A bearing friction dynamics response analysis method integrating quasi-statics and dynamics

Abstract

The present invention discloses a bearing friction dynamics response analysis method integrating pseudo-statics and dynamics, which belongs to the field of friction dynamics of mechanical engineering technology. A reference coordinate system is established according to the installation direction and rotation direction of a solid-lubricated angular contact ball bearing, and a pseudo-static equilibrium equation and a dynamic differential equation of the solid-lubricated ball bearing with consistent displacement variables are established according to the reference coordinate system; according to the relationship between the contact angle and the contact parameter, the friction force and the friction torque in the pseudo-static equilibrium equation are calculated; according to the friction force and the friction torque, the pseudo-static equilibrium equation is solved to obtain a steady-state analysis result, and the steady-state analysis result is used as the initial value of the dynamic differential equation to solve the dynamic differential equation to obtain the friction dynamics response of the bearing. The present invention can effectively and accurately analyze the tribological and dynamic characteristics of the solid-lubricated ball bearing under stable or time-varying working conditions, and provide guidance for the tribological analysis and design of the solid-lubricated ball bearing.

Description

Technical Field

[0001] The invention belongs to the field of friction dynamics of mechanical engineering technology, and specifically relates to a bearing friction dynamics response analysis method that integrates pseudo-statics and dynamics. Background Art

[0002] Solid-lubricated ball bearings are widely used in high-end equipment such as aerospace and are key basic components of the equipment rotor system. Especially in liquid rocket turbopumps, facing extreme working conditions of high axial force, high radial force, high speed and solid lubrication, the tribological and vibration characteristics of solid-lubricated bearings such as friction slip, spin and vibration acceleration seriously affect the stable operation and service life of the engine rotor system. As the basis for the safe, stable and long-life operation of rotating equipment under extreme working conditions, the accurate analysis of the friction dynamics characteristics of solid-lubricated ball bearings is an important prerequisite for the research and design of system tribology and structure. The existing rolling bearing analysis technology mainly focuses on the modeling of oil-lubricated bearings, which is difficult to apply to solid-lubricated bearings; and the few solid-lubricated ball bearing analysis models are also limited to static load analysis, which cannot meet the needs of friction dynamics analysis under time-varying conditions such as variable working conditions / variable speeds in engineering practice, which brings certain challenges to the design and analysis of bearings in key components.

[0003] The existing analysis model for solid-lubricated ball bearings can only consider axial loads, which simplifies the model too much and has limited analysis capabilities. In addition, the current technology does not involve the dynamic modeling and analysis of solid-lubricated ball bearings. The main reason is that the model is relatively complex, and both modeling and solving are difficult. Therefore, it cannot meet the tribological and dynamic analysis of solid-lubricated ball bearings under multi-load and variable load conditions in actual engineering applications. Summary of the invention

[0004] The purpose of the present invention is to address the problem that the current rolling bearing analysis technology cannot meet the friction dynamics analysis of solid-lubricated ball bearings in engineering, and to propose a bearing friction dynamics response analysis method that integrates quasi-statics and dynamics. The method first establishes a reference coordinate system according to the installation direction and rotation direction of the ball bearing, and on this basis establishes quasi-statics and dynamics equations with consistent displacement variables; secondly, the friction force of the solid-lubricated angular contact ball bearing is quickly solved according to the relationship between the contact angle and the contact parameter, and then the force and torque in the quasi-statics and dynamics equations are calculated; finally, the quasi-static equilibrium equation is solved, and according to the consistency of the displacement variables in the quasi-statics equation and the dynamics equation, the steady-state solution is mapped to the initial value of the dynamic model and the dynamic differential equation is solved to obtain the friction dynamics response of the bearing.

[0005] The technical solutions adopted by the present invention to achieve the above-mentioned purpose are as follows:

[0006] A bearing friction dynamics response analysis method integrating pseudo-statics and dynamics includes the following steps:

[0007] Step 1: First, a reference coordinate system is established according to the installation direction and rotation direction of the solid-lubricated angular contact ball bearing, and a pseudo-static equilibrium equation and a dynamic differential equation of the solid-lubricated ball bearing with consistent displacement variables are established according to the reference coordinate system;

[0008] Step 2: According to the relationship between the contact angle and the contact parameter, the friction force and friction torque in the pseudo-static equilibrium equation are calculated;

[0009] Step 3: According to the friction force and friction torque, solve the quasi-static equilibrium equation established in step 1 to obtain the steady-state analysis results. Then use the steady-state analysis results as the initial values ​​of the dynamic differential equation, and use the fourth-order Runge-Kutta numerical iterative algorithm to solve the dynamic differential equation to obtain the friction dynamics response of the bearing.

[0010] Furthermore, according to the installation direction and rotation direction of the solid lubrication angular contact ball bearing, a reference coordinate system is established, including:

[0011] 1) Inertial coordinate system O-XYZ, set the positive direction of the Z axis to the positive direction of the angular contact bearing;

[0012] 2) Steel ball moving coordinate system o b -x b y b z b , where z b The y axis is consistent with the Z axis of the inertial coordinate system. b The axis points away from the axis;

[0013] 3) Local contact coordinate system between the steel ball and the inner and outer raceways i -x i y i z i and o -x o y o z o , where z i Axis and z o The axis is toward the center of the steel ball, x i and x o The axes are the rolling directions of the steel ball at the inner contact and outer contact points, respectively. It should be noted that the rolling directions of the steel ball relative to the inner and outer raceways are always opposite;

[0014] 4) Cage pocket coordinate system o c -x c y c z c , where o cLocated at the geometric center of each pocket of the cage, the three axes are parallel to the steel ball moving coordinate system.

[0015] Furthermore, the quasi-static equilibrium equation includes the quasi-static equilibrium equation of the steel ball and the equilibrium equation of the inner ring. The quasi-static equilibrium equation of the steel ball is:

[0016] F ball =0 (20)

[0017] In the formula, F ball The resultant force on the steel ball is expressed as follows:

[0018]

[0019] In the formula, and They represent the steel ball at x b ,y b and z b The net force in the direction and the force around x b ,y b 、z b Resultant moment about the shaft, T xi 、T yi and M i They are respectively the direction of the steel ball along x at the inner contact point i The longitudinal friction of the axis and the friction along the y i The lateral friction of the axis and the friction around z i Friction torque of the shaft, T xo 、T yo and M o They are respectively the direction of the steel ball along x at the external contact point o The longitudinal friction of the axis and the friction along the y o The lateral friction of the axis and the friction around z o Friction torque of the shaft, Q i , Q o are the contact forces between the steel ball and the inner and outer raceways, α i and α o are the inner contact angle and outer contact angle of the steel ball after deformation, F c is the centrifugal force of the steel ball, M gx 、M gy and M gz They are respectively the moment of the steel ball gyro in the follower coordinate system along x b ,y b and z b Directional component;

[0020] When the ball bearing is subjected to a pure axial force F a When , the equilibrium equation of the inner circle is:

[0021] F a =Nb (Q i sinα i +T yi cosα i ) (twenty two)

[0022] Where N b is the number of rolling elements;

[0023] When the inner ring is subjected to a 5-DOF load, the equilibrium equation of the inner ring is:

[0024] T bik F i b +F=0 (23)

[0025] Among them, T bik is the matrix that converts the force applied by the moving reference frame to the inertial reference frame; F i b is the force exerted by the ball on the inner raceway in the follower coordinate system; F is the 5-DOF load acting on the inner ring.

[0026] Furthermore, the matrix that converts the force applied by the moving reference frame into the inertial reference frame is as follows:

[0027]

[0028] Furthermore, the force exerted by the ball on the inner raceway in the follower coordinate system is as follows:

[0029]

[0030] Furthermore, the 5-DOF load acting on the inner ring is as follows:

[0031] F=(F x ,F y ,F z ,M x ,M y ) T (26)

[0032] in, is the orbital position of the kth ball, R i Indicates the pitch radius of the inner ring, F ix b 、F iy b and F iz b are the forces along the X, Y and Z directions applied by the i-th steel ball to the inner ring in the follower coordinate system, α ik is the internal contact angle of the i-th steel ball, F x 、F y and F zare the thrust loads of the inner ring along the X, Y and Z directions, M and M y are the torque loads of the inner ring around the X and Y axes respectively;

[0033]

[0034] Where M b and I b denote the mass and polar moment of inertia of the steel ball, θ bx ,θ bx and θ bx are the angles of rotation of the steel ball around the three coordinate axes of the moving coordinate system, C b It is the damping of the translational motion of the steel ball.

[0035] Furthermore, the differential equation of the translational motion of the inner circle is:

[0036]

[0037] In the formula, and Respectively represent x i The first and second derivatives of and Respectively represent y i The first and second derivatives of and Respectively represent z i The first and second derivatives of M i is the mass of the inner ring, K i and C i are the stiffness and damping of the inner ring translation motion, and are the resultant forces exerted by all the steel balls on the inner ring.

[0038] Furthermore, the differential equation of motion of the cage is:

[0039]

[0040] In the formula, I c is the cage polar inertia moment, θ c is the angular displacement of the cage.

[0041] Furthermore, the contact parameters include the inner and outer contact angles of the steel ball, the contact curvature and the contact curvature difference, wherein the inner and outer contact angles of the steel ball are:

[0042]

[0043]

[0044] In the formula, α i and αo are the inner contact angle and outer contact angle of the steel ball after deformation, A i 0 and A o 0 They represent the initial distances from the geometric center of the steel ball to the inner channel curvature center and the outer channel curvature center, α 0 is the initial contact angle, z i is the displacement of the inner ring after deformation due to load in the steel ball follower coordinate system along the z axis b Component of the axis, y b The steel ball along y b The amount of translational motion of the axis;

[0045] The contact curvature and the difference in contact curvature are expressed as:

[0046]

[0047]

[0048]

[0049]

[0050] In the formula, ∑ρ i ,∑ρ o 、F(ρ) i and F(ρ) o They represent the sum of the inner contact curvature and the outer contact curvature, the difference of the inner contact curvature and the difference of the outer contact curvature, respectively. i ,d o and r o are the curvature radii of the inner ring, outer ring and steel ball respectively.

[0051] Furthermore, the sliding rate and spin rate of the steel ball relative to the inner / outer raceway are:

[0052]

[0053]

[0054]

[0055]

[0056]

[0057]

[0058] In the formula, ξ ix , iy and are the longitudinal sliding rate, lateral sliding rate and spin rate of the steel ball relative to the inner raceway, ξ ox , oy and are the longitudinal sliding rate, lateral sliding rate and spin rate of the steel ball relative to the outer raceway, r b is the radius of the steel ball, r p is the pitch radius of the steel ball, δ i and δ o are the normal contact deformation of the steel ball at the inner contact and outer contact, ω i and ω o are the angular velocities of the inner ring and outer ring relative to the cage;

[0059] According to the longitudinal sliding rate, lateral sliding rate and spin rate of the steel ball relative to the outer raceway, the friction force and friction torque of the steel ball are obtained by FASTSIM algorithm.

[0060] Compared with the prior art, the present invention has the following beneficial effects:

[0061] In the present invention, the same displacement variable is used for the establishment of quasi-static and dynamic differential equations. On the one hand, it is convenient for technicians to select models according to needs in actual engineering applications. On the other hand, it also enables the solution of quasi-statics to be easily converted into the initial value of the dynamic equation, thereby improving the efficiency of solving the dynamic model; the contact angle of the rolling body is represented by a consistent displacement variable, and the contact parameters corresponding to different contact angles are calculated in advance according to the corresponding relationship between the contact angle and the contact parameter. In the process of solving the equation, the contact force and friction force can be directly calculated quickly by interpolation, which can save a lot of calculation time; finally, the analysis of the friction dynamic response of the bearing is realized. The present invention can effectively and accurately analyze the tribological and dynamic characteristics of solid-lubricated ball bearings under stable or time-varying conditions, and provide guidance for the tribological analysis and design of solid-lubricated ball bearings.

[0062] Furthermore, the friction force exerted on the steel balls in the solid-lubricated ball bearings is mainly rolling friction. Different from the existing dynamic models based on oil lubrication, the present invention introduces the FASTSIM algorithm of rolling contact friction to calculate the friction force and friction torque exerted on the steel balls. The proposed dynamic model has higher analysis accuracy for solid-lubricated ball bearings.

[0063] Furthermore, the proposed quasi-static model and dynamic model can simultaneously consider the loading conditions of axial force, radial force and overturning moment, and the analysis scope is wider. The proposed quasi-static model load iteration solution framework also ensures that the model can effectively converge under complex working conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Figure 1The reference coordinate system used to establish the equilibrium equation; (a) is the inertial coordinate system, (b) is the steel ball follower coordinate system, and (c) is the cage contact coordinate system;

[0065] Figure 2 Schematic diagram of the forces on the inner ring and the steel ball; (a) is a schematic diagram of the forces on the inner ring, and (b) is a schematic diagram of the forces on the steel ball;

[0066] Figure 3 Schematic diagram of the contact area; (a) is a description of the contact area, and (b) is the finite element mesh division of the contact area when the rolling contact FASTSIM algorithm is used;

[0067] Figure 4 A flow chart of the comprehensive solution strategy for the quasi-static model;

[0068] Figure 5 The steady-state results of contact force of quasi-static analysis and the comparison with the Jones-Harris classic method; (a) is the contact force between the steel ball and the inner raceway (N), and (b) is the contact force between the steel ball and the outer raceway (N).

[0069] Figure 6 are the vibration accelerations of the inner ring in the X and Y directions obtained from the dynamic analysis; where (a) is the vibration acceleration of the inner ring in the X direction, and (b) is the vibration acceleration of the inner ring in the Y direction.

[0070] Figure 7 The dynamic response of the steel ball angular velocity and the calculation result of the dynamic friction torque obtained by dynamic analysis. Among them, (a) is the dynamic response of the steel ball angular velocity, and (b) is the calculation result of the dynamic friction torque. DETAILED DESCRIPTION

[0071] In order to more clearly describe the specific implementation process of the present invention, the implementation process is described in detail below with reference to the accompanying drawings.

[0072] A bearing friction dynamics response analysis method integrating pseudo-statics and dynamics of the present invention comprises the following steps:

[0073] Step 1: First, a reference coordinate system is established according to the installation direction and rotation direction of the ball bearing. On this basis, the quasi-static equilibrium equation and dynamic differential equation of the solid-lubricated ball bearing with consistent displacement variables are established.

[0074] According to the installation direction and rotation direction of the solid lubricated angular contact ball bearing, establish Figure 1 The four reference coordinate systems shown in (a), (b) and (c) include:

[0075] 1) Inertial coordinate system O-XYZ, set the positive direction of the Z axis to the positive direction of the angular contact bearing;

[0076] 2) Steel ball moving coordinate system o b -x b y b z b , where z b The y axis is consistent with the Z axis of the inertial coordinate system. b The axis points away from the axis;

[0077] 3) Local contact coordinate system between the steel ball and the inner and outer raceways i -x i y i z i and o -x o y o z o , where z i Axis and z o The axis is toward the center of the steel ball, x i and x o The axes are the rolling directions of the steel ball at the inner contact and outer contact points, respectively. It should be noted that the rolling directions of the steel ball relative to the inner and outer raceways are always opposite;

[0078] 4) Cage pocket coordinate system o c -x c y c z c , where o c Located at the geometric center of each pocket of the cage, the three axes are parallel to the steel ball moving coordinate system.

[0079] Based on the inertial coordinate system and the follower coordinate system, the displacement variables of the steel ball and the inner ring are defined as the unified variables of the pseudo-static and dynamic models. b Axis and z b The translational motion variable y of the axis b ,z b And around z b The angular velocity variable of the axis rotation ω bm ; Define the three angular velocity variables of the steel ball using the steel ball following coordinate system as the rectangular coordinate system: b Axis angular velocity ω bx , around y b Axis angular velocity ω by and around z b Axis angular velocity ω bz Define the inner circle's translational motion variable x along the X axis in the inertial coordinate system i , translational motion variable y along the Y axis i and the translational motion variable z along the Z axis i , and the rotation angle variable θ around the X axisix and the rotation angle variable θ around the Y axis iy .

[0080] The steel ball is kept in a state of force equilibrium by contact force, friction force, centrifugal force, friction torque and gyroscopic torque, and these forces and torques can be calculated from the variables defined above. Figure 2 In the force diagrams shown in (a) and (b), the quasi-static equilibrium equation of the steel ball is expressed as:

[0081] F ball =0 (20)

[0082] In the formula, F ball The resultant force on the steel ball is expressed as follows:

[0083]

[0084] In the formula, and They represent the steel ball at x b ,y b and z b The net force in the direction and the force around x b ,y b 、z b Resultant moment about the shaft, T xi 、T yi and M i They are respectively the direction of the steel ball along x at the inner contact point i The longitudinal friction of the axis and the friction along the y i The lateral friction of the axis and the friction around z i Friction torque of the shaft, T xo 、T yo and M o They are respectively the direction of the steel ball along x at the external contact point o The longitudinal friction of the axis and the friction along the y o The lateral friction of the axis and the friction around z o Friction torque of the shaft, Q i , Q o are the contact forces between the steel ball and the inner and outer raceways, α i and α o are the inner contact angle and outer contact angle of the steel ball after deformation, F c is the centrifugal force of the steel ball, M gx 、M gy and M gz They are respectively the moment of the steel ball gyro in the follower coordinate system along x b ,y b and z b Direction component.

[0085] When the ball bearing is subjected to a pure axial force Fa , the inner ring will only have axial displacement, so the inner ring has only one equilibrium equation:

[0086] F a =N b (Q i sinα i +T yi cosα i ) (twenty two)

[0087] Where N b is the number of rolling elements. If the inner ring is subjected to a 5-DOF load, the inner ring will move in five directions (x, y, z, θ) along the coordinate axis. x ,θ y ) move, such as Figure 2 As shown in (a) and (b), the equilibrium equation of the inner ring with 5 degrees of freedom can be expressed in vector form:

[0088] T bik F i b +F=0 (23)

[0089] Among them, T bik is the matrix that converts the force applied by the moving reference frame to the inertial reference frame; F i b is the force of the ball on the inner raceway in the follower coordinate system. F is the 5-DOF load acting on the inner ring. These vectors are expressed as follows:

[0090]

[0091]

[0092] F=(F x ,F y ,F z ,M x ,M y ) T (26)

[0093] in, is the orbital position of the kth ball, R i Indicates the pitch radius of the inner ring, F ix b 、F iy b and F iz b are the forces along the X, Y and Z directions applied by the i-th steel ball to the inner ring in the follower coordinate system, α ik is the internal contact angle of the i-th steel ball, F x 、F y and F zare the thrust loads of the inner ring along the X, Y and Z directions, M 和 M y are the torque loads on the inner ring about the X and Y axes respectively.

[0094] The above has completed the quasi-static modeling of the bearing, and the next step is to establish the dynamic equation.

[0095] In the dynamic equation, the outer ring of the bearing is fixed, the inner ring is simplified to two degrees of freedom, namely the translational motion in the X and Y directions, and the cage is simplified to have one degree of rotational freedom, such as Figure 1 As shown in (c) in the figure. Since the cage pocket has a certain gap, the elastic force is only generated when the steel ball contacts the pocket wall. Assume that the rotational displacement of a steel ball is θ bm , the corresponding rotational displacement of the cage pocket is θ c , the angle difference between the two is Δθ. Based on the small deformation assumption, the contact force on the cage pocket from the steel ball is:

[0096]

[0097] In the formula, K is the contact force from the steel ball on the kth pocket of the cage, cb and C cb are the contact stiffness and damping between the steel ball and the cage pocket, r p is the bearing pitch radius, θ cl is the cage pocket clearance, which is given by the bearing parameters. Then the cage is subjected to the total driving torque M from the steel ball c for:

[0098]

[0099] Where N b is the number of steel balls, F cb k is the contact force on the kth pocket, and k is the number of contact forces on the pocket. Based on the above interaction forces and moments, the dynamic differential equation of the steel ball is established as follows:

[0100]

[0101] Where M b and I b denote the mass and polar moment of inertia of the steel ball, θ bx ,θ bx and θ bx are the angles of rotation of the steel ball around the three coordinate axes of the moving coordinate system, C b is the translational motion damping of the steel ball, which is given by the bearing parameters. The differential equation for the inner ring translational motion is:

[0102]

[0103] In the formula, and Respectively represent x i The first and second derivatives of and Similarly, M i is the mass of the inner ring, K i and C i are the stiffness and damping of the inner ring translation motion, both given by the bearing parameters, and are the combined forces exerted by all the steel balls on the inner ring, which can be calculated according to formula (21). The main function of the cage is to constrain the circumferential position of each steel ball and reflect the acceleration process of the bearing. It has one rotational degree of freedom, and the differential equation of motion is:

[0104]

[0105] In the formula, I c is the cage polar inertia moment, θ c is the angular displacement of the cage.

[0106] Step 2: Based on the relationship between the contact angle and the contact parameters, the friction force of the solid lubrication angular contact ball bearing is quickly solved, and then the forces and moments in the pseudo-static and dynamic equations are calculated.

[0107] Based on the 11 variables of the steel ball and the inner ring proposed in step 1, the contact angle is derived, and the contact parameters are calculated based on the contact angle. The inner and outer contact angles of the steel ball are expressed as follows based on the kinematic displacement relationship:

[0108]

[0109]

[0110] In the formula, α i and α o are the inner contact angle and outer contact angle of the steel ball after deformation, A i 0 and A o 0 They represent the initial distances from the geometric center of the steel ball to the inner channel curvature center and the outer channel curvature center, α 0 is the initial contact angle, z i is the displacement of the inner ring after deformation due to load in the steel ball follower coordinate system along the z axis b Component of the axis, y b The steel ball along y b The amount of translational motion of the axis.

[0111] The contact angle is closely related to the major semi-axis a, minor semi-axis b, contact stiffness κ, and ellipticity k of the contact spot, and the calculation formulas are:

[0112] a=[6k 2 εRQ / (πE')] 1 / 3 (34)

[0113] b=[6εRQ / (πkE')] 1 / 3 (35)

[0114]

[0115] Where ε is the second kind of elliptic integral of ellipticity k, Q = Q i or Q o , Q is the contact force, E' is the equivalent elastic modulus, R = R i or R o , is the equivalent radius of curvature, expressed as follows:

[0116]

[0117] In the formula, ρ b , ir and ρ or are the curvatures of the steel ball and the inner and outer rings respectively. The subscripts “1” and “2” represent the longitudinal and transverse directions along the rolling direction respectively.

[0118] For the ellipticity k, it can be iteratively calculated using the following formula:

[0119]

[0120] In the formula, n and (n+1) represent the number of iterations, F n is the first kind elliptic integral of k, F(ρ) is the curvature difference, ε n The contact curvature and the curvature difference are expressed as:

[0121]

[0122]

[0123]

[0124]

[0125] In the formula, ∑ρ i ,∑ρ o 、F(ρ) i and F(ρ) o They represent the sum of the inner contact curvature and the outer contact curvature, the difference of the inner contact curvature and the difference of the outer contact curvature, respectively. i,d o and r o are the curvature radii of the inner ring, outer ring and steel ball respectively.

[0126] According to equations (34) to (42), a, b, and κ are all related to the ellipticity k, and the ellipticity is related to the contact angle. When the contact angle changes, the ellipticity must be solved by an iterative method. In order to avoid wasting time by repeatedly solving the ellipticity in the process of solving the equation, the ellipticity corresponding to the contact angle in the range of 0 to 90° can be calculated in advance. Then, when solving the equation group, it is only necessary to interpolate the different contact angles to quickly obtain the ellipticity, and then calculate the values ​​of a, b, and κ.

[0127] The sliding rate and spin rate indicate the relative sliding and spinning rate of the steel ball relative to the inner and outer raceways, and are the basic conditions for calculating the friction of the steel ball in solid lubrication ball bearings. Based on the direction specified by the coordinate system, the sliding rate and spin rate of the steel ball relative to the inner / outer raceways are:

[0128]

[0129]

[0130]

[0131]

[0132]

[0133]

[0134] In the formula, ξ ix , iy and are the longitudinal sliding rate, lateral sliding rate and spin rate of the steel ball relative to the inner raceway, ξ ox , oy and are the longitudinal sliding rate, lateral sliding rate and spin rate of the steel ball relative to the outer raceway, r b is the radius of the steel ball, r p is the pitch radius of the steel ball, δ i and δ o are the normal contact deformation of the steel ball at the inner contact and outer contact, ω i and ω o are the angular velocities of the inner ring and outer ring relative to the cage.

[0135] Using the six sliding rates and spin rates calculated by formulas (43)-(48) and combined with the material parameters of the bearing, the friction force and friction torque of the steel ball can be obtained using the rolling contact friction algorithm FASTSIM, as follows:

[0136] The algorithm assumes that the sliding of the front edge of the rolling contact area is zero, that is, the friction at the front edge is zero, and then only needs to be gradually recursively calculated from the front edge to the back edge to find the friction of each particle, and finally sum up to get the total friction of the entire contact area. The contact area is elliptical, including both the sliding area and the adhesion area, such as Figure 3 As shown in (a) in the figure. To facilitate calculation, the contact area is meshed by finite element and the nodes are numbered, as shown in Figure 3 In order to improve the accuracy and versatility of the calculation program, some parameters are first dimensionally processed:

[0137]

[0138] P0=Q i / (2πab / 3) (50)

[0139] U X =3G b C 11 ξ x / (8μP0) (51)

[0140] U Y =3G b C 22 ξ y / (8μP0) (52)

[0141]

[0142] In the above formula, X and Y are the dimensionless coordinates of the particle in the contact area, the origin of the coordinate is at the exact center of the ellipse, x and y are the dimensionless coordinates of the particle, P0 is the dimensionless normal contact force, a and b are the major and minor semi-axes of the contact ellipse, respectively, and U is X and U Y are the dimensionless longitudinal creep rate and dimensionless transverse creep rate, G b is the rigidity modulus, μ is the friction coefficient, ξ x and y denote the longitudinal creep rate and the transverse creep rate respectively, ξ x Take xi or xo , similarly y Take yi or yo , and is the dimensionless spin rate, is the dimensionless spin rate, Pick or The subscript i or o indicates the contact between the steel ball and the inner raceway or the outer raceway, respectively. According to equations (49)-(53), the dimensionless sliding amount can be obtained as:

[0143]

[0144] In the formula, C X and C Y They are the dimensionless sliding amounts in the longitudinal direction (along the x direction) and the transverse direction (along the y direction). The friction force of each particle in the contact area can be recursively calculated using the sliding amount:

[0145]

[0146] In the formula, and represents the friction force of the particle number (i,j) along the x and y directions, ΔX and ΔY are the dimensionless distances between the particle (i,j) and the particle (i+1,j) along the x and y directions. After obtaining the friction force of each particle in the contact area, the dimensionless total friction force and friction torque in the contact area are calculated by summing:

[0147]

[0148] Where, T x 、T y and M z They are the longitudinal friction force along the x-direction, the lateral friction force along the y-direction, and the friction torque rotating around the z-axis. Since the contact area of ​​rolling contact includes both the sliding area and the adhesion area, such as Figure 3 As shown in (a), Coulomb's friction law can be used to determine the sliding state of a particle. That is, when the friction force of a particle is not greater than the maximum static friction force, it is considered to be in a sticking state (static friction), and the friction force calculated by recursion is valid; if the friction force calculated by a particle is greater than the maximum static friction force, it is considered to be in a sliding state (sliding friction), and the friction force at this time is equal to the maximum static friction force. The formula for the above judgment criteria is as follows:

[0149]

[0150] Where P0(i,j) represents the normal contact force at position particle (i,j).

[0151] The friction force and friction torque between the steel ball and the inner and outer raceways are calculated by the FASTSIM algorithm, and the six friction forces and friction torques of the steel ball are finally obtained as shown in equation (21). The friction force will also cause an additional torque on the steel ball, corresponding to the longitudinal friction force T xi 、T xo and the lateral friction force T yi 、Tyo The torque is calculated as follows:

[0152]

[0153] Where M xi / xo corresponds to the longitudinal friction force T xi 、T xo The torque, M yi / yo Corresponding to the lateral friction force T yi 、T yo The normal contact force between the steel ball and the inner / outer raceway can be calculated based on the Hertz contact elastic deformation and contact stiffness:

[0154]

[0155] In the formula, κ i and κ o are the contact stiffness coefficients of the inner contact and the outer contact, δ i ,δ o are the contact deformation between the steel ball and the inner and outer raceways, Q i , Q o are the contact forces between the steel ball and the inner and outer raceways, respectively. They can be calculated based on the Harris rolling bearing theory using parameters such as the contact angle. The contact deformation is also calculated using the displacement variable in step 1:

[0156]

[0157] In order to make the calculation more accurate, the inertia effect of the steel ball should also be considered. The centrifugal force F of the steel ball c for:

[0158]

[0159] In the formula, m b is the mass of the steel ball, ω bm is the angular velocity of the steel ball, which is one of the six independent variables of the steel ball. The components of the steel ball gyro torque in the three directions of the moving coordinate system are:

[0160]

[0161] In the formula, I b is the polar moment of inertia of the steel ball, M gx 、M gy and M gz They are respectively the moment of the steel ball gyro in the follower coordinate system along x b ,y b and z b Direction component.

[0162] Step 3: First, use the load iteration solution framework to solve the quasi-static equilibrium equation established in step 2 to obtain the steady-state analysis result. Then use the solution of the quasi-static equilibrium equation (i.e., the steady-state analysis result) as the initial value of the dynamic differential equation, and use the fourth-order Runge-Kutta numerical iteration algorithm to solve the differential equation to obtain the friction dynamic response of the bearing. The specific process is:

[0163] First, solve the quasi-static equations. When the bearing is only subjected to axial load, the quasi-static equations should be Equations (20) and (22), a total of 7 nonlinear equilibrium equations. The initial values ​​of the equations are given by the classic Jones-Harris model and solved using the trust region algorithm.

[0164] When the bearing is subjected to other loads such as radial force in addition to the axial force, each steel ball is loaded differently. Each steel ball needs to establish its own set of equilibrium equations shown in equation (20), plus the 5-degree-of-freedom motion of the inner ring, a total of 6N b +5(N b represents the number of steel balls) equilibrium equations. Since the equations are complex and very sensitive to the initial values, an initial value close to the solution must be given to make the equation converge. In this regard, a load iteration solution method is proposed, that is, firstly, other loads except the axial force are ignored, and the first modeling method (i.e., the 7-equation method) is used to calculate the solution of the first step, and then other loads are gradually increased from zero, and the second modeling method (i.e., (6N b +5) Equation system method) Step by step solve the equation system under complex loads. In this process, the solution of the previous load is always used as the initial value of the next load until the given load is reached.

[0165] For problems with a large number of equations that are difficult to solve at one time, a modular solution method is used. That is, the equilibrium equations of each steel ball are solved one by one, and only 6 equations need to be solved at a time. When all steel balls are solved, the equilibrium equations of the inner ring are finally solved, and only 5 equations need to be solved at a time. Repeated iterations are performed in each load step until the inner ring and all steel balls converge, then the calculation of this load step is stopped. The detailed process of the above solution strategy is as follows Figure 4As shown: ① First, determine whether the load category is pure axial load or combined load. For pure axial load, it can be solved directly by processing seven equilibrium equations. In addition, the initial value of the equation group can be initially obtained from the classic JH model, which does not consider the friction effect. ② Secondly, an incremental solution strategy is adopted for the combined load. Since the combined load has a great influence on the inner ring displacement, the convergence is very sensitive to the initial value, so the radial load and moment load must be gradually increased from zero, and the initial value comes from the pure axial load model. In the iterative increase of the load, the solution obtained under the previous load F should be iteratively used as the initial value of the increased load F' until the increased load is equal to the given load F0. ③ For the combined load balance equation, the solution process is divided into two stages: the ball balance stage and the ring balance stage. In the first stage, the balls are solved one by one, so only six equations need to be processed in one step. In the second stage, after all the steel balls are balanced, the balance equation of the inner ring is solved, so that only five equations need to be solved in one step.

[0166] After solving the quasi-static equations, we can finally get: the displacement y of each steel ball b / z b , angular velocity ω bm , angular velocity ω bx / ω by / ω bz And the inner race displacement x i / y i / z i / θ ix / θ iy The initial value of the dynamic equation is written as:

[0167]

[0168] In the formula, x0 i 、x0 o 、x0 c 、x0 b They represent the inner ring, outer ring, cage and steel ball parts in the dynamic initial value respectively, and the superscript bi represents the i-th steel ball. They are respectively expressed as:

[0169]

[0170]

[0171]

[0172]

[0173] In the formula, ω m The average angular velocity of all the steel balls is: Using these variables, the steady-state characteristics of each component, including contact angle, contact force, friction force, etc., can be calculated according to equations (56) to (62) in step 2. The friction torque of the inner ring can also be calculated using the following equation:

[0174]

[0175] Where N b is the number of steel balls, the subscript k represents the steel ball number, T xi is the tangential friction force of the steel ball, r pk is the pitch radius of the steel ball, r b is the radius of the steel ball, M zi k is the contact friction torque inside the steel ball, α ik is the internal contact angle of the steel ball.

[0176] The variables obtained by quasi-statics are used as initial values ​​and substituted into equations (29) to (31). The fourth-order fixed-step Runge-Kutta integration method is used to solve the dynamic differential equations, and finally the dynamic results of the variables changing with time can be obtained. Compared with the quasi-statics model, the dynamic model can calculate the analysis results under variable load conditions (such as variable speed, variable axial force, etc.), and can be used for the full process analysis of acceleration, stability, and deceleration of bearing equipment, so it has a wider range of applications. The variable time series obtained above can be used using equations (56) to (62) to finally calculate the dynamic responses of the bearing state angle, friction, contact force, vibration acceleration, etc.

[0177] The friction dynamics analysis model and method established above can be used to realize the tribological and dynamic analysis of solid-lubricated ball bearings under different types of loads (axial force, radial force, overturning moment, steady state, dynamic) in engineering practice, and meet the analysis, design and optimization needs of bearing systems in tribology and structural mechanics under real working conditions.

[0178] Example 1

[0179] This embodiment uses the B7218 angular contact ball bearing parameters shown in Table 1 for calculation, which verifies the effectiveness of the present invention. Obviously, the embodiments developed in the present invention are not all embodiments, but are only representative embodiments. Unless creative results are achieved, the embodiments obtained by those skilled in the art based on the present invention are all within the scope of protection of the present invention. Under the combined load shown in Table 1, the quasi-statics of the bearing uses equations (20) and (23) to model, with a total of 16 steel balls, each with 6 equilibrium equations, and the inner ring with 5 equilibrium equations, for a total of 101 equilibrium equations for the system. According to Figure 4The solution process shown in the figure adopts the trust region method to solve the problem. The variables finally obtained are shown in Table 2, where only the variables of steel balls No. 1, No. 5 and No. 9 are listed. According to the quasi-static results, the steady-state analysis results of the bearing can be obtained by combining equations (56)-(62). Among them, the contact force calculation results of the steel ball with the inner and outer raceways during one revolution are compared with the results of the classic Jones-Harris quasi-static calculation. Figure 5 As shown in (a) and (b), it can be found from the figure that the maximum contact load occurs at the positive axis position of X, corresponding to the radial force loading direction, and the calculation results of the model of the present invention are basically consistent with the JH model, which proves the effectiveness of the model of the present invention. In addition, the steady-state friction torque of the inner ring calculated by formula (68) is 4.43Nm.

[0180] Table 1 Angular contact ball bearing parameters and working conditions used in Example 1

[0181] parameter Numeric parameter Numeric <![CDATA[Number of balls N b > 16 <![CDATA[Axial force F a / N]]> 40000 <![CDATA[Ball diameter D b / m]]> <![CDATA[22.23×10 -3 ]]> <![CDATA[Radial force F r / N]]> 10000 <![CDATA[Inner diameter D i / m]]> <![CDATA[102.79×10 -3 ]]> Overturning moment M / Nm 0 <![CDATA[Outer diameter D o / m]]> <![CDATA[147.73×10 -3 ]]> Friction coefficient μ 0.1 <![CDATA[Contact angle α 0 / (°)]]> 40 <![CDATA[Inner ring stiffness K i / (N / m)]]> 0 Elastic modulus E / MPa <![CDATA[2.06×10 11 ]]> <![CDATA[Inner ring damping C i / N / (m / s)]]> 30000 Poisson's ratio 0.3 <![CDATA[Steel ball damping C b / N / (m / s)]]> 5000 <![CDATA[Density ρ / Kg / m 3 > <![CDATA[7.8×10 3 ]]> <![CDATA[Cage contact damping C c / N / (m / s)]]> 10000

[0182] Table 2 Solution results of the quasi-static equations of the bearing in Example 1

[0183] Inner circle variables — Steel ball variable <![CDATA[N b =1]]> <![CDATA[N b =5]]> <![CDATA[N b =9]]> <![CDATA[x i / m]]> 1.67e-05 <![CDATA[y b / m]]> 1.19e-05 3.51e-06 -4.75e-06 <![CDATA[y i / m]]> 0 <![CDATA[z b / m]]> 2.51e-05 2.53e-05 2.54e-05 <![CDATA[z i / m]]> 5.90e-05 <![CDATA[ω bm / rpm]]> 433.75 434.32 434.93 <![CDATA[θ ix / rad]]> 0 <![CDATA[ω bx / rpm]]> 1.43 1.28 1.12 <![CDATA[θ iy / rad]]> 0 <![CDATA[ω by / rpm]]> 1985.06 2017.48 2044.63 — — <![CDATA[ω bz / rpm]]> -1949.16 -1917.43 -1889.93

[0184] Dynamic calculation: Set the dynamic speed of the bearing as follows: 0 to 0.5 seconds inner ring idle speed (speed stabilized at 1000rpm); then increase the speed evenly from 1000rpm to 10000rpm in the time period from 0.5 to 1.4 seconds; finally stabilize at 10000rpm after 1.4 seconds, and the total dynamic calculation time is 2.5 seconds. Figure 5 The quasi-static solution shown is used as the initial value of the dynamic equation, and the differential equation is solved by the fourth-order fixed-step Runge-Kutta method with a step size of 10 -5 sec. The vibration acceleration response of the inner ring in the X and Y directions is calculated as Figure 6 As shown in (a) and (b), the inner ring vibration acceleration increases significantly during acceleration and reaches dynamic stability after the speed stabilizes. The dynamic response of the angular velocity components of the steel ball in three directions is shown in Figure 7 As shown in (a), the friction torque is calculated using formula (68) for the variables in each time step, and the dynamic friction torque of the bearing during the entire speed change process is finally obtained as follows: Figure 7 As shown in (b). Figure 7 In (b), we can first find that the friction torque is positively correlated with the rotational speed and increases significantly under acceleration (between 0.5 and 1.4 s).

Claims

1. A bearing friction dynamics response analysis method integrating pseudo-statics and dynamics, characterized in that: The following steps are involved: Step 1: First, a reference coordinate system is established according to the installation direction and rotation direction of the solid-lubricated angular contact ball bearing, and a pseudo-static equilibrium equation and a dynamic differential equation of the solid-lubricated ball bearing with consistent displacement variables are established according to the reference coordinate system; Step 2: According to the relationship between the contact angle and the contact parameter, the friction force and friction torque in the pseudo-static equilibrium equation are calculated; Step 3: According to the friction force and friction torque, solve the quasi-static equilibrium equation established in step 1 to obtain the steady-state analysis results. Then use the steady-state analysis results as the initial values ​​of the dynamic differential equation, and use the fourth-order Runge-Kutta numerical iterative algorithm to solve the dynamic differential equation to obtain the friction dynamics response of the bearing.

2. The bearing friction dynamics response analysis method integrating pseudo-statics and dynamics according to claim 1 is characterized in that: According to the installation direction and rotation direction of the solid lubrication angular contact ball bearing, a reference coordinate system is established, including: 1) Inertial coordinate system O-XYZ, set the positive direction of the Z axis to the positive direction of the angular contact bearing; 2) Steel ball moving coordinate system o b -x b y b z b , where z b The y axis is consistent with the Z axis of the inertial coordinate system. b The axis points away from the axis; 3) Local contact coordinate system between the steel ball and the inner and outer raceways i -x i y i z i and o -x o y o z o , where z i Axis and z o The axis is toward the center of the steel ball, x i and x o The axes are the rolling directions of the steel ball at the inner contact and outer contact points, respectively. It should be noted that the rolling directions of the steel ball relative to the inner and outer raceways are always opposite; 4) Cage pocket coordinate system o c -x c y c z c , where o c Located at the geometric center of each pocket of the cage, the three axes are parallel to the steel ball moving coordinate system.

3. The bearing friction dynamics response analysis method integrating pseudo-statics and dynamics according to claim 1 is characterized in that: The quasi-static equilibrium equation includes the quasi-static equilibrium equation of the steel ball and the equilibrium equation of the inner ring. The quasi-static equilibrium equation of the steel ball is: F ball ＝0 (20) In the formula, F ball The resultant force on the steel ball is expressed as follows: In the formula, and They represent the steel ball at x b ,y b and z b The net force in the direction and the force around x b ,y b 、z b Resultant moment about the shaft, T xi and T yi They are respectively the direction of the steel ball along x at the inner contact point i The longitudinal friction of the axis and the friction along the y i Transverse friction of the shaft, T xo and T yo They are respectively the direction of the steel ball along x at the external contact point o The longitudinal friction of the axis and the friction along the y o Transverse friction of the shaft, Q i , Q o are the contact forces between the steel ball and the inner and outer raceways, α i and α o are the inner contact angle and outer contact angle of the steel ball after deformation, F c is the centrifugal force of the steel ball, M gx and M gy They are respectively the moment of the steel ball gyro in the follower coordinate system along x b and b Directional component; When the ball bearing is subjected to a pure axial force F a When , the equilibrium equation of the inner circle is: F a =N b (Q i sinα i +T yi cosα i ) (22) Where N b is the number of rolling elements; When the inner ring is subjected to a 5-DOF load, the equilibrium equation of the inner ring is: Among them, T bik is the matrix that converts the force applied by the moving reference frame to the inertial reference frame; F i b is the force exerted by the ball on the inner raceway in the follower coordinate system; F is the 5-DOF load acting on the inner ring.

4. The bearing friction dynamics response analysis method integrating pseudo-statics and dynamics according to claim 3 is characterized in that: The matrix that converts the force applied by the moving reference frame to the inertial reference frame is as follows:

5. The bearing friction dynamics response analysis method integrating pseudo-statics and dynamics according to claim 3 is characterized in that: The force exerted by the ball on the inner raceway in the follower coordinate system is as follows:

6. The bearing friction dynamics response analysis method integrating pseudo-statics and dynamics according to claim 5 is characterized in that: The 5-DOF loads acting on the inner ring are as follows: F=(F x ,F y ,F z ,M x ,M y ) T (26) in, is the orbital position of the kth ball, R i Indicates the pitch radius of the inner ring, F ix b 、F iy b and F iz b are the forces along the X, Y and Z directions applied by the i-th steel ball to the inner ring in the follower coordinate system, α ik is the internal contact angle of the i-th steel ball, F x 、F y and F z are the thrust loads of the inner ring along the X, Y and Z directions, M and M y are the torque loads of the inner ring around the X and Y axes respectively; Where M b and I b denote the mass and polar moment of inertia of the steel ball, θ bx ,θ by and θ bz are the angles of rotation of the steel ball around the three coordinate axes of the moving coordinate system, C b It is the damping of the translational motion of the steel ball.

7. The bearing friction dynamics response analysis method integrating pseudo-statics and dynamics according to claim 2 is characterized in that: The differential equation for the translational motion of the inner ring is: In the formula, and Respectively represent x i The first and second derivatives of and Respectively represent y i The first and second derivatives of and Respectively represent z i The first and second derivatives of M i is the mass of the inner ring, K i and C i are the stiffness and damping of the inner ring translation motion, and are the resultant forces exerted by all the steel balls on the inner ring.

8. The bearing friction dynamics response analysis method integrating pseudo-statics and dynamics according to claim 2 is characterized in that: The differential equation of motion of the cage is: In the formula, I c is the cage polar inertia moment, θ c is the angular displacement of the cage.

9. The bearing friction dynamics response analysis method integrating pseudo-statics and dynamics according to claim 1 is characterized in that: The contact parameters include the inner and outer contact angles of the steel ball, the contact curvature and the contact curvature difference, where the inner and outer contact angles of the steel ball are: In the formula, α i and α o are the inner contact angle and outer contact angle of the steel ball after deformation, A i 0 and A o 0 They represent the initial distances from the geometric center of the steel ball to the inner channel curvature center and the outer channel curvature center, α 0 is the initial contact angle, z i is the displacement of the inner ring after deformation due to load in the steel ball follower coordinate system along the z axis b Component of the axis, y b The steel ball along y b The amount of translational motion of the axis; The contact curvature and the difference in contact curvature are expressed as: In the formula, ∑ρ i ,∑ρ o 、F(ρ) i and F(ρ) o They represent the sum of the inner contact curvature and the outer contact curvature, the difference of the inner contact curvature and the difference of the outer contact curvature, respectively. i ,d o and r o are the curvature radii of the inner ring, outer ring and steel ball respectively.

10. The bearing friction dynamics response analysis method integrating pseudo-statics and dynamics according to claim 9, characterized in that: The sliding rate and spin rate of the steel ball relative to the inner / outer raceway are: In the formula, ξ ix , iy and are the longitudinal sliding rate, lateral sliding rate and spin rate of the steel ball relative to the inner raceway, ξ ox , oy and are the longitudinal sliding rate, lateral sliding rate and spin rate of the steel ball relative to the outer raceway, r b is the radius of the steel ball, r p is the pitch radius of the steel ball, δ i and δ o are the normal contact deformation of the steel ball at the inner contact and outer contact, ω i and ω o are the angular velocities of the inner ring and outer ring relative to the cage; According to the longitudinal sliding rate, lateral sliding rate and spin rate of the steel ball relative to the outer raceway, the friction force and friction torque of the steel ball are obtained by FASTSIM algorithm.

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