CN117629631A - Heterogeneous ball bearing contact characteristic analysis method based on equivalent impurity clamping method - Google Patents
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Abstract
The invention provides a heterogeneous ball bearing contact characteristic analysis method based on an equivalent impurity clamping method, and relates to the technical field of bearing detection. The heterogeneous ball bearing contact characteristic analysis method based on the equivalent impurity clamping method comprises the following steps of S1, modeling the contact characteristic of the impurity-containing ball bearing; s2, analyzing the contact characteristics of the ball bearings under different inclusion shapes; and S3, analyzing the contact characteristics of the ball bearing in the presence of multiple impurities. The prolate ellipsoid has a greater impact on contact characteristics than the flat ellipsoid; the larger the volume of cubic impurities, the greater the contact surface stress; the deeper the impurity is located, i.e. the farther from the contact surface, the less the impurity affects the bearing contact characteristics; the height of the cylindrical impurities has larger influence on contact characteristics than the radius, and the characteristic stress generated by the impurities is larger; the smaller the total impurity volume differs from the matrix volume, the smaller the maximum stress of the bearing contact surface.
Description
Technical Field
The invention relates to the technical field of bearing detection, in particular to a heterogeneous ball bearing contact characteristic analysis method based on an equivalent hybrid method.
Background
Impurities inside bearings can be divided into two categories: the method mainly comprises the steps of mixing molten steel high-temperature resistant materials, slag, dust particles and the like, wherein the foreign impurities have the characteristics of large particle volume, irregular appearance, irregular distribution and the like, and the factors such as the type, shape, size, quantity, distribution rule and the like of the impurities in the bearing influence the contact characteristics of the bearing, so that the bearing contact characteristics are researched by considering the typical working condition and the influence of the internal heterogeneous impurities of the bearing, the research of the contact characteristics of the bearing is carried out, the research of the influence rule of each attribute of the impurities on the contact characteristics of the heterogeneous bearing has important engineering value, and basic theoretical support can be provided for the analysis and the life prediction of the contact characteristics of the heterogeneous bearing.
The ship ball bearing is a core part of an engine, is also a key ring for guaranteeing stable operation of a power system, is directly related to whether the engine can stably operate with long service life, and the coupling effect of severe heavy load and high temperature working conditions and impurities in the bearing is an important cause for causing bearing damage, and the existence of the impurities has non-negligible influence on the contact characteristics of the bearing.
Disclosure of Invention
Aiming at the defects of the prior art, the invention provides a heterogeneous ball bearing contact characteristic analysis method based on an equivalent impurity clamping method, which solves the problem of difficult detection of the heterogeneous interface contact characteristic of the ship ball bearing.
In order to achieve the above purpose, the invention is realized by the following technical scheme: a heterogeneous ball bearing contact characteristic analysis method based on an equivalent hybrid method comprises the following steps:
s1, modeling contact characteristics of an impurity-containing ball bearing:
establishing an actual contact model of the impurity-containing matrix, and defining the matrix, wherein the elastic modulus is defined as C ijkl Distinguishing impurities, the modulus of elasticity of the material isW represents the contact load on the active matrix, all impurities are transformed into inclusions by the equivalent inclusion method, i.e. the elastic modulus of the original impurity region +.>For modulus of elasticity C with the matrix ijkl Simultaneously introducing a characteristic strain into the equivalent model, and simultaneously adopting a ball contact model calculation by an equivalent impurity clamping method;
s2, ball bearing contact characteristic analysis under different inclusion shapes:
creating an equivalent contact model of the ball bearing containing impurities, wherein the equivalent contact model comprises balls, inner rings, outer rings and inclusions, and analyzing the contact characteristics of the ball bearing containing impurities in different shapes;
S3, ball bearing contact characteristic analysis under the condition of multi-impurity existence:
and determining the interaction radius of the inclusions, and simultaneously analyzing bearing stress and displacement distribution under the condition of existence of a plurality of impurities based on a ball bearing contact model to explore the quantity, distribution rule and distribution density of the impurities for bearing contact characteristics.
Preferably, the step S1 includes the following specific steps:
s1-1, defining a ball contact model based on an equivalent clutter method:
based on a semi-analytical model based on an equivalent clutter clipping method, dispersing an ellipsoid and a semi-infinite contact area into N x ×N y ×N z The lengths of each unit are deltax, deltay and deltaz, the stress and strain fields in each unit can be regarded as uniform distribution and are equal to the value at the center point of the unit, and for the situation that no analyzed random-shaped impurity is distributed at random, a numerical calculation method is used for determining parameters in the model;
s1-2, model verification and exploration of impurity-to-ball contact characteristics:
based on the half-space inclusion solution, the tangential response of the contact point is considered, the problem of partial sliding contact of the non-uniform material is solved, and the influence coefficient of the calculated characteristic stress is reduced from four three-dimensional convolution to three-dimensional convolution.
Preferably, the step S2 includes the following specific steps:
S2-1, establishing an equivalent contact model of the ball bearing containing impurities:
establishing a ball bearing contact model with impurities, carrying out equivalence on ball bearing contact, converting the contact problem of a ball and an inner ring into the contact problem of an ellipsoid and a semi-infinite plane by utilizing a geometric principle, and carrying out equivalence on the contact problem in x j O j z j And y j O j z j On the plane, the radius of the spherical rolling bodies is the same, and a calculation formula of the size of the spherical rolling bodies is obtained;
s2-2, ball bearing contact characteristic analysis of impurities with different shapes:
and (3) depending on the established ball bearing contact model, exploring the influence of the size and shape of single ellipsoidal impurities in the inner ring of the bearing on the contact characteristics of the ball bearing, and analyzing the contact characteristics of the ball bearing containing ellipsoidal impurities and the contact characteristics of the ball bearing containing cubic impurities.
Preferably, the step S3 includes the following specific steps:
s3-1, determining the interaction radius of inclusions:
based on the equivalent impurity inclusion method, the characteristic strain field of one impurity (impurity i) is disturbed by the other impurity (impurity j), the characteristic strain change of the impurity is smaller than a specific value (set as 1%), and the influence radius of the impurity i relative to the impurity j is defined by the distance between the impurity and the impurityWhen the distance between the two inclusions is larger than the influence radius +. >The strain disturbance caused by the inclusion will be greater than 1% for the inclusion, and for the randomly distributed impurities in the bearing or equivalent inclusions, the characteristic strain in an inclusion will be affected by all inclusions within their relative radius of influence, defined as +.>And obtaining formula (37);
s3-2, analyzing contact characteristics of the ball bearing containing multiple impurities:
based on a ball bearing contact model, bearing stress and displacement distribution under the condition of existence of a plurality of impurities are analyzed, and the quantity, distribution rule and distribution density of the impurities are explored for bearing contact characteristics, including comprehensive analysis of dual-impurity ball bearing contact characteristics, three-impurity ball bearing contact characteristics and regular-distribution impurity ball bearing contact characteristics.
Preferably, the step S1-1 comprises the following specific steps:
s1-1-1, construction of inclusion equation sets:
based on the equivalent inclusion principle, a formula (1) is constructed, and each impurity unit satisfies an equivalent equation of the formula (1), wherein the formula (1) is as follows:
the left and right sides of the equation are respectively the total stress field in the impurity and the total stress field of the equivalent inclusion,-the unit is elastically strained in the absence of impurities; epsilon ij -perturbing the strain; />-a characteristic strain; c (C) ijkl ，/>-material coefficients of matrix and impurities; alpha, beta, gamma-grid cell number;
Wherein the method comprises the steps ofC ijkl Andthe subscript ij indicates tensor sign in relation to the modulus of elasticity of the matrix and the impurity material, the equivalent characteristic stress changes the surface pressure distribution, further changing the surface deformation of the semi-infinite plane, whereas the elastic strain is determined by the surface contact pressure and the surface shape, so the elastic strain is the equivalent characteristic strain->The functional relationship between the two is expressed as the coupling relationship between the impurity and the surface contact pressure, and the following formula is given: />
The stress is expressed as:
the elastic stress and the characteristic stress at the impurity element constitute the total stress field at this element, wherein the elastic stress satisfies hooke's law:
the relationship of the characteristic stress to the disturbance strain is as follows:
the formula for actually calculating the characteristic stress is as follows:
mu-shear modulus of the matrix material; v—poisson ratio of the matrix material; z—distance of the cell from the surface;-an influence coefficient, this formula indicating that the characteristic stress of each cell is the sum of the characteristic stresses produced at that cell by all cells containing the characteristic strain in the calculated region, formula (6) can be abbreviated as:
there are two equivalent representations of the characteristic stress at a cell:
the above deformation can obtain the relation between the disturbance strain and the equivalent characteristic strain of the unit:
After the relationship between the elastic strain and the disturbance strain and the equivalent characteristic strain is obtained from the above, the equivalent equation (1) can be rewritten as a system of equations with respect to unknowns:
neglecting the coupling relation of the impurity and the surface contact pressure (i.e. implicit function f in the above i ) The contact pressure distribution of the heterogeneous semi-space surface conforms to the hertz point contact formula, wherein the elastic strain becomes constant, and formula (9) is degenerated into a linear system of equations:
wherein equations (7) and (8) are a system of equations for equivalent characteristic strain.
S1-1-2, solving characteristic stress and characteristic displacement:
obtaining a formula (11), which is a formula for solving the characteristic stress, wherein the influence coefficient means a value of the characteristic stress generated in the target cell when the original cell contains the unit characteristic strain, and if there is the characteristic strain in a plurality of cells in the calculation region, the total characteristic stress in a certain cell is a sum of the characteristic stresses of all cells with non-zero characteristic strain in the calculation region in the target cell, and the influence coefficient between each pair of cells is a sum of the characteristic stresses generated in the functions of the cell size, the inter-cell distance and the mechanical parameters of the matrix material, so that when the divided grid parameters are fixed, the influence coefficient is also unchanged, comprising the following formula for solving:
Omega-space containing characteristic strain; the processing method comprises the steps of (a) the step of (a),-a vector associated with the potential function; x—the spatial position of the target point; x is x 3 -the distance between the target point and the half-space surface; />-a characteristic strain; equations (7) and (8) respectively represent the characteristic stress values of points outside and inside the region, and the influence coefficient U of the source unit at the target unit can be obtained by integrating the right side of the equation i The influence coefficients of the characteristic displacement and the characteristic strain are also used in the process of solving the characteristic displacement, and the specific formula is as follows:
the characteristic displacement within a cell can be expressed as:
s1-1-3, solving the surface contact pressure and the total displacement:
in the impurity bearing contact model, discretized surface contact stress distribution is obtained by solving an equation (11):
Δx, Δy—the side lengths of the cells in the x and y directions, the multiplication is the cell area; alpha, beta-unit numbering; p—surface contact pressure profile; w-the total normal load applied to the press head; h-the total distance between the two contact surfaces; h is a 0 -the distance between the two contact surfaces without any deformation of the surfaces; omega-rigid body displacement of the ram; u (u) 3 -the sum of the surface normal elastic displacement and the characteristic displacement; i c -a collection of cells located within the contact area; i g -computing a set of all surface units within the region; the conjugate gradient method is used in the solving process, and the g of the current two contact surfaces is defined by the following formula [α,β] The gap distribution is:
g [α,β] ＝-u 3[α,β] -h [α,β] [α,β]∈I g (16)
calculating the average value in the contact area, and adjusting:
in N c Is the number of cells located within the contact area;
obtaining a new conjugation direction t [α,β] ：
Wherein the initial value of the auxiliary parameter delta is zero, G old An initial value of 1;
storing the current parameters for the next cycle step, G old ＝G，Continuing to calculate intermediate variables:
the influence coefficient of the surface contact pressure and the elastic displacement is calculated and adjusted after the influence coefficient is obtained:
the step size used to calculate the conjugate direction after the update:
calculating a new contact pressure distribution according to the step size and the direction:
p [α,β] ←p [α,β] -τ·t [α,β] [α,β]∈I c (24)
all points where the contact pressure is less than zero are assigned to zero, the set of non-contact points where two contact surface overlaps occur can be determined, namely:
I ol ＝{[α,β]∈I g :p [α,β] ＝0,g [α,β] <0} (25)
correcting the pressure at the overlapping point:
p [α,β] ←p [α,β] -τ·g [α,β] ,[α,β]∈I ol (26)
the total load on the contact surface is calculated:
the contact pressure distribution is corrected so that the total load satisfies the load balancing condition:
calculating the error between adjacent iterations:
if the error is smaller than the set value, stopping iteration, otherwise, returning to the first step to continue iteration, and for the contact behavior of the heterogeneous material, the surface deformation is mainly caused by the contact pressure and impurities, so that the formula for calculating the total displacement is as follows:
The total deformation is the surface deformation caused by contact pressure and the characteristic displacement caused by impurities, and only the deformation caused by pressure is calculated:
similarly, K is the surface pressure and the coefficient of influence of the pressure causing the surface deformation.
S1-1-4, solving flow:
dividing the contact area into 64X 64 grids, setting the grid size according to the contact area, defining two contact bodies and internal impurity parameters, setting additional load, setting initial gap of the given contact surface, solving the characteristic strain according to the EIM-based Conjugate Gradient Method (CGM) by utilizing Hertz theory and the initial surface morphology of the two contact surfaces, solving the surface characteristic displacement and the displacement caused by pressure according to the characteristic strain, updating the surface morphology information, judging whether the pressure is converged or not, if so, ending the iteration, outputting a result, otherwise, repeating the whole process.
Preferably, the step S2-1 comprises the following specific steps:
at x j O j z j And y j O j z j On the plane, the radius of the spherical rolling bodies is the same, and the calculation formula of the size is as follows:
bearing inner race is different from rolling element and is x j And y j Radius of curvature in direction is different, at x j O j z j Plane and y j O j z j The radius of the plane needs to be calculated respectively, and the formula is as follows:
Solving the equivalent radius of the equivalent ellipsoid: at x j In the direction, the equivalent radius of the convex contact of the rolling body and the inner ring isIn y j In the direction, the equivalent radius of the contact between the rolling body and the inner ring is +.>The calculation formula is as follows:
wherein D is W Is spherical diameter, f i Is the curvature coefficient alpha j For initial contact angle, D m Is the pitch diameter.
The step S2-2 comprises the following specific steps:
s2-2-1, ball bearing contact characteristic analysis of the ellipsoidal impurity:
the influence of the size and the shape of a single ellipsoidal impurity in the inner ring of the bearing on the contact characteristics of the ball bearing is explored by depending on the established ball bearing contact model, and the ellipsoidal impurity is introduced into a semi-infinite plane;
s2-2-2, ball bearing contact characteristic analysis of the ball bearing containing the cubic impurities:
according to the established equivalent model, taking cubic impurities as an example, researching the influence of the volume and the central position of the impurities on the contact characteristics of the bearing, wherein parameters such as the elastic modulus, the poisson ratio and the load of the impurities are unchanged;
s2-2-3, analysis of contact characteristics of the ball bearing containing the cylindrical impurities:
the method starts from the distribution of internal stress and surface pressure of the bearing with the cylindrical impurities, and explores the rule of influence of the geometric parameters of the cylindrical impurities on the contact characteristics of the bearing.
Preferably, the step S3-1 further comprises the steps of:
The influence radius of the defined inclusions i isThe following formula may be established: />
AndWherein the characteristic strain obtained after considering the influence of all inclusions on inclusion i is +.>Considering only the characteristic strain of i obtained after the influence of impurities in the influence radius on the impurity i, if n impurities are distributed in the bearing in total, the equivalent influence radius of the distributed impurities can be defined as: />
The step S3-2 further comprises the following steps:
s3-2-1, analysis of contact characteristics of ball bearings containing double impurities:
taking a double-inclusion ball bearing as an example, introducing two cubic impurities with the same volume, wherein the elastic modulus is 420GPa, the elastic modulus of a matrix is 210GPa, and researching the influence of the impurities on the contact characteristic of the ball bearing by changing the interval of the impurities;
s3-2-2, and analysis of contact characteristics of ball bearings containing three impurities:
developing a double-impurity model into a three-impurity model, wherein other bearings and impurity parameters are unchanged, and exploring the influence rule of the three-impurity spacing on the stress, displacement and characteristic displacement of the ball bearing based on the three-impurity ball bearing contact model;
s3-2-3, wherein the contact characteristics of the ball bearing containing the regularly distributed impurities are as follows:
the distribution diagram of the subsurface layers of the ball bearings with different densities and evenly distributed impurities is provided, the impurities are small cubes, the elastic modulus of the impurities is 420GPa, and the additional load and the bearing parameters are unchanged;
S3-2-4, ball bearing contact characteristic analysis of randomly distributed impurities:
starting from subsurface stress, characteristic displacement, surface pressure and the like during bearing contact, a ball bearing belt random distribution bearing contact model is developed, bearing material parameters and impurity elastic modulus 420GPa are kept unchanged, and the influence rule of the distribution density of random distribution impurities in the bearing on the ball bearing contact characteristics is studied.
The invention provides a heterogeneous ball bearing contact characteristic analysis method based on an equivalent impurity clamping method.
The beneficial effects are as follows:
1. according to the invention, through verifying the correctness of the model, a conclusion is further drawn that the presence of the hard impurity can not only increase the maximum value of the surface contact pressure, but also generally increase the surface contact pressure of the impurity-existing region, the larger the characteristic stress is, the larger the characteristic displacement is, the influence of ellipsoidal impurities, cubic impurities and cylindrical impurities on the contact characteristics of the ball bearing is respectively explored, and the result shows that the influence of the ellipsoidal impurities on the contact characteristics is larger compared with the flat ellipsoidal impurities, the larger the cubic impurities are, the larger the contact surface stress is, the deeper the depth of the impurities is, namely the farther the impurities are away from the contact surface, the smaller the influence of the impurities on the bearing contact characteristics is, and the influence of the height of the cylindrical impurities on the contact characteristics is larger than the influence of the radius.
2. According to the invention, the influence of various distribution characteristics of double impurities, three impurities, regularly distributed multiple impurities and randomly distributed impurities on the contact characteristic of the ball bearing is explored, and the result shows that the influence of the double impurities on each other is reduced along with the increase of the impurity spacing, when the double impurity spacing is large enough, the maximum stress of the contact surface is basically not influenced, the characteristic stress and the characteristic displacement are reduced along with the decrease of the impurity spacing, the degree of increase is large, under the condition that the three impurities exist, the maximum stress of the double impurities is relieved by the impurities at the middle position, the smaller the three impurity spacing is, the larger the characteristic displacement generated on the ball bearing is, the distribution density of the multiple distributed impurities is larger, the characteristic stress generated by the impurities is larger, and the maximum stress of the contact surface of the bearing is smaller when the total volume of the impurities is smaller than the volume of the matrix.
Drawings
FIG. 1 is a schematic diagram of the equivalent doping method conversion of the present invention;
FIG. 2 is a schematic diagram of solving for surface contact pressure in accordance with the present invention;
FIG. 3 is a schematic diagram of a solution flow according to the present invention;
FIG. 4 is a schematic diagram of a verification model of the present invention;
FIG. 5 is a stress contrast diagram of the present invention;
FIG. 6 is a comparative diagram of stress and displacement distribution according to the present invention;
FIG. 7 is a schematic diagram of an equivalent model of the contact between the rolling bodies and the inner ring of the ball bearing;
FIG. 8 is a schematic diagram of the contact transformation of the ball bearing with impurities according to the present invention;
FIG. 9 is a schematic view of a contact model of an ellipsoidal impurity ball bearing of the present invention;
FIG. 10 is a schematic diagram showing the effect of the shape of the oblong impurity on the xyz-plane stress of the ball bearing according to the present invention;
FIG. 11 is a schematic illustration of the impact of the impurity size of the oblate ellipsoid on bearing characteristic stress and characteristic displacement according to the present invention;
FIG. 12 is a schematic view of a contact model of a cubic impurity bearing of the present invention;
FIG. 13 is a schematic representation of the effect of depth of cubic impurities on surface pressure in accordance with the present invention;
FIG. 14 is a schematic view of a contact model of a impurity bearing with a cylindrical shape according to the present invention;
FIG. 15 is a schematic view of the effect of cylindrical impurity radius on ball bearing contact stress in accordance with the present invention;
FIG. 16 is a schematic view of a contact model of a ball bearing containing dual impurities according to the present invention;
FIG. 17 is a schematic diagram showing the effect of bearing dual impurity spacing on characteristic stress and characteristic displacement in the x-direction according to the present invention;
FIG. 18 is a schematic view of a contact model of a ball bearing with three impurities according to the present invention;
FIG. 19 is a graph showing the effect of impurity spacing on the surface pressure and characteristic displacement of a ball bearing containing three impurities in accordance with the present invention;
FIG. 20 is a graph showing the effect of impurity distribution density on the stress of a ball bearing containing regularly distributed impurities according to the present invention.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
Examples:
the embodiment of the invention provides a heterogeneous ball bearing contact characteristic analysis method based on an equivalent impurity clamping method, which comprises the following steps of:
s1, modeling contact characteristics of an impurity-containing ball bearing:
the basic idea of the equivalent inclusion theory is that impurities are equivalently replaced by inclusions, namely an impurity solution can be obtained by adding a homogeneous material solution and a corresponding inclusion solution, the essence of the theory is that an equivalent relation between disturbance solution containing the impurities and the inclusion solution is established, the influence caused by the inclusions is equal to the influence caused by the impurities by selecting proper characteristic strain distribution, and a material containing the impurities in a matrix D-omega is called as heterogeneous material; the impurity refers to subregion omega which is different from the matrix property (elastic modulus) in the material, the impurity also refers to a subregion in the material, the material property is the same as that of the matrix, but the characteristic stress caused by the characteristic strain is generated in the material, the characteristic strain is the total name of inelastic strain including dislocation strain, thermal expansion, plastic strain load and the like, the equivalent impurity clamping method is as shown in the equivalent figure 1, the left part of the figure 1 is an actual contact model of the matrix containing the impurity, wherein the gray region is the matrix, the elastic modulus is C ijkl The red region represents impurities and has an elastic modulus ofW represents the contact load on the active matrix, the right part of FIG. 1 is the equivalent model after transformation, all impurities are transformed into inclusions by the equivalent inclusion method, i.e. the elastic modulus of the original impurity regionFor modulus of elasticity C with the matrix ijkl Simultaneously introducing a characteristic strain into the equivalent model, and simultaneously adopting a ball contact model calculation of an equivalent impurity clamping method, wherein the method comprises the following specific steps of:
s1-1, defining a ball contact model based on an equivalent clutter method:
based on a semi-analytical model based on an equivalent clutter clipping method, dispersing an ellipsoid and a semi-infinite contact area into N x ×N y ×N z The length of each cell is deltax, deltay, deltaz, the stress and strain field inside each cell can be regarded as uniform distribution and equal to the value at the center point of the cell, according to the semi-analytic meaning that the stress-strain relationship between each pair of cells in the model is determined by an analytic formula, and for the case of any shape of any distribution without analytic, the parameters in the model are determined by using a numerical calculation method, comprising the following specific steps:
s1-1-1, construction of inclusion equation sets:
based on the equivalent inclusion principle, a formula (1) is constructed, and each impurity unit satisfies an equivalent equation of the formula (1), wherein the formula (1) is as follows:
The left and right sides of the equation are respectively the total stress field in the impurity and the total stress field of the equivalent inclusion,-the unit is elastically strained in the absence of impurities; epsilon ij -perturbing the strain; />-a characteristic strain; c (C) ijkl ，/>-material coefficients of matrix and impurities; alpha, beta, gamma-grid cell number;
wherein C is ijkl Andthe subscript ij indicates the tension in relation to the modulus of elasticity of the matrix and the impurity materialThe sign of the measurement indicates that the equivalent characteristic stress changes the surface pressure distribution, further changes the surface deformation of the semi-infinite plane, and the elastic strain is determined by the surface contact pressure and the surface shape, so the elastic strain +.>Is equivalent characteristic strain->Is expressed as f 1 I.e. the coupling relation of the impurity and the surface contact pressure, the following formula is given:
the stress is expressed as:
the elastic stress and the characteristic stress at the impurity element constitute the total stress field at this element, wherein the elastic stress satisfies hooke's law:
the relationship of the characteristic stress to the disturbance strain is as follows:
the formula for actually calculating the characteristic stress is as follows:
mu-shear modulus of the matrix material; v—poisson ratio of the matrix material; z- (z)-distance of the unit from the surface;-an influence coefficient, this formula indicating that the characteristic stress of each cell is the sum of the characteristic stresses produced at that cell by all cells containing the characteristic strain in the calculated region, formula (6) can be abbreviated as:
There are two equivalent representations of the characteristic stress at a cell:
the above deformation can obtain the relation between the disturbance strain and the equivalent characteristic strain of the unit:
after the relationship between the elastic strain and the disturbance strain and the equivalent characteristic strain is obtained from the above, the equivalent equation (1) can be rewritten as a system of equations with respect to unknowns:
neglecting the coupling relation of the impurity and the surface contact pressure (i.e. implicit function f in the above i ) The contact pressure distribution of the heterogeneous semi-space surface conforms to the hertz point contact formula, wherein the elastic strain becomes constant, and formula (9) is degenerated into a linear system of equations:
wherein equations (7) and (8) are a system of equations for equivalent characteristic strain.
S1-1-2, solving characteristic stress and characteristic displacement:
obtaining a formula (11), which is a formula for solving the characteristic stress, wherein the influence coefficient means a value of the characteristic stress generated in the target cell when the original cell contains the unit characteristic strain, and if there is the characteristic strain in a plurality of cells in the calculation region, the total characteristic stress in a certain cell is a sum of the characteristic stresses of all cells with non-zero characteristic strain in the calculation region in the target cell, and the influence coefficient between each pair of cells is a sum of the characteristic stresses generated in the functions of the cell size, the inter-cell distance and the mechanical parameters of the matrix material, so that when the divided grid parameters are fixed, the influence coefficient is also unchanged, comprising the following formula for solving:
Omega-space containing characteristic strain; the processing method comprises the steps of (a) the step of (a),-a vector associated with the potential function; x—the spatial position of the target point; x is x 3 -the distance between the target point and the half-space surface; />-a characteristic strain; equations (7) and (8) respectively represent the characteristic stress values of points outside and inside the region, and the influence coefficient U of the source unit at the target unit can be obtained by integrating the right side of the equation i The influence coefficients of the characteristic displacement and the characteristic strain are also used in the process of solving the characteristic displacement, and the specific formula is as follows:
the characteristic displacement within a cell can be expressed as:
s1-1-3, solving the surface contact pressure and the total displacement:
in the impurity bearing contact model, discretized surface contact stress distribution is obtained by solving an equation (11):
Δx, Δy—the side lengths of the cells in the x and y directions, the multiplication is the cell area; alpha, beta-unit numbering; p—surface contact pressure profile; w-the total normal load applied to the press head; h-the total distance between the two contact surfaces; h is a 0 -the distance between the two contact surfaces without any deformation of the surfaces; omega-rigid body displacement of the ram; u (u) 3 -the sum of the surface normal elastic displacement and the characteristic displacement; i c -a collection of cells located within the contact area; i g -computing a set of all surface units within the region; the conjugate gradient method is used in the solving process, and the g of the current two contact surfaces is defined by the following formula [α,β] The gap distribution is:
g [α,β] ＝-u 3[α,β] -h [α,β] [α,β]∈I g (16)
calculating the average value in the contact area, and adjusting:
in N c Is the number of cells located within the contact area;
obtaining a new conjugation direction:
wherein the initial value of the auxiliary parameter is zero, G old An initial value of 1;
storing the current parameters for the next cycle step, G old ＝G，Continuing to calculate intermediate variables:
wherein the influence coefficient of the surface contact pressure and the elastic displacement is obtained by [α,β] After that, the average value calculated and adjusted:
the step size used to calculate the conjugate direction after the update:
calculating a new contact pressure distribution according to the step size and the direction:
p [α,β] ←p [α,β] -τ·t [α,β] [α,β]∈I c (24)
all points where the contact pressure is less than zero are assigned to zero, the set of non-contact points where two contact surface overlaps occur can be determined, namely:
I ol ＝{[α,β]∈I g :p [α,β] ＝0,g [α,β] <0} (25)
correcting the pressure at the overlapping point:
p [α,β] ←p [α,β] -τ·g [α,β] ,[α,β]∈I ol (26)
the total load on the contact surface is calculated:
the contact pressure distribution is corrected so that the total load satisfies the load balancing condition:
calculating the error between adjacent iterations:
if the error is smaller than the set value, stopping iteration, otherwise, returning to the first step to continue iteration, and for the contact behavior of the heterogeneous material, the surface deformation is mainly caused by the contact pressure and impurities, so that the formula for calculating the total displacement is as follows:
The total deformation is the surface deformation caused by contact pressure and the characteristic displacement caused by impurities, and only the deformation caused by pressure is calculated:
similarly, K is the surface pressure and the coefficient of influence of the pressure causing the surface deformation.
S1-1-4, solving flow:
fig. 3 is an algorithm flow chart of a semi-analytical model, which is more efficient and more versatile than conventional finite element and analytical methods, divides the contact area into 64 x 64 grids, sets the grid size according to the contact area, defines two contacts and internal impurity parameters, sets additional loads, sets the initial gap of the contact surface, solving the characteristic strain according to an EIM-based Conjugate Gradient Method (CGM) by utilizing Hertz theory and the initial surface morphology of the two contact surfaces, updating surface morphology information after solving surface characteristic displacement and displacement caused by pressure according to the characteristic strain, judging whether the pressure is converged, if so, ending iteration, outputting a result, otherwise, repeating the whole process;
s1-2, model verification and exploration of impurity-to-ball contact characteristics:
based on half-space inclusion solution, the tangential response of contact points is considered, the partial sliding contact problem of non-uniform materials is solved, the influence coefficient of characteristic stress is reduced from four three-dimensional convolution to three-dimensional convolution, a cube impurity is introduced into a matrix contacted with a rigid body ball and added with load as shown in figure 4, the obtained result is compared with a reference, specific input parameters are consistent with the reference, for example, as shown in figure 5, left graphs (a), (b) and (c) are references, respectively represent the force caused by pressure alone, the stress caused by inclusion alone and the stress caused by the interaction of the two, right graphs (d), (e) and (f) are stress distribution obtained by a program used in the text and correspond to the left reference content one by one, the maximum Hertz stress corresponding to contact of homogeneous materials is 7345.67MPa, the Hertz contact radius is 1mm, the pressure and displacement in the analysis result are dimensionless through the maximum Hertz stress and the contact radius, it can be seen that the obtained result of the research method is highly consistent with the obtained result of the reference document, the correctness and feasibility of the program are directly demonstrated, FIG. 5 shows that the presence of inclusions has obvious influence on the stress distribution in the contact body, especially at the interface of inclusions and matrix, obvious discontinuous jump occurs due to the different properties of the inclusions and matrix materials, and the obtained distribution of stress and two stress components in z direction and displacement distribution of the surface in x direction are given below for further proving the feasibility of the program, in FIG. 6 (a), (b) The stress along the z direction and the displacement distribution along the x direction in the references respectively, (c) and (d) are the results obtained in the procedure, and it can be seen that the values and trends of the curves are highly consistent with the references, further demonstrating the feasibility of the procedure, in order to explore the effect of impurities on the contact characteristics of the spherical contact.
S2, ball bearing contact characteristic analysis under different inclusion shapes:
creating an equivalent contact model of the ball bearing containing impurities, wherein the equivalent contact model comprises balls, inner rings, outer rings and inclusions, and analyzing the contact characteristics of the ball bearing containing impurities with different shapes, and the equivalent contact model comprises the following specific steps: the method comprises the following specific steps:
s2-1, establishing an equivalent contact model of the ball bearing containing impurities: establishing a ball bearing contact model with impurities, carrying out equivalence on ball bearing contact, converting the contact problem of a ball and an inner ring into the contact problem of an ellipsoid and a semi-infinite plane by utilizing a geometric principle, and carrying out equivalence on the contact point in x as shown in figure 7 j O j z j And y j O j z j On the plane, the radius of the spherical rolling bodies is the same, and a calculation formula of the size of the spherical rolling bodies is obtained, and the method comprises the following specific steps:
at x j O j z j And y j O j z j On the plane, the radius of the spherical rolling bodies is the same, and the calculation formula of the size is as follows:
bearing inner race is different from rolling element and is x j And y j Radius of curvature in direction is different, at x j O j z j Plane and y j O j z j The radius of the plane needs to be calculated respectively, and the formula is as follows:
solving the equivalent radius of the equivalent ellipsoid: at x j In the direction, the equivalent radius of the convex contact of the rolling body and the inner ring isIn the direction, the equivalent radius of the contact of the rolling bodies with the inner ring is +. >The calculation formula is as follows:
wherein D is W Is spherical diameter, f i Is the curvature coefficient alpha j For initial contact angle, D m For pitch diameter, according to the principle of equivalent inclusion, fig. 8 shows that the ball bearing ball is contacted with the inner ring with impurities and converted into ellipsoid and contacted with the semi-infinite large plane with impurities, based on the above-mentioned geometrical equivalent principle and formula, the radius of the equivalent ellipsoid of the ball bearing in x direction is 20.4mm, the radius in y direction is 848.6mm, according to the hertz theory, the contact area between the ellipsoid and the semi-infinite plane is an ellipse, the radius of the ellipse is calculated to be 0.43mm and 4.64mm, the dividing grid number of the calculated area is 64 multiplied by 64, and in order to make the obtained result more accurate, the dividing grid area covers the contact ellipse, so the size of each grid is set to 63.5 mu m multiplied by 150 mu m multiplied by 63.5 mu m, and the size of the dividing grid area is 4mm multiplied by 9.6mm multiplied by 4mm;
s2-2, ball bearing contact characteristic analysis of impurities with different shapes:
depending on the established ball bearing contact model, the influence of the size and the shape of single ellipsoidal impurities in the inner ring of the bearing on the contact characteristics of the ball bearing is explored, and the contact characteristics of the ball bearing containing ellipsoidal impurities and the contact characteristics of the ball bearing containing cubic impurities are analyzed, and the method comprises the following specific steps:
S2-2-1, ball bearing contact characteristic analysis of the ellipsoidal impurity:
depending on the established ball bearing contact model, exploring the influence of the size and shape of single ellipsoidal impurities in the inner ring of the bearing on the contact characteristic of the ball bearing, introducing ellipsoidal impurities in a semi-infinite plane, as shown in fig. 9, according to the shape characteristics of the ellipsoidal impurities, the graph (a) is called a prolate ellipsoid, the graph (c) is called a flat ellipsoid, changing the radius of the ellipsoidal shape in the x and z directions, researching the influence of the ellipsoidal shape on the stress distribution of a subsurface layer, and in order to make the exploration result more obvious, setting the length of a short half shaft of the flat ellipsoid and the length of the prolate ellipsoid to be 0.5mm, changing the size of the long half shaft on the basis, and obtaining the total stress distribution of the subsurface layer of the prolate ellipsoid as shown in fig. 10;
as shown in fig. 10, the radius of the oblong ellipsoid in the z direction is increased from 0.5mm to 1.25mm, the maximum stress in the bearing is increased from 1630MPa to 1890MPa, the length of the ellipsoidal impurity is increased, the stress disturbance of the boundary of the impurity is aggravated, especially when approaching to the contact surface, the stress concentration is obvious, the degree of the stress increase is increased along with the increase of the size, the reason is that the volume of the impurity is increased, the contact characteristic of the bearing is directly affected, and on the other hand, the distance between the boundary of the impurity and the contact surface of the bearing is reduced, the stress is aggravated, and the influence of the ellipsoidal impurity on the bearing is better studied; FIG. 11 shows the characteristic stress and characteristic displacement of the flat ellipsoidal impurities with different sizes in the x direction, and it can be seen that the characteristic stress and characteristic displacement of the impurities on the ball bearing increase with the increase of the impurity size, and the increase degree is in a weakening trend, which further verifies the conclusion; the result shows that the long ellipsoid impurities have larger influence on the contact characteristics of the bearing, the generated stress concentration phenomenon is more serious, compared with the volume of the impurities, the influence of the distance between the impurities and the contact surface on the contact characteristics of the bearing is more remarkable, and more emphasis is placed on avoiding the generation of the long ellipsoid impurities in engineering;
S2-2-2, ball bearing contact characteristic analysis of the ball bearing containing the cubic impurities:
taking cubic impurities as an example, researching the influence of the volume and the central position of the impurities on the contact characteristics of the bearing according to the established equivalent model, wherein parameters such as the elastic modulus, the poisson ratio and the load of the impurities are unchanged, and the model is shown in figure 12; FIG. 13 shows the surface pressure distribution at different depths of the impurity, and it can be seen that the depth where the cubic impurity is located has a great influence on the surface pressure, the closer the impurity is to the contact surface, the greater the degree of pressure change, and when the impurity surface is coincident with the contact surface, the maximum pressure reaches 4000N, which is extremely disadvantageous for the bearing;
s2-2-3, analysis of contact characteristics of the ball bearing containing the cylindrical impurities:
starting from the distribution of internal stress and surface pressure of the bearing with the cylindrical impurities, exploring the influence rule of the geometric parameters of the cylindrical impurities on the contact characteristics of the bearing, wherein the central coordinates of the cylindrical impurities are (0, 2 mm), the material parameters and grid information of the bearing are kept unchanged, and a cylindrical impurity model is shown in figure 20;
FIG. 15 shows the distribution of stress in the bearing in the presence of cylindrical impurities with different radiuses, the height of the cylindrical impurities is kept to be 1.5mm, the radius of the cylindrical impurities is changed from 0.5mm to 1.25mm, and the stress distribution is obtained as follows, and as the radius of the cylindrical impurities is increased, the stress distribution does not show similar linear change as before, when the radius of the cylindrical impurities is changed from 0.5mm to 1.25mm, the volume of the impurities is increased, the stress distribution is slightly reduced, the maximum stress is changed from 1875MPa to 1865MPa, at the moment, the influence of the shape of the impurities on the stress is stronger than the influence of the volume on the stress, and the influence of the shape of the impurities on the contact characteristics of the bearing is proved to be more non-negligible;
S3, ball bearing contact characteristic analysis under the condition of multi-impurity existence:
determining the interaction radius of inclusions, simultaneously analyzing bearing stress and displacement distribution under the condition of a plurality of impurities based on a ball bearing contact model, and exploring the quantity, distribution rule and distribution density of the impurities to bearing contact characteristics, wherein the method comprises the following specific steps:
s3-1, determining the interaction radius of inclusions:
based on the equivalent impurity inclusion method, the characteristic strain field of one impurity (impurity i) is disturbed by the other impurity (impurity j), the characteristic strain change of the impurity is smaller than a specific value (set as 1%), and the influence radius of the impurity i relative to the impurity j is defined by the distance between the impurity and the impurityWhen the distance between the two inclusions is larger than the influence radius +.>The strain disturbance caused by the inclusion will be greater than 1% for the inclusion, and for the randomly distributed impurities in the bearing or equivalent inclusions, the characteristic strain in an inclusion will be affected by all inclusions within their relative radius of influence, defined as +.>And obtaining equation (37), further comprising the steps of:
defining the radius of influence of inclusions i asThe following formula may be established: / >
AndWherein the characteristic strain obtained after considering the influence of all inclusions on inclusion i is +.>The characteristic strain obtained by considering the influence of impurities in the influence radius on impurities is considered, if n impurities are distributed in the bearing in total, the equivalent influence radius of the distributed impurities can be determinedThe meaning is as follows: />
S3-2, analyzing contact characteristics of the ball bearing containing multiple impurities:
based on a ball bearing contact model, bearing stress and displacement distribution under the condition of a plurality of impurities are analyzed, the quantity, distribution rule and distribution density of the impurities are explored, and the comprehensive analysis of the double-impurity ball bearing contact characteristics, the three-impurity ball bearing contact characteristics and the regular-distribution impurity ball bearing contact characteristics is included, and the method further comprises the following steps:
s3-2-1, analysis of contact characteristics of ball bearings containing double impurities:
taking a double-inclusion ball bearing as an example in the section, introducing two cubic impurities with the same volume as a model shown in fig. 16, and researching the influence of the two cubic impurities on the contact characteristics of the ball bearing by changing the interval of the impurities; FIG. 17 further shows the law of influence of the double-impurity spacing on the characteristic stress and the characteristic displacement of the ball bearing, and the result shows that when the impurity spacing is 1.0mm, the characteristic stress and the characteristic displacement are obviously larger than those of other groups, when the impurity spacing is 1.5mm, the maximum characteristic stress generated by the impurities is half of that generated by 1.0mm, when the impurity spacing is gradually increased, the characteristic stress and the characteristic displacement caused by the impurities on the bearing are reduced and the degree of reduction is reduced, so that two convex parts of the region where the impurities are located can be seen, and the characteristic stress and the characteristic displacement of the position where the impurities are located are larger;
S3-2-2, and analysis of contact characteristics of ball bearings containing three impurities:
developing a double-impurity model to a three-impurity model as shown in fig. 18, keeping the volume of three impurities, keeping the additional load at 15000N, keeping other bearings and impurity parameters unchanged, and exploring the influence rule of the spacing of three impurities on the stress, displacement and characteristic displacement of the ball bearing based on the three-impurity ball bearing contact model; introducing new impurities into the centers of the double impurities within a certain range of intervals can reduce the maximum stress, and when the interval between the double impurities is large enough, the mutual influence among the impurities becomes small until the mutual influence is similar to the existence condition of single impurities; fig. 19 (a) shows the surface pressure in the y direction when the three impurity pitches are in contact with the bearing, three protruding parts are obviously seen, when the impurity pitch is 0.5mm, the protruding parts are relatively concentrated, the influence of the impurity pitch on the surface pressure in the central position is not great, as the pitch is increased, the protruding degree of the surface pressure at the position where the impurity is located is smaller, and fig. (b) shows the characteristic displacement distribution of the impurity at different pitches in the x direction, and it can be seen that the characteristic displacement is obviously increased only when the impurity pitch distance is small, namely 0.5mm, and the characteristic displacement change caused by the impurity in the x direction is not great when the impurity pitch is increased;
S3-2-3, wherein the contact characteristics of the ball bearing containing the regularly distributed impurities are as follows: FIG. 20 shows the distribution diagram of the subsurface layers of a ball bearing with uniformly distributed impurities, the impurities are small cubes, the elastic modulus is 420GPa, the additional load and the bearing parameters are unchanged, and the maximum stress is increased when the impurity density is increased from 5X 5 to 11X 11, so that the influence of the impurity distribution density on the bearing is more intuitively observed;
s3-2-4, ball bearing contact characteristic analysis of randomly distributed impurities: in a real bearing material, the internal impurities are often large in quantity and random in distribution, in order to explore the contact characteristics of a ball bearing containing the random distributed impurities, the method starts from subsurface stress, characteristic displacement, surface pressure and the like during bearing contact, develops a ball bearing belt random distributed bearing contact model, keeps the parameters of the bearing material, the external load 15000N and the elastic modulus of the impurities 420GPa unchanged, and researches the influence rule of the distribution density of the random distributed impurities in the bearing on the contact characteristics of the ball bearing.
Although embodiments of the present invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made therein without departing from the principles and spirit of the invention, the scope of which is defined in the appended claims and their equivalents.
Claims (8)
1. The heterogeneous ball bearing contact characteristic analysis method based on the equivalent impurity clamping method is characterized by comprising the following steps of:
s1, modeling contact characteristics of an impurity-containing ball bearing:
establishing an actual contact model of the impurity-containing matrix, and defining the matrix, wherein the elastic modulus is defined as C ijkl Distinguishing impurities, the modulus of elasticity of the material isW represents the contact load on the active matrix, all impurities are transformed into inclusions by the equivalent inclusion method, i.e. the modulus of elasticity of the original impurity region +.>Becomes the elastic modulus C with the matrix ijkl Simultaneously introducing a characteristic strain into the equivalent model, and simultaneously adopting a ball contact model calculation by an equivalent impurity clamping method;
s2, ball bearing contact characteristic analysis under different inclusion shapes:
creating an equivalent contact model of the ball bearing containing impurities, wherein the equivalent contact model comprises balls, inner rings, outer rings and inclusions, and analyzing the contact characteristics of the ball bearing containing impurities in different shapes;
s3, ball bearing contact characteristic analysis under the condition of multi-impurity existence:
determining the interaction radius of inclusions, and simultaneously analyzing bearing stress and displacement distribution under the condition of the existence of a plurality of impurities based on a ball bearing contact model, and exploring the quantity, distribution rule and distribution density of the impurities for bearing contact characteristics;
S4, analysis of contact characteristics of the coated ball bearing based on an equivalent hybrid method:
based on the contact model of the ball bearing containing the impurities, the contact model is expanded to a contact model with a surface coating, the obtained distribution of stress, characteristic displacement and total displacement is analyzed, the influence of the type, thickness and distribution of the impurity coating on the contact characteristic of the ball bearing is confirmed, and the influence rule of the external load and friction on the ball bearing containing the surface coating impurity is confirmed.
2. The method for analyzing the contact characteristics of the heterogeneous ball bearing based on the equivalent hybrid method according to claim 1, wherein the step S1 comprises the following specific steps:
s1-1, defining a ball contact model based on an equivalent clutter method:
based on a semi-analytical model based on an equivalent clutter clipping method, dispersing an ellipsoid and a semi-infinite contact area into N x ×N y ×N z The lengths of each unit are deltax, deltay and deltaz, the stress and strain fields in each unit can be regarded as uniform distribution and are equal to the value at the center point of the unit, and for the situation that no analyzed random-shaped impurity is distributed at random, a numerical calculation method is used for determining parameters in the model;
s1-2, model verification and exploration of impurity-to-ball contact characteristics:
Based on the half-space inclusion solution, the tangential response of the contact point is considered, the problem of partial sliding contact of the non-uniform material is solved, and the influence coefficient of the calculated characteristic stress is reduced from four three-dimensional convolution to three-dimensional convolution.
3. The method for analyzing the contact characteristics of the heterogeneous ball bearing based on the equivalent hybrid method according to claim 1, wherein the step S2 comprises the following specific steps:
s2-1, establishing an equivalent contact model of the ball bearing containing impurities:
establishing a ball bearing contact model with impurities, carrying out equivalence on ball bearing contact, converting the contact problem of a ball and an inner ring into the contact problem of an ellipsoid and a semi-infinite plane by utilizing a geometric principle, and carrying out equivalence on the contact problem in x j O j z j And y j O j z j On the plane, the radius of the spherical rolling bodies is the same, and a calculation formula of the size of the spherical rolling bodies is obtained;
s2-2, ball bearing contact characteristic analysis of impurities with different shapes:
and (3) depending on the established ball bearing contact model, exploring the influence of the size and shape of single ellipsoidal impurities in the inner ring of the bearing on the contact characteristics of the ball bearing, and analyzing the contact characteristics of the ball bearing containing ellipsoidal impurities and the contact characteristics of the ball bearing containing cubic impurities.
4. The method for analyzing the contact characteristics of the heterogeneous ball bearing based on the equivalent hybrid method according to claim 1, wherein the step S3 comprises the following specific steps:
S3-1, determining the interaction radius of inclusions:
based on the equivalent impurity inclusion method, the characteristic strain field of one impurity (impurity i) is disturbed by the other impurity (impurity j), the characteristic strain change of the impurity i is smaller than a specific value (set as 1%), and the influence radius of the impurity i relative to the impurity j is defined by the distance between the impurity and the impurityWhen the distance between two inclusions i and j is greater than the influencing radius +.>When the disturbance of the strain caused by the inclusion j to the inclusion i will be greater than 1%, the characteristic strain in an inclusion will be affected by all inclusions within their relative radius of influence, defined as +.>And obtaining formula (37);
s3-2, analyzing contact characteristics of the ball bearing containing multiple impurities:
based on a ball bearing contact model, bearing stress and displacement distribution under the condition of existence of a plurality of impurities are analyzed, and the quantity, distribution rule and distribution density of the impurities are explored for bearing contact characteristics, including comprehensive analysis of dual-impurity ball bearing contact characteristics, three-impurity ball bearing contact characteristics and regular-distribution impurity ball bearing contact characteristics.
5. The method for analyzing the contact characteristics of the heterogeneous ball bearing based on the equivalent hybrid method according to claim 1, wherein the step S4 comprises the following specific steps:
S4-1, analyzing the contact characteristics of the ball bearing by the elastic modulus of the coating:
analyzing the contact characteristics of the ball bearing through the elastic modulus of the coating, and simultaneously analyzing the contact characteristics of the ball bearing according to the thickness of the coating;
s4-2, analysis of contact characteristics of the double-layer coating on the ball bearing:
changing the position and the elastic modulus of the inner layer impurity, observing the change of the internal stress and the surface displacement of the bearing, maintaining the bearing parameters unchanged by a ball bearing contact model with double impurity coatings, defining the elastic modulus of a matrix, an additional load, the thickness of an outer layer coating, the elastic modulus, the inner layer soft impurity coating and the thickness, and changing the central position of the inner layer impurity coating to obtain the subsurface stress of the bearing;
s4-3, considering the influence of load and friction on the contact characteristics of the ball bearing with the impurity coating:
and the contact characteristics of the bearing under different normal loads are researched through analyzing the obtained characteristic displacement and total stress, and meanwhile, the contact characteristics of the coated ball bearing are analyzed aiming at friction.
6. The method for analyzing the contact characteristics of the heterogeneous ball bearing based on the equivalent hybrid method according to claim 2, wherein the step S1-1 comprises the following specific steps:
s1-1-1, construction of inclusion equation sets:
Based on the equivalent inclusion principle, a formula (1) is constructed, each impurity unit satisfies the formula (1), and an equivalent equation is obtained, wherein the formula (1) is as follows:
the left and right sides of the equation are respectively the total stress field in the impurity and the total stress field of the equivalent inclusion,-the unit is elastically strained in the absence of impurities; epsilon ij -perturbing the strain; />-a characteristic strain; c (C) ijkl ，/>-material coefficients of matrix and impurities; alpha, beta, gamma-grid cell number;
wherein C is ijkl Andthe subscript ij indicates tensor sign in relation to the modulus of elasticity of the matrix and the impurity material, the equivalent characteristic stress changes the surface pressure distribution, further changes the surface deformation of the semi-infinite plane, and the elastic strain is determined by the surface contact pressure and the surface shape, so the elastic strain +.>Is equivalent characteristic strain->Is expressed as f 1 I.e. the coupling relation of the impurity and the surface contact pressure, the following formula is given:
the stress is expressed as:
the elastic stress and the characteristic stress at the impurity element constitute the total stress field at this element, wherein the elastic stress satisfies hooke's law:
the relationship of the characteristic stress to the disturbance strain is as follows:
the formula for actually calculating the characteristic stress is as follows:
mu-shear modulus of the matrix material; v—poisson ratio of the matrix material; z—distance of the cell from the surface; -an influence coefficient, this formula indicating that the characteristic stress of each cell is the sum of the characteristic stresses produced at that cell by all cells containing the characteristic strain in the calculated region, formula (6) can be abbreviated as:
there are two equivalent representations of the characteristic stress at a cell:
the above deformation can obtain the relation between the disturbance strain and the equivalent characteristic strain of the unit:
after the relationship between the elastic strain and the disturbance strain and the equivalent characteristic strain is obtained from the above, the equivalent equation (1) can be rewritten as a system of equations with respect to unknowns:
neglecting the coupling relation of the impurity and the surface contact pressure (i.e. implicit function f in the above i ) The contact pressure distribution of the heterogeneous semi-space surface conforms to the hertz point contact formula, wherein the elastic strain becomes constant, and formula (9) is degenerated into a linear system of equations:
wherein equations (7) and (8) are a system of equations for equivalent characteristic strain.
S1-1-2, solving characteristic stress and characteristic displacement:
obtaining a formula (11), which is a formula for solving the characteristic stress, wherein the influence coefficient means a value of the characteristic stress generated in the target cell when the original cell contains the unit characteristic strain, and if there is the characteristic strain in a plurality of cells in the calculation region, the total characteristic stress in a certain cell is a sum of the characteristic stresses of all cells with non-zero characteristic strain in the calculation region in the target cell, and the influence coefficient between each pair of cells is a sum of the characteristic stresses generated in the functions of the cell size, the inter-cell distance and the mechanical parameters of the matrix material, so that when the divided grid parameters are fixed, the influence coefficient is also unchanged, comprising the following formula for solving:
Omega-space containing characteristic strain; the processing method comprises the steps of (a) the step of (a),-a vector associated with the potential function; x—the spatial position of the target point; x is x 3 -the distance between the target point and the half-space surface; />-a characteristic strain; equations (7) and (8) respectively represent the characteristic stress values of points outside and inside the region, and the influence coefficient U of the source unit at the target unit can be obtained by integrating the right side of the equation i The influence coefficients of the characteristic displacement and the characteristic strain are also used in the process of solving the characteristic displacement, and the specific formula is as follows:
the characteristic displacement within a cell can be expressed as:
s1-1-3, solving the surface contact pressure and the total displacement:
in the impurity bearing contact model, discretized surface contact stress distribution is obtained by solving an equation (11):
Δx, Δy—the side lengths of the cells in the x and y directions, the multiplication is the cell area; alpha, beta-unit numbering; p—surface contact pressure profile; w-the total normal load applied to the press head; h-the total distance between the two contact surfaces; h is a 0 -the distance between the two contact surfaces without any deformation of the surfaces; omega-rigid body displacement of the ram;u 3 -the sum of the surface normal elastic displacement and the characteristic displacement; i c -a collection of cells located within the contact area; i g -computing a set of all surface units within the region; the conjugate gradient method is used in the solving process, and the g of the current two contact surfaces is defined by the following formula [α,β] The gap distribution is:
g [α,β] ＝-u 3[α,β] -h [α,β] [α,β]∈I g (16)
calculating g in contact area [α,β] Average of (d), and for g [α,β] And (3) adjusting:
in N c Is the number of cells located within the contact area;
obtaining a new conjugation direction t [α,β] ：
Wherein the initial value of the auxiliary parameter delta is zero, G old An initial value of 1;
storing the current parameters for the next cycle step, G old ＝G，Continuing to calculate intermediate variables:
wherein K is the effect of surface contact pressure and elastic displacementThe response coefficient is obtained by [α,β] After that, the average value r is calculated [α,β] And adjusting:
the step size used to calculate the conjugate direction after the update:
calculating a new contact pressure distribution according to the step size and the direction:
p [α,β] ←p [α,β] -τ·t [α,β] [α,β]∈I c (24)
all points where the contact pressure is less than zero are assigned to zero, the set of non-contact points where two contact surface overlaps occur can be determined, namely:
I ol ＝{[α,β]∈I g :p [α,β] ＝0,g [α,β] <0} (25)
correcting the pressure at the overlapping point:
p [α,β] ←p [α,β] -τ·g [α,β] ,[α,β]∈I ol (26)
the total load on the contact surface is calculated:
the contact pressure distribution is corrected so that the total load satisfies the load balancing condition:
calculating the error between adjacent iterations:
if the error is smaller than the set value, stopping iteration, otherwise, returning to the first step to continue iteration, and for the contact behavior of the heterogeneous material, the surface deformation is mainly caused by the contact pressure and impurities, so that the formula for calculating the total displacement is as follows:
The total deformation is the surface deformation caused by contact pressure and the characteristic displacement caused by impurities, and only the deformation caused by pressure is calculated:
similarly, K is the surface pressure and the coefficient of influence of the pressure causing the surface deformation.
S1-1-4, solving flow:
dividing the contact area into 64X 64 grids, setting the grid size according to the contact area, defining two contact bodies and internal impurity parameters, setting additional load, setting initial gap of the given contact surface, solving the characteristic strain according to the EIM-based Conjugate Gradient Method (CGM) by utilizing Hertz theory and the initial surface morphology of the two contact surfaces, solving the surface characteristic displacement and the displacement caused by pressure according to the characteristic strain, updating the surface morphology information, judging whether the pressure is converged or not, if so, ending the iteration, outputting a result, otherwise, repeating the whole process.
7. The method for analyzing the contact characteristics of the heterogeneous ball bearing based on the equivalent hybrid method according to claim 3, wherein the step S2-1 comprises the following specific steps:
at x j O j z j And y j O j z j On the plane, the radius of the spherical rolling bodies is the same, and the calculation formula of the size is as follows:
bearing inner race is different from rolling element and is x j And y j Radius of curvature in direction is different, at x j O j z j Plane and y j O j z j The radius of the plane needs to be calculated respectively, and the formula is as follows:
solving the equivalent radius of the equivalent ellipsoid: at x j In the direction, the equivalent radius of the convex contact of the rolling body and the inner ring isIn y j In the direction, the equivalent radius of the contact between the rolling body and the inner ring is +.>The calculation formula is as follows:
wherein D is W Is spherical diameter, f i Is the curvature coefficient alpha j For initial contact angle, D m Is the pitch diameter.
The step S2-2 comprises the following specific steps:
s2-2-1, ball bearing contact characteristic analysis of the ellipsoidal impurity:
the influence of the size and the shape of a single ellipsoidal impurity in the inner ring of the bearing on the contact characteristics of the ball bearing is explored by depending on the established ball bearing contact model, and the ellipsoidal impurity is introduced into a semi-infinite plane;
s2-2-2, ball bearing contact characteristic analysis of the ball bearing containing the cubic impurities:
according to the established equivalent model, taking cubic impurities as an example, researching the influence of the volume and the central position of the impurities on the contact characteristics of the bearing, wherein the elastic modulus, the poisson ratio and the load parameters of the impurities are unchanged;
s2-2-3, analysis of contact characteristics of the ball bearing containing the cylindrical impurities:
the method starts from the distribution of internal stress and surface pressure of the bearing with the cylindrical impurities, and explores the rule of influence of the geometric parameters of the cylindrical impurities on the contact characteristics of the bearing.
8. The method for analyzing contact characteristics of a heterogeneous ball bearing based on an equivalent hybrid method according to claim 4, wherein the step S3-1 further comprises the steps of:
the influence radius of the defined inclusions i isThe following formula may be established:
andWherein the characteristic strain of i is obtained by considering the influence of all inclusions on inclusion i,/i>Considering only the characteristic strain of i obtained after the influence of impurities in the influence radius on the impurity i, if n impurities are distributed in the bearing in total, the equivalent influence radius of the distributed impurities can be defined as:
the step S3-2 further comprises the following steps:
s3-2-1, analysis of contact characteristics of ball bearings containing double impurities:
taking a double-inclusion ball bearing as an example, introducing two cubic impurities with the same volume, wherein the elastic modulus is 420GPa, the elastic modulus of a matrix is 210GPa, and researching the influence of the impurities on the contact characteristic of the ball bearing by changing the interval of the impurities;
s3-2-2, and analysis of contact characteristics of ball bearings containing three impurities:
developing a double-impurity model into a three-impurity model, wherein other bearings and impurity parameters are unchanged, and exploring the influence rule of the three-impurity spacing on the stress, displacement and characteristic displacement of the ball bearing based on the three-impurity ball bearing contact model;
S3-2-3, wherein the contact characteristics of the ball bearing containing the regularly distributed impurities are as follows:
the distribution diagram of the subsurface layers of the ball bearings with different densities and evenly distributed impurities is provided, the impurities are small cubes, the elastic modulus of the impurities is 420GPa, and the additional load and the bearing parameters are unchanged;
s3-2-4, ball bearing contact characteristic analysis of randomly distributed impurities:
starting from subsurface stress, characteristic displacement, surface pressure and the like during bearing contact, a ball bearing belt random distribution bearing contact model is developed, bearing material parameters and impurity elastic modulus 420GPa are kept unchanged, and the influence rule of the distribution density of random distribution impurities in the bearing on the ball bearing contact characteristics is studied.
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