A method for dynamic modeling of a thermally coupled deep groove ball bearing wave-shaped cage

Abstract

The application provides a kind of thermal coupling deep groove ball bearing wave-shaped retainer dynamics modeling method, and the technical scheme adopted by the application is: based on initial parameters to establish the quasi-statics and friction heat generation model of deep groove ball bearing, calculate the average heat generation rate and solve the steady temperature field by combining the heat network method;According to the steady temperature field, calculate the thermal expansion deformation of the component, and dynamically update the internal working clearance of the bearing and the lubricant kinematic viscosity parameters combined with the centrifugal expansion effect;The updated gap and other parameters are substituted back into the quasi-static model to solve again, and the accurate contact load and contact angle under the steady temperature field are obtained as boundary conditions;Based on the contact parameters of the re-solution, the nonlinear dynamics equation of the steel ball and the wave-shaped retainer under the action of thermal coupling is established and solved, so as to obtain the real dynamic response result of the retainer.The application can solve the technical problem that the existing pure mechanics model uses isothermal assumption, which leads to inaccurate prediction of the dynamic behavior of the retainer.

Description

Technical Field

[0001] This invention relates to the field of rolling bearing dynamics analysis technology, specifically to a dynamic modeling method for a thermo-coupled deep groove ball bearing with a wave-shaped cage. Background Technology

[0002] Deep groove ball bearings, as core support components of high-speed rotating equipment, are widely used in engineering, especially those with corrugated cages. Unlike traditional ring-guided cages, corrugated cages lack radial support and employ a special steel ball guiding mechanism, relying entirely on the collision and pushing of the steel balls for operation. This unique mechanism means that the bearing's internal geometric parameters, such as pocket clearance and radial clearance, directly determine the collision strength between the steel balls and the cage, making it highly susceptible to nonlinear dynamic instability.

[0003] Under high-speed operating conditions, the intense friction between the components inside the bearing generates a large amount of heat, causing thermal expansion of the components and a continuous reduction in the internal working clearance. This reduction in clearance, in turn, exacerbates the instability of the steel ball's movement, further worsening frictional heat generation, thus forming a positive feedback loop of frictional heat generation – clearance reduction – increased friction. This bidirectional coupling of heat and force makes the actual operating state of the cage extremely complex.

[0004] However, traditional bearing dynamics analysis methods generally employ the isothermal assumption, separating temperature changes from mechanical motion and neglecting the thermal expansion of components caused by intense frictional heat generation during high-speed operation, as well as the centrifugal expansion effect. This leads to existing models often ignoring the dynamic reduction of pocket clearance and radial clearance during calculations, resulting in significant deviations in the prediction of the cage's stress state and motion. Therefore, existing purely mechanical models are insufficient to accurately describe the true dynamic behavior of this type of special cage under complex and variable service conditions, exhibiting obvious limitations.

[0005] Therefore, how to incorporate temperature changes and mechanical motion into a unified solution framework and accurately predict the true dynamic behavior of wave-shaped cages under complex working conditions is an urgent problem to be solved in this field. Summary of the Invention

[0006] To address the shortcomings of existing technologies, this invention proposes a dynamic modeling method for the wave-shaped cage of a thermo-coupled deep groove ball bearing, in order to solve the technical problem of inaccurate prediction of cage dynamic behavior caused by the use of isothermal assumptions in existing pure mechanical models.

[0007] The technical solution adopted in this invention is as follows: First, a quasi-static and frictional heat generation model of a deep groove ball bearing is established based on initial parameters. The average heat generation rate is calculated, and the steady-state temperature field is obtained by combining the thermal network method. Then, the thermal expansion deformation of the components is calculated based on the steady-state temperature field, and the internal working clearance and lubricant kinematic viscosity parameters of the bearing are dynamically updated by combining the centrifugal expansion effect. Next, the updated clearance and other parameters are substituted back into the quasi-static model for re-solving to obtain the accurate contact load and contact angle under the steady-state temperature field as boundary conditions. Finally, based on the re-solved contact parameters, the nonlinear dynamic equations of the steel ball and the corrugated cage under thermo-mechanical coupling are established and solved to obtain the true dynamic response results of the cage.

[0008] In the first feasible approach, a dynamic modeling method for the corrugated cage of a thermo-coupled deep groove ball bearing is provided, comprising the following steps: Obtain the initial parameters of the deep groove ball bearing, including geometric parameters, material parameters, operating conditions, and lubrication parameters; Based on the initial parameters, a quasi-static model and a frictional heat generation model for the deep groove ball bearing are established. The average heat generation rate of the bearing in one operating cycle is calculated, and the steady-state temperature field of the bearing is solved iteratively using the thermal network method. The thermal expansion deformation of each component of the bearing is calculated based on the steady-state temperature field, and the internal working clearance and lubricant kinematic viscosity parameters of the bearing are updated in combination with the centrifugal expansion effect. Substitute the updated internal working clearance into the quasi-static model to obtain the contact load and contact angle between the steel ball and the inner and outer raceways under the steady-state temperature field; Based on the contact load and contact angle, a nonlinear dynamic equation for the steel ball and the corrugated cage under thermo-mechanical coupling is established to obtain the dynamic response results of the corrugated cage.

[0009] Furthermore, based on the initial parameters, a quasi-static model and a frictional heat generation model for the deep groove ball bearing are established. The average heat generation rate of the bearing within one operating cycle is calculated, and the steady-state temperature field of the bearing is iteratively solved using the thermal network method. Specific steps include: Calculate transient frictional torque based on quasi-static model The transient frictional torque includes elastic hysteresis frictional torque. Differential sliding friction torque Spin sliding friction torque Friction torque due to lubricant drag ; Regarding the transient frictional torque and rotational speed The product of these factors is integrated over time to obtain the average heat generation rate over one operating cycle T. The average heat generation rate is calculated using the formula for allocating heat to the inner and outer nodes:

[0010] In the formula, The average heat generation rate over one operating cycle T, where T is the operating cycle. () represents the transient frictional torque. Rotational speed; Using the average heat generation rate distributed to the inner and outer ring nodes as a constant heat source input, a bearing thermal network model based on Kirchhoff's energy balance law is established. The model is iteratively solved until the temperature residual satisfies the set convergence tolerance, thus obtaining the steady-state temperature field.

[0011] Furthermore, in the bearing thermal network model based on Kirchhoff's energy balance law, the transient thermal balance differential equation at any node i is expressed as:

[0012] In the formula, and The instantaneous temperature of adjacent nodes. Thermal response time Distribute the average heat generation rate to the nodes heat source, The thermal resistance varies with temperature. For nodes The heat capacity at that point.

[0013] Furthermore, based on the steady-state temperature field, the thermal expansion deformation of each component of the bearing is calculated, and the internal working clearance and lubricant kinematic viscosity parameters of the bearing are updated in conjunction with the centrifugal expansion effect, specifically including: Calculate the centrifugal change in the fitting diameter caused by bearing speed and the thermal expansion change in the fitting diameter caused by temperature change, and then determine the reduction in radial clearance caused by the interference fit of the outer ring. Reduction in radial clearance caused by interference fit with inner ring ; Calculate the relative thermal deformation of the average diameters of the inner and outer rings of the bearing. The transient radial clearance increment is obtained by combining the reduction in radial clearance. ; Based on the transient radial clearance increment Calculate the actual radial working clearance at the current temperature. And based on this, calculate the actual axial working clearance. This is to update the internal work gaps; The Walther equation is used to describe the change in kinematic viscosity of the lubricant with temperature, based on the lubricant temperature in the steady-state temperature field. Calculate and update lubricant kinematic viscosity parameters :

[0014] In the formula, A, B, and C are parameters related to the lubricating grease.

[0015] Furthermore, the updated internal working clearance is substituted into the quasi-static model to obtain the contact load and contact angle between the steel ball and the inner and outer raceways under a steady-state temperature field, specifically including: Update the actual radial working clearance Actual axial working clearance Substituting the geometric relationship between the relative displacements of the inner and outer rings of the bearing, the deformation compatibility equation of the steel balls in the bearing's load-bearing and non-load-bearing areas is modified. The modified deformation compatibility equations of the steel ball, the force balance equations of the steel ball, and the force balance equations of the inner ring of the deep groove ball bearing are combined and the discretization method is used to solve for a complete operating cycle. The contact loads between all steel balls and the inner and outer raceways at different locations under steady-state temperature field were calculated. , and contact angle , This provides steady-state boundary conditions for solving the dynamics of the wave-shaped cage.

[0016] Furthermore, the nonlinear dynamic equations of the steel ball and the wave-shaped cage include the differential equations of the steel ball's motion; When establishing the differential equations of motion for the steel ball, a rotating coordinate system for the steel ball is established with the center of mass of the steel ball as the origin. Taking into account the contact force between the steel ball and the inner and outer raceways, the contact drag force, the fluid turbulence resistance, the centrifugal force on the steel ball, and the gyroscopic torque, the differential equations of motion for the steel ball in the directions of rotation and revolution around its own coordinate axis are established.

[0017] Furthermore, when establishing the nonlinear dynamic equations for the steel ball and the corrugated cage under the aforementioned thermo-coupling effect, based on Hertzian point contact theory and damping effects, the nonlinear contact force between the corrugated cage pocket and the steel ball is established. The model, and its calculation formula is:

[0018] In the formula, This refers to the deformation stiffness at the contact point between the steel ball and the cage pocket. This is the damping coefficient at the contact point between the steel ball and the cage pocket. Let be the contact deformation between the nth steel ball and the inner wall of the cage pocket at a single point of contact. The relative velocity of contact deformation.

[0019] Furthermore, the nonlinear dynamic equations of the steel ball and the corrugated cage under thermo-mechanical coupling are established, including the differential equations of motion of the corrugated cage:

[0020]

[0021]

[0022] In the formula, , For the rotational inertia and mass of the wave-shaped cage, , , To maintain the angular acceleration and translational acceleration of the frame, To maintain the transient position vector from the center of mass of the frame to the contact point of the nth steel ball, To maintain the reaction force of the bracket pocket on the contact force of the steel ball, To maintain the reaction force of the cage on the frictional force of the steel ball, , and , These are the projected components of the corresponding force on the x-axis and y-axis.

[0023] Furthermore, the dynamic response results of the wave-shaped cage are obtained, specifically including: At a given time step, the contact load and contact angle of the steel ball at the current moment are obtained by linear interpolation based on the current azimuth angle of the steel ball. Substituting the contact load and contact angle into the nonlinear dynamic equation, and using the fourth-order Runge-Kutta algorithm for numerical integration, the dynamic response results of the corrugated cage are obtained.

[0024] In conjunction with the first feasible approach, a second feasible approach provides a dynamic modeling system for the corrugated cage of a thermo-coupled deep groove ball bearing, the system comprising: The initial parameter acquisition module is used to acquire the initial parameters of the deep groove ball bearing, including geometric parameters, material parameters, operating conditions and lubrication parameters. The temperature field calculation module is used to establish a quasi-static model and a frictional heat generation model of the deep groove ball bearing based on the initial parameters, calculate the average heat generation rate of the bearing in one operating cycle, and iteratively solve the steady-state temperature field of the bearing using the thermal network method. The parameter update module is used to calculate the thermal expansion deformation of each component of the bearing based on the steady-state temperature field, and update the internal working clearance and lubricant kinematic viscosity parameters of the bearing in combination with the centrifugal expansion effect. The contact state re-solution module is used to substitute the updated internal working gap into the quasi-static model to obtain the contact load and contact angle between the steel ball and the inner and outer raceways under the steady-state temperature field. The dynamics solution module is used to establish the nonlinear dynamic equations of the steel ball and the corrugated cage under thermo-mechanical coupling based on the contact load and contact angle, and to obtain the dynamic response results of the corrugated cage.

[0025] As can be seen from the above technical solution, the beneficial technical effects of the present invention are as follows: 1. By incorporating the dynamic changes in clearance caused by frictional heat generation, the changes in lubrication parameters, and the nonlinear collision friction of internal components into a unified solution framework, this thermo-mechanical bidirectional coupling mechanism can describe the complex contact friction state inside the bearing in real time and accurately. It effectively eliminates the local disorder and unrealistic displacement mutation phenomena that occur in traditional pure mechanical models, making the calculated trajectory and time-domain waveform more consistent with reality, and effectively improving the stability prediction accuracy under complex working conditions.

[0026] 2. By establishing a thermo-mechanical coupling model, the deep coupling mechanism between heat-induced changes in internal clearance dimensions and external macroscopic operating conditions is revealed. This model can clearly present the thermo-dynamic instability limit and complex evolution path of the system under ultra-high-speed service, providing a reliable theoretical model basis for the optimization of internal parameters and anti-instability design of high-speed deep groove ball bearings. Attached Figure Description

[0027] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the accompanying drawings used in the description of the specific embodiments or the prior art will be briefly introduced below. In all the drawings, similar elements or parts are generally identified by similar reference numerals. In the drawings, the elements or parts are not necessarily drawn to scale.

[0028] Figure 1 This is a schematic diagram of the fixed coordinate system, the rotating coordinate system of the steel ball, the pocket coordinate system, and the contact coordinate system in Embodiment 1 of the present invention; Figure 2 is a schematic diagram of the geometric relationship and deformation coordination of the deep groove ball bearing under different states in Embodiment 1 of the present invention. In this figure, Figure 2(a) is a geometric relationship diagram of the transformation from the initial state to the initial contact state, Figure 2(b) is a schematic diagram of the deformation coordination relationship of the steel ball in the load-bearing area, and Figure 2(c) is a schematic diagram of the deformation coordination relationship of the steel ball in the non-load-bearing area. Figure 3 This is a schematic diagram of the force analysis of the steel ball in the three-dimensional coordinate plane according to Embodiment 1 of the present invention; Figure 4 is an analysis diagram of the sliding speed of the steel ball in contact with the raceway in Embodiment 1 of the present invention. In Figure 4(a), it is a schematic diagram of the outer raceway contact, and Figure 4(b) is a schematic diagram of the inner raceway contact. Figure 5 is a schematic diagram of the evolution of the contact state between the steel ball and the cage pocket in Embodiment 1 of the present invention. In Figure 5(a), it is a diagram describing the displacement difference between the steel ball and the pocket. Figure 5(b) is a diagram of the initial state. Figure 5(c) is a diagram of the state in which the displacement difference occurs but there is no contact. Figure 5(d) is a diagram of the state in which a single point contact occurs. Figure 5(e) is a diagram of the state in which critical clamping occurs. Figure 6 is a schematic diagram of the deep groove ball bearing and the corrugated cage structure and stress of Embodiment 1 of the present invention. In Figure 6(a), it is a schematic diagram of the actual assembly of the bearing and the corrugated cage, and Figure 6(b) is a stress analysis diagram of the corrugated cage. Figure 7 This is a schematic diagram of the bearing temperature node division and location distribution in Embodiment 1 of the present invention; Figure 8 This is a schematic diagram of a bearing thermal network model based on temperature nodes in Embodiment 1 of the present invention; Figure 9 This is a flowchart of the solution process for the dynamic model of a deep groove ball bearing under thermo-mechanical coupling in Embodiment 1 of the present invention. Figure 10 This is a verification curve showing the comparison between the measured temperature and the simulated temperature rise of the bearing in Embodiment 1 of the present invention. Figure 11 This is a curve comparing the calculated cage rotational speed with the theoretical rotational speed in Embodiment 1 of the present invention; Figure 12 is a comparative analysis diagram of the nonlinear dynamic response of the cage in Embodiment 1 of the present invention. In Figure 12(a), the center of mass trajectory, time-domain waveform in the y-direction, FFT spectrum and Poincaré cross section of the cage are shown without considering thermo-coupling. Figure 12(b) shows the center of mass trajectory, time-domain waveform in the y-direction, FFT spectrum and Poincaré cross section of the cage with thermo-coupling considered. Figure 13 This is a system structure diagram of Embodiment 2 of the present invention.

[0029] Figure label: 2-A dynamic modeling system for a thermo-coupled deep groove ball bearing corrugated cage, 21-Initial parameter acquisition module, 22-Temperature field calculation module, 23-Parameter update module, 24-Contact state re-solution module, 25-Dynamic solution module. Detailed Implementation

[0030] The embodiments of the technical solution of the present invention will now be described in detail with reference to the accompanying drawings. These embodiments are merely illustrative of the technical solution of the present invention and are therefore intended to limit the scope of protection of the present invention.

[0031] It should be noted that, unless otherwise stated, the technical or scientific terms used in this application should have the ordinary meaning as understood by one of ordinary skill in the art to which this invention pertains.

[0032] Example 1

[0033] This embodiment provides a dynamic modeling method for the corrugated cage of a thermo-coupled deep groove ball bearing. The working principle of Embodiment 1 is explained in detail below: This embodiment includes the following steps: Obtain the initial parameters of the deep groove ball bearing, including geometric parameters, material parameters, operating conditions, and lubrication parameters; Based on the initial parameters, a quasi-static model and a frictional heat generation model for the deep groove ball bearing are established. The average heat generation rate of the bearing in one operating cycle is calculated, and the steady-state temperature field of the bearing is solved iteratively using the thermal network method. The thermal expansion deformation of each component of the bearing is calculated based on the steady-state temperature field, and the internal working clearance and lubricant kinematic viscosity parameters of the bearing are updated in combination with the centrifugal expansion effect. Substitute the updated internal working clearance into the quasi-static model to obtain the contact load and contact angle between the steel ball and the inner and outer raceways under the steady-state temperature field; Based on the contact load and contact angle, a nonlinear dynamic equation for the steel ball and the corrugated cage under thermo-mechanical coupling is established to obtain the dynamic response results of the corrugated cage.

[0034] In this embodiment, the following basic assumptions are adopted when establishing the dynamic model of the deep groove ball bearing: In a bearing, the centers of mass and centroid of the steel balls, inner ring, outer ring, and cage coincide. The flexible deformation of each component of the bearing is not considered; they are all treated as rigid bodies. The contact between the steel ball and the raceway and cage is considered as an elastic contact between rigid bodies and satisfies the Hertzian contact condition.

[0035] Four types of coordinate systems are used when establishing the bearing dynamics model, such as Figure 1 As shown: (1) Fixed coordinate system O-xyz: The origin O is located at the geometric center of the bearing, the y-axis coincides with the direction of gravity, the z-axis coincides with the center line of the outer ring rotation, and the x-axis is determined by the right-hand screw rule; (2) Steel ball rotating coordinate system Ob-x1y1z1: fixed on the steel ball, with the origin Ob located at the center of the steel ball, the z1 axis parallel to the z axis of the fixed coordinate system, the y1 axis outward along the direction of the steel ball's revolution, and the x1 axis determined by the right-hand screw rule; (3) Contact coordinate system Oh-x2y2z2: The center of the contact area is Oh, the x2 axis always coincides with the major axis of the contact ellipse, the y2 axis always coincides with the minor axis of the contact ellipse, and the z2 axis points to the object being compressed; (4) Pocket coordinate system Op-x3y3z3: The origin Op coincides with the geometric center of the pocket, the z3 axis is parallel to the z-axis of the fixed coordinate system, the y3 axis is outward along the direction of the cage's revolution, and the x3 axis is determined by the right-hand screw rule.

[0036] In its initial state, a deep groove ball bearing has radial clearance. After the inner ring moves a certain distance axially (half of the axial clearance, i.e., 0.5Sa), the steel balls just come into contact with the inner and outer raceways without generating contact force; this is the initial contact state of the deep groove ball bearing. The theoretical contact angle between the steel balls and the raceways at this point is... Represented as:

[0037] In the formula, For the radial clearance of a deep groove ball bearing, and These represent the curvature coefficients of the inner and outer raceways, respectively. The diameter of the ball bearing is [value].

[0038] With the outer ring of a deep groove ball bearing fixed, the vibration displacement of the inner ring relative to the outer ring. It can be represented as:

[0039] In the formula, , , This represents the translational displacement of the inner ring relative to the outer ring along the x, y, and z axes. , , ω represents the angular displacement around the x, y, and z axes.

[0040] During operation, deep groove ball bearings have internal load-bearing and non-load-bearing zones, and their geometric relationship is shown in Figure 2. The axial and radial distances Aan' and Arn' between the center of curvature Oo of the outer raceway and the center of curvature Oi'' of the inner raceway are respectively expressed as:

[0041]

[0042] In the formula, Let be the time-varying azimuth angle of the nth steel ball. This represents the actual axial working clearance at temperature T. This represents the actual radial working clearance at temperature T. The radial distance between the inner raceway curvature center Oi and the bearing center is expressed as:

[0043] In the formula, This refers to the bearing pitch circle diameter.

[0044] Contact angle between the steel ball and the inner and outer raceways and They can be represented as follows:

[0045]

[0046] In the formula, and These represent the axial and radial distances between the center of curvature Oo of the outer raceway of the bearing and the center of the ball Ob'', respectively. and These represent the contact deformation between the steel ball and the inner and outer raceways, respectively.

[0047] According to the Pythagorean theorem, the deformation compatibility equation for the steel ball located in the load-bearing zone is:

[0048] When the steel ball is located in the non-load-bearing zone, its deformation compatibility relationship satisfies the following inequality:

[0049] The force balance equation for the steel ball located in the load-bearing area is expressed as:

[0050] In the formula, This represents the average coefficient of friction between the steel ball and the inner and outer raceways. and This represents the contact force between the steel ball and the inner and outer raceways. For contact stiffness, This represents the centrifugal force acting on the steel ball. This represents the gyroscopic torque acting on the steel ball.

[0051] The force balance equations located in the non-load-bearing zone are: .

[0052] The force balance equation for the inner ring of a deep groove ball bearing is expressed as:

[0053] In the formula, , and The external forces acting on the inner ring along the x, y, and z axes. and The external torques about the x and y axes, Let n be the lever arm of the contact force between the nth steel ball and the inner raceway. The total number of steel balls.

[0054] In this embodiment, the force analysis of the steel ball in the rotating coordinate system Ob-x1y1z1 is further as follows: Figure 3 As shown. The differential equations of motion for the nth steel ball in its rotation and revolution directions are:

[0055] In the formula, and They are steel balls wrapped around Gyroscopic torques on the y-axis and y-axis Let be the moment of inertia of the steel ball about the z-axis. Let be the moment of inertia of the steel ball about its own axis. The mass of a single steel ball.

[0056] The formula for calculating the turbulence resistance of the steel ball caused by the lubricant is:

[0057] In the formula, The drag coefficient, For effective lubricant density.

[0058] and Let x2 and y2 be the components of the drag force between the steel ball and the inner ring. and The components of the drag force between the steel ball and the outer ring in x2 and y2 are calculated as follows: , In the formula and The coefficient of contact friction is determined based on the sliding-rolling ratio. and Sure:

[0059] The contact situation between the steel ball and the outer and inner raceways is shown in Figure 4. The sliding velocity between the steel ball and the outer raceway is expressed as:

[0060] The sliding velocity between the steel ball and the inner raceway is expressed as:

[0061] In the formula, and Let be the rotational speeds of the inner and outer rings relative to the coordinate system Ob-x1y1z1. and Let be the radius of curvature of the elliptical deformation caused by the contact of the inner and outer rings. and It is the major semi-axis of the inner and outer contact ellipse.

[0062] The formula for calculating the average surface velocity at the contact point is:

[0063] In the formula, and The parameters are dimensionless, namely: , .

[0064] In this embodiment, the contact state between the steel ball and the corrugated cage pocket is further shown in Figure 5. The normal contact force between the steel ball and the cage... There are three states: no contact, single-point contact, and critical clamping multi-point contact.

[0065] In the formula, This represents the total number of actual contact points. For the single-sided gap of the pocket, for The amount of projection in the radial direction, for The spatial length can be defined by the following formula:

[0066] In the formula, the center of mass of the steel ball Geometry of pockets Their coordinates in the O-xyz coordinate system are as follows:

[0067]

[0068] In the formula, and To maintain the position of the rack's center of gravity, and These are the azimuth angle of the steel ball and the position angle of the pocket, respectively.

[0069] The maximum permissible limit for cage displacement. The formula for calculating the cage width is:

[0070] The force direction from the point of contact to the steel ball is a unit vector. Defined as: .

[0071] Based on Hertzian point contact theory and taking into account damping effects, contact force and Represented as:

[0072] In the formula, The contact damping coefficient is... For contact deformation stiffness.

[0073] The formula for calculating deformation stiffness is:

[0074] The relevant geometric and elliptic parameters are as follows: , , , , , .

[0075] Single-point contact deformation Represented as:

[0076] When the cage offset reaches When clamping occurs, the steel ball and the four corner points of the pocket , , , Contact deformation Represented as:

[0077] The criterion for the occurrence of contact is .

[0078] The position vectors of the four boundary corner points are determined by the following formula:

[0079] In the formula, and The unit vectors for the tangential and radial directions of the pocket. The equivalent diameter of the inner side of the pocket. The radial half-height of the wave bridge section. and These are the outer and inner diameters of the cage, respectively.

[0080] Point of application arrive The direction vector is: .

[0081] Projected contact force in the direction of revolution: The tangential vector of the steel ball's revolution is: Friction between the steel ball and the pocket In contact coordinate system components and satisfy:

[0082] In the formula, the coefficient of contact friction is The value is 0.05, and the direction of the frictional force is opposite to the sliding speed of the steel ball relative to the pocket.

[0083] The structural features and force diagram of the corrugated cage are shown in Figure 6. (The text then abruptly shifts to discussing the mass of the corrugated cage.) The calculation formula is:

[0084] In the formula, To maintain shelf density, For the thickness of stamped steel, The equivalent diameter of the outer side of the pocket. The moment of inertia of the cage is... .

[0085] Establish the differential equations of motion for the wave-shaped cage:

[0086] In the formula, To maintain the transient position vector from the center of mass of the frame to the contact point of the nth steel ball, The z-axis unit vector. and These are the reaction forces of the cage pocket on the steel ball, namely the contact force and the frictional force. , and , These are the projection components of each force on the x-axis and y-axis, respectively.

[0087] In this embodiment, further, the total frictional torque of the bearing... This is the sum of all irreversible energy losses within the system. ,in: Elastic hysteresis friction torque Rolling resistance , It is the elastic hysteresis coefficient; Differential sliding friction torque ,in and The differential sliding friction heat generated between the nth steel ball and the inner and outer raceways along the y2 direction of the contact ellipse is represented as:

[0088]

[0089] The normal contact stress distribution is as follows:

[0090] The frictional torque caused by spin sliding is expressed as:

[0091] in, Let be the spin velocity of the nth steel ball relative to the inner and outer rings, and let the spin friction generate heat. Represented as:

[0092] Lubricant viscous drag friction torque :when hour, ;when hour, In the formula, grease-lubricated deep groove ball bearings... (Oil lubrication is selected based on the bearing). Kinematic viscosity, The value is the rotational speed.

[0093] The average heat generation rate of the bearing within one cycle T is .

[0094] The heat flow allocated to the inner and outer nodes is as follows:

[0095] These are the frictional losses caused by elastic hysteresis, differential sliding, spin sliding, and lubricant drag, respectively.

[0096] Establish as Figure 7 The bearing thermal network model shown contains 7 temperature nodes: steel ball T b The contact surface T between the steel ball and the inner and outer rings bi(o) Lubricating Grease T r Inner and outer rings T i(o) Bearing housing center T A With spindle center T z And the environment T0, the nodes are connected into a thermal grid with different forms of contact thermal resistance, according to Figure 7 Temperature nodes are used to establish a bearing thermal mesh model, such as Figure 8 As shown.

[0097] This paper adopts grease lubrication. Since the grease forms a thin oil film between the steel balls and the inner and outer rings, it is assumed that heat conduction occurs between the bearing and the oil film, and no heat convection occurs. The heat generation inside the bearing is mainly due to frictional losses between the steel balls and the inner and outer rings. The arrows indicate the direction of heat transfer, and R represents the contact thermal resistance between each component. The calculation formulas for the thermal resistance of each component are shown in Table 1. Table 1. Calculation of thermal resistance of internal components of the bearing.

[0098] In the formula, , , , , The corresponding component has a thermal conductivity coefficient; D and d are the outer diameter and inner diameter, respectively. and B is the diameter of the inner and outer raceway bottoms; B is the bearing width. , , , , The convective heat transfer coefficient; and The characteristic length; This refers to the outer diameter of the bearing housing.

[0099] The heat network equations based on Kirchhoff's energy balance law are as follows:

[0100] The transient thermal equilibrium differential equation at any node i is:

[0101] Node heat capacity .

[0102] After time-domain discretization, the backward difference approximation is used:

[0103] Iterate using the Newton-Raphson method until... It reaches thermal steady state.

[0104] In this embodiment, the changes in the fit state caused by the high-speed operation and temperature rise of the bearing are further as follows: Centrifugal change in fitting diameter due to rotational speed:

[0105]

[0106] Changes in the diameter of the mating parts due to temperature variations:

[0107]

[0108] The reduction in radial clearance caused by interference fit:

[0109]

[0110] In the formula, and This refers to the interference between the outer and inner rings. and These are the average values ​​of the outer and inner ring flange diameters and the groove bottom diameter, respectively.

[0111] Update on component free thermal expansion and operating clearance:

[0112]

[0113]

[0114]

[0115] Transient radial clearance increment: .

[0116] The actual radial working clearance is updated as follows: ; The actual axial working clearance is updated as follows: .

[0117] Update the kinematic viscosity of the lubricant using the Walther equation With density :

[0118] In this implementation, the model solution process is further as follows: Figure 9 As shown: First, the geometry and clearance are updated based on steady-state thermal equilibrium. , The contact forces between all steel balls and the raceway during the period were calculated again using a quasi-static model. , With contact angle , Then, linear interpolation is performed based on the real-time azimuth angle of the steel ball, and the result is passed to the steel ball and cage dynamics module. Numerical integration is then performed using the fourth-order Runge-Kutta algorithm.

[0119] To verify the accuracy of the model, the HRB-6210 deep groove ball bearing was tested on a high-speed bearing life enhancement testing machine. The test condition was: radial load. The spindle speed is 4200 rpm. Specific bearing parameters are shown in Table 2. Table 2 Geometric and material parameters of the tested bearing

[0120] The experimental and simulation comparison results are as follows: Temperature rise characteristic verification as follows Figure 10 As shown, the temperature rises exponentially in the first 50 minutes, and reaches thermal steady state after about 150 minutes of operation. The experimental temperature range is 68℃-73℃, while the simulated temperature range is 65℃-72℃, and the temperature rise rate is in high agreement with the extreme values.

[0121] Cage speed slippage verification as follows Figure 11 As shown, during the startup phase, the cage speed rapidly increases and then stabilizes. Looking at the details in the magnified view, regardless of whether the bearing is thermo-coupled or uncoupled, the actual cage speed consistently fluctuates slightly below the theoretical speed of 180 rad / s. This characteristic of the actual speed being slightly lower than the theoretical speed is precisely the cage slippage phenomenon commonly observed in deep groove ball bearings during actual operation. Overall, the simulated speed not only closely matches the theoretical speed but also accurately reflects the macroscopic slippage characteristics of the cage, effectively verifying the correctness of the established wave-shaped cage dynamic model.

[0122] Figure 12 shows a comparison of the results for the cases without and with thermal coupling. Comparing the centroid trajectories in Figure 12(a) without thermal coupling and Figure 12(b) with thermal coupling, it can be observed that without thermal coupling, the centroid trajectory exhibits obvious local disorder and chaos, but the cage does not experience global instability; instead, it fluctuates violently relative to the stable trajectory in Figure 12(b). After introducing the thermal coupling mechanism, this chaotic dispersion phenomenon is significantly suppressed, and the trajectory becomes a highly stable, clear closed curve.

[0123] Observing the time-domain waveform in the y-direction, it can be seen that, without considering thermo-coupling, as shown in Figure 12(a), the cage displacement waveform exhibits strong non-uniformity, accompanied by significant amplitude abrupt changes and impact phenomena. After introducing the thermo-coupling mechanism, as shown in Figure 12(b), the original displacement abrupt changes and high-frequency impact characteristics basically disappear, and the waveform recovers to a smooth periodic oscillation. This improvement in time-domain characteristics is highly consistent with the convergence phenomenon of the aforementioned trajectory.

[0124] The aforementioned dynamic evolution laws are also corroborated in the frequency domain and phase space. In the FFT spectrum without thermal coupling, Figure 12(a), besides the cage's characteristic frequency fc and its harmonics 2fc and 3fc, the bottom of the spectrum is filled with a large amount of continuous noise. Correspondingly, in Figure 12(a), although the Poincaré map points are roughly distributed in one region, they exhibit obvious discreteness and disorder. However, after considering thermal coupling, in Figure 12(b), the Poincaré map highly converges to an isolated point, the noise component in the spectrum of Figure 12(b) is significantly weakened, and the system exhibits periodic motion characteristics.

[0125] In summary, thermo-mechanical coupling can cause an increase in internal bearing temperature and thermal expansion of components, leading to a reduction in internal clearance. This mechanism effectively enhances the kinematic constraint on the corrugated cage, significantly altering its dynamic behavior. Therefore, it is essential to fully consider this crucial thermo-mechanical coupling mechanism when conducting dynamic analysis of corrugated cages in deep groove ball bearings.

[0126] Example 2

[0127] In conjunction with the method in Embodiment 1, this embodiment provides a thermo-coupled deep groove ball bearing corrugated cage dynamic modeling system 2, the system structure diagram of which is shown below. Figure 13 As shown, it includes: The initial parameter acquisition module 21 is used to acquire the initial parameters of the deep groove ball bearing, including geometric parameters, material parameters, operating conditions and lubrication parameters. Temperature field calculation module 22 is used to establish a quasi-static model and a frictional heat generation model of the deep groove ball bearing based on the initial parameters, calculate the average heat generation rate of the bearing in one operating cycle, and iteratively solve the steady-state temperature field of the bearing using the thermal network method. Parameter update module 23 is used to calculate the thermal expansion deformation of each component of the bearing based on the steady-state temperature field, and update the internal working clearance and lubricant kinematic viscosity parameters of the bearing in combination with the centrifugal expansion effect. The contact state re-solution module 24 is used to substitute the updated internal working gap into the quasi-static model to obtain the contact load and contact angle between the steel ball and the inner and outer raceways under the steady-state temperature field. The dynamics solution module 25 is used to establish the nonlinear dynamic equations of the steel ball and the corrugated cage under thermo-mechanical coupling based on the contact load and contact angle, and to obtain the dynamic response results of the corrugated cage.

[0128] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention, and they should all be covered within the scope of the claims and specification of the present invention.

Claims

1. A method for dynamic modeling of the corrugated cage of a thermo-coupled deep groove ball bearing, characterized in that, Includes the following steps: Obtain the initial parameters of the deep groove ball bearing, including geometric parameters, material parameters, operating conditions, and lubrication parameters; Based on the initial parameters, a quasi-static model and a frictional heat generation model for the deep groove ball bearing are established. The average heat generation rate of the bearing in one operating cycle is calculated, and the steady-state temperature field of the bearing is solved iteratively using the thermal network method. The thermal expansion deformation of each component of the bearing is calculated based on the steady-state temperature field, and the internal working clearance and lubricant kinematic viscosity parameters of the bearing are updated in combination with the centrifugal expansion effect. Substitute the updated internal working clearance into the quasi-static model to obtain the contact load and contact angle between the steel ball and the inner and outer raceways under the steady-state temperature field; Based on the contact load and contact angle, a nonlinear dynamic equation for the steel ball and the corrugated cage under thermo-mechanical coupling is established to obtain the dynamic response results of the corrugated cage.

2. The method for dynamic modeling of the corrugated cage of a thermo-coupled deep groove ball bearing according to claim 1, characterized in that, Based on the initial parameters, a quasi-static model and a frictional heat generation model for the deep groove ball bearing are established. The average heat generation rate of the bearing during one operating cycle is calculated, and the steady-state temperature field of the bearing is iteratively solved using the thermal network method. The specific steps include: Calculate transient frictional torque based on quasi-static model The transient frictional torque includes elastic hysteresis frictional torque. Differential sliding friction torque Spin sliding friction torque Friction torque due to lubricant drag ; Regarding the transient frictional torque and rotational speed The product of these factors is integrated over time to obtain the average heat generation rate over one operating cycle T. The average heat generation rate is calculated using the formula for allocating heat to the inner and outer nodes: In the formula, The average heat generation rate over one operating cycle T, where T is the operating cycle. () represents the transient frictional torque. Rotational speed; Using the average heat generation rate distributed to the inner and outer ring nodes as a constant heat source input, a bearing thermal network model based on Kirchhoff's energy balance law is established. The model is iteratively solved until the temperature residual satisfies the set convergence tolerance, thus obtaining the steady-state temperature field.

3. The method for dynamic modeling of a thermo-coupled deep groove ball bearing corrugated cage according to claim 2, characterized in that, In the bearing thermal network model based on Kirchhoff's energy balance law, the transient thermal balance differential equation at any node i is expressed as: In the formula, and The instantaneous temperature of adjacent nodes. Thermal response time Distribute the average heat generation rate to the nodes heat source, The thermal resistance varies with temperature. For nodes The heat capacity at that point.

4. The method for dynamic modeling of the corrugated cage of a thermo-coupled deep groove ball bearing according to claim 1, characterized in that, The thermal expansion deformation of each component of the bearing is calculated based on the steady-state temperature field, and the internal working clearance and lubricant kinematic viscosity parameters of the bearing are updated in conjunction with the centrifugal expansion effect. Specifically, this includes: Calculate the centrifugal change in the fitting diameter caused by bearing speed and the thermal expansion change in the fitting diameter caused by temperature change, and then determine the reduction in radial clearance caused by the interference fit of the outer ring. Reduction in radial clearance caused by interference fit with inner ring ; Calculate the relative thermal deformation of the average diameters of the inner and outer rings of the bearing. The transient radial clearance increment is obtained by combining the reduction in radial clearance. ; Based on the transient radial clearance increment Calculate the actual radial working clearance at the current temperature. And based on this, calculate the actual axial working clearance. This is to update the internal work gaps; The Walther equation is used to describe the change in kinematic viscosity of the lubricant with temperature, based on the lubricant temperature in the steady-state temperature field. Calculate and update lubricant kinematic viscosity parameters : In the formula, A, B, and C are parameters related to the lubricating grease.

5. The method for dynamic modeling of the corrugated cage of a thermo-coupled deep groove ball bearing according to claim 1, characterized in that, Substituting the updated internal working clearance into the quasi-static model, the contact load and contact angle between the steel ball and the inner and outer raceways under a steady-state temperature field are obtained, specifically including: Update the actual radial working clearance Actual axial working clearance Substituting the geometric relationship between the relative displacements of the inner and outer rings of the bearing, the deformation compatibility equation of the steel balls in the bearing's load-bearing and non-load-bearing areas is modified. The modified deformation compatibility equations of the steel ball, the force balance equations of the steel ball, and the force balance equations of the inner ring of the deep groove ball bearing are combined and the discretization method is used to solve for a complete operating cycle. The contact loads between all steel balls and the inner and outer raceways at different locations under steady-state temperature field were calculated. , and contact angle , This provides steady-state boundary conditions for solving the dynamics of the wave-shaped cage.

6. The method for dynamic modeling of the corrugated cage of a thermo-coupled deep groove ball bearing according to claim 1, characterized in that, The nonlinear dynamic equations of the steel ball and the corrugated cage include the differential equations of the steel ball's motion; When establishing the differential equations of motion for the steel ball, a rotating coordinate system for the steel ball is established with the center of mass of the steel ball as the origin. Taking into account the contact force between the steel ball and the inner and outer raceways, the contact drag force, the fluid turbulence resistance, the centrifugal force on the steel ball, and the gyroscopic torque, the differential equations of motion for the steel ball in the directions of rotation and revolution around its own coordinate axis are established.

7. The method for dynamic modeling of a thermo-coupled deep groove ball bearing corrugated cage according to claim 6, characterized in that, When establishing the nonlinear dynamic equations for the steel ball and the corrugated cage under the aforementioned thermo-coupling effect, based on Hertzian point contact theory and damping effects, the nonlinear contact force between the corrugated cage pocket and the steel ball is established. The model, and its calculation formula is: In the formula, This refers to the deformation stiffness at the contact point between the steel ball and the cage pocket. This is the damping coefficient at the contact point between the steel ball and the cage pocket. Let be the contact deformation between the nth steel ball and the inner wall of the cage pocket at a single point of contact. The relative velocity of contact deformation.

8. The method for dynamic modeling of the corrugated cage of a thermo-coupled deep groove ball bearing according to claim 7, characterized in that, Establish the nonlinear dynamic equations of the steel ball and the corrugated cage under thermo-mechanical coupling, including the differential equations of motion of the corrugated cage: In the formula, , For the rotational inertia and mass of the wave-shaped cage, , , To maintain the angular acceleration and translational acceleration of the frame, To maintain the transient position vector from the center of mass of the frame to the contact point of the nth steel ball, To maintain the reaction force of the bracket pocket on the contact force of the steel ball, To maintain the reaction force of the cage on the frictional force of the steel ball, , and , These are the projected components of the corresponding force on the x-axis and y-axis.

9. The method for dynamic modeling of the corrugated cage of a thermo-coupled deep groove ball bearing according to claim 1, characterized in that, Obtain the dynamic response results of the wave-shaped cage, specifically including: At a given time step, the contact load and contact angle of the steel ball at the current moment are obtained by linear interpolation based on the current azimuth angle of the steel ball. Substituting the contact load and contact angle into the nonlinear dynamic equation, and using the fourth-order Runge-Kutta algorithm for numerical integration, the dynamic response results of the corrugated cage are obtained.

10. A thermo-coupled deep groove ball bearing corrugated cage dynamic modeling system, applied to the method described in any one of claims 1 to 9, characterized in that, The system includes: The initial parameter acquisition module is used to acquire the initial parameters of the deep groove ball bearing, including geometric parameters, material parameters, operating conditions and lubrication parameters. The temperature field calculation module is used to establish a quasi-static model and a frictional heat generation model of the deep groove ball bearing based on the initial parameters, calculate the average heat generation rate of the bearing in one operating cycle, and iteratively solve the steady-state temperature field of the bearing using the thermal network method. The parameter update module is used to calculate the thermal expansion deformation of each component of the bearing based on the steady-state temperature field, and update the internal working clearance and lubricant kinematic viscosity parameters of the bearing in combination with the centrifugal expansion effect. The contact state re-solution module is used to substitute the updated internal working gap into the quasi-static model to obtain the contact load and contact angle between the steel ball and the inner and outer raceways under the steady-state temperature field. The dynamics solution module is used to establish the nonlinear dynamic equations of the steel ball and the corrugated cage under thermo-mechanical coupling based on the contact load and contact angle, and to obtain the dynamic response results of the corrugated cage.

Images