Bevel gear pair time-varying meshing stiffness calculation method considering tooth surface pitting corrosion and gluing damage
By establishing a gear cantilever beam model and considering pitting and scuffing damage on the tooth surface, cutting it into an equivalent spur gear, calculating the oil film thickness and temperature safety factor, and correcting the stiffness calculation method, the problem of low accuracy of time-varying meshing stiffness of spur bevel gear pairs was solved, and more accurate prediction of meshing stiffness was achieved.
Patent Information
- Application Number
- CN202511932565.8
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-12-19
- Publication Date
- 2026-03-20
AI Technical Summary
Existing methods fail to accurately calculate the time-varying meshing stiffness of spur bevel gear pairs, neglecting the influence of pitting and scuffing damage on the contact state of the gear pair, resulting in low calculation accuracy.
A gear cantilever beam model was established, considering pitting and scuffing damage on the tooth surface. By cutting the teeth of the spur bevel gear into an equivalent spur gear, the oil film thickness and temperature safety factor were calculated. The bending, shearing, and radial compression stiffness were corrected. Combined with the lubrication state and contact stiffness, the normal contact stiffness of the meshing tooth surface was calculated.
The accuracy of time-varying meshing stiffness calculation for straight bevel gears has been improved, the influence of tooth surface damage on meshing stiffness has been considered, and the dynamic performance of gear transmission systems can be accurately predicted.
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Figure CN121706286A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of gear mechanics calculation, specifically to a method for calculating the time-varying meshing stiffness of a straight bevel gear pair that considers pitting and scuffing damage on the tooth surface. Background Technology
[0002] Because the contact ratio of meshing gears is not an integer, the number of gear teeth involved in meshing changes periodically over time during the meshing process. This time-varying effect in meshing stiffness is one of the main factors contributing to vibration and noise during gear meshing. The magnitude and variation law of the time-varying stiffness effect in gear transmission systems are closely related to the operational stability and reliability of the gear transmission. Therefore, accurately calculating the time-varying meshing stiffness of gear transmission systems is crucial for effectively predicting their dynamic performance.
[0003] Currently, the main methods for calculating the time-varying meshing stiffness of gear pairs include the standard method, analytical method, finite element method, and experimental method. The standard method is simple and fast, but has low accuracy and can only calculate the maximum meshing stiffness during single and double tooth meshing, failing to calculate precise values of meshing stiffness at different meshing positions. The finite element method offers high accuracy, but requires a long time to establish a finite element analysis model of the meshing gear pair, and its accuracy is closely related to the number and quality of the mesh, resulting in long modeling cycles, low computational efficiency, and poor convergence stability of the calculation results. The experimental method uses strain gauges to measure the elastic deformation of the tooth surface to invert the meshing stiffness, but is limited by high experimental equipment costs, complex test preparation, and unavoidable human error and instrument system error, leading to insufficient data repeatability and reliability.
[0004] In comparison, analytical methods exhibit significant advantages in computational efficiency, accuracy, and universality. Among them, the potential energy method based on the cantilever beam model has become the most widely used analytical method in engineering practice due to its clear physical meaning and outstanding computational accuracy. However, the potential energy method still uses an approximate cross-section of the gear in the calculation of the meshing stiffness of straight bevel gears, rather than the actual cross-sectional shape, which reduces the calculation accuracy of the time-varying meshing stiffness of straight bevel gears. In addition, existing methods ignore the influence of tooth surface damage such as pitting and scuffing on the contact state of the gear pair, and cannot accurately calculate the time-varying meshing stiffness of the gear pair under different lubrication conditions and tooth surface damage states such as pitting and scuffing, making it difficult to effectively predict the dynamic performance of the gear transmission system under real working conditions. Summary of the Invention
[0005] To address the problem that the current method for calculating the time-varying meshing stiffness of spur bevel gear pairs does not consider the actual cross-sectional shape of the gears and neglects the influence of pitting and scuffing damage on the contact state of the gear pairs, resulting in low accuracy in calculating the time-varying meshing stiffness of spur bevel gear pairs, this invention proposes a method for calculating the time-varying meshing stiffness of bevel gear pairs that considers pitting and scuffing damage.
[0006] The technical solution adopted in this invention is:
[0007] It includes the following steps:
[0008] S1. Establish a gear cantilever beam model, which includes the sliding friction between the straight bevel gear pairs.
[0009] S2. Solve for the meshing force expression at the meshing point of the straight bevel gear tooth surface based on the gear cantilever beam model;
[0010] S3. Cut the teeth of the spur bevel gear along the direction of the pitch cone generatrix. For equivalent spur gears with the same thickness, the time-varying meshing stiffness of a spur bevel gear pair is defined as the sum of the meshing stiffnesses of all equivalent spur gears.
[0011] S4. Calculate the pitch circle radius, addendum circle radius, dedendum circle radius, base circle radius, number of teeth, and module of each equivalent spur gear based on the geometric dimensional relationship of the spur bevel gear teeth.
[0012] S5. Calculate the moment of inertia and cross-sectional area of the equivalent spur gear tooth surface based on the geometric dimensional relationship of the spur bevel gear teeth.
[0013] S6. Establish a fractal characterization model for tooth surfaces with pitting and adhesive damage;
[0014] S7. Calculate the thickness of the elastohydrodynamic lubricating oil film for each equivalent spur gear linear contact using the Dowson-Higginson formula.
[0015] S8. Establish the Reynolds equation, oil film thickness equation, and load balance equation for the line contact of each equivalent spur gear tooth surface, and solve the oil film pressure of each equivalent spur gear tooth surface by simultaneously solving the Reynolds equation, oil film thickness equation, and load balance equation.
[0016] S9. Based on S3, obtain any set of equivalent spur gear pairs. According to the oil film thickness of each equivalent spur gear in the equivalent spur gear pair calculated in S7, calculate the oil film safety factor between the meshing tooth surfaces of the equivalent spur gear pair in combination with the profile arithmetic mean deviation of the surface roughness of the spur bevel gear.
[0017] Meanwhile, the tooth surface temperature safety factor is calculated based on the temperature at which thermal adhesion occurs on the tooth surface of the straight bevel gear and the maximum meshing temperature of the tooth surface;
[0018] Based on the oil film safety factor and the tooth surface temperature safety factor, determine whether there is pitting and thermal adhesion damage on the tooth surface of each equivalent spur gear during the meshing process of the equivalent spur gear pair;
[0019] S10. Based on the oil film thickness calculated in S7 and the oil film pressure calculated in S8, and combined with the basic parameters of the gear, calculate the normal oil film stiffness of the equivalent spur gear.
[0020] S11. Based on the fractal characterization model of the tooth surface established in S6, calculate the normal contact stiffness of the multi-ellipsoidal micro-protrusions on the equivalent spur gear tooth surface containing pitting and thermal adhesive damage.
[0021] S12. Based on S3, obtain any set of equivalent spur gear pairs, and determine the lubrication state of the equivalent spur gear pairs during meshing according to the oil film safety factor and tooth surface temperature safety factor calculated in S9. The lubrication state includes full film lubrication, mixed lubrication and boundary lubrication.
[0022] Based on the lubrication condition, the normal oil film stiffness of each equivalent spur gear in the equivalent spur gear pair calculated in S10, and the normal contact stiffness of the multiple ellipsoidal micro-protrusions on the tooth surface of each equivalent spur gear in the equivalent spur gear pair calculated in S11, the normal contact stiffness of the meshing tooth surface of the equivalent spur gear pair under conditions of no pitting and thermal scuffing damage, the normal contact stiffness of the meshing tooth surface of the equivalent spur gear pair under conditions of pitting but no thermal scuffing damage, and the normal contact stiffness of the meshing tooth surface of the equivalent spur gear pair under conditions of only thermal scuffing damage.
[0023] S13. Based on S5, correct the bending stiffness, shear stiffness and radial compressive stiffness of each equivalent spur gear in each equivalent spur gear pair.
[0024] The axial bending stiffness of each equivalent spur gear in each equivalent spur gear pair is calculated based on S2.
[0025] The elastic deformation stiffness of the gear matrix is calculated based on the theoretical expression of the elastic deformation stiffness of the gear matrix.
[0026] S14. Solve for the single tooth stiffness of each equivalent spur gear based on S12 and S13;
[0027] S15. Solve for the time-varying meshing stiffness of a spur bevel gear pair, considering pitting and thermal scuffing damage, based on the single-tooth stiffness of the equivalent spur gear.
[0028] The beneficial effects of this invention are as follows:
[0029] This invention establishes a cantilever beam model of a straight bevel gear, considering the influence of inter-tooth friction during gear meshing. The straight bevel gear is sliced to obtain an equivalent spur gear. The time-varying meshing stiffness of the straight bevel gear pair is defined as the sum of the meshing stiffnesses of all equivalent spur gears. The meshing stiffness of the equivalent spur gear is determined by calculating the forces acting on it, its geometric parameters, moment of inertia, cross-sectional area, oil film thickness, oil film pressure, normal oil film stiffness, and the normal contact stiffness of the multiple ellipsoidal micro-protrusions on the tooth surface of the equivalent spur gear, thus obtaining the time-varying meshing stiffness of the straight bevel gear pair.
[0030] This invention calculates the oil film safety factor and tooth surface temperature safety factor based on the oil film thickness of the equivalent spur gear, the temperature at which thermal scuffing occurs on the tooth surface, and the maximum meshing temperature of the tooth surface. This provides the criteria for determining whether pitting and thermal scuffing damage exist on the tooth surface of the spur bevel gear. Based on these criteria, the lubrication state during the meshing process of the equivalent spur gear pair is determined. According to the lubrication state, the normal oil film stiffness of the equivalent spur gear, and the normal contact stiffness of the multi-ellipsoidal micro-convexity, the normal contact stiffness of the meshing tooth surface is calculated under three states: no pitting and thermal scuffing damage, pitting but no thermal scuffing damage, and only thermal scuffing damage. This solves the problem in existing methods that neglect the influence of tooth surface damage such as pitting and scuffing on the contact state of the gear pair, and cannot accurately calculate the time-varying meshing stiffness of the spur bevel gear pair under different lubrication states and tooth surface damage conditions such as pitting and scuffing.
[0031] This invention considers the influence of axial force on the time-varying meshing stiffness of the gear pair during the meshing process of spur bevel gears. By adding axial bending potential energy to the traditional potential energy method, a calculation formula for the axial bending stiffness of the equivalent spur gear is derived, thus improving the calculation accuracy of the time-varying meshing stiffness of spur bevel gears. Simultaneously, the influence of the changes in the moment of inertia and cross-sectional area of any equivalent spur gear on the pitch cone generatrix of the spur bevel gear on the time-varying meshing stiffness is considered, thereby correcting the calculation expressions for bending stiffness, shear stiffness, and compressive stiffness in the traditional potential energy method, further improving the calculation accuracy of the time-varying meshing stiffness of spur bevel gears. Finally, by combining the bending stiffness, shear stiffness, radial compressive stiffness, axial bending stiffness, and elastic deformation stiffness of the base material of the equivalent spur gear with the normal contact stiffness of the meshing tooth surface under the three states, an analytical expression for the meshing stiffness of the equivalent spur gear is derived. Attached Figure Description
[0032] Figure 1 This is a force analysis diagram of the meshing point on the tooth surface of a straight bevel gear;
[0033] Figure 2 yes Figure 1 A side sectional view;
[0034] Figure 3 This is a diagram of a slice;
[0035] Figure 4 yes Figure 3 A side sectional view;
[0036] Figure 5 This is a diagram showing the geometric relationship between the teeth of a straight bevel gear.
[0037] Figure 6 This is a schematic diagram of the meshing force acting on the teeth of a straight bevel gear;
[0038] Figure 7 It is the projection of the meshing force onto the equivalent spur gear at the midpoint of the tooth segment;
[0039] Figure 8 This is a schematic diagram showing the geometric dimensions of the tooth segment; Detailed Implementation
[0040] Specific implementation method one: Combining Figures 1-8 This embodiment describes a method for calculating the time-varying meshing stiffness of a bevel gear pair, considering pitting and scuffing damage. The method includes the following steps:
[0041] S1. Establish a gear cantilever beam model, which includes the sliding friction between the straight bevel gear pairs.
[0042] S2. Solve for the expressions of the meshing forces at the meshing point of the spur bevel gear tooth surface in the tangential, radial, and axial directions based on the gear cantilever beam model:
[0043] (1)
[0044] in, The normal meshing force on the tooth surface, for Tangential force decomposed in the X direction, for Axial force decomposed in the Y direction, for Radial force decomposed in the Z direction, The normal pressure angle on the pitch circle of a straight bevel gear. The cone angle of the pitch cone line of a straight bevel gear. for and The combined force. For example... Figure 1 and Figure 2 As shown, where, This is the intersection of the axis of the spur bevel gear and the tip of the cone. Center of the equivalent gear on the large end face The reference point for the projection of the axis of the straight bevel gear onto a plane. The tooth tip cone angle, The pitch circle radius of the gear on the large end face. It is the root circle radius of the large end of the equivalent spur gear.
[0045] S3, such as Figure 3 and Figure 4 As shown, based on the slicing theory, the teeth of a straight bevel gear are cut along the generatrix of the pitch cone into... Given equivalent spur gears of the same thickness, the time-varying meshing stiffness of a spur bevel gear pair is defined as the sum of the meshing stiffnesses of all equivalent spur gears.
[0046] S4, such as Figure 5 As shown, since the tooth profile parameters of a spur bevel gear can be approximated by multiple equivalent spur gears, this invention calculates the pitch circle radius of each equivalent spur gear based on the geometric dimensional relationship of the spur bevel gear teeth. Tooth tip circle radius Root circle radius Base circle radius Number of teeth and modulus :
[0047] (2)
[0048] in, The pitch circle radius of the large end of the equivalent spur gear. The radius of the addendum circle at the large end of the equivalent spur gear is... The base circle radius of the equivalent spur gear's large end. This is the distance from the meshing point to the large end face of the equivalent spur gear. The tooth tip angle of a straight bevel gear. The root angle of a straight bevel gear. This refers to the tooth width of a spur bevel gear. The tooth base angle of a straight bevel gear can be calculated using the following formula:
[0049] (3)
[0050] in, This is the length of the generatrix of the pitch cone of the spur bevel gear.
[0051] Number of teeth of equivalent spur gear for:
[0052] (4)
[0053] in, The module of the equivalent spur gear is calculated using the following formula:
[0054] (5)
[0055] in, For the module of a straight bevel gear, This is the face width factor. . Figure 5 middle, Let be the center of the equivalent gear pitch circle at any point on the pitch cone of the bevel gear (corresponding to the meshing point). , , , , , For the positioning point, Let be the pitch circle radius at the meshing point of the bevel gear. Let be the pitch circle radius of the small end face of the bevel gear. This is the projection of the distance from the engagement point to the large end onto the axis. The equivalent gear pitch circle radius is the radius of any point (corresponding to the meshing point) on the pitch cone of the bevel gear.
[0056] S5, such as Figure 6 and Figure 7 As shown, based on the geometric dimensions of the spur bevel gear teeth, calculate the moment of inertia and cross-sectional area of the equivalent spur gear tooth surface:
[0057] (6)
[0058] in, For rotational inertia, The height of the trapezoidal section of the equivalent spur gear. Let be the distance from any point on the tooth profile of the large end face of the equivalent spur gear to the center line of the equivalent spur gear. The distance from any point on the tooth profile of the small end face of the equivalent spur gear to the center line of the equivalent spur gear can be expressed as:
[0059] (7)
[0060] in, It is the half-angle of the base tooth on the base circle radius of the equivalent spur gear. The meshing angle of the equivalent spur gear. The base circle radius corresponding to the large end face of the equivalent spur gear. The base circle radius corresponding to the small end face of the equivalent spur gear can be calculated using the following formula:
[0061] (8)
[0062] in, This represents the number of teeth at the cross section at the pitch circle of an equivalent spur gear. The thickness is the equivalent of a spur gear.
[0063] Figure 6 middle, Let be the base circle radius of the equivalent spur gear at the midpoint of the gear teeth. To determine the position of the moment of inertia of the gear tooth section. Figure 7 middle, This is the distance from the point of engagement to the tooth root. M represents the distance between any point of contact and the center line of the gear teeth. The resulting bending moment.
[0064] Depend on Figure 8 It can be seen that when dividing the bevel gear teeth into N equal parts, the larger the value of N, the smaller the stiffness change caused by the triangle, while also ensuring better accuracy in stiffness calculation. Therefore, to improve the calculation efficiency of bevel gear meshing stiffness, the change caused by the triangle is ignored. Figure 8 The stiffness change caused by the triangle formed by the three red dots, namely the stiffness change caused by the tooth tip angle and tooth base angle, is then in equation (6). .
[0065] (9)
[0066] in, It represents the cross-sectional area.
[0067] S6. Based on fractal theory and straight bevel gears, establish a fractal characterization model for tooth surfaces with pitting and scuffing damage, wherein the fractal dimension is... With characteristic coefficients All are derived from the arithmetic mean deviation of the gear surface roughness profile. Calculation. This invention improves upon existing gear surface fractal characterization models by superimposing the fractal curves of the gear tooth surface profile in the tooth width and tooth profile directions, thus constructing a tooth surface fractal characterization model that can simulate the characteristics of micro-pitting and scuffing damage on the tooth surface:
[0068] (10)
[0069] (11)
[0070] (12)
[0071] in, The tooth profile height, The sampling length is in the tooth profile direction. The sampling length is in the tooth width direction. Let fractal dimension be the number of teeth along the tooth profile. Let f be the fractal dimension in the tooth width direction. , or , represents the characteristic coefficient in the tooth profile direction. is the characteristic coefficient in the tooth width direction. , For the scaling parameters of the equivalent spur gear, , This is the scaling parameter in the tooth profile direction. This is the scaling parameter in the tooth width direction. The highest frequency index, , , This is the sampling cutoff length. The lowest frequency index, , It is a random phase, and its value range is... .
[0072] S7. Calculate the thickness of the elastohydrodynamic lubricating oil film for each equivalent spur gear using the Dowson-Higginson formula.
[0073] (13)
[0074] In the formula, The minimum oil film thickness for an equivalent spur gear. This refers to the pressure viscosity coefficient of the lubricating oil. The dynamic viscosity of the lubricating oil. The tooth surface entrainment speed, The equivalent radius of curvature of the master and driven equivalent spur gears. The sampling length is the equivalent spur gear. The modulus of elasticity is determined based on the gear material. This is the normal load borne by the tooth surface of the equivalent spur gear.
[0075] S8. Based on the elastohydrodynamic lubrication theory, establish the Reynolds equation, oil film thickness equation, and load balance equation for the line contact of each equivalent spur gear tooth surface. Solve the oil film pressure of each equivalent spur gear tooth surface by simultaneously solving the Reynolds equation, oil film thickness equation, and load balance equation.
[0076] 1) The Reynolds equation is:
[0077] (14)
[0078] in, For the density of lubricating oil, This is the equivalent oil film thickness on the spur gear surface. The viscosity of the lubricating oil. This is the equivalent oil film pressure on the spur gear surface. The coordinates of the direction of movement of the tooth surface meshing point are: The average entrainment speed of the tooth surface. , These are the tangential velocities (m / s) of the driving and driven equivalent spur gears along the tooth profile at the meshing point.
[0079] The boundary conditions for the Reynolds equation are: at the inlet of the lubrication region... At this point, the first derivative of the oil film pressure At the lubrication area outlet At this point, the oil film pressure gradient .
[0080] 2) The equation for oil film thickness is:
[0081] (15)
[0082] in, To account for the roughness of the oil film thickness, The initial center oil film thickness on the tooth surface. For elastic deformation, For surface roughness function, It is the equivalent radius of curvature.
[0083] 3) The load balance equation is:
[0084] (16)
[0085] in, This refers to the oil film pressure at the meshing point of the tooth surfaces.
[0086] S9. Based on S3, obtain any set of equivalent spur gear pairs. Calculate the oil film thickness of each equivalent spur gear in the equivalent spur gear pair according to S7, and combine this with the arithmetic mean deviation of the surface roughness of the spur bevel gear. Calculate the oil film safety factor between the meshing tooth surfaces of the equivalent spur gear pair;
[0087] Meanwhile, the tooth surface temperature safety factor is calculated based on the temperature at which thermal adhesion occurs on the tooth surface of the straight bevel gear and the maximum meshing temperature of the tooth surface;
[0088] The presence of pitting and thermal adhesion damage on the tooth surface of each equivalent spur gear during the meshing process is determined based on the oil film safety factor and the tooth surface temperature safety factor.
[0089] 1) There is no expression for pitting and thermal bonding damage:
[0090] (17)
[0091] in, For the oil film safety factor, , The arithmetic mean deviation of the surface roughness of the master and driven gears.
[0092] 2) The expression for pitting corrosion but no thermal bonding damage:
[0093] (18)
[0094] in, For the tooth surface temperature safety factor, This refers to the temperature at which thermal bonding damage occurs on the tooth surface. This is the maximum meshing temperature of the tooth surface. This is the minimum specified value for the safety factor of tooth surface temperature, under normal circumstances. In high reliability cases .
[0095] 3) The expression for only thermal bonding damage:
[0096] (19)
[0097] S10. Based on the oil film thickness calculated in S7 and the oil film pressure calculated in S8, and combined with the basic gear parameters, calculate the normal oil film stiffness of the equivalent spur gear. The basic gear parameters include the gear module, number of teeth, pressure angle, and pitch circle radius. The specific process is as follows:
[0098] Normal oil film stiffness Theoretical formula:
[0099] (20)
[0100] in, This represents the change in normal meshing force on the tooth surface. This represents the change in normal displacement of the tooth surface.
[0101] This invention combines mesh generation to improve the normal oil film stiffness at individual nodes on the mesh. for:
[0102] (twenty one)
[0103] in, For the first Change in normal load at each node For the first Change in normal displacement at each node For the first The width of the oil film at each node, For the first The width of the oil film at each node, For the first The axial coordinate position of each node. For the first The axial coordinate position of each node. For the first Oil film pressure at each node, For the first Oil film pressure at each node, The first The upper and lower bounds of the oil film thickness for each node grid length. For grid length, For the first The node and the first The change in oil film pressure at each node for The change in quantity.
[0104] At a single node, the total normal oil film stiffness of the contact area of the meshing tooth surface is:
[0105] (twenty two)
[0106] in, For the first Dimensionless oil film pressure at each node For the first Dimensionless oil film thickness at each node For maximum Hertz pressure, For contact half-width.
[0107] S11. Based on the fractal characterization model of the tooth surface established in S6, calculate the normal contact stiffness of each multi-ellipsoidal micro-protrusion on the equivalent spur gear tooth surface containing pitting and thermal adhesion damage. The specific process is as follows:
[0108] For the normal contact stiffness of multi-ellipsoidal micro-protrusions on gear surfaces containing pitting and thermal adhesion damage. The calculation can be performed by calculating the actual contact area of the damaged surface of the spur bevel gear during the fully elastic deformation stage, the fully plastic deformation stage, the first elastic-plastic deformation stage, and the second elastic-plastic deformation stage of the ellipsoidal micro-protrusion, and then determining the normal contact stiffness of the damaged surface of the spur bevel gear at different deformation stages.
[0109] According to contact mechanics theory:
[0110] a. The actual deformation of the ellipsoidal micro-protrusion on the damaged gear tooth surface Less than the critical elastic deformation of the ellipsoidal micro-convex body At that time, the ellipsoidal micro-protrusions on the damaged gear tooth surface are in the fully elastic deformation stage, and the normal contact stiffness of the multi-ellipsoidal micro-protrusions is... for:
[0111] (twenty three)
[0112] in, This represents the maximum contact area of a single ellipsoidal micro-protrusion. This refers to the critical contact area during the fully elastic deformation stage of a single ellipsoidal micro-convex body. Let be the contact area distribution function of the ellipsoidal micro-protrusions on the damaged gear tooth surface. is the contact coefficient of the ellipsoidal micro-convexity. , The expression is:
[0113] (twenty four)
[0114] (25)
[0115] in, and Let be the first and second complete elliptic integrals of the kind for an ellipsoidal microconvex body, respectively, and their expressions are:
[0116] (26)
[0117] (27)
[0118] in, The ellipticity of a single ellipsoidal micro-convex body, with a value range of... , The integral angle variable.
[0119] and The expressions are as follows:
[0120] (28)
[0121] (29)
[0122] in, This represents the contact area of a single ellipsoidal micro-protrusion.
[0123] b. When At that time, the ellipsoidal micro-protrusions on the damaged gear tooth surface are in the elastoplastic stage, and the elastoplastic deformation range is divided into two stages:
[0124] when At that time, the deformation of the ellipsoidal micro-protrusions on the damaged gear tooth surface is in the first elastoplastic deformation stage, and the normal contact stiffness of the multi-ellipsoidal micro-protrusions is... for:
[0125] (30)
[0126] in, , , and Based on the KE model, these are the contact coefficients for ductile metallic materials when the ellipsoidal micro-protrusions with rough surfaces are in the first elastoplastic deformation stage. , , , , The normal contact load is for the elastic deformation of a single ellipsoidal micro-convex body. This represents the critical elastoplastic contact area during the elastoplastic deformation stage of a single ellipsoidal micro-protrusion.
[0127] when At that time, the deformation of the ellipsoidal micro-protrusions on the damaged gear tooth surface is in the second elastoplastic deformation stage, and the normal contact stiffness of the multi-ellipsoidal micro-protrusions is... for:
[0128] (31)
[0129] in, , , and Based on the KE model, these are the contact coefficients for ductile metallic materials when the ellipsoidal micro-protrusions with rough surfaces are in the second elastoplastic deformation stage. , , , , It represents the critical plastic contact area when a single ellipsoidal micro-protrusion undergoes complete plastic deformation.
[0130] c. When When the deformation of the ellipsoidal micro-protrusions on the damaged gear tooth surface is in the fully plastic deformation stage, the normal contact stiffness of the multi-ellipsoidal micro-protrusions is... for:
[0131] (32)
[0132] Therefore, since equations (23), (30), and (31) all contain... Therefore, formula (29) is introduced into the tooth surface fractal characterization model (formula 10) established by S6 to obtain the normal contact stiffness of each multi-ellipsoidal micro-protrusion on the equivalent spur gear tooth surface containing pitting and thermal bonding damage. The calculation is represented as follows:
[0133] (33)
[0134] S12. Based on S3, obtain any set of equivalent spur gear pairs. According to the oil film safety factor and tooth surface temperature safety factor calculated in S9, determine the lubrication state of the equivalent spur gear pairs during meshing. The lubrication state includes full film lubrication, mixed lubrication and boundary lubrication.
[0135] Based on the lubrication condition, the normal oil film stiffness of each equivalent spur gear in the equivalent spur gear pair calculated in S10, and the normal contact stiffness of the multiple ellipsoidal micro-protrusions on the tooth surface of each equivalent spur gear in the equivalent spur gear pair calculated in S11, the normal contact stiffness of the meshing tooth surface of the equivalent spur gear pair under conditions of no pitting and thermal scuffing damage, the normal contact stiffness of the meshing tooth surface of the equivalent spur gear pair under conditions of pitting but no thermal scuffing damage, and the normal contact stiffness of the meshing tooth surface of the equivalent spur gear pair under conditions of only thermal scuffing damage are calculated. The specific process is as follows:
[0136] (1) When When there is no pitting or thermal adhesion damage on the meshing tooth surfaces of the equivalent spur gear pair, the lubrication state between the meshing tooth surfaces is full film lubrication. The meshing tooth surfaces are completely filled with oil film, there is no ellipsoidal micro-protrusion contact, and the load is entirely borne by the oil film. Therefore, the contact potential energy between the meshing tooth surfaces of the equivalent spur gear pair is... for:
[0137] (34)
[0138] In full film lubrication, the normal contact stiffness of the meshing tooth surface of the equivalent spur gear pair is equal to the normal oil film stiffness, i.e. .
[0139] (2) When ,and ≥1 (normal conditions) or (Under the specified conditions), when pitting occurs on the meshing tooth surfaces of the equivalent spur gear pair, but thermal scuffing damage does not occur, the lubrication state between the meshing tooth surfaces is mixed lubrication. The meshing tooth surfaces mainly have partial contact with ellipsoidal micro-protrusions, and the load is borne jointly by the oil film and the ellipsoidal micro-protrusions. Therefore, the contact potential energy is borne jointly by the oil film contact potential energy and the ellipsoidal micro-protrusion contact potential energy. Thus, the contact potential energy between the meshing tooth surfaces of the equivalent spur gear pair is... for:
[0140] (35)
[0141] In mixed lubrication, the normal contact stiffness of the meshing tooth surface of the equivalent spur gear pair is equal to the sum of the normal oil film stiffness and the normal contact stiffness of the multi-ellipsoidal micro-protrusions, i.e. .
[0142] (3) When ,and (Under normal conditions) or (Under the given conditions), when only thermal adhesion damage exists on the meshing tooth surfaces of the equivalent spur gear pair, the lubrication state between the meshing tooth surfaces is boundary lubrication. The oil film between the meshing tooth surfaces ruptures, and the ellipsoidal micro-protrusions are in direct contact. Almost all the load is borne by the ellipsoidal micro-protrusions. Therefore, the contact potential energy between the meshing tooth surfaces of the equivalent spur gear pair is... for:
[0143] (36)
[0144] Under boundary lubrication, the normal contact stiffness of the meshing tooth surface of the equivalent spur gear pair is equal to the normal contact stiffness of the multi-ellipsoidal micro-convexity, i.e. .
[0145] Therefore, the normal contact stiffness of the meshing tooth surface of an equivalent spur gear pair under three states—no pitting and thermal scuffing damage, pitting but no thermal scuffing damage, and only thermal scuffing damage—can be represented by a piecewise function:
[0146] (37)
[0147] Similarly, the normal contact stiffness of the meshing tooth surface of each equivalent spur gear pair is obtained. .
[0148] S13. A straight bevel gear can be considered as an approximate equivalent of multiple equivalent spur gears linearly distributed along the generatrix of a cone. Therefore, the total potential energy stored in a pair of meshing straight bevel gears can be calculated by summing the potential energies of the multiple equivalent spur gears linearly distributed along the generatrix of the cone. According to the principle of the potential energy method, the potential energy of each equivalent spur gear can be divided into six components: meshing tooth surface contact potential energy... Bending potential energy Shear potential Radial compressive potential energy , matrix potential energy and axial bending potential energy Based on the six potential energy components and the equivalent spur gear pitch circle radius obtained from S4. Tooth tip circle radius Root circle radius Base circle radius Number of teeth and modulus Calculate the normal contact stiffness of the meshing tooth surfaces of the meshing gears. Elastic deformation stiffness of the matrix and axial bending stiffness .
[0149] Based on S5, this invention considers the influence of the rotational inertia and cross-sectional area of the equivalent spur gear tooth surface on the meshing stiffness, and uses this to correct the calculation expressions for bending stiffness, shear stiffness, and compressive stiffness in the traditional potential energy method. The corrected bending stiffness of each equivalent spur gear in each equivalent spur gear pair is... Shear stiffness and radial compressive stiffness They are respectively:
[0150] (38)
[0151] (39)
[0152] (40)
[0153] in, The moment of inertia of the equivalent trapezoidal cross-section of a spur gear. It is an angular displacement. It is angular displacement The function, i.e. , The base circle radius of the equivalent spur gear. For the equivalent elastic modulus, The tooth root is a rounded half-angle. The angle between the normal meshing force of the gear and the direction perpendicular to the gear centerline is denoted as . Poisson's ratio, The area of the trapezoidal cross-section of the equivalent spur gear. It is angular displacement The function, i.e. .
[0154] This invention introduces the influence of the axial force generated during the meshing of straight bevel gears on the time-varying meshing stiffness of straight bevel gear pairs. By introducing axial bending potential energy, it improves the existing calculation method for the time-varying meshing stiffness of straight bevel gear pairs, further enhancing the solution accuracy. Based on the meshing force expression obtained in S2, the axial bending potential energy of each equivalent spur gear in each equivalent spur gear pair is calculated. :
[0155] (41)
[0156] in, Let be the axial bending stiffness of an equivalent spur gear. The radius of the tooth root circle, This is the distance between the meshing point of the gear teeth and the center of the equivalent spur gear. The torque applied as an axial force to the equivalent spur gear base can be expressed as:
[0157] (42)
[0158] in, The horizontal distance from the slice of the matrix to the center of the equivalent spur gear is given. Let be the horizontal distance from the gear meshing point to the center of the equivalent spur gear. Based on the properties of the involute curve... It can be represented as:
[0159] (43)
[0160] Finally, the formula for the axial bending stiffness of each equivalent spur gear in each equivalent spur gear pair is derived as follows:
[0161] (44)
[0162] in, The base circle radius of the gear. The helix angle of the gear. This represents the lateral displacement at the meshing point.
[0163] Besides tooth deformation, gear body deformation also affects gear stiffness, according to the theoretical expression for the elastic deformation stiffness of the gear base:
[0164] (45)
[0165] (46)
[0166] (47)
[0167] in, This represents the amount of tooth volume deformation. The force acting on the gear teeth. For the root radius, , , , Equation (48) is approximately approximated as:
[0168] (48)
[0169] in, , Let be the inner radius of the gear body. Representative coefficient , , , , These are fixed coefficients, and their values are shown in Table 1.
[0170] Table 1
[0171]
[0172] Therefore, the base elastic deformation stiffness of each set of equivalent spur gears is:
[0173] (49)
[0174] Substituting equation (47) into equation (49), we get:
[0175] (50)
[0176] (51)
[0177] The theory of the base elastic deformation stiffness of spur gears is now extended to the calculation of the base elastic deformation stiffness of spur bevel gears. However, for spur bevel gears, different gear geometries, such as the base circle radius, correspond to different meshing points. This will cause the bevel gear to exhibit variations along the tooth width direction. and The changes in parameters divide the matrix into equal parts along the tooth width direction. When the width of the equivalent spur gear is sufficiently small, the base elastic deformation stiffness of the equivalent spur gear is approximately calculated using the parameters at the midpoint of the equivalent spur gear.
[0178] Based on equation (50), the matrix elastic deformation stiffness of each equivalent spur gear in the equivalent spur gear pair can be obtained as follows:
[0179] (52)
[0180] in, This indicates the number of parts into which the matrix is divided along the tooth width direction. This represents the tooth width of the nth tooth segment. It can be represented as:
[0181] (53)
[0182] in, and Let each represent a dimension at the midpoint of the nth tooth segment, which can be expressed as:
[0183] (54)
[0184] in, This represents the root circle radius of the equivalent gear at the midpoint of the nth tooth segment.
[0185] S14. Based on the normal contact stiffness of the meshing tooth surface of the equivalent spur gear pair without pitting and thermal scuffing damage calculated in S12, the normal contact stiffness of the meshing tooth surface of the equivalent spur gear pair with pitting but without thermal scuffing damage, and the normal contact stiffness of the meshing tooth surface of the equivalent spur gear pair with only thermal scuffing damage, as well as the bending stiffness, shear stiffness, radial compression stiffness, axial bending stiffness, and matrix elastic deformation stiffness obtained in S13, solve for the single tooth stiffness of each equivalent spur gear.
[0186] (55)
[0187] S15. Based on the single-tooth stiffness of S14, the time-varying meshing stiffness of the spur bevel gear pair considering pitting and thermal scuffing damage is solved according to the overlap ratio:
[0188] Total potential energy stored in a pair of meshing teeth The contact potential energy can be derived from the normal contact energy of the meshing tooth surfaces. Bending potential energy Shear potential Radial compressive potential energy , matrix potential energy and axial bending potential energy Summing yields, i.e.
[0189] (56)
[0190] in, These represent the bending potential energy, shear potential energy, radial compression potential energy, matrix potential energy, and axial bending potential energy of the driven gear in a meshing bevel gear. These are the bending potential energy, shear potential energy, radial compression potential energy, base potential energy, and axial bending potential energy of the driving gear in a meshing bevel gear. This is the total meshing stiffness of a pair of meshing gear teeth, i.e.:
[0191] (57)
[0192] in, For the normal contact stiffness of the meshing tooth surface, These are the bending stiffness, shear stiffness, radial compressive stiffness, matrix elastic deformation stiffness, and axial bending stiffness of the driving gear in a pair of meshing gears. These are the bending stiffness, shear stiffness, radial compressive stiffness, matrix elastic deformation stiffness, and axial bending stiffness of the driven gear in a pair of meshing gears.
[0193] Based on equation (57), the time-varying meshing stiffness of the straight bevel gear pair considering pitting and thermal scuffing damage is:
[0194] (58)
[0195] in, The total number of equivalent spur gear pairs after cutting the master and driven gears. , For the first An equivalent spur gear pair For the first Normal contact stiffness of the meshing tooth surface of an equivalent spur gear pair For the first Bending stiffness of the driving gear in an equivalent spur gear pair For the first Shear stiffness of the driving gear in an equivalent spur gear pair For the first The compressive stiffness of the driving gear in an equivalent spur gear pair For the first The base elastic deformation stiffness of the driving gear in an equivalent spur gear pair For the first Axial bending stiffness of the driving gear in an equivalent spur gear pair For the first The bending stiffness of the driven gear in an equivalent spur gear pair For the first Shear stiffness of the driven gear in an equivalent spur gear pair For the first The compressive stiffness of the driven gear in an equivalent spur gear pair For the first The base elastic deformation stiffness of the driven gear in an equivalent spur gear pair For the first Axial bending stiffness of the driven gear in an equivalent spur gear pair.
[0196] The above examples of the present invention are merely illustrative of the computational model and process of the present invention, and are not intended to limit the implementation of the present invention. Those skilled in the art will recognize that other variations or modifications can be made based on the above description. It is impossible to exhaustively list all possible implementations here. Any obvious variations or modifications derived from the technical solutions of the present invention are still within the scope of protection of the present invention.
Claims
1. A method for calculating the time-varying meshing stiffness of bevel gear pairs considering pitting and scuffing damage, characterized in that: It includes the following steps: S1. Establish a gear cantilever beam model, which includes the sliding friction between the straight bevel gear pairs. S2. Solve for the meshing force expression at the meshing point of the straight bevel gear tooth surface based on the gear cantilever beam model; S3. Cut the teeth of the spur bevel gear along the direction of the pitch cone generatrix. For equivalent spur gears with the same thickness, the time-varying meshing stiffness of a spur bevel gear pair is defined as the sum of the meshing stiffnesses of all equivalent spur gears. S4. Calculate the pitch circle radius, addendum circle radius, dedendum circle radius, base circle radius, number of teeth, and module of each equivalent spur gear based on the geometric dimensional relationship of the spur bevel gear teeth. S5. Calculate the moment of inertia and cross-sectional area of the equivalent spur gear tooth surface based on the geometric dimensional relationship of the spur bevel gear teeth. S6. Establish a fractal characterization model for tooth surfaces with pitting and adhesive damage; S7. Calculate the thickness of the elastohydrodynamic lubricating oil film for each equivalent spur gear linear contact using the Dowson-Higginson formula. S8. Establish the Reynolds equation, oil film thickness equation, and load balance equation for the line contact of each equivalent spur gear tooth surface, and solve the oil film pressure of each equivalent spur gear tooth surface by simultaneously solving the Reynolds equation, oil film thickness equation, and load balance equation. S9. Based on S3, obtain any set of equivalent spur gear pairs. According to the oil film thickness of each equivalent spur gear in the equivalent spur gear pair calculated in S7, calculate the oil film safety factor between the meshing tooth surfaces of the equivalent spur gear pair in combination with the profile arithmetic mean deviation of the surface roughness of the spur bevel gear. Meanwhile, the tooth surface temperature safety factor is calculated based on the temperature at which thermal adhesion occurs on the tooth surface of the straight bevel gear and the maximum meshing temperature of the tooth surface; Based on the oil film safety factor and the tooth surface temperature safety factor, determine whether there is pitting and thermal adhesion damage on the tooth surface of each equivalent spur gear during the meshing process of the equivalent spur gear pair; S10. Based on the oil film thickness calculated in S7 and the oil film pressure calculated in S8, and combined with the basic parameters of the gear, calculate the normal oil film stiffness of the equivalent spur gear. S11. Based on the fractal characterization model of the tooth surface established in S6, calculate the normal contact stiffness of the multi-ellipsoidal micro-protrusions on the equivalent spur gear tooth surface containing pitting and thermal adhesive damage. S12. Based on S3, obtain any set of equivalent spur gear pairs, and determine the lubrication state of the equivalent spur gear pairs during meshing according to the oil film safety factor and tooth surface temperature safety factor calculated in S9. The lubrication state includes full film lubrication, mixed lubrication and boundary lubrication. Based on the lubrication condition, the normal oil film stiffness of each equivalent spur gear in the equivalent spur gear pair calculated in S10, and the normal contact stiffness of the multiple ellipsoidal micro-protrusions on the tooth surface of each equivalent spur gear in the equivalent spur gear pair calculated in S11, the normal contact stiffness of the meshing tooth surface of the equivalent spur gear pair under conditions of no pitting and thermal scuffing damage, the normal contact stiffness of the meshing tooth surface of the equivalent spur gear pair under conditions of pitting but no thermal scuffing damage, and the normal contact stiffness of the meshing tooth surface of the equivalent spur gear pair under conditions of only thermal scuffing damage. S13. Based on S5, correct the bending stiffness, shear stiffness and radial compressive stiffness of each equivalent spur gear in each equivalent spur gear pair. The axial bending stiffness of each equivalent spur gear in each equivalent spur gear pair is calculated based on S2. The elastic deformation stiffness of the gear matrix is calculated based on the theoretical expression of the elastic deformation stiffness of the gear matrix. S14. Solve for the single tooth stiffness of each equivalent spur gear based on S12 and S13; S15. Solve for the time-varying meshing stiffness of a spur bevel gear pair, considering pitting and thermal scuffing damage, based on the single-tooth stiffness of the equivalent spur gear.
2. The method for calculating the time-varying meshing stiffness of bevel gear pairs considering pitting and scuffing damage according to claim 1, characterized in that: The moment of inertia in S5 is: （1） in, For rotational inertia, The height of the trapezoidal section of the equivalent spur gear. Let be the distance from any point on the tooth profile of the large end face of the equivalent spur gear to the center line of the equivalent spur gear. It is the distance from any point on the tooth profile of the small end face of the equivalent spur gear to the center line of the equivalent spur gear; The cross-sectional area is: （2） in, For cross-sectional area, The thickness is the equivalent of a spur gear.
3. The method for calculating the time-varying meshing stiffness of bevel gear pairs considering pitting and scuffing damage according to claim 1, characterized in that: The specific process of S7 is as follows: （3） in, The minimum oil film thickness for an equivalent spur gear. This refers to the pressure viscosity coefficient of the lubricating oil. The dynamic viscosity of the lubricating oil. The tooth surface entrainment speed, The equivalent radius of curvature of the master and driven equivalent spur gears. The sampling length is the equivalent spur gear. To measure the overall elastic modulus, This is the normal load borne by the tooth surface of the equivalent spur gear.
4. The method for calculating the time-varying meshing stiffness of bevel gear pairs considering pitting and scuffing damage according to claim 1, characterized in that: The specific process of S8 is as follows: 1) The Reynolds equation is: （4） in, For the density of lubricating oil, This is the equivalent oil film thickness on the spur gear surface. The viscosity of the lubricating oil. This is the equivalent oil film pressure on the spur gear surface. The coordinates of the direction of movement of the tooth surface meshing point are: The average entrainment speed of the tooth surface. , These are the tangential velocities along the tooth profile of the driving and driven equivalent spur gears at the meshing point, respectively. The boundary conditions for the Reynolds equation are: at the inlet of the lubrication region... At this point, the first derivative of the oil film pressure At the lubrication area outlet At this point, the oil film pressure gradient ; 2) The equation for oil film thickness is: （5） in, To account for the roughness of the oil film thickness, The initial center oil film thickness on the tooth surface. For elastic deformation, For surface roughness function, The equivalent radius of curvature; 3) The load balance equation is: （6） in, This refers to the oil film pressure at the meshing point of the tooth surfaces.
5. The method for calculating the time-varying meshing stiffness of bevel gear pairs considering pitting and scuffing damage according to claim 1, characterized in that: The specific process of S9 is as follows: 1) No pitting or thermal bonding damage exists: （7） in, For the oil film safety factor, , The arithmetic mean deviation of the surface roughness of the master and driven wheels; 2) Pitting corrosion exists, but thermal bonding damage will not occur: （8） in, For the tooth surface temperature safety factor, This refers to the temperature at which thermal bonding damage occurs on the tooth surface. This is the maximum meshing temperature of the tooth surface. This is the minimum specified value for the safety factor of tooth surface temperature; 3) Only thermal bonding damage exists: （9）。 6. The method for calculating the time-varying meshing stiffness of bevel gear pairs considering pitting and scuffing damage according to claim 5, characterized in that: The specific process of S10 is as follows: By combining mesh generation, the normal oil film stiffness at individual nodes on the mesh is increased. for: （10） in, For the first Change in normal load at each node For the first Change in normal displacement at each node For the first The width of the oil film at each node, For the first The width of the oil film at each node, For the first The axial coordinate position of each node. For the first The axial coordinate position of each node. For the first Oil film pressure at each node, For the first Oil film pressure at each node, The first The upper and lower bounds of the oil film thickness for each node grid length. For grid length, For the first The node and the first The change in oil film pressure at each node for The change in; At a single node, the normal oil film stiffness of the equivalent spur gear is: （11） in, This refers to the tooth width of a spur bevel gear. For the first Dimensionless oil film pressure at each node For the first Dimensionless oil film thickness at each node For maximum Hertz pressure, For contact half-width.
7. The method for calculating the time-varying meshing stiffness of bevel gear pairs considering pitting and scuffing damage according to claim 1, characterized in that: The specific process of S11 is as follows: First, the fractal characterization model of the tooth surface with pitting and adhesive damage in S6 is as follows: （12） in, The tooth profile height, The sampling length is in the tooth profile direction. The sampling length is in the tooth width direction. Let f(x) be the fractal dimension in the tooth profile direction. Let f(x) be the fractal dimension in the tooth width direction. , For fractal dimension, , This represents the arithmetic mean deviation of the gear surface roughness profile. represents the characteristic coefficient in the tooth profile direction. is the characteristic coefficient in the tooth width direction. , For characteristic coefficients, , For the scaling parameters of the equivalent spur gear, , This is the scaling parameter in the tooth profile direction. This is the scaling parameter in the tooth width direction. The highest frequency index, The lowest frequency index, , It is a random phase, with a value range of 1. ; a. The actual deformation of the ellipsoidal micro-protrusion on the damaged gear tooth surface Less than the critical elastic deformation of the ellipsoidal micro-convex body At that time, the ellipsoidal micro-protrusions on the damaged gear tooth surface are in the fully elastic deformation stage, and the normal contact stiffness of the multi-ellipsoidal micro-protrusions is... for: （13） in, This represents the maximum contact area of a single ellipsoidal micro-protrusion. This refers to the critical contact area during the fully elastic deformation stage of a single ellipsoidal micro-convex body. Let be the contact area distribution function of the ellipsoidal micro-protrusions on the damaged gear tooth surface. The contact coefficient of the ellipsoidal micro-convexity. , The expression is: （14） （15） in, and These are the first and second complete elliptic integrals of the kind for an ellipsoidal microconvex body, respectively: （16） （17） in, The ellipticity of a single ellipsoidal micro-convex body, with a value range of... , The integral angle variable; and The expressions are as follows: （18） （19） in, The contact area of a single ellipsoidal micro-protrusion; b. When At that time, the ellipsoidal micro-protrusions on the damaged gear tooth surface are in the elastoplastic stage, and the elastoplastic deformation range is divided into two stages: when At that time, the deformation of the ellipsoidal micro-protrusions on the damaged gear tooth surface is in the first elastoplastic deformation stage, and the normal contact stiffness of the multi-ellipsoidal micro-protrusions is... for: （20） in, , , , , The normal contact load is for the elastic deformation of a single ellipsoidal micro-convex body. This represents the critical elastoplastic contact area during the elastoplastic deformation stage of a single ellipsoidal micro-protrusion. when At that time, the deformation of the ellipsoidal micro-protrusions on the damaged gear tooth surface is in the second elastoplastic deformation stage, and the normal contact stiffness of the multi-ellipsoidal micro-protrusions is... for: （21） in, , , , , The critical plastic contact area when a single ellipsoidal micro-protrusion undergoes complete plastic deformation; c. When When the deformation of the ellipsoidal micro-protrusions on the damaged gear tooth surface is in the fully plastic deformation stage, the normal contact stiffness of the multi-ellipsoidal micro-protrusions is... for: （22） Adding formula (19) to formula (12) yields the normal contact stiffness of each multi-ellipsoidal micro-protrusion on the equivalent spur gear tooth surface, which includes pitting and thermal bonding damage. : （23）。 8. The method for calculating the time-varying meshing stiffness of bevel gear pairs considering pitting and scuffing damage according to claim 7, characterized in that: The specific process of S12 is as follows: (1) When At this time, there is no pitting or thermal adhesion damage on the meshing tooth surfaces of the equivalent spur gear pair, and the lubrication state between the meshing tooth surfaces is full film lubrication. The normal contact stiffness of the meshing tooth surfaces of the equivalent spur gear pair is equal to the normal oil film stiffness, that is... ; (2) When ,and or At this time, pitting occurs on the meshing tooth surfaces of the equivalent spur gear pair, but thermal scuffing damage does not occur. The lubrication state between the meshing tooth surfaces is mixed lubrication. The normal contact stiffness of the meshing tooth surfaces of the equivalent spur gear pair is equal to the sum of the normal oil film stiffness and the normal contact stiffness of the multi-ellipsoidal micro-protrusions, that is... ; (3) When ,and or At this time, the meshing tooth surfaces of the equivalent spur gear pair only suffer from thermal scuffing damage, and the lubrication state between the meshing tooth surfaces is boundary lubrication. The normal contact stiffness of the meshing tooth surfaces of the equivalent spur gear pair is equal to the normal contact stiffness of the multi-ellipsoidal micro-convexity, i.e. ; Similarly, the normal contact stiffness of the meshing tooth surface of each equivalent spur gear pair is obtained. .
9. The method for calculating the time-varying meshing stiffness of bevel gear pairs considering pitting and scuffing damage according to claim 8, characterized in that: The specific process of S13 is as follows: Based on the rotational inertia and cross-sectional area of S5, the bending stiffness of each equivalent spur gear in each equivalent spur gear pair is corrected. Shear stiffness and radial compressive stiffness : （24） （25） （26） in, The base circle radius of the equivalent spur gear. For the equivalent elastic modulus, The tooth root is a rounded half-angle. The angle between the normal meshing force of the gear and the direction perpendicular to the gear centerline is denoted as . For angular displacement, The moment of inertia is the equivalent trapezoidal cross-section of a spur gear. It is angular displacement The function, Poisson's ratio, The area of the trapezoidal cross-section of the equivalent spur gear. It is angular displacement The function; Calculate the axial bending stiffness of each equivalent spur gear in each equivalent spur gear pair based on the meshing force expression obtained from S2. : （27） in, The radius of the tooth root circle, The base circle radius of the gear. The helix angle of the gear. This represents the lateral displacement at the engagement point. This is the distance between the meshing point of the tooth surface and the center of the equivalent spur gear; The elastic deformation stiffness of the gear matrix is calculated based on the theoretical expression for the elastic deformation stiffness of the gear matrix in each equivalent spur gear pair. : （28） in, This indicates the number of parts into which the matrix is divided along the tooth width direction. This represents the tooth width of the nth base tooth segment. for: （29） in, and Let represent the dimensional parameters at the midpoint of the nth base tooth segment. , , , Representative coefficient.
10. The method for calculating the time-varying meshing stiffness of bevel gear pairs considering pitting and scuffing damage according to claim 9, characterized in that: The specific process of S15 is as follows: The single-tooth stiffness of each equivalent spur gear in S14 is: （30） The single-tooth meshing stiffness of each equivalent spur gear pair is: （31） in, For equivalent spur gear pairs, This is the normal contact stiffness of the meshing tooth surface of the equivalent spur gear pair. These are the bending stiffness, shear stiffness, radial compressive stiffness, matrix elastic deformation stiffness, and axial bending stiffness of the driving gear in an equivalent spur gear pair. These are the bending stiffness, shear stiffness, radial compressive stiffness, base elastic deformation stiffness, and axial bending stiffness of the driven gear in an equivalent spur gear pair, respectively. Based on equation (30), the time-varying meshing stiffness of the straight bevel gear pair considering pitting and thermal scuffing damage is: （32） in, The total number of equivalent spur gear pairs after cutting the master and driven gears. , For the first An equivalent spur gear pair, For the first Normal contact stiffness of the meshing tooth surface of an equivalent spur gear pair For the first The bending stiffness of the driving gear in an equivalent spur gear pair For the first Shear stiffness of the driving gear in an equivalent spur gear pair For the first The compressive stiffness of the driving gear in an equivalent spur gear pair For the first The base elastic deformation stiffness of the driving gear in an equivalent spur gear pair For the first Axial bending stiffness of the driving gear in an equivalent spur gear pair For the first The bending stiffness of the driven gear in an equivalent spur gear pair For the first Shear stiffness of the driven gear in an equivalent spur gear pair For the first The compressive stiffness of the driven gear in an equivalent spur gear pair For the first The base elastic deformation stiffness of the driven gear in an equivalent spur gear pair For the first Axial bending stiffness of the driven gear in an equivalent spur gear pair.