Method and system for predicting load distribution of high-end machine tool feeding system

Abstract

The invention provides a method and a system for predicting load distribution of a high-end machine tool feeding system. The method is used for predicting load distribution. The method comprises the following steps of S1, establishing a coupling statics model; s2, a contact force coordinate transformation and decomposition step; s3, analyzing the stress of the workbench and the lead screw shaft; s4, calculating the bending deformation of the axis of the lead screw; s5, a workbench structure balance equation is established; s6, a load distribution obtaining step: determining the central position of a lead screw raceway and the central position of a nut raceway, establishing an expression of the axial displacement variation of a ball corresponding to the raceway center, and establishing a nonlinear equation set for describing the statics behavior of the whole ball screw feeding structure by combining an axial displacement variation equation and a statics equilibrium equation. And solving the nonlinear equation set through a numerical iteration method to obtain load distribution.

Description

Technical Field

[0001] This invention relates to the technical field of high-end CNC machine tools, and specifically to a method and system for predicting the load distribution of a high-end machine tool feed system. Background Technology

[0002] High-end CNC machine tools are the "mother machines" of the equipment manufacturing industry, and their performance directly affects the machining quality and efficiency of key components in high-tech fields such as aerospace, precision instruments, and automobiles. As a core functional component of CNC machine tools, the feed system undertakes the crucial task of converting rotary motion into precise linear positioning, and its performance is one of the core indicators for evaluating the overall level of a machine tool. A typical feed system usually consists of components such as a servo drive motor, coupling, ball screw pair, support bearings, guide rails, and a worktable.

[0003] During actual machine tool operation, the feed system bears complex loads from the cutting process, gravity, and friction. These loads are ultimately transmitted and distributed through the worktable, guide rails, leadscrew nut, leadscrew shaft, and support bearings, forming the load distribution within the system. The uniformity and rationality of the load distribution are fundamental factors affecting the positioning accuracy, dynamic stability, stiffness, and service life of the feed system. The traditional "prototype-test-improvement" cycle is costly and time-consuming, and can no longer meet the needs of rapid iterative development of modern machine tools. Therefore, there is an urgent need to develop advanced digital prediction methods. Thus, accurately predicting the load distribution of the feed system during the machine tool design phase is of paramount importance.

[0004] Calculate system stiffness: Load distribution directly determines the tensile and compressive deformation, torsional deformation, and contact deformation of the lead screw, which are the core elements constituting the overall stiffness of the system.

[0005] Predicted positioning accuracy: Uneven load distribution can cause uneven elastic deformation of the lead screw, resulting in positioning errors of the worktable, especially at different positions of the stroke, where the error varies.

[0006] Service life prediction: The fatigue life of ball screws and linear guides is directly related to the contact stress they bear, and the contact stress depends on the load distribution.

[0007] Currently, there are several main types of methods for load analysis of machine tool feed systems, but each has significant limitations:

[0008] 1. Static Simplified Calculation Method

[0009] Technical Description: This method, based on the principles of materials mechanics and static equilibrium, simplifies the feed system into a rigid system or a simple beam model. Typically, only the maximum working load is considered, and it is assumed that the load is uniformly distributed on key contact pairs (such as the lead screw nut and guide rail slider). Average stress, deformation, etc., are calculated using simple formulas.

[0010] Existing defects:

[0011] (1) Low accuracy and dependent on assumptions: The load distribution prediction of the feed system is based on overly idealized assumptions and cannot accurately calculate the nonlinear distribution of the load of each rolling element under complex torque loads.

[0012] (2) Lack of coupling effect: This method fails to establish a unified model to consider the mutual influence between screw deformation and guide rail force. For example, the radial bending of the screw will change the force angle of the nut, thus affecting the ball load, and will also cause the worktable to tilt, redistributing the load on the guide rail. This kind of coupled deformation effect of the system is difficult to reflect in the empirical formula method.

[0013] (3) Ignoring key deformations: The influence of the axial, radial and torsional deformation of the lead screw shaft on the input torque of the nut, and the resulting changes in the ball load, are often ignored or simplified in traditional analysis, resulting in distorted prediction of load distribution between the drive end and the non-drive end.

[0014] 2. Linear Static Finite Element Method in Commercial Software

[0015] Technical Description: A 3D model is created using CAD software and imported into finite element analysis software for linear static analysis. By applying fixed constraints and force loads, the overall stress and deformation are calculated.

[0016] Existing defects:

[0017] (1) Lack of contact nonlinearity handling: Linear static analysis assumes that the components are "bound" in contact or completely separated, which cannot realistically simulate the highly nonlinear mechanical behavior of "Hertz contact" in ball screw pairs and guide rail slider pairs. Ignoring contact nonlinearity will lead to serious inaccuracies in the prediction of stress and deformation (i.e. contact stiffness) in the contact area, thereby distorting the calculation results of deformation and load distribution of the entire system.

[0018] (2) High modeling complexity and weak specificity: Although FEA is powerful, building an accurate model containing a large number of contact pairs is still cumbersome in preprocessing and computationally intensive. More importantly, general-purpose FEA software is not designed for the specific object of the feed system. Its analysis process fails to make full use of the force transmission mechanism of the feed system, making the analysis process like a "black box". Engineers find it difficult to intuitively understand and control the inherent laws of load transmission.

[0019] In summary, existing technologies are either overly simplified, leading to unreliable predictions, or their complex models and low computational efficiency make them unsuitable for rapid engineering design cycles. They generally lack a dedicated method that can accurately reveal the spatial distribution of loads at the system level while ensuring computational efficiency. Therefore, there is an urgent need in this field for a new method for predicting load distribution in feed systems to overcome these shortcomings and provide an effective tool for the innovative design and performance optimization of high-end machine tool feed systems. Summary of the Invention

[0020] This invention is made to solve the above-mentioned problems, and aims to provide a method and system for predicting the load distribution of a high-end machine tool feed system.

[0021] This invention provides a method for predicting load distribution in a high-end machine tool feed system. The method, characterized by the following steps: S1: Coupled static model establishment step. A static model of the ball screw feed structure, including a ball screw nut pair, a linear guide pair, and a worktable, is established. The worktable, nut, and slider assembly are considered rigid bodies, and the screw is modeled as an elastic beam representing axial, radial bending, and torsional deformation. The cutting force acting on the center of the worktable and its components and moments in spatial degrees of freedom are determined. A global coordinate system and a coordinate system fixed to the worktable are defined. S2: Contact force coordinate transformation and decomposition step. For each ball in the ball screw nut pair, its contact points on the screw raceway and nut raceway are determined. Multiple associated coordinate systems are established with the screw axis as the reference. The coordinate relationship of the contact points in different coordinate systems is established through multiple coordinate transformations. The contact force of the ball is decomposed into a coordinate system with the screw axis as the reference to obtain the contact force components. S3: Worktable and screw axis force analysis step. Based on the contact force components, the contact force on the worktable is calculated. The following steps are performed: S4: Calculation of bending deformation of the screw axis, based on the forces and additional torques acting on the screw axis, and establishing the axial load balance equation and axial displacement variation equation of the ball screw nut pair; S5: Establishment of the table structure balance equation, within the framework of the static model, establishing a table structure model including the table, guide rail assembly, slider assembly and rolling elements, determining the center position of the slider raceway and the guide rail raceway, establishing expressions for the contact force and contact angle between the slider raceway and the guide rail raceway, and establishing the static balance equation of the table structure under external load; S6: Obtaining the load distribution, determining the center position of the screw raceway and the nut raceway, establishing an expression for the axial displacement variation of the ball corresponding to the raceway center, and combining the axial displacement variation equation and the static balance equation to establish a nonlinear equation set describing the static behavior of the entire ball screw feed structure, solving the nonlinear equation set by numerical iteration method to obtain the load distribution.

[0022] The method for predicting the load distribution of a high-end machine tool feed system provided by this invention may also have the following feature: The cutting force F in S1 can be decomposed into three component forces Fx, Fy, and Fz along the x, y, and z axes, and three moments Mx, My, and Mz about the x, y, and z axes, in the global coordinate system. The origin is fixed at the angular contact ball bearing, its x-axis is perpendicular to the ideal worktable surface, and its z-axis coincides with the lead screw axis. This coordinate system is fixed to the worktable. The axes of the coordinate system and the global coordinate system Keep parallel.

[0023] The method for predicting the load distribution of a high-end machine tool feed system provided by this invention may also have the following feature: wherein, the multiple associated coordinate systems in S2 include: coordinate system The origin is established at the position of the lead screw axis corresponding to the ball contact point A, and its axes are parallel to the axes of the global coordinate system. The intermediate coordinate system... The origin is located at the center of the ball bearing, and its axes are perpendicular to the global coordinate system. The axes are parallel; Frenet-Serret coordinate system Origin and They share the same origin, and The axis is perpendicular to the lead screw axis. The direction of the axis is the tangential direction in which the balls move along the helical raceway. The angle between the axis and z is the helix angle. Contact coordinate system and The origins are located at contact points A and B between the ball and the screw raceway and the nut raceway, respectively. Axis perpendicular to - flat, The axis points to the center of the ball, and the contact coordinate system is... and The axes of the two are parallel and have the same direction, and their z-axis and z-axis are parallel. The angle between the negative directions of the axis is the initial contact angle. Coordinate system and The origins are located at ball contact points A and B, respectively, and each axis is perpendicular to the global coordinate system. All axes are parallel.

[0024] The method for predicting the load distribution of a high-end machine tool feed system provided by this invention may also have the following feature: Specifically, the method for obtaining the contact force component in S2 involves: performing three coordinate transformations to establish the coordinate relationship of the contact point in different coordinate systems, decomposing the contact force into a coordinate system with the lead screw axis as the reference; the first transformation: determining the coordinate system... To coordinate system homogeneous coordinate transformation matrix coordinate system The origin is shifted to the center of the ball; the second transformation: determining the origin from... arrive homogeneous coordinate transformation matrix Through coordinate system Around Rotation This makes it shaft and The axes coincide and their direction is perpendicular to the lead screw axis. Based on this, then... Axis rotation ( ), making axis, The shafts are respectively and axis, Axis coincidence; Third transformation: Determine the coordinate system from the intermediate coordinate system To contact coordinate system and homogeneous coordinate transformation matrix and coordinate system Around Axis rotation Angle, making shaft and The axes are in the same direction and parallel; based on this, then... Axis rotation Angle, making shaft and The shafts are respectively with shaft and With the axes parallel, the origin of the current coordinate system is finally translated to the ball contact points A and B respectively, thus adjusting the contact force. Transform to coordinate system The equivalent force in the equation yields the contact force components, expressed as:

[0025] (8)

[0026] Where: azimuth angle From Rotate the axis clockwise to Angle of the axis; The helix angle of the ball screw nut assembly; It is the contact angle between the ball and the lead screw raceway or nut raceway.

[0027] The method for predicting the load distribution of a high-end machine tool feed system provided by this invention may also have the following feature: The calculation method for the force and additional torque exerted by the contact force on the worktable and on the lead screw axis in S3 is as follows:

[0028] S3-1: Calculate the force and additional torque exerted by the contact force on the worktable, specifically:

[0029] The contact force of each ball is translated to the geometric center of the worktable. According to equation (8), the contact force of the ball affects the worktable along the edge. and The forces acting in different directions are as follows:

[0030] (9)

[0031] (10)

[0032] In the formula: This refers to the number of load-bearing balls in the ball screw nut assembly.

[0033] The contact force of the balls on the worktable , and The additional torques on the shafts are as follows:

[0034] (11)

[0035] (12)

[0036] (13)

[0037] In the formula: The distance between the geometric center of the worktable and the centerline of the nut in the x-axis direction; For contact point B in coordinate system The axial coordinate of is expressed as:

[0038] (14)

[0039] In the formula: This is the axial distance between the first ball and the last ball. The angle between adjacent balls.

[0040] S3-2: Calculate the additional torque of the contact force on the leadscrew axis, specifically:

[0041] In coordinate system The balls in the ball joint are equivalently translated to the center line of the lead screw shaft. Assuming the cross-sectional area of ​​the lead screw shaft is rigid, the additional torques about the x1, y1, and z1 axes generated by the equivalent translation are:

[0042] (17).

[0043] The method for predicting the load distribution of a high-end machine tool feed system provided by this invention may also have the following feature: Specifically, the method for establishing the axial load balance equation and axial displacement variation equation of the ball screw nut pair in S3 is as follows:

[0044] From equation (8), the axial load balance equation for the ball screw nut pair is:

[0045] (15)

[0046] The relative position of the raceway center corresponding to the i-th ball is determined, and the axial displacement change between the raceway centers corresponding to the adjacent balls is:

[0047] (16)

[0048] In the formula: This represents the relative axial displacement between the raceway centers corresponding to the i-th ball; Let be the axial distance between the i-th ball and the (i+1)-th ball; and These are the effective cross-sectional areas of the lead screw shaft and the nut, respectively.

[0049] The method for predicting the load distribution of a high-end machine tool feed system provided by this invention may also have the following feature: Specifically, the calculation method for the bending deformation of the lead screw axis over its entire length in S4 is as follows:

[0050] In radial force , and additional torque , Under the action of the ball contact point A, the bending deformation of the screw axis along the x and y axes is as follows:

[0051] (18)

[0052] (19)

[0053] (20)

[0054] From equations (18) to (20), the bending deformations of the screw axis corresponding to the i-th ball contact point A along the x and y directions are respectively:

[0055] (twenty one)

[0056] In radial force , and additional torque , Under the action of the ball bearing, the bending deformation of the screw axis along the x and y directions between the angular contact ball bearing and the first ball contact point A are as follows:

[0057] (twenty two)

[0058] (twenty three)

[0059] (twenty four)

[0060] From equations (22) to (24), the bending deformations of the lead screw axis on the left side of the nut along the x and y directions are respectively:

[0061] (25)

[0062] In radial force , and additional torque , Under the action of the ball bearing, the bending deformation of the screw axis along the x and y directions between the last ball contact point A and the deep groove ball bearing are as follows:

[0063] (26)

[0064] (27)

[0065] (28)

[0066] From equations (26) to (28), the bending deformations of the lead screw axis on the right side of the nut along the x and y directions are respectively:

[0067] (29).

[0068] This invention also provides a prediction system for load distribution in a high-end machine tool feed system, characterized by comprising: a coupled static model establishment module, which establishes a static model of the ball screw feed structure including a ball screw nut pair, a linear guide pair, and a worktable, treating the worktable, nut, and slider as rigid bodies, and modeling the screw as an elastic beam characterizing axial, radial bending, and torsional deformation, determining the cutting force acting on the center of the worktable and its components and moments in spatial degrees of freedom, and defining a global coordinate system and a coordinate system fixed to the worktable; a contact force coordinate transformation and decomposition module, which, for each ball in the ball screw nut pair, determines its contact point on the screw raceway and nut raceway, and establishes multiple associated coordinate systems based on the screw axis, establishing the coordinate relationship of the contact point in different coordinate systems through multiple coordinate transformations, and decomposing the contact force of the ball into a coordinate system based on the screw axis to obtain the contact force components; and a worktable and screw shaft force analysis module, which, based on the contact force components, calculates the force exerted by the contact force on the worktable and the attached force. The system employs several modules: A first module calculates the applied torque and the additional torque on the lead screw axis, establishing the axial load balance equation and axial displacement variation equation for the ball screw and nut pair; a second module calculates the bending deformation of the lead screw axis along its entire length based on the applied force and additional torque; a third module establishes the table structure balance equation, creating a static model of the table, guide rails, slider, and rolling elements; it determines the center positions of the slider and guide rail raceways, establishes expressions for the contact force and contact angle between the slider and guide rail raceways, and establishes the static balance equation for the table structure under external loads; and a fourth module obtains the load distribution, determining the center positions of the lead screw and nut raceways, establishing expressions for the axial displacement variation of the balls corresponding to the raceway centers, and combining the axial displacement variation equation and the static balance equation to establish a set of nonlinear equations describing the static behavior of the entire ball screw feed structure. The load distribution is obtained by solving these nonlinear equations using a numerical iteration method.

[0069] The role and effect of invention

[0070] The method and system for predicting load distribution in a high-end machine tool feed system according to the present invention have the following beneficial effects:

[0071] 1. High prediction accuracy, overcoming the limitations of idealized assumptions: This invention establishes a static model that accurately reflects the coupling relationships of key components such as the ball screw nut pair and linear guide pair, abandoning the idealized assumptions of treating the system as a rigid body or uniformly loaded as in traditional simplification methods. The method of this invention can accurately calculate the nonlinear distribution of loads on each rolling element in the nut and guide assembly under the combined action of complex torques and axial forces, thus providing a reliable basis for accuracy traceability and life prediction.

[0072] 2. Achieving an optimal balance between computational efficiency and accuracy, with strong engineering applicability: The modeling and solution strategy adopted in this invention significantly reduces computational complexity and resource consumption while ensuring prediction accuracy. Compared to the traditional finite element method, which requires building a large model and is difficult and time-consuming to solve, the method of this invention has a fast computation speed, enabling rapid parametric analysis and design iteration, greatly improving R&D efficiency, and is very suitable for widespread application in the engineering design and optimization stages.

[0073] 3. Transparent Mechanism and Strong Guidance: This invention not only provides the final load distribution data, but also clearly reveals the evolution law of the load within the feed structure. This helps designers to deeply understand the weak links in the performance of high-end machine tool feed systems, and provides direct and clear theoretical guidance for stiffness matching, bearing selection and configuration, structural lightweighting, and performance improvement of high-end machine tool feed systems.

[0074] 4. Wide applicability and good versatility: The prediction model established by this invention has good versatility. By adjusting the model parameters, it can be applied to high-end machine tool feed systems of different types (such as horizontal and vertical) and under different working conditions, and has wide applicability. Attached Figure Description

[0075] Figure 1 This is a flowchart of a method for predicting the load distribution of a high-end machine tool feed system in an embodiment of the present invention.

[0076] Figure 2 This is a schematic diagram of the ball screw feed structure in an embodiment of the present invention.

[0077] Figure 3 This is a schematic diagram of different coordinate systems for the ball screw nut pair in an embodiment of the present invention.

[0078] Figure 4 This is a schematic diagram of the force exerted by the ball screw nut pair on the worktable in an embodiment of the present invention.

[0079] Figure 5 This is a mechanical model diagram of the lead screw axis in an embodiment of the present invention.

[0080] Figure 6 This is a model diagram of guide rail assembly error in an embodiment of the present invention.

[0081] Figure 7 This is a geometric diagram of the raceway center in an embodiment of the present invention.

[0082] Figure 8 In this embodiment of the invention, the centers of the lead screw raceway and the nut raceway are in the coordinate system. The coordinate graph.

[0083] Figure 9This is a flowchart of the load distribution calculation for the ball screw feed structure in an embodiment of the present invention.

[0084] Figure 10 This is a comparison diagram of the load distribution of the ball screw nut pair between the model of the present invention and the existing model in this embodiment of the invention.

[0085] Figure 11 This is a bending deformation diagram of the lead screw shaft under axial and radial loads in an embodiment of the present invention.

[0086] Figure 12 This is a load distribution diagram of the ball screw nut pair under axial and radial loads in an embodiment of the present invention.

[0087] Figure 13 This is a diagram showing the distribution of loaded balls along the spiral raceway in an embodiment of the present invention.

[0088] Figure 14 This is a load distribution diagram of the slider-guide rail structure under axial and radial loads in an embodiment of the present invention.

[0089] Figure 15 This is a diagram showing the bending deformation of the lead screw shaft caused by the vertical deviation of the guide rail in an embodiment of the present invention.

[0090] Figure 16 This describes the load distribution of the ball screw nut pair under vertical deviation of the guide rail in this embodiment of the invention.

[0091] Figure 17 This is a load distribution diagram of the slider-guide rail structure under guide rail verticality deviation in an embodiment of the present invention.

[0092] Figure 18 This is a diagram showing the bending deformation of the lead screw shaft caused by the horizontal deviation of the guide rail in an embodiment of the present invention.

[0093] Figure 19 This is a load distribution diagram of the ball screw nut pair under horizontal deviation of the guide rail in an embodiment of the present invention.

[0094] Figure 20 This is a load distribution diagram of the slider-guide rail structure under horizontal deviation of the guide rail in an embodiment of the present invention.

[0095] Explanation of symbols for main components:

[0096] In the diagram: 1. Servo motor; 2. Coupling; 3. Lead screw; 4. Worktable; 5. Angular contact ball bearing; 6. Deep groove ball bearing; 7. Slider assembly; I. First slider; II. Second slider; III. Third slider; IV. Fourth slider; 8. Nut; 9. Guide rail assembly; 901. First guide rail; 902. Second guide rail. Detailed Implementation

[0097] In the description of this application, it should be noted that, unless otherwise expressly specified and limited, the terms "installation," "connection," and "linking" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection, an electrical connection, or a connection that allows communication between them; they can refer to a direct connection or an indirect connection through an intermediate medium; they can refer to the internal communication between two components or the interaction between two components. Those skilled in the art can understand the specific meaning of the above terms in this application according to the specific circumstances.

[0098] To make the technical means, creative features, objectives and effects of this invention easy to understand, the following embodiments, in conjunction with the accompanying drawings, specifically illustrate the method and system for predicting load distribution in the high-end machine tool feed system of this invention.

[0099] Example 1

[0100] This embodiment of a method for predicting load distribution in a high-end machine tool feed system includes the following steps:

[0101] Figure 1 This is a flowchart of a method for predicting the load distribution of a high-end machine tool feed system in an embodiment of the present invention. Figure 2 This is a schematic diagram of the ball screw feed structure in an embodiment of the present invention.

[0102] like Figure 1-2 As shown, step S1 is the coupled static model establishment step. A static model of the ball screw feed structure, including the ball screw nut pair, linear guide pair, and worktable 4, is established. The worktable 4, nut 8, and slider assembly 7 are considered as rigid bodies, and the screw 3 is modeled as an elastic beam representing axial, radial bending, and torsional deformation. The cutting force acting on the center of the worktable 4 and its components and moments in spatial degrees of freedom are determined. A global coordinate system and a coordinate system fixed to the worktable are defined, specifically:

[0103] The ball screw and nut assembly includes a screw 3, a nut 8, and balls. A servo motor 1 is mounted on one end of the screw 3 and connected to it via a coupling 2. The left side of the screw 3 (based on...) Figure 2 (From a certain perspective) An angular contact ball bearing 5 is installed on the right side, and a deep groove ball bearing 6 is installed on the right side. The lead screw 3 is rotatably connected to the nut 8, and the balls are located between the lead screw 3 and the nut 8. The nut 8 is connected to the worktable 4.

[0104] The linear guide pair includes a slider assembly 7 and a guide rail assembly 9. The slider assembly 7 can drive the worktable 4 to slide along the guide rail assembly 9. Ball bearings are provided between the slider assembly 7 and the guide rail assembly 9. The guide rail assembly 9 includes a first guide rail 901 and a second guide rail 902. The slider assembly 7 includes a first slider I, a second slider II, a third slider III, and a fourth slider IV.

[0105] The bending deformation of the lead screw shaft is derived by simulating the lead screw 3 as an Euler-Bernoulli beam. The cutting force F acting on the center of the worktable 4 can be decomposed into three components along the x, y, and z axes: Fx, Fy, and Fz, and three moments about the x, y, and z axes: Mx, My, and Mz.

[0106] Global coordinate system The origin is fixed at the angular contact ball bearing 5, its x-axis is perpendicular to the surface of the ideal worktable 4, and its z-axis coincides with the lead screw axis, fixed in the coordinate system of the worktable 4. Its axes are perpendicular to the global coordinate system. Keep parallel.

[0107] Step S2 is the contact force coordinate transformation and decomposition step. For each ball in the ball screw and nut assembly, its contact point on the screw raceway and nut raceway is determined, and multiple associated coordinate systems are established with the screw axis as the reference. Through multiple coordinate transformations, the coordinate relationship of the contact point in different coordinate systems is established, and the contact force of the ball is decomposed into a coordinate system with the screw axis as the reference to obtain the contact force components, specifically:

[0108] Figure 3 This is a schematic diagram of different coordinate systems for the ball screw nut pair in an embodiment of the present invention.

[0109] like Figure 3 As shown, to facilitate the study of the effect of contact force on the deformation of the leadscrew axis, the coordinate system is... The origin is established at the position of the leadscrew axis corresponding to the ball contact point A, and its axes are parallel to the axes of the global coordinate system. The intermediate coordinate system... The origin is located at the center of the ball bearing, and its axes are perpendicular to the global coordinate system. All axes are parallel.

[0110] Frenet-Serret coordinate system The origin and They share the same origin, and The axis is perpendicular to the lead screw axis. The direction of the axis is the tangential direction in which the balls move along the helical raceway. The angle between the axis and z is the helix angle. .

[0111] Furthermore, the contact points between the balls and the screw raceway and nut raceway are located at... - In the plane. Contact coordinate system. and The origins are located at the contact points A and B between the ball and the lead screw raceway and the nut raceway, respectively. Axis perpendicular to - flat, The axis points to the center of the ball, coordinate system and The axes of each other are parallel and have the same direction, and their z-axis and The angle between the negative directions of the axis is the initial contact angle. Coordinate system and The origins are located at ball contact points A and B, respectively, and their axes are perpendicular to the global coordinate system. All axes are parallel.

[0112] To achieve contact force decomposition, three coordinate transformations are required to establish the coordinate relationship of the contact point in different coordinate systems, and decompose the contact force into a coordinate system with the lead screw axis as the reference, as follows:

[0113] First transformation: Determine the coordinate system To coordinate system homogeneous coordinate transformation matrix . Coordinate system If the origin is shifted to the center of the ball, the homogeneous coordinate transformation matrix is:

[0114] (1)

[0115] Where: azimuth angle From Rotate the axis clockwise to Angle of the axis; Let be the helix angle of the ball screw nut assembly, and its expression is:

[0116] (2)

[0117] In the formula: The pitch of the ball screw nut assembly.

[0118] Second transformation: Determine from arrive homogeneous coordinate transformation matrix Through coordinate system Around Rotation This makes it shaft and The axes coincide and their direction is perpendicular to the lead screw axis. Based on this, then... Axis rotation ( ), making axis, The shafts are respectively and axis, If the axes coincide, then the homogeneous coordinate transformation matrix is:

[0119] (3)

[0120] Third transformation: Determine the intermediate coordinate system To contact coordinate system and homogeneous coordinate transformation matrix and . Coordinate system Around Axis rotation Angle, making shaft and The axes are in the same direction and parallel. Based on this, then... Axis rotation Angle, making shaft and The shafts are respectively with shaft and The axes are parallel. Finally, the origin of the current coordinate system is translated to the ball contact points A and B respectively. The homogeneous coordinate transformation matrices are as follows:

[0121] (4)

[0122] (5)

[0123] In the formula: It is the contact angle between the ball and the lead screw raceway or nut raceway.

[0124] From equations (1), (3), (4) and (5), it can be determined that and , The conversion relationship between them means that the contact points A and B between the ball and the screw raceway and the nut raceway are in the coordinate system. The coordinates can be represented as follows:

[0125] (6)

[0126] (7)

[0127] To analyze the axial, radial, and torsional elastic deformation of the lead screw axis under contact force, the contact force... Transform to coordinate system The equivalent force, the expression for the contact force component, is:

[0128] (8).

[0129] Step S3 is the force analysis step for the worktable 4 and the lead screw shaft. Based on the contact force components, the force and additional torque exerted by the contact force on the worktable 4, as well as the additional torque on the lead screw axis, are calculated respectively. The axial load balance equation and axial displacement variation equation for the ball screw nut pair are then established, specifically:

[0130] S3-1: Calculate the force and additional torque exerted by the contact force on the worktable 4, specifically:

[0131] Figure 4 This is a schematic diagram of the force exerted by the ball screw nut pair on the worktable in an embodiment of the present invention.

[0132] like Figure 4 The coordinate system positions of the ball screw and nut assembly are shown. Under the action of external load and assembly error, the motion error of the worktable 4 and the contact force between the screw and nut balls influence each other. To derive the force exerted by the contact force of the balls on the nut raceway on the worktable 4, the contact force of each ball is translated to the geometric center of the worktable 4. According to equation (8), the contact force of the balls on the worktable 4 along the raceway... and The forces acting in different directions are as follows:

[0133] (9)

[0134] (10)

[0135] In the formula: This refers to the number of load-bearing balls in the ball screw nut assembly.

[0136] The contact force of the ball bearings on the nut raceway affects the 4-axis rotation of the worktable. , and The additional torques on the shafts are as follows:

[0137] (11)

[0138] (12)

[0139] (13)

[0140] In the formula: The distance between the geometric center of the worktable and the centerline of the nut in the x-axis direction; For contact point B in coordinate system The axial coordinate of is expressed as:

[0141] (14)

[0142] In the formula: This is the axial distance between the first ball and the last ball. It is the included angle between adjacent balls.

[0143] From equation (8), the axial load balance equation for the ball screw nut pair is:

[0144] (15)

[0145] Under the combined effects of external load and assembly error, the absolute positions of the raceway centers of the lead screw and nut are difficult to determine. Therefore, to calculate the axial deformation of the lead screw 3 and nut 8, the relative position of the raceway center corresponding to the i-th ball is determined, and the axial displacement change between the raceway centers of adjacent balls and the corresponding raceway centers is:

[0146] (16)

[0147] In the formula: This represents the relative axial displacement between the raceway centers corresponding to the i-th ball; Let be the axial distance between the i-th ball and the (i+1)-th ball; and These are the effective cross-sectional areas of the lead screw shaft and the nut, respectively.

[0148] S3-2: Calculate the additional torque of the contact force on the leadscrew axis, specifically:

[0149] In coordinate system The balls in the ball joint are equivalently translated to the center line of the lead screw shaft. Assuming the cross-sectional area of ​​the lead screw shaft is rigid, the additional torques about the x1, y1, and z1 axes generated by the equivalent translation are:

[0150] (17).

[0151] Step S4 is the calculation step for the bending deformation of the leadscrew axis. Based on the force and additional torque applied to the leadscrew axis, the bending deformation of the leadscrew axis over its entire length is calculated, specifically as follows:

[0152] Figure 5 This is a mechanical model diagram of the lead screw axis in an embodiment of the present invention.

[0153] Since the diameter of the nut is slightly smaller than its length, the lead screw 3 and nut 8 are simulated as an Euler-Bernoulli beam and a rigid body, respectively. Figure 5 As shown, the lead screw axis is subjected to radial force. , Axial force Additional torque , and additional torque The effect of additional torque. The deformation of the lead screw shaft centerline is not affected; the axial force is ignored here. This causes axial deformation of the leadscrew axis. Therefore, under radial force... , and additional torque , Under the action of the ball contact point A, the bending deformation of the screw axis along the x and y axes is as follows:

[0154] (18)

[0155] (19)

[0156] (20)

[0157] From equations (18) to (20), the bending deformations of the screw axis corresponding to the i-th ball contact point A along the x and y directions are respectively:

[0158] (twenty one)

[0159] In radial force , and additional torque , Under the action of the ball bearing 5 (left end bearing) and the first ball contact point A, the bending deformation of the screw axis along the x and y directions are as follows:

[0160] (twenty two)

[0161] (twenty three)

[0162] (twenty four)

[0163] From equations (22) to (24), the bending deformations of the lead screw axis on the left side of nut 8 along the x and y directions are respectively:

[0164] (25)

[0165] In radial force , and additional torque , Under the action of the ball bearing, the bending deformation of the screw axis along the x and y directions between the last ball contact point A and the deep groove ball bearing 6 (right end bearing) are as follows:

[0166] (26)

[0167] (27)

[0168] (28)

[0169] From equations (26) to (28), the bending deformations of the lead screw axis on the right side of nut 8 along the x and y directions are respectively:

[0170] (29).

[0171] Step S5 is the step of establishing the equilibrium equations for the worktable structure. Within the framework of the static model, a worktable structure model is established, including the worktable 4, guide rail assembly 9, slider assembly 7, and rolling elements. The center positions of the slider raceway and guide rail raceway are determined, expressions for the contact force and contact angle between the slider raceway and guide rail raceway are established, and the static equilibrium equations for the worktable structure under external loads are established, specifically as follows:

[0172] S5-1: Determine the center position of the slider raceway and the center position of the guide rail raceway, specifically:

[0173] Figure 6 This is a model diagram of guide rail assembly error in an embodiment of the present invention.

[0174] like Figure 6 As shown, within the framework of the static model of the ball screw feed structure, a worktable structure model is established, including the worktable 4, guide rail assembly 9, slider assembly 7, and rolling elements. Coordinate system. and Position such as Figure 6 As shown. When the worktable structure is in an ideal state, according to the homogeneous coordinate transformation, the raceway centers of each slider and each guide rail are in the coordinate system. The coordinates are:

[0175] (30)

[0176] In the formula: , and Coordinate systems The origin along the x, y, and z directions in the coordinate system The coordinates.

[0177] When there is an assembly error in the guide rail assembly 9, the position of the center line of curvature of the slider raceway changes with the pose of the worktable 4. Under the influence of the assembly error, the worktable 4 has five types of motion errors, including two types of straightness errors ( , ) and three types of angular errors ( , , ).like Figure 6 As shown, let the theoretical correct coordinate system and the error coordinate system of worktable 4 be respectively... and Since the error angle of worktable 4 is a micro-angle, then , , , Therefore, the error matrix at the center of the workbench Represented as:

[0178] (31)

[0179] Through homogeneous transformation matrix In the m-th slider, the center of the k-th ball in the j-th raceway corresponds to the center of the slider raceway in the coordinate system. The coordinates are:

[0180] (32)

[0181] To establish an assembly error model for guide rail assembly 9, it is assumed that the second guide rail 902 has a positional deviation while the first guide rail 901 remains in an ideal position. For example... Figure 6 As shown, establish the correct coordinate system at the left end of the second guide rail 902. Sum of error coordinates Then the center of the guide rail raceway is in the coordinate system. The coordinates are:

[0182] (33)

[0183] In the formula: From arrive The homogeneous coordinate transformation matrix; From coordinate system arrive The homogeneous coordinate transformation matrix; , and The centers of the second guide rail raceway along the x, y, and z directions in the coordinate system are respectively... The coordinates.

[0184] S5-2: Establish the expressions for the contact force and contact angle between the slider raceway and the guide rail raceway, specifically:

[0185] From equations (32) and (33), the distance between the centers of the slider raceway and the guide rail raceway is:

[0186] (34)

[0187] Assembly errors of guide rail assembly 9 will affect the motion error of worktable 4, causing the center distance of the slider-guide rail structure raceway to be less than the original distance. In this case, the contact force is zero. Based on whether the ball is under load, and according to Hertzian contact theory, the expression for the contact force corresponding to the i-th ball in the slider-guide structure is derived as follows:

[0188] (35)

[0189] In the formula: and These are the Hertzian contact constants between the ball and slider assembly 7 and the guide rail assembly 9, respectively. This represents the initial distance between the raceway centers under preload.

[0190] Based on the geometric relationship of the raceway center, the expression for the contact angle is:

[0191] (36)

[0192] S5-3: Establish the static equilibrium equations of the workbench structure under external loads, specifically:

[0193] The external load on the worktable 4 is the cutting force acting on the workpiece. According to equations (35) and (36), the resultant force and resultant moment functions of the worktable structure are:

[0194] (37)

[0195] In the formula: , and For the contact point between the ball and the slider raceway at The coordinates in the diagram.

[0196] In two forces , and three torques , , Under the action of the action, the static equilibrium equation of the workbench structure is:

[0197] (38).

[0198] Step S6 is the load distribution acquisition step. Determine the center positions of the screw raceway and the nut raceway. Establish an expression for the axial displacement change of the balls corresponding to the raceway centers, and combine the axial displacement change equation and the static equilibrium equation to establish a set of nonlinear equations describing the static behavior of the entire ball screw feed structure. Solve the nonlinear equations using a numerical iteration method to obtain the load distribution, specifically:

[0199] S6-1: Determine the center position of the lead screw raceway and the center position of the nut raceway, specifically:

[0200] Figure 7 This is a geometric diagram of the raceway center in an embodiment of the present invention. Figure 8 In this embodiment of the invention, the centers of the lead screw raceway and the nut raceway are in the coordinate system. The coordinate graph.

[0201] like Figure 7-8 As shown, due to the elastic deformation of the balls under load, the center of the balls and the center of the raceways will... , , Move to , , To calculate the radial distance from the raceway center, the coordinates of the middle raceway center must be determined. According to the homogeneous transformation matrix, the raceway center of the nut corresponding to the i-th ball lies in the coordinate system... The coordinates are:

[0202] (39)

[0203] In the formula: and Coordinate systems The origin and contact point A in the coordinate system and The z-coordinate in the middle, The expression is:

[0204] (40)

[0205] The center of the screw raceway corresponding to the i-th ball is in the coordinate system. The coordinates are:

[0206] (41)

[0207] S6-2: Establish the expression for the change in axial displacement of the ball corresponding to the raceway center, specifically:

[0208] From equations (39) and (41), the radial distance between the centers of the lead screw and nut raceways is:

[0209] (42)

[0210] according to Figure 7 The geometric relationship of the raceway center is shown, and the contact angle function is:

[0211] (43)

[0212] In the formula: The initial distance between the centers of the lead screw and nut raceways.

[0213] The distance between the centers of the lead screw and nut raceways is:

[0214] (44)

[0215] In the formula: and These are the contact deformations of the ball raceway and the screw raceway, and the nut raceway, respectively.

[0216] The axial displacement change of the i-th ball corresponding to the raceway center is:

[0217] (45)

[0218] Figure 9 This is a flowchart of the load distribution calculation for the ball screw feed structure in an embodiment of the present invention.

[0219] Substituting equation (45) into equation (16), and combining equations (15), (37), and (43), a set of nonlinear equations concerning the motion error of the worktable 4, the contact load of the ball screw and nut pair, and the contact angle were established. The Newton-Raphson numerical iteration method was used to solve the established static equilibrium equations, thereby solving for the numerical values ​​of the motion error of the worktable 4, the contact force and contact angle of the ball screw and nut pair, the bending deformation of the screw axis, and the contact force of the slider-guide rail structure. Figure 9 The diagram shows the calculation process for the load distribution of the ball screw feed structure, specifically:

[0220] Input the geometric parameters of the ball screw nut pair, linear guide pair, and worktable, as well as the external load and assembly error of guide assembly 9. Input the calculation accuracy ACC, the initial values ​​of the contact force and contact angle of the ball screw nut pair, and the motion error of worktable 4. .

[0221] Calculate the bending deformation of the lead screw shaft [Equations (21), (25), and (29)]. Calculate the distance between the center of the slider raceway and the center of the guide rail raceway [Equation (34)], and determine whether the distance between the center of the slider-guide rail raceway is less than the original distance. When When calculating the contact force and contact angle of the slider-guide rail structure [Equations (23) and (24)], the equilibrium equation of the worktable structure is established [Equation (25)]. When the contact force between the slider and the guide rail is zero [Equation (35)], the equilibrium equation of the worktable structure is established [Equation (25)]. The nonlinear equations of the contact force and contact angle of the ball screw nut pair are established [Equations (43) and (44)].

[0222] The established static equilibrium equations [Equations (15), (16), (37), and (43)] are solved using the Newton-Raphson numerical iteration method. If the error f(x) of the current iterative solution ≤ ACC, the numerical solution is considered sufficiently accurate and meets the convergence condition. The iteration can then be terminated and the following values ​​can be output: the contact force and contact angle of the ball joint of the lead screw and nut, the motion error of the worktable 4, the bending deformation of the lead screw axis, and the contact force of the slider-guide rail structure. If f(x) ≥ ACC, the initial values ​​of the iteration are updated with the current solution, and the calculation continues until the convergence condition is met.

[0223] This embodiment also provides a load distribution prediction system for high-end machine tool feed systems, including:

[0224] The coupled static model building module is used to implement step S1, namely: to build a static model of the ball screw feed structure, including the ball screw nut pair, the linear guide pair, and the worktable. The worktable, nut, and slider are regarded as rigid bodies, and the screw is modeled as an elastic beam representing axial, radial bending, and torsional deformation. The cutting force acting on the center of the worktable and its components and moments in spatial degrees of freedom are determined, and a global coordinate system and a coordinate system fixed to the worktable are defined.

[0225] The contact force coordinate transformation and decomposition module is used to implement step S2, which involves determining the contact point of each ball in the ball screw and nut assembly on the screw raceway and establishing multiple associated coordinate systems with the screw axis as the reference. Through multiple coordinate transformations, the coordinate relationship of the contact point in different coordinate systems is established, and the contact force of the ball is decomposed into a coordinate system with the screw axis as the reference to obtain the contact force components.

[0226] The worktable and lead screw axis force analysis module is used to implement step S3, namely: based on the contact force components, calculate the force and additional torque of the contact force on the worktable, as well as the additional torque on the lead screw axis, and establish the axial load balance equation and axial displacement change equation of the ball screw nut pair.

[0227] The lead screw axis bending deformation calculation module is used to implement step S4, namely: based on the force and additional torque applied to the lead screw axis, calculate the bending deformation of the lead screw axis over its entire length.

[0228] The module for establishing the equilibrium equations of the worktable structure is used to implement step S5, namely: within the framework of the static model, establishing a worktable structural model including the worktable, guide rail, slider, and rolling elements. It determines the center positions of the slider raceway and the guide rail raceway, establishes expressions for the contact force and contact angle between the slider raceway and the guide rail raceway, and establishes the static equilibrium equations of the worktable structure under external loads.

[0229] The load distribution acquisition module is used to implement step S6, namely: determining the center positions of the ball screw raceway and the nut raceway. It establishes an expression for the axial displacement change of the balls corresponding to the raceway centers, and combines the axial displacement change equation and the static equilibrium equation to establish a set of nonlinear equations describing the static behavior of the entire ball screw feed structure. The load distribution is obtained by solving the nonlinear equations using a numerical iteration method.

[0230] Example 2

[0231] This embodiment examines the accuracy of the model of the present invention.

[0232] The load distribution of the ball screw nut pair is difficult to measure directly in experiments. To verify the accuracy of the model proposed in this invention, the load distribution results of the proposed model and existing models are compared. The geometric parameters of the ball screw nut pair and the worktable structure are shown in Table 1 and Table 2, respectively.

[0233] Figure 10 This is a comparison diagram of the load distribution of the ball screw nut pair between the model of the present invention and the existing model in this embodiment of the invention.

[0234] like Figure 10 As shown, this illustrates the effect of axial load F z Under a load of 5000 N, the load distribution of the ball screw in this invention model is compared with that of existing models. The load distribution curves of the model in this invention are consistent with those of the Lin model, but the contact load of the model in this invention is slightly smaller than that of the Lin model, with a maximum error of 7.62%. This is mainly because the Lin model assumes that the ball-raceway contact angle remains constant at 45°. The load distribution of the Mei model shows a monotonically decreasing trend, because it only considers the axial deformation of the screw 3 and nut 8. The comparison of the three models verifies the accuracy of the model in this invention and further demonstrates that the bending deformation of the screw shaft is the main factor affecting the load distribution of the ball screw nut pair.

[0235] Table 1 Relevant parameters of ball screw nut pair

[0236]

[0237] Table 2 Parameters of the workbench structure

[0238]

[0239] References:

[0240] [1] Mei X, Tsutsumi M, Tao T, et al. Study on the load distribution of ball screws with errors [J]. Mechanism and Machine Theory, 2003, 38(11):1257-1269.

[0241] [2] Lin B, Okwudire CE, Wou J S. Low order static load distribution model for ball screw mechanisms including effects of lateral deformation and geometric errors [J]. ASME Journal of Mechanical Design, 2018, 140: 022301.

[0242] Example 3

[0243] This embodiment examines the influence of axial and radial loads on load distribution. In this embodiment, unless otherwise specified, the first slider I refers to the first slider I-guide rail structure, the second slider II refers to the second slider II-guide rail structure, the third slider III refers to the third slider III-guide rail structure, and the fourth slider IV refers to the fourth slider IV-guide rail structure.

[0244] Load distribution analysis of ball screw nut pair:

[0245] The ball screw feed structure may be subjected to axial and radial loads simultaneously during operation. Therefore, it is necessary to study the influence of the axial and radial loads on the worktable 4 on the overall load distribution of the feed structure.

[0246] Figure 11 This is a bending deformation diagram of the lead screw shaft under axial and radial loads in an embodiment of the present invention. Figure 12 This is a load distribution diagram of the ball screw nut pair under axial and radial loads in an embodiment of the present invention. Figure 13 This is a diagram showing the distribution of loaded balls along the spiral raceway in an embodiment of the present invention.

[0247] When the worktable 4 is located at the midpoint of the lead screw shaft, the load distribution law of the ball screw nut pair can be revealed by analyzing the bending deformation of the lead screw shaft. For example... Figure 11 As shown, this is under axial load F z = 5000 N, radial load F x With Fy Under operating conditions varying from -5000 N to 5000 N, the lead screw shaft undergoes bending deformation under two types of loads. Radial load changes the position of the lead screw shaft; when it increases by the same magnitude in both positive and negative directions, the lead screw shaft produces the same displacement in the corresponding direction. However, as the radial load changes, the overall shape of the load distribution curve remains essentially unchanged, as shown below. Figure 12 As shown. This is mainly because the degree of bending of the screw shaft at the screw-nut interface remains almost constant, resulting in the degree of ball compression in the helical raceway being largely unaffected by radial load. Furthermore, as... Figure 13 As shown, influenced by the distribution of loaded balls in the ball screw nut assembly, the bending deformation of the screw shaft in the y-axis direction is significantly greater than that in the x-axis direction, indicating that y-axis bending plays a dominant role in the load distribution pattern. Under the action of y-axis bending of the screw shaft, the compression degree of the 5th and 29th balls decreases, while the compression degree of the 13th and 21st balls increases. The above analysis shows that the load distribution pattern of the ball screw nut assembly is mainly determined by the bending deformation of the screw shaft, while the radial load has a relatively small impact on the distribution curve shape.

[0248] Load distribution analysis of linear guide pairs:

[0249] Figure 14 This is a load distribution diagram of the slider-guide rail structure under axial and radial loads in an embodiment of the present invention.

[0250] To analyze the influence of radial load on the load distribution of the four sets of slider-guide rail structures, such as Figure 14 As shown, the load distribution curves of this structure under the same horizontal and vertical radial loads are illustrated. With increasing radial load (F... x =F y As the load increases from -5000 N to 5000 N, the contact load of the first row of balls gradually decreases to zero, while the contact load of the third row of balls gradually increases from zero. The contact loads of the second and fourth rows of balls change very little. This is because under a positive vertical load F... x Under the action of the load, the compression degree of the third and fourth rows of balls increases; under the positive horizontal load F y Under the influence of the radial load, the compression of the second and third rows of balls increases, causing the contact load on the third row of balls to rise with the increase of the positive radial load. Similarly, when the negative radial load increases, the contact load on the first row of balls gradually increases. Axial load F zThe axial component of the contact load with the ball screw nut pair generates a counterclockwise torque about the y-axis, causing the compression of the first and second rows of balls in the first slider I and second slider II to gradually intensify (with the smallest and largest increases in the first and last balls, respectively). Similarly, in the third slider III and fourth slider IV, the compression of the third and fourth rows of balls gradually intensifies (with the largest and smallest increases in the first and last balls, respectively). This ultimately results in an uneven distribution of the contact load on the balls within each raceway. Therefore, in the first slider I and second slider II, the contact load on the first row of balls under a negative radial load is greater than the contact load on the third row of balls under a positive radial load; while in the third slider III and fourth slider IV, the relationship is exactly the opposite. Furthermore, the positive and negative horizontal loads and the horizontal components of the contact load between the slider and guide rail structure generate clockwise and counterclockwise torques about the z-axis, respectively. Under a positive horizontal load, the contact load on the third and fourth rows of balls in the second slider II and the third slider III increases; under a negative horizontal load, the contact load on the first and second rows of balls in the second slider II and the third slider III increases. This is compared by examining the load distribution of the ball screw under radial load (see...). Figure 12 As can be seen, the radial load is mainly balanced by the contact load in the slider-guide structure.

[0251] Example 4

[0252] This embodiment examines the impact of assembly errors on load distribution. In this embodiment, unless otherwise specified, the first slider I refers to the first slider I-guide rail structure, the second slider II refers to the second slider II-guide rail structure, the third slider III refers to the third slider III-guide rail structure, and the fourth slider IV refers to the fourth slider IV-guide rail structure.

[0253] The effect of the offset of guide rail assembly 9 in the x-axis direction on load distribution:

[0254] Figure 15 This is a diagram showing the bending deformation of the lead screw shaft caused by the vertical deviation of the guide rail in an embodiment of the present invention. Figure 16 This describes the load distribution of the ball screw nut pair under vertical deviation of the guide rail in this embodiment of the invention.

[0255] Assembly errors in guide rail assembly 9 can affect the posture of worktable 4, thereby altering the load distribution of the ball screw nut pair and the slider-guide rail structure. When the right end of the second guide rail 902 shifts in the x-axis direction, the bending curve of the screw axis is as follows: Figure 15As shown. Under the influence of deviations in the positive and negative x-axis directions, the position of the lead screw axis moves along the positive and negative x-axis directions respectively, and the distance moved along the positive and negative directions is the same under the same positive and negative deviations. However, the position of the lead screw axis in the y-axis direction remains unchanged. As the positive vertical deviation (positive x-axis direction) increases, the compression degree of the 1st and 25th balls increases, while the compression degree of the 9th and 32nd balls decreases. Conversely, as the negative vertical deviation (negative x-axis direction) increases, the compression degree of the 1st and 25th balls decreases, while the compression degree of the 9th and 32nd balls increases. Therefore, when the x-axis assembly error increases from -300 μm to 300 μm, the contact load of the 1st and 25th balls gradually increases, while the contact load of the 9th and 32nd balls gradually decreases, as shown. Figure 16 As shown, because the degree of compression of the first and last balls relative to the other balls varies greatly with the vertical deviation, the fluctuation of their contact load is also the most significant.

[0256] Figure 17 This is a load distribution diagram of the slider-guide rail structure under guide rail verticality deviation in an embodiment of the present invention.

[0257] like Figure 17 As shown, the load distribution curve of the slider-guide rail structure varies with the deviation of the right end of the second guide rail 902 in the x-axis direction. Under the influence of positive vertical deviation, the worktable 4 generates clockwise torques around the y-axis and z-axis, respectively. Therefore, the contact loads of the first and second rows of balls in the second slider II and the fourth slider IV gradually increase, with the contact load of the first row of balls being greater than that of the second row; while the contact loads of the third and fourth rows of balls in the first slider I and the third slider III gradually increase, with the contact load of the third row of balls being greater than that of the fourth row. Correspondingly, under the action of negative vertical deviation, the worktable 4 generates counterclockwise torques around the y-axis and z-axis. At this time, the contact loads of the third and fourth rows of balls in the second slider II and the fourth slider IV gradually increase; the contact loads of the first and second rows of balls in the first slider I and the third slider III gradually increase. In addition, under the influence of the torque around the y-axis, the contact load distribution of each row of raceways is not equal. Ultimately, under positive vertical deviation: the contact load of the third row of balls in the first slider I and the third slider III is greater than that of the fourth row of balls; the contact load of the first row of balls in the second slider II and the fourth slider IV is greater than that of the second row of balls. Under negative vertical deviation: the contact load of the second row of balls in the first slider I and the third slider III is greater than that of the first row of balls; the contact load of the fourth row of balls in the second slider II and the fourth slider IV is greater than that of the third row of balls.

[0258] The effect of the y-axis offset of guide rail assembly 9 on load distribution:

[0259] Figure 18 This is a diagram showing the bending deformation of the lead screw shaft caused by the horizontal deviation of the guide rail in an embodiment of the present invention. Figure 19 This is a load distribution diagram of the ball screw nut pair under horizontal deviation of the guide rail in an embodiment of the present invention. Figure 20 This is a load distribution diagram of the slider-guide rail structure under horizontal deviation of the guide rail in an embodiment of the present invention.

[0260] like Figure 18 As shown, the bending deformation curves of the lead screw shaft are displayed when the right end of the second guide rail 902 shifts in the positive and negative y-axis directions. It can be seen that the position of the lead screw shaft moves along the y-axis direction, but remains unchanged in the x-axis direction. This indicates that the assembly error of the guide rail assembly 9 only causes a change in the position of the lead screw shaft in the corresponding direction. Furthermore, as... Figure 19 As shown, under the action of positive and negative horizontal deviations (positive and negative directions of the y-axis), the worktable 4 generates counterclockwise and clockwise torques around the x-axis, respectively. Therefore, as the positive horizontal deviation increases, the contact load of the 5th and 29th balls gradually increases, while the contact load of the 13th and 21st balls gradually decreases; as the negative horizontal deviation increases, the contact load of the 5th and 29th balls gradually decreases, while the contact load of the 13th and 21st balls gradually increases. Figure 20 As shown, under the action of positive and negative deviations in the y-direction, the compression of the balls in the yz plane of the ball screw nut pair is opposite, resulting in a symmetrical load distribution curve. The load distribution curve of the slider-guide structure is shown when the y-axis deviation changes from -300 μm to 300 μm. With the increase of positive horizontal deviation, the contact load of the second and third rows of balls in the first slider I and the fourth slider IV increases, while the contact load of the first and fourth rows of balls in the second slider II and the third slider III increases. With the increase of negative horizontal deviation, the contact load of the first and fourth rows of balls in the first slider I and the fourth slider IV increases, while the contact load of the second and third rows of balls in the second slider II and the third slider III increases. Furthermore, with the change of horizontal deviation, the contact loads of the first and fourth rows of balls are basically equal, and the contact loads of the second and third rows of balls are also basically equal. This is because the worktable only generates torque around the x-axis, and not torque around the y and z axes.

[0261] The role and effect of the embodiments

[0262] The method and system for predicting load distribution in a high-end machine tool feed system according to the present invention have the following beneficial effects:

[0263] 1. High prediction accuracy, overcoming the limitations of idealized assumptions: This invention establishes a static model that accurately reflects the coupling relationships of key components such as the ball screw nut pair and linear guide pair, abandoning the idealized assumptions of treating the system as a rigid body or uniformly loaded as in traditional simplification methods. The method of this invention can accurately calculate the nonlinear distribution of loads on each ball within the nut 8 and on each rolling element of the guide assembly 9 under the combined action of complex torques and axial forces. This provides a reliable basis for accuracy tracing and life prediction.

[0264] 2. Achieving an optimal balance between computational efficiency and accuracy, with strong engineering applicability: Within the scope of statics, this invention employs a modeling and solution strategy to achieve high computational efficiency at an engineering-practical level. While maintaining prediction accuracy close to that of complex simulation models, it significantly reduces computational complexity and resource consumption. Compared to the traditional finite element method, which requires building large models and is difficult and time-consuming to solve, this invention offers faster computation speed, enabling rapid parametric analysis and design iteration, greatly improving R&D efficiency and making it highly suitable for widespread application in engineering design and optimization stages.

[0265] 3. Transparent Mechanism and Strong Guidance: This invention not only provides the final load distribution data but also clearly reveals the evolution law of the load within the feed structure. Furthermore, it explicitly and comprehensively considers the influence of axial deformation of the leadscrew shaft, as well as bending and torsional deformation—often neglected in traditional methods—on load distribution. This helps designers gain a deeper understanding of the weak points in the performance of high-end machine tool feed systems, providing direct and clear theoretical guidance for stiffness matching, bearing selection and configuration, structural lightweighting, and performance improvement of high-end machine tool feed systems.

[0266] 4. Wide applicability and good versatility: The prediction model established by this invention has good versatility. By adjusting the model parameters, it can be applied to high-end machine tool feed systems of different types (such as horizontal and vertical) and under different working conditions, and has wide applicability.

[0267] Those skilled in the art should understand that this invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to this invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the invention as claimed. The scope of protection of this invention is defined by the appended claims and their equivalents.

Claims

1. A method for predicting load distribution in a high-end machine tool feed system, used to predict load distribution, characterized in that, Includes the following steps: S1: Coupled static model establishment steps: Establish a static model of the ball screw feed structure including the ball screw nut pair, linear guide pair and worktable. Treat the worktable, nut and slider assembly as rigid bodies, and model the screw as an elastic beam representing axial, radial bending and torsional deformation. Determine the cutting force acting on the center of the worktable and its components and moments in the spatial degrees of freedom. Define the global coordinate system and the coordinate system fixed to the worktable. S2: Contact force coordinate transformation and decomposition step: For each ball in the ball screw and nut pair, determine its contact point on the screw raceway and nut raceway, and establish multiple associated coordinate systems with the screw axis as the reference. Through multiple coordinate transformations, establish the coordinate relationship of the contact point under different coordinate systems, and decompose the contact force of the ball into a coordinate system with the screw axis as the reference to obtain the contact force components. S3: Force analysis steps for the worktable and lead screw shaft. Based on the contact force components, calculate the force and additional torque of the contact force on the worktable and the additional torque on the lead screw shaft, and establish the axial load balance equation and axial displacement change equation of the ball screw nut pair. S4: Steps for calculating the bending deformation of the lead screw axis: Based on the force and the additional torque applied to the lead screw axis, calculate the bending deformation of the lead screw axis over its entire length. S5: Steps for establishing the equilibrium equation of the worktable structure: Under the framework of the static model, establish a worktable structure model including the worktable, guide rail assembly, slider assembly and rolling elements, determine the center position of the slider raceway and the center position of the guide rail raceway, establish the expressions for the contact force and contact angle between the slider raceway and the guide rail raceway, and establish the static equilibrium equation of the worktable structure under external load. S6: Load distribution acquisition step: Determine the center position of the ball screw raceway and the center position of the nut raceway, establish an expression for the axial displacement change of the ball corresponding to the raceway center, and combine the axial displacement change equation and the static equilibrium equation to establish a set of nonlinear equations describing the static behavior of the entire ball screw feed structure. Solve the set of nonlinear equations by numerical iteration to obtain the load distribution.

2. The method for predicting load distribution in a high-end machine tool feed system according to claim 1, characterized in that: in, The cutting force F in S1 can be decomposed into three components Fx, Fy, and Fz along the x, y, and z axes, and three moments Mx, My, and Mz about the x, y, and z axes. The global coordinate system The origin is fixed at the angular contact ball bearing, its x-axis is perpendicular to the ideal worktable surface, and its z-axis coincides with the lead screw axis. The coordinate system fixed to the worktable The directions of each axis and the global coordinate system Keep parallel.

3. The method for predicting load distribution in a high-end machine tool feed system according to claim 2, characterized in that: in, The plurality of associated coordinate systems in S2 include: coordinate system The origin is established at the position of the lead screw axis corresponding to the ball contact point A, and its axes are parallel to the axes of the global coordinate system. The intermediate coordinate system... The origin is located at the center of the ball bearing, and its axes are perpendicular to the global coordinate system. The axes are parallel; Frenet-Serret coordinate system Origin and They share the same origin, and The axis is perpendicular to the lead screw axis. The direction of the axis is the tangential direction in which the balls move along the helical raceway. The angle between the axis and z is the helix angle. ; Contact coordinate system and The origins are located at contact points A and B between the ball and the screw raceway and the nut raceway, respectively. Axis perpendicular to - flat, The axis points to the center of the ball, and the contact coordinate system and The axes of the two are parallel and have the same direction, and their z-axis and z-axis are parallel. The angle between the negative directions of the axis is the initial contact angle. ; coordinate system and The origins are located at ball contact points A and B, respectively, and each of its axes is perpendicular to the global coordinate system. All axes are parallel.

4. The method for predicting load distribution in a high-end machine tool feed system according to claim 3, characterized in that: in, The method for obtaining the contact force component in S2 is as follows: Three coordinate transformations were performed to establish the coordinate relationship of the contact point in different coordinate systems, and the contact force was decomposed into a coordinate system with the lead screw axis as the reference. First transformation: Determine the coordinate system To coordinate system homogeneous coordinate transformation matrix coordinate system The origin point is shifted to the center of the ball; Second transformation: Determine from arrive homogeneous coordinate transformation matrix Through coordinate system Around Rotation This makes it shaft and The axes coincide and their direction is perpendicular to the lead screw axis. Based on this, then... Axis rotation ( ), making axis, The shafts are respectively and axis, Axis coincidence; Third transformation: Determine the intermediate coordinate system To contact coordinate system and homogeneous coordinate transformation matrix and coordinate system Around Axis rotation Angle, making shaft and The axes are in the same direction and parallel; based on this, then... Axis rotation Angle, making shaft and The shafts are respectively with shaft and With the axes parallel, the origin of the current coordinate system is finally translated to the ball contact points A and B respectively. Contact force Transform to coordinate system The equivalent force in the equation yields the contact force components, expressed as: (8)， Where: azimuth angle From Rotate the axis clockwise to Angle of the axis; The helix angle of the ball screw nut assembly; It is the contact angle between the ball and the lead screw raceway or nut raceway.

5. The method for predicting load distribution in a high-end machine tool feed system according to claim 4, characterized in that: in, The calculation method for the contact force exerted on the worktable and the additional torque, as well as the contact force exerted on the lead screw axis, in S3 is as follows: S3-1: Calculate the force and additional torque exerted by the contact force on the worktable, specifically: The contact force of each ball is translated to the geometric center of the worktable. According to equation (8), the contact force of the ball affects the worktable along the edge. and The forces acting in different directions are as follows: (9)， (10)， In the formula: This refers to the number of load-bearing balls in the ball screw nut assembly. The contact force of the ball bearings on the worktable , and The additional torques on the shafts are as follows: (11)， (12)， (13)， In the formula: The distance between the geometric center of the worktable and the centerline of the nut in the x-axis direction; For contact point B in coordinate system The axial coordinate of is expressed as: (14)， In the formula: This is the axial distance between the first ball and the last ball. The angle between adjacent balls. S3-2: Calculate the additional torque of the contact force on the leadscrew axis, specifically: In coordinate system The balls in the ball joint are equivalently translated to the center line of the lead screw shaft. Assuming the cross-sectional area of ​​the lead screw shaft is rigid, the additional torques about the x1, y1, and z1 axes generated by the equivalent translation are: (17)。 6. The method for predicting load distribution in a high-end machine tool feed system according to claim 5, characterized in that: in, The specific method for establishing the axial load balance equation and axial displacement variation equation of the ball screw nut pair in S3 is as follows: From equation (8), the axial load balance equation for the ball screw nut pair is: (15)， The relative position of the raceway center corresponding to the i-th ball is determined, and the axial displacement change between the raceway centers corresponding to the adjacent balls is: (16)， In the formula: This represents the relative axial displacement between the raceway centers corresponding to the i-th ball; Let be the axial distance between the i-th ball and the (i+1)-th ball; and These are the effective cross-sectional areas of the lead screw shaft and the nut, respectively.

7. The method for predicting load distribution in a high-end machine tool feed system according to claim 6, characterized in that: in, The specific method for calculating the bending deformation of the lead screw axis over its entire length in S4 is as follows: In radial force , and additional torque , Under the action of the ball contact point A, the bending deformation of the screw axis along the x and y axes is as follows: (18)， (19)， (20)， From equations (18) to (20), the bending deformations of the screw axis corresponding to the i-th ball contact point A along the x and y directions are respectively: (21)， In radial force , and additional torque , Under the action of the ball bearing, the bending deformation of the screw axis along the x and y directions between the angular contact ball bearing and the first ball contact point A are as follows: (22)， (23)， (24)， From equations (22) to (24), the bending deformations of the lead screw axis on the left side of the nut along the x and y directions are respectively: (25)， In radial force , and additional torque , Under the action of the ball bearing, the bending deformation of the screw axis along the x and y directions between the last ball contact point A and the deep groove ball bearing are as follows: (26)， (27)， (28)， From equations (26) to (28), the bending deformations of the lead screw axis on the right side of the nut along the x and y directions are respectively: (29)。 8. A prediction system for load distribution in a high-end machine tool feed system, characterized in that, include: The coupled static model establishment module establishes a static model of the ball screw feed structure, which includes a ball screw nut pair, a linear guide pair, and a worktable. The worktable, nut, and slider are regarded as rigid bodies, and the screw is modeled as an elastic beam representing axial, radial bending, and torsional deformation. The cutting force acting on the center of the worktable and its components and moments in the spatial degrees of freedom are determined. A global coordinate system and a coordinate system fixed to the worktable are defined. The contact force coordinate transformation and decomposition module determines the contact point of each ball in the ball screw and nut assembly on the screw raceway and nut raceway, and establishes multiple associated coordinate systems with the screw axis as the reference. Through multiple coordinate transformations, the coordinate relationship of the contact point under different coordinate systems is established, and the contact force of the ball is decomposed into the coordinate system with the screw axis as the reference to obtain the contact force components. The worktable and lead screw shaft force analysis module calculates the force and additional torque of the contact force on the worktable and the additional torque on the lead screw shaft based on the contact force components, and establishes the axial load balance equation and axial displacement change equation of the ball screw nut pair. The lead screw axis bending deformation calculation module calculates the bending deformation of the lead screw axis over its entire length based on the force and the additional torque applied to the lead screw axis. The workbench structure equilibrium equation establishment module establishes a workbench structure model including the workbench, guide rail, slider and rolling elements within the framework of the static model. It determines the center position of the slider raceway and the center position of the guide rail raceway, establishes the expressions for the contact force and contact angle between the slider raceway and the guide rail raceway, and establishes the static equilibrium equation of the workbench structure under external load. The load distribution acquisition module determines the center position of the ball screw raceway and the center position of the nut raceway, establishes an expression for the axial displacement change of the ball corresponding to the raceway center, and establishes a set of nonlinear equations describing the static behavior of the entire ball screw feed structure by combining the axial displacement change equation and the static equilibrium equation. The load distribution is obtained by solving the set of nonlinear equations through numerical iteration.

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