Ball screw pair mixed lubrication wear modeling method based on measured raceway topography

Abstract

The application discloses a kind of based on the mixed lubrication wear modeling method of real track topography ball screw pair.This application method is first obtained ball screw pair parameter and carries out contact kinematics modeling;Real rough topography is obtained using replica material and laser confocal microscope;The measured topography is introduced into mixed thermal elastohydrodynamic lubrication model, simultaneously solving total pressure distribution, oil film thickness distribution and contact area average temperature rise distribution;Fluid bearing load and micro asperity bearing load are calculated, and micro asperity load ratio is obtained;According to the time-varying wear coefficient of lubrication film thickness ratio;The wear depth of screw and nut track is calculated;Finally, the geometric parameters are updated according to the wear depth and iteratively calculated, and the wear evolution result varying with service revolution is obtained.The application considers the time-varying influence of real track topography and lubrication state on wear, and realizes the accurate prediction of ball screw pair wear evolution process through closed loop iteration.

Description

Technical Field

[0001] This invention belongs to the field of mechanical transmission system performance prediction and reliability assessment technology. Specifically, it is a method for modeling the hybrid lubrication wear of ball screw pairs based on measured raceway morphology. It can be used to predict the wear evolution, accuracy retention and life of ball screw pairs under long-term service conditions, and provide a theoretical basis for the design optimization and reliability analysis of ball screw pairs. Background Technology

[0002] A ball screw assembly is a highly efficient transmission component that converts rotary motion into linear motion. It utilizes the rolling contact of balls between the screw and nut raceways to transmit motion, offering significant advantages such as high transmission efficiency, low frictional resistance, high positioning accuracy, and sensitive response. Public data shows that the driving torque required by a ball screw assembly is only about one-third that of a traditional sliding screw. Currently, ball screw assemblies are widely used in high-end manufacturing equipment such as CNC machine tools, industrial robots, and automated production lines, and are continuously expanding into fields with extremely stringent precision and reliability requirements, such as aerospace, new energy vehicles, and semiconductor manufacturing equipment.

[0003] However, under complex operating conditions such as high speed, heavy load, frequent start-stop, and alternating loads, the contact interface between the balls and raceways of ball screw assemblies inevitably experiences cumulative wear. This wear directly leads to a decrease in preload, an increase in axial clearance, a reduction in contact stiffness, and an increase in frictional heat, resulting in a series of accuracy degradation problems, including decreased positioning accuracy, poor repeatability, and deterioration of servo tracking performance. In extreme cases, severe wear can even induce raceway surface spalling, ball jamming, or even complete machine failure. Therefore, establishing an accurate ball screw assembly wear prediction model is of significant engineering importance for revealing its accuracy degradation mechanism and guiding product design and maintenance decisions.

[0004] Currently, research on wear of ball screw pairs can be mainly categorized into the following technical approaches.

[0005] Firstly, there are wear calculation methods based on the classic Archard wear model. These methods proportionally determine the wear volume between the ball and raceway to the contact load and sliding distance, and inversely to the material hardness. Existing techniques have attempted to apply this model to predict ball screw wear, obtaining the ball contact load and relative sliding speed through contact mechanics and kinematic analysis, and then calculating the wear amount. Some studies further consider the influence of time-varying loads and dynamically fluctuating feed speeds on wear, establishing stochastic wear models. However, a common limitation of these methods is that the wear coefficient is usually treated as a constant, failing to reflect the nonlinear influence of changes in lubrication state, surface morphology, and contact state over service time on the wear rate. Furthermore, existing research largely focuses on the unidirectional mapping from time-varying loads to wear amount, lacking a unified theoretical framework to describe the redistribution of contact parameters and the feedback evolution of the wear rate caused by the continuous change in raceway contact state after wear.

[0006] Secondly, there is the ball screw pair lubrication state analysis method based on elastohydrodynamic lubrication theory. By constructing an elastohydrodynamic lubrication model of the rough surface of the ball screw pair, the influence of different raceway surface textures and material parameters on lubrication characteristics is studied. This method can accurately obtain the oil film thickness and oil film pressure distribution in the contact area, providing an effective means for quantitative assessment of the lubrication state. However, most existing lubrication analysis studies stop at the calculation of lubrication parameters, failing to organically couple the time-varying evolution of the lubrication state with the wear process. In other words, the lubrication analysis results are not effectively fed back into the determination of the wear coefficient and the calculation of wear amount, thus the regulatory role of lubrication on wear is not fully characterized.

[0007] Thirdly, consider wear and preload decay analysis methods based on microscopic contact characteristics. In recent years, some studies have attempted to establish wear models for ball screw pairs from a microscopic contact perspective. For example, fractal theory is used to characterize the microscopic morphology of the raceway surface, calculate the contact area and contact load of micro-protrusions, and thus establish a wear calculation model based on microscopic contact mechanics. Such methods, to some extent, compensate for the shortcomings of the macroscopic Archard model in characterizing surface morphology. However, related research still mainly focuses on the calculation of wear amount itself, making it difficult to truly reflect the physical essence of the co-evolution of surface morphology and lubrication state during the wear process.

[0008] Fourth, there are methods for analyzing the wear and tribothermal effects of ball screw pairs. These studies incorporate the influence of temperature on material hardness and wear rate, expanding the applicable operating conditions of the wear model. However, existing studies on thermal effects mainly focus on macroscopic temperature field analysis and have not yet effectively coupled the thermo-elastohydrodynamic lubrication effect in the contact area with the wear evolution process.

[0009] In summary, the existing technologies still have significant shortcomings in the following aspects: (1) In terms of obtaining and characterizing the true morphology of the raceway surface, most studies use theoretical assumptions or statistical parameters to describe the rough surface, which fails to reflect the true three-dimensional morphological characteristics of the actual processed raceway, and lacks technical means to effectively introduce the measured morphological data into the lubrication and wear coupling model; (2) In terms of determining the wear coefficient under mixed lubrication conditions, existing methods mostly regard the wear coefficient as an empirical constant related to the material, ignoring the dynamic influence of lubrication state parameters such as the lubrication film thickness ratio and the micro-protrusion bearing ratio on the wear rate, resulting in significant deviations in the wear prediction results under long-term service conditions; (3) In terms of the coupling feedback mechanism between wear and lubrication, existing studies have failed to establish a complete closed-loop iterative framework of "wear changes geometry → geometry changes contact load and lubrication state → lubrication state changes wear coefficient → wear coefficient changes subsequent wear rate", making it difficult to accurately predict the nonlinear wear evolution law of the ball screw pair during long-term service. Summary of the Invention

[0010] In view of this, the purpose of this invention is to provide a method for modeling the hybrid lubrication wear of ball screw pairs based on measured raceway morphology. By introducing measured morphology, thermo-elastohydrodynamic lubrication analysis and wear closed-loop iteration, the method can accurately predict wear evolution and accuracy degradation under service conditions.

[0011] To achieve the above objectives, the present invention provides the following technical solution: A method for modeling mixed lubrication wear of ball screw pairs based on measured raceway morphology includes the following steps: S1: Obtain the geometric parameters, material parameters, operating condition parameters and lubrication parameters of the ball screw pair, and perform contact kinematic modeling to determine the principal curvature, relative sliding speed and entrainment speed of the contact points between the balls and the screw raceway and the nut raceway. S2: Obtain a surface replica of the ball screw raceway using a replicating material, obtain three-dimensional height data of the surface replica using a laser confocal microscope, and perform mirror restoration and macroscopic benchmark morphology removal on the three-dimensional height data to obtain the true rough morphology of the raceway. S3: Introduce the actual rough morphology of the raceway into the hybrid thermo-elasto-fluidic lubrication model of the ball and the raceway, and solve the Reynolds equation, energy equation, film thickness equation, contact deformation equation, lubricant viscosity equation, lubricant density equation and load balance equation simultaneously to obtain the total pressure distribution, oil film thickness distribution and average temperature rise distribution in the contact area. The total pressure distribution includes the pressure distribution in the oil film area and the pressure distribution in the micro-protrusion contact area. S4: Integrate the pressure distribution in the oil film region to obtain the fluid load; integrate the pressure distribution in the micro-protrusion contact region to obtain the micro-protrusion load; and calculate the micro-protrusion load ratio based on the fluid load, the micro-protrusion load, and the total normal load. S5: Determine the time-varying wear coefficient based on the lubricating film thickness ratio; S6: Calculate the wear depth of the lead screw raceway and nut raceway based on the time-varying wear coefficient, the load ratio of the micro-protrusion, the contact pressure, and the relative sliding distance. S7: Update the geometric parameters of the ball screw pair according to the calculated wear depth, and repeat steps S1 to S6 to recalculate the contact load, lubrication state, micro-protrusion load ratio and time-varying wear coefficient to obtain the wear evolution results of the ball screw pair as the service speed changes.

[0012] Furthermore, in step S1, when the ball contacts the lead screw raceway, the lead screw raceway... Axial direction and The principal curvatures along the axial direction are expressed as follows:

[0013] in: For ball bearing edge Principal curvature in the axial direction; For ball bearing edge Principal curvature in the axial direction; For the lead screw raceway edge Principal curvature in the axial direction; For the lead screw raceway edge Principal curvature in the axial direction; The radius of the circle containing the center of the ball is the pitch circle radius; The radius of the ball bearing; Where is the raceway radius; The contact angle between the ball and the lead screw. The helix angle; When the ball contacts the nut raceway, the nut raceway... Axial direction and The principal curvatures along the axial direction are expressed as follows:

[0014] in: For the nut raceway edge Principal curvature in the axial direction; For the nut raceway edge Principal curvature in the axial direction; The contact angle between the ball and the nut.

[0015] Furthermore, in step S1, the relative sliding speed of the lead screw side... Represented as:

[0016] Relative sliding speed on the nut side Represented as:

[0017] in: The speed of the lead screw rotation; The angular velocity of the ball bearings' revolution; The rotational speed of the nut; The angular velocity of the ball's rotation; For the ball's rotational angular velocity at Components along the axial direction; The radius of the pitch circle; The radius of the ball bearing; The contact angle between the ball and the lead screw; The contact angle between the ball and the nut; It is the helix angle.

[0018] Furthermore, in step S1, the lead screw side winding speed... Represented as:

[0019] Nut side suction speed Represented as:

[0020] in: The relative rotational angular velocity of the lead screw with respect to the balls; The relative rotational angular velocity of the nut with respect to the balls; The rotational speed of the nut; The angular velocity of the ball's rotation; For the ball's rotational angular velocity at Components along the axial direction; The radius of the pitch circle; The radius of the ball bearing; The contact angle between the ball and the lead screw; The contact angle between the ball and the nut; It is the helix angle.

[0021] Furthermore, in step S2, the actual roughness morphology of the raceway... Represented as:

[0022] in: This is the raceway height matrix obtained through mirror reconstruction; This is the macroscopic baseline morphology obtained by fitting the original replication height data; Indicates the lead screw side. Indicates the nut side; This indicates the number of revolutions the ball screw pair operates in.

[0023] Furthermore, in step S3, the Reynolds equation is expressed as:

[0024] in: The density of the lubricant; Oil film thickness; For lubricant viscosity; Oil film pressure; The entrainment speed on the raceway side. Indicates the lead screw side. Indicates the nut side; For time; The energy equation is expressed as:

[0025] in: Specific heat capacity; For temperature; , and They are respectively , and The velocity component in the direction; Thermal conductivity; The coordinates are along the film thickness direction; The film thickness equation is expressed as:

[0026] in: This represents the initial oil film thickness. Along the contact point The equivalent radius of curvature in the direction; Along the contact point The equivalent radius of curvature in the direction; This represents the amount of elastic deformation. The composite deterministic roughness morphology of the ball and raceway is given, and:

[0027] in: The height of the ball surface morphology in the local contact coordinate system after coordinate transformation; This represents the true rough morphology of the raceway; The contact deformation equation is expressed as:

[0028] Wherein: equivalent elastic modulus , and These are the elastic modulus and Poisson's ratio of the ball bearing, respectively. and These are the elastic modulus and Poisson's ratio of the raceway, respectively. For contact pressure distribution, The contact area; The viscosity equation for the lubricant is expressed as:

[0029] in: This represents the initial environmental viscosity. Pressure coefficient; Temperature coefficient; For temperature; Ambient temperature; The lubricant density equation is expressed as follows:

[0030] in: For environmental density; It is the density temperature coefficient; The load balance equation is expressed as:

[0031] in: This is the total normal load at the contact point between the ball and the raceway; This represents the total pressure distribution. Oil film lubrication area; This is the contact area of ​​the micro-protrusion.

[0032] Furthermore, in step S4, the load ratio of the micro-protrusions... Represented as:

[0033] Wherein: fluid bearing load Micro-protrusions bear loads ; This represents the total pressure distribution. To run to Turning point The total normal load borne by each ball; This indicates the number of revolutions during operation.

[0034] Furthermore, in step S5, the thickness ratio of the lubricating film is... Defined as the ratio of the nominal minimum oil film thickness to the composite root mean square roughness:

[0035] in: Minimum oil film thickness, Oil film thickness; The root mean square roughness is the combined roughness of the ball and raceway. The composite root mean square roughness Represented as:

[0036] in: denoted as the root mean square roughness of the ball surface; For the raceway contact surface during cumulative operation to Root mean square roughness of the surface during rotation.

[0037] Furthermore, in step S5, the time-varying wear coefficient Represented as:

[0038] in: The ratio of the thickness of the lubricating film; Let be the wear coefficient under boundary lubrication, and:

[0039]

[0040] in: These are dimensionless load parameters; These are dimensionless material parameters; These are dimensionless raceway surface morphology parameters; The overall elastic modulus of the contact body; The equivalent elastic modulus of the contact between the ball and the raceway of the screw / nut; Extending the contact area between the balls and the lead screw / nut raceway The equivalent radius of curvature in the direction; The viscosity-pressure coefficient of the lubricant; The root mean square roughness is the combined roughness of the ball and raceway.

[0041] Furthermore, in step S6, the depth of mixed lubrication wear generated by a single ball in the contact area on the screw side / nut side Represented as:

[0042] in: This represents the amount of wear per unit time under mixed lubrication conditions. Contact area; The wear coefficient is time-varying. The percentage of load on the micro-protrusions; For the normal contact load of a single ball on the screw side / nut side; The relative sliding speed of a single ball relative to the screw side / nut side; For the number of revolutions; Material hardness; The angular velocity of the lead screw; For the semi-major axis of the contact ellipse; The effective contact length of the ball on the raceway; The helix angle; The cumulative wear depth on the raceway between the ball and the screw / nut Represented as:

[0043] in: The number of balls that can be effectively carried in one raceway; Wear depth of ball screw assembly Represented as:

[0044] in: This represents the cumulative wear depth between the ball bearings and the lead screw. This represents the cumulative wear depth on the raceway between the ball and the nut.

[0045] The beneficial effects of this invention are as follows: This invention, based on a measured raceway morphology, provides a method for modeling the mixed lubrication wear of ball screw pairs, achieving the following technical effects: (1) Improve the accuracy and physical reality of wear prediction: Compared with the existing Archard model that treats the wear coefficient as a constant, this invention is based on the measured raceway morphology and obtains the real three-dimensional micro morphology through laser confocalization and benchmark removal, avoiding the errors caused by theoretical assumptions or statistical parameters, and making wear calculation leap from "morphological approximation" to "morphological reality". (2) Achieving dynamic coupling of lubrication state with wear rate: This invention solves the oil film pressure, micro-protrusion contact pressure and load ratio through a hybrid thermo-elastohydrodynamic lubrication model, and determines the time-varying wear coefficient in real time according to the film thickness ratio, so that the wear coefficient is dynamically adjusted with the lubrication state, overcoming the problem that the constant wear coefficient deviates significantly under long-term service conditions. (3) Constructing a closed-loop iterative framework of “wear-morphology-lubrication-wear”: This invention updates the raceway geometric parameters by feeding back the calculated wear depth, and recalculates the contact load, lubrication state and time-varying wear coefficient, fully characterizing the nonlinear evolution process of wear changing morphology, morphology changing lubrication, and lubrication changing wear rate, breaking through the limitations of the unidirectional transmission model; (4) Significant engineering application value: This invention can obtain the wear evolution results of ball screw pairs as the service speed changes, providing direct data support for accuracy retention prediction, remaining life assessment and preload decay analysis, and effectively guiding product design and maintenance decisions. Attached Figure Description

[0046] To make the objectives, technical solutions, and beneficial effects of this invention clearer, the following figures are provided for illustration: Figure 1 This is a flowchart of the method for modeling mixed lubrication wear of ball screw pairs based on measured raceway morphology according to the present invention. Figure 2 This is a schematic diagram of the contact interface between the ball and the raceway and the local contact coordinate system; Figure 3 A schematic diagram of the acquisition of raceway surface morphology data for a ball screw pair; Figure 4 The distribution diagram of the measured three-dimensional surface morphology height of the ball screw raceway; Figure 5 Comparison of the morphology of the ball screw raceway replica before and after mirror restoration and datum removal: (a) before removal; (b) after removal. Figure 6 This is a diagram showing the distribution of oil film pressure at the contact points of the ball screw pair. Figure 7 This is a distribution diagram of the oil film thickness at the contact points of the ball screw pair; Figure 8 This is a distribution diagram of the average temperature rise of the oil film at the contact points of the ball screw pair. Figure 9 This is a diagram showing the load percentage of micro-protrusions. Figure 10 This is a time-varying wear coefficient diagram; Figure 11 This is a diagram showing the total wear depth of the raceways in the ball screw assembly. Detailed Implementation

[0047] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can better understand and implement the present invention. However, the embodiments described are not intended to limit the present invention.

[0048] This embodiment aims to propose a method for modeling hybrid lubrication wear of ball screw pairs based on measured raceway morphology. First, the geometric parameters, material parameters, operating condition parameters, and lubrication parameters of the ball screw pair are obtained, followed by contact kinematic modeling. A surface replica of the ball screw raceway is obtained using a replica material. The three-dimensional height data of the surface replica is obtained through a laser confocal microscope, and the three-dimensional height data is then mirrored and the macroscopic reference morphology is removed to obtain the true roughness morphology of the raceway. This true roughness morphology is then introduced into a hybrid thermo-elasto-fluidic lubrication model of the balls and raceway to solve for oil film pressure, oil film thickness, and contact wear. The system calculates the contact temperature and micro-protrusion contact pressure; it obtains the fluid load based on the oil film pressure integral and the micro-protrusion load based on the micro-protrusion contact pressure integral, and calculates the micro-protrusion load ratio; it determines the time-varying wear coefficient based on the lubricating film thickness ratio; it calculates the wear depth of the screw raceway and nut raceway based on the time-varying wear coefficient, micro-protrusion load ratio, contact pressure, and relative sliding distance; it then recalculates the ball screw pair geometric parameters based on the calculated wear depth, and recalculates the contact load, lubrication state, micro-protrusion load ratio, and time-varying wear coefficient to obtain the wear evolution results of the ball screw pair as the service speed changes.

[0049] This embodiment takes the 3210 model ball screw pair as the research object and provides a more detailed description of the specific implementation method of the mixed lubrication wear modeling method for ball screw pairs based on measured raceway morphology. The relevant parameters of this model ball screw pair are shown in Table 1, the lubricating oil parameters in Table 2, and the material parameters in Table 3.

[0050] Table 1 Ball Screw Pair Parameters

[0051] Table 2 Lubricating Oil Parameters

[0052] Table 3 Material Parameters

[0053] Specifically, such as Figure 1 As shown, this embodiment describes a method for modeling the mixed lubrication wear of ball screw pairs based on measured raceway morphology, which includes the following steps.

[0054] S1, Contact Kinematics Modeling. Obtain the geometric parameters, material parameters, operating condition parameters, and lubrication parameters of the ball screw pair, and perform contact kinematics modeling to determine the principal curvature, relative sliding speed, and entrainment speed at the contact points between the balls and the screw raceway and nut raceway.

[0055] (1) Calculation of principal curvature at the contact point between the ball and the raceway.

[0056] When the ball contacts the screw raceway, the screw raceway... Axial direction and The principal curvatures along the axial direction are expressed as follows:

[0057] in: For ball bearing edge Principal curvature in the axial direction; For ball bearing edge Principal curvature in the axial direction; For the lead screw raceway edge Principal curvature in the axial direction; For the lead screw raceway edge Principal curvature in the axial direction; The radius of the circle containing the center of the ball is the pitch circle radius; The radius of the ball bearing; The radius of the raceway (the radius of the arc of the lead screw raceway or nut raceway); The contact angle between the ball and the lead screw. It is the helix angle, that is, the helix angle of the ball's spiral trajectory.

[0058] When the ball contacts the nut raceway, the nut raceway... Axial direction and The principal curvatures along the axial direction are expressed as follows:

[0059] in: For the nut raceway edge Principal curvature in the axial direction; For the nut raceway edge Principal curvature in the axial direction; The contact angle between the ball and the nut.

[0060] (2) Calculation of the angular velocity of the ball's revolution and rotation.

[0061] At the pure rolling radius, the balls and raceways have equal velocities in the rolling direction and at the center of the contact ellipse. For example... Figure 2 As shown, in the lead screw drive transmission method, the lead screw is... The ball rotates at an angular velocity of [value missing], and the ball's revolution angular velocity is [value missing]. The nut moves only in a straight line within an absolute reference frame. Under these conditions, assuming the center of the ball remains fixed in space, the rotational speed of the leadscrew can be denoted as... The rotational speed of the nut This occurs in the opposite direction. The angular velocity of the ball's revolution... and the angular velocity of the ball's rotation They can be represented as follows:

[0062] in: The angular velocity of the ball bearings' revolution; The angular velocity of the ball's rotation; The speed of the lead screw rotation; The pure rolling radius of the ball and the screw raceway; The pure rolling radius of the raceway between the ball and the nut; The radius of the pitch circle; For lead; The pitch angle; Yaw angle; The contact angle between the ball and the lead screw; The contact angle between the ball and the nut; It is the helix angle.

[0063] The sliding of the contact interface of the ball screw pair can no longer be simply understood as relative sliding in a single rolling direction, but should be understood as the result of the combined action of sliding in the rolling direction and lateral sliding.

[0064] Relative sliding speed of the lead screw side Represented as:

[0065] Relative sliding speed on the nut side Represented as:

[0066] in: The speed of the lead screw rotation; The angular velocity of the ball bearings' revolution; The rotational speed of the nut; The angular velocity of the ball's rotation; For the ball's rotational angular velocity at Components along the axial direction; The radius of the pitch circle; The radius of the ball bearing; The contact angle between the ball and the lead screw; The contact angle between the ball and the nut; It is the helix angle.

[0067] (3) Sucking speed.

[0068] Specifically, the lead screw side winding speed Represented as:

[0069] Nut side suction speed Represented as:

[0070] in: The relative rotational angular velocity of the lead screw with respect to the balls; The relative rotational angular velocity of the nut with respect to the balls; The rotational speed of the nut; The angular velocity of the ball's rotation; For the ball's rotational angular velocity at Components along the axial direction; The radius of the pitch circle; The radius of the ball bearing; The contact angle between the ball and the lead screw; The contact angle between the ball and the nut; It is the helix angle.

[0071] (4) Calculation of ball contact load.

[0072] When the nut in a ball screw assembly is subjected to an axial external load, a force is generated between the internal screw and nut raceways and the balls, which can be expressed as:

[0073] in: Indicates axial external load; Indicates the number of thread starts; Indicates the number of cycles; This indicates the number of balls that are effectively supported in one raceway. The ball position angle; For the ball bearings and screw / nut raceways during operation The actual contact angle during rotation.

[0074] S2, Obtaining the true roughness morphology of the raceway. A surface replica of the ball screw raceway is obtained using a replicating material. The three-dimensional height data of the surface replica is obtained through a laser confocal microscope. The three-dimensional height data is then mirrored and the macroscopic reference morphology is removed to obtain the true roughness morphology of the raceway.

[0075] (1) Three-dimensional height data acquisition.

[0076] In this step, it is necessary to obtain the actual surface morphology data of the ball screw pair raceway and set repeatable positioning marks on the screw raceway and nut raceway. For example... Figure 3 As shown, the raceway surface was cleaned before measurement. A replica mold was pressed into the raceway area to be measured. After the replica material cured, it was peeled off to form a surface replica. The replica was scanned using a laser confocal microscope to obtain the original height matrix in the replica's coordinate system. The true height matrix of the raceway was then recovered through mirroring. ,like Figure 4 As shown. Among them, , Indicates the lead screw side. This represents the nut side. The mirrored raceway height field macroscopically exhibits a continuous arc-shaped undulation, higher on both sides and lower in the middle. The original three-dimensional height data simultaneously contains the macroscopic baseline morphology formed by the raceway's circular curvature and the microscopic peak and valley information superimposed upon it. Therefore, Figure 4 The height matrix shown cannot be directly used as the roughness input in the hybrid lubrication model. It is necessary to further remove the macroscopic reference morphology to avoid the overall curvature of the raceway being repeatedly included in the local film thickness equation.

[0077] (2) Macro-benchmark morphology removal.

[0078] Because the raceway height matrix reconstructed through mirroring is affected by macroscopic geometric factors such as the raceway's own curvature and the measurement posture tilt, the true 3D height data of the raceway surface needs to be fitted and removed using a macroscopic benchmark morphology to eliminate the influence of macroscopic shape factors such as raceway curvature and measurement posture. This process yields the true raceway surface morphology used to characterize the microscopic peaks and valleys of the raceway. ,like Figure 5 As shown. Figure 5 (a) The 3D topography before removal still retains a clear macroscopic surface trend, with microscopic roughness features superimposed on this trend; after baseline removal... Figure 5 In (b), the height fluctuations are basically distributed around the zero reference, the macroscopic curvature is weakened, while the local peaks, valleys and processing textures are preserved. The resulting true roughness morphology can be introduced into the film thickness equation as part of the composite deterministic roughness morphology, so that the subsequent oil film thickness, local pressure and micro-protrusion contact state change with the measured peak and valley positions on the surface.

[0079] Specifically, the actual roughness of the raceway Represented as:

[0080] in: This is the raceway height matrix obtained through mirror reconstruction; This is the macroscopic baseline morphology obtained by fitting the original replication height data; Indicates the lead screw side. Indicates the nut side; This indicates the number of revolutions the ball screw pair operates in.

[0081] (3) Composite roughness morphology and root mean square roughness.

[0082] Composite deterministic roughness morphology of balls and raceways It can be written as:

[0083] in: The height of the ball bearing surface morphology in the local contact coordinate system after coordinate transformation.

[0084] The surface roughness of the balls can be obtained through direct microscopic measurement. The combined root-mean-square roughness of the balls and raceways. for:

[0085] in, This indicates the root mean square roughness of the ball surface; Indicates the raceway contact surface after cumulative operation to Root mean square roughness of the surface during rotation.

[0086] S3, Solving the Hybrid Thermo-Elasto-Hydraulic Lubrication Model. The actual roughness morphology of the raceway is introduced into the hybrid thermo-elasto-hydraulic lubrication model of the ball and raceway. The Reynolds equation, energy equation, film thickness equation, contact deformation equation, lubricant viscosity equation, lubricant density equation, and load balance equation are solved simultaneously to obtain the total pressure distribution, oil film thickness distribution, and average temperature rise distribution in the contact area. The total pressure distribution includes the pressure distribution in the oil film region and the pressure distribution in the micro-protrusion contact region, such as... Figure 6-9 As shown. There are certain local undulations at the top of the high-pressure zone, such as... Figure 6 As shown, this reflects the disturbance of measured roughness morphology on local film thickness and pressure gradient. The oil film load and micro-protrusion contact load are obtained by integrating the local pressure in different load-bearing areas, and their sum satisfies the overall load balance. The oil film thickness is smaller near the contact center and gradually increases towards both sides along the contact direction, exhibiting an overall distribution characteristic of being thinner at the center and thicker at the periphery, as shown in the figure. Figure 7 As shown. The minimum oil film thickness near the contact center is used in conjunction with the composite root mean square roughness to calculate the lubrication film thickness ratio, thereby determining whether the ball and raceway are in boundary lubrication, mixed lubrication, or full oil film lubrication state, and providing state parameters for determining the time-varying wear coefficient. The average temperature rise in the contact area has a strong correlation with the high-pressure load area. The temperature rise is significantly higher in the main contact area, decreases rapidly after leaving the contact area, and there are local fluctuations near the outlet, such as... Figure 8 As shown in the figure. The results demonstrate the coupling effect of pressure, oil film shear, and heat transfer processes. The obtained temperature field is further fed back into the calculation of lubricant viscosity and density, enabling the thermo-elasto-fluid coupling solution of pressure, film thickness, and temperature.

[0087] Given the significant coupling effect between the pressure, film thickness, and temperature rise of the lubricating medium in the contact area between the ball and the raceway, and the dynamic changes in viscosity, density, and elastic deformation with surface morphology and local geometric conditions, this embodiment further introduces thermo-elastohydrodynamic lubrication analysis to solve for the oil film pressure, film thickness distribution, and average temperature rise in the contact area, and extracts state parameters such as film thickness ratio, providing a basis for subsequent wear coefficient correction and cumulative wear evolution calculation.

[0088] (1) Reynolds equation.

[0089] In this embodiment, the Reynolds equation is expressed as:

[0090] in: The density of the lubricant; Oil film thickness; For lubricant viscosity; Oil film pressure; The entrainment speed on the raceway side. Indicates the lead screw side. Indicates the nut side; For time.

[0091] In this embodiment, the lead screw side winding speed Represented as:

[0092] In this embodiment, the nut-side winding speed Represented as:

[0093] in: The relative rotational angular velocity of the lead screw with respect to the balls; The relative rotational angular velocity of the nut with respect to the balls; The rotational speed of the nut; The angular velocity of the ball's rotation; For the ball's rotational angular velocity at Components along the axial direction; The radius of the pitch circle; The radius of the ball bearing; The contact angle between the ball and the lead screw; The contact angle between the ball and the nut; It is the helix angle.

[0094] (2) Energy equation.

[0095] In this embodiment, the energy equation is expressed as:

[0096] in: Specific heat capacity; For temperature; , and They are respectively , and The velocity component in the direction; Thermal conductivity; The coordinates are along the film thickness direction.

[0097] The temperature boundary conditions for the upper and lower surfaces of the contact surface are:

[0098] in: The interface between the oil film and the ball ( Temperature at (location); The oil film and the raceway contact interface ( Temperature at (location); The thermal conductivity coefficient of the lubricant; The density of the ball bearing material; The specific heat capacity of the ball bearing material; The thermal conductivity coefficient of the ball bearing material; For the ball surface at The velocity component in the direction; The density of the raceway material; The specific heat capacity of the raceway material; The thermal conductivity coefficient of the raceway material; For the raceway surface in The velocity component in the direction; and The first one in the contact area grid nodes and coordinate; Let be the integral variable, representing the direction along... The position coordinates of the direction; The ambient temperature.

[0099] (3) Film thickness equation.

[0100] In this embodiment, the film thickness equation is expressed as:

[0101] in: This represents the initial oil film thickness. Along the contact point The equivalent radius of curvature in the direction, and ; Along the contact point The equivalent radius of curvature in the direction, and ; This represents the amount of elastic deformation. This represents the composite deterministic rough morphology of the ball and raceway.

[0102] In this embodiment, the combined deterministic roughness morphology of the ball and raceway is described. for:

[0103] in: The height of the ball surface morphology in the local contact coordinate system after coordinate transformation; This represents the true rough morphology of the raceway; (4) Contact deformation equation.

[0104] In this embodiment, the contact deformation equation is expressed as:

[0105] Wherein: equivalent elastic modulus , and These are the elastic modulus and Poisson's ratio of the ball bearing, respectively. and These are the elastic modulus and Poisson's ratio of the raceway, respectively. For contact pressure distribution, This refers to the contact area.

[0106] (5) Lubricant viscosity equation.

[0107] In this embodiment, the lubricant viscosity equation is expressed as:

[0108] in: This represents the initial environmental viscosity. Pressure coefficient; Temperature coefficient; For temperature; The ambient temperature.

[0109] (6) Lubricant density equation.

[0110] In this embodiment, the lubricant density equation is expressed as:

[0111] in: For environmental density; This is the density coefficient.

[0112] (7) Oil film pressure load balance equation in the contact area.

[0113] In this embodiment, the load balance equation is expressed as:

[0114] in: This is the total normal load at the contact point between the ball and the raceway; This represents the total pressure distribution. This is the oil film lubrication area; This is the contact area of ​​the micro-protrusion.

[0115] S4, Calculation of micro-protrusion load percentage. Integrate the pressure distribution in the oil film region to obtain the fluid load; integrate the pressure distribution in the micro-protrusion contact region to obtain the micro-protrusion load; and calculate the micro-protrusion load percentage based on the fluid load, the micro-protrusion load, and the total normal load, as follows: Figure 10 As shown.

[0116] In this embodiment, the micro-protrusion load ratio is defined as the ratio of the load borne by the micro-protrusion to the total normal load:

[0117] Wherein: fluid bearing load Micro-protrusions bear loads ; This represents the total pressure distribution. To run to Turning point The total normal load borne by each ball; This indicates the number of revolutions during operation.

[0118] As the operating speed increases, the proportion of micro-protrusion load shows a trend of first decreasing rapidly and then decreasing slowly, such as... Figure 9 As shown, during the initial break-in phase, a significant portion of the load is borne by the surface micro-protrusions. As the break-in period progresses and the surface condition is updated, the proportion of load directly borne by the micro-protrusions decreases, while the proportion of load borne by the fluid oil film increases accordingly. This micro-protrusion load percentage serves as a load correction factor in the wear model, reflecting the share of load truly involved in solid contact wear under mixed lubrication conditions.

[0119] S5, Determine the time-varying wear coefficient. The time-varying wear coefficient is determined based on the lubricating film thickness ratio.

[0120] To characterize the temporal evolution of the lubrication state in the contact area and to provide directly callable state parameters for subsequent wear models, this embodiment introduces a time-varying film thickness ratio. As a characterization index of lubrication status. It is usually defined as the ratio of the nominal minimum oil film thickness to the composite root mean square roughness height, i.e. To facilitate characterizing the lubrication state of the ball screw pair contact interface, the film thickness ratio engineering criterion is referenced. Operating conditions with a value ≤0.5 are characterized as boundary lubrication conditions, while those with a value <0.5 are considered boundary lubrication conditions. Operating conditions ≤4 are characterized as mixed lubrication states. A roughness greater than 4 indicates a fully lubricated oil film condition. Composite roughness is typically the resultant of the second norm of the RMS roughness of the two contact surfaces. Therefore, for ball screw pairs under axial loads... Thermo-elasto-hydrodynamic (TEHL) solution under action, lubricating film thickness ratio Defined as the ratio of the nominal minimum oil film thickness to the composite root mean square roughness:

[0121] in: The minimum oil film thickness is calculated using TEHL when the lead screw has run a certain number of revolutions; Oil film thickness; The root mean square roughness is the combined roughness of the ball and raceway.

[0122] Running-in leads to gradual changes in lubrication quality, and wear mainly occurs at the prominent peaks on the surface rather than in the middle or valley areas. The RMS index alone has relatively limited ability to reflect the flattening of these peaks; therefore, it is necessary to introduce [a specific indicator] into wear modeling. This reflects the degree of wear and tear.

[0123] In this embodiment, the time-varying wear coefficient Represented as:

[0124] in: The ratio of the thickness of the lubricating film; Let be the wear coefficient under boundary lubrication, and:

[0125]

[0126] in: These are dimensionless load parameters; These are dimensionless material parameters; These are dimensionless raceway surface morphology parameters; The overall elastic modulus of the contact body; The equivalent elastic modulus of the contact between the ball and the raceway of the screw / nut; Extending the contact area between the balls and the lead screw / nut raceway The equivalent radius of curvature in the direction; The viscosity-pressure coefficient of the lubricant; The root mean square roughness is the combined roughness of the ball and raceway.

[0127] The time-varying wear coefficient shows a significant decreasing trend with increasing operating speed, such as... Figure 10 As shown, during the initial running-in period, the surface peaks have strong contact, resulting in a high wear coefficient. As the surface roughness and oil film conditions change, the wear coefficient gradually decreases and tends to stabilize. Using this time-varying wear coefficient avoids the long-term wear rate distortion caused by treating the wear coefficient as a fixed constant in the traditional Arcard model.

[0128] S6, Wear Depth Calculation. Based on the time-varying wear coefficient, the load ratio of the micro-protrusions, the contact pressure, and the relative sliding distance, the wear depth of the screw raceway and the nut raceway are calculated respectively. Figure 11 As shown.

[0129] The classic Archard wear model describes the fundamental relationship between the total wear volume and the normal load, sliding distance, and material hardness during relative sliding. It is suitable for quantifying the average wear behavior under steady-state conditions. Its core idea is that the heavier the contact and the longer the sliding, the larger the volume of material removed, and the harder the material, the smaller the wear. Existing research mostly treats wear accumulation under time-varying loads and lubrication state analysis separately, lacking a unified study on the time-varying wear evolution of ball screw pairs. In particular, it lacks a systematic study on how the evolution of lubrication state over time affects the wear coefficient and how changes in the geometric parameters of the ball screw pair caused by complex raceway morphology are fed back into the wear process.

[0130] The traditional Archard wear model can be expressed as:

[0131] in: This represents the wear volume; The wear coefficient; This represents the normal contact load experienced at the wear contact point. This is the sliding distance; Let the material hardness be denoted by . Therefore, the wear volume generated by a single ball in the contact area on the screw / nut side of the ball screw assembly can be expressed as:

[0132] The sliding distance of a single ball relative to the screw side / nut side can be expressed as:

[0133] in: This indicates the relative sliding speed of the ball screw assembly on the screw side / nut side. This indicates the number of revolutions of the ball screw assembly. This represents the angular velocity of the lead screw. A new formula for calculating the wear volume of a single ball in the contact area on the screw / nut side of a ball screw pair:

[0134] During machine tool operation, the ball screw assembly operates under continuous lubrication. During machining, the ball screw assembly frequently starts and stops to drive the column. When the nut flange is subjected to axial load from the column, the contact portion between the balls and raceways is borne by the fluid, while the other portion is borne by the micro-protrusions. Existing research on ball screw assembly wear mainly focuses on dry contact or direct micro-protrusion contact, without considering the lubricant load at the contact point. Under mixed elastohydrodynamic lubrication, by considering the proportion of micro-protrusion load to the total normal load, a wear model for the ball screw assembly per unit time under mixed elastohydrodynamic lubrication is established, which can be specifically expressed as:

[0135] in: The amount of wear per unit time under mixed lubrication conditions. This represents the percentage of micro-protrusion load on the contact surface between the ball and the raceway.

[0136] Since the wear depth of a single ball in the contact area on the screw / nut side under mixed elastohydrodynamic lubrication conditions can be expressed as the ratio of wear amount to contact area, its expression is:

[0137] in: This represents the wear depth per unit time under mixed elastohydrodynamic lubrication conditions.

[0138] Since the traditional Archard wear model typically accumulates wear based on continuous sliding contact, while in ball screw pairs, the balls only experience primary wear near the local contact ellipse of the screw raceway and nut raceway, and this wear area is repeatedly distributed within the effective stroke as the nut reciprocates axially, a correction factor is needed to transform the ideal full-contact sliding wear into an equivalent wear amount that conforms to the actual local contact characteristics. The corrected wear depth of a single ball in the contact area on the screw / nut side can be expressed as:

[0139] in: This represents the amount of wear per unit time under mixed lubrication conditions. Contact area; The wear coefficient is time-varying. The percentage of load on the micro-protrusions; For the normal contact load of a single ball on the screw side / nut side; The relative sliding speed of a single ball relative to the screw side / nut side; For the number of revolutions; Material hardness; The angular velocity of the lead screw; For the semi-major axis of the contact ellipse; The effective contact length of the ball on the raceway; It is the helix angle.

[0140] Therefore, the cumulative wear depth on the raceway between the ball and the screw / nut can be obtained, and its expression is:

[0141] in: This refers to the number of balls that can be effectively supported in one raceway.

[0142] When the total number of revolutions reaches The wear depth of the ball screw assembly can be expressed as:

[0143] in: This represents the cumulative wear depth between the ball bearings and the lead screw. This represents the cumulative wear depth on the raceway between the ball and the nut.

[0144] The wear depth of the ball screw assembly increases continuously with the operating speed, with a rapid increase in the initial stage, followed by a gradual slowdown in the rate of increase. Figure 11 As shown in the figure, this trend is consistent with the decrease in the proportion of micro-protrusion load and the decay of the time-varying wear coefficient, indicating that in this embodiment, although the cumulative wear depth continues to increase as the running-in process proceeds, the amount of new wear per unit revolution gradually decreases.

[0145] S7: Update the geometric parameters of the ball screw pair according to the calculated wear depth, and repeat steps S1 to S6 to recalculate the contact load, lubrication state, micro-protrusion load ratio and time-varying wear coefficient to obtain the wear evolution results of the ball screw pair as the service speed changes.

[0146] The above-described embodiments are merely preferred embodiments provided to fully illustrate the present invention, and the scope of protection of the present invention is not limited thereto. Equivalent substitutions or modifications made by those skilled in the art based on the present invention are all within the scope of protection of the present invention. The scope of protection of the present invention is defined by the claims.

Claims

1. A method for modeling hybrid lubrication wear of ball screw pairs based on measured raceway morphology, characterized in that, Includes the following steps: S1: Obtain the geometric parameters, material parameters, operating condition parameters and lubrication parameters of the ball screw pair, and perform contact kinematic modeling to determine the principal curvature, relative sliding speed and entrainment speed of the contact points between the balls and the screw raceway and the nut raceway. S2: Obtain a surface replica of the ball screw raceway using a replicating material, obtain three-dimensional height data of the surface replica using a laser confocal microscope, and perform mirror restoration and macroscopic benchmark morphology removal on the three-dimensional height data to obtain the true rough morphology of the raceway. S3: Introduce the actual rough morphology of the raceway into the hybrid thermo-elasto-fluidic lubrication model of the ball and the raceway, and solve the Reynolds equation, energy equation, film thickness equation, contact deformation equation, lubricant viscosity equation, lubricant density equation and load balance equation simultaneously to obtain the total pressure distribution, oil film thickness distribution and average temperature rise distribution in the contact area. The total pressure distribution includes the pressure distribution in the oil film area and the pressure distribution in the micro-protrusion contact area. S4: Integrate the pressure distribution in the oil film region to obtain the fluid load; integrate the pressure distribution in the micro-protrusion contact region to obtain the micro-protrusion load; and calculate the micro-protrusion load ratio based on the fluid load, the micro-protrusion load, and the total normal load. S5: Determine the time-varying wear coefficient based on the lubricating film thickness ratio; S6: Calculate the wear depth of the lead screw raceway and nut raceway based on the time-varying wear coefficient, the load ratio of the micro-protrusion, the contact pressure, and the relative sliding distance. S7: Update the geometric parameters of the ball screw pair according to the calculated wear depth, and repeat steps S1 to S6 to recalculate the contact load, lubrication state, micro-protrusion load ratio and time-varying wear coefficient to obtain the wear evolution results of the ball screw pair as the service speed changes.

2. The method for modeling mixed lubrication wear of ball screw pairs based on measured raceway morphology according to claim 1, characterized in that, In step S1, when the ball contacts the screw raceway, the screw raceway moves along... Axial direction and The principal curvatures along the axial direction are expressed as follows: in: For ball bearing edge Principal curvature in the axial direction; For ball bearing edge Principal curvature in the axial direction; For the lead screw raceway edge Principal curvature in the axial direction; For the lead screw raceway edge Principal curvature in the axial direction; The radius of the circle containing the center of the ball is the pitch circle radius; The radius of the ball bearing; Where is the raceway radius; The contact angle between the ball and the lead screw. The helix angle; When the ball contacts the nut raceway, the nut raceway... Axial direction and The principal curvatures along the axial direction are expressed as follows: in: For the nut raceway edge Principal curvature in the axial direction; For the nut raceway edge Principal curvature in the axial direction; The contact angle between the ball and the nut.

3. The method for modeling mixed lubrication wear of ball screw pairs based on measured raceway morphology according to claim 1, characterized in that, In step S1, the relative sliding speed of the lead screw side Represented as: Relative sliding speed on the nut side Represented as: in: The speed of the lead screw rotation; The angular velocity of the ball bearings' revolution; The rotational speed of the nut; The angular velocity of the ball's rotation; For the ball's rotational angular velocity at Components along the axial direction; The radius of the pitch circle; The radius of the ball bearing; The contact angle between the ball and the lead screw; The contact angle between the ball and the nut; It is the helix angle.

4. The method for modeling mixed lubrication wear of ball screw pairs based on measured raceway morphology according to claim 1, characterized in that, In step S1, the lead screw side winding speed Represented as: Nut side suction speed Represented as: in: The relative rotational angular velocity of the lead screw with respect to the balls; The relative rotational angular velocity of the nut with respect to the balls; The rotational speed of the nut; The angular velocity of the ball's rotation; For the ball's rotational angular velocity at Components along the axial direction; The radius of the pitch circle; The radius of the ball bearing; The contact angle between the ball and the lead screw; The contact angle between the ball and the nut; It is the helix angle.

5. The method for modeling mixed lubrication wear of ball screw pairs based on measured raceway morphology according to claim 1, characterized in that, In step S2, the actual roughness morphology of the raceway is determined. Represented as: in: This is the raceway height matrix obtained through mirror reconstruction; This is the macroscopic baseline morphology obtained by fitting the original replication height data; Indicates the lead screw side. Indicates the nut side; This indicates the number of revolutions the ball screw pair operates in.

6. The method for modeling hybrid lubrication wear of ball screw pairs based on measured raceway morphology according to claim 1, characterized in that, In step S3, the Reynolds equation is expressed as: in: The density of the lubricant; Oil film thickness; For lubricant viscosity; Oil film pressure; The entrainment speed on the raceway side. Indicates the lead screw side. Indicates the nut side; For time; The energy equation is expressed as: in: Specific heat capacity; For temperature; , and They are respectively , and The velocity component in the direction; Thermal conductivity; The coordinates are along the film thickness direction; The film thickness equation is expressed as: in: This represents the initial oil film thickness. Along the contact point The equivalent radius of curvature in the direction; Along the contact point The equivalent radius of curvature in the direction; This represents the amount of elastic deformation. The composite deterministic roughness morphology of the ball and raceway is given, and: in: The height of the ball surface morphology in the local contact coordinate system after coordinate transformation; This represents the true rough morphology of the raceway; The contact deformation equation is expressed as: Wherein: equivalent elastic modulus , and These are the elastic modulus and Poisson's ratio of the ball bearing, respectively. and These are the elastic modulus and Poisson's ratio of the raceway, respectively. For contact pressure distribution, The contact area; The viscosity equation for the lubricant is expressed as: in: This represents the initial environmental viscosity. Pressure coefficient; Temperature coefficient; For temperature; Ambient temperature; The lubricant density equation is expressed as follows: in: For environmental density; It is the density temperature coefficient; The load balance equation is expressed as: in: This is the total normal load at the contact point between the ball and the raceway; This represents the total pressure distribution. Oil film lubrication area; This is the contact area of ​​the micro-protrusion.

7. The method for modeling hybrid lubrication wear of ball screw pairs based on measured raceway morphology according to claim 1, characterized in that, In step S4, the micro-protrusion load ratio Represented as: Wherein: fluid bearing load Micro-protrusions bear loads ; This represents the total pressure distribution. To run to Turning point The total normal load borne by each ball; This indicates the number of revolutions during operation.

8. The method for modeling mixed lubrication wear of ball screw pairs based on measured raceway morphology according to claim 1, characterized in that, In step S5, the thickness ratio of the lubricating film is... Defined as the ratio of the nominal minimum oil film thickness to the composite root mean square roughness: in: Minimum oil film thickness, Oil film thickness; The root mean square roughness is the combined roughness of the ball and raceway. The composite root mean square roughness Represented as: in: denoted as the root mean square roughness of the ball surface; For the raceway contact surface during cumulative operation to Root mean square roughness of the surface during rotation.

9. The method for modeling mixed lubrication wear of ball screw pairs based on measured raceway morphology according to claim 1, characterized in that, In step S5, the time-varying wear coefficient Represented as: in: The ratio of the thickness of the lubricating film; Let be the wear coefficient under boundary lubrication, and: in: These are dimensionless load parameters; These are dimensionless material parameters; These are dimensionless raceway surface morphology parameters; The overall elastic modulus of the contact body; The equivalent elastic modulus of the contact between the ball and the raceway of the screw / nut; Extending the contact area between the balls and the lead screw / nut raceway The equivalent radius of curvature in the direction; The viscosity-pressure coefficient of the lubricant; The root mean square roughness is the combined roughness of the ball and raceway.

10. The method for modeling hybrid lubrication wear of ball screw pairs based on measured raceway morphology according to claim 1, characterized in that, In step S6, the depth of mixed lubrication wear generated by a single ball in the contact area on the screw side / nut side Represented as: in: This represents the amount of wear per unit time under mixed lubrication conditions. Contact area; The wear coefficient is time-varying. The percentage of load on the micro-protrusions; For the normal contact load of a single ball on the screw side / nut side; The relative sliding speed of a single ball relative to the screw side / nut side; For the number of revolutions; Material hardness; The angular velocity of the lead screw; For the semi-major axis of the contact ellipse; The effective contact length of the ball on the raceway; The helix angle; The cumulative wear depth on the raceway between the ball and the screw / nut Represented as: in: The number of balls that can be effectively carried in one raceway; Wear depth of ball screw assembly Represented as: in: This represents the cumulative wear depth between the ball bearings and the lead screw. This represents the cumulative wear depth on the raceway between the ball and the nut.

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