A rolling bearing slip state analysis method considering raceway waviness

Abstract

The present application relates to the technical field of rolling bearing, in particular to a rolling bearing slip state analysis method considering raceway waviness, comprising the following steps: S1: establishing a waviness model of a wavy surface with time-varying curvature radius; S2: calculating time-varying displacement excitation based on contact geometry kinematics theory; S3: calculating time-varying stiffness based on Hertz contact theory and nonlinear characteristics of contact stiffness; S4: calculating displacement relationship and acting force of each bearing component; S5: calculating bearing nonlinear dynamics differential equation; S6: analyzing the influence of wavy amplitude and wavy order on slip through a bearing slip effect calculation method of time-varying displacement and time-varying stiffness coupling based on kinematics relationship, contact mechanics characteristics and multi-body dynamics; the present application provides a wavy bearing slip dynamics model based on time-varying displacement and time-varying stiffness coupling.

Description

Technical Field

[0001] This invention relates to the field of rolling bearing technology, and more specifically to a method for analyzing the slippage state of rolling bearings considering raceway waviness. Background Technology

[0002] Rolling bearings are widely used transmission components in mechanical systems, and their performance directly affects the operational stability and service life of equipment. However, due to the complex dynamic behavior of bearings during operation, especially the slippage between the rolling elements and the raceway, frictional losses, temperature rises, and fatigue damage can occur, significantly reducing the reliability of the bearings.

[0003] Raceway waviness is an unavoidable micro-geometric error in the manufacturing process of rolling bearings. It not only affects the contact state between the rolling elements and the raceway but also significantly influences slippage behavior. However, existing bearing dynamics analysis methods typically neglect the influence of raceway waviness, failing to accurately describe the slippage state and its patterns. Therefore, there is an urgent need for a rolling bearing slippage analysis method that considers raceway waviness to reveal the mechanism by which waviness affects slippage motion and provide a theoretical basis for bearing design and optimization. Summary of the Invention

[0004] In response to the problems raised in the background art, the present invention provides a method for analyzing the slippage state of rolling bearings that takes into account raceway waviness. The present invention will be further described below.

[0005] A method for analyzing the slippage state of rolling bearings considering raceway waviness includes the following steps:

[0006] S1: Based on the surface topography geometric modeling theory, a corrugated surface waviness model with a time-varying curvature radius is established. The raceway corrugation is assumed to be a sine wave, and its characteristics can be described by amplitude, wavelength and initial phase.

[0007] S2: Calculate time-varying displacement excitation based on contact geometry kinematics theory;

[0008] S3: Calculate time-varying stiffness based on Hertz contact theory and nonlinear characteristics of contact stiffness;

[0009] S4: Establish a bearing slippage dynamic model based on rigid body kinematics theory and Hertzian contact theory, and calculate the displacement relationship and force of each component of the bearing;

[0010] S5: Based on Newton's second law, establish a slip dynamic model of a waviness bearing coupled with time-varying displacement and time-varying stiffness, and calculate the nonlinear dynamic differential equation of the bearing;

[0011] S6: Based on kinematic relationships, contact mechanics characteristics, and multibody dynamics, this paper analyzes the influence of ripple amplitude and ripple order on slippage by using a bearing slippage effect calculation method that couples time-varying displacement and time-varying stiffness.

[0012] Preferably, in step S1, the basic parameters of the raceway waviness model include: inner raceway waviness amplitude, outer raceway waviness amplitude, waviness order, raceway angular position, inner raceway radius, outer raceway radius, pitch circle radius, number of rolling elements, revolution angle of rolling elements, and inner raceway rotation speed.

[0013] Preferably, step S1 specifically includes the following steps:

[0014] S101. Based on the basic parameters of the raceway waviness model, calculate the amplitude corresponding to any position of the inner and outer raceway waviness. , The corresponding function expression is as follows:

[0015]

[0016] In the formula, Nw is the wave number of the raceway, Πwis and Πwos are the maximum amplitude values ​​corresponding to the inner and outer raceway waviness, respectively, and λwis and λwos are the average lengths corresponding to a single wave in the inner and outer raceways, respectively.

[0017] S102. Based on the basic parameters of the raceway waviness model, calculate the average length corresponding to a single waviness of the inner and outer raceways. , The corresponding function expression is as follows:

[0018]

[0019] In the formula, φws is the radian angle corresponding to the s-th ripple relative to the geometric center of the bearing;

[0020] S103. Based on the basic parameters of the raceway waviness model, calculate the radian angle corresponding to the s-th waviness relative to the bearing geometric center. The corresponding function expression is as follows:

[0021]

[0022] S104. Based on the basic parameters of the raceway waviness model, calculate Lwis and Lwos at any position of the rolling element and raceway waviness. The corresponding function expressions are as follows:

[0023]

[0024] In the formula, θdij and θdoj represent the relative position angles between the j-th rolling element and the inner and outer raceways, respectively;

[0025] S105. Based on the basic parameters of the raceway waviness model, calculate the relative position angle between the j-th rolling element and the inner and outer raceways. , The corresponding function expression is as follows:

[0026]

[0027] In the formula, ωi is the rotational speed of the inner raceway; The number of rolling elements; Let be the revolution angle of the j-th rolling element.

[0028] Preferably, step S2 specifically includes the following steps:

[0029] S201. Using the basic parameters of the raceway waviness model described in S1, calculate the time-varying displacement excitations Пi and Пo of the rolling element relative to the inner and outer raceway waviness. The corresponding function expressions are as follows:

[0030]

[0031] In the formula, Nw is the raceway ripple number; Пwi and Пwo are the maximum ripple amplitudes of the inner and outer raceways, respectively; П0 is the initial ripple amplitude, П0=0; λwis and λwos are the average lengths of a single ripple in the inner and outer raceways, respectively; Lws represents any position, and the rolling element at any position in the inner and outer raceway ripples are represented by Lwis and Lwos, respectively.

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

[0033] S301. Let the rolling element be contact body a, and the inner and outer raceways be contact bodies b. Define the normal direction of the contact surface between the rolling element and the raceway as the first principal plane, denoted by subscript "1," and define the first principal plane as parallel to the radial plane of the bearing. Define the plane passing through the center of the ball and parallel to the axial direction as the second plane, denoted by subscript "2." Calculate the principal curvature of the contact pair between the ball bearing and the inner raceway. The corresponding function expression is as follows:

[0034]

[0035]

[0036]

[0037]

[0038] In the formula, The inner radius of the inner raceway; The inner raceway radius; Where is the radius of the rolling element;

[0039] S302. Similarly, calculate the principal curvature of the contact pair between the ball bearing and the outer raceway. The corresponding function expression is as follows:

[0040]

[0041]

[0042]

[0043]

[0044] In the formula, The inner radius of the inner raceway; The inner raceway radius; Where is the radius of the rolling element;

[0045] S303. Based on the principal curvature results of the contact pair between the ball bearing and the inner and outer ring raceways calculated in S301 and S302, obtain the sum of the curvatures of the contact pair between the rolling elements and the raceways, as follows:

[0046]

[0047]

[0048] S304. The load on the bearing under actual operating conditions is much smaller than the yield strength of its material. Therefore, it is assumed that the bearing does not undergo plastic deformation, and only the elastic deformation of the material is considered. Furthermore, it is assumed that the rolling elements and raceways are smooth elastic components. Based on Hertz's elastic contact theory, the contact area size between the ball bearing and the raceway (a), the contact deformation (b), and the contact pressure are... The calculation is as follows:

[0049]

[0050]

[0051]

[0052] In the formula, F is the external load on the bearing, k, Γ and Σ are the elliptic parameters of the contact area between the ball bearing and the raceway, the first kind of elliptic integral and the second kind of elliptic integral, respectively.

[0053] S305. Based on the calculation results of the sum of the curvatures of the contact pair between the rolling element and the raceway in S303, calculate the elliptic parameter k, the first elliptic integral Γ, and the second elliptic integral Σ of the contact area between the ball and the raceway. The function expressions are as follows:

[0054]

[0055]

[0056]

[0057] S306. Calculate the equivalent elastic modulus E' of the rolling elements and raceways. The function expression is as follows:

[0058]

[0059] In the formula, E1 and E2 are the elastic moduli of the rolling element and the raceway material, respectively, and ν1 and ν2 are the Poisson's ratios of the rolling element and the raceway material, respectively.

[0060] S307. Based on the calculation results of S303, S304, S305, and S306, the contact stiffness coefficient between the rolling element and the raceway can be obtained. The calculation is as follows:

[0061]

[0062] S308. The radius of curvature of the rolling element changes continuously at different positions of the waviness on the raceway surface. This causes the contact stiffness between the rolling element and the raceway to change over time. Therefore, when calculating the contact stiffness between the rolling element and the raceway, it is necessary to calculate the time-varying contact stiffness. For the radius of curvature formed by the waviness of the inner and outer raceways, the time-varying curvature value at any position Lws of the inner and outer raceways can be obtained by calculating the time-varying displacement excitations Пi and Пo of the rolling element relative to the waviness of the inner and outer raceways according to S201. The calculation is as follows:

[0063]

[0064] S309. Obtain the radius of curvature of the inner and outer raceway waviness of the rolling element at any position. , The function expression is as follows:

[0065]

[0066] S310. When the inner raceway has waviness, the curvature of the contact pair between the rolling element and the inner raceway can be expressed as:

[0067]

[0068]

[0069]

[0070]

[0071] S311. When the outer raceway has waviness, the curvature of the contact pair between the rolling element and the outer raceway can be expressed as:

[0072]

[0073]

[0074]

[0075]

[0076] S312. Based on S310 and S311, the contact pair curvatures between the rolling element and the inner and outer raceways when the raceway has waviness are obtained. Substituting these curvatures into S307, the time-varying contact stiffness coefficients Kwi and Kwo between the rolling element and the inner and outer raceways when the raceway has waviness can be calculated.

[0077] Preferably, step S4 specifically includes the following steps:

[0078] S401. Considering the radial clearance of the bearing, the contact deformation between the j-th rolling element and the inner ring can be expressed as:

[0079]

[0080] In the formula, Pd is the radial clearance of the rolling bearing; Пi represents the displacement excitation function caused by the waviness of the inner raceway, and Пi=0 when there is no waviness in the raceway; φj represents the position angle of the rolling element;

[0081] S402, the position angle of the rolling element can be expressed as:

[0082]

[0083] In the formula, Nb represents the number of rolling elements, and θmj is the revolution angle of the rollers;

[0084] S403. The contact deformation between the rolling element and the bearing outer ring is related to the radial displacement and radial clearance of the rolling element, and its expression is:

[0085]

[0086] In the formula, Пo represents the displacement excitation function caused by the waviness of the outer raceway. When there is no waviness in the raceway, Пo=0.

[0087] S404. The contact force between the rolling element and the raceway depends on the contact deformation, the stiffness of the material itself, and the contact form. Based on Hertz theory, the contact force between the rolling element and the inner and outer raceways can be expressed as:

[0088]

[0089] In the formula, Ki and Ko represent the contact stiffness coefficients between the rolling element and the inner and outer raceways, respectively, and their values ​​depend on the geometric and material parameters of the bearing; n represents the contact index, n=3 / 2 for ball bearings (point contact) and n=10 / 9 for roller bearings (line contact); λij and λoj represent the judgment coefficients for the contact between the rolling element and the inner and outer raceways, respectively.

[0090] S405, the expression for the judgment coefficient of contact between the rolling element and the inner and outer raceways is:

[0091]

[0092] S406. Under external load, the bearing undergoes geometric deformation, forming a load-bearing area and a non-load-bearing area. The rolling elements then periodically alternate between these areas. Due to factors such as load, friction, and resistance, the rotational speed of the rolling elements varies significantly. Therefore, the cage plays a crucial role in ensuring the rolling elements operate normally within the bearing without mutual interference. A linear spring is used to simulate the contact force and deformation relationship between the rolling elements and the cage. The interaction forces between the rolling elements and the front and rear walls of the cage pockets are calculated and can be expressed as:

[0093]

[0094]

[0095] In the formula, θc is the angle of rotation of the cage around its own geometric center, θmj is the revolution angle of the j-th rolling element; Kc is the stiffness coefficient of the contact between the rolling element and the front and rear walls of the cage, with a value of Kc=108N / m, Rm is the bearing pitch circle radius; cp is the pocket clearance between the rolling element and the cage, assuming cp=0mm;

[0096] S407. The movement of the rolling elements is driven by the frictional force of the inner and outer raceways. Therefore, friction plays a crucial role in the movement of each component of the bearing. Based on a semi-empirical calculation model, a four-parameter model is used to calculate the traction friction coefficient between the rolling bearing and the inner and outer raceways. The expression is as follows:

[0097]

[0098] In the formula, A, B, C, and D are the lubricating oil coefficients; Δv is the relative slippage speed between the rolling element and the inner and outer raceways.

[0099] S408, the expression for the sliding velocity of the j-th rolling element with respect to the inner and outer raceways is:

[0100]

[0101] In the formula, Rm is the bearing pitch circle radius; Rr is the rolling element radius; Let J be the rotational speed of the j-th rolling element; Let be the revolution speed of the j-th rolling element;

[0102] S409. Based on the contact force calculation results between the rolling element and the raceway in S404, the frictional force between the j-th rolling element and the inner and outer raceways is calculated as follows:

[0103]

[0104] S410. Based on the calculation results of the interaction forces between the rolling elements and the front and rear walls of the cage pockets according to S406, the rolling elements...

[0105] The frictional force between the moving body and the front and rear walls of the cage pocket is expressed as:

[0106]

[0107] In the formula, μc is the coefficient of sliding friction between the rolling element and the cage pocket, and μc = 0.3 is taken.

[0108] Preferably, step S5 specifically includes the following steps:

[0109] S501. According to Newton's second law, the differential equations of motion for the radial displacement, revolution, and rotation of the rolling element are as follows:

[0110]

[0111]

[0112]

[0113] In the formula, Jr represents the moment of inertia of the rolling element about its own axis; g is the acceleration due to gravity (g=9.8m / s2); mr is the mass of the rolling element; and c is the system damping coefficient. Let $j$ be the radial velocity of the j-th rolling element. Let j be the acceleration of the j-th rolling element; Let Fj be the orbital acceleration of the rolling element; Fwj be the centrifugal force of the j-th roller;

[0114] S502. According to Newton's second law, the differential equation of motion for the cage is:

[0115]

[0116] In the formula, Jc is the cage acceleration; Jc is the cage moment of inertia.

[0117] S503. According to Newton's second law, the differential equations of motion for the inner circle in the x and y directions are:

[0118]

[0119]

[0120] In the formula, mi is the inner ring mass; , These represent the displacement velocities of the bearing inner ring in the x and y directions, respectively. , These are the accelerations of the bearing inner ring in the x and y directions, respectively.

[0121] S504. The fixed-step fourth-order Runge-Kutta algorithm is used to solve the motion differential equations in S501, S502, and S503. The iteration is based on the time step, with a time step of Δt = 5 × 10⁻⁶ s. The iteration variables are transferred in a loop of “geometry (position) → mechanics (deformation-contact force) → motion (state update)”. Finally, through step-by-step calculation and storage, the dynamic response of the bearing in the entire time domain [0, T] is obtained.

[0122] Preferably, step S6 specifically includes the following steps: based on a corrugated surface waviness model with a time-varying curvature radius based on S1, S2, and S3, a waviness bearing slippage dynamic model based on the coupling of time-varying displacement and time-varying stiffness is established, and the motion differential equation is solved using the fixed-step fourth-order Runge-Kutta algorithm, and the influence of waviness amplitude and waviness order on the slippage effect is analyzed.

[0123] Beneficial effects: Compared with the prior art, the present invention:

[0124] This invention provides a waviness model for a corrugated surface with a time-varying radius of curvature;

[0125] The present invention also provides a method for calculating time-varying displacement excitation when the raceway surface has waviness;

[0126] The present invention also provides a method for calculating time-varying stiffness when the raceway surface has waviness;

[0127] The present invention also provides a slip dynamic model for a waviness bearing based on the coupling of time-varying displacement and time-varying stiffness;

[0128] This invention also provides a method for calculating bearing slippage effect based on the coupling of time-varying displacement and time-varying stiffness, and analyzes the influence of ripple amplitude and ripple order on slippage. Attached Figure Description

[0129] Figure 1 Flowchart of a method for analyzing the slippage state of rolling bearings considering raceway waviness according to the present invention;

[0130] Figure 2This invention provides a method for analyzing the slippage state of rolling bearings considering raceway waviness, with a raceway waviness model diagram.

[0131] Figure 3 This invention provides a simplified schematic diagram of a rolling bearing slippage state analysis method that considers raceway waviness.

[0132] Figure 4 This invention provides a method for analyzing the slippage state of rolling bearings, taking into account raceway waviness, and the positional relationship between the rolling elements and the raceway.

[0133] Figure 5 This invention provides a method for analyzing the slippage state of rolling bearings, taking into account raceway waviness, and the positional relationship between the rolling elements and the cage.

[0134] Figure 6 This invention provides a dynamic model and force diagram of the rolling element in a rolling bearing slippage analysis method considering raceway waviness.

[0135] Figure 7 This invention provides a dynamic model and cage force diagram for analyzing the slippage state of rolling bearings considering raceway waviness.

[0136] Figure 8 This invention provides a method for analyzing the slippage state of rolling bearings considering raceway waviness, with a dynamic model showing the force diagram of the raceway inside the model. Detailed Implementation

[0137] Next, we will combine the appendix Figure 1-8 A specific embodiment of the present invention will be described in detail below.

[0138] A method for analyzing the slippage state of rolling bearings considering raceway waviness includes the following steps:

[0139] S1: Based on the surface topography geometric modeling theory, a corrugated surface waviness model with a time-varying curvature radius is established. The raceway corrugation is assumed to be a sine wave, and its characteristics can be described by amplitude, wavelength and initial phase.

[0140] S2: Calculate time-varying displacement excitation based on contact geometry kinematics theory;

[0141] S3: Calculate time-varying stiffness based on Hertz contact theory and nonlinear characteristics of contact stiffness;

[0142] S4: Establish a bearing slippage dynamic model based on rigid body kinematics theory and Hertzian contact theory, and calculate the displacement relationship and force of each component of the bearing;

[0143] S5: Based on Newton's second law, establish a slip dynamic model of a waviness bearing coupled with time-varying displacement and time-varying stiffness, and calculate the bearing's nonlinear dynamic differential equation.

[0144] S6: Based on kinematic relationships, contact mechanics characteristics, and multibody dynamics, this paper analyzes the influence of ripple amplitude and ripple order on slippage by using a bearing slippage effect calculation method that couples time-varying displacement and time-varying stiffness.

[0145] As described above, the rolling bearing slippage state analysis method considering raceway waviness includes the following basic parameters in step S1: inner raceway waviness amplitude, outer raceway waviness amplitude, waviness order, raceway angular position, inner raceway radius, outer raceway radius, pitch circle radius, number of rolling elements, rolling element revolution angle, and inner raceway rotational speed.

[0146] The above-described method for analyzing the slippage state of rolling bearings considering raceway waviness includes the following steps:

[0147] S1 specifically includes the following steps:

[0148] S101. Based on the basic parameters of the raceway waviness model, calculate the amplitude corresponding to any position of the inner and outer raceway waviness. , The corresponding function expression is as follows:

[0149]

[0150] In the formula, Nw is the wave number of the raceway, Πwis and Πwos are the maximum amplitude values ​​corresponding to the inner and outer raceway waviness, respectively, and λwis and λwos are the average lengths corresponding to a single wave in the inner and outer raceways, respectively.

[0151] S102. Based on the basic parameters of the raceway waviness model, calculate the average length corresponding to a single waviness of the inner and outer raceways. , The corresponding function expression is as follows:

[0152]

[0153] In the formula, φws is the radian angle corresponding to the s-th ripple relative to the geometric center of the bearing.

[0154] S103. Based on the basic parameters of the raceway waviness model, calculate the radian angle corresponding to the s-th waviness relative to the bearing geometric center. The corresponding function expression is as follows:

[0155]

[0156] S104. Based on the basic parameters of the raceway waviness model, calculate Lwis and Lwos at any position of the rolling element and raceway waviness. The corresponding function expressions are as follows:

[0157]

[0158] In the formula, θdij and θdoj represent the relative position angles between the j-th rolling element and the inner and outer raceways, respectively.

[0159] S105. Based on the basic parameters of the raceway waviness model, calculate the relative position angle between the j-th rolling element and the inner and outer raceways. , The corresponding function expression is as follows:

[0160]

[0161] In the formula, ωi is the rotational speed of the inner raceway; The number of rolling elements; Let be the revolution angle of the j-th rolling element.

[0162] The above-described method for analyzing the slippage state of rolling bearings considering raceway waviness includes the following steps:

[0163] S2 specifically includes the following steps:

[0164] S201. Using the basic parameters of the raceway waviness model described in S1, calculate the time-varying displacement excitations Пi and Пo of the rolling element relative to the inner and outer raceway waviness. The corresponding function expressions are as follows:

[0165]

[0166] In the formula, Nw is the raceway ripple number; Пwi and Пwo are the maximum ripple amplitudes of the inner and outer raceways, respectively; П0 is the initial ripple amplitude, П0=0; λwis and λwos are the average lengths of a single ripple in the inner and outer raceways, respectively; Lws represents any position, and the rolling element at any position in the inner and outer raceway ripples are represented by Lwis and Lwos, respectively.

[0167] As described above, the method for analyzing the slippage state of rolling bearings considering raceway waviness includes the following specific steps in step S3:

[0168] S301. Let the rolling element be contact body a, and the inner and outer raceways be contact bodies b. Define the normal direction of the contact surface between the rolling element and the raceway as the first principal plane, denoted by subscript "1," and define the first principal plane as parallel to the radial plane of the bearing. Define the plane passing through the center of the ball and parallel to the axial direction as the second plane, denoted by subscript "2." Calculate the principal curvature of the contact pair between the ball bearing and the inner raceway. The corresponding function expression is as follows:

[0169]

[0170]

[0171]

[0172]

[0173] In the formula, The inner radius of the inner raceway; The inner raceway radius; Where is the radius of the rolling element.

[0174] S302. Similarly, calculate the principal curvature of the contact pair between the ball bearing and the outer raceway. The corresponding function expression is as follows:

[0175]

[0176]

[0177]

[0178]

[0179] In the formula, The inner radius of the inner raceway; The inner raceway radius; Where is the radius of the rolling element.

[0180] S303. Based on the principal curvature results of the contact pair between the ball bearing and the inner and outer ring raceways calculated in S301 and S302, obtain the sum of the curvatures of the contact pair between the rolling elements and the raceways, as follows:

[0181]

[0182]

[0183] S304. The load on the bearing under actual operating conditions is much smaller than the yield strength of its material. Therefore, it is assumed that the bearing does not undergo plastic deformation, and only the elastic deformation of the material is considered. Furthermore, it is assumed that the rolling elements and raceways are smooth elastic components. Based on Hertz's elastic contact theory, the contact area size between the ball bearing and the raceway (a), the contact deformation (b), and the contact pressure are... The calculation is as follows:

[0184]

[0185]

[0186]

[0187] In the formula, F is the external load on the bearing, k, Γ and Σ are the elliptic parameters of the contact area between the ball bearing and the raceway, the first kind of elliptic integral and the second kind of elliptic integral, respectively.

[0188] S305. Based on the calculation results of the sum of the curvatures of the contact pair between the rolling element and the raceway in S303, calculate the elliptic parameter k, the first elliptic integral Γ, and the second elliptic integral Σ of the contact area between the ball and the raceway. The function expressions are as follows:

[0189]

[0190]

[0191]

[0192] S306. Calculate the equivalent elastic modulus E' of the rolling elements and raceways. The function expression is as follows:

[0193]

[0194] In the formula, E1 and E2 are the elastic moduli of the rolling element and the raceway material, respectively, and ν1 and ν2 are the Poisson's ratios of the rolling element and the raceway material, respectively.

[0195] S307. Based on the calculation results of S303, S304, S305, and S306, the contact stiffness coefficient between the rolling element and the raceway can be obtained. The calculation is as follows:

[0196]

[0197] S308. The radius of curvature of the rolling element changes continuously at different positions of the waviness on the raceway surface. This causes the contact stiffness between the rolling element and the raceway to change over time. Therefore, when calculating the contact stiffness between the rolling element and the raceway, it is necessary to calculate the time-varying contact stiffness. For the radius of curvature formed by the waviness of the inner and outer raceways, the time-varying curvature value at any position Lws of the inner and outer raceways can be obtained by calculating the time-varying displacement excitations Пi and Пo of the rolling element relative to the waviness of the inner and outer raceways according to S201. The calculation is as follows:

[0198]

[0199] S309. Obtain the radius of curvature of the inner and outer raceway waviness of the rolling element at any position. , The function expression is as follows:

[0200]

[0201] S310. When the inner raceway has waviness, the curvature of the contact pair between the rolling element and the inner raceway can be expressed as:

[0202]

[0203]

[0204]

[0205]

[0206] S311. When the outer raceway has waviness, the curvature of the contact pair between the rolling element and the outer raceway can be expressed as:

[0207]

[0208]

[0209]

[0210]

[0211] S312. Based on S310 and S311, the contact pair curvatures between the rolling element and the inner and outer raceways when the raceway has waviness are obtained. Substituting these curvatures into S307, the time-varying contact stiffness coefficients Kwi and Kwo between the rolling element and the inner and outer raceways when the raceway has waviness can be calculated.

[0212] The above-described method for analyzing the slippage state of rolling bearings considering raceway waviness includes the following steps:

[0213] S4 specifically includes the following steps:

[0214] S401. Considering the radial clearance of the bearing, the contact deformation between the j-th rolling element and the inner ring can be expressed as:

[0215]

[0216] In the formula, Pd is the radial clearance of the rolling bearing; Пi represents the displacement excitation function caused by the waviness of the inner raceway, and Пi=0 when there is no waviness in the raceway; φj represents the position angle of the rolling element.

[0217] S402, the position angle of the rolling element can be expressed as:

[0218]

[0219] In the formula, Nb represents the number of rolling elements, and θmj is the revolution angle of the rollers.

[0220] S403. The contact deformation between the rolling element and the bearing outer ring is related to the radial displacement and radial clearance of the rolling element, and its expression is:

[0221]

[0222] In the formula, Пo represents the displacement excitation function caused by the waviness of the outer raceway. When there is no waviness in the raceway, Пo=0.

[0223] S404. The contact force between the rolling element and the raceway depends on the contact deformation, the stiffness of the material itself, and the contact form. Based on Hertz theory, the contact force between the rolling element and the inner and outer raceways can be expressed as:

[0224]

[0225] In the formula, Ki and Ko represent the contact stiffness coefficients between the rolling element and the inner and outer raceways, respectively, and their values ​​depend on the geometric and material parameters of the bearing; n represents the contact index, n=3 / 2 for ball bearings (point contact) and n=10 / 9 for roller bearings (line contact); λij and λoj represent the judgment coefficients for the contact between the rolling element and the inner and outer raceways, respectively.

[0226] S405, the expression for the judgment coefficient of contact between the rolling element and the inner and outer raceways is:

[0227]

[0228] S406. Under external load, the bearing undergoes geometric deformation, forming a load-bearing area and a non-load-bearing area. The rolling elements then periodically alternate between these areas. Due to factors such as load, friction, and resistance, the rotational speed of the rolling elements varies significantly. Therefore, the cage plays a crucial role in ensuring the rolling elements operate normally within the bearing without mutual interference. A linear spring is used to simulate the contact force and deformation relationship between the rolling elements and the cage. The interaction forces between the rolling elements and the front and rear walls of the cage pockets are calculated and can be expressed as:

[0229]

[0230]

[0231] In the formula, θc is the angle of rotation of the cage around its geometric center, θmj is the revolution angle of the j-th rolling element; Kc is the stiffness coefficient of the contact between the rolling element and the front and rear walls of the cage, with a value of Kc=108N / m, Rm is the bearing pitch circle radius; cp is the pocket clearance between the rolling element and the cage, assuming cp=0mm.

[0232] S407. The movement of the rolling elements is driven by the frictional force of the inner and outer raceways. Therefore, friction plays a crucial role in the movement of each component of the bearing. Based on a semi-empirical calculation model, a four-parameter model is used to calculate the traction friction coefficient between the rolling bearing and the inner and outer raceways. The expression is as follows:

[0233]

[0234] In the formula, A, B, C, and D are the lubricating oil coefficients; Δv is the relative slippage speed between the rolling element and the inner and outer raceways.

[0235] S408, the expression for the sliding velocity of the j-th rolling element with respect to the inner and outer raceways is:

[0236]

[0237] In the formula, Rm is the bearing pitch circle radius; Rr is the rolling element radius; Let J be the rotational speed of the j-th rolling element; Let be the revolution speed of the j-th rolling element.

[0238] S409. Based on the contact force calculation results between the rolling element and the raceway in S404, the frictional force between the j-th rolling element and the inner and outer raceways is calculated as follows:

[0239]

[0240] S410. Based on the calculation results of the interaction forces between the rolling elements and the front and rear walls of the cage pockets according to S406, the rolling elements...

[0241] The frictional force between the moving body and the front and rear walls of the cage pocket is expressed as:

[0242]

[0243] In the formula, μc is the coefficient of sliding friction between the rolling element and the cage pocket, and μc = 0.3 is taken.

[0244] The above-described method for analyzing the slippage state of rolling bearings considering raceway waviness includes the following steps:

[0245] S5 specifically includes the following steps:

[0246] S501. According to Newton's second law, the differential equations of motion for the radial displacement, revolution, and rotation of the rolling element are as follows:

[0247]

[0248]

[0249]

[0250] In the formula, Jr represents the moment of inertia of the rolling element about its own axis; g is the acceleration due to gravity (g=9.8m / s2); mr is the mass of the rolling element; and c is the system damping coefficient. Let $j$ be the radial velocity of the j-th rolling element. Let j be the acceleration of the j-th rolling element; Fj is the revolution acceleration of the rolling element; Fwj is the centrifugal force of the j-th roller.

[0251] S502. According to Newton's second law, the differential equation of motion for the cage is:

[0252]

[0253] In the formula, Jc is the cage acceleration; Jc is the cage moment of inertia.

[0254] S503. According to Newton's second law, the differential equations of motion for the inner circle in the x and y directions are:

[0255]

[0256]

[0257] In the formula, mi is the inner ring mass; , These represent the displacement velocities of the bearing inner ring in the x and y directions, respectively. , These are the accelerations of the bearing inner ring in the x and y directions, respectively.

[0258] S504. The fixed-step fourth-order Runge-Kutta algorithm is used to solve the motion differential equations in S501, S502, and S503. The iteration is based on the time step, with a time step of Δt = 5 × 10⁻⁶ s. The iteration variables are transferred in a loop of “geometry (position) → mechanics (deformation-contact force) → motion (state update)”. Finally, through step-by-step calculation and storage, the dynamic response of the bearing in the entire time domain [0, T] is obtained.

[0259] The above-described method for analyzing the slippage state of rolling bearings considering raceway waviness includes the following steps:

[0260] S6 specifically includes the following steps:

[0261] S601. Based on S1, S2, and S3, a corrugated surface waviness model with a time-varying curvature radius is established. Then, a waviness bearing slippage dynamic model based on the coupling of time-varying displacement and time-varying stiffness is established, and the motion differential equation is solved by the fixed-step fourth-order Runge-Kutta algorithm. The influence of waviness amplitude and waviness order on slippage effect is analyzed.

[0262] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for analyzing the slippage state of rolling bearings considering raceway waviness, characterized in that: Includes the following steps: S1: Based on the surface topography geometric modeling theory, a corrugated surface waviness model with a time-varying curvature radius is established. The raceway corrugation is assumed to be a sine wave, and its characteristics can be described by amplitude, wavelength and initial phase. S2: Calculate time-varying displacement excitation based on contact geometry kinematics theory; S3: Calculate time-varying stiffness based on Hertz contact theory and nonlinear characteristics of contact stiffness; S4: Establish a bearing slippage dynamic model based on rigid body kinematics theory and Hertzian contact theory, and calculate the displacement relationship and force of each component of the bearing; S5: Based on Newton's second law, establish a slip dynamic model of a waviness bearing coupled with time-varying displacement and time-varying stiffness, and calculate the nonlinear dynamic differential equation of the bearing; S6: Based on kinematic relationships, contact mechanics characteristics, and multibody dynamics, this paper analyzes the influence of ripple amplitude and ripple order on slippage by using a bearing slippage effect calculation method that couples time-varying displacement and time-varying stiffness.

2. The method for analyzing the slippage state of rolling bearings considering raceway waviness according to claim 1, characterized in that: In step S1, the basic parameters of the raceway waviness model include: inner raceway waviness amplitude, outer raceway waviness amplitude, waviness order, raceway angular position, inner raceway radius, outer raceway radius, pitch circle radius, number of rolling elements, revolution angle of rolling elements, and inner raceway rotation speed.

3. The method for analyzing the slippage state of rolling bearings considering raceway waviness according to claim 1, characterized in that: Step S1 specifically includes the following steps: S101. Based on the basic parameters of the raceway waviness model, calculate the amplitude corresponding to any position of the inner and outer raceway waviness. , The corresponding function expression is as follows: In the formula, Nw is the wave number of the raceway, Πwis and Πwos are the maximum amplitude values ​​corresponding to the inner and outer raceway wave degree, respectively, and λwis and λwos are the average lengths corresponding to a single wave in the inner and outer raceways, respectively. S102. Based on the basic parameters of the raceway waviness model, calculate the average length corresponding to a single waviness of the inner and outer raceways. , The corresponding function expression is as follows: In the formula, φws is the radian angle corresponding to the s-th ripple relative to the geometric center of the bearing; S103. Based on the basic parameters of the raceway waviness model, calculate the radian angle corresponding to the s-th waviness relative to the bearing geometric center. The corresponding function expression is as follows: S104. Based on the basic parameters of the raceway waviness model, calculate Lwis and Lwos at any position of the rolling element and raceway waviness. The corresponding function expressions are as follows: In the formula, θdij and θdoj represent the relative position angles between the j-th rolling element and the inner and outer raceways, respectively; S105. Based on the basic parameters of the raceway waviness model, calculate the relative position angle between the j-th rolling element and the inner and outer raceways. , The corresponding function expression is as follows: In the formula, ωi is the rotational speed of the inner raceway; The number of rolling elements; Let be the revolution angle of the j-th rolling element.

4. The method for analyzing the slippage state of rolling bearings considering raceway waviness according to claim 1, characterized in that: Step S2 specifically includes the following steps: S201. Using the basic parameters of the raceway waviness model described in S1, calculate the time-varying displacement excitations Пi and Пo of the rolling element relative to the inner and outer raceway waviness. The corresponding function expressions are as follows: In the formula, Nw is the raceway ripple number; Пwi and Пwo are the maximum ripple amplitudes of the inner and outer raceways, respectively; П0 is the initial ripple amplitude, П0=0; λwis and λwos are the average lengths of a single ripple in the inner and outer raceways, respectively; Lws represents any position, and the rolling element at any position in the inner and outer raceway ripples are represented by Lwis and Lwos, respectively.

5. The method for analyzing the slippage state of rolling bearings considering raceway waviness according to claim 4, characterized in that: Step S3 specifically includes the following steps: S301. Let the rolling element be contact body a, and the inner and outer raceways be contact bodies b. Define the normal direction of the contact surface between the rolling element and the raceway as the first principal plane, denoted by subscript "1," and define the first principal plane as parallel to the radial plane of the bearing. Define the plane passing through the center of the ball and parallel to the axial direction as the second plane, denoted by subscript "2." Calculate the principal curvature of the contact pair between the ball bearing and the inner raceway. The corresponding function expression is as follows: In the formula, The inner radius of the inner raceway; The inner raceway radius; Where is the radius of the rolling element; S302. Similarly, calculate the principal curvature of the contact pair between the ball bearing and the outer raceway. The corresponding function expression is as follows: In the formula, The inner radius of the inner raceway; The inner raceway radius; Where is the radius of the rolling element; S303. Based on the principal curvature results of the contact pair between the ball bearing and the inner and outer ring raceways calculated in S301 and S302, obtain the sum of the curvatures of the contact pair between the rolling elements and the raceways, as follows: S304. The load on the bearing under actual operating conditions is much smaller than the yield strength of its material. Therefore, it is assumed that the bearing does not undergo plastic deformation, and only the elastic deformation of the material is considered. Furthermore, it is assumed that the rolling elements and raceways are smooth elastic components. Based on Hertz's elastic contact theory, the contact area size between the ball bearing and the raceway (a), the contact deformation (b), and the contact pressure are... The calculation is as follows: In the formula, F is the external load on the bearing, k, Γ and Σ are the elliptic parameters of the contact area between the ball bearing and the raceway, the first kind of elliptic integral and the second kind of elliptic integral, respectively. S305. Based on the calculation results of the sum of the curvatures of the contact pair between the rolling element and the raceway in S303, calculate the elliptic parameter k, the first elliptic integral Γ, and the second elliptic integral Σ of the contact area between the ball and the raceway. The function expressions are as follows: S306. Calculate the equivalent elastic modulus E' of the rolling elements and raceways. The function expression is as follows: In the formula, E1 and E2 are the elastic moduli of the rolling element and the raceway material, respectively, and ν1 and ν2 are the Poisson's ratios of the rolling element and the raceway material, respectively. S307. Based on the calculation results of S303, S304, S305, and S306, the contact stiffness coefficient between the rolling element and the raceway can be obtained. The calculation is as follows: S308. The radius of curvature of the rolling element changes continuously at different positions of the waviness on the raceway surface. This causes the contact stiffness between the rolling element and the raceway to change over time. Therefore, when calculating the contact stiffness between the rolling element and the raceway, it is necessary to calculate the time-varying contact stiffness. For the radius of curvature formed by the waviness of the inner and outer raceways, the time-varying curvature value at any position Lws of the inner and outer raceways can be obtained by calculating the time-varying displacement excitations Пi and Пo of the rolling element relative to the waviness of the inner and outer raceways according to S201. The calculation is as follows: S309. Obtain the radius of curvature of the inner and outer raceway waviness of the rolling element at any position. , The function expression is as follows: S310. When the inner raceway has waviness, the curvature of the contact pair between the rolling element and the inner raceway can be expressed as: S311. When the outer raceway has waviness, the curvature of the contact pair between the rolling element and the outer raceway can be expressed as: S312. Based on S310 and S311, the contact pair curvatures between the rolling element and the inner and outer raceways when the raceway has waviness are obtained. Substituting these curvatures into S307, the time-varying contact stiffness coefficients Kwi and Kwo between the rolling element and the inner and outer raceways when the raceway has waviness can be calculated.

6. The method for analyzing the slippage state of rolling bearings considering raceway waviness according to claim 1, characterized in that: Step S4 specifically includes the following steps: S401. Considering the radial clearance of the bearing, the contact deformation between the j-th rolling element and the inner ring can be expressed as: In the formula, Pd is the radial clearance of the rolling bearing; Пi represents the displacement excitation function caused by the waviness of the inner raceway, and Пi=0 when there is no waviness in the raceway; φj represents the position angle of the rolling element; S402, the position angle of the rolling element can be expressed as: In the formula, Nb represents the number of rolling elements, and θmj is the revolution angle of the rollers; S403. The contact deformation between the rolling element and the bearing outer ring is related to the radial displacement and radial clearance of the rolling element, and its expression is: In the formula, Пo represents the displacement excitation function caused by the waviness of the outer raceway. When there is no waviness in the raceway, Пo=0. S404. The contact force between the rolling element and the raceway depends on the contact deformation, the stiffness of the material itself, and the contact form. Based on Hertz theory, the contact force between the rolling element and the inner and outer raceways can be expressed as: In the formula, Ki and Ko represent the contact stiffness coefficients between the rolling element and the inner and outer raceways, respectively, and their values ​​depend on the geometric and material parameters of the bearing; n represents the contact index, n=3 / 2 for ball bearings (point contact) and n=10 / 9 for roller bearings (line contact); λij and λoj represent the judgment coefficients for the contact between the rolling element and the inner and outer raceways, respectively. S405, the expression for the judgment coefficient of contact between the rolling element and the inner and outer raceways is: S406. Under external load, the bearing undergoes geometric deformation, forming a load-bearing area and a non-load-bearing area. The rolling elements then periodically alternate between these areas. Due to factors such as load, friction, and resistance, the rotational speed of the rolling elements varies significantly. Therefore, the cage plays a crucial role in ensuring the rolling elements operate normally within the bearing without mutual interference. A linear spring is used to simulate the contact force and deformation relationship between the rolling elements and the cage. The interaction forces between the rolling elements and the front and rear walls of the cage pockets are calculated and can be expressed as: In the formula, θc is the angle of rotation of the cage around its own geometric center, θmj is the revolution angle of the j-th rolling element; Kc is the stiffness coefficient of the contact between the rolling element and the front and rear walls of the cage, with a value of Kc=108N / m, Rm is the bearing pitch circle radius; cp is the pocket clearance between the rolling element and the cage, assuming cp=0mm; S407. The movement of the rolling elements is driven by the frictional force of the inner and outer raceways. Therefore, friction plays a crucial role in the movement of each component of the bearing. Based on a semi-empirical calculation model, a four-parameter model is used to calculate the traction friction coefficient between the rolling bearing and the inner and outer raceways. The expression is as follows: In the formula, A, B, C, and D are the lubricating oil coefficients; Δv is the relative slippage speed between the rolling element and the inner and outer raceways. S408, the expression for the sliding velocity of the j-th rolling element with respect to the inner and outer raceways is: In the formula, Rm is the bearing pitch circle radius; Rr is the rolling element radius; Let J be the rotational speed of the j-th rolling element; Let be the revolution speed of the j-th rolling element; S409. Based on the contact force calculation results between the rolling element and the raceway in S404, the frictional force between the j-th rolling element and the inner and outer raceways is calculated as follows: S410. Based on the calculation results of the interaction forces between the rolling elements and the front and rear walls of the cage pockets according to S406, the rolling elements... The frictional force between the moving body and the front and rear walls of the cage pocket is expressed as: In the formula, μc is the coefficient of sliding friction between the rolling element and the cage pocket, and μc = 0.3 is taken.

7. The method for analyzing the slippage state of rolling bearings considering raceway waviness according to claim 1, characterized in that: Step S5 specifically includes the following steps: S501. According to Newton's second law, the differential equations of motion for the radial displacement, revolution, and rotation of the rolling element are as follows: In the formula, Jr represents the moment of inertia of the rolling element about its own axis; g is the acceleration due to gravity (g=9.8m / s2); mr is the mass of the rolling element; and c is the system damping coefficient. Let $j$ be the radial velocity of the j-th rolling element. Let j be the acceleration of the j-th rolling element; Let Fj be the orbital acceleration of the rolling element; Fwj be the centrifugal force of the j-th roller; S502. According to Newton's second law, the differential equation of motion for the cage is: In the formula, Jc is the cage acceleration; Jc is the cage moment of inertia. S503. According to Newton's second law, the differential equations of motion for the inner circle in the x and y directions are: In the formula, mi is the inner ring mass; , These represent the displacement velocities of the bearing inner ring in the x and y directions, respectively. , These are the accelerations of the bearing inner ring in the x and y directions, respectively. S504. The fixed-step fourth-order Runge-Kutta algorithm is used to solve the motion differential equations in S501, S502, and S503. The iteration is periodic with a time step, and the time step is Δt = 5 × 10⁻⁶ s. The iteration variables are cyclically recursively transferred in the "geometry (position) → mechanics (deformation - contact force) → motion (state update)" chain. Finally, through step-by-step calculation and storage, the dynamic response of the bearing in the entire time domain [0, T] is obtained.

8. The method for analyzing the slippage state of rolling bearings considering raceway waviness according to claim 1, characterized in that: Step S6 specifically includes the following steps: Based on a corrugated surface waviness model with a time-varying curvature radius as described in S1, S2, and S3, a waviness bearing slippage dynamic model based on the coupling of time-varying displacement and time-varying stiffness is established, and the motion differential equation is solved using a fixed-step fourth-order Runge-Kutta algorithm. The influence of waviness amplitude and waviness order on the slippage effect is analyzed.

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