CN109933950B - Guide rail pair abrasion analysis and prediction method based on multi-scale bridge domain method - Google Patents
Guide rail pair abrasion analysis and prediction method based on multi-scale bridge domain method Download PDFInfo
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Abstract
The invention discloses a guide rail pair abrasion analysis and prediction method based on a multi-scale bridge domain method, which comprises the steps of firstly, carrying out error precision analysis on a guide rail pair; and then establishing a multiscale bridge domain abrasion index model, dividing the guide rail pair into a local continuous region and a non-local atomic region according to a multiscale bridge domain method, and introducing index parameters to describe the influence of the non-local atomic region. Thirdly, performing a wear experiment, measuring the linear accuracy of the guide rail, combining the distribution condition of a multi-scale bridge domain wear index model, and performing data correction; and finally, establishing a machine tool guide rail pair precision maintaining model. And (3) combining the linear precision measurement data of the guide rail, establishing a multi-scale bridge domain wear index model, and establishing a machine tool guide rail pair precision maintenance model. The method for judging the linear precision loss of the guide rail based on the multi-scale bridge domain principle realizes the description of the linear precision loss process of the guide rail, thereby improving the precision maintainability of the guide rail pair of the machine tool.
Description
Technical Field
The invention provides a guide rail pair wear analysis and prediction method based on a multi-scale bridge domain method, and belongs to the field of machine tool precision design.
Background
The numerical control machine tool is taken as basic equipment of advanced manufacturing technology, is an important mark for measuring the state industrial development level, and the manufacturing precision of parts of the current domestic precision machine tool, the precision retention and reliability of the whole machine are still to be improved, and the development capability of new products cannot completely meet the demands of users. As a core component of a feeding system of a precision machining machine tool, a guide rail pair directly determines feeding precision and precision retention of the machine tool to a certain extent, so that it is significant to study the precision and the retention of the guide rail pair.
At present, the simulation and optimization of the rail wear are all based on an elastoplastic mechanical model established by macroscopic assumptions of continuous media, isotropy and the like, but aiming at the problem of the precision retention of the rail under longer working time, the process of describing the precision loss only depends on the material performance index defined by macroscopic assumption conditions due to the fluctuation of the friction surface morphology and the microscopic characteristics of the material. The wear process of the guide rail pair can be visually reflected through the contact surface microscopic morphology before and after the wear occurs, the frictional contact process has multiscale due to the microscopic characteristics of the contact surface of the friction pair, and meanwhile, the microscopic performance index of the friction surface is complex and is difficult to directly measure, so that the macroscopic performance is often considered in the common wear amount prediction and ignored. In combination with past experience of friction and wear research, when a friction pair operates for a short time, the wear prediction model which pays attention to macroscopic expression has better accuracy; when the running time of the friction pair is longer, the prediction accuracy of the wear prediction model neglecting microscopic expression is fast in reduction, and certain limitation exists in practical application.
Disclosure of Invention
The invention aims at improving the precision retention of the machine tool guide rail pair, establishes a description model of the micro-convex body on the surface of the guide rail, and realizes the description of the linear precision loss process of the guide rail based on a multi-scale analysis method. The theoretical model established by the invention can provide a theoretical basis for accurately describing the abrasion process of the guide rail pair.
The invention discloses a guide rail pair wear analysis and prediction method based on a multi-scale bridge domain method, which specifically comprises the following steps:
step 1: analyzing the precision of the linear motion unit;
establishing a rectangular coordinate system O-xyz on a lathe bed of the abrasion test platform, and establishing a rectangular coordinate system O on a sliding table 1 -x 1 y 1 z 1 Wherein O-xyz is a reference coordinate system, the origin is fixed on the lathe bed, and the x axis of the O-xyz is in the ideal movement direction of the guide rail pairIn the machine tool, the y-axis is usually in the direction of the radial feed motion of the tool, and the z-axis direction can be determined by the right-hand rule. O (O) 1 -x 1 y 1 z 1 In order to move the coordinate system, the origin is fixed on the sliding table and moves along with the sliding table, wherein x is 1 、y 1 、z 1 The axes are respectively parallel to the x, y and z axes of the lathe bed coordinate system O-xyz, and the directions are the same. In an initial state, i.e. when the slipway has not moved, the coordinate system O is moved 1 -x 1 y 1 z 1 Origin O of (2) 1 The homogeneous coordinates in the bed coordinate system O-xyz are denoted as [ a, b, c,1] T 。
Assuming that one point on the sliding table moves along with the sliding table on one side of the guide rail, six errors are generated, namely displacement errors alpha along the x, y and z axis directions Lx 、α Ly 、α Lz And an angular error beta of rotation about the x, y, z axes Lx 、β Ly 、β Lz The distance that the error term follower point moves on the guide rail is different, so that the error term follower point is a function of the moving distance x of the sliding table, namely the ideal moving direction of the guide rail pair, and six errors are expressed as alpha Lx (x)、α Ly (x)、α Lz (x)、β Lx (x)、β Ly (x)、β Lz (x)。
Step 2: guide rail pair abrasion model based on multi-scale bridge domain method;
the bridge domain method is a trans-scale simulation theory, the core idea is that two models of finite element and molecular dynamics are established in the same area at the same time, material analysis is divided into a macro scale and an atomic scale, in the macro scale model, nonlinear elastic deformation of the material is described by adopting a finite element method, and a nonlinear constitutive equation is determined by a Cauchy-born rule; in the atomic scale model, the material is described as a crystal, the deformation of which conforms to classical newton's law, while the total displacement is decomposed into the sum of the coarse-scale and fine-scale displacement fields, and the sum of the atomic energy and the continuous medium energy is taken as the total energy of the system.
Step 2.1, rail strain energy distribution based on the influence of bridge domain theoretical deformation;
in the bridge domain theory, the concept of deformation gradient is introduced, so that the strain energy density function of the continuous medium is related to the interatomic potential function, and the load born by the models with different scales has equivalence. When the deformation gradient of the material is single and uniform, the cauchy-born criterion can accurately describe the deformation condition of the material lattice; when the deformation is severe and the deformation gradient is large, the higher-order cauchy-born criterion can be considered:
let the coordinate before deformation be X i The coordinate after deformation is x i The deformed coordinates are regarded as a function of the coordinates before deformation, i.e. x i ＝x i (X i ) Defining the partial derivative F of the current coordinates of the object to the deformed coordinates:
x i called deformation gradient, X i Is an asymmetric second order tensor describing the deformation around object point X. In order to facilitate analysis of the wear problem of the rail pair, it is assumed that the unit deformation gradients in the unidirectional loaded rail pair system are distributed in order from small to large and a deformation gradient map is obtained.
For a continuous medium region with small deformation, the change rate of the total deformation gradient among the units is relatively small and is a certain value, and assuming that the change is linear and the gradient is tan theta, theta represents the inclination angle of the deformation gradient curve, the maximum deformation gradient of the elastic part unit of the continuous region line is M 1 + (1-w) utanθ; for an atomic region, because the calculation of the interatomic interaction potential energy is complex, in order to embody the action of an atomic part and simplify analysis, the atomic part is described by adopting exponential function approximation, and the deformation gradient simplification relation of the atomic region is obtained according to the sum of the interatomic interaction potential energy of the atomic region of the simplified guide rail subsystem:
wherein W represents a constant term coefficient of an atomic region index equation, n w And an atomic ratio index representing an atomic region index equation. M is M 1 Representing the minimum deformation gradient of the elastic portion of the line, i.e. the continuous area unit, M 2 Representing the maximum deformation gradient of the non-local atomic region unit, M 3 The junction of the continuous region and the non-local atomic region is represented by u, the total strain energy deformation gradient of the rail subsystem is represented by u, and the proportionality coefficient of the atomic region with larger deformation is represented by w.
The deformation gradient distribution of the formula (2) only considers the index influence of the deformation of the linear elastic part on the atomic region, and the deformation of the ideal elastic solid is instantaneous reversible deformation within the limit stress range according to the classical elastic theory, and the deformation amount is small and has no time dependence. Therefore, when the external load is removed, the deformation displacement u is completely recovered, and n is w =0. However, the abrasion process of the guide rail pair is not completely a linear elastic deformation process, and the influence on the atomic region of the guide rail pair system is not only a linear elastic factor, so parameters other than the linear elasticity should be introduced for description.
In actual wear deformation, as load time increases, not only the inter-atomic distance but also the atomic arrangement sequence changes due to the influence of molecular or inter-atomic forces, at this time, after the load is unloaded, the deformation displacement amount cannot be fully recovered to the initial state, and a certain time is required for the atomic arrangement to recover from the current state to the new equilibrium state, which can be explained by the lattice relaxation in the quantization theory, the time elapsed for the system to recover to the equilibrium state becomes the relaxation time, and this process follows the law of exponential change. The gradient of the coupling region is larger than that of the continuous medium region, so that the deformation gradient curve of the atomic region is extended to the position with the gradient of 0, and the virtual elongation rate is r to represent the lattice relaxation proportion of the atomic part, and the relaxation proportion index of the atomic part is n r 。
At this time, when the length of the atomic region is rwu, the deformation gradient of the atomic region is expressed by the simplified relationship:
wherein W represents a constant term coefficient of an atomic region index equation, n w Atomic ratio index, n, representing an atomic region index equation r And (3) expressing the relaxation ratio index of the atomic region index equation, and r expressing the virtual elongation of the deformation gradient curve.
Step 2.2 rail Strain energy distribution based on grain boundary diffusion loss
In the process of wearing the guide rail pair material, as for the microstructure, the internal atoms of the microprotrusions are diffused at the grain boundaries during transient contact, namely grain boundary diffusion (Grain Boundary Diffusion), and the activation energy required by the grain boundary diffusion is small, so that the grain boundary diffusion is often generated under the condition of low temperature, and the diffusion coefficient is set as D.
In this case, when the length of the atomic region is Drwu, the deformation gradient reduction relational expression of the atomic region is:
wherein W represents a constant term coefficient of an atomic region index equation, n w Atomic ratio index, n, representing an atomic region index equation r Relaxation ratio index, n, representing atomic region index equation d Representing the diffusion ratio index of the atomic region index equation.
The cross section of the original region and the loaded part of the system perpendicular to u is A 1 The cross-sectional area of the continuous region perpendicular to u is kA 1 K is a proportionality coefficient, and the simplified energy expression of the continuous medium area and the atomic area of the system is obtained by the geometric relationship:
according to the virtual work principle, the balance point displacement of the system unit is determined by the following formula:
obtaining:
based on binomial theorem, the method is simplified and divided into:
f i u i the sum of the work performed by the external load is represented, and the loss of energy in the elastic region of the line hardly affects the exponential distribution of the atomic region, mainly affects the constant term coefficient, because the proportion of loss in the continuous portion is relatively large. In summary, based on the multi-scale analysis principle and the index approximation assumption, the wear amount of the sliding guide rail pair varies exponentially with the accumulated wear number in the service process. Assuming that the number of times of stable contact wear of the sliding guide rail pair is m, obtaining a general relation between the local wear h of the sliding guide rail pair multi-scale system and the number of times of contact wear m:
h＝A n m n +A I m (9)
wherein A is n The index part atomic region abrasion coefficient of the sliding guide rail pair is represented; a is that I Representing the abrasion coefficient of the elastic part of the sliding guide rail pair; n is the wear coefficient.
Step 3: a linear accuracy index model based on a multi-scale wear model;
step 3.1, straightness error measurement data and data processing;
the linear precision measurement is respectively carried out on the left guide rail and the right guide rail of the double-guide rail abrasion test bed, the load is applied on one side, the load is not applied on the other side, the straightness error data of the guide rails on the two sides along the XX axis moving direction are obtained, the straightness error data are shown in a table 1, and the Z-direction index data of the left guide rail and the right guide rail after data processing are shown in a table 2.
Step 3.2, a linear precision index model based on a multi-scale analysis model;
and taking the logarithm of the absolute value of the representative coordinate of the linear precision measurement value and the linear precision measurement value, wherein the slope +1 of the log graph after linear regression is the corresponding wear index. Thereby obtaining the left guide rail unloaded wear index n rz =0.5625, loaded wear indexRight rail unloaded wear index n lz = 0.8427, load wear index +.>And (3) converting the representative coordinate quantity into the total feed quantity L to be substituted by considering that the feed frequency of the machine tool is not 1, and setting the linear precision expression quantity influenced by the abrasion quantity as h to obtain a linear precision index model of the left guide rail and the right guide rail of the linear motion experimental platform.
Left guide rail unloaded linear accuracy index model:
h lz ＝0.0001616L 0.5625 (10)
left guide rail loaded straight line precision index model:
right guide rail unloaded linear accuracy index model:
h lz ＝0.0003242L 0.8427 (12)
right guide rail loaded straight line precision index model:
the left rail is under an external load of 1000N when not loaded, 2000N when loaded, 1000N when not loaded, 4000N when loaded.
Step 3.3 Linear guide precision maintenance model based on wear index model
The deviation analysis is carried out by combining the wear index model of the double guide rails, the running-in operation of the guide rails is considered for 3 months, and the straightness loss only related to Z-direction wear of the guide rails can be obtained: non-loaded Z-direction straightness A of left guide rail lz 0.002778mm loaded Z-direction straightnessThe unloaded Z-direction straightness A of the right guide rail is 0.005592mm rz 0.001771mm loaded Z-direction straightness +.>For 0.00457mm, the guide rail running time is set as t, and the linear guide rail precision retentivity model obtained by substituting calculation is as follows:
left rail unloaded precision retention index model:
A lz ＝0.001498t 0.5625 (14)
left rail loaded precision retention index model:
right rail unloaded precision retention index model:
A lz ＝0.0007015t 0.8427 (16)
right rail loaded precision retention index model:
when the failure precision caused by Z-direction abrasion is 0.02mm, the theoretical effective life of 1000N loaded on the left guide rail is about 100 months, the theoretical effective life of 2000N loaded on the left guide rail is about 11 months, the theoretical effective life of 1000N loaded on the right guide rail is about 53 months, and the theoretical effective life of 4000N loaded on the right guide rail is about 14 months.
Drawings
FIG. 1 is a rail pair coordinate system of a wear test stand;
FIG. 2 is a schematic diagram of a gradient distribution of cell deformation within a multi-scale bridge domain system;
FIG. 3 is a schematic diagram of a deformation gradient distribution of a multi-scale bridge domain system under the influence of lattice relaxation;
FIG. 4 is a schematic diagram of deformation gradient distribution of a multi-scale bridge domain system under the influence of grain boundary diffusion;
fig. 5 is a schematic flow chart of the implementation of the method.
Detailed Description
Step 1: precision analysis of a linear motion unit
When the rigid body moves, any point on the rigid body is taken as a base point, and the movement process of the rigid body can be regarded as that the rigid body rotates around the base point while translating along with the base point. The homogeneous coordinate transformation method (HTM) can relate the motion, transformation and mapping matrix operation of the rigid body, so that the relative position and direction of each coordinate system in the rigid body motion process can be conveniently and accurately described, and therefore, the homogeneous coordinate transformation method is widely applied to rigid body kinematics analysis. The seal adopts the homogeneous coordinate transformation theory to model the precision of the guide rail pair of the abrasion test bed.
As shown in FIG. 1, a rectangular coordinate system O-xyz is established on the bed of the abrasion test platform, and a rectangular coordinate system O is established on the sliding table 1 -x 1 y 1 z 1 Wherein O-xyz is a reference coordinate system, an origin is fixed on the lathe bed, and an x-axis of the O-xyz is in an ideal movement direction of the guide rail pair, a y-axis of the O-xyz is in a radial feeding movement direction of a tool in a machine tool, and a z-axis direction can be determined by a right-hand rule. O (O) 1 -x 1 y 1 z 1 In order to move the coordinate system, the origin is fixed on the sliding table and moves along with the sliding table, wherein x is 1 、y 1 、z 1 The axes are respectively parallel to the x, y and z axes of the lathe bed coordinate system O-xyz, and the directions are the same. In an initial state, i.e. when the slipway has not moved, the coordinate system O is moved 1 -x 1 y 1 z 1 Origin O of (2) 1 The homogeneous coordinates in the bed coordinate system O-xyz are denoted as [ a, b, c,1] T
Assume that a point on the sliding table is along with the sliding tableThe guide rail on one side moves, which generates 6 errors, namely displacement errors alpha along the x, y and z axes Lx 、α Ly 、α Lz And an angular error beta of rotation about the x, y, z axes Lx 、β Ly 、β Lz The distance that the error term follower points move on the guide rail is different, so that the error term follower points are functions of the movement distance x of the sliding table, and then 6 errors are expressed as alpha Lx (x)、α Ly (x)、α Lz (x)、β Lx (x)、β Ly (x)、β Lz (x)。
Step 2: guide rail pair abrasion model based on multi-scale bridge domain method;
the bridge domain method is a trans-scale simulation theory, the core idea is that two models of finite element and molecular dynamics are established in the same area at the same time, material analysis is divided into a macro scale and an atomic scale, in the macro scale model, nonlinear elastic deformation of the material is described by adopting a finite element method, and a nonlinear constitutive equation is determined by a Cauchy-born rule; in the atomic scale model, the material is described as a crystal, the deformation of which conforms to classical newton's law, while the total displacement is decomposed into the sum of the coarse-scale and fine-scale displacement fields, and the sum of the atomic energy and the continuous medium energy is taken as the total energy of the system.
Step 2.1 Cauchy-Boen criterion;
the Cauchy-Born criterion, also called Cauchy-Born criterion, is the basic theory of multi-scale analysis methods such as bridge domain method, quasi-continuous medium method, etc., and is suitable for describing the approximate uniform deformation of space block crystals. For a material with a relatively uniform local deformation, the cauchy-born criterion can describe the deformation condition of the material lattice relatively accurately, but when the deformation is relatively severe, the atomic displacement can only be described approximately, and then the high-order cauchy-born criterion can be considered. The core idea of the cauchy-born criterion is that the change rule of lattice vector of the crystal in the process of uniform deformation is assumed to be the same as the vector change rule of the continuous medium in the process of uniform deformation, so that the microstructure deformation of atomic scale and constitutive relation of the continuous medium are effectively related, the criterion introduces the concept of deformation gradient, thus the strain energy density function of the continuous medium is related with interatomic potential function, the load born by different scale models is equivalent, and the stress born by corresponding units can be deduced according to different interatomic potential functions.
In continuous medium theory, the infinite small line element vector dX in the reference configuration before deformation and the infinite small line element vector dX in the post-deformation configuration are related by the deformation gradient F, and thus have the relation:
dx＝F·dX (1)
thus, the cauchy-born criterion regards the lattice vector of the crystal as an infinitely small vector, assuming that the lattice at a point of the material in the continuous region is the same as the deformation state of that point, the deformation gradient F can be similarly used to describe.
r＝F·R (2)
Where R represents a lattice vector in the deformed crystal structure and R represents a lattice vector of R before deformation. Because the local deformation of the material is relatively uniform, the lattice unit body is uniformly changed according to the deformation gradient, and the atomic potential energy of one atom and the total number of atoms in the unit body can be multiplied to obtain the total atomic potential energy E in the unit body 0 The strain energy density function can be determined by the atomic potential energy E in the unit body 0 Divided by the volume V of the lattice unit 0 And (3) obtaining:
the total potential energy of the crystal is the sum of the potential energy of the lattice cells, namely:
wherein V is i Representing the volume of the ith lattice cell. Thus, the calculated amount is greatly reduced, and the calculation efficiency is remarkably improved.
Step 2.2, rail strain energy distribution based on the influence of bridge domain theoretical deformation;
from the above, in the bridge domain theory, the concept of deformation gradient is introduced, so that the strain energy density function and the interatomic potential function of the continuous medium are related, and the load applied between models with different scales is equivalent. When the deformation gradient of the material is single and uniform, the cauchy-born criterion can describe the deformation condition of the material lattice more accurately; when the deformation is severe and the deformation gradient is large, the higher-order cauchy-Barn criterion can be considered
Let the coordinate before deformation be X i The coordinate after deformation is x i The deformed coordinates are regarded as a function of the coordinates before deformation, i.e. x i ＝x i (X i ) Defining the partial derivative of the current coordinates of the object to the deformed coordinates:
known as deformation gradient, is an asymmetric second order tensor describing the deformation near object point X. In order to facilitate analysis of the rail pair wear problem, it is assumed that the cell deformation gradients within the unidirectionally loaded rail pair system of the present invention are arranged in a descending order, as shown in fig. 2.
In FIG. 2, the abscissa represents the deformation displacement, and can also be considered as strain energy distribution of the rail subsystem, the ordinate represents the deformation gradient, u represents the total strain energy deformation gradient of the rail subsystem, and w represents the proportionality coefficient of the atomic region with larger deformation, i.e. the atomic region length in the total strain energy deformation gradient is wu, M 1 Representing the minimum deformation gradient of the line elastic portion, i.e. the continuous medium region unit, M 2 Representing the maximum deformation gradient of atomic region units, M 3 Representing the junction of the continuous dielectric region and the atomic region, M 3 The area of the left part of the abscissa (1-w) u is the sum of the linear elastic stress energy of the continuous medium region, and the area of the right part is the sum of the interatomic interaction potential energy of the atomic region.
For a continuous medium region with less deformation, the overall deformation gradient between units changesThe ratio is relatively small and is a constant value, and assuming that the ratio is linearly changed and the slope is tan theta, theta represents the inclination angle of the deformation gradient curve, the maximum deformation gradient of the elastic part unit of the continuous area line is M 1 + (1-w) utanθ; for the atomic region, because the calculation of the interatomic interaction potential energy is complex, in order to conveniently embody the action of the atomic part and simplify the analysis, the invention adopts exponential function approximation to describe the atomic part, and the area of the dotted line region on the right side of (1-w) u in fig. 2 is the sum of the interatomic interaction potential energy of the atomic region of the simplified rear rail subsystem, so as to obtain a deformation gradient simplified relational expression of the atomic region:
wherein W represents a constant term coefficient of an atomic region index equation, n w And an atomic ratio index representing an atomic region index equation. M is M 1 Representing the minimum deformation gradient of the elastic portion of the line, i.e. the continuous area unit, M 2 Representing the maximum deformation gradient of the non-local atomic region unit, M 3 Indicating the intersection of the continuous region with the non-localized atomic region.
The deformation gradient distribution of the above formula only considers the exponential influence of the deformation of the linear elastic part on the atomic region, and according to the classical elastic theory, the deformation of the ideal elastic solid is the instantaneous reversible deformation within the limit stress range, and the deformation amount is small and has no time dependence. Therefore, when the external load is removed, the deformation displacement u is completely recovered, and n is w =0. However, the abrasion process of the guide rail pair is not completely a linear elastic deformation process, and the influence on the atomic region of the guide rail pair system is not only a linear elastic factor, so parameters other than the linear elasticity should be introduced for description.
In actual wear deformation, not only the inter-atomic distance but also the atomic arrangement order changes due to the influence of molecular or inter-atomic forces as the load time increases, and at this time, the deformation displacement amount cannot be fully recovered after the load is unloadedTo the initial state, and the atomic arrangement is restored to a new equilibrium state from the current state, a phenomenon which can be explained by using lattice relaxation in the theory of quantization, the time elapsed for the system to restore the equilibrium state becomes a relaxation time, and the process follows an exponential change law. Referring to FIG. 2, the gradient of the coupling region is greater than the gradient of the continuous dielectric region, so that the atomic region deformation gradient curve is extended to a point where the gradient is 0, and the virtual elongation is r, which represents the lattice relaxation ratio of the atomic portion, and the relaxation ratio index of the atomic portion is n r . As shown in fig. 3.
At this time, when the length of the atomic region is rwu, the deformation gradient of the atomic region is expressed by the simplified relationship:
wherein W represents a constant term coefficient of an atomic region index equation, n w Atomic ratio index, n, representing an atomic region index equation r And represents the relaxation ratio index of the atomic region index equation.
Step 2.3 guide rail Strain energy distribution based on grain boundary diffusion loss
In the process of wearing the guide rail pair material, as for the microstructure, the internal atoms of the microprotrusions are diffused at the grain boundaries during transient contact, namely grain boundary diffusion (Grain Boundary Diffusion), and the activation energy required for the grain boundary diffusion is small, so that the grain boundary diffusion tends to occur at a low temperature. The morphology of the crystal at the grain boundaries affects the diffusion process to some extent.
Or alternatively
Wherein D is vol Coefficient of bulk diffusion
Delta C-atomic concentration difference of constituent elements
X ave Average thickness of grains
D GB Grain boundary diffusion coefficient
Delta grain boundary width
d-transverse diameter of grain
R-ratio of grain boundary thickness to average grain thickness, i.eAnd->For body diffusion flux, +.>The flux is diffused for the grain boundaries.
Since the activation energy required for grain boundary diffusion is much smaller than the bulk diffusion, the grain boundary diffusion flux is much greater than its bulk diffusion flux, i.e., the total diffusion flux may be approximately equal to the grain boundary diffusion flux:
from equiaxed grains, it is approximated that:
d≈X ave (11)
thus, formula (10) can be expressed as:
or alternatively
The rate of increase of the diffusion flux is:
the average thickness of the grains can be obtained as a function of time by separating the variables and integrating the above formulae:
as can be seen from the growth rate equation, in the grain boundary diffusion model, when R and δ are constants, the average thickness of the crystal grains is in a cubic curve relationship with time, and the diffusion rate index thereof is 0.333. In the grain boundary diffusion model, the grain morphology parameters and diffusion process are different, and the grain boundary has different diffusion rate indexes. Therefore, the index curve of the continuous dielectric region will shrink back due to grain boundary diffusion loss in the atomic region, assuming a diffusion coefficient D, as shown in fig. 4.
In this case, when the length of the atomic region is Drwu, the deformation gradient reduction relational expression of the atomic region is:
wherein W represents a constant term coefficient of an atomic region index equation, n w Atomic ratio index, n, representing an atomic region index equation r Relaxation ratio index, n, representing atomic region index equation d Representing the diffusion ratio index of the atomic region index equation.
The cross section of the original region and the loaded part of the system perpendicular to u is A 1 The cross-sectional area of the continuous region perpendicular to u is kA 1 K is a proportionality coefficient, and the simplified energy expression of the continuous medium area and the atomic area of the system is obtained by the geometric relationship:
according to the virtual work principle, the balance point displacement of the system unit is determined by the following formula:
the method can obtain:
the method is based on binomial theorem and can be obtained after simplification and general classification:
since the proportion of loss in the continuous portion is relatively large, the loss of energy in the linear elastic region hardly affects the exponential distribution of the atomic region, mainly affecting the constant term coefficient. In summary, based on the multi-scale analysis principle and the index approximation assumption, the wear amount of the sliding guide rail pair varies exponentially with the accumulated wear number in the service process. Assuming that the number of times of stable contact wear of the sliding guide rail pair is m, a general relation between the local wear amount and the number of times of contact wear of the multi-scale bridge domain system of the sliding guide rail pair can be obtained:
h＝A n m n +A I m (21)
wherein A is n The index part atomic region abrasion coefficient of the sliding guide rail pair is represented; a is that I Representing the abrasion coefficient of the elastic part of the sliding guide rail pair; n is the wear coefficient.
Step 3: linear accuracy index model step 3.1 straightness error measurement data and data processing based on multi-scale abrasion model
The linear precision measurement is respectively carried out on the left guide rail and the right guide rail of the double-guide rail abrasion test bed, the load is applied on one side, the load is not applied on the other side, the straightness error data of the guide rails on the two sides along the XX axis moving direction are obtained, the straightness error data are shown in a table 1, and the Z-direction index data of the left guide rail and the right guide rail after data processing are shown in a table 2.
Step 3.2 Linear precision index model based on Multi-scale abrasion model
And taking the logarithm of the absolute value of the representative coordinate of the linear precision measurement value and the linear precision measurement value, wherein the slope +1 of the log graph after linear regression is the corresponding wear index. The left guide rail non-loading abrasion index n can be obtained rz =0.5625, loaded wear indexRight rail unloaded wear index n lz = 0.8427, load wear index +.>The representative coordinate quantity is converted into the total feed quantity L to be substituted, and the linear accuracy expression quantity influenced by the abrasion quantity is set as h, so that a linear accuracy index model of the left guide rail and the right guide rail of the linear motion experimental platform can be obtained:
left guide rail unloaded linear accuracy index model:
h lz ＝0.0001616L 0.5625 (22)
left guide rail loaded straight line precision index model:
right guide rail unloaded linear accuracy index model:
h lz ＝0.0003242L 0.8427 (24)
right guide rail loaded straight line precision index model:
the left rail is under an external load of 1000N when not loaded, 2000N when loaded, 1000N when not loaded, 4000N when loaded.
Step 3.3 Linear guide precision maintenance model based on wear index model
Deviation analysis is carried out by combining with a wear index model of the double guide rails, and guide rail running-in is consideredThe straightness loss only related to Z-direction abrasion of the guide rail can be obtained after 3 months of operation: non-loaded Z-direction straightness A of left guide rail lz 0.002778mm loaded Z-direction straightnessThe unloaded Z-direction straightness A of the right guide rail is 0.005592mm rz 0.001771mm loaded Z-direction straightness +.>For 0.00457mm, the guide rail running time is set as t, and the linear guide rail precision retentivity model obtained by substituting calculation is as follows:
left rail unloaded precision retention index model:
A lz ＝0.001498t 0.5625 (26)
left rail loaded precision retention index model:
right rail unloaded precision retention index model:
A lz ＝0.0007015t 0.8427 (28)
right rail loaded precision retention index model:
when the failure precision caused by Z-direction abrasion is 0.02mm, the theoretical effective life of the left guide rail loaded 1000N is about 100 months, the theoretical effective life of the left guide rail loaded 2000N is about 11 months, the theoretical effective life of the right guide rail loaded 1000N is about 53 months, and the theoretical effective life of the right guide rail loaded 4000N is about 14 months.
Table 1 measurement of Z-direction straight line accuracy of left and right guide rails
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TABLE 2 correction values for Z-direction index data of left and right guide rails
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The invention will be more clearly understood from the above description taken in conjunction with the accompanying drawings which illustrate the method and practice of the invention.
Claims (1)
1. A guide rail pair abrasion analysis and prediction method based on a multi-scale bridge domain method is characterized by comprising the following steps of: specifically comprises the following steps of,
step 1: analyzing the precision of the linear motion unit;
establishing a rectangular coordinate system O-xyz on a lathe bed of the abrasion test platform, and establishing a rectangular coordinate system O on a sliding table 1 -x 1 y 1 z 1 Wherein O-xyz is a reference coordinate system, an origin is fixed on a lathe bed, an x-axis of the O-xyz is in an ideal movement direction of a guide rail pair, a y-axis of the O-xyz is in a radial feeding movement direction of a cutter in a machine tool, and a z-axis direction is determined by a right-hand rule; o (O) 1 -x 1 y 1 z 1 In order to move the coordinate system, the origin is fixed on the sliding table and moves along with the sliding table, wherein x is 1 、y 1 、z 1 The axes are respectively parallel to the x, y and z axes of the lathe bed coordinate system O-xyz, and the directions are the same; in an initial state, i.e. when the slipway has not moved, the coordinate system O is moved 1 -x 1 y 1 z 1 Origin O of (2) 1 The homogeneous coordinates in the bed coordinate system O-xyz are denoted as [ a, b, c,1] T ；
Assuming that one point on the sliding table moves along with the sliding table on one side of the guide rail, six errors are generated, namely displacement errors alpha along the x, y and z axis directions Lx 、α Ly 、α Lz And an angular error beta of rotation about the x, y, z axes Lx 、β Ly 、β Lz The distance that the error term follower point moves on the guide rail is different, so that the error term follower point is a function of the moving distance x of the sliding table, namely the ideal moving direction of the guide rail pair, and six errors are expressed as alpha Lx (x)、α Ly (x)、α Lz (x)、β Lx (x)、β Ly (x)、β Lz (x)；
Step 2: guide rail pair abrasion model based on multi-scale bridge domain method;
in a macro scale model, nonlinear elastic deformation of a material is described by adopting a finite element method; in the atomic scale model, the material is described as a crystal, the deformation accords with classical Newton's law, the total displacement is decomposed into the sum of a coarse scale displacement field and a fine scale displacement field, and the sum of atomic energy and continuous medium energy is used as the total energy of the system;
step 2.1, rail strain energy distribution based on the influence of bridge domain theoretical deformation;
introducing a concept of deformation gradient, so that a strain energy density function and an interatomic potential function of a continuous medium are related, and the load born by models with different scales has equivalence;
let the coordinate before deformation be X i The coordinate after deformation is x i The deformed coordinates are regarded as a function of the coordinates before deformation, i.e. x i ＝x i (X i ) Defining the partial derivative F of the current coordinates of the object to the deformed coordinates:
x i called deformation gradient, X i Is an asymmetric second order tensor describing the deformation near object point X;
for less deformed continuous mediaA region whose rate of change of the total deformation gradient between the units is relatively small and has a constant value, and assuming that the linear change is made and the slope is tan θ, θ represents the inclination angle of the deformation gradient curve, the maximum deformation gradient of the elastic portion unit of the continuous region line is M 1 ++ (1-w) u tan θ; for an atomic region, because the calculation of the interatomic interaction potential energy is complex, in order to embody the action of an atomic part and simplify analysis, the atomic part is described by adopting exponential function approximation, and the deformation gradient simplification relation of the atomic region is obtained according to the sum of the interatomic interaction potential energy of the atomic region of the simplified guide rail subsystem:
wherein W represents a constant term coefficient of an atomic region index equation, n w An atomic ratio index representing an atomic region index equation; m is M 1 Representing the minimum deformation gradient of the elastic portion of the line, i.e. the continuous area unit, M 2 The maximum deformation gradient of the non-local atomic region unit is represented, u represents the total strain energy deformation gradient of the guide rail subsystem, and w represents the proportionality coefficient of the atomic region with larger deformation;
the deformation gradient distribution of the formula (2) only considers the index influence of the deformation of the linear elastic part on the atomic region, and the deformation of the ideal elastic solid is instantaneous reversible deformation within the limit stress range according to the classical elastic theory, so that the deformation amount is small and the time dependence is avoided; when the external load is removed, the deformation displacement u is completely recovered, and n is w =0; however, the abrasion process of the guide rail pair is not completely a linear elastic deformation process, and the influence on the atomic region of the guide rail pair system is not only a linear elastic factor, so parameters except the linear elasticity should be introduced for description;
extending the deformation gradient curve of the atomic region to a position with a slope of 0, setting a virtual elongation rate of r to represent the relaxation proportion of the lattice of the atomic part, and setting the relaxation proportion index of the atomic part to be n r ；
At this time, when the length of the atomic region is rwu, the deformation gradient of the atomic region is expressed by the simplified relationship:
wherein W represents a constant term coefficient of an atomic region index equation, n w Atomic ratio index, n, representing an atomic region index equation r A relaxation ratio index of an atomic region index equation is represented, and r represents a virtual elongation of a deformation gradient curve;
step 2.2 rail Strain energy distribution based on grain boundary diffusion loss
In the abrasion process of the guide rail pair material, as for the microstructure, internal atoms of the microprotrusions are diffused at grain boundaries during transient contact, namely grain boundary diffusion, and the activation energy required by the grain boundary diffusion is smaller, so that the grain boundary diffusion is often generated under the condition of lower temperature, and the diffusion coefficient is set as D;
in this case, when the length of the atomic region is Drwu, the deformation gradient reduction relational expression of the atomic region is:
wherein W represents a constant term coefficient of an atomic region index equation, n w Atomic ratio index, n, representing an atomic region index equation r Relaxation ratio index, n, representing atomic region index equation d A diffusion ratio index representing an atomic region index equation;
the cross section of the original region and the loaded part of the system perpendicular to u is A 1 The cross-sectional area of the continuous region perpendicular to u is kA 1 K is a proportionality coefficient, and the simplified energy expression of the continuous medium area and the atomic area of the system is obtained by the geometric relationship:
according to the virtual work principle, the balance point displacement of the system unit is determined by the following formula:
obtaining:
based on binomial theorem, the method is simplified and divided into:
f i u i the total sum of the acting of the external load is represented, and the loss ratio of the continuous part is relatively large, but the energy loss of the linear elastic region hardly affects the exponential distribution of the atomic region, and affects the constant term coefficient;
based on a multi-scale analysis principle and an index approximation assumption, the abrasion loss of the sliding guide rail pair is exponentially changed along with the accumulated abrasion frequency in the service process; assuming that the number of times of stable contact wear of the sliding guide rail pair is m, obtaining a general relation between the local wear h of the sliding guide rail pair multi-scale system and the number of times of contact wear m:
h＝A n m n +A I m (9)
wherein A is n The index part atomic region abrasion coefficient of the sliding guide rail pair is represented; a is that I Representing the abrasion coefficient of the elastic part of the sliding guide rail pair; n is the wear coefficient;
step 3: a linear accuracy index model based on a multi-scale wear model;
step 3.1, straightness error measurement data and data processing;
respectively measuring the linear precision of the left and right guide rails of the double-guide rail abrasion test bed, and applying a load on one side and not applying a load on the other side to obtain straightness error data of the guide rails on the two sides along the X-axis movement direction and Z-direction index data of the left and right guide rails after data processing;
step 3.2, a linear precision index model based on a multi-scale analysis model;
taking the logarithm of the absolute value of the representative coordinate of the linear precision measurement value and the linear precision measurement value, wherein the slope +1 of the log graph after linear regression is the corresponding wear index; thereby obtaining the left guide rail unloaded wear index n rz Load wear indexRight rail unloaded wear index n lz Load wear index->Considering that the feeding frequency of the machine tool is not 1, converting the representative coordinate quantity into the total feeding quantity L, substituting the total feeding quantity L, and setting the linear precision expression quantity influenced by the abrasion quantity as h to obtain a linear precision index model of the left guide rail and the right guide rail of the linear motion experimental platform;
step 3.3 Linear guide precision maintenance model based on wear index model
And carrying out deviation analysis by combining the wear index model of the double guide rails to obtain straightness loss only related to Z-directional wear of the guide rails.
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