A method for calculating time-varying meshing stiffness of a fault gear considering crack position

Abstract

The application relates to the technical field of mechanical dynamics, in particular to a fault gear time-varying meshing stiffness calculation method considering crack positions, which comprises the following steps: establishing a gear fault calculation model; determining corresponding crack category information, and pre-storing crack gear section division rules corresponding to the crack category information; detecting a crack position on the gear that has occurred a fault, adopting the crack gear section division rule to divide the crack position into sections to obtain a plurality of crack sub-sections; obtaining the bending stiffness and the shear stiffness of each crack sub-section for each crack sub-section; and summing the bending stiffness and the shear stiffness of each crack sub-section to obtain the bending stiffness and the shear stiffness of the fault gear. The bending stiffness and the shear stiffness of the fault gear can be obtained by inputting the parameters of the crack, and the calculation efficiency is higher.

Description

TECHNICAL FIELD

[0001] The application relates to the technical field of mechanical dynamics, in particular to a calculation method for time-varying meshing stiffness of a fault gear considering crack position. BACKGROUND

[0002] Gear transmission is one of the most important industrial basic components, which is widely used in aerospace, ship transportation, vehicle transportation, engineering machinery, energy and chemical industry, rail transportation and many other industrial fields, and is a core device for realizing mechanical transmission and power transmission. As a core component of mechanical equipment, the running performance of the gear transmission directly affects the quality and reliability of the whole equipment.

[0003] Crack is one of the most common gear faults. During the crack propagation process, not only the depth and angle of the crack change, but also the position of the crack changes along the tooth width direction, so most of the actual tooth cracks are spatial cracks. Since the depth, angle and position of the spatial crack change along the tooth width direction, the meshing stiffness of the whole tooth cannot be directly solved, so the solution of the spatial crack usually needs to combine the potential energy method and the slice method.

[0004] As can be seen from the patent CN107420523A-a calculation method for meshing stiffness of a helical gear pair with tooth surface crack defect, after sufficient slicing, the crack length, crack angle and crack position in each tooth slice can be regarded as fixed, and the crack parameters of each tooth slice are different, so the meshing stiffness of each tooth slice needs to be solved, and the meshing stiffness of all slices is superimposed to obtain the meshing stiffness of the whole tooth. As can be seen, the key step is to use a calculation method containing three parameters of crack length, crack angle and crack position to solve the stiffness of each slice. However, the existing literature and patents only propose a calculation method for meshing stiffness considering crack length and crack angle, ignoring the change of crack position, and do not propose a calculation method for meshing stiffness containing crack length, crack angle and crack position, which cannot accurately obtain the meshing stiffness of each slice, resulting in inaccurate calculation results of the crack gear stiffness. SUMMARY

[0005] Therefore, it is necessary to provide a calculation method for time-varying meshing stiffness of a fault gear considering crack position to solve the above technical problems. The method comprises the following steps:

[0006] A gear fault calculation model is established; in the gear fault calculation model, the fault position is divided, the corresponding crack category information is determined for the fault position where the fault occurs, and the crack gear section division rule corresponding to the crack category information is pre-stored; the crack position where the fault occurs on the gear is detected, the crack category information is determined according to the gear fault calculation model and the crack position, the crack position is divided into sections by using the crack gear section division rule, and a plurality of crack sub-sections are obtained; for each crack sub-section, the load distance from the tooth profile to the central axis at the load part of the crack gear, the cross-sectional area at the load part of the crack gear and the moment of inertia at the load part of the crack gear are obtained, the load distance, the cross-sectional area and the moment of inertia are input into the bending stiffness calculation model and the shear stiffness calculation model of the normal gear, and the bending stiffness and the shear stiffness of each crack sub-section are obtained; the bending stiffness and the shear stiffness of each crack sub-section are summed, and the bending stiffness and the shear stiffness of the fault gear are obtained.

[0007] The above-mentioned fault gear time-varying meshing stiffness calculation method considering the crack position, the stiffness of the crack gear is calculated according to the parameter data of the crack, the gear fault calculation model divides the fault position, a plurality of groups of crack category information are set correspondingly, and the crack gear section division rule corresponding to the category information is established, so that the calculation model of the load distance, the cross-sectional area and the moment of inertia of the load part of the gear at a plurality of crack sub-sections can be obtained after the parameters of the crack are input, the calculation model of the load distance, the cross-sectional area and the moment of inertia at the load part of the gear is calculated and substituted into the bending stiffness calculation model and the shear stiffness calculation model of the normal gear to obtain the bending stiffness and the shear stiffness of each crack sub-section of the crack gear, and finally the bending stiffness and the shear stiffness of each crack sub-section are summed to obtain the bending stiffness and the shear stiffness of the fault gear. The calculation efficiency is higher. BRIEF DESCRIPTION OF DRAWINGS

[0008] Figure 1 It is a variable cross-section cantilever beam model of the involute spur gear tooth in one embodiment;

[0009] Figure 2 It is a geometric parameter of the wheel body deformation in one embodiment;

[0010] Figure 3 It is two types of crack gear models in one embodiment;

[0011] Figure 4 It is a model in which the crack starting point is in the fillet section of the tooth root and the crack endpoint is inside the tooth root in one embodiment;

[0012] Figure 5 It is a model when the third section is divided in one embodiment;

[0013] Figure 6Model of one embodiment in which the crack initiation point is in the fillet section and the crack endpoint is in the fillet section;

[0014] Figure 7 Model of one embodiment in which the sixth section is divided;

[0015] Figure 8 Model of one embodiment in which the crack initiation point is in the involute section and the crack endpoint is in the inside of the root;

[0016] Figure 9 Model of one embodiment in which the crack initiation point is in the involute section and the crack endpoint is in the fillet section;

[0017] Figure 10 Model of one embodiment in which the crack initiation point is in the involute section and the crack endpoint is in the involute section;

[0018] Figure 11 Flowchart of one embodiment in which the bending stiffness and shear stiffness of the cracked gear are calculated;

[0019] Figure 12 Finite element model of one embodiment in which the stiffness of the cracked gear is calculated;

[0020] Figure 13 Finite element model data graph of one embodiment in which the change rule of the meshing stiffness of the gear with the crack depth q1 is shown; wherein (a) is the change rule of the meshing stiffness calculation result of the theoretical model with the crack depth q1; (b) is the change rule of the meshing stiffness calculation result of the finite element model with the crack depth q1;

[0021] Figure 14 Finite element model data graph of one embodiment in which the change rule of the meshing stiffness of the gear with the crack position ΔR cs is shown; wherein (a) is the change rule of the meshing stiffness calculation result of the theoretical model with the crack position ΔR cs ; (b) is the change rule of the meshing stiffness calculation result of the finite element model with the crack position ΔR cs ;

[0022] Figure 15 Schematic diagram of the slicing method in the prior art. DETAILED DESCRIPTION

[0023] In order to make the purpose, technical scheme and advantages of the present application clearer, the present application will be further described in detail below with reference to the drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application and are not used to limit the present application.

[0024] Since the depth, angle and position of the spatial crack vary along the tooth width direction, the meshing stiffness of the entire gear tooth cannot be directly solved. At present, the solution of the spatial crack is usually based on the potential energy method and the slice method, such as a method for calculating the meshing stiffness of a helical gear pair with a tooth surface crack defect, as shown in Figure 15 It can be seen that, after sufficient slicing, the crack length, crack angle and crack position in each gear tooth slice can be regarded as fixed, so the meshing stiffness of each gear tooth slice only needs to be solved, and then the meshing stiffness of the entire gear tooth can be obtained by superimposing the meshing stiffness of all the slices. It can be seen that the key step is to use a calculation method containing the three crack parameters of crack length, crack angle and crack position to solve the stiffness of each slice. However, the existing and the patent only classify and solve different shapes of spatial crack surfaces based on the slice method, but do not propose a meshing stiffness calculation method containing the three parameters of depth, angle and position. After sufficient slicing, the calculation model of the slice stiffness is lacking, which leads to the inability to obtain an accurate gear meshing stiffness. The present application is to perform high-precision stiffness calculation on each slice after sufficient slicing.

[0025] In one embodiment, as shown in Figures 1-14 a method for calculating the time-varying meshing stiffness of a fault gear considering the crack position is provided, the method comprising:

[0026] A gear fault calculation model is established for calculating the shear stiffness and bending stiffness of each slice of the fault gear. In the gear fault calculation model, the fault position is divided, mainly according to the positions of the crack starting point and the crack end point in the tooth root circular arc section, the involute section and the tooth root interior. For the fault position where the fault occurs, the corresponding crack category information is determined, and the crack gear section division rule corresponding to the crack category information is pre-stored, i.e. according to the division of the crack, it is determined which crack category the crack belongs to, and then the crack sub-section division is performed according to the pre-stored crack gear section division rule corresponding to the crack category, so as to calculate the shear stiffness and bending stiffness of each crack sub-section.

[0027] In the gear fault calculation model, the fault position is divided, and different crack category information is determined according to different fault positions. The gear fault calculation model can calculate the prediction of the gear meshing stiffness according to different crack position parameters. By pre-storing the crack gear section division rule corresponding to the crack category information, after the crack position parameters are input into the gear fault model, the gear meshing stiffness calculation can be directly performed in the corresponding crack category information mode according to the gear section division rule, thereby improving the calculation efficiency and accuracy.

[0028] For specific use of the gear fault calculation model: detect the crack position of the gear fault, determine the crack category information according to the gear fault calculation model and the crack position, perform section division on the crack position by adopting a crack gear section division rule, and obtain a plurality of crack sub-sections; for each crack sub-section, the loaded distance of the tooth profile to the central axis at the loaded part of the crack gear, the cross-sectional area at the loaded part of the crack gear, and the moment of inertia at the loaded part of the crack gear are obtained, the loaded distance, the cross-sectional area, and the moment of inertia are input into the bending stiffness calculation model and the shear stiffness calculation model of the normal gear, and the bending stiffness and the shear stiffness of each crack sub-section are obtained; by dividing the crack position into a plurality of crack sub-sections, and obtaining the loaded distance, the cross-sectional area, and the moment of inertia at the loaded part of the gear for each crack sub-section, the calculation accuracy can be effectively improved by calculating different sections. Since the stress state of the crack gear tooth is consistent with that of the normal gear, the loaded distance, the cross-sectional area, and the moment of inertia are directly input into the bending stiffness calculation model and the shear stiffness calculation model of the normal gear, so that the bending stiffness and the shear stiffness of each crack sub-section of each slice of the crack gear can be obtained, and the accuracy of the stiffness calculation can be ensured; finally, the bending stiffness and the shear stiffness of each crack sub-section are summed up, and the bending stiffness and the shear stiffness of each slice of the fault gear are obtained. By dividing the crack into crack sub-sections, and calculating the bending stiffness and the shear stiffness of each crack sub-section, and finally summing up the bending stiffness and the shear stiffness of each crack sub-section, the accuracy of the stiffness calculation can be improved.

[0029] The present application calculates the stiffness of the crack gear according to the length, angle and position parameters of the crack, the gear fault calculation model divides the fault position, a plurality of sets of crack category information are correspondingly set, and the crack gear section division rule corresponding to the category information is established, so that the calculation model of the loaded distance, the cross-sectional area and the moment of inertia of the loaded part of the gear at a plurality of crack sub-sections can be obtained after the length, angle and position parameters of the crack are input, the calculation model of the loaded distance, the cross-sectional area and the moment of inertia at the loaded part of the gear is calculated, and the bending stiffness and the shear stiffness of each crack sub-section of the crack gear are obtained by substituting the calculation model into the bending stiffness calculation model and the shear stiffness calculation model of the normal gear, and finally the bending stiffness and the shear stiffness of each slice of the fault gear are obtained by summing up the bending stiffness and the shear stiffness of each crack sub-section. The present application has higher calculation efficiency than the finite element method, and a meshing stiffness calculation method containing three parameters of depth, angle and position is proposed.

[0030] The existing literature and patents propose numerous feasible mesh stiffness calculation models for tooth root crack faults, which can quickly and efficiently predict the mesh stiffness of gears under the influence of tooth root cracks. In actual operation, cracks can not only occur at the tooth root. On the tooth surface, the meshing position of the gear tooth is under heavy load and can generate a large stress, which can easily produce fatigue cracks with an increase in the number of stress cycles. In addition to the reference circle, cracks can occur at any position of the tooth profile in practice. However, most of the existing crack gear models are for tooth root cracks, and there is a lack of mesh stiffness calculation models for non-tooth root cracks, so further exploration of the mesh stiffness modeling method under different crack positions is still needed.

[0031] During the crack propagation process, not only the depth (input q1 in the flowchart) and angle (input γ in the flowchart) of the crack change, but also the position of the crack (input Rcs in the flowchart) changes along the tooth width direction, so most of the actual gear tooth cracks are spatial cracks. Since the depth, angle, and position of the spatial crack change along the tooth width direction, the mesh stiffness of the entire gear tooth cannot be directly solved, so the solution of the spatial crack is usually based on the potential energy method and the slicing method (such as patent CN107420523A - A Mesh Stiffness Calculation Method for Helical Gear Pairs with Tooth Surface Crack Defects). As can be seen, after sufficient slicing, the crack length, crack angle, and crack position in each gear tooth slice can be considered fixed, so the mesh stiffness of each gear tooth slice only needs to be solved, and the mesh stiffness of all slices is then superimposed to obtain the mesh stiffness of the entire gear tooth. As can be seen, the key step is to use a calculation method that includes the crack length, crack angle, and crack position to solve the stiffness of each slice. However, patent CN107420523A - A Mesh Stiffness Calculation Method for Helical Gear Pairs with Tooth Surface Crack Defects only classifies and solves spatial crack surfaces of different shapes based on the slicing method, but does not propose a mesh stiffness calculation method that includes the depth, angle, and position of the crack, and the steps lack corresponding model methods.

[0032] When calculating the mesh stiffness of a gear using the potential energy method, it is usually assumed that the gear tooth is a two-dimensional variable cross-section cantilever beam, as shown in Figure 1 . The tooth profile of the gear tooth is composed of three parts, the AB segment is the circular arc of the tooth root fillet, the BC segment is the involute tooth profile, and the CD segment is the tooth top circle arc. Point E is the intersection of the involute and the base circle; O1 is the center of the transition arc, with coordinates (x t ,y t ); A is the starting point of the transition arc on the tooth root circle, with coordinates (x r ,h r); B is the end point of the transition arc. d is the equivalent length of the cantilever beam, representing the distance from the meshing point to the dedendum; F is the force acting perpendicular to the meshing surface; a1 is the angle between the force and the vertical direction (the tooth thickness direction); h is the distance from the tooth profile at the meshing point to the center axis; h x is the distance from the tooth profile at x to the center axis; h a is the distance from the tooth profile at the addendum circle to the center axis. R r is the dedendum radius, R b is the base circle radius, R te is the radius at which the end point B of the transition arc is located. p t is the fillet radius of the dedendum, which can be calculated according to the following formula, formula calculation,

[0033]

[0034] where a0 = 20°, c * = 0.25, p t ≈ 0.38 m, which is consistent with the recommended value of the fillet radius of the dedendum given in the national standard "GB1356-88 Involute Cylindrical Gear Basic Tooth Profile".

[0035] Figure 1 The base circle radius R b in the formula is greater than the dedendum radius R r . In fact, since the fillet of the dedendum always exists regardless of whether the base circle is greater than the dedendum, and the fillet radius is independent of the relative size of R b and R r , it is not necessary to divide the model into two cases of R r > R b and R r < R b . The theoretical derivation of the time-varying stiffness of the gear is completely consistent in these two cases. Therefore, Figure 1 the base circle in the formula only serves as an auxiliary line.

[0036] Under the action of the meshing force F, the deformation at the meshing point is mainly composed of three parts: the deformation of the gear tooth (cantilever beam), the elastic contact deformation of the contact tooth surface, and the deformation of the gear body.

[0037] The derivation process of the calculation expression of the sub-meshing stiffness corresponding to the gear tooth deformation is as follows:

[0038] The deformation of the gear tooth is further divided into bending, shearing and axial compression deformation, and the deformation of the gear tooth is regarded as the deformation of an equivalent spring along the direction of the meshing force. The elastic potential energy stored in the spring due to the three deformations is

[0039]

[0040] where: k bBending stiffness / N-m -1 ; k s Shear stiffness / N-m -1 ; k a Axial compression stiffness / N-m -1 ; P b , P s , P a are the equivalent elastic potential energy of gear tooth due to bending, shear and axial compression deformation, respectively.

[0041] According to the beam deformation theory, the bending, shear and axial compression deformation potential energy of gear tooth under the action of meshing force are

[0042]

[0043] In the formula: E is the elastic modulus / N-m -2 ; G is the shear modulus / N-m -2 ; I x and A x are the cross-sectional moment of inertia and area at x, respectively.

[0044] From the force analysis, it can be seen that

[0045] F b = F cos α1 (4)

[0046] F a = F sin α1 (5)

[0047] M = F b (d-x)-F a h (6)

[0048] Substitute formulas (4), (5) and (6) into formula (3), and combine with formula (2), to obtain

[0049]

[0050]

[0051]

[0052] From the geometric properties of involute and transition curve, the following expressions can be obtained

[0053] A x = 2h x L (10)

[0054]

[0055] h = R b [(α1+α2)cosα1-sinα1] (12)

[0056]

[0057] d = R b [(α1+α2)sinα1+cosα1]-x r (14)

[0058] wherein: L is the tooth width / mm; α2 is the angle corresponding to the half tooth width at the base circle, which is determined by the following formula

[0059]

[0060] Substituting the formula (10) to (14) into the formula (7), (8) and (9), the bending stiffness k b , the shear stiffness k s , and the axial compression stiffness k a of the gear tooth can be obtained respectively as

[0061]

[0062]

[0063]

[0064] The equivalent stiffness k t corresponding to the deformation of the gear tooth can be defined by integrating the bending stiffness, the shear stiffness and the axial compression stiffness as follows

[0065]

[0066] The process of the calculation expression of the sub-meshing stiffness corresponding to the elastic contact deformation of the gear contact surface is as follows:

[0067] In the meshing process of the gear, the contact surface will be deformed elastically, and the corresponding Hertz contact stiffness calculation method is as follows:

[0068]

[0069] wherein: F i is the meshing force of the ith gear pair / N; LSR is the load sharing ratio. The Hertz contact stiffness obtained is not a constant, and is related to the contact force.

[0070] The process of the calculation expression of the sub-meshing stiffness corresponding to the deformation of the gear base body of the gear contact surface is as follows:

[0071] Under the action of the meshing force, the gear body will also be deformed. The corresponding gear body stiffness expression is as follows

[0072]

[0073] In the formula: α m u f and S f like Figure 2 As shown. Coefficient L * M * ,P * Q * Fitted by the following polynomial

[0074]

[0075] Where: h fi =r f / r int r f r int and θ f like Figure 2 As shown. Polynomial coefficients A i B i C i D i E i and F i See Table 1.

[0076]

[0077] The values ​​of the coefficients in the formulas in Table 1

[0078] Based on the above sub-stiffnesses, the single-tooth pair meshing stiffness k of the i-th pair of gear teeth is... (i) The calculation formula is

[0079]

[0080] In the formula: 1 and 2 represent the driving wheel and the driven wheel, respectively.

[0081] Normally, when the overlap ratio of gears is greater than 1, two or more pairs of teeth will mesh simultaneously during the meshing process.

[0082] The expression for the meshing stiffness of multiple pairs of teeth is:

[0083]

[0084] In the formula: λ is the correction coefficient for wheel stiffness; k tooth The total stiffness of all meshing gear pairs (excluding gear body stiffness) is expressed as follows:

[0085]

[0086] In the formula: The total stiffness of the i-th gear pair (excluding gear body stiffness) / N·m -1 .

[0087] Through the calculation model of the time-varying stiffness of the healthy gear, it is known that the presence of the crack will mainly affect the cross-sectional area and the moment of inertia of the loadable part of the gear tooth, and will not affect the geometric parameters and the involute characteristics of the gear tooth. In order to simplify the calculation process, the present application assumes that the crack is a straight line, penetrates the entire tooth width, and the crack depth is constant along the tooth width direction. The crack only affects the bending stiffness and the shear stiffness, and the gear tooth can still be loaded in the axial direction, so the crack does not affect the axial compression stiffness, the Hertz contact stiffness and the gear body stiffness. Therefore, the present patent only re-derives the calculation formula of the bending stiffness and the shear stiffness of the cracked gear tooth.

[0088] The above-mentioned calculation model of the stiffness of the healthy gear contains the tooth root fillet section, but the position of the crack is generally not limited to the tooth root fillet section. Considering the different positions of the crack, the crack can appear in the tooth root fillet section or in the involute tooth profile section. Therefore, the present patent determines the crack category information according to the position of the crack, which is specifically divided into five categories of crack category information, which are respectively:

[0089] The crack starting point is in the tooth root fillet section, and the crack end point is in the tooth root interior;

[0090] The crack starting point is in the tooth root fillet section, and the crack end point is in the tooth root fillet section;

[0091] The crack starting point is in the involute section, and the crack end point is in the tooth root interior;

[0092] The crack starting point is in the involute section, and the crack end point is in the tooth root fillet section;

[0093] The crack starting point is in the involute section, and the crack end point is in the involute section;

[0094] As shown in Figure 3 . In the figure, the circular arc I is the tooth root circle Rr, the circular arc II is the circular arc R cs on which the crack starting point F is located, and the circular arc III is the circular arc tooth root circle R te on which the tooth root fillet end B is located. In the figure, AB is the tooth root fillet section, BC is the involute, and FG is the crack. In the left model, the crack starting position is on the tooth root fillet section, that is, the circular arc radius R cs on which the crack starting point F is located is smaller than the circular arc radius R te on which the tooth root fillet end B is located; in the right model, the crack starting position is on the involute, that is, the circular arc radius R cs on which the crack starting point F is located is greater than the circular arc radius R te on which the tooth root fillet end B is located.

[0095] When the crack starting point is in the tooth root fillet section, that is, R cs < R teAt that time, the crack depth is set to q1, and the angle between the crack and the gear's central axis (referred to as the crack angle) is γ, such as... Figure 4 or Figure 6 As shown in the figure. Based on the geometric relationships in the figure, the coordinates (x, y) of the crack initiation point F are... cs ,h cs It can be determined by the following relationship.

[0096]

[0097] From this we can obtain

[0098] d cs =x cs -x r (27)

[0099] h c =h cs -q1sinγ (28)

[0100] In the formula: d cs The distance from the crack initiation point to the tooth root is in mm.

[0101] When the crack tip is inside the tooth root, such as Figure 4 As shown, the crack tip G has extended to the left of the dashed line AH, at which point d cs <q1cosγ;

[0102] At this point, considering the degree of crack propagation, it is necessary to divide the gear into different segments according to the cracked gear segmentation rules, and then solve for the bending stiffness and shear stiffness of the cracked gear teeth based on the different segment divisions. Specifically:

[0103] First, we need to obtain the distance h between the crack centerline and the crack axis. c α, the angle between the crack initiation points cs α, the angle between the crack ends c And the pre-stored distance h between the center lines of the tooth root fillets te , distance h from the centerline of the tooth tip a and the angle α1 between the forces;

[0104] Wherein, the distance from the crack centerline to h c The vertical distance from the crack tip G to the gear's central axis; the crack initiation angle α. cs The point of tangency between the crack initiation point and the base circle, and the angle between this point of tangency, the gear center, and the central axis; the crack endpoint angle α. c Let h be the projection point of the crack endpoint onto the tooth profile along the central axis, the point of tangency between the projection point and the base circle, and the angle between the connection between this point of tangency and the center of the gear circle and the central axis; the distance h from the central axis to the tooth root fillet. te The distance from the endpoint of the root fillet away from the tooth root to the centerline (i.e., half the tooth width at the endpoint B of the root fillet); the distance h from the centerline of the tooth tip.a is the distance from the addendum point to the center axis (i.e. the half addendum width).

[0105] When the root fillet center axis distance > crack center axis distance > addendum center axis distance, and the force angle > crack end point angle (i.e. h te > h c > h a & a1> a c时 is divided into the first section;

[0106] At this time, the cross-sectional area and the moment of inertia of the loaded part of the tooth are

[0107]

[0108]

[0109] In the formula: h x determined by formula (13), h c determined by formula (28).

[0110] The stress state of the cracked tooth is consistent with that of the normal gear. From the stress analysis, the bending stiffness k b,crack and shear stiffness k s,crack of the cracked tooth can be obtained, and the calculation expressions are

[0111]

[0112]

[0113] According to the expressions of h x , A x and I x in different crack sub-sections, the integral interval in formulas (31) and (32) can be divided into the following three crack sub-sections:

[0114] a) [0, d1]. In this interval, h x , A x and I x are respectively

[0115]

[0116] A x = (h c +h x )L (33)

[0117]

[0118] b) [d1, d c ]. In this interval, h x , A x and Ix respectively

[0119] h x = R b [(α2- α)cosα + sinα]

[0120] A x = (h c +h x )L (34)

[0121]

[0122] c)[d c ,d]. In this interval h x , A x and I x are respectively

[0123] h x = R b [(α2- α)cosα + sinα]

[0124] A x = 2h x L (35)

[0125]

[0126] Substituting formula (33), (34) and (35) into formula (31) and (32) and summing the three parts of integral, the bending stiffness and shear stiffness of each slice of the faulty gear are obtained as Figure 15

[0127]

[0128]

[0129] When the crack central axis distance < the tooth top central axis distance, or, when the tooth root fillet central axis distance > the crack central axis distance > the tooth top central axis distance, and the force angle < the crack end point angle (i.e. h c < h a or h te > h c > h a & α1< α c ) is the second section division;

[0130] At this time, the cross-sectional area and the moment of inertia of the loaded part of the gear tooth are

[0131] A x = (h c +h x )L (38)​

[0132]

[0133] Where: h x h is determined by formula (13). c It is determined by formula (28).

[0134] According to h x A x and I x In the expressions for different crack sub-segments, the integral intervals in the formulas can be divided into the following two parts:

[0135] a) [0, d1]. Within this interval h x A x and I x The expression is the same as formula (33);

[0136] b)[d1,d]. Within this interval h x A x and I x The expression is the same as formula (34).

[0137] Similarly, the bending stiffness and shear stiffness of each slice of the faulty gear can be obtained as follows:

[0138]

[0139]

[0140] When the distance from the crack centerline is greater than the distance from the tooth root fillet centerline (i.e., h) c h te This is the third section division.

[0141] At this time, when the crack is small or in the initial propagation stage, it will appear Figure 5 In the case shown, the length d of the tooth affected by the crack c For a length d1 less than the root fillet, the formulas for calculating the cross-sectional area and moment of inertia of the gear teeth are the same as those for formulas (29) and (30), except that d1 is less than the length d1. c Different, d at this time c Determined by the following two formulas

[0142]

[0143] d c =x c -x r (43)

[0144] According to h x A x and I xIn the expression of different crack sub-sections, the integral interval in formula (31) and (32) can be divided into the following three parts:

[0145] a) [0, d c ]. In this interval, h x , A x and I x expressions are the same as formula (33);

[0146] b) [d c , d1]. In this interval, h x , A x and I x are respectively

[0147]

[0148] A x = 2h x L(44)

[0149]

[0150] c) [d1, d]. In this interval, h x , A x and I x expressions are the same as formula (35).

[0151] The integral of the three parts is integrated, and the bending stiffness and shear stiffness of the cracked gear tooth are respectively

[0152]

[0153]

[0154] When the crack end point is in the tooth root fillet section, the crack model is shown in Figure 6 , the crack end point G is not expanded to the left side of the dashed line AH, and at this time d cs > q1cosγ;

[0155] At this time, when the tooth root fillet axis distance > crack axis distance > tooth top axis distance, and the force angle > crack end point angle (that is, h te > h c > h a &α1>α c ) is the fourth section division;

[0156] At this time, the cross-sectional area and the moment of inertia of the loaded part of the gear tooth are

[0157]

[0158]

[0159] where h x is determined by equation (13). c is determined by equation (28).

[0160] According to h x , A x and I x , the expressions of different crack sub-sections can divide the integral interval in equations (31) and (32) into the following four parts:

[0161] a) [0, d cs -q1cosγ]. In this interval, h x , A x and I x expressions are the same as equation (44);

[0162] b) [d cs -q1cosγ, d1]. In this interval, h x , A x and I x expressions are the same as equation (33);

[0163] c) [d1, d c ]. In this interval, h x , A x and I x expressions are the same as equation (34);

[0164] d) [d c , d]. In this interval, h x , A x and I x expressions are the same as equation (35).

[0165] Similarly, the bending stiffness and shear stiffness of each slice of the faulty gear can be obtained as

[0166]

[0167]

[0168] When the crack central axis distance < the addendum central axis distance, or, when the dedendum fillet central axis distance > the crack central axis distance > the addendum central axis distance, and the force angle < the crack end point angle (i.e. h c <h a or h te > h c > h a &α1<α c ) is the fifth section division.

[0169] At this time, the cross-sectional area and the moment of inertia of the cracked gear tooth are

[0170]

[0171]

[0172] where h x is determined by equation (13). c is determined by equation (28).

[0173] According to h x , A x and I x , the integral interval in equations (31) and (32) can be divided into three parts as follows:

[0174] a) [0, d cs -q1cosγ]. In this interval, h x , A x and I x are determined by equation (44).

[0175] b) [d cs -q1cosγ, d1]. In this interval, h x , A x and I x are determined by equation (33).

[0176] c) [d1, d]. In this interval, h x , A x and I x are determined by equation (34).

[0177] The bending stiffness and shear stiffness of each slice of the faulty gear are respectively

[0178]

[0179]

[0180] When the distance between the crack axis and the pitch circle is greater than the distance between the crack axis and the root fillet (i.e. h c >h te ), the sixth section is divided.

[0181] At this time, the crack model is shown in Figure 7 . The cross-sectional area and the moment of inertia of the loaded part of the gear tooth are calculated by equations (29) and (30), and the length of the gear tooth affected by the crack is d c determined by equations (42) and (43).

[0182] According to h x , A x and I xIn the expression of different crack sub-sections, the integral interval in formula (31) and (32) can be divided into the following four parts:

[0183] a) [0, d cs -q1cosγ]. In this interval, h x , A x and I x expressions are the same as formula (44);

[0184] b) [d cs -q1cosγ, d c ]. In this interval, h x , A x and I x expressions are the same as formula (33);

[0185] c) [d c , d1]. In this interval, h x , A x and I x expressions are the same as formula (44);

[0186] d) [d1, d]. In this interval, h x , A x and I x expressions are the same as formula (35).

[0187] Thus, the calculation formula of bending stiffness and shear stiffness of each slice of the fault gear is obtained

[0188]

[0189]

[0190] When the crack initiation point is in the involute section, that is, the crack appears on the involute (R cs >R te )

[0191] As Figures 8-10 shown in the figure. According to the geometric relationship in the figure, the coordinates (x cs , h cs ) of the crack initiation point F can be determined by the following relationship

[0192]

[0193] Thus, we have

[0194] h c = h cs -q1sinγ (58)

[0195] d cs =x cs -xr (59)

[0196] Three types of crack category information are classified according to the crack propagation depth, namely:

[0197] The crack starting point is in the involute section, and the crack end point is inside the tooth root;

[0198] The crack starting point is in the involute section, and the crack end point is in the tooth root fillet section;

[0199] The crack starting point is in the involute section, and the crack end point is in the involute section:

[0200] At this time, considering the crack propagation degree, different section divisions need to be obtained according to the crack gear section division rules, and the bending stiffness and shear stiffness of the cracked tooth are solved according to different section divisions. Specifically:

[0201] When the crack starting point is in the involute section and the crack end point is inside the tooth root; the crack has extended below the tooth root. As Figure 8 shown, the crack end point G has extended to the left of the dotted line AH. At this time, d cs <q1cosγ;

[0202] Figure 8 This is the cracked tooth model at this time. It can be seen that the model at this time is the same as that of R cs <R te And when d cs <q1cosγ, the cracked tooth model is basically the same. The only difference is the half-tooth width h cs at the crack starting point F and the perpendicular distance h c from the crack end point G to the gear central axis. Therefore, the calculation formulas for the bending stiffness and shear stiffness are the same as those of R cs <R te And when d cs <q1cosγ, only the expressions of h cs and h c are different. Similarly, according to the crack propagation degree, the bending stiffness and shear stiffness of the cracked tooth are solved by the following three section divisions.

[0203] When the crack central axis distance > the tooth tip central axis distance and the acting force angle is greater than the crack starting point angle (i.e., h c >h a &α1>α c ), it is the seventh section division;

[0204] The calculation formulas for the cross-sectional area and moment of inertia of the tooth are the same as those in formulas (29) and (30). The calculation formulas for the bending stiffness and shear stiffness of each slice of the faulty gear are the same as those in formulas (36) and (37), and will not be repeated here.

[0205] When the crack central axis distance < the addendum central axis distance, and the force included angle > the crack start point included angle, or when the crack central axis distance > the addendum central axis distance, and the crack start point included angle < the force included angle < the crack end point included angle (i.e. h c <h a &α1>α cs orh c >h a &α cs <α1<α c , the eighth section is divided;

[0206] The cross-sectional area and the moment of inertia of the tooth are calculated by the same formula (29) and formula (30), and the bending stiffness and the shear stiffness of each slice of the fault gear are calculated by the same formula (40) and formula (41), which are not repeated here.

[0207] When the crack end point included angle < the crack start point included angle (i.e. h cs , the ninth section is divided.

[0208] At this time, the tooth engagement point is between BF, and the loaded part of the tooth can still bear the bending moment and the shear force, and the crack does not affect the bending stiffness and the shear stiffness. Therefore, the calculation formula of the bending stiffness and the shear stiffness is the same as that of the healthy tooth, i.e. formula (16) and formula (17).

[0209] When the crack start point is in the involute section, and the crack end point is in the root fillet section; the crack has not yet expanded below the root, but has expanded into the root fillet section, as shown in Figure 9 , the crack end point G has not yet expanded to the left side of the dashed line AH, but has expanded in the AB section, at this time d cs >q1cosγ and d cs -q1cosγ<d1;

[0210] Figure 9 This is the crack tooth model at this time. It can also be found that the model at this time is basically the same as the crack tooth model when R cs <R te and d cs >q1cosγ, the only difference is that the half-tooth width h cs at the crack start point F and the vertical distance h c from the crack end point G to the gear central axis. Therefore, the calculation formula of the bending stiffness and the shear stiffness is the same as that when R cs <R te and d cs >q1cosγ, only the expressions of h cs and h c are different.

[0211] When the distance between the crack centerline and the tooth tip centerline is greater than the distance between the crack tip and the tooth tip centerline, and the angle of the applied force is greater than the angle of the crack tip (i.e., h) c h a &α1>α c (Time) is the tenth segment division;

[0212] The formulas for calculating the bending stiffness and shear stiffness of each slice of the faulty gear are the same as those for formulas (53) and (54), h cs and h c It is determined by formulas (57) and (58).

[0213] When the distance from the crack centerline is less than the distance from the tooth tip centerline, and the angle of the applied force is greater than the angle at the crack initiation point; or when the distance from the crack centerline is greater than the distance from the tooth tip centerline, and the angle at the crack initiation point is less than the angle of the applied force and less than the angle at the crack endpoint (i.e., h... c <h a &α1>α cs orh c h a &α cs <α1<α c (Time) is the eleventh segment division;

[0214] The formulas for calculating the bending stiffness and shear stiffness of each slice of the faulty gear are the same as those for formulas (53) and (54), h cs and h c It is determined by formulas (57) and (58).

[0215] When the included angle at the crack endpoint is less than the included angle at the crack initiation (i.e., α1 < α) cs (Time) is designated as the twelfth section;

[0216] Similarly, the calculation formulas for bending stiffness and shear stiffness are the same as those for healthy gear teeth, namely formula (16) and formula (17).

[0217] When the crack has not yet extended into the tooth root fillet section, such as Figure 10 As shown, at this time d cs -q1cosγ>d1.

[0218] At this point, the crack has not yet extended to the tooth root fillet section, such as Figure 10 As shown. Considering the extent of crack propagation, the bending stiffness and shear stiffness of the cracked gear teeth need to be solved according to the following segment division.

[0219] When the distance between the crack centerline and the tooth tip centerline is greater than the distance between the crack tip and the tooth tip centerline, and the angle of the applied force is greater than the angle of the crack tip (i.e., h) c h a &α1>α c (Time) is the thirteenth section;

[0220] At this time, the cross-sectional area and the moment of inertia of the gear tooth are

[0221]

[0222]

[0223] wherein h x is determined from equation (13) and h c is determined from equation (58).

[0224] According to h x , A x and I x , the integral intervals in equations (31) and (32) can be divided into the following four parts:

[0225] a) [0, d1]. In this interval, h x , A x and I x are expressed as equation (44);

[0226] b) [d1, d cs -q1cosγ]. In this interval, h x , A x and I x are expressed as equation (35);

[0227] c) [d cs -q1cosγ, d c ]. In this interval, h x , A x and I x are expressed as equation (34);

[0228] d) [d c , d]. In this interval, h x , A x and I x are expressed as equation (35).

[0229] The bending stiffness and the shear stiffness of each slice of the faulty gear are

[0230]

[0231]

[0232] When the axial distance of the crack is less than the axial distance of the tooth top, and the angle of the force is greater than the angle of the crack starting point, or when the axial distance of the crack is greater than the axial distance of the tooth top, and the angle of the crack starting point is less than the angle of the force and greater than the angle of the crack end point (i.e. h c < h a & α1> αcs orh c >h a &α cs <α1<α c when) is the fourteenth section division;

[0233] At this time, the cross-sectional area and the moment of inertia of the gear tooth are

[0234]

[0235]

[0236] where: h x h c is determined by equation (58).

[0237] According to h x , A x and I x expressions in different crack sub-sections, the integral interval in equations (31) and (32) can be divided into the following three parts:

[0238] a) [0, d1]. In this interval, h x , A x and I x expressions are the same as equation (44);

[0239] b) [d1, d cs -q1cosγ]. In this interval, h x , A x and I x expressions are the same as equation (35);

[0240] c) [d cs -q1cosγ, d]. In this interval, h x , A x and I x expressions are the same as equation (34);

[0241] Thus, the bending stiffness and shear stiffness of each slice of the faulty gear can be obtained as

[0242]

[0243]

[0244] When the crack end point included angle < crack start point included angle (i.e. α1<α csc when) is the fifteenth section division.

[0245] The calculation formulas of bending stiffness and shear stiffness are the same as those of the healthy gear tooth, i.e. equation (16) and equation (17).

[0246] Comprehensive cracks appear on the fillet (R cs <R te ) 2 cracks category information of 6 different section classification, with cracks appear on the involute (R cs >R te ) 3 cracks category information and 9 different section classification, a total of 5 cracks category information, 15 different section classification of mesh stiffness calculation formula. Figure 11 Comprehensive all the above conditions and the calculation process of the crack tooth bending stiffness k b,crack And shear stiffness k s,crack The schematic diagram.

[0247] In the acquisition of crack tooth each slice of bending stiffness k b,crack And shear stiffness k s,crack , the equivalent stiffness k t,crack Corresponding to the deformation of a single tooth can be obtained as follows

[0248]

[0249] Then the single tooth with crack mesh stiffness The formula (assuming in the tooth pair in the mesh, the active wheel with the rth tooth crack)

[0250]

[0251] 1 and 2 represent the active wheel and driven wheel, respectively.

[0252] Similar to the normal gear mesh stiffness calculation method, the expression is

[0253]

[0254] In the formula: k tooth Indicates the total stiffness of all participating meshing tooth pairs (excluding wheel stiffness), and its expression is

[0255]

[0256] In the formula: The total stiffness of the ith pair of healthy gear pair (excluding wheel stiffness) / N·m -1 ; The total stiffness of the crack gear pair (excluding wheel stiffness) / N·m -1 .

[0257] The present application also provides a solution to the time-varying mesh stiffness of the crack gear, and analyzes the influence of crack parameters on the mesh stiffness. At the same time, by comparing the calculation results of the finite element model and the theoretical model, the accuracy of the theoretical model is verified.

[0258] The basic parameters of the spur gear pair are shown in Table 2. The crack is determined by three parameters: crack depth q1, crack angle γ and the radius R cs of the crack initiation point F cs For convenience, the crack initiation point radius R r is defined as the radius difference ΔR cs between the crack initiation position and the root circle radius R

[0259] ΔR cs = R cs - R r (72)

[0260] Assuming that the crack occurs on the driving gear tooth and only one tooth contains the crack, the crack angle γ is 70°. The finite element model of the spur gear pair is shown in Figure 12 Pro / E is used to establish the three-dimensional entity model of the gear pair, and different sizes of cracks are generated in the entity model by segmentation. Solid186 three-dimensional entity elements are selected in ANSYS to mesh the gear entity, and the mesh is refined on the meshing surface and the crack surface. The face-to-face contact type (Conta174 and Targe170) is used to set the tooth surfaces of the driving gear and the driven gear that come into contact with each other as the contact surface (Conta174) and the target surface (Targe170).

[0261]

[0262] Table 2 Gear parameters

[0263] Assuming that the crack occurs at the root of the driving gear tooth, ΔR cs = 0.5 mm. Figure 13 (a) and (b) are the calculation results of the mesh stiffness of the theoretical model and the finite element model, respectively, with the change of the crack depth q1. In the single tooth meshing period shown in the figure, 53 meshing points are selected when the finite element model is calculated. It can be seen that the presence of the crack reduces the carrying capacity of the gear tooth, resulting in a decrease in the mesh stiffness, and the decrease will increase with the increase of q1. Since the crack occurs at the root of the driving gear tooth, when the angle is small, the tooth is just in the meshing state, at this time the mesh stiffness is not much affected by the crack. As the angle gradually increases, the meshing point gradually approaches the addendum position of the driving gear tooth, and the influence of the crack on the tooth stiffness gradually increases, so the decrease in the mesh stiffness increases with the increase of the angle. Comparing Figure 13(a) and (b) are the meshing stiffness curves of the theoretical model for different q1, which are in good agreement with the results of the finite element model. Table 3 gives the comparison of the two methods at A and B, taking the results of the finite element model as the benchmark, the average meshing stiffness deviation of the theoretical model and the meshing stiffness deviation at A and B are calculated. Among them, the average meshing stiffness deviation is calculated according to the following formula

[0264]

[0265] In the formula: k 1,i is the meshing stiffness of the i th meshing point of the theoretical model / N·m -1 ; k 2,j is the meshing stiffness of the j th meshing point of the finite element model / N·m -1 ; n1 is the total number of meshing points of the theoretical model in a single tooth meshing period; n2 is the total number of meshing points of the finite element model in a single tooth meshing period.

[0266] From Table 3, the maximum deviation of the theoretical model at A is 4.41%, the maximum deviation at B is 0.97%, and the maximum deviation of the average meshing stiffness is 1.60%. Obviously, the results obtained by the theoretical model and the finite element model are highly consistent, which verifies the accuracy of the theoretical model proposed in this chapter.

[0267]

[0268] Table 3 Comparison of the results of the two models

[0269] The influence of crack position on meshing stiffness is assumed that the crack depth is 3mm, that is, q1=3.0mm. Figure 14 (a) and (b) are the change rules of the meshing stiffness calculation results of the theoretical model and the finite element model respectively with the crack position ΔR cs . Unlike the phenomenon in Figure 13 , when the meshing point is close to the tooth root (when the rotation angle is small), the meshing point is still below the crack position (in the direction of tooth height), at this time the crack does not affect the stiffness at the meshing point, only when the meshing point passes the initial position of the crack (when the rotation angle is greater than a certain angle), the meshing stiffness will be affected by the crack, as shown in Figure 14 (a). When the meshing point gradually moves from the tooth root to the tooth top, before the initial position of the crack, the cross-sectional area and the moment of inertia of the loaded part of the gear tooth are equal to those of the healthy gear tooth, and the meshing stiffness of the gear tooth is equal to that of the healthy gear tooth; after passing the initial position of the crack, the cross-sectional area and the moment of inertia of the loaded part of the gear tooth are affected by the crack and decrease, so the carrying capacity of the gear tooth decreases, resulting in the decrease of the meshing stiffness.

[0270] In Figure 14(b) In the results of the finite element model, similar phenomenon can also be seen, i.e. the mesh stiffness decreases from the initial crack position. Comparing Figure 14 (a) and (b), it can be found that the initial crack position and the stiffness change trend predicted by the theoretical model are basically consistent with the finite element results, which proves the reliability of the proposed theoretical model.

[0271] Table 4 gives the comparison of the two models at A and B times and the comparison of the average mesh stiffness. The maximum deviation of the theoretical model at A time is 1.48%, and the maximum deviation at B time is 5.70%. The maximum deviation of the average mesh stiffness is 3.49%, which further verifies the accuracy of the theoretical model.

[0272]

[0273] Table 4 Comparison of the results of the two models

[0274] The technical features of the above embodiments can be combined in any manner. In order to make the description simple, all possible combinations of the technical features in the above embodiments are not described, however, as long as the combinations of the technical features do not contradict, they should be considered as the scope of the description.

[0275] The above embodiments only express several implementation manners of the present application, and the description is specific and detailed, but it should not be understood as a limitation on the scope of the patent. It should be pointed out that for ordinary skilled in the art, without departing from the concept of the present application, some modifications and improvements can be made, which are all within the protection scope of the present application. Therefore, the protection scope of the patent of the present application should be subject to the appended claims.

Claims

1. A method for calculating the time-varying meshing stiffness of a faulty gear considering the location of a crack, characterized in that, The method includes: A gear fault calculation model is established; the fault location is divided in the gear fault calculation model, and for the fault location where the fault occurs, the corresponding crack category information is determined, and the crack gear segment division rules corresponding to the crack category information are pre-stored. The location of the crack on the gear is detected. Based on the gear fault calculation model and the crack location, the crack category information is determined. The crack location is divided into multiple crack sub-segments using the crack gear segmentation rule. For each cracked sub-segment, the load distance from the tooth profile to the central axis at the load-bearing part of the cracked gear, the cross-sectional area at the load-bearing part of the cracked gear, and the moment of inertia at the load-bearing part of the cracked gear are obtained. The load distance, cross-sectional area, and moment of inertia are then input into the bending stiffness calculation model and shear stiffness calculation model of the normal gear to obtain the bending stiffness and shear stiffness of each cracked sub-segment. The bending stiffness and shear stiffness of the faulty gear are obtained by summing the bending stiffness and shear stiffness of each crack sub-segment.

2. The method according to claim 1, characterized in that, The fault locations include: the tooth root fillet section and the involute section; The steps to determine the corresponding crack category information include: Obtain the crack initiation point and crack endpoint of the fault, and determine the crack category information based on the positional relationship between the crack initiation point and crack endpoint and the fault location; When the crack initiation point is in the tooth root fillet section, the crack endpoint is inside the tooth root or in the tooth root fillet section; When the crack initiation point is in the involute section, the crack endpoint is inside the tooth root, in the tooth root fillet section, or in the involute section.

3. The method according to claim 2, characterized in that, The distance of the crack centerline, the crack initiation angle, the crack end angle, and the pre-stored distance of the tooth root fillet centerline, the distance of the tooth tip centerline, and the force angle are obtained, and the crack gear segment division rules are determined according to the crack category information. Wherein, the distance along the crack axis is the distance from the crack endpoint to the axis; the crack initiation angle is the angle between the crack initiation point and the base circle, and the connection between this tangent point and the gear center and the axis; the crack endpoint angle is the projection point of the crack endpoint onto the tooth profile along the axis, the tangent point of this projection point and the base circle, and the angle between this tangent point and the gear center and the axis; the distance along the tooth root fillet axis is the distance from the endpoint of the tooth root fillet away from the tooth root to the axis; the distance along the tooth tip axis is the distance from the tooth tip to the axis; and the force angle is the angle between the force and the vertical direction.

4. The method according to claim 3, characterized in that, The steps for determining the segmentation rules for cracked gears include: When the crack initiation point is in the tooth root fillet section, and the crack endpoint is inside the tooth root; The first segment is defined when the distance from the centerline of the tooth root fillet is greater than the distance from the centerline of the crack, which is greater than the distance from the centerline of the tooth tip, and the angle of the applied force is greater than the angle of the crack end point. The second segment is defined when the distance from the crack centerline is less than the distance from the tooth tip centerline, or when the distance from the tooth root fillet centerline is greater than the distance from the crack centerline and greater than the distance from the tooth tip centerline, and the angle of the applied force is less than the angle at the crack end. When the distance from the crack centerline is greater than the distance from the tooth root fillet centerline, it is divided into the third section.

5. The method according to claim 3, characterized in that, The steps for determining the segmentation rules for cracked gears include: When the crack initiation point is in the tooth root fillet section, and the crack endpoint is in the tooth root fillet section; When the distance from the centerline of the tooth root fillet is greater than the distance from the centerline of the crack, and the distance from the centerline of the tooth tip is greater than the angle of the applied force, it is divided into the fourth section. The fifth segment is defined when the distance from the crack centerline is less than the distance from the tooth tip centerline, or when the distance from the tooth root fillet centerline is greater than the distance from the crack centerline and greater than the distance from the tooth tip centerline, and the angle of the applied force is less than the angle at the crack end. When the distance from the crack centerline is greater than the distance from the tooth root fillet centerline, it is divided into the sixth section.

6. The method according to claim 3, characterized in that, The steps for determining the segmentation rules for cracked gears include: When the crack initiation point is in the involute section, the crack endpoint is inside the tooth root; When the distance between the crack centerline and the tooth tip centerline is greater than the distance between the crack centerline and the force angle is greater than the crack initiation angle, it is divided into the seventh section. When the distance from the crack centerline is less than the distance from the tooth tip centerline, and the angle of the applied force is greater than the angle of the crack initiation, or when the distance from the crack centerline is greater than the distance from the tooth tip centerline, and the angle of the crack initiation is less than the angle of the applied force and less than the angle of the crack end, the eighth section is defined. When the included angle at the crack endpoint is less than the included angle at the crack initiation, it is divided into the ninth section.

7. The method according to claim 3, characterized in that, The steps for determining the segmentation rules for cracked gears include: When the crack initiation point is in the involute section, the crack endpoint is in the tooth root fillet section; When the distance between the crack centerline and the tooth tip centerline is greater than the distance between the crack centerline and the force angle is greater than the crack tip angle, it is divided into the tenth section. When the distance from the crack centerline is less than the distance from the tooth tip centerline, and the angle of the applied force is greater than the angle of the crack initiation, or when the distance from the crack centerline is greater than the distance from the tooth tip centerline, and the angle of the crack initiation is less than the angle of the applied force and less than the angle of the crack end, the eleventh segment is defined. When the included angle at the crack endpoint is less than the included angle at the crack initiation, it is divided into the twelfth segment.

8. The method according to claim 3, characterized in that, The steps for determining the segmentation rules for cracked gears include: When the crack initiation point is in the involute section, the crack endpoint is in the involute section; When the distance between the crack centerline and the tooth tip centerline is greater than the distance between the crack tip and the tooth tip centerline, and the angle of the applied force is greater than the angle of the crack tip, it is divided into the thirteenth section. When the distance from the crack centerline is less than the distance from the tooth tip centerline, and the angle of the applied force is greater than the angle of the crack initiation, or when the distance from the crack centerline is greater than the distance from the tooth tip centerline, and the angle of the crack initiation is less than the angle of the applied force and less than the angle of the crack end, it is the fourteenth segment division; When the included angle at the crack endpoint is less than the included angle at the crack initiation, it is the fifteenth segment division.

9. The method according to any one of claims 1-8, characterized in that, It also includes obtaining the meshing stiffness of a single tooth pair; establishing a single tooth pair meshing stiffness model, and inputting the obtained bending stiffness and shear stiffness of the faulty gear into the single tooth pair meshing stiffness model to obtain the meshing stiffness of a cracked single tooth pair.

10. The method according to claim 9, characterized in that, It also includes obtaining the multi-tooth meshing stiffness; substituting the obtained single-tooth pair meshing stiffness with cracks into the multi-tooth meshing stiffness calculation model of normal gears to obtain the multi-tooth meshing stiffness.

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