to consider the tolerances of other parts in the gearbox.

[0004] Regarding the influence of the tolerances of other parts on the circumferential backlash during assembly, a three-dimensional tolerance analysis method can be used. Compared with the dimension chain method, it can take into account the geometric deflections caused by form and position tolerances and shaft-hole clearances. The three-dimensional tolerance analysis methods developed at home and abroad include the matrix approach, the unified Jacobian-Torsor model, the Vector Loop method, the Direct Linearization Method (DLM), etc. Some of the mature methods have been successfully applied to commercial tolerance analysis software such as 3DCS, VisVSA, and Cetol-6σ. The unified Jacobian-Torsor model is a linear kinematic model for tolerance analysis. The torsor model part is suitable for expressing dimensional and form and position tolerances, and the Jacobian model part is suitable for expressing the influence of the tolerances of other parts of the assembly on the geometric position of the target part. Its combination with the Monte Carlo statistical simulation method can be used for the transfer calculation of tolerances with a given statistical distribution. Through the calculation using the unified Jacobian-Torsor model, the statistical distribution of the center distance tolerance of the gear axis and the statistical distribution of the axis angle tolerance can be obtained, but the circumferential backlash cannot be directly calculated.

[0005] In addition to being affected by the position and angular tolerance of the gear tooth axis, the circumferential backlash tolerance is also affected by the pitch, profile, and helix angle tolerances of the gear teeth, as well as the circumferential runout tolerance of the gear teeth. At the same time, the meshing angle and helix angle of the helical gear also affect the calculation of the circumferential backlash. The standard DIN3967-1978 "Principles of Tooth Thickness Deviation and Tooth Thickness Tolerance for Backlash" provides an empirical calculation formula for the circumferential backlash of helical gears, which takes into account factors such as pitch, profile, helix angle, center distance, axis non-parallelism, and component shape and size deviations, and obtains the circumferential backlash tolerance through the synthesis method of independent random variables:

[0006]

[0007]

[0008] Although the above formula mentions three-dimensional geometric variables such as center distance, axis non-parallelism, and component shape and size, it does not give a method for calculating these variables, so the circumferential backlash tolerance still cannot be accurately obtained. Summary of the Invention

[0009] Aiming at the problem of reduced calculation accuracy of the circumferential backlash in the prior art, the present invention provides a calculation method for the circumferential backlash deviation distribution of helical gears based on the Jacobian screw model, which can build a bridge between the Jacobian screw model and the empirical formula of the circumferential backlash, realize the calculation of the distribution range of the circumferential backlash of helical gears under the tolerances of various parts of a given assembly, realize the simulation calculation of the assembly accuracy, help to pre-estimate the circumferential backlash before trial assembly, guide gear processing, and shorten the assembly and grinding time.

[0010] To achieve the above object, the present invention provides the following technical solutions:

[0011] A calculation method for the circumferential backlash deviation distribution of helical gears based on the Jacobian screw model, specifically including the following steps:

[0012] S1: According to the assembly model, define the planes or cylindrical surfaces involved in assembly contact and tolerance marking in each part as feature surfaces, and then obtain the feature information, tolerance information, and assembly information of the feature surfaces;

[0013] S2: Use the Jacobian screw model to calculate the deviation screw in the six-degree-of-freedom direction of the gear tooth axis;

[0014] S3: Combine the deviation screw in the six-degree-of-freedom direction of the gear tooth axis in S2, and use the improved circumferential backlash formula of helical gears to calculate the circumferential backlash deviation distribution.

[0015] Preferably, in S1, the parts include sun gear, output shaft, left end cover, ring gear, connecting ring, right support, input shaft, planet carrier, planet shaft, planet gear, output shaft bearing, input shaft bearing, and planet gear bearing.

[0016] Preferably, in S1, the feature information includes the position and orientation of the feature surface in the coordinates of the assembly model; the tolerance information includes the tolerance type and tolerance parameters; the assembly information includes the assembly contact and tolerance marking relationship between the feature surfaces, denoted as an assembly pair.

[0017] Preferably, S2 includes the following steps:

[0018] S2-1: According to the tolerance information and assembly information, randomly generate the deviation screw of the assembly pair within the tolerance range of each feature surface;

[0019] S2-2: Generate the Jacobian matrix of each assembly pair according to the feature information;

[0020] S2-3: Multiply the deviation screw of the assembly pair in S2-1 by the Jacobian matrix of the assembly pair in S2-2 and superimpose and calculate M times to obtain the deviation screw of the six degrees of freedom direction of the tooth axis.

[0021] Preferably, in S2-1, the deviation screw of the assembly pair is a 6×1 vector, containing six components of three translations and three rotations, and the expression is as follows:

[0022] T = [u, v, w, α, β, γ] T (1)

[0023] In formula (1), u, v, w are the translational deviations along the local coordinate axes x, y, z, and α, β, γ are the rotational deviations around the local coordinate axes x, y, z.

[0024] Preferably, in S2-2, the Jacobian matrix is a 6×6 matrix defined as follows:

[0025]

[0026]

[0027]

[0028] In formula (2), represents the Jacobian matrix of the i-th deviation source, FE represents the geometric feature; n represents the target feature, is the direction matrix of the coordinate system of the i-th deviation source relative to the global coordinate system, which can be calculated from the direction cosines of the three local coordinate axes in the global coordinates; is the direction matrix of the target feature coordinate system relative to the coordinate system of the i-th deviation source; dk n 、dk i (k = x, y, z) are the values of the k-axis of the target feature sub-coordinate system and the coordinate system of the i-th deviation source in the global coordinate system.

[0029] Preferably, in the step S2-3, the deviation screw of the tooth axis in six degrees of freedom is as follows:

[0030] T FR =[u FR ,v FR ,w FR ,α FR ,β FR ,γ FR T =[[J] FE1 … [J] FEn [T FEi … T FEn T (3)

[0031] In formula (3), T FR represents the deviation screw between the axes of the two meshing teeth, and FR represents the output result; u FR , v FR , w FR represent the translational deviations along the local coordinate axes x, y, z; α FR , β FR , γ FR represent the rotational deviations around the local coordinate axes x, y, z; the superscript T in the upper right corner represents the transpose matrix; represents the Jacobian matrix of the first deviation source; represents the Jacobian matrix of the nth deviation source; represents the deviation screw of the ith deviation source (i = 1, 2, …, n).

[0032] Preferably, the step S3 includes the following steps:

[0033] S3-1: Combining the deviation screw obtained in S2, calculate the backlash correction caused by the center distance tolerance and the backlash correction caused by the non-parallelism of the hole axes;

[0034] S3-2: Substitute the tooth surface tolerance information into the improved helical gear circular backlash formula to calculate the maximum circular backlash and the minimum circular backlash.

[0035] Preferably, in the step S3-1, the formula for calculating the backlash correction Δj a caused by the center distance tolerance is:

[0036]

[0037] The formula for calculating the backlash correction Δj ∑β caused by the non-parallelism of the hole axes is:

[0038]

[0039] ​​In Formulas (4) and (5), A a represents the maximum radial displacement of the axis; α n represents the engagement angle; f ∑β represents the axis skew on the L G length, and L G represents the separation of the bearing centers on the shaft, and b represents the tooth width.

[0040] Preferably, in the S3-2,

[0041] The improved circumferential backlash formula for helical gears is as follows:

[0042]

[0043]

[0044] In Formula (6), j tmin represents the minimum value of the circumferential backlash of the helical gear; j tmax represents the maximum value of the circumferential backlash of the helical gear; ∑A ste is the sum of the upper deviations of the tooth thickness on the end face, A sne1 represents the upper deviation of one of the gears in the engagement pair; A sne2 represents the upper deviation of the other gear in the engagement pair; β represents the helix angle; ΣA sti is the sum of the lower deviations of the tooth thickness on the end face, A sni1 represents the lower deviation of one of the gears in the engagement pair; A sni2 represents the lower deviation of the other gear in the engagement pair; Δj a represents the backlash correction caused by the center distance tolerance, A a represents the maximum radial displacement of the axis; α n represents the engagement angle; Δj Σβ represents the backlash correction caused by the non-parallelism of the hole axes, f ∑β represents the axis skew on the L G length, and L G represents the separation of the bearing centers on the shaft, and b represents the tooth width; Δj F1 represents the backlash correction caused by the tooth deviation of Gear 1, Δj F2 represents the backlash correction caused by the tooth deviation of Gear 2, F β represents the tooth direction deviation, F f represents the tooth profile deviation, f pt is the single pitch deviation, α t represents the end face pressure angle, Δj BIndicates the backlash correction caused by the dimensions and geometric and positional deviations of the dimensional chain, which is obtained by estimation in the standard. This software adopts the circumferential backlash caused by the radial runout of the gear teeth. F″ i1 Indicates the radial runout of Gear 1, F″ i2 Indicates the radial runout of Gear 2.

[0045] In summary, due to the adoption of the above technical solutions, compared with the prior art, the present invention has at least the following beneficial effects:

[0046] 1. Combines the tolerances of the components of the gearbox and the pitch, profile, and helix angle tolerances of the gears, accurately calculates the distribution range of the circumferential backlash of the helical gears under the tolerances of each part of the given assembly, realizes the simulation calculation of the assembly accuracy, helps to pre-estimate the circumferential backlash before trial assembly, helps to guide gear grinding, and shortens the assembly and grinding time.

[0047] 2. Realizes the statistical tolerance calculation of the circumferential backlash under the influence of three-dimensional tolerances, can calculate according to the part tolerance data before assembly, and realizes the function of predicting the circumferential backlash from the dimensional and geometric tolerances.

[0048] 3. Based on the concise expression of matrix multiplication, it is very suitable for computer programming-assisted analysis, and provides a mathematical basis for developing the corresponding functional software platform. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] Figure 1 It is a schematic flow chart of a method for calculating the circumferential backlash deviation distribution of a helical gear based on the Jacobian twist model according to an exemplary embodiment of the present invention.

[0050] Figure 2 It is a schematic diagram of the twist distribution of the axis deviation of the gearbox according to an exemplary embodiment of the present invention.

[0051] Figure 3 It is a schematic diagram of the circumferential backlash distribution of the gearbox according to an exemplary embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0052] The present invention will be further described in detail below in conjunction with the embodiments and the specific implementation manners. However, this should not be construed as limiting the scope of the above-mentioned subject matter of the present invention to the following embodiments. All technologies implemented based on the content of the present invention belong to the scope of the present invention.

[0053] In the description of the present invention, it should be understood that the orientation or positional relationship indicated by the terms "longitudinal", "transverse", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", etc. is based on the orientation or positional relationship shown in the drawings. It is only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore should not be construed as a limitation to the present invention.

[0054] As Figure 1 shown, the present invention provides a method for calculating the circumferential backlash deviation distribution of helical gears based on the Jacobian screw model, including the following steps:

[0055] S1: Obtain the feature information, tolerance information, and assembly information of various parts in the assembly: According to the assembly model (i.e., the designed assembly drawing), define the planes or cylindrical surfaces involved in assembly contact and tolerance marking in each part, denoted as feature surfaces; then obtain the feature information of the feature surfaces, where the feature information includes the position and direction of the feature surfaces in the coordinate of the assembly model; the tolerance information includes the tolerance type and tolerance parameters (which have been marked on the assembly model according to the design); the assembly information includes the assembly contact and tolerance marking relationship between the feature surfaces, denoted as an assembly pair. As shown in Table 1.

[0056] Table 1. Feature information, tolerance information, and assembly information of various parts in the assembly

[0057]

[0058]

[0059] In this embodiment, the meaning of the assembly pair is a general term for the contact pair formed by two contacting feature surfaces and the tolerance pair formed by two tolerance - marked feature surfaces. An assembly pair can be the same part or different parts. An assembly pair is a deviation source, introducing a deviation screw, and the specific definition can refer to the relevant literature on Jacobian screws.

[0060] For example, the connection line between the left - end - cover tooth and the left - end - cover shaft on the left - end cover is an assembly pair (the same part); the connection line between the left - left - end - cover tooth on the left - end cover and the left - end of the tooth ring on the tooth ring is an assembly pair (different parts).

[0061] S2: Use the Jacobian screw model to calculate the statistical distribution of the deviation screws in the six - degree - of - freedom directions of the gear tooth axis.

[0062] S2 - 1: According to the tolerance information and assembly information in S1, randomly generate the deviation screws of the assembly pairs within the tolerance range of each feature surface.

[0063] In this embodiment, the manufacturing deviation of the cylindrical surface geometric feature or the planar geometric feature is described by a deviation screw. Taking the center of the cylinder as the origin and the axial direction of the cylinder as the x-direction to establish a local coordinate axis, the actual cylindrical surface with manufacturing deviation can be obtained by translating and rotating the ideal cylindrical surface relative to the cylindrical surface. Taking the center of the plane as the origin and the normal direction of the plane as the x-direction to establish a local coordinate axis, the actual plane with manufacturing deviation can be obtained by translating and rotating the ideal plane relative to the cylindrical surface. The ZYX Euler angle convention is adopted as the rule for continuous planar rotation.

[0064] In this embodiment, the deviation screw is a 6×1 vector, which contains six components of three translations and three rotations, and the expression is as follows:

[0065] T = [u, v, w, α, β, γ] T (1)

[0066] In formula (1), u, v, and w are translational deviations along the local coordinate axes x, y, and z, and α, β, and γ are rotational deviations around the local coordinate axes x, y, and z. Each assembly pair will introduce a deviation source, and the resulting deviation is the translation and rotation in the local coordinates. The six-degree-of-freedom deviation screw T is used to express this deviation.

[0067] The components of the deviation screw are restricted by the tolerance, so that the feature surface after translation and rotation remains within the space of the tolerance domain. After the tolerance domain is given, the six components of the deviation screw are randomly selected within the tolerance domain inequality constraints according to the normal distribution 3σ principle. The method of repeated random selection is the Monte Carlo method. The definition of the tolerance domain inequality refers to the relevant literature on Jacobian screws (for example, Chinese Patent Publication No.: CN113779735A).

[0068] S2-2: Generate the Jacobian matrix of each assembly pair according to the feature information obtained in S1.

[0069] In this embodiment, a local coordinate system is established at the center of all geometric features serving as deviation sources, and a fixed global coordinate system is arbitrarily established. A local coordinate system (target feature coordinate system) is also established at the center of the geometric feature (target feature) where the deviation needs to be measured, and the establishment method is the same as that of other local coordinate systems.

[0070] Then the generated Jacobian matrix is a 6×6 matrix defined as follows:

[0071]

[0072]

[0073]

[0074] In formula (2), The Jacobian matrix representing the i-th deviation source, FE represents the geometric feature; n represents the target feature. It is the direction matrix of the coordinate system of the i-th deviation source relative to the global coordinate system, which can be calculated from the direction cosines of the three local coordinate axes in the global coordinates. It is the direction matrix of the target feature coordinate system relative to the coordinate system of the i-th deviation source; dk n 、dk i (k = x, y, z) are the values of the k-axis of the target feature coordinate system and the coordinate system of the i-th deviation source in the global coordinate system.

[0075] S2-3: Multiply the deviation screw of the assembly pair in S2-1 by the Jacobian matrix of the assembly pair in S2-2 and sum them M times (preferably 300 times) to calculate the deviation screw of the six degrees of freedom direction of the tooth axis:

[0076] T FR = [u FR , v FR , w FR , α FR , β FR , γ FR T = [[J] FE1 … [J] FEn [T FEi … T FEn T (3)

[0077] In formula (3), T FR represents the deviation screw between the axes of the two meshing teeth, FR represents the output result; u FR 、v FR 、w FR represent the translational deviations along the local coordinate axes x, y, z; α FR 、β FR 、γ FR represent the rotational deviations about the local coordinate axes x, y, z; the superscript T in the upper right corner represents the transpose matrix; represents the Jacobian matrix of the first deviation source; represents the Jacobian matrix of the n-th deviation source; represents the deviation screw of the i-th deviation source (i = 1, 2, …, n).

[0078] In this embodiment, after summing 300 times, the result of the deviation screw of the six degrees of freedom direction of the tooth axis is as Figure 2 shown.

[0079] S3: Use the improved circumferential backlash formula of the helical gear to calculate and obtain the circumferential backlash distribution.

[0080] S3-1: Calculate the backlash correction Δj caused by the center distance tolerance in combination with the deviation screw obtained in S2 a and the backlash correction Δj caused by the non-parallelism of the hole axes ∑β .

[0081] In this embodiment, the calculation formula for the backlash correction caused by the center distance tolerance is as follows:

[0082]

[0083] The calculation formula for the backlash correction caused by the non-parallelism of the hole axes is as follows:

[0084]

[0085] In formulas (4) and (5), A a represents the maximum radial displacement of the axis; α n represents the pressure angle; f ∑β represents the axis deflection amount on the length of L G where L G represents the separation amount of the bearing centers on the shaft, and b represents the tooth width;

[0086] S3-2: Substitute the tooth surface tolerance information into the improved helical gear circular backlash formula to calculate and obtain the maximum circular backlash and the minimum circular backlash.

[0087] In this embodiment, the improved helical gear circular backlash formula is as follows:

[0088]

[0089]

[0090] In formula (6), j tmin represents the minimum value of the helical gear circular backlash; j tmax represents the maximum value of the helical gear circular backlash; ∑A ste is the sum of the upper deviations of the tooth thickness on the end face, A sne1 represents the upper deviation of one of the gears in the meshing pair; A sne2 represents the upper deviation of the other gear in the meshing pair; β represents the helix angle; ∑A sti is the sum of the lower deviations of the tooth thickness on the end face, A sni1 represents the lower deviation of one of the gears in the meshing pair; A sni2 represents the lower deviation of the other gear in the meshing pair; Δj a represents the backlash correction caused by the center distance tolerance, A a represents the maximum radial displacement of the axis; α n represents the pressure angle; Δj ∑βIndicates the side clearance correction caused by the non - parallelism of the hole axis f ∑β Indicates L G The amount of axis skew in the length direction, L G Indicates the separation amount of the bearing centers on the shaft, b represents the tooth width; Δj F1 Indicates the side clearance correction caused by the tooth deviation of gear 1, Δj F2 Indicates the side clearance correction caused by the tooth deviation of gear 2

[0091] F β Indicates the tooth - direction deviation, F f Indicates the tooth - profile deviation, f pt Is the single - pitch deviation, α t Indicates the transverse pressure angle Δj B Indicates the side clearance correction caused by the dimensional and form - position deviations of the dimension chain, which is obtained by estimation in the standard. This software adopts the circumferential side clearance caused by the radial run - out of the teeth F″ i1 Indicates the radial run - out of gear 1, F″ i2 Indicates the radial run - out of gear 2; for the calculation of j tmax If the value inside the absolute value is negative, a plus sign is used before the square root, otherwise a minus sign is used

[0092] In this embodiment, the tooth - surface tolerance information of the gearbox includes tooth width, pressure angle, helix angle, lower limit of tooth thickness, upper limit of tooth thickness, pitch tolerance, tooth - direction tolerance, tooth - profile tolerance, and tooth run - out, etc., as shown in Table 2

[0093] Table 2. Tooth - surface tolerance information of the gearbox is as follows (unit: mm):

[0094]

[0095]

[0096] In this embodiment, for 3000 simulation results, formula (6) is repeatedly calculated 3000 times, and the minimum circumferential side - clearance distribution calculated for the sun gear and planet gear according to the least - material requirement is as Figure 3 shown, which is a truncated normal distribution, with a maximum of 0.035 mm

[0097] Those of ordinary skill in the art can understand that the above - mentioned embodiments are specific embodiments for implementing the present invention, and in actual applications, various changes can be made in form and details without departing from the spirit and scope of the present invention

Claims

1. A calculation method for the circumferential backlash deviation distribution of helical gears based on the Jacobian spinor model, characterized in that Specifically, it includes the following steps: S1: According to the assembly model, define the planes or cylindrical surfaces in each part that involve assembly contact and tolerance marking as feature surfaces, and then obtain the feature information, tolerance information, and assembly information of the feature surfaces; S2: Use the Jacobian screw model to calculate the deviation screw of the six degrees of freedom direction of the gear tooth axis; S3: Combine the deviation screw of the six degrees of freedom direction of the gear tooth axis in S2 and use the improved circumferential backlash formula of helical gears to calculate the circumferential backlash deviation distribution; The S3 includes the following steps: S3-1: Combine the deviation screw obtained in S2 to calculate the backlash correction caused by the center distance tolerance and the backlash correction caused by the non-parallelism of the hole axis; In the above S3-1, the side clearance correction Δj caused by the center distance tolerance a is calculated by the following formula: The side clearance correction Δj caused by the non-parallelism of the hole axes ∑β The calculation formula is as follows: In Formulas (4) and (5), A a represents the maximum radial displacement of the axis; α n represents the pressure angle; β represents the helix angle; v FR , w FR represent the translational deviations along the local coordinate axes y and z; f ∑β represents the axis deflection amount over the length of L G , where L G represents the separation amount between the bearing centers on the shaft, b represents the tooth width; β FR , γ FR represent the rotational deviations about the local coordinate axes y and z; S3-2: Substitute the tooth surface tolerance information into the improved circumferential backlash formula for helical gears to calculate and obtain the maximum circumferential backlash and the minimum circumferential backlash, that is, the circumferential backlash Δj caused by the radial runout of the tooth B : F″ i1 represents the radial runout of gear 1, F″ i2 represents the radial runout of gear 2, α t represents the transverse pressure angle, 2. The method for calculating the circumferential backlash deviation distribution of a helical gear based on the Jacobian screw model according to claim 1, wherein In the S1, the parts include sun gear, output shaft, left end cover, ring gear, connecting ring, right support, input shaft, planet carrier, planet shaft, planet gear, output shaft bearing, input shaft bearing, and planet gear bearing.

3. The calculation method for the circumferential backlash deviation distribution of helical gears based on the Jacobian screw model according to claim 1, characterized in that In the S1, the feature information includes the position and direction of the feature surface in the coordinate of the assembly model; the tolerance information includes the tolerance type and tolerance parameters; the assembly information includes the assembly contact and tolerance marking relationship between the feature surfaces, denoted as an assembly pair.

4. The method for calculating the circumferential backlash deviation distribution of a helical gear based on the Jacobian screw model according to claim 1, characterized in that The S2 includes the following steps: S2-1: Randomly generate the deviation screw of the assembly pair within the tolerance range of each feature surface according to the tolerance information and assembly information; S2-2: Generate the Jacobian matrix of each assembly pair according to the feature information; S2-3: Multiply the deviation screw of the assembly pair in S2-1 by the Jacobian matrix of the assembly pair in S2-2 and superimpose it M times to calculate the deviation screw of the six degrees of freedom direction of the gear tooth axis.

5. The calculation method for the circumferential backlash deviation distribution of helical gears based on the Jacobian screw model according to claim 4, wherein In the S2-1, the deviation screw of the assembly pair is a 6×1 vector, containing six components of three translations and three rotations, and the expression is as follows: T = [u, v, w, α, β, γ] T (1) In formula (1), u, v, w are the translational deviations along the local coordinate axes x, y, z, and α, β, γ are the rotational deviations around the local coordinate axes x, y, z.

6. The calculation method for the circumferential backlash deviation distribution of helical gears based on the Jacobian spinor model according to claim 4, characterized in that, In the S2-2, the Jacobian matrix is a 6×6 matrix defined as follows: In Formula (2), represents the Jacobian matrix of the i-th deviation source, and FE represents the geometric feature; n represents the target feature, is the direction matrix of the coordinate system of the i-th deviation source relative to the global coordinate system, which can be calculated from the direction cosines of the three local coordinate axes in the global coordinates; is the direction matrix of the target feature coordinate system relative to the coordinate system of the i-th deviation source; dk n and dk i are the values of the k-axis of the target feature sub-coordinate system and the coordinate system of the i-th deviation source in the global coordinate system, where k = x, y, z.

7. The calculation method for the circumferential backlash deviation distribution of helical gears based on the Jacobian screw model according to claim 4, characterized in that In the S2-3, the deviation screw of the six degrees of freedom direction of the gear tooth axis is: T FR = [u FR , v FR , w FR , α FR , β FR , γ FR T = [[J] FE1 …[J] FEn [T FEi … T FEn T (3)​​ In formula (3), T FR represents the deviation screw between the axes of two meshing gear teeth, and FR represents the output result; u FR , v FR , w FR represent the translational deviations along the local coordinate axes x, y, z; α FR , β FR , γ FR represent the rotational deviations about the local coordinate axes x, y, z; the superscript T in the upper right corner represents the transpose matrix; represents the Jacobian matrix of the first deviation source; represents the Jacobian matrix of the nth deviation source; represents the deviation screw of the ith deviation source, where i = 1, 2, …, n.

8. The calculation method for the circumferential backlash deviation distribution of helical gears based on the Jacobian screw model according to claim 1, characterized in that In the S3-2, The improved circumferential backlash formula of helical gears is as follows: In formula (6), j tmin represents the minimum circumferential backlash of the helical gear; j tmax represents the maximum circumferential backlash of the helical gear; ∑A ste is the sum of the upper deviations of the tooth thickness on the end face, A sne1 represents the upper deviation of one of the gears in the meshing pair; A sne2 represents the upper deviation of the other gear in the meshing pair; β represents the helix angle; ∑A sti is the sum of the lower deviations of the tooth thickness on the end face, A sni1 represents the lower deviation of one of the gears in the meshing pair; A sni2 represents the lower deviation of the other gear in the meshing pair; Δj a represents the backlash correction generated by the center distance tolerance, A a represents the maximum radial displacement of the axis; α n represents the pressure angle; Δj ∑β represents the backlash correction generated by the non-parallelism of the hole axes, f ∑β represents the axis skew on the length of L G L G represents the separation of the bearing centers on the shaft, b represents the tooth width; Δj F1 represents the backlash correction generated by the tooth deviation of Gear 1, Δj F2 represents the backlash correction generated by the tooth deviation of Gear 2, F β represents the helix deviation, F f represents the profile deviation, f pt is the single pitch deviation, α t represents the transverse pressure angle Δj B represents the backlash correction caused by the dimensional and geometric deviations of the dimension chain

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