Non-circular planetary gear random pitting meshing stiffness calculation method

Abstract

The invention discloses a non-circular planetary gear random pitting meshing stiffness calculation method which comprises the following steps: (1) constructing a non-circular planetary gear configuration, and solving a time-varying pressure angle; (2) establishing a tooth width equation under random pitting; (3) establishing a non-circular planetary gear time-varying meshing stiffness analytical calculation model under random pitting corrosion; (4) the influence characteristics of pitting corrosion factors on the non-circular gear meshing rigidity are explored; the method has the beneficial effects that the non-circular planetary gear random pitting meshing rigidity can be solved, the blank of research on the non-circular planetary gear random pitting meshing rigidity at present is filled, and the method has good universality.

Description

Technical Field

[0001] The present invention relates to the technical field of gear design and analysis, and in particular to a method for calculating the random pitting meshing stiffness of non-circular planetary gears. Background Art

[0002] Non-circular planetary gear transmission is a new type of transmission method. Compared with traditional planetary gear transmission, it has the characteristics of variable center distance and variable transmission ratio. It is suitable for specific applications under complex working conditions and is widely used in automobiles, robotics, aerospace, agricultural machinery and other fields. Non-circular gears are the key components of this transmission system. Their unique pitch curve shape breaks through the limitations of traditional gears in special application scenarios and demonstrates their unique advantages. The stiffness of non-circular planetary gears is an important characteristic that characterizes the transmission accuracy and stability of the transmission system; random pitting is the most common form of gear failure in the early stage of gear failure. Therefore, exploring the random pitting meshing stiffness characteristics of non-circular planetary gears is of great significance for improving the stability of the transmission system and providing a theoretical basis for the optimal design of non-circular planetary gear transmission systems. Due to the non-ratio transmission characteristics of non-circular gears and the complex gear structure, there is currently a lack of research on the calculation method of the random pitting stiffness of non-circular planetary gears.

[0003] To address the above-mentioned issues, the present invention proposes a method for calculating the random pitting mesh stiffness of non-circular planetary gears. This method can accurately and effectively calculate the time-varying mesh stiffness characteristics of non-circular planetary gear transmission systems, filling the technical gap in the calculation of time-varying mesh stiffness of non-circular planetary gear transmission systems, promoting the development of engineering technology, and generating significant social and economic benefits. Summary of the Invention

[0004] In order to overcome the shortcomings of the existing technology and fill the gaps in the relevant technology, the present invention provides a method for calculating the random pitting mesh stiffness of non-circular planetary gears. This method fully considers the characteristics of variable center distance and variable pressure angle of the non-circular planetary gear structure, and can design different system configurations according to experimental requirements. At the same time, the random pitting tooth width equation is introduced to establish a mathematical model of the random pitting mesh stiffness of non-circular planetary gears. By solving the equation, the random pitting mesh stiffness of non-circular planetary gears under the influence of factors such as pitting degree and pitting location is obtained. This method has a certain degree of universality.

[0005] The technical solution adopted by the present invention to solve the technical problem is as follows: a method for calculating the random pitting meshing stiffness of non-circular planetary gears, characterized by comprising the following steps:

[0006] Step (1): Construct a non-circular planetary gear configuration and solve the time-varying pressure angle;

[0007] The pitch curve of the inner gear ring is a high-order elliptic curve, and its equation is:

[0008]

[0009] Where: r r is the pitch curve radius of the inner gear ring, A is the radius of the major axis of the ellipse, e is the eccentricity, N r is the high-order elliptical order of the inner gear ring, is the polar angle of the inner ring gear;

[0010] According to the motion relationship between the inner gear ring and the sun gear, the equation of the sun gear pitch curve is obtained:

[0011]

[0012] Where: r s 、r r 、r p They are the pitch curve radius of the inner gear ring, the pitch curve radius of the sun gear, and the pitch circle radius of the planet gear. are the polar angles corresponding to the inner gear ring and the sun gear respectively, u1 is the angle between the positive direction of the tangent of the inner gear ring pitch curve at the meshing point and the radial direction, For r r The derivative of is the derivative of u1;

[0013] Boundary conditions for uniform distribution of teeth on the pitch curve of non-circular gears:

[0014]

[0015] Where: Z s , Z r are the number of teeth of the sun gear and the inner ring gear respectively, m is the gear module, r r is the pitch curve diameter of the inner gear ring, N s 、N r are the pitch curve orders of the sun gear and the ring gear, is the polar angle corresponding to the inner ring gear;

[0016] The curvature radius of the sun gear pitch curve ρ s and the curvature radius ρ of the inner gear ring pitch curve r They are:

[0017]

[0018] Where: r j is the pitch curve radius of the sun gear and the ring gear (j = s, r);

[0019] The theoretical center distances of the meshing pairs of the planetary gear and the sun gear, and the planetary gear and the internal gear ring are:

[0020]

[0021] Where: a sis the theoretical center distance between the planetary gear and the sun gear meshing pair at time t, a r is the theoretical center distance between the planetary gear and the inner ring gear meshing pair at time t; r p is the planet gear pitch radius;

[0022] Use equivalent circular gears to replace non-circular gears, and discretize non-circular gears into different equivalent circular gears. The number of teeth of the internal gear ring equivalent circular gear is Z. rv and the number of teeth of the sun gear equivalent circular gear Z sv Expressed as:

[0023]

[0024] Equivalent center distance a between planetary gear and internal gear ring rv and the equivalent center distance a between the planetary gear and the sun gear sv Expressed as:

[0025]

[0026] Where: m is the gear module, Z rv , Z sv are the equivalent circular gear teeth numbers of the internal ring gear and the sun gear respectively;

[0027] Assume that at the initial meshing position, the pressure angle is α0 = 20°, and the theoretical center distances between the planetary gear, the sun gear and the inner ring gear are a and b. sv0 、a rv0 At any time, the center distance between the planetary gear and the sun gear is a s ′, the center distance between the planetary gear and the inner ring gear is a r ′, then the sun gear time-varying pressure angle α s and the ring gear pressure angle α r The value of can be expressed as:

[0028]

[0029] Step (2): Establish the gear tooth width equation under random pitting; assume that the circular pitting radius is r, which is distributed near the pitch line, and the coordinates of the center of the pitting point obey the normal distribution. Its distribution area is the entire tooth surface, and the coordinates of the center of the pitting point are:

[0030]

[0031] In the formula x n is the X coordinate of the center of the pitting point, y n is the Y coordinate of the center of the pitting point, μ x is the mean value of the X coordinate of the center of the pitting point, μ y is the mean value of the Y coordinate of the center of the pitting point, σ x is the standard deviation of the X coordinate of the center of the pitting point, σy is the standard deviation of the Y coordinate of the center of the pitting point, B is the tooth width, and H is the tooth height;

[0032] In order to calculate the gear stiffness after pitting, the tooth surface is discretized into micro rectangles. Due to the coincidence of pitting points and the length of pitting in the differential area is not the diameter, let ξ be the pitting coincidence coefficient and Q be the number of pitting points; then the length of pitting points in each rectangle is ΔB x for:

[0033]

[0034] Remaining tooth width B x For: B x =B-△B x ;

[0035] Where: r is the circular pitting radius, μ x is the mean value of the X coordinate of the center of the pitting point, σ x is the standard deviation of the X coordinate of the center of the pitting point, and B is the tooth width;

[0036] Step (3): Establish an analytical calculation model for the time-varying meshing stiffness of non-circular planetary gears under random pitting; use the tooth number reduction method to approximate the tooth profile of the non-circular planetary gear, discretize the non-circular gear into an equivalent circular gear to calculate the time-varying meshing stiffness of the gear; calculate and analyze the time-varying meshing stiffness of the non-circular planetary gear based on the potential energy method; decompose the gear meshing force F into the gear plane coordinate system to obtain:

[0037]

[0038] Where: β is the angle between the gear meshing force F and the Y axis, S x is the half tooth thickness at point A on the tooth profile, h is the tooth height at the point of application, x is the X-axis coordinate of point A on the tooth profile, and M is the bending moment on the gear;

[0039] Compression potential energy U of non-circular planetary gear a , bending potential energy U b , Hertzian contact strain energy U h , gear shear potential energy U s It can be expressed as:

[0040]

[0041] Where: F y is the component of F in the Y direction, F x is the component of F in the X direction, S x is half the tooth thickness at point A on the tooth profile, B is the tooth width; E is the elastic modulus; M is the bending moment on the gear, G is the shear modulus; k a is the axial compression stiffness; k b is the bending stiffness; k sis the shear stiffness, k h is the Hertzian stiffness;

[0042] Substituting the tooth width formula into the equation, we get:

[0043]

[0044]

[0045] Where: is the base circle tooth half angle of the equivalent circular gear, β is the angle between the gear meshing force F and the Y axis, x is the pressure angle at point A on the tooth profile, ξ is the pitting coincidence coefficient, Q is the number of pitting points, r is the circular pitting radius, μ x is the mean value of the X coordinate of the center of the pitting point, σ x is the standard deviation of the X coordinate of the center of the pitting point, A=[γ x sinβ x -cosβ x ]cosβ,r fv 、r bv are the root circle radius and base circle radius of the equivalent circular gear respectively, ν is the Poisson's ratio, k f is the matrix stiffness, L * 、M * 、P * , Q * is a constant value, h f S is the tooth height at the intersection of the extended line of the force and the tooth centerline, f The tooth root is rounded and thick;

[0046] The time-varying mesh stiffness of the sun gear-planet gear meshing pair in non-circular planetary gears can be divided into single-tooth mesh stiffness and double-tooth mesh stiffness. Since the two adjacent teeth of the sun gear are different, the i-th double-tooth meshing is different from the i+1-th double-tooth meshing, which is different from circular gears. The time-varying mesh stiffness of single and double teeth can be expressed as:

[0047]

[0048] Where: k i is the meshing stiffness of the i-th single tooth of the sun gear-planet gear meshing pair, k a1,i is the axial compressive stiffness of gear 1 in the sun gear-planet gear meshing pair, k a2,i is the axial compressive stiffness of the i-th gear 2 in the sun gear-planet gear meshing pair, k b1,i is the bending stiffness of the i-th gear 1 in the sun gear-planet gear meshing pair, k b2,i is the i-th bending stiffness of gear 2 in the sun gear-planet gear meshing pair, k f1,i is the base stiffness of gear 1 in the sun gear-planet gear meshing pair, kf2,i is the base stiffness of gear 2 in the sun gear-planet gear meshing pair, k s1,i is the i-th shear stiffness of gear 1 in the sun gear-planet gear meshing pair, k s2,i is the i-th shear stiffness of gear 2 in the sun gear-planet gear meshing pair, k h,i is the Hertzian stiffness of the sun gear-planet gear meshing pair, k mi is the meshing stiffness of the i-th double tooth of the sun gear-planet gear meshing pair;

[0049] Step (4): Investigate the influence characteristics of random pitting factors on the meshing stiffness of non-circular gears; select the number of pitting Q and the mean value of the Y coordinate of the pitting point center μ y As the influencing parameters, the changing states of the meshing stiffness of non-circular planetary gears under different parameters are solved respectively.

[0050] Compared with the prior art, the present invention has the following beneficial effects: a mathematical model of the time-varying mesh stiffness of non-circular planetary gears is established under the premise of considering structural factors such as the variable pressure angle and variable center distance of non-circular planetary gears; a pitting tooth surface equation is established by comprehensively considering the random pitting characteristics of the tooth surface, and the pitting factor is introduced into the stiffness calculation of non-circular planetary gears, filling the technical gap in the calculation of the random pitting mesh stiffness of non-circular planetary gears, and providing a reference for the structural optimization design and fault detection of non-circular planetary gear transmissions. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] Figure 1 This is a flow chart of the calculation method for the random pitting mesh stiffness of non-circular planetary gears;

[0052] Figure 2 It is a three-dimensional configuration of a non-circular planetary gear transmission system;

[0053] Figure 3 It is a time-varying mesh stiffness model for non-circular planetary gears;

[0054] Figure 4 is the time-varying mesh stiffness of the sun gear-planet gear meshing pair under random pitting and healthy conditions. DETAILED DESCRIPTION

[0055] The embodiments of the present invention are described below with reference to the accompanying drawings. Figures 1-4 The specific embodiments of the present invention are described in detail.

[0056] like Figure 1 The figure shows a flow chart of the method for calculating the mesh stiffness of non-circular planetary gears with random pitting, which includes the following steps:

[0057] Step (1): Construct a non-circular planetary gear configuration, such as Figure 2 As shown, solve the time-varying pressure angle of the non-circular planetary gear meshing pair;

[0058] The pitch curve of the inner gear ring is a high-order elliptic curve, and its equation is:

[0059]

[0060] Where: r r is the pitch curve radius of the inner gear ring, A is the radius of the major axis of the ellipse, e is the eccentricity, N r is the high-order elliptical order of the inner gear ring, is the polar angle of the inner ring gear;

[0061] According to the motion relationship between the inner gear ring and the sun gear, the equation of the sun gear pitch curve is obtained:

[0062]

[0063] Where: r s 、r r 、r p They are the pitch curve radius of the inner gear ring, the pitch curve radius of the sun gear, and the pitch circle radius of the planet gear. are the polar angles corresponding to the inner gear ring and the sun gear respectively, u1 is the angle between the positive direction of the tangent of the inner gear ring pitch curve at the meshing point and the radial direction, For r r The derivative of is the derivative of u1;

[0064] Boundary conditions for uniform distribution of teeth on the pitch curve of non-circular gears:

[0065]

[0066] Where: Z s , Z r are the number of teeth of the sun gear and the inner ring gear respectively, m is the gear module, r r is the pitch curve diameter of the inner gear ring, N s 、N r are the pitch curve orders of the sun gear and the ring gear, is the polar angle corresponding to the inner ring gear;

[0067] The curvature radius of the sun gear pitch curve ρ s and the curvature radius ρ of the inner gear ring pitch curve r They are:

[0068]

[0069] Where: r j is the pitch curve radius of the sun gear and the ring gear (j = s, r);

[0070] The theoretical center distances of the meshing pairs of the planetary gear and the sun gear, and the planetary gear and the internal gear ring are:

[0071]

[0072] Where: a s is the theoretical center distance between the planetary gear and the sun gear meshing pair at time t, a r is the theoretical center distance between the planetary gear and the inner ring gear meshing pair at time t; r p is the planet gear pitch radius;

[0073] Use equivalent circular gears to replace non-circular gears, and discretize non-circular gears into different equivalent circular gears. The number of teeth of the internal gear ring equivalent circular gear is Z. rv and the number of teeth of the sun gear equivalent circular gear Z sv Expressed as:

[0074]

[0075] Equivalent center distance a between planetary gear and internal gear ring rv and the equivalent center distance a between the planetary gear and the sun gear sv Expressed as:

[0076]

[0077] Where: m is the gear module, Z rv , Z sv are the equivalent circular gear teeth numbers of the inner ring gear and the sun gear respectively; ρ s , ρ r are the curvature radii of the pitch curves of the sun gear and the inner ring gear respectively; r p is the planet gear pitch radius;

[0078] Assume that at the initial meshing position, the pressure angle is α0 = 20°, and the theoretical center distances between the planetary gear, the sun gear and the inner ring gear are a and b. sv0 、a rv0 At any time, the center distance between the planetary gear and the sun gear is a s ′, the center distance between the planetary gear and the inner ring gear is a r ′, then the sun gear time-varying pressure angle α s and the ring gear pressure angle α r The value of can be expressed as:

[0079]

[0080] Step (2): Establish the gear tooth width equation under random pitting; assume that the circular pitting radius is r, which is distributed near the pitch line, and the coordinates of the center of the pitting point obey the normal distribution. Its distribution area is the entire tooth surface, and the coordinates of the center of the pitting point are:

[0081]

[0082] In the formula xn is the X coordinate of the center of the pitting point, y n is the Y coordinate of the center of the pitting point, μ x is the mean value of the X coordinate of the center of the pitting point, μ y is the mean value of the Y coordinate of the center of the pitting point, σ x is the standard deviation of the X coordinate of the center of the pitting point, σ y is the standard deviation of the Y coordinate of the center of the pitting point, B is the tooth width, and H is the tooth height;

[0083] In order to calculate the gear stiffness after pitting, the tooth surface is discretized into micro rectangles. Due to the coincidence of pitting points and the length of pitting in the differential area is not the diameter, let ξ be the pitting coincidence coefficient and Q be the number of pitting points; then the length of pitting points in each rectangle is ΔB x for:

[0084]

[0085] Remaining tooth width B x For: B x =B-△B x ;

[0086] Where: r is the circular pitting radius, μ x is the mean value of the X coordinate of the center of the pitting point, σ x is the standard deviation of the X coordinate of the center of the pitting point, and B is the tooth width;

[0087] Step (3): Establish an analytical calculation model for the time-varying meshing stiffness of non-circular planetary gears under random pitting; Figure 3 As shown in the figure, the position of the non-circular planetary gear teeth is determined by the tooth number reduction method, and the non-circular gear tooth profile is generated to obtain the non-circular planetary gear model; in order to facilitate the time-varying meshing stiffness of each meshing pair of the non-circular planetary gear, as shown in the figure, Figure 3 As shown in the figure, the non-circular gear is discretized into equivalent circular gears to calculate the time-varying meshing stiffness of the gear meshing pair. Based on the equivalent tooth profile, the potential energy method and the slicing method are used to obtain the cantilever beam calculation model of the non-circular gear, and the time-varying meshing stiffness of the non-circular planetary gear is dynamically analyzed. The tooth meshing force F is decomposed into the tooth plane coordinate system to obtain:

[0088]

[0089] Where: β is the angle between the gear meshing force F and the Y axis, S x is the half tooth thickness at point A on the tooth profile, h is the tooth height at the point of application, x is the X-axis coordinate of point A on the tooth profile, and M is the bending moment on the gear;

[0090] Compression potential energy U of non-circular planetary gear a , bending potential energy U b , Hertzian contact strain energy U h , gear shear potential energy Us It can be expressed as:

[0091]

[0092] Where: F y is the component of F in the Y direction, F x is the component of F in the X direction, S x is half the tooth thickness at point A on the tooth profile, B is the tooth width; E is the elastic modulus; M is the bending moment on the gear, G is the shear modulus; k a is the axial compression stiffness; k b is the bending stiffness; k s is the shear stiffness, k h is the Hertzian stiffness;

[0093] Substituting the tooth width formula into the equation, we get:

[0094]

[0095] Where: is the base circle tooth half angle of the equivalent circular gear, β is the angle between the gear meshing force F and the Y axis, x is the pressure angle at point A on the tooth profile, ξ is the pitting coincidence coefficient, Q is the number of pitting points, r is the circular pitting radius, μ x is the mean value of the X coordinate of the center of the pitting point, σ x is the standard deviation of the X coordinate of the center of the pitting point, A=[γ x sinβ x -cosβ x ]cosβ,r fv 、r bv are the root circle radius and base circle radius of the equivalent circular gear respectively, ν is the Poisson's ratio, k f is the matrix stiffness, L * 、M * 、P * , Q * is a constant value, h f S is the tooth height at the intersection of the extended line of the force and the tooth centerline, f The tooth root is rounded and thick;

[0096] The time-varying mesh stiffness of the sun gear-planet gear meshing pair in non-circular planetary gears can be divided into single-tooth mesh stiffness and double-tooth mesh stiffness. Since the two adjacent teeth of the sun gear are different, the i-th double-tooth meshing is different from the i+1-th double-tooth meshing, which is different from circular gears. The time-varying mesh stiffness of single and double teeth can be expressed as:

[0097]

[0098] Where: k iis the meshing stiffness of the i-th single tooth of the sun gear-planet gear meshing pair, k a1,i is the axial compressive stiffness of gear 1 in the sun gear-planet gear meshing pair, k a2,i is the axial compressive stiffness of the i-th gear 2 in the sun gear-planet gear meshing pair, k b1,i is the bending stiffness of the i-th gear 1 in the sun gear-planet gear meshing pair, k b2,i is the i-th bending stiffness of gear 2 in the sun gear-planet gear meshing pair, k f1,i is the base stiffness of gear 1 in the sun gear-planet gear meshing pair, k f2,i is the base stiffness of gear 2 in the sun gear-planet gear meshing pair, k s1,i is the i-th shear stiffness of gear 1 in the sun gear-planet gear meshing pair, k s2,i is the i-th shear stiffness of gear 2 in the sun gear-planet gear meshing pair, k h,i is the Hertzian stiffness of the sun gear-planet gear meshing pair, k mi is the meshing stiffness of the i-th double tooth of the sun gear-planet gear meshing pair;

[0099] Step (4): Investigate the influence characteristics of random pitting factors on the meshing stiffness of non-circular gears; select the number of pitting Q and the mean value of the Y coordinate of the pitting point center μ y As the influencing parameters, the changing states of the meshing stiffness of non-circular planetary gears under different parameters are solved respectively.

[0100] In the example, the parameters are shown in Table 1. Taking the sun gear-planet gear pair as an example, the above method is used to calculate the time-varying meshing stiffness of the gear pair under healthy and random pitting conditions. The results are shown in Table 1. Figure 4 shown.

[0101] Figure 4 The results show that the time-varying meshing stiffness of the sun gear-planet gear pair is in a periodic state, and pitting corrosion will significantly reduce the stiffness of the gear meshing pair. The pitting corrosion of the system can be detected by detecting the change in gear stiffness.

[0102] The above description is only a preferred embodiment of the invention and does not limit the invention in any way. Any modifications, changes and equivalent changes made to the above embodiments based on the essence of the invention shall still fall within the scope of protection of the technology of the invention.

[0103] Table 1 Basic parameters of non-circular planetary gears

[0104]

Claims

1. A method for calculating the meshing stiffness of non-circular planetary gears with random pitting, characterized in that: The following steps are involved: Step (1): Construct a non-circular planetary gear configuration and solve the time-varying pressure angle; The pitch curve of the inner gear ring is a high-order elliptic curve, and its equation is: Where: r r is the pitch curve radius of the inner gear ring, A is the radius of the major axis of the ellipse, e is the eccentricity, N r is the high-order elliptical order of the inner gear ring, is the polar angle of the inner ring gear; According to the motion relationship between the inner gear ring and the sun gear, the equation of the sun gear pitch curve is obtained: Where: r s 、r r 、r p They are the pitch curve radius of the inner gear ring, the pitch curve radius of the sun gear, and the pitch circle radius of the planet gear. are the polar angles corresponding to the inner gear ring and the sun gear respectively, u1 is the angle between the positive direction of the tangent direction of the inner gear ring pitch curve at the meshing point and the radial direction, For r r The derivative of is the derivative of u1; Boundary conditions for uniform distribution of teeth on the pitch curve of non-circular gears: Where: Z s , Z r are the number of teeth of the sun gear and the inner ring gear respectively, m is the gear module, r r Pitch curve diameter of inner gear ring, N s 、N r are the pitch curve orders of the sun gear and the ring gear, is the polar angle corresponding to the inner ring gear; The curvature radius of the sun gear pitch curve ρ s and the curvature radius ρ of the inner gear ring pitch curve r They are: Where: r j is the pitch curve radius of the sun gear and the ring gear (j = s, r); The theoretical center distances of the meshing pairs of the planetary gear and the sun gear, and the planetary gear and the internal gear ring are: Where: a s is the theoretical center distance between the planetary gear and the sun gear meshing pair at time t, a r is the theoretical center distance of the meshing pair of the planetary gear and the internal gear ring at time t; Use equivalent circular gears to replace non-circular gears, and discretize non-circular gears into different equivalent circular gears. The number of teeth of the internal gear ring equivalent circular gear is Z. rv and the number of teeth of the sun gear equivalent circular gear Z sv Expressed as: Equivalent center distance a between planetary gear and internal gear ring rv and the equivalent center distance a between the planetary gear and the sun gear sv Expressed as: Where: m is the gear module, Z rv 、Z sv are the equivalent circular gear teeth numbers of the internal ring gear and the sun gear respectively; Assume that at the initial meshing position, the pressure angle is α0 = 20°, and the center distance between the planetary gear and the sun gear at any time is a s ′, the center distance between the planetary gear and the inner ring gear is a r ′, then the sun gear time-varying pressure angle α s and the ring gear pressure angle α r The value of can be expressed as: Step (2): Establish the gear tooth width equation under random pitting; assume that the circular pitting radius is r, which is distributed near the pitch line, and the coordinates of the center of the pitting point obey the normal distribution. Its distribution area is the entire tooth surface, and the coordinates of the center of the pitting point are: In the formula x n is the X coordinate of the center of the pitting point, y n is the Y coordinate of the center of the pitting point, μ x is the mean value of the X coordinate of the center of the pitting point, μ y is the mean value of the Y coordinate of the center of the pitting point, σ x is the standard deviation of the X coordinate of the center of the pitting point, σ y is the standard deviation of the Y coordinate of the center of the pitting point, B is the tooth width, and H is the tooth height; In order to calculate the gear stiffness after pitting, the tooth surface is discretized into micro rectangles. Due to the coincidence of pitting points and the length of pitting in the differential area is not the diameter, let ξ be the pitting coincidence coefficient and Q be the number of pitting points; then the length of pitting points in each rectangle is ΔB x for: Remaining tooth width B x For: B x =B-△B x ; Where: r is the circular pitting radius, μ x is the mean value of the X coordinate of the center of the pitting point, σ x is the standard deviation of the X coordinate of the center of the pitting point, and B is the tooth width; Step (3): Establish an analytical calculation model for the time-varying mesh stiffness of non-circular planetary gears under random pitting; use the tooth number reduction method to approximate the tooth profile of the non-circular planetary gear, discretize the non-circular gear into an equivalent circular gear to calculate the time-varying mesh stiffness of the gear; The time-varying meshing stiffness of non-circular planetary gears is calculated and analyzed based on the potential energy method; the tooth meshing force F is decomposed into the tooth plane coordinate system to obtain: Where: β is the angle between the gear meshing force F and the Y axis, S x is the half tooth thickness at point A on the tooth profile, h is the tooth height at the point of application, x is the X-axis coordinate of point A on the tooth profile, and M is the bending moment on the gear; Compression potential energy U of non-circular planetary gear a , bending potential energy U b , Hertzian contact strain energy U h , gear shear potential energy U s It can be expressed as: Where: F y is the component of F in the Y direction, F x is the component of F in the X direction, S x is half the tooth thickness at point A on the tooth profile, B is the tooth width; E is the elastic modulus; G is the shear modulus; k a is the axial compression stiffness; k b is the bending stiffness; k s is the shear stiffness, k h is the Hertzian stiffness; Substituting the tooth width formula into the equation, we get: Where: is the base circle tooth half angle of the equivalent circular gear, β is the angle between the meshing force F and the Y axis, β x is the pressure angle at point A on the tooth profile, ξ is the pitting coincidence coefficient, Q is the number of pitting points, r is the circular pitting radius, μ x is the mean value of the X coordinate of the center of the pitting point, σ x is the standard deviation of the X coordinate of the center of the pitting point, A=[γ x sinβ x -cosβ x ]cosβ,r fv 、r bv are the root circle radius and base circle radius of the equivalent circular gear respectively, ν is the Poisson's ratio, k f is the matrix stiffness, L * 、M * 、P * , Q * is a constant value, h f S is the tooth height at the intersection of the extended line of the force and the tooth centerline, f The tooth root is rounded and thick; The time-varying mesh stiffness of the sun gear-planet gear meshing pair in non-circular planetary gears can be divided into single-tooth mesh stiffness and double-tooth mesh stiffness. Since the two adjacent teeth of the sun gear are different, the i-th double-tooth meshing is different from the i+1-th double-tooth meshing, which is different from circular gears. The time-varying mesh stiffness of single and double teeth can be expressed as: Where: k i is the meshing stiffness of the i-th single tooth of the sun gear-planet gear meshing pair, k a1,i is the axial compressive stiffness of gear 1 in the sun gear-planet gear meshing pair, k a2,i is the axial compressive stiffness of the i-th gear 2 in the sun gear-planet gear meshing pair, k b1,i is the bending stiffness of the i-th gear 1 in the sun gear-planet gear meshing pair, k b2,i is the i-th bending stiffness of gear 2 in the sun gear-planet gear meshing pair, k f1,i is the base stiffness of gear 1 in the sun gear-planet gear meshing pair, k f2,i is the base stiffness of gear 2 in the sun gear-planet gear meshing pair, k s1,i is the i-th shear stiffness of gear 1 in the sun gear-planet gear meshing pair, k s2,i is the i-th shear stiffness of gear 2 in the sun gear-planet gear meshing pair, k h,i is the Hertzian stiffness of the sun gear-planet gear meshing pair, k mi is the meshing stiffness of the i-th double tooth of the sun gear-planet gear meshing pair; Step (4): Investigate the influence characteristics of random pitting factors on the meshing stiffness of non-circular gears; select the number of pitting Q and the mean value of the Y coordinate of the pitting point center μ y As the influencing parameters, the changing states of the meshing stiffness of non-circular planetary gears under different parameters are solved respectively.

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