A method for calculating the impact dynamic load of a floating support friction plate with gear tooth cracks

Abstract

The present invention discloses a method for calculating the impact dynamic load of a floating support friction plate with a gear tooth crack, comprising: S1: obtaining the parameters of the floating support friction plate assembly to be calculated and the crack parameters, wherein the friction plate assembly comprises a friction plate and an inner hub, and the friction plate has a defective gear tooth with a crack; S2: calculating the comprehensive stiffness K of the healthy friction plate gear tooth H ; S3. Calculate the comprehensive stiffness K of the friction plate tooth with root crack C ; S4. Calculate the eccentricity caused by the friction plate under the action of gravity; S5. Determine whether the friction plate collides with the inner hub, and determine the number of teeth colliding between the inner hub and the friction plate and the position of the random collision teeth; S6. Calculate the impact rebound coefficient and hysteresis damping coefficient of the collision between the friction plate and the inner hub gear teeth; S7. Calculate the collision force between the friction plate and each contact tooth of the inner hub; S8. Based on the construction of the two-mass impact dynamic load dynamic model of the inner hub and the friction plate, calculate the impact dynamic load of the floating gear ring friction plate assembly containing gear tooth cracks.

Description

Technical Field

[0001] The invention relates to the technical field of friction plates, and in particular to a method for calculating the impact dynamic load of a floating support friction plate with gear tooth cracks. Background Art

[0002] The friction plate is a key core component in power transmission that integrates the safety, operability and comfort of vehicles and ships. It can realize the engagement and separation of transmission parts, transmit torque and movement on the coaxial line, and is widely used in the transmission mechanism of various engineering machinery such as automobiles and aircraft. During the operation of the floating support friction plate, the speed fluctuation of the inner hub caused by the torsional vibration of the transmission mechanism causes the friction plate and the tooth surface of the inner hub to impact and collide, generating a large impact load, causing fatigue damage to the tooth surface and tooth root of the friction plate gear teeth, and causing cracks at the tooth root of the friction plate.

[0003] The existing technology mainly studies the calculation of the impact dynamic load of the external meshing gear pair with tooth root cracks, including analytical method, experimental method and finite element method. The experimental method is more accurate but requires a lot of time and good experimental conditions; the finite element method is difficult to set the boundary conditions close to the actual situation and requires a long calculation time; the analytical method has a fast calculation speed, but the calculation method of the impact dynamic load of the internal meshing friction plate considering the tooth root crack has not yet been found. Summary of the invention

[0004] In view of the problem of low efficiency in calculating the impact dynamic load of a floating support friction plate with tooth cracks in the prior art, the present invention proposes a method for calculating the impact dynamic load of a floating support friction plate with tooth cracks, which fully considers the influence of the position of the friction plate tooth cracks, the number of friction plate tooth cracks, etc. on the impact load of the floating gear ring friction plate assembly, and ensures high calculation efficiency while ensuring high calculation accuracy.

[0005] In order to achieve the above object, the present invention provides the following technical solutions:

[0006] A method for calculating the impact dynamic load of a floating support friction plate with a gear tooth crack comprises the following steps:

[0007] S1: Obtaining parameters of a floating support friction plate assembly and crack parameters to be calculated, wherein the friction plate assembly comprises a friction plate and an inner hub, and the friction plate has a defective gear tooth with a crack;

[0008] S2. Calculate the comprehensive stiffness K of the healthy friction plate gear teeth based on the basic parameters of the friction plate assembly H ;

[0009] S3. Based on the basic parameters of the friction plate assembly, calculate the comprehensive stiffness K of the friction plate tooth with root crack C ;

[0010] S4. Based on the basic parameters of the friction plate assembly, calculate the eccentricity caused by the friction plate under the action of gravity;

[0011] S5. judging whether the friction plate and the inner hub collide with each other based on the eccentricity and relative motion relationship between the friction plate and the inner hub determined in S3, and determining the number of teeth colliding between the inner hub and the friction plate and the position of the random colliding teeth;

[0012] S6. Based on the energy balance theory and the relative motion relationship between the inner hub and the friction plate, calculate the impact rebound coefficient and hysteresis damping coefficient of the collision between the friction plate and the inner hub gear teeth;

[0013] S7, calculating the collision force between the friction plate and each contact tooth of the inner hub according to the comprehensive stiffness of the friction plate teeth with or without cracks determined in S2, the eccentricity between the friction plate and the inner hub determined in S3, the number of teeth of the inner hub and the friction plate colliding with each other determined in S4, and the impact rebound coefficient and hysteresis damping coefficient between the friction plate and the inner hub teeth determined in S5;

[0014] S8. According to the collision force between the inner hub and the friction plate determined in S6, a two-mass impact dynamic load dynamic model of the inner hub and the friction plate is constructed, and the impact dynamic load of the floating gear ring friction plate assembly containing gear tooth cracks is output.

[0015] Preferably, in S1, the basic parameters of the floating support friction plate assembly include: the number of teeth, module, tooth width, pitch circle pressure angle, elastic modulus, Poisson's ratio, mass of the friction plate and the inner hub, moment of inertia, tooth side clearance, speed of the inner hub and speed fluctuation frequency and amplitude; the crack parameters include crack position angle, crack length and crack direction.

[0016] Preferably, S2 comprises the following steps:

[0017] S201. Calculate the bending stiffness k of the single tooth pair of the inner hub or the friction plate by using the potential energy method according to the parameters of the friction plate assembly. bj , shear stiffness k sj and axial compressive stiffness k aj , the corresponding relationship is as follows:

[0018]

[0019]

[0020]

[0021] In formula (1), j = 1 represents the inner hub; j = 2 represents the friction plate; U aj , U bj and U sj are the elastic potential energy stored in the axial compression, bending and shear deformation of the inner hub gear teeth or friction plate gear teeth respectively; Fj Indicates the contact load at the meshing point of the inner hub gear teeth or friction plate gear teeth; F aj E is the horizontal component of the meshing force of the inner hub gear teeth or friction plate gear teeth; j Indicates the elastic modulus of the inner hub gear teeth or friction plate gear teeth; A xj It represents the area of ​​the cross section at the fixed end x of the inner hub gear or friction plate gear; d j Indicates the distance from the inner hub gear or friction plate gear meshing position to the tooth root circle; M j I is the bending moment at the fixed end x from the inner hub gear teeth or friction plate gear teeth; xj It represents the moment of inertia of the cross section at the fixed end x of the inner hub gear or friction plate gear; F bj G is the vertical component of the meshing force of the inner hub gear teeth or friction plate gear teeth; j Indicates the shear modulus of the inner hub gear teeth or friction plate gear teeth;

[0022] S202. Based on the parameters of the friction plate assembly, the O'Donnell deformation theory is used to calculate the matrix stiffness k of a single tooth pair of gear teeth. fj , the expression is as follows:

[0023]

[0024] In formula (2), j = 1 represents the inner hub; j = 2 represents the friction plate; θ fj M is the deformation of the gear tooth matrix of the inner hub or friction plate; fj V is the unit bending moment at the fixed end x of the inner hub gear teeth or friction plate gear teeth; j It represents the unit shear force at the fixed end x of the inner hub gear teeth or friction plate gear teeth; E j Indicates the elastic modulus of the inner hub gear teeth or friction plate gear teeth; υ j Indicates the Poisson's ratio of the inner hub gear teeth or friction plate gear teeth; h j’ Indicates the cross section at the fixed end x of the inner hub gear or friction plate gear; F j Indicates the contact load at the meshing point of the inner hub gear teeth or friction plate gear teeth;

[0025] S203, according to the bending stiffness k of the inner hub and friction plate teeth in S201 bj , shear stiffness k sj , axial compression stiffness k aj and the matrix stiffness k in S202 fj , obtain the comprehensive stiffness K of the single tooth pair contact of the friction plate assembly H , calculated as follows:

[0026]

[0027] Preferably, S3 comprises the following steps:

[0028] S301. Assuming that the crack is parallel to the line connecting the center of the circle and the midpoint of the tooth top, and the length of the crack is q, the meshing stiffness is divided into two regions, Ⅰ and Ⅱ, and their bending stiffness, shear stiffness and axial compression stiffness are:

[0029]

[0030]

[0031]

[0032] In formula (4), k Cb Indicates the bending stiffness of the friction plate with tooth root cracks; k Cs Indicates the shear stiffness of the friction plate with tooth root cracks; k Ca Indicates the axial compression stiffness of the friction plate with tooth root cracks; k CbⅠ represents the bending stiffness of the friction plate with tooth root crack in region I; k CbⅡ k represents the bending stiffness of the friction plate with tooth root crack in zone II; CsⅠ represents the shear stiffness of the friction plate with tooth root crack in region I; k CsⅡ represents the shear stiffness of the friction plate with tooth root crack in region II; k CaⅠ represents the axial compressive stiffness of the friction plate with tooth root crack in zone I; k CaⅡ represents the axial compressive stiffness of the friction plate with tooth root crack in zone II;

[0033] S302. The calculation of the bending stiffness, shear stiffness and axial compression stiffness of the first region is the same as that of the healthy tooth. Therefore, the bending stiffness, shear stiffness and axial compression stiffness of the first region of the friction plate gear tooth are expressed as:

[0034] k CbⅠ =k b2 , k CsⅠ =k s2 , k CaⅠ =k a2 (5)

[0035] In formula (5), k b2 k is the bending stiffness of the healthy teeth of the friction plate; s2 k is the shear stiffness of the healthy teeth of the friction plate; a2 is the axial compression stiffness of the healthy teeth of the friction plate;

[0036] S303. Establish a calculation model for the stiffness of a cracked friction plate tooth and define a rectangular coordinate system. The X-axis coincides with the center line of the friction plate tooth, the Y-axis is the line connecting the straight line AB, the intersection of the X-axis and the Y-axis is O1, point A is the starting position of the friction plate tooth root crack, point D is the ending position of the friction plate tooth root crack, point B is the intersection point of the tooth profile and the root circle of the adjacent teeth of the friction plate; point E is the intersection point of the friction plate tooth root circle and the X-axis; point C is the intersection point of the straight line BE and the straight line AD; point M is the position of the distance d from the meshing force application point to the fixed part of the root circle; the straight line BC is perpendicular to MG and intersects at point G; then the coordinates of each point are M(-d,-h);

[0037] The equations of lines BC and MG are:

[0038]

[0039] l MG :xtanθ-y+dtanθ-h=0 (6)

[0040] In formula (6), R rr is the friction plate tooth root circle radius; is the semi-tooth angle at the root of the friction plate; d is the horizontal coordinate of point M, that is, the distance from the meshing force application point to the fixed part of the root circle; h is the vertical coordinate of point M, that is, the distance from the meshing force application point to the x-axis; θ is the angle of ∠ABC;

[0041] According to the similarity of triangles, point E is the midpoint of line segment BC, so the coordinates of point E are According to the distance formula from a point to a straight line, l and e are obtained, that is,

[0042]

[0043]

[0044] In formula (7), l is the distance from point M to line BC; e is the distance from point E to line MG;

[0045] S304. The calculation expressions of the cross-sectional inertia moment and cross-sectional area of ​​the second region at the rotation angle θ are:

[0046]

[0047] In formula (8), I θ A represents the moment of inertia of the section at the rotation angle θ in the second region; θ It represents the cross-sectional area at the turning angle θ in the second region; W is the tooth width of the friction plate; is the half tooth angle at the root of the friction plate; θ is the angle of ∠ABC; R rr is the friction plate tooth root circle radius;

[0048] S305. Calculate the bending stiffness K of the friction plate with tooth root crack in region II CbⅡ , shear stiffness K CsⅡ and axial compressive stiffness K CaⅡ , the calculation expression is:

[0049]

[0050] In formula (9), γ is the distance from the crack position of the friction plate to the top position of the adjacent gear tooth, that is, the distance from point B to the straight line AD; e is the distance from point E to the straight line MG; G represents the shear modulus of the friction plate gear tooth; E is the elastic modulus of the gear tooth with the crack of the friction plate; α1 is the pressure angle at the meshing point of the friction plate; θ is the angle of ∠ABC;

[0051] S306. Using the finite element method, the matrix stiffness of the cracked gear teeth of the friction plate is corrected, and the matrix stiffness coefficient λ caused by the crack is given, which is expressed as:

[0052] K Cf =λK f (10)

[0053] In formula (10), K Cf Indicates the matrix stiffness of the cracked gear teeth of the friction plate, k f represents the stiffness of the friction plate matrix; λ represents the matrix stiffness coefficient caused by cracks;

[0054] S307, Comprehensive stiffness K of a single pair of teeth meshing between the friction plate with cracked gear teeth and the inner hub of healthy gear teeth C It is expressed as:

[0055]

[0056] In formula (11), k b1 represents the bending stiffness of the inner hub gear teeth, k s1 represents the shear stiffness of the inner hub gear teeth, k a1 represents the axial compressive stiffness of the inner hub gear and k fs represents the inner hub gear tooth matrix stiffness, k Cb Indicates the bending stiffness of the friction plate with tooth root crack; k Cs Indicates the shear stiffness of the friction plate at the root of the tooth; K Cf Indicates the matrix stiffness of the cracked gear teeth of the friction plate; K Ca Indicates the axial compression stiffness of the friction plate with tooth root crack;

[0057] S308. Assume that the number of cracks in the friction plate gear teeth is Cn, and the position of the cracked teeth is Cl = [C1, C2 ... Cn]; when the number of teeth n = Cl, the gear tooth stiffness is K = K C , otherwise the gear tooth stiffness is K = K H .

[0058] Preferably, S4 comprises the following steps:

[0059] The eccentricity generated by the first contact of the working tooth surface of the friction plate under the action of gravity is calculated as follows:

[0060]

[0061] In formula (12), a R It represents the eccentricity caused by the contact between the working tooth surface of the friction plate under the action of gravity; abs(…) represents the absolute value; min(…) represents the minimum value; c represents the single-sided tooth side clearance between the friction plate and the gear teeth of the inner hub in the non-eccentric state; α0 represents the pressure angle; Z represents the number of teeth of the friction plate and the inner hub, and n represents the tooth sequence number;

[0062] The eccentricity caused by the non-working tooth surface of the friction plate contacting first under the action of gravity is calculated as follows:

[0063]

[0064] In formula (13), a L It represents the eccentricity caused by the contact between the working tooth surface of the friction plate under the action of gravity; abs(…) represents the absolute value; min(…) represents the minimum value; c represents the single-sided tooth side clearance between the friction plate and the gear teeth of the inner hub in the non-eccentric state; α0 represents the pressure angle; Z represents the number of teeth of the friction plate and the inner hub, and n represents the tooth sequence number;

[0065] Then the initial eccentricity a of the friction plate under the action of gravity is expressed as:

[0066] a=(a L +a R ) / 2.

[0067] Preferably, the S5 comprises the following steps:

[0068] S501. Using the basic parameters of the inner hub and the friction plate described in S1, calculate the tooth side clearance between the working tooth surface and the non-working tooth surface of the nth tooth of the inner hub and the friction plate under the action of self-eccentricity. The calculation form is as follows:

[0069]

[0070] In formula (14), c Rn Indicates the tooth side clearance of the working tooth surface of the nth tooth in the self-eccentric state; c Ln represents the tooth side clearance of the non-working tooth surface of the nth tooth in the self-eccentric state; c represents the single-sided tooth side clearance between the friction plate and the gear teeth of the inner hub in the non-eccentric state; a represents the initial self-eccentricity generated by the friction plate under the action of gravity; α0 represents the pressure angle; Z represents the number of teeth of the friction plate and the inner hub; n represents the tooth sequence number;

[0071] S502. Considering the relative displacement of the friction pair components under the action of torque, the deformation of the working tooth surface and the non-working tooth surface of the inner hub and the friction plate is calculated. The calculation form is as follows:

[0072] δ Rn =R(θ1-θ2)cosα0-c Rn , δ Ln =R(θ2-θ1)cosα0-c Ln (15)

[0073] In formula (15), δ Rn Indicates the deformation of the working tooth surface of the nth tooth of the inner hub and the friction plate; δ Ln represents the deformation of the non-working tooth surface of the nth tooth of the inner hub and the friction plate; R is the pitch circle radius; θ1 is the angular displacement of the inner hub; θ2 is the angular displacement of the friction plate; α0 is the pressure angle; c Rn Indicates the tooth side clearance of the working tooth surface of the nth tooth in the self-eccentric state; c Ln Indicates the tooth side clearance of the non-working tooth surface of the nth tooth in the self-eccentric state;

[0074] S503, according to the deformation amount δ of the working tooth surface in S502 Rn and non-working tooth surface deformation δ Ln The calculation results are used for collision judgment:

[0075] If δ Rn > 0, the friction plate and the nth working tooth surface of the inner hub collide positively, denoted by C Rn =1; if δ Ln > 0, the friction plate and the nth non-working tooth surface of the inner hub collide with each other, denoted by C Ln =1; if δ Rn ≤0 or δ Ln ≤0, the friction plate does not collide with the nth tooth of the inner hub, and C Rn =0 or C Ln =0;

[0076] S504. According to the judgment result of S403, the collision teeth numbers of the inner hub and the friction plate in positive collision and negative collision are obtained respectively, which are expressed as:

[0077]

[0078] In formula (16), Z is the number of teeth of the friction plate and the inner hub; n is the tooth number of the nth tooth;

[0079] Then the total number of teeth colliding between the inner hub and the friction plate teeth is:

[0080] N=N R +N L .

[0081] Preferably, the S6 comprises the following steps:

[0082] S601. Based on Hertz contact theory, the nonlinear impact collision force F is expressed as:

[0083]

[0084] In formula (17), Kδ m represents the elastic contact force part; is the impact damping force part, δ is the relative normal deformation of the tooth; is the relative velocity during the collision process; m is the nonlinear index, m=1.5; μ is the hysteresis damping coefficient;

[0085] S602, the energy loss ΔT from the start of the gear teeth collision to the end of the collision separation is expressed as:

[0086]

[0087] In formula (18), m1 is the mass of the inner hub; m2 is the mass of the friction plate; is the relative speed of the gear teeth before collision; e is the impact rebound coefficient;

[0088] The impact rebound coefficient e is defined as the ratio of the relative approach speed of two objects before the collision to the relative separation speed after the collision, expressed as:

[0089]

[0090] In formula (19), is the speed of the inner hub when the collision ends and separation occurs; It is the speed of the friction plate when the collision ends and the separation occurs; is the speed of the inner hub when the collision contact begins; is the speed of the friction plate when the collision starts;

[0091] S603, when the gear teeth collide with the damping force The energy dissipation ΔT in the form of work is expressed as:

[0092]

[0093] In formula (20), δ0 is the maximum deformation of gear tooth collision;

[0094] S604. According to the energy conservation theorem, the hysteresis damping coefficient μ is obtained by combining equation (18) and equation (20), which is expressed as:

[0095]

[0096] In formula (21), e is the impact rebound coefficient; K represents the comprehensive stiffness of the single tooth pair contact of the friction plate assembly; is the relative speed of the gear teeth before collision.

[0097] Preferably, the collision force between the inner hub and the friction plate in S7 is calculated as:

[0098]

[0099] In formula (20), F n * It represents the collision force between the friction plate and the nth tooth of the inner hub; is the relative velocity during the collision; δ Rn Indicates the deformation of the inner hub and the friction plate at the nth working tooth surface; δ Ln It represents the deformation of the inner hub and the friction plate at the nth non-working tooth surface; p = μ / K, μ is the hysteresis damping coefficient; K is the comprehensive stiffness of the single tooth pair contact of the gear. If the friction plate gear teeth are healthy teeth, K = K H ; If the friction plate teeth are cracked teeth, K = K C ; is the inner hub angular velocity; is the angular acceleration of the friction plate.

[0100] Preferably, the S8 comprises the following steps:

[0101] S801. Due to the rotational speed fluctuation of the friction plate caused by the torsional vibration of the inner hub, the input angular velocity ω and input angular acceleration α of the inner hub are expressed as

[0102] ω=ω0+Asin(2πft), α=2πfAcos(2πft) (21)

[0103] In formula (21), ω represents the inner hub speed fluctuation, ω0 represents the constant speed; A is the speed fluctuation amplitude; f is the speed fluctuation frequency; t represents the time t; α represents the inner hub input angular acceleration;

[0104] S802. Establish a dynamic model of the inner hub and friction plate with two-mass impact loads, which is in the following form:

[0105]

[0106] In formula (22), J1 represents the moment of inertia of the inner hub; J2 represents the moment of inertia of the friction plate; is the inner hub angular acceleration; is the friction plate angular acceleration; α represents the inner hub input angular acceleration; F total is the total impact load of all teeth on the inner hub and friction plate at each moment; R b is the base circle radius; is the impact load of the nth tooth of the inner hub and the friction plate; n represents the nth tooth; Z is the total number of gear teeth;

[0107] S803, iteratively calculate the two-mass impact dynamic load model of the inner hub and the friction plate constructed in S802 within time t, and output the total impact load of all teeth of the inner hub and the friction plate and the impact load of a single gear tooth at each moment.

[0108] In summary, due to the adoption of the above technical solution, compared with the prior art, the present invention has at least the following beneficial effects:

[0109] The present invention provides a method for calculating the stiffness of a gear tooth with a tooth root crack in a friction plate by using an energy method;

[0110] The present invention also provides a method for calculating the eccentricity generated by the friction plate under the action of gravity when there is a tooth side clearance between the friction plate and the inner hub;

[0111] The present invention also provides a method for determining whether the inner hub and the tooth surface of the friction plate with or without cracks have positive or negative collisions when there is a tooth side clearance and eccentricity between the gear teeth of the friction plate and the inner hub, and a method for calculating the number of collision teeth and the random collision tooth position;

[0112] The present invention also provides a method for determining the impact rebound coefficient before and after the friction plate collides with the inner hub and the friction plate gear teeth with or without cracks;

[0113] The present invention also provides a method for calculating the tooth collision force of each gear tooth of the inner hub and the friction plate with or without cracks when positive collision or negative collision occurs;

[0114] The present invention also provides a two-mass impact dynamic load dynamics model calculation algorithm for the inner hub and the friction plate, which determines the impact load of the inner hub and the friction plate with / without cracks at each moment, ensuring high calculation efficiency while ensuring high calculation accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0115] Figure 1 It is a schematic diagram of a method for calculating the impact dynamic load of a floating support friction plate with a gear tooth crack according to an exemplary embodiment of the present invention.

[0116] Figure 2 Schematic diagram of the position of cracks in the friction plate teeth according to an exemplary embodiment of the present invention.

[0117] Figure 3 Schematic diagram of crack geometry parameters according to an exemplary embodiment of the present invention.

[0118] Figure 4 Schematic diagram of a crack rectangular coordinate system according to an exemplary embodiment of the present invention. DETAILED DESCRIPTION

[0119] The present invention is further described in detail below in conjunction with the examples and specific implementation methods. However, this should not be understood as the scope of the above subject matter of the present invention being limited to the following examples, and all technologies realized based on the content of the present invention belong to the scope of the present invention.

[0120] In the description of the present invention, it is necessary to understand that the terms "longitudinal", "lateral", "up", "down", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inside", "outside", etc., indicating the orientation or position relationship, are based on the orientation or position relationship shown in the accompanying drawings, and are 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 cannot be understood as a limitation on the present invention.

[0121] like Figure 1 As shown, the present invention provides a method for calculating the impact dynamic load of a floating support friction plate with a gear tooth crack, comprising the following steps:

[0122] S1. Obtaining parameters of a floating support friction plate assembly to be calculated, the friction plate assembly parameters including structural parameters, material parameters, operating condition parameters, and crack parameters. The friction plate assembly includes a friction plate and an inner hub, and the friction plate has a defective gear tooth with a crack.

[0123] In this embodiment, the basic parameters of the floating support friction plate assembly include: the number of teeth, module, tooth width, pitch circle pressure angle, elastic modulus, Poisson's ratio, mass of the friction plate and the inner hub, moment of inertia, tooth side clearance, speed of the inner hub and speed fluctuation frequency and amplitude; the crack parameters include: crack position angle, crack length and crack direction.

[0124] S2. Calculate the comprehensive stiffness K of the healthy friction plate gear teeth based on the basic parameters of the friction plate assembly H .

[0125] S201. Calculate the bending stiffness k of the single tooth pair of the inner hub or the friction plate by using the potential energy method according to the parameters of the friction plate assembly. bj , shear stiffness k sj and axial compressive stiffness k aj , the corresponding relationship is as follows:

[0126]

[0127]

[0128]

[0129] In formula (1), j = 1 represents the inner hub; j = 2 represents the friction plate; Uaj , U bj and U sj are the elastic potential energy stored in the axial compression, bending and shear deformation of the inner hub gear teeth or friction plate gear teeth respectively; F j Indicates the contact load at the meshing point of the inner hub gear or friction plate gear; F aj E is the horizontal component of the meshing force of the inner hub gear teeth or friction plate gear teeth; j Indicates the elastic modulus of the inner hub gear teeth or friction plate gear teeth; A xj Indicates the area of ​​the cross section at the fixed end x of the inner hub gear or friction plate gear; d j Indicates the distance from the inner hub gear or friction plate gear meshing position to the tooth root circle; M j I is the bending moment at the fixed end x from the inner hub gear teeth or friction plate gear teeth; xj It represents the moment of inertia of the cross section at the fixed end x of the inner hub gear or friction plate gear; F bj G is the vertical component of the meshing force of the inner hub gear teeth or friction plate gear teeth; j Indicates the shear modulus of the inner hub gear teeth or friction plate gear teeth;

[0130] S202. Based on the parameters of the friction plate assembly, the O'Donnell deformation theory is used to calculate the matrix stiffness k of a single tooth pair of gear teeth. fj , the expression is as follows:

[0131]

[0132] In formula (2), j = 1 represents the inner hub; j = 2 represents the friction plate; θ fj M is the deformation of the gear tooth matrix of the inner hub or friction plate; fj V is the unit bending moment at the fixed end x of the inner hub gear teeth or friction plate gear teeth; j It represents the unit shear force at the fixed end x of the inner hub gear teeth or friction plate gear teeth; E j Indicates the elastic modulus of the inner hub gear teeth or friction plate gear teeth; υ j Indicates the Poisson's ratio of the inner hub gear teeth or friction plate gear teeth; h j’ Indicates the cross section at the fixed end x of the inner hub gear or friction plate gear; F j Indicates the contact load at the meshing point of the inner hub gear teeth or friction plate gear teeth;

[0133] S203, according to the bending stiffness k of the inner hub and friction plate teeth in S201 bj , shear stiffness k sj , axial compression stiffness k aj and the matrix stiffness k in S202 fj , obtain the comprehensive stiffness K of the single tooth pair contact of the friction plate assembly H , calculated as follows:

[0134]

[0135] S3. Based on the basic parameters of the friction plate assembly, calculate the comprehensive stiffness Kc of the friction plate tooth with root crack.

[0136] S301. Assuming that the crack is parallel to the line connecting the center of the circle and the midpoint of the tooth top, and the length of the crack is q, the meshing stiffness is divided into two regions, I and II. Figure 2 As shown, its bending stiffness, shear stiffness and axial compression stiffness are:

[0137]

[0138]

[0139]

[0140] In formula (4), k Cb Indicates the bending stiffness of the friction plate with tooth root cracks; k Cs Indicates the shear stiffness of the friction plate with tooth root cracks; k Ca Indicates the axial compression stiffness of the friction plate with tooth root cracks; k CbⅠ represents the bending stiffness of the friction plate with tooth root crack in region I; k CbⅡ k represents the bending stiffness of the friction plate with tooth root crack in zone II; CsⅠ represents the shear stiffness of the friction plate with tooth root crack in region I; k CsⅡ represents the shear stiffness of the friction plate with tooth root crack in region II; k CaⅠ represents the axial compressive stiffness of the friction plate with tooth root crack in zone I; k CaⅡ Represents the axial compressive stiffness of the friction plate with tooth root crack in zone II.

[0141] S302. The calculation of the bending stiffness, shear stiffness and axial compression stiffness of the first region is the same as that of the healthy tooth, and can be calculated by formula (1). Therefore, the bending stiffness, shear stiffness and axial compression stiffness of the first region of the friction plate gear tooth can be expressed as:

[0142] k CbⅠ =k b2 , k CsⅠ =k s2 , k CaⅠ =k a2 (5)

[0143] In formula (5), k b2 k is the bending stiffness of the healthy teeth of the friction plate; s2 k is the shear stiffness of the healthy teeth of the friction plate; a2 is the axial compressive stiffness of the healthy tooth of the friction plate.

[0144] S303. The bending stiffness, shear stiffness and axial compression stiffness of the second region are derived. The calculation model of the friction plate crack tooth stiffness is as follows: Figure 3 shown.

[0145] like Figure 4 As shown, a rectangular coordinate system is defined, the X axis coincides with the center line of the friction plate gear teeth, the Y axis is the line connecting the straight line AB, the intersection of the X axis and the Y axis is O1, point A is the starting position of the friction plate tooth root crack, point D is the end point of the friction plate tooth root crack, point B is the intersection of the tooth profile and the root circle of the adjacent teeth of the friction plate; point E is the intersection of the friction plate gear tooth root circle and the X axis; point C is the intersection of straight line BE and straight line AD; point M is the distance d from the meshing force action point to the fixed part of the root circle; straight line BC is perpendicular to MG and intersects at point G. The coordinates of each point are M(-d,-h).

[0146] In this embodiment, the equations of the straight lines BC and MG are:

[0147]

[0148] l MG :xtanθ-y+dtanθ-h=0 (6)

[0149] In formula (6), R rr is the friction plate tooth root circle radius; is the semi-tooth angle of the friction plate tooth root; d is the horizontal coordinate of point M, that is, the distance from the meshing force application point to the fixed part of the root circle; h is the vertical coordinate of point M, that is, the distance from the meshing force application point to the x-axis; θ is the angle of ∠ABC.

[0150] According to the similarity of triangles, point E is the midpoint of line segment BC, so the coordinates of point E are According to the distance formula from a point to a straight line, l and e can be calculated, that is,

[0151]

[0152]

[0153] In formula (7), l is the distance from point M to line BC; e is the distance from point E to line MG.

[0154] S304. The calculation expressions of the cross-sectional inertia moment and cross-sectional area of ​​the second region at the rotation angle θ are:

[0155]

[0156] In formula (8), I θA represents the moment of inertia of the section at the rotation angle θ in the second region; θ It represents the cross-sectional area at the turning angle θ in the second region; W is the tooth width of the friction plate; is the half tooth angle at the root of the friction plate; θ is the angle of ∠ABC; R rr is the friction plate tooth root circle radius.

[0157] S305. Calculate the bending stiffness K of the friction plate with tooth root crack in region II CbⅡ , shear stiffness K CsⅡ and axial compressive stiffness K CaⅡ , the calculation expression is:

[0158]

[0159] In formula (9), γ is the distance from the crack position of the friction plate to the top position of the adjacent gear tooth, that is, the distance from point B to straight line AD; e is the distance from point E to straight line MG; G represents the shear modulus of the friction plate gear tooth; E is the elastic modulus of the gear tooth with the crack in the friction plate; α1 is the pressure angle at the meshing point of the friction plate; and θ is the angle of ∠ABC.

[0160] S306. Using the finite element method, the matrix stiffness of the cracked gear teeth of the friction plate is corrected, and the matrix stiffness coefficient λ caused by the crack is given, which can be expressed as:

[0161] K Cf =λK f (10)

[0162] In formula (10), K Cf Indicates the matrix stiffness of the cracked gear teeth of the friction plate, k f represents the stiffness of the friction plate matrix; λ represents the matrix stiffness coefficient caused by cracks.

[0163] S307, Comprehensive stiffness K of a single pair of teeth meshing between the friction plate with cracked gear teeth and the inner hub of healthy gear teeth C It can be expressed as:

[0164]

[0165] In formula (11), k b1 represents the bending stiffness of the inner hub gear teeth, k s1 represents the shear stiffness of the inner hub gear teeth, k a1 represents the axial compressive stiffness of the inner hub gear and k fs represents the inner hub gear tooth matrix stiffness, k Cb Indicates the bending stiffness of the friction plate with tooth root crack; K Cs Indicates the shear stiffness of the friction plate at the root of the tooth; k Cf Indicates the matrix stiffness of the cracked gear teeth of the friction plate; k CaIndicates the axial compressive stiffness of the friction plate with tooth root crack.

[0166] S308. Assume that the number of cracks in the friction plate gear teeth is Cn, and the position of the cracked teeth (calculated clockwise from the positive direction of the x-axis) is Cl = [C1, C2 ... Cn]; when the number of teeth n = Cl, the gear tooth stiffness is K = K C , otherwise the gear tooth stiffness is K = K H .

[0167] S4. Based on the basic parameters of the friction plate assembly, calculate the eccentricity caused by the friction plate under the action of gravity.

[0168] In this embodiment, there is a unilateral tooth side clearance c between the friction plate and the gear teeth of the inner hub, so the friction plate will generate an initial self-eccentricity a under the action of gravity. The eccentricity generated by the first contact of the right tooth (working tooth surface) of the friction plate under the action of gravity is calculated as follows:

[0169]

[0170] In formula (12), a R It represents the eccentricity caused by the contact of the right teeth (working tooth surface) of the friction plate under the action of gravity; abs(…) represents the absolute value; min(…) represents the minimum value; c represents the single-sided tooth side clearance between the friction plate and the gear teeth of the inner hub in the non-eccentric state; α0 represents the pressure angle; Z represents the number of teeth of the friction plate and the inner hub, and n represents the tooth sequence number;

[0171] In this embodiment, the eccentricity generated by the first contact of the left teeth (non-working tooth surface) of the friction plate under the action of gravity is calculated as follows:

[0172]

[0173] In formula (13), a L It represents the eccentricity caused by the contact of the left teeth (working tooth surface) of the friction plate under the action of gravity; abs(…) represents the absolute value; min(…) represents the minimum value; c represents the single-sided tooth side clearance between the friction plate and the gear teeth of the inner hub in the non-eccentric state; α0 represents the pressure angle; Z represents the number of teeth of the friction plate and the inner hub, and n represents the tooth sequence number.

[0174] Then the initial self-eccentricity a of the friction plate under the action of gravity can be expressed as:

[0175] a=(a L +a R ) / 2.

[0176] S5. Based on the eccentricity and relative motion relationship between the friction plate and the inner hub determined in S3, determine whether the friction plate and the inner hub collide, and determine the number of teeth colliding between the inner hub and the friction plate and the position of the random collision teeth.

[0177] S501. Using the basic parameters of the inner hub and the friction plate described in S1, calculate the tooth side clearance between the working tooth surface and the non-working tooth surface of the nth tooth of the inner hub and the friction plate under the action of self-eccentricity. The calculation form is as follows:

[0178]

[0179] In formula (14), c Rn Indicates the tooth side clearance of the working tooth surface of the nth tooth in the self-eccentric state; c Ln It represents the tooth side clearance of the non-working tooth surface of the nth tooth in the self-eccentric state; c represents the single-sided tooth side clearance between the friction plate and the gear teeth of the inner hub in the non-eccentric state; a represents the initial self-eccentricity generated by the friction plate under the action of gravity; α0 represents the pressure angle; Z is the number of teeth of the friction plate and the inner hub; n is the tooth sequence number.

[0180] S502. Considering the relative displacement of the friction pair components under the action of torque, the deformation of the working tooth surface and the non-working tooth surface of the inner hub and the friction plate is calculated. The calculation form is as follows:

[0181] δ Rn =R(θ1-θ2)cosα0-c Rn , δ Ln =R(θ2-θ1)cosα0-c Ln (15)

[0182] In formula (15), δ Rn Indicates the deformation of the working tooth surface of the nth tooth of the inner hub and the friction plate; δ Ln represents the deformation of the non-working tooth surface of the nth tooth of the inner hub and the friction plate; R is the pitch circle radius; θ1 is the angular displacement of the inner hub; θ2 is the angular displacement of the friction plate; α0 is the pressure angle; c Rn Indicates the tooth side clearance of the working tooth surface of the nth tooth in the self-eccentric state; c Ln Indicates the tooth side clearance of the non-working tooth surface of the nth tooth in the self-eccentric state.

[0183] S503, according to the deformation amount δ of the working tooth surface in S502 Rn and non-working tooth surface deformation δ Ln The calculation results are used for collision judgment:

[0184] If δ Rn > 0, the friction plate collides with the nth working tooth surface of the inner hub (positive collision), denoted by C Rn =1; if δ Ln >0, the friction plate collides (collides against) the nth non-working tooth surface of the inner hub, and is recorded as C Ln =1; if δ Rn ≤0 or δ Ln ≤0, the friction plate does not collide with the nth tooth of the inner hub, and CRn =0 or C Ln = 0, through C Rn and C Ln The collision tooth position can be determined.

[0185] S504. According to the calculation result of S403, the number of collision teeth of the inner hub and the friction plate in positive collision and negative collision can be obtained respectively, which can be expressed as:

[0186]

[0187] In formula (16), Z is the number of teeth of the friction plate and the inner hub; n is the tooth number of the nth tooth;

[0188] Then the total number of teeth colliding between the inner hub and the friction plate teeth is:

[0189] N=N R +N L .

[0190] S6. Based on the energy balance theory and the relative motion relationship between the inner hub and the friction plate, calculate the impact rebound coefficient and hysteresis damping coefficient of the collision between the friction plate and the inner hub gear teeth.

[0191] S601. Based on Hertz contact theory, the nonlinear impact collision force can be expressed as:

[0192]

[0193] In formula (17), Kδ m represents the elastic contact force part; is the impact damping force part, δ is the relative normal deformation of the tooth; is the relative velocity during the collision process; m is the nonlinear index, m=1.5; μ is the hysteresis damping coefficient;

[0194] S602, the energy loss ΔT from the start of the gear teeth collision to the end of the collision separation is expressed as:

[0195]

[0196] In formula (18), m1 is the mass of the inner hub; m2 is the mass of the friction plate; is the relative speed of the gear teeth before collision; e is the impact rebound coefficient.

[0197] The impact rebound coefficient e is defined as the ratio of the relative approach speed of two objects before the collision to the relative separation speed after the collision, expressed as:

[0198]

[0199] In formula (19), is the speed of the inner hub when the collision ends and separation occurs; It is the speed of the friction plate when the collision ends and the separation occurs; is the speed of the inner hub when the collision contact begins; It is the speed of the friction plate when the collision starts.

[0200] S603, when the gear teeth collide with the damping force The energy dissipation ΔT in the form of work can be expressed as:

[0201]

[0202] In formula (20), δ0 is the maximum deformation of gear tooth collision.

[0203] S604. According to the law of conservation of energy, the hysteresis damping coefficient μ can be obtained by combining equations (18) and (20), which can be expressed as:

[0204]

[0205] In formula (21), e is the impact rebound coefficient; K represents the comprehensive stiffness of the single tooth pair contact of the friction plate assembly; is the relative speed of the gear teeth before collision.

[0206] S7. Based on the Hertz contact theory, calculate the collision force between the friction plate and each contact tooth of the inner hub according to the comprehensive stiffness of the friction plate teeth with / without cracks determined in S2, the eccentricity between the friction plate and the inner hub determined in S3, the number of colliding teeth between the inner hub and the friction plate determined in S4, and the impact rebound coefficient and hysteresis damping coefficient between the friction plate and the inner hub teeth determined in S5.

[0207] In this embodiment, the combined impact force of the gear teeth is transmitted by the collision between the inner hub gear teeth and the friction plate gear teeth. The Hertz contact mechanics model can be used to describe the nonlinear impact collision between the contact surfaces. Based on the Hertz contact theory, the collision force between the inner hub and the working tooth surface (positive collision) and the non-working tooth surface (negative collision) of the friction plate can be calculated as:

[0208]

[0209] In formula (20), F n * It represents the collision force between the friction plate and the nth tooth of the inner hub; is the relative velocity during the collision; δ Rn Indicates the deformation of the inner hub and the friction plate at the nth working tooth surface; δ Ln It represents the deformation of the inner hub and the friction plate at the nth non-working tooth surface; p = μ / K, μ is the hysteresis damping coefficient; K is the comprehensive stiffness of the single tooth pair contact of the gear. If the friction plate gear teeth are healthy teeth, K = K H; If the friction plate gear teeth are cracked teeth, then K = K C ; is the inner hub angular velocity; is the angular acceleration of the friction plate.

[0210] S8. According to the collision force between the inner hub and the friction plate determined in S6 and based on Newtonian mechanics theory, a two-mass impact dynamic load dynamic model of the inner hub and the friction plate is constructed to calculate the impact dynamic load of the floating gear ring friction plate assembly with / without gear tooth cracks.

[0211] S801. Due to the rotational speed fluctuation of the friction plate caused by the torsional vibration of the inner hub, the input rotational speed ω and the input angular acceleration α of the inner hub can be expressed as

[0212] ω=ω0+Asin(2πft), α=2πfAcos(2πft) (21)

[0213] In formula (21), ω represents the inner hub speed fluctuation, ω0 represents the constant speed; A is the speed fluctuation amplitude; f is the speed fluctuation frequency; t represents the time t; α represents the input angular acceleration.

[0214] S802. Establish a dynamic model of the inner hub and friction plate with two-mass impact loads, which is in the following form:

[0215]

[0216] In formula (22), J1 represents the moment of inertia of the inner hub; J2 represents the moment of inertia of the friction plate; is the inner hub angular acceleration; is the friction plate angular acceleration; α represents the inner hub input acceleration; F total is the total impact load of all teeth on the inner hub and friction plate at each moment; R b is the base circle radius; is the impact load of the nth tooth between the inner hub and the friction plate; n represents the nth tooth; Z is the number of teeth (the total number of gear teeth);

[0217] S803. Use the Runge-Kutta method to iteratively calculate the two-mass impact dynamic load model of the inner hub and the friction plate constructed in S802 within time t. If t≤t1, update θ1, θ2, x, y, as well as the time-varying eccentricity and time-varying eccentricity phase, and set t+1, repeat S2-S7; if t>t1, output the total impact load of all teeth of the inner hub and the friction plate and the impact load of a single gear tooth at each moment.

[0218] Those skilled in the art will appreciate that the above-mentioned embodiments are specific examples for implementing the present invention, and in actual applications, various changes may be made thereto in form and detail without departing from the spirit and scope of the present invention.

Claims

1. A method for calculating the impact dynamic load of a floating support friction plate with a gear tooth crack, characterized in that: The specific steps include: S1: Obtaining parameters of a floating support friction plate assembly and crack parameters to be calculated, wherein the friction plate assembly comprises a friction plate and an inner hub, and the friction plate has a defective gear tooth with a crack; S2. Calculate the comprehensive stiffness K of the healthy friction plate gear teeth based on the basic parameters of the friction plate assembly H ; S3. Based on the basic parameters of the friction plate assembly, calculate the comprehensive stiffness K of the friction plate gear with root cracks C ; S4. Based on the basic parameters of the friction plate assembly, calculate the eccentricity caused by the friction plate under the action of gravity; S5. judging whether the friction plate and the inner hub collide with each other based on the eccentricity and relative motion relationship between the friction plate and the inner hub determined in S3, and determining the number of teeth colliding between the inner hub and the friction plate and the position of the random colliding teeth; S6. Based on the energy balance theory and the relative motion relationship between the inner hub and the friction plate, calculate the impact rebound coefficient and hysteresis damping coefficient of the collision between the friction plate and the inner hub gear teeth; S7, calculating the collision force between the friction plate and each contact tooth of the inner hub according to the comprehensive stiffness of the friction plate teeth with or without cracks determined in S2, the eccentricity between the friction plate and the inner hub determined in S3, the number of teeth of the inner hub and the friction plate colliding with each other determined in S4, and the impact rebound coefficient and hysteresis damping coefficient between the friction plate and the inner hub teeth determined in S5; S8, constructing a two-mass impact dynamic load dynamics model of the inner hub and the friction plate according to the collision force between the inner hub and the friction plate determined in S6, and outputting the impact dynamic load of the floating gear ring friction plate assembly containing gear tooth cracks; The S8 comprises the following steps: S801. Due to the rotational speed fluctuation of the friction plate caused by the torsional vibration of the inner hub, the input angular velocity ω and input angular acceleration α of the inner hub are expressed as ω=ω0+Asin(2πft), α=2πfAcos(2πft) (21) In formula (21), ω represents the inner hub speed fluctuation, ω0 represents the constant speed; A is the speed fluctuation amplitude; f is the speed fluctuation frequency; t represents the time t; α represents the inner hub input angular acceleration; S802. Establish a dynamic model of the inner hub and friction plate with two-mass impact loads, which is in the following form: In formula (22), J1 represents the moment of inertia of the inner hub; J2 represents the moment of inertia of the friction plate; is the inner hub angular acceleration; is the friction plate angular acceleration; α represents the inner hub input angular acceleration; F total is the total impact load of all teeth on the inner hub and friction plate at each moment; R b is the base circle radius; is the impact load of the nth tooth of the inner hub and the friction plate; n represents the nth tooth; Z is the total number of gear teeth; S803, iteratively calculate the two-mass impact dynamic load model of the inner hub and the friction plate constructed in S802 within time t, and output the total impact load of all teeth of the inner hub and the friction plate and the impact load of a single gear tooth at each moment.

2. A method for calculating the impact dynamic load of a floating support friction plate with a gear tooth crack as claimed in claim 1, characterized in that: In S1, the basic parameters of the floating support friction plate assembly include: the number of teeth, module, tooth width, pitch circle pressure angle, elastic modulus, Poisson's ratio, mass of the friction plate and the inner hub, moment of inertia, tooth side clearance, speed of the inner hub and speed fluctuation frequency and amplitude; the crack parameters include crack position angle, crack length and crack direction.

3. The method for calculating the impact dynamic load of a floating support friction plate with a gear tooth crack according to claim 1, characterized in that: The S2 comprises the following steps: S201. Calculate the bending stiffness k of the single tooth pair of the inner hub or the friction plate by using the potential energy method according to the parameters of the friction plate assembly. bj , shear stiffness k sj and axial compressive stiffness k aj , the corresponding relationship is as follows: In formula (1), j = 1 represents the inner hub; j = 2 represents the friction plate; U aj , U bj and U sj are the elastic potential energy stored in the axial compression, bending and shear deformation of the inner hub gear teeth or friction plate gear teeth respectively; F j Indicates the contact load at the meshing point of the inner hub gear teeth or friction plate gear teeth; F aj E is the horizontal component of the meshing force of the inner hub gear teeth or friction plate gear teeth; j Indicates the elastic modulus of the inner hub gear teeth or friction plate gear teeth; A xj It represents the area of ​​the cross section at the fixed end x of the inner hub gear or friction plate gear; d j Indicates the distance from the inner hub gear or friction plate gear meshing position to the tooth root circle; M j I is the bending moment at the fixed end x from the inner hub gear teeth or friction plate gear teeth; xj It represents the moment of inertia of the cross section at the fixed end x of the inner hub gear or friction plate gear; F bj G is the vertical component of the meshing force of the inner hub gear teeth or friction plate gear teeth; j Indicates the shear modulus of the inner hub gear teeth or friction plate gear teeth; S202. Based on the parameters of the friction plate assembly, the O'Donnell deformation theory is used to calculate the matrix stiffness k of a single tooth pair of gear teeth. fj , the expression is as follows: In formula (2), j = 1 represents the inner hub; j = 2 represents the friction plate; θ fj M is the deformation of the gear tooth matrix of the inner hub or friction plate; fj V is the unit bending moment at the fixed end x of the inner hub gear teeth or friction plate gear teeth; j It represents the unit shear force at the fixed end x of the inner hub gear teeth or friction plate gear teeth; E j Indicates the elastic modulus of the inner hub gear teeth or friction plate gear teeth; υ j Indicates the Poisson's ratio of the inner hub gear teeth or friction plate gear teeth; h j’ Indicates the cross section at the fixed end x of the inner hub gear or friction plate gear; F j Indicates the contact load at the meshing point of the inner hub gear teeth or friction plate gear teeth; S203, according to the bending stiffness k of the inner hub and friction plate teeth in S201 bj , shear stiffness k sj , axial compression stiffness k aj and the matrix stiffness k in S202 fj , obtain the comprehensive stiffness K of the single tooth pair contact of the friction plate assembly H , calculated as follows:

4. The method for calculating the impact dynamic load of a floating support friction plate with a gear tooth crack according to claim 1, characterized in that: The S3 comprises the following steps: S301. Assuming that the crack is parallel to the line connecting the center of the circle and the midpoint of the tooth top, and the length of the crack is q, the meshing stiffness is divided into two regions, Ⅰ and Ⅱ, and their bending stiffness, shear stiffness and axial compression stiffness are: In formula (4), k Cb Indicates the bending stiffness of the friction plate with tooth root cracks; k Cs Indicates the shear stiffness of the friction plate with tooth root cracks; k Ca Indicates the axial compression stiffness of the friction plate with tooth root cracks; k CbⅠ represents the bending stiffness of the friction plate with tooth root crack in region I; k CbⅡ k represents the bending stiffness of the friction plate with tooth root crack in zone II; CsⅠ represents the shear stiffness of the friction plate with tooth root crack in region I; k CsⅡ represents the shear stiffness of the friction plate with tooth root crack in region II; k CaⅠ represents the axial compressive stiffness of the friction plate with tooth root crack in zone I; k CaⅡ represents the axial compressive stiffness of the friction plate with tooth root crack in zone II; S302. The calculation of the bending stiffness, shear stiffness and axial compression stiffness of the first region is the same as that of the healthy tooth. Therefore, the bending stiffness, shear stiffness and axial compression stiffness of the first region of the friction plate gear tooth are expressed as: k CbⅠ =k b2 , k CsⅠ =k s2 , k CaⅠ =k a2 (5) In formula (5), k b2 k is the bending stiffness of the healthy teeth of the friction plate; s2 k is the shear stiffness of the healthy teeth of the friction plate; a2 is the axial compression stiffness of the healthy teeth of the friction plate; S303. Establish a calculation model for the stiffness of a cracked friction plate tooth and define a rectangular coordinate system. The X-axis coincides with the center line of the friction plate tooth. The Y-axis is the line connecting the straight line AB. The intersection of the X-axis and the Y-axis is O1. Point A is the starting position of the friction plate tooth root crack. Point D is the ending position of the friction plate tooth root crack. Point B is the intersection point of the tooth profile and the root circle of the adjacent teeth of the friction plate. Point E is the intersection point of the friction plate tooth root circle and the X-axis. Point C is the intersection point of the straight line BE and the straight line AD. Point M is the distance d from the meshing force action point to the fixed part of the root circle. The straight line BC is perpendicular to MG and intersects at point G. The coordinates of each point are: M(-d,-h); The equations of lines BC and MG are: lBC: l MG :xtanθ-y+dtanθ-h=0 (6) In formula (6), R rr is the friction plate tooth root circle radius; is the semi-tooth angle of the friction plate tooth root; d is the horizontal coordinate of point M, that is, the distance from the meshing force action point to the fixed part of the tooth root circle; h is the vertical coordinate of point M, that is, the distance from the meshing force action point to the x-axis; θ is the angle of ∠ABC; According to the similarity of triangles, point E is the midpoint of line segment BC, so the coordinates of point E are According to the distance formula from a point to a straight line, l and e are obtained, that is, In formula (7), l is the distance from point M to line BC; e is the distance from point E to line MG; S304. The calculation expressions of the cross-sectional inertia moment and cross-sectional area of ​​the second region at the rotation angle θ are: In formula (8), I θ A represents the moment of inertia of the section at the rotation angle θ in the second region; θ It represents the cross-sectional area at the turning angle θ in the second region; W is the tooth width of the friction plate; is the half tooth angle at the root of the friction plate; θ is the angle of ∠ABC; R rr is the friction plate tooth root circle radius; S305. Calculate the bending stiffness K of the friction plate with tooth root crack in region II CbⅡ , shear stiffness K CsⅡ and axial compressive stiffness K CaⅡ , the calculation expression is: In formula (9), γ is the distance from the crack position of the friction plate to the top position of the adjacent gear tooth, that is, the distance from point B to the straight line AD; e is the distance from point E to the straight line MG; G represents the shear modulus of the friction plate gear tooth; E is the elastic modulus of the gear tooth with the crack of the friction plate; α1 is the pressure angle at the meshing point of the friction plate; θ is the angle of ∠ABC; S306. Using the finite element method, the matrix stiffness of the cracked gear teeth of the friction plate is corrected, and the matrix stiffness coefficient λ caused by the crack is given, which is expressed as: K Cf =λK f (10) In formula (10), K Cf Indicates the matrix stiffness of the cracked gear teeth of the friction plate, k f represents the stiffness of the friction plate matrix; λ represents the matrix stiffness coefficient caused by cracks; S307, Comprehensive stiffness K of a single pair of teeth meshing between the friction plate with cracked gear teeth and the inner hub of healthy gear teeth C It is expressed as: In formula (11), k b1 represents the bending stiffness of the inner hub gear teeth, k s1 represents the shear stiffness of the inner hub gear teeth, k a1 represents the axial compressive stiffness of the inner hub gear and k fs represents the inner hub gear tooth matrix stiffness, k Cb Indicates the bending stiffness of the friction plate with tooth root crack; k Cs Indicates the shear stiffness of the friction plate at the root of the tooth; k Cf Indicates the matrix stiffness of the cracked gear teeth of the friction plate; k Ca It represents the axial compression stiffness of the friction plate with tooth root crack; S308. Assume that the number of cracks in the friction plate gear teeth is Cn, and the position of the cracked teeth is Cl = [C1, C2 ... Cn]; when the number of teeth n = Cl, the gear tooth stiffness is K = K C , otherwise the gear tooth stiffness is K = K H .

5. The method for calculating the impact dynamic load of a floating support friction plate with a gear tooth crack as claimed in claim 1, characterized in that: The S4 The following steps are involved: The eccentricity generated by the first contact of the working tooth surface of the friction plate under the action of gravity is calculated as follows: In formula (12), a R It represents the eccentricity caused by the contact between the working tooth surface of the friction plate under the action of gravity; abs(…) represents the absolute value; min(…) represents the minimum value; c represents the single-sided tooth side clearance between the friction plate and the gear teeth of the inner hub in the non-eccentric state; α0 represents the pressure angle; Z represents the number of teeth of the friction plate and the inner hub, and n represents the tooth sequence number; The eccentricity caused by the non-working tooth surface of the friction plate contacting first under the action of gravity is calculated as follows: In formula (13), a L It represents the eccentricity caused by the contact between the working tooth surface of the friction plate under the action of gravity; abs(…) represents the absolute value; min(…) represents the minimum value; c represents the single-sided tooth side clearance between the friction plate and the gear teeth of the inner hub in the non-eccentric state; α0 represents the pressure angle; Z represents the number of teeth of the friction plate and the inner hub, and n represents the tooth sequence number; Then the initial eccentricity a of the friction plate under the action of gravity is expressed as: a=(a L +a R ) / 2。 6. The method for calculating the impact dynamic load of a floating support friction plate with a gear tooth crack as claimed in claim 1, characterized in that: The S5 comprises the following steps: S501. Using the basic parameters of the inner hub and the friction plate described in S1, calculate the tooth side clearance between the working tooth surface and the non-working tooth surface of the nth tooth of the inner hub and the friction plate under the action of self-eccentricity. The calculation form is as follows: In formula (14), c Rn Indicates the tooth side clearance of the working tooth surface of the nth tooth in the self-eccentric state; c Ln represents the tooth side clearance of the non-working tooth surface of the nth tooth in the self-eccentric state; c represents the single-sided tooth side clearance between the friction plate and the gear teeth of the inner hub in the non-eccentric state; a represents the initial self-eccentricity generated by the friction plate under the action of gravity; α0 represents the pressure angle; Z represents the number of teeth of the friction plate and the inner hub; n represents the tooth sequence number; S502. Considering the relative displacement of the friction pair components under the action of torque, the deformation of the working tooth surface and the non-working tooth surface of the inner hub and the friction plate is calculated. The calculation form is as follows: d Rn =R(θ1-θ2)cosα0-c Rn ,d Ln =R(θ2-θ1)cosα0-c Ln (15) In formula (15), δ Rn Indicates the deformation of the working tooth surface of the nth tooth of the inner hub and the friction plate; δ Ln represents the deformation of the non-working tooth surface of the nth tooth of the inner hub and the friction plate; R is the pitch circle radius; θ1 is the angular displacement of the inner hub; θ2 is the angular displacement of the friction plate; α0 is the pressure angle; c Rn Indicates the tooth side clearance of the working tooth surface of the nth tooth in the self-eccentric state; c Ln Indicates the tooth side clearance of the non-working tooth surface of the nth tooth in the self-eccentric state; S503, according to the deformation amount δ of the working tooth surface in S502 Rn and non-working tooth surface deformation δ Ln The calculation results are used for collision judgment: If δ Rn > 0, the friction plate and the nth working tooth surface of the inner hub collide positively, denoted by C Rn =1; if δ Ln > 0, the friction plate and the nth non-working tooth surface of the inner hub collide with each other, denoted by C Ln =1; if δ Rn ≤0 or δ Ln ≤0, the friction plate does not collide with the nth tooth of the inner hub, and C Rn =0 or C Ln =0; S504. According to the judgment result of S403, the collision teeth numbers of the inner hub and the friction plate in positive collision and negative collision are obtained respectively, which are expressed as: In formula (16), Z is the number of teeth of the friction plate and the inner hub; n is the tooth number of the nth tooth; Then the total number of teeth colliding between the inner hub and the friction plate teeth is: N=N R +N L 。 7. The method for calculating the impact dynamic load of a floating support friction plate with a gear tooth crack as claimed in claim 1, characterized in that: The S6 comprises the following steps: S601. Based on Hertz contact theory, the nonlinear impact collision force F is expressed as: In formula (17), Kδ m represents the elastic contact force part; is the impact damping force part, δ is the relative normal deformation of the tooth; is the relative velocity during the collision process; m is the nonlinear index, m=1.5; μ is the hysteresis damping coefficient; S602, the energy loss ΔT from the start of the gear teeth collision to the end of the collision separation is expressed as: In formula (18), m1 is the mass of the inner hub; m2 is the mass of the friction plate; is the relative speed of the gear teeth before collision; e is the impact rebound coefficient; The impact rebound coefficient e is defined as the ratio of the relative approach speed of two objects before the collision to the relative separation speed after the collision, expressed as: In formula (19), is the speed of the inner hub when the collision ends and separation occurs; It is the speed of the friction plate when the collision ends and the separation occurs; is the speed of the inner hub when the collision contact begins; is the speed of the friction plate when the collision starts; S603, when the gear teeth collide with the damping force The energy dissipation ΔT in the form of work is expressed as: In formula (20), δ0 is the maximum deformation of gear tooth collision; S604. According to the energy conservation theorem, the hysteresis damping coefficient μ is obtained by combining equation (18) and equation (20), which is expressed as: In formula (21), e is the impact rebound coefficient; K represents the comprehensive stiffness of the single tooth pair contact of the friction plate assembly; is the relative speed of the gear teeth before collision.

8. The method for calculating the impact dynamic load of a floating support friction plate with a gear tooth crack as claimed in claim 1, characterized in that: The collision force between the inner hub and the friction plate in S7 is calculated as: In formula (20), F n * It represents the collision force between the friction plate and the nth tooth of the inner hub; is the relative velocity during the collision; δ Rn Indicates the deformation of the inner hub and the friction plate at the nth working tooth surface; δ Ln It represents the deformation of the inner hub and the friction plate at the nth non-working tooth surface; p = μ / K, μ is the hysteresis damping coefficient; K is the comprehensive stiffness of the single tooth pair contact of the gear. If the friction plate gear teeth are healthy teeth, K = K H ; If the friction plate teeth are cracked teeth, K = K C ; is the inner hub angular velocity; is the angular acceleration of the friction plate.

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