Modeling method of multi-information fusion model of grinding wheel considering random characteristics of abrasive particles

Abstract

The present invention discloses a modeling method of a grinding wheel multi-information fusion model considering the random characteristics of abrasive particles. Firstly, the conjugate contact relationship between the grinding surface of the grinding wheel and the grinding surface of the workpiece is used to obtain the grinding wheel grinding surface profile equation through the workpiece grinding surface profile equation. Then, the grinding wheel grinding surface is unfolded into a two-dimensional annular plane through axial side view, and the traditional three-dimensional expression of the abrasive particle position is converted into a two-dimensional expression of the abrasive particle position. Finally, the two-dimensional annular plane is unfolded into an isosceles trapezoidal plane along the circumferential direction, and a square matrix is ​​constructed with the isosceles trapezoidal plane. The two-dimensional expression of the abrasive particle position is converted into a matrix expression, and a grinding wheel matrix is ​​constructed. Specifically, each element in the grinding wheel matrix represents the position of an abrasive particle. According to the needs of the actual application scenario, the numerical value of each element in the matrix can represent different information such as the protruding height, shape and orientation of the corresponding abrasive particle, thereby realizing the multi-information fusion of the grinding wheel.

Description

Technical Field

[0001] The invention belongs to the technical field of grinding processing, and specifically relates to a method for building a grinding wheel multi-information fusion model taking into account the random characteristics of abrasive particles. Background Art

[0002] The grinding wheel is the tool for grinding, and the abrasive grains on the working surface of the grinding wheel are the cutting edges that remove the workpiece material. Therefore, the macro-micro modeling of the grinding wheel is the starting step to establish the grinding process and the grinding quality model. The accuracy and efficiency of the grinding wheel model are of great significance to the modeling of the grinding process. Especially for form grinding, according to its processing principle, the grinding surface of the grinding wheel is in conjugate contact with the workpiece gear. In actual processing, the grinding surface of the grinding wheel is first trimmed to the shape corresponding to the designed gear with a diamond dressing wheel, and then the grinding process is performed to make the gear tooth surface have the same shape as the grinding surface profile of the grinding wheel. Therefore, the accuracy of the grinding surface profile of the grinding wheel determines the processing accuracy of the gear.

[0003] The grinding wheel model is the calculation and expression of the macroscopic geometry and microscopic abrasive information of the grinding wheel. It is also the basis for establishing the grinding process and grinding quality model. The grinding wheel model information will be repeatedly used in the subsequent calculation and modeling process. Therefore, a more accurate and efficient grinding wheel model can improve the accuracy of the subsequent model and reduce its complexity. However, due to the random characteristics of the grinding wheel abrasives themselves, the microscopic geometric information such as the position, size, and shape of any abrasive is difficult to directly and accurately express. In addition, due to the rich microscopic information of the abrasives, the microscopic geometric parameters and distribution frequency of the abrasives usually require multiple matrices of different categories to jointly express, resulting in a complex model, a large amount of calculated data, and low accuracy. The abrasive positions in the traditional grinding wheel model usually use three position matrices to represent the X, Y, and Z coordinate values ​​of the abrasives respectively. Assuming that the length of each position matrix is ​​the total number of abrasives N, the spatial position coordinate distribution of the abrasives can be expressed by three [1×N] position matrices, or the three position matrices can be formed into a new [3×N] or [1×3N] position matrix by data splicing. This method is sensitive to the position and posture of the grinding wheel coordinate system. In the subsequent calculation, the changes in the grinding wheel position and posture and the mutual conversion relationship between the grinding wheel and other spatial coordinate systems need to be considered. At the same time, the abrasive spatial position matrix established by this method has low data utilization, large model data volume, and time-consuming calculation. In addition, since the profiled gear grinding wheel is a special double-beveled grinding wheel, the macroscopic geometry of its grinding surface is a special geometric shape such as an involute, and the traditional grinding wheel modeling method for plane / external cylindrical grinding is not applicable. Summary of the invention

[0004] In view of this, in order to solve the shortcomings of the existing grinding wheel model, the purpose of the present invention is to provide a grinding wheel multi-information fusion model modeling method taking into account the random characteristics of abrasive particles.

[0005] In order to achieve the above object, the present invention provides the following technical solutions:

[0006] A method for modeling a grinding wheel multi-information fusion model considering the random characteristics of abrasive particles comprises the following steps:

[0007] Step 1: Construct the macro-geometric model of the grinding wheel grinding surface

[0008] The conjugate contact relationship between the grinding surface of the grinding wheel and the grinding surface of the workpiece is used, and the grinding surface profile equation of the workpiece is used to obtain the grinding surface profile equation of the grinding wheel, and the macro-geometric model of the grinding surface of the grinding wheel is established;

[0009] Step 2: Construct a micro-geometric model of the grinding wheel abrasive

[0010] 21) Using the macro-geometric model of the grinding wheel grinding surface, establish a three-dimensional expression of the grinding wheel grinding surface profile, and perform a three-dimensional expression of any abrasive grain position on the grinding wheel grinding surface;

[0011] 22) Expand the grinding wheel grinding surface into a two-dimensional annular plane along the axial side view, convert the three-dimensional expression of the grinding wheel grinding surface profile into a two-dimensional expression of the grinding wheel grinding surface side view, and express any abrasive grain position on the grinding wheel grinding surface in two dimensions; wherein, the difference between the radius of any point A on the two-dimensional annular plane and the inner circle radius of the two-dimensional annular plane is equal to the length of the curve segment between point A and the inner end point of the curve intercepted by the plane passing through the grinding wheel grinding surface point A′ and the grinding wheel axis on the grinding wheel grinding surface, and any point A on the two-dimensional annular plane is obtained by expanding the point A′ on the grinding wheel grinding surface along the axial side view;

[0012] 23) Expand the two-dimensional annular plane into an isosceles trapezoidal plane along the circumferential direction and perform numerical discretization, and convert the two-dimensional side view expression of the grinding wheel grinding surface into a grinding wheel matrix expression;

[0013] Step 3: Construct the micro-geometric parameter model of the grinding wheel abrasive

[0014] Elements are filled in the grinding wheel matrix according to the random distribution law of abrasive particles. The actual spatial position of the abrasive particles is expressed by the number of rows and columns of the elements in the grinding wheel matrix. The abrasive particle information is expressed by the size of the elements. The abrasive particle information includes the protruding height of the abrasive particles, the shape of the abrasive particles and the direction of the abrasive particles.

[0015] Furthermore, the workpiece is an involute spur gear, and the grinding surface of the involute spur gear is the tooth surface. The grinding surface profile equation of the grinding wheel obtained by the tooth surface is:

[0016]

[0017] in: is the involute angle parameter, and are the upper and lower limits of the parameters respectively; r b is the radius of the gear base circle: r b =mzcosα

[0018] Where: m is the gear module, z is the number of gear teeth, and α is the pressure angle.

[0019] Further, in step 21), the three-dimensional expression W1 of any abrasive grain position on the grinding surface of the grinding wheel is:

[0020]

[0021] in: is the gear involute angle parameter corresponding to the abrasive position; θ is the rotation angle parameter of the abrasive position on the circumference of the grinding wheel.

[0022] Further, in step 22), the two-dimensional expression W2 of the position of any abrasive grain on the grinding surface of the grinding wheel is:

[0023] W2=f(R,θ)

[0024] Where R is the radius of the abrasive particle in the two-dimensional annular plane; and:

[0025]

[0026] Where: Y D Y is the distance between the grinding wheel center and the origin of the gear coordinate system in the Y direction; end is the Y coordinate value of the tooth top; is the involute angle parameter corresponding to the starting point of the involute; and:

[0027]

[0028] in: is the involute angle parameter of the tooth addendum.

[0029] Further, in step 23), the conversion relationship between the grinding wheel grinding surface side view two-dimensional expression W2 and the grinding wheel matrix expression W3 is:

[0030]

[0031] Where: i w and j w Respectively represent the number of rows and columns of the grinding wheel matrix; Δi w and Δj w are the row spacing and column spacing between adjacent abrasive particles, respectively, which are determined by the number of abrasive particles per unit area; δ(i w ) is the distance between the grinding wheel matrix element boundary and the maximum width position of the matrix; and:

[0032]

[0033] The maximum number of rows in the grinding wheel matrix is:

[0034] i wmax =1+r b (((z+2) / zcosα) 2 ―1) / 2Δi w

[0035] For the solved i wmax Perform rounding;

[0036] The maximum number of columns in the grinding wheel matrix is:

[0037]

[0038] For the solved j wmax Round up.

[0039] Further, in step 3, when filling elements in the grinding wheel matrix, from i w =1,j w =1 starts filling the matrix elements;

[0040] If satisfied:

[0041] δ(i w ) / Δj w ≤j w ≤j wmax ―2δ(i w ) / Δj w

[0042] Then there are abrasive particles at the position corresponding to the corresponding matrix element, otherwise there are no actual abrasive particles at the position corresponding to the corresponding matrix element.

[0043] Furthermore, the protrusion height of the abrasive particles obeys the Gaussian distribution:

[0044]

[0045] Where: h i is the protrusion height of any abrasive grain; a, μ and σ are the peak value of the Gaussian distribution curve, the protrusion height value corresponding to the peak value and the standard deviation, respectively, which are obtained by measuring the microscopic morphology of the working surface of the grinding wheel.

[0046] Furthermore, the abrasive grains are oriented at an angle The scope meets:

[0047]

[0048] in: and The abrasive particle orientation angles are minimum and maximum values ​​of ; and:

[0049]

[0050]

[0051] Where: x g ,y g is the coordinate point of the abrasive particle space position; and are the coordinates of the two intersection points of a circle with the spatial position of the abrasive grain as the center and the protruding height of the abrasive grain as the radius and the tooth profile involute, which are obtained by combining the grinding wheel grinding surface profile equation and the parametric equation of the abrasive grain radius circle, and the parametric equation of the abrasive grain radius circle is:

[0052]

[0053] Where: x oc ,y oc is the coordinate point on the abrasive particle radius circle; is the parameter variable of the equation of the wear particle radius circle, and and They are The minimum and maximum values ​​of .

[0054] Furthermore, the shapes of the abrasive particles are conical and quadrangular pyramidal, the cross section of the abrasive particles is triangular, and the vertex of the triangle is located on the abrasive particle radius circle, and its position is determined by the abrasive particle orientation:

[0055] x tv =x g +h i cosθ i

[0056] y tv =y g +h i sinθ i

[0057] Among them, x tv and tv The vertex coordinates of the abrasive cross-section triangle; θ i is the orientation angle of any abrasive particle; h i is the protrusion height of any abrasive grain;

[0058] The upper boundary line of the triangular cross-section abrasive is represented by the abrasive vertex angle and the abrasive orientation direction:

[0059]

[0060] Where: x tt and tt are the coordinate values ​​of the boundary lines on the triangular cross section; t tt is the parameter variable of the upper boundary line equation, t tt_max t tt The maximum value of is the direction vector of the upper boundary line; and

[0061]

[0062] The lower boundary line of the triangular cross-section abrasive is represented by the abrasive vertex angle and the abrasive orientation direction:

[0063]

[0064] Where: x tb and tb are the coordinates of the lower boundary line of the triangular cross section; t tb is the parameter variable of the lower boundary line equation, t tb_max t tb The maximum value of is the direction vector of the lower boundary line, and:

[0065]

[0066] Where: θ v is the vertex angle of the triangular cross section.

[0067] The beneficial effects of the present invention are:

[0068] The present invention provides a grinding wheel multi-information fusion modeling method that considers the random characteristics of abrasive particles. First, the conjugate contact relationship between the grinding surface of the grinding wheel and the grinding surface of the workpiece is used to obtain the grinding surface profile equation of the grinding wheel through the grinding surface profile equation of the workpiece. Then, the grinding surface of the grinding wheel is expanded into a two-dimensional annular plane through axial side view, and the traditional three-dimensional expression of the abrasive particle position is converted into a two-dimensional expression of the abrasive particle position. Finally, the two-dimensional annular plane is expanded into an isosceles trapezoidal plane along the circumferential direction, and a square matrix is ​​constructed with the isosceles trapezoidal plane. The two-dimensional expression of the abrasive particle position is converted into a matrix expression, and a grinding wheel matrix is ​​constructed. Specifically, each element in the grinding wheel matrix represents the position of an abrasive particle. According to the needs of the actual application scenario, the numerical value of each element in the matrix can represent different information such as the protruding height, shape and orientation of the corresponding abrasive particle, thereby realizing the multi-information fusion of the grinding wheel and having the following advantages:

[0069] 1) By unfolding the grinding surface of the grinding wheel into a two-dimensional annular plane from the axial side view, for special grinding wheel grinding surface geometry, the grinding wheel abrasive grains and the radial distance increment can satisfy a linear relationship, thereby meeting the modeling requirements of the grinding surface of the grinding wheel with special geometry; that is, the grinding wheel multi-information fusion model modeling method considering the random characteristics of the abrasive grains of the present invention is not affected by the grinding surface profile of the grinding wheel, and can be extended to more complex curved grinding wheels;

[0070] 2) In the constructed grinding wheel multi-information fusion model, the element position of the grinding wheel matrix can directly express the position of the grinding wheel abrasive grains, and the size of the grinding wheel matrix element can express the random characteristics of the micro-geometric characteristics of the abrasive grains, such as the random characteristics of the normal protrusion height, orientation, shape, size, etc. of the abrasive grains, making the model more accurate;

[0071] 3) Without increasing the matrix size and the amount of element data, the simultaneous expression of multiple types of wear particle information is achieved, which reduces the amount of model data and improves the calculation efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0072] In order to make the purpose, technical solution and beneficial effects of the present invention clearer, the present invention provides the following drawings for illustration:

[0073] Figure 1 It is a schematic diagram of the grinding surface of the involute grinding wheel;

[0074] Figure 2 This is the process diagram of the conversion between the three-dimensional position information of the grinding wheel and the number of rows and columns of the matrix;

[0075] Figure 3 It is a schematic diagram of the three-dimensional expression of the grinding wheel;

[0076] Figure 4 It is a schematic diagram of the conversion between the three-dimensional expression and the axial side view expression of the grinding wheel;

[0077] Figure 5 It is a schematic diagram of the conversion between the grinding wheel side view expression and the matrix expression;

[0078] Figure 6 Fill the step flow chart for the grinding wheel matrix element;

[0079] Figure 7 It is a schematic diagram of the wear particle position-height fusion matrix;

[0080] Figure 8 The decomposition and influence diagram of wear particle orientation vector;

[0081] Fig. 9 Schematic diagram of abrasive grain orientation, local coordinate system and abrasive grain orientation circle on the involute;

[0082] Fig.10 Schematic diagram of the abrasive geometry with circular and triangular cross sections;

[0083] Fig.11 Schematic diagram of the main forms of abrasive wear;

[0084] Fig.12 The geometric diagram of the grinding wheel abrasive grains;

[0085] Fig.13 This is the calculation process diagram of abrasive wear rate;

[0086] Fig.14 The measurement and fitting results of wear particle protrusion and residual height distribution;

[0087] Fig.15 It is the numerical distribution of abrasive protrusion height, residual height and wear height;

[0088] Fig.16 is the calculation result of each abrasive wear height;

[0089] Fig.17 To calculate the abrasion wear rate results multiple times;

[0090] Fig.18 This is a schematic diagram of the calculation of the material removal volume of gears and grinding wheels;

[0091] Fig.19 Schematic diagram of abrasive material removal;

[0092] Fig. 20 Schematic diagram of the grinding wheel modeling process;

[0093] Fig.21 Equipment and arrangement for grinding wheel profile measurement;

[0094] Fig. 22 This is the normal height measurement diagram of the grinding wheel;

[0095] Fig.23 The measurement results of the wear particle protrusion height distribution;

[0096] Fig.24 is the distribution frequency of abrasive particle height in different measurement ranges;

[0097] Fig.25 It is the three-dimensional measurement result of the grinding wheel morphology;

[0098] Fig.26 Schematic diagram of the measurement of abrasive particle inclination angle;

[0099] Fig. 27 It is the calculation of wear particle protrusion height fitting and height series;

[0100] Fig.28 Comparison of surface roughness between measured and simulated results. DETAILED DESCRIPTION

[0101] The present invention is further described below in conjunction with the accompanying drawings and specific embodiments so that those skilled in the art can better understand the present invention and implement it, but the embodiments are not intended to limit the present invention.

[0102] 1. Grinding wheel multi-information fusion model

[0103] The method for building a grinding wheel multi-information fusion model considering the random characteristics of abrasive particles in this embodiment includes the following steps:

[0104] Step 1: Construct a macro-geometric model of the grinding wheel grinding surface: Utilize the conjugate contact relationship between the grinding wheel grinding surface and the workpiece grinding surface, use the workpiece grinding surface profile equation to obtain the grinding wheel grinding surface profile equation, and establish the grinding wheel grinding surface macro-geometric model.

[0105] This embodiment takes an involute spur gear as an example, that is, the workpiece ground by the grinding wheel is an involute spur gear. Forming gear grinding is a typical line contact processing method, in which the grinding surface of the grinding wheel and the gear tooth surface are in full tooth height meshing conjugate contact, and each contact point between the grinding wheel and the workpiece tooth surface meets the conjugate contact condition:

[0106] n·v=0

[0107] Among them: n is the normal vector of the contact point, and v is the relative movement speed of the grinding wheel at the contact point to the workpiece surface.

[0108] The conjugate contact line between the grinding wheel and the gear can be regarded as the basic element of the grinding wheel and the workpiece at the same time. The grinding wheel grinding surface is formed by rotating the conjugate contact line around the grinding wheel axis. At the same time, the gear tooth surface is formed by translating the contact line along the gear helix line. Therefore, according to the conjugate grinding processing principle, the grinding wheel grinding surface profile can be deduced from the gear end face profile.

[0109] First, determine the gear end profile based on the gear design parameters. Taking the involute cylindrical gear as an example, the common design parameters include the normal module m n , number of teeth z, helix angle β, displacement coefficient x, gear width b, etc. The tooth profile curve can be regarded as a multi-end curve connected, including an involute segment, a tooth root transition arc segment, and a connecting segment of two adjacent teeth. This embodiment mainly studies the process of forming gear grinding. Therefore, the macroscopic modeling of the grinding wheel mainly considers the involute part, that is, the grinding surface profile of the grinding wheel. Taking the involute external meshing standard spur gear as an example, the parametric equation is determined according to the basic properties of the involute, and the grinding surface profile equation of the grinding wheel is obtained by the tooth surface:

[0110]

[0111] in: is the involute angle parameter, and are the upper and lower limits of the parameters respectively; r b is the base circle radius of the gear:

[0112] r b =mz cosα

[0113] Where: m is the gear module, z is the number of gear teeth, and α is the pressure angle.

[0114] Involute angle parameter The upper and lower limits can be determined according to the starting and ending points of the effective involute of the gear:

[0115]

[0116] Where: r Nf and r Na They are respectively the radii of the starting and ending points of the effective involute of the gear.

[0117] Usually, the effective involute end point is located on the gear tooth addendum circle, and the effective involute end point radius is the same as the tooth addendum circle radius. The effective involute start point position is determined according to the gear design meshing requirements. For standard meshing cylindrical gears, the effective involute start point is located on the tooth root circle, and the effective involute start point radius is the same as the tooth root circle radius. The calculation method of the tooth addendum circle and tooth root circle radius will not be repeated here, and you can refer to the relevant books on mechanical principles.

[0118] The cross-sectional profile is solved according to the involute parameter equation of the tooth profile, and the rotating surface obtained by rotating along the axis of the grinding wheel is the macroscopic shape of the grinding surface of the grinding wheel, such as Figure 1 shown.

[0119] Step 2: Construct a micro-geometric model of the grinding wheel abrasive

[0120] Grinding wheel micro-modeling is the expression of the micro-geometric parameters of the grinding wheel abrasive particles, mainly including: abrasive particle position, abrasive particle height, abrasive particle shape, abrasive particle size, abrasive particle orientation, etc. The abrasive particles of the double-beveled grinding wheel for forming gear grinding are distributed on a three-dimensional surface, and the position information expression and calculation are more complicated. In addition, the abrasive particle geometric information is rich and has random characteristics, requiring a large amount of data to express the microscopic characteristics of the abrasive particles. Therefore, it is necessary to propose a more efficient and accurate method for expressing the information of grinding wheel abrasive particles to provide theoretical support for subsequent calculations.

[0121] Abrasive position modeling is an important part of expressing the microscopic modeling of the grinding wheel. The position matrix is ​​used to express the position of the abrasive in the grinding wheel space. The abrasive position in the traditional grinding wheel model usually uses three position matrices to represent the X, Y, and Z coordinate values ​​of the abrasive. Assuming that the length of each position matrix is ​​the total number of abrasives N, the spatial position coordinate distribution of the abrasive can be expressed by three [1×N] position matrices, or the three position matrices can be formed into a new [3×N] or [1×3N] position matrix by data splicing. This method is sensitive to the position and posture of the grinding wheel coordinate system. In subsequent calculations, the changes in the position and posture of the grinding wheel and the mutual conversion relationship between the grinding wheel and other coordinate systems in space need to be considered. At the same time, the abrasive spatial position matrix established by this method has low data utilization, large model data volume, and time-consuming calculation.

[0122] Therefore, this embodiment proposes a micro-geometric modeling method for abrasive grains of double-beveled grinding wheels for forming gear grinding, establishes an abrasive grain micro-information fusion matrix, calculates the position of the grinding wheel abrasive grains by the number of rows and columns of the matrix, and characterizes other geometric information of the abrasive grains by the corresponding matrix element size, so as to realize the simultaneous expression of multiple types of abrasive grain micro-information and improve the calculation efficiency. The number of matrix rows and columns corresponding to the spatial position of the abrasive grains is calculated as follows: Figure 2 shown.

[0123] 21) Using the macro-geometric model of the grinding wheel grinding surface, a three-dimensional expression of the grinding wheel grinding surface profile is established, and the position of any abrasive grains on the grinding wheel grinding surface is expressed in three dimensions.

[0124] Specifically, according to the principle of profile grinding, the grinding wheel grinding surface profile is the same as the gear tooth profile. The three-dimensional expression of the grinding wheel grinding surface profile can be established by referring to the involute parameter equation, such as Figure 3 Therefore, for any abrasive grain at any position on the grinding surface of the grinding wheel, its three-dimensional expression W1 is:

[0125]

[0126] in: is the gear involute angle parameter corresponding to the abrasive position; θ is the rotation angle parameter of the abrasive position on the circumference of the grinding wheel.

[0127] Any abrasive grain position on the grinding surface of the grinding wheel can be expressed by three-dimensional expression, but the involute angle of the three-dimensional expression parameter of the grinding wheel There is a nonlinear relationship between the increment of and the radial distance increment between the grinding wheel abrasive grains and the axis. In the numerical model, when the involute angle increases uniformly, the spacing between the abrasive grains increases nonlinearly, which affects the calculation accuracy of the numerical model and cannot meet the requirements of the subsequent calculation for the grinding wheel model. In addition, the three-dimensional geometric shape of the grinding wheel grinding surface is complex and difficult to directly convert into a plane matrix. Therefore, it is necessary to convert the three-dimensional expression of the grinding wheel W1 into the side view expression of the grinding wheel W2 through the geometric relationship of the grinding wheel grinding surface.

[0128] 22) Expand the grinding wheel grinding surface into a two-dimensional annular plane along the axial side view, and convert the three-dimensional expression of the grinding wheel grinding surface profile into a two-dimensional expression of the grinding wheel grinding surface side view, such as Figure 4 As shown, the position of any abrasive grain on the grinding surface of the grinding wheel is expressed in two dimensions; wherein, the difference between the radius of any point A on the two-dimensional annular plane and the inner circle radius of the two-dimensional annular plane is equal to the length of the curve segment between point A and the inner end point of the curve intercepted by the plane passing through point A′ on the grinding surface of the grinding wheel and the axis of the grinding wheel on the grinding surface of the grinding wheel, and any point A on the two-dimensional annular plane is obtained by unfolding point A′ on the grinding surface of the grinding wheel along the axial side view.

[0129] Depend on Figure 4It can be seen that the transformation from three-dimensional geometric figures to two-dimensional plane figures can be achieved by unfolding the grinding wheel side view. The two-dimensional expression W2 of any abrasive grain position on the grinding surface of the grinding wheel is:

[0130] W2=f(R,θ)

[0131] Where R is the radius of the position of the abrasive particle in the two-dimensional annular plane;

[0132] In both the three-dimensional expression W1 and the side view expression W2 of the grinding wheel, the rotation angle parameter θ of the abrasive position on the circumference of the grinding wheel remains unchanged. Therefore, only the involute angle needs to be calculated. The conversion relationship between the radius R of the abrasive particle in the side view of the gear.

[0133] The radius R of any point in the side view of the grinding wheel can be calculated from the inner diameter of the grinding wheel:

[0134] R=R1+ΔR=R1+ΔS

[0135] Among them: ΔR is the distance between the abrasive position and the inner diameter of the grinding wheel in the side view expression; ΔS is the involute length distance between the abrasive point and the starting point of the effective involute in the three-dimensional expression of the grinding wheel; R1 is the inner diameter of the grinding wheel side view. According to the position relationship between the grinding wheel and the gear in the form grinding process:

[0136] R1=Y D ―Y end

[0137] Where Y D Y is the distance between the center of the grinding wheel and the origin of the gear coordinate system in the Y direction of the gear and grinding wheel coordinate system; end is the Y coordinate value of the effective involute end point of the gear in the gear coordinate system. The involute angle parameter of the tooth tip circle can be Substitute the involute equation into the calculation, according to the geometric characteristics of the involute gear:

[0138]

[0139] Where z is the number of gear teeth; α is the pressure angle of the upper end face of the gear pitch circle, usually 20°.

[0140] According to the involute parametric equation, the length increment of the involute is:

[0141]

[0142] In the formula, Incremental value for the involute angle parameter.

[0143] Combine the grinding wheel grinding surface profile equation and the involute length increment equation to solve the involute length of the grinding wheel grinding surface:

[0144]

[0145] In the formula, It is the involute angle parameter corresponding to the effective involute starting point.

[0146] From this we get:

[0147]

[0148] Where: Y D Y is the distance between the grinding wheel center and the origin of the gear coordinate system in the Y direction; end is the Y coordinate value of the tooth top; is the involute angle parameter corresponding to the starting point of the involute; and:

[0149]

[0150] in: is the involute angle parameter of the tooth addendum.

[0151] According to the above calculation, the involute angle parameter in the three-dimensional expression of the grinding wheel is Converted to the radius parameter R in the side view expression, the distance between adjacent abrasive grains changes from nonlinear growth in the three-dimensional expression to linear growth in the side view expression. At the same time, this transformation also realizes the transformation of the grinding wheel grinding surface from a complex curved surface to a circular annular plane.

[0152] 23) The two-dimensional circular plane is expanded into an isosceles trapezoidal plane along the circumferential direction and numerically discretized, and the two-dimensional side view expression of the grinding wheel grinding surface is converted into a grinding wheel matrix expression.

[0153] In order to further optimize the model and form a more efficient abrasive position information expression model, the grinding wheel side view expression is further expanded from the perspective of graphics and numerically discretized, and the grinding wheel side view expression W2 is converted into the grinding wheel matrix expression W3, as shown in Figure 5 In the matrix expression of the grinding wheel abrasive position information, any abrasive position variable is the row number i of the abrasive in the matrix. w and the number of columns j w :

[0154] W3=(i w ,j w )

[0155] The side view of the grinding wheel is a trapezoid after unfolding. First, according to the conservation principle of the length of the corresponding line segment before and after the side view of the grinding wheel is unfolded, the geometric dimensions of the trapezoid are calculated to determine the matrix boundary and the critical values ​​of the number of rows and columns of the matrix:

[0156] L bot =2πR1

[0157] L top=2πR2=Δj w (j wmax ―1)

[0158] Where, L bot and L top Respectively represent the lengths of the two parallel sides of the trapezoid; Δj w j is the distance between adjacent rows of abrasive particles; wmax is the maximum number of columns in the grinding wheel matrix; R2 is the outer diameter of the grinding wheel side view, through the grinding wheel geometric relationship:

[0159] R2=R1+ΔS total

[0160] In the formula, ΔS total is the total length of the involute of the grinding wheel grinding surface, which can be calculated based on the involute angle corresponding to the end point of the effective involute of the grinding wheel. In this way, we can get:

[0161]

[0162] Similarly, the height H of the grinding wheel matrix corresponds to the difference between the inner and outer diameters of the grinding wheel:

[0163] H=ΔR total =ΔS total =Δi w (i wmax ―1)

[0164] In the formula, Δi w is the distance between adjacent rows of abrasive particles; i wmax is the maximum number of rows in the grinding wheel matrix, and we can obtain wmax for:

[0165]

[0166] Since the calculation result may be a non-integer, and the number of rows and columns of the grinding wheel are integers, the maximum number of rows and columns obtained are rounded:

[0167]

[0168] The matrices in mathematical expressions are all "rectangular", while the matrix of the profile grinding wheel is a "trapezoidal" matrix. The inside of the "trapezoidal" is the grinding wheel abrasive grains, and the part between the "rectangular" and the "trapezoidal" is the part without abrasive grains, but due to the form of the matrix itself, this part is still unavoidable. Therefore, the boundary line between the "rectangular" and the "trapezoidal" is the dividing line between the grinding wheel matrix abrasive grain element area and the element-free area.

[0169] The abrasive grain with radius R in the side view of the grinding wheel corresponds to the i-th w The distance between the row element boundary and the "rectangular" matrix boundary is δ(i w)for:

[0170] δ(i w )=π(R2―R)

[0171] Finally, the number of rows and columns of the abrasive grain position in the grinding wheel matrix is ​​calculated according to the abrasive grain position radius parameter R in the grinding wheel side view matrix. In the grinding wheel matrix expression, R can be determined according to the number of rows of the abrasive grain position:

[0172] R=R1+(i w ―1)Δi w

[0173] In addition, according to the grinding wheel matrix geometry:

[0174] Δj w (j w ―1)=δ(i w )+Rθ

[0175] but:

[0176]

[0177] The expression of the number of rows and columns of any abrasive grains of the grinding wheel can be obtained:

[0178]

[0179]

[0180] The conversion relationship between the side view two-dimensional expression W2 of the grinding wheel grinding surface and the grinding wheel matrix expression W3 is obtained as follows:

[0181]

[0182] Where: i w and j w Respectively represent the number of rows and columns of the grinding wheel matrix; Δi w and Δj w are the row spacing and column spacing between adjacent abrasive particles, respectively, which are determined by the number of abrasive particles per unit area; δ(i w ) is the distance between the grinding wheel matrix element boundary and the maximum width position of the matrix.

[0183] Through the above-mentioned grinding wheel geometric parameter conversion relationship calculation, any set of parameters in the three-dimensional expression of the gear grinding wheel and θ can be solved in the grinding wheel matrix expression, realizing the three-dimensional expression of the grinding wheel The side view expression of the grinding wheel W2(R,θ) and the matrix expression of the grinding wheel W3(i w ,j w )’s mutual conversion.

[0184] Step 3: Construct a micro-geometric parameter model of grinding wheel abrasives: fill elements in the grinding wheel matrix according to the random distribution law of abrasives, express the actual spatial position of the abrasives with the number of rows and columns of elements in the grinding wheel matrix, and express the abrasive information with the size of the elements, and the abrasive information includes the protruding height of the abrasives, the shape of the abrasives and the direction of the abrasives.

[0185] The expression of the grinding wheel matrix contains regions with or without elements. When constructing the grinding wheel matrix, it is necessary to distinguish the abrasive element and non-abrasive element regions according to the boundary calculation results of the side-view unfolded graph, and only fill the corresponding values ​​in the regions containing abrasive elements. The discrete element filling process is as follows: Figure 6 When filling elements in the grinding wheel matrix, from i w =1,j w =1 starts filling the matrix elements;

[0186] If satisfied:

[0187] δ(i w ) / Δj w ≤j w ≤j wmax ―2δ(i w ) / Δj w

[0188] Then there are abrasive particles at the position corresponding to the corresponding matrix element, otherwise there are no actual abrasive particles at the position corresponding to the corresponding matrix element.

[0189] Compared with the three-dimensional expression and side-view expression of the grinding wheel, the grinding wheel matrix expression directly uses the rows and columns of the grinding wheel matrix as parameter variables. The position information of the grinding wheel is converted from specific values ​​to the number of rows and columns of the grinding wheel matrix. The elements in the matrix can be used to express other abrasive information. By establishing the abrasive micro-information fusion matrix, the abrasive multi-information fusion expression is realized without increasing the original matrix size and data length, which improves the calculation efficiency of the model and provides theoretical support for subsequent calculations.

[0190] Due to the random characteristics of the grinding wheel, the microscopic geometric information such as the height and size of any abrasive grain is difficult to measure or calculate accurately. However, according to research, the normal height of the abrasive grains on the grinding wheel surface satisfies the Gaussian distribution. Therefore, the height h of any abrasive grain on the grinding wheel surface is expressed by the Gaussian probability density function: i The probability density of is:

[0191]

[0192] Where: h i is the protrusion height of any abrasive grain; a, μ and σ are the peak value of the Gaussian distribution curve, the protrusion height value corresponding to the peak value and the standard deviation, respectively, which are obtained by measuring the microscopic morphology of the working surface of the grinding wheel.

[0193] According to the Gaussian density function, a sequence of abrasive protrusion heights that obeys the distribution can be generated by mathematical methods. The total length of the sequence is the total number of abrasive particles. The element values ​​in the sequence represent any abrasive protrusion height, and the production of the element values ​​satisfies the Gaussian distribution.

[0194] According to the calculation and modeling of the abrasive position in the previous article, the number of rows and columns of the grinding wheel matrix represent the position of the abrasive, and the value of the element can represent other abrasive information. Therefore, on the basis of establishing the abrasive protrusion height sequence, taking the abrasive protrusion height as an example, an abrasive position-height fusion matrix is ​​established. In the process of establishing the abrasive height matrix, when filling elements according to the calculated abrasive area boundary, the established abrasive height sequence elements are filled into the abrasive area of ​​the grinding wheel matrix in sequence. At the same time, since the spacing between adjacent abrasives is uneven, there are also some unfilled areas in the abrasive element area. The abrasive spacing is distinguished by the size of the area, and then the height element value is assigned according to the abrasive protrusion height sequence, such as Figure 7 shown.

[0195] As can be seen from the figure, the element value in the abrasive position-height fusion matrix represents the protruding height of the abrasive. When there is no abrasive at any position, the element size corresponding to the matrix position is 0, thereby realizing the calculation of different abrasive spacing. The abrasive position and protruding height fusion matrix model establishes the basic properties of the grinding wheel abrasive, meeting the basic requirements of the grinding process calculation simulation model for the tool model.

[0196] However, for the calculation of grinding process and related parameters, other microscopic information such as abrasive grain orientation needs to be considered to achieve accurate modeling of grinding. Therefore, based on the model of arbitrary abrasive grain position and protrusion height, the abrasive grain orientation is calculated to supplement the microscopic geometric model of grinding wheel abrasive grains.

[0197] Any direction vector in space can be decomposed into two direction vectors located in mutually perpendicular planes. Therefore, the normal direction vector of the abrasive is decomposed into a plane parallel to the grinding wheel feed direction and a plane perpendicular to the grinding wheel feed direction, as shown in Figure 8 (a) is shown. The influence of the angle change on the plane perpendicular to the feed direction and the angle change on the cutting parameters are different. Taking the grinding force as an example, the influence of the angle of different planes on the direction of the grinding force is shown in Figure 8 (b) and Figure 8 (c) As shown. However, for the grinding wheel grinding process, the rotation of the grinding wheel around its own axis during the processing will also produce an angle change in the plane parallel to the feed direction, such as Figure 8 (d), and this angle changes frequently and is usually not considered when calculating the cutting force of grinding. Therefore, this embodiment only considers the orientation vector of the abrasive decomposition perpendicular to the feed direction. The angle between the orientation direction of the abrasive perpendicular to the feed direction of the grinding wheel and the normal direction of the grinding wheel plane at the abrasive position is defined as the abrasive orientation angle.

[0198] First, calculate the normal vector direction of any point on the grinding wheel surface. The grinding wheel used for profile grinding is a double-bevel grinding wheel, and its grinding surface is an involute surface. The normal vector direction at different positions is different, such as Fig. 9 As shown in (a), the tangential direction and normal direction at any position on the involute are calculated as:

[0199] γ n +γ k =π / 2

[0200]

[0201] We can get:

[0202]

[0203] Taking the triangular cross-section abrasive as an example, the motion trajectories of abrasives with different orientations are different during the grinding process, such as Fig. 9 (b) As shown. According to the abrasive position and the normal vector direction of the grinding wheel at the abrasive position, the abrasive local coordinate system (LCSY) is established. It can be seen that the influence of the abrasive orientation on the normal and tangential grinding forces is as follows: Fig. 9 (c) as shown.

[0204] The direction angle vector range of the abrasive grain is an arc with the abrasive grain position point as the center, the abrasive grain protrusion height as the radius, and the grinding wheel surface as the boundary, such as Fig. 9 (d), the parametric equation of this arc is:

[0205]

[0206] In the formula, x oc and oc is the coordinate variable of the abrasive particle toward the circle in the X and Y directions; g and g h is the coordinate of the abrasive particle position, which is calculated by the involute angle corresponding to the abrasive particle position; i is the protruding height of the abrasive grains; is the wear particle circle equation parameter, and its critical range and It can be solved by simultaneously solving the involute equation and the equation of the abrasive particle's direction circle.

[0207] The critical value of the abrasive particle direction vector corresponding to the critical point of the abrasive particle circle is:

[0208]

[0209]

[0210]

[0211] In the formula, Indicates the abrasive grain orientation angle; x g ,y g is the coordinate point of the abrasive particle space position; and They are the coordinate values ​​of the two intersection points of a circle with the spatial position of the abrasive as the center and the protruding height of the abrasive as the radius and the involute of the tooth profile.

[0212] According to the above calculation, the calculation range of the abrasive particle orientation circle can be determined, and the orientation vector of the abrasive particle is the position of any point within the arc segment.

[0213] The real shape of abrasive particles is irregular and difficult to accurately express with simple geometric bodies. However, in order to achieve unified modeling of abrasive particles and facilitate expression and calculation, it is usually assumed that the shape of abrasive particles is some simple geometric bodies. Commonly used geometric bodies include sphere, cone, tetrahedron, truncated cone, prism, etc. The grinding process is the process of abrasive particles rotating around the axis of the grinding wheel. The geometric shape of abrasive particles is often expressed by the cross-sectional shape of abrasive particles. Therefore, on the basis of modeling the position, protrusion height, and orientation of abrasive particles, the shape of abrasive particles is further expressed and calculated, and the microscopic model of abrasive particles is supplemented, such as Fig.10 shown.

[0214] The cross section of a spherical abrasive particle is a circle, and the radius is the protruding height of the abrasive particle. Since the circle is an isotropic geometric figure, the orientation of the abrasive particle has no effect on the geometric modeling of the circle. Therefore, the geometric equations of the circular cross-section abrasive particles in various orientations are the same as those of the abrasive particle orientation circle, such as Fig.10 (a) As shown. The cross-sections of conical and tetrahedral abrasive grains are both isosceles triangles, and the vertices of the triangles are located on the abrasive grain orientation circle. The specific position is determined by the orientation angle, such as Fig.10 (b) In addition, the abrasive grain vertex angle also affects the geometric shape. The positions of the upper and lower edges of the cross section of abrasive grains with the same orientation will change with the vertex angle parameters, such as Fig.10 (c) as shown.

[0215] In this embodiment, it is assumed that the shape of the abrasive particles is conical and quadrangular pyramidal, the cross section of the abrasive particles is triangular, and the vertex of the triangle is located on the radius circle of the abrasive particles, and its position is determined by the orientation of the abrasive particles:

[0216] x tv =x g +h i cosθ i

[0217] y tv =y g +h i sinθ i

[0218] Among them, x tv and tvThe vertex coordinates of the abrasive cross-section triangle; θ i is the orientation angle of any abrasive particle; h i is the protrusion height of any abrasive particle.

[0219] The upper boundary line of the triangular cross-section abrasive can be expressed by the abrasive vertex angle and the abrasive orientation direction:

[0220]

[0221] Where: x tt and tt are the coordinate values ​​of the boundary lines on the triangular cross section; t tt is the parameter variable of the upper boundary line equation, t tt_max t tt The maximum value of is the direction vector of the upper boundary line; and

[0222]

[0223] The lower boundary line of the triangular cross-section abrasive is represented by the abrasive vertex angle and the abrasive orientation direction:

[0224]

[0225] Where: x tb and tb are the coordinates of the lower boundary line of the triangular cross section; t tb is the parameter variable of the lower boundary line equation, t tb_max t tb The maximum value of is the direction vector of the lower boundary line, and:

[0226]

[0227] Where: θ v is the vertex angle of the triangular cross section.

[0228] In the grinding wheel abrasive modeling, each abrasive grain does not interfere with each other. The abrasive grains will form an independent space according to the abrasive grain direction circle. The upper and lower boundaries of the abrasive grain cross section should not exceed this area, otherwise it will be regarded as interference with adjacent abrasive grains and invalid abrasive grains. Fig.10 (e) and Fig.10As shown in (f), when the vertex angle of the abrasive is too large or the orientation of the abrasive is too close to the critical orientation, the upper and lower boundary lines of the triangular cross section will exceed the effective range, forming invalid abrasives. The geometric relationship between the critical vertex angle value and the orientation of the abrasive needs to be guaranteed: the angle between the line connecting the upper and lower critical points of the abrasive circle to the vertex of the triangle and the line connecting the abrasive position point to the vertex of the triangle is not less than half of the vertex angle of the triangle. The cross-sections of truncated cone abrasives and prism-shaped abrasives are both isosceles trapezoids, which can be regarded as the situation where the vertex position of the isosceles triangle is worn. Its parameters are similar to the parameters of the triangular cross section and will not be repeated.

[0229] 2. Grinding wheel wear rate

[0230] like Fig.11 As shown, the abrasive grains of the grinding wheel are connected to other abrasive grains or the grinding wheel matrix through a binder. The wear of the grinding wheel during the grinding process mainly includes oxidation wear, adhesive wear, abrasive grain shedding, abrasive grain crushing and abrasive grain abrasion wear. Among the various wear forms of the grinding wheel, adhesive wear has a limited effect on the microscopic geometric parameters of the abrasive grains, abrasive grain shedding rarely occurs under normal working conditions, and abrasive grain crushing and fracture will improve the self-sharpening of the grinding wheel to a certain extent and restore the grinding performance of the grinding wheel, while abrasion wear dominates the wear of the grinding wheel and is the main factor affecting the surface quality of the workpiece. Therefore, this embodiment mainly studies the abrasion wear of the grinding wheel.

[0231] In order to quantitatively express the abrasive wear and tear, considering the random characteristics of the grinding wheel abrasives and the uncontrollable factors in the wear process, this embodiment makes the following assumptions: ① The abrasive wear and tear occurs in the axial direction of the abrasive cone; ② The geometric shape of the unworn part of the abrasive remains unchanged.

[0232] like Fig.12 The bottom surface of the frustum abrasive is the contact surface between the abrasive and the workpiece during grinding. When the grinding wheel is worn, the height of the abrasive becomes smaller and the contact bottom surface becomes larger. The height of the abrasive before the grinding wheel is worn is the protrusion height, the remaining height after wear is defined as the residual height, and the height of the part removed by grinding is defined as the wear height. The abrasive wear rate ε is defined as the ratio of the abrasive wear height to the protrusion height of the abrasive before grinding:

[0233]

[0234] In the formula, h w Indicates the abrasive wear height, h i Indicates the protruding height of abrasive particles before wear, h r Represents the residual height of the abrasive grain after wear. The wear height of the abrasive grain is calculated by the protruding height of the abrasive grain before grinding and the residual height of the abrasive grain after grinding. In addition to the residual height of the abrasive grain, other geometric parameters can also be solved according to the abrasive wear conditions:

[0235] L2=L1+2h wtanα g

[0236] Where L2 is the diameter of the contact bottom surface after grinding, L1 represents the diameter of the initial wear plane of the abrasive before processing, which is generally 2%-3% of the protruding height of the abrasive before grinding, and α g Indicates the grinding front angle of the abrasive grain. w Expressed in terms of directly measurable height values:

[0237] L2=L1+2(h i ―h r )tanα g

[0238] It can be seen from the formula that the abrasive wear rate can be determined by measuring the residual height distribution of the grinding wheel after grinding; the geometric dimensions of the abrasive in the current wear state can be determined by the abrasive wear rate.

[0239] Specifically, in this embodiment, measuring the grinding wheel abrasion wear rate includes the following steps.

[0240] (1) Before the grinding wheel is worn, the distribution law of the protruding height of the grinding wheel abrasive is fitted; after the grinding wheel is worn, the distribution law of the residual height of the grinding wheel abrasive is fitted.

[0241] Similar to the protrusion height before grinding, the distribution frequency of the residual height of the abrasive particles after grinding obeys a certain distribution. The abrasive wear height can be solved by the protrusion height before grinding and the residual height after grinding. However, the direct calculation between the probability density functions of the protrusion height before grinding and the residual height after grinding needs to consider the independence of the two events, and the independence of the height of the grinding wheel abrasive particles before and after grinding is difficult to directly prove. Therefore, according to the height distribution and the total number of abrasive particles, the abrasive particle protrusion height sequence and the abrasive particle residual height sequence matrix are established respectively. The abrasive wear height is solved by calculating the corresponding elements in the matrix sequence, avoiding the discussion of the independence between the probability density functions. The calculation process is as follows: Fig.13 shown.

[0242] In order to explain the entire calculation process and steps more clearly, a set of data measured in the experiment is used as an example to calculate the wear rate. The abrasive height distribution of the grinding wheel grinding surface before and after wear is measured, and the Gaussian curve is used to fit the measurement results and calculate the fitting residual, such as Fig.14 As shown. The Gaussian distribution fitting effect of the protrusion height distribution before grinding and the residual height distribution after grinding is good, and the fitting coefficient R2 is 0.997 and 0.995 respectively. The residual value is very small, which proves the effectiveness of using the Gaussian probability density function to fit the abrasive height. That is, the protrusion height of the grinding wheel abrasive grains obeys the Gaussian distribution:

[0243]

[0244] Where: h iis the protrusion height of any abrasive grain; a, μ and σ are the peak value of the Gaussian distribution curve, the protrusion height value corresponding to the peak value and the standard deviation, respectively, which are obtained by measuring the microscopic morphology of the working surface of the grinding wheel.

[0245] The residual height of the grinding wheel abrasive grains follows the Gaussian distribution:

[0246]

[0247] Where: h r is the residual height of any abrasive particle.

[0248] (2) According to the distribution law of the protruding height of the grinding wheel abrasive grains, elements are filled in the grinding wheel matrix to obtain the protruding height distribution matrix of the grinding wheel abrasive grains; according to the distribution law of the residual height of the grinding wheel abrasive grains, elements are filled in the grinding wheel matrix to obtain the residual height distribution matrix of the grinding wheel abrasive grains.

[0249] (3) The difference between the protruding height distribution matrix of the grinding wheel abrasive grains and the residual height distribution matrix of the grinding wheel abrasive grains is used to obtain the grinding wheel abrasive grain wear height distribution matrix:

[0250] [h w ]=[h i ]―[h r ]

[0251] In the formula, [h i ] represents the protruding height distribution matrix of grinding wheel abrasive grains; [h r ] represents the residual height distribution matrix of grinding wheel abrasive particles; [h w ] represents the grinding wheel wear height matrix.

[0252] (4) The grinding wheel abrasive wear height distribution matrix is ​​used to solve the grinding wheel wear rate:

[0253] ε=mean(([h i ]―[h r ]) / [h i ])

[0254] Where ε represents the wear rate of the grinding wheel; mean(·) represents the mean function; [h i ] represents the protruding height distribution matrix of grinding wheel abrasive grains; [h r ] represents the grinding wheel abrasive residual height distribution matrix.

[0255] In order to simplify the calculation, this embodiment assumes that the total number of abrasive grains does not change before and after grinding. Therefore, the total number of abrasive grains on the grinding wheel is represented by the number of grinding wheels measured before grinding:

[0256] d g =15.2M ―1

[0257] Where, d g It represents the average diameter of the abrasive particles in mm, and M represents the total number of abrasive particles in the area.

[0258] (5) Repeat steps 22) to 24) to calculate the grinding wheel wear rate respectively;

[0259] (6) The average value of all the grinding wheel wear rates calculated is taken as the final grinding wheel wear rate.

[0260] The elements in each sequence are classified according to the numerical interval and the total number of wear particles in each interval is counted, such as Fig.15 As shown. It can be seen from the figure that the maximum value in the protrusion height sequence before grinding is about 90μm, and the highest frequency intervals are 30-40μm and 40-50μm. In contrast, the maximum value in the residual height sequence after grinding is 60μm, and the highest frequency interval is 20-40μm. The distribution of values ​​in the protrusion height sequence before grinding is relatively more even, while the values ​​in the residual height sequence after grinding are more concentrated. Except for the highest frequency interval, the frequencies of other intervals are less than half of the highest interval frequency. Therefore, after grinding, the size of the abrasive particles tends to be below a certain threshold, the number of abrasive particles exceeding the critical value drops sharply, and the number of abrasive particles in the interval near the critical value will be more than before grinding.

[0261] At the same time, when generating the respective sequences, it cannot be guaranteed that each element value in the protrusion height sequence is greater than the corresponding value in the residual height sequence. The elements in the wear and tear sequence obtained by subtracting the two sequences will have negative values. This element only represents the calculated value in the statistical probability, not the actual value of the wear and tear, and does not affect the calculation of the wear and tear rate. The calculation results are as follows: Fig.16 As shown. Fig.17 It can be seen that under this working condition, the abrasive wear is 17.2μm and the wear rate ε is 36.7%. In addition, in order to minimize the impact of randomness in data generation, the same algorithm is used to repeat the calculation multiple times and calculate the average value, such as Fig.17 shown.

[0262] (7) Use the grinding wheel wear rate to solve the contact bottom surface size after the grinding wheel abrasive wear:

[0263] [L2]=[L1]+2ε[h i ]tanα g

[0264] Wherein, [L2] represents the diameter distribution matrix of the contact bottom surface of the grinding wheel after abrasive wear; [L1] represents the diameter distribution matrix of the initial wear plane of the abrasive before processing; α g Indicates the grinding front angle of the abrasive.

[0265] 3. Grinding wheel grinding ratio measurement

[0266] The grinding ratio defined in traditional grinding theory, also abbreviated as G value, is a commonly used parameter for evaluating grinding wheel wear. It represents the ratio of the amount of material removed from the workpiece to the amount of wear on the grinding wheel. The grinding ratio represents the grinding efficiency of the grinding wheel. The larger the grinding ratio G, the more workpiece material can be removed under the same amount of grinding wheel wear, which means the better the wear resistance of the grinding wheel. The grinding ratio varies greatly under different grinding wheels and different processing conditions. Under extreme conditions, the grinding ratio G may be less than 1 or greater than 60,000.

[0267] According to the definition of grinding ratio, its calculation method can be known:

[0268]

[0269] Where V w Indicates the volume of workpiece material removed, V S Indicates the volume of grinding wheel material removed.

[0270] like Fig.18 As shown in (a), in the fine grinding of profile grinding, the volume of workpiece material removed is:

[0271] V w =C1C2nbL ai a p

[0272] Where, L ai The total effective involute length of the gear machining is:

[0273]

[0274] In the formula, and Respectively represent the involute angles corresponding to the effective involute end point and starting point.

[0275] The calculation method of material removal volume of grinding wheel in plane / external cylindrical grinding is similar to that of gear material removal volume. Fig.18 As shown in (b), ignoring the grinding wheel corner, the volume of material removed by the grinding wheel is:

[0276] V S =πb s (2r s ―Δr)Δr

[0277] Where b s is the surface grinding wheel width, r s is the radius of the grinding wheel before grinding, and Δr is the normal wear of the grinding wheel.

[0278] Therefore, for surface grinding, the volume of material removed by the grinding wheel can be determined by measuring the outer diameter of the grinding wheel before and after wear, and the grinding ratio can be calculated. However, for the grinding wheel used in profile grinding, Fig.18 As shown in (c), the normal vector direction of each position of the grinding wheel has an angle with the radial direction of the grinding wheel, making it difficult to determine the corresponding position of the same part of the grinding wheel before and after wear, and it is impossible to measure or calculate the normal wear amount Δr at any position of the grinding wheel.

[0279] On the other hand, the grinding wheel is composed of abrasive grains and a matrix, and both the abrasive grains and the matrix will be consumed during grinding. When calculating the grinding ratio, the macroscopic wear of the grinding wheel is considered as a whole, and the consumed matrix will also be included in the volume of the removed grinding wheel material, but the grinding wheel matrix only serves as a carrier of the abrasive grains and does not participate in the grinding process and material removal. Similarly, the pores in the grinding wheel are not eliminated during the calculation, resulting in the calculated grinding wheel removal material volume including the grinding wheel material that does not participate in cutting except for the abrasive grains. Therefore, the traditional method of calculating the grinding ratio of the grinding wheel cannot fully reflect the wear resistance of the abrasive material, such as Fig.19 (a) shown.

[0280] In view of this, this embodiment improves the measurement of the grinding ratio of the grinding wheel based on the grinding wheel multi-information fusion model. In this embodiment, the steps of measuring the grinding ratio of the grinding wheel are as follows.

[0281] (1) According to the effective grinding area of ​​the grinding wheel in contact with the workpiece during grinding, the elements corresponding to the effective grinding area of ​​the grinding wheel are extracted from the grinding wheel matrix to form a grinding wheel grinding matrix. The effective grinding area of ​​the grinding wheel is an annular area located within the grinding surface of the grinding wheel, and the inner radius and outer radius of the effective grinding area of ​​the grinding wheel are respectively:

[0282]

[0283]

[0284] Where: r′ Nf and r′ Na Respectively represent the inner radius and outer radius of the effective grinding area of ​​the grinding wheel; d Nf With d Na Respectively represent the diameters of the starting and ending points of the effective involute of the gear; Y D is the center distance between the gear and the grinding wheel in the radial direction;

[0285] The grinding wheel grinding matrix is ​​formed by extracting the grinding wheel matrix elements located between the inner circle radius and the outer circle radius of the effective grinding area of ​​the grinding wheel.

[0286] (2) Before the grinding wheel is worn, the distribution law of the protruding height of the grinding wheel abrasive grains is fitted; after the grinding wheel is worn, the distribution law of the residual height of the grinding wheel abrasive grains is fitted. According to the above, the distribution laws of the protruding height of the grinding wheel abrasive grains and the residual height of the grinding wheel abrasive grains both conform to the Gaussian distribution. The protruding height of the grinding wheel abrasive grains obeys the Gaussian distribution:

[0287]

[0288] Where: h i is the protrusion height of any abrasive grain; a, μ and σ are the peak value of the Gaussian distribution curve, the protrusion height value corresponding to the peak value and the standard deviation, respectively, which are obtained by measuring the microscopic morphology of the working surface of the grinding wheel.

[0289] The residual height of the grinding wheel abrasive grains follows the Gaussian distribution:

[0290]

[0291] Where: h r is the residual height of any abrasive particle.

[0292] (3) According to the distribution law of the protruding height of the grinding wheel abrasive grains, elements are filled in the grinding wheel grinding matrix to obtain the protruding height distribution matrix of the grinding wheel abrasive grains; according to the distribution law of the residual height of the grinding wheel abrasive grains, elements are filled in the grinding wheel grinding matrix to obtain the residual height distribution matrix of the grinding wheel abrasive grains.

[0293] (4) The difference between the protruding height distribution matrix of the grinding wheel abrasive grains and the residual height distribution matrix of the grinding wheel abrasive grains is used to obtain the wear height distribution matrix of the grinding wheel abrasive grains:

[0294] [h w ]=[h i ]―[h r ]

[0295] Where: [h w ] represents the residual height distribution matrix of grinding wheel abrasive particles; [h i ] represents the protruding height distribution matrix of grinding wheel abrasive grains; [h r ] represents the abrasive wear height distribution matrix of grinding wheel.

[0296] (5) Solve the wear volume of the abrasive corresponding to each element in the grinding wheel grinding matrix, obtain the sum of the wear volumes of all abrasives in the grinding wheel grinding matrix, and obtain the grinding wheel grinding ratio.

[0297] The wear volume of the abrasive particles corresponding to any element is:

[0298] V′ s_i =f(h w ,L1,α g )

[0299] Where: V′ s_i represents the wear volume of any abrasive particle; h w represents the abrasive wear height; L1 represents the diameter of the initial wear plane of the abrasive; α g represents the grinding front angle of the abrasive particle; f(·) represents the abrasive wear volume solution function.

[0300] like Fig.19 As shown in (b), according to the defined geometric shape of the abrasive and the abrasive wear hypothesis, the part of the abrasive worn away is a part of the frustum geometry. The volume of the worn part can be solved by the wear height of the abrasive and the sum of all the abrasives on the grinding surface of the grinding wheel:

[0301]

[0302] Where L1 and L2 are the contact bottom diameters of the abrasive before and after grinding, respectively.

[0303]

[0304] Where: G' represents the grinding ratio of the grinding wheel; V w Represents the workpiece material removal volume; V′ s It represents the sum of the wear volumes of all abrasive particles in the grinding matrix of the grinding wheel.

[0305] (6) looping through steps 23) to 25) to calculate the grinding ratio of the grinding wheel;

[0306] (7) The average value of all grinding wheel grinding ratios calculated is taken as the final grinding wheel grinding ratio.

[0307] Due to the random characteristics of the grinding wheel abrasive, it is difficult to determine the geometric shape parameters of a specific abrasive, but the protruding height and cross-sectional bevel angle of all abrasives obey a certain distribution. Therefore, similar to the calculation process of the grinding wheel wear rate, the corresponding geometric parameter sequence is established according to the distribution frequency of the measured microscopic geometric information of the abrasive. The parameters of any abrasive are represented and calculated by the elements in the abrasive geometric parameter sequence. The calculation of a sequence will also be repeated many times (more than 100 times) and the average value is calculated to reduce the error generated when generating random numbers.

[0308] 4. Double bevel grinding wheel surface topography measurement and model verification

[0309] Through the macro-geometric modeling of the grinding wheel, the geometric contour of the grinding surface of the grinding wheel is established. On this basis, the abrasive information of the grinding wheel is gradually input to realize the macro-micro modeling of the grinding wheel, as shown in 20. For the convenience of drawing, the abrasive particles are distributed on a line in the figure, which does not represent the actual abrasive particles and the abrasive particle position distribution in the model. A multi-information fusion matrix of the grinding wheel abrasive particles is established, but the information such as the height, shape, and orientation of the abrasive particles in the matrix is ​​only the result of parametric modeling, and the specific values ​​are identified by measuring the actual grinding wheel.

[0310] The grinding wheel micro-measurement equipment is the multifunctional friction and wear tester MFT-5000 of Rtec Company, using the white light interferometer module of the equipment and the lens with 10X magnification, such as Fig.21 As shown. The measured grinding wheel is a SG (Seeded Gel) abrasive grinding wheel, and the grinding wheel brand is PSX350*25*127 3SG 60 45m / s. The grinding surface of the grinding wheel is a complex curved surface. Therefore, a height pad is used to adjust the grinding wheel measurement posture to make the measurement point as horizontal as possible in the tangent direction of the involute surface. The surface morphology measurement result data of the grinding wheel is processed and analyzed using the software Gwyddion.

[0311] 4.1 Measurement results of normal protrusion height distribution of abrasive particles

[0312] The normal height distribution measurement results of the grinding wheel surface are as follows: Fig. 22 As shown. The measurement range is 0.9mm×1.2mm. Due to the randomness of the grinding wheel itself, there is a certain deviation between the measurement results. There may be different measurement results with more abrasive particles or more substrates. Multiple areas are measured and the measurement results of the same involute position are averaged. The abrasive particle height is obtained based on the measurement results, and the distribution frequency of the abrasive particle protrusion height is obtained through static analysis, as shown in Fig.23 As shown in the figure, the measurement results of different measuring points of the grinding wheel are similar. The fitting parameters and determination coefficient results based on the mathematical relationship of Gaussian distribution are shown in Table 1.

[0313] Table 1 Measurement results and parameter fitting

[0314]

[0315] The distribution frequency of the protrusion height of the abrasive particles measured in a smaller area is similar to the global distribution frequency within the measurement range of 0.9 mm × 1.2 mm, such as Fig.24 It is proved that the measurement results in a smaller measurement area can be extended to all abrasive particles in the measurement area. Therefore, the rules obtained within the measurement range can be extended to the entire grinding surface of the grinding wheel.

[0316] 4.2 Measurement results of abrasive particle orientation and geometric parameters

[0317] The three-dimensional morphology measurement results of the grinding wheel surface, such as Fig.25(a)-(b) are shown. The orientation of the abrasive grain is the same as the direction of the measured line of sight, indicating that the orientation of the abrasive grain is the normal direction of the measuring surface of the grinding wheel. The grinding surface of the profile grinding wheel is an involute surface, and the normal direction vectors at different involute angle positions are different. Therefore, although the orientation of the abrasive grain is the normal direction of the abrasive grain position, the orientation of the abrasive grains at different positions is different. In addition, in the local magnified view, the abrasive grains are screened from the measurement results based on the abrasive grain height and height change rate. In addition, according to the local magnified view of the grinding wheel morphology, the microscopic geometric shape of the abrasive grain is approximately a truncated cone geometry, as shown in FIG. Fig.25 (b) The main parameters of the frustum are the size of the contact bottom surface, the height of the abrasive protrusion and the top angle of the frustum section. The bevel angle α of the frustum-shaped abrasive g The measured normal height change rate is calculated. According to the local measurement results of the normal height, the abrasive particle apex and the abrasive particle bottom position are connected and the abrasive particle bevel angle is calculated by the height difference and the distance difference in the horizontal direction, such as Fig.26 shown.

[0318] As can be seen from the figure, similar to the distribution of the protrusion height of the abrasive grains, the distribution of the cross-sectional bevel angle of the frustum abrasive grains also obeys the Gaussian density function distribution. Therefore, the cross-sectional bevel angle sequence of the abrasive grains is established using a method similar to the protrusion height.

[0319] 4.3 Grinding wheel model verification

[0320] Some parameters in the abrasive model are determined by the grinding wheel measurement results, and the results are filled into the abrasive grinding wheel abrasive multi-information fusion matrix. The abrasive height sequence is established based on the measured normal height distribution of the abrasive, and then the height elements in the sequence are filled into the fusion matrix. The grinding wheel model is compared with the measurement results in terms of the distribution density of the abrasive height elements and the surface roughness of the grinding wheel to verify the effectiveness and accuracy of the model. For the abrasive protrusion height distribution: Taking a group of experimental grinding wheel height distribution measurement results as an example, Gaussian fitting is performed and the fitting residual is calculated. The abrasive protrusion height sequence is generated according to the fitting results, such as Fig. 27 shown.

[0321] As can be seen from the figure, the distribution frequency of the abrasive protrusion height is highly consistent with the Gaussian probability density function, and the fitting determination coefficient R2 is 0.996, which reflects the effectiveness and reliability of the fitting. According to the proposed height sequence generation method, the abrasive protrusion height sequence is established, and the distribution frequency of the sequence elements is obtained through static analysis. The distribution frequency of the abrasive height sequence has a high degree of overlap with the Gaussian probability density function obtained by fitting, and the data trends are the same, which is consistent with the fitting results, reflecting the effectiveness of the height sequence generation method. The effectiveness of the grinding wheel model is proved from the aspect of the abrasive protrusion height. For the surface morphology of the grinding wheel: According to the definition of surface roughness in ISO4287-1997 (corresponding to the national standard GB / T3505-2009), the surface average roughness of the established grinding wheel model is calculated and compared with the measurement results, as shown in the figure. Fig.28 shown.

[0322] As can be seen from the figure, the simulation and measurement results are close in the extreme values ​​of normal height on the sampling length, and the peak-to-valley frequencies are similar. The surface roughness of the simulation model is Ra5.1, and the surface roughness measurement result of the grinding wheel is Ra4.8. The results are close, which can reflect the surface roughness state of the grinding wheel and prove the effectiveness of the grinding wheel model from the aspect of the grinding wheel surface morphology.

[0323] 5. Conclusion

[0324] This embodiment carries out macro-micro geometric modeling of double-beveled grinding wheels considering abrasive wear. First, a macro-geometric model of the grinding surface of the grinding wheel based on the involute cylindrical gear is established; then, for the micro-geometric model of the grinding wheel abrasive particles, the three-dimensional position information of the abrasive particles is first converted into discrete two-dimensional side view information, and then converted into a mathematical matrix expression, and other micro-geometric parameters such as the normal protrusion height, orientation, shape, and size of the abrasive particles are considered, and a multi-information fusion matrix calculation method for the abrasive particles is proposed. Without increasing the matrix size and the amount of element data, the simultaneous expression of multiple types of abrasive particle information is achieved, the amount of model data is reduced, the calculation efficiency is improved, and the micro-geometry and distribution of the abrasive particles are efficiently and accurately expressed. In addition, in terms of grinding wheel wear, a quantitative calculation method for abrasive wear rate was proposed based on the micro-geometric model of abrasive particles, and a normal height matrix sequence of abrasive particles before and after wear was established, avoiding the discussion of the independence of probability density function; an improved grinding ratio calculation method based on the volume of material removed by abrasive particles was proposed, which does not require the measurement of the macro-normal wear of double-beveled grinding wheels, and also avoids the influence of grinding wheel matrix materials and pores on the calculation results. Compared with the traditional grinding ratio calculation method, it can more intuitively reflect the anti-wear performance of grinding wheel abrasive particles. Finally, the normal vector direction, geometric shape, height distribution frequency and number of abrasive particles were determined through grinding wheel measurement tests, and the normal height distribution frequency of abrasive particles and surface roughness were compared with the measurement results to verify the effectiveness and accuracy of the grinding wheel model. Compared with other grinding wheel models, the proposed model contains more comprehensive micro-geometric information of abrasive particles, can reflect the random characteristics of grinding wheel abrasive particles, has higher accuracy, and reduces the amount of model data through multi-information fusion matrix, and improves the calculation efficiency of the model.

[0325] The above-described embodiments are only preferred embodiments for fully illustrating the present invention, and the protection scope of the present invention is not limited thereto. Equivalent substitutions or changes made by those skilled in the art based on the present invention are within the protection scope of the present invention. The protection scope of the present invention shall be subject to the claims.

Claims

1. A method for modeling a grinding wheel multi-information fusion model considering the random characteristics of abrasive particles, characterized by: The steps include: Step 1: Construct the macro-geometric model of the grinding wheel grinding surface The conjugate contact relationship between the grinding surface of the grinding wheel and the grinding surface of the workpiece is used, and the grinding surface profile equation of the workpiece is used to obtain the grinding surface profile equation of the grinding wheel, and the macro-geometric model of the grinding surface of the grinding wheel is established; Step 2: Construct a micro-geometric model of the grinding wheel abrasive 21) Using the macro-geometric model of the grinding wheel grinding surface, establish a three-dimensional expression of the grinding wheel grinding surface profile, and perform a three-dimensional expression of any abrasive grain position on the grinding wheel grinding surface; 22) Expand the grinding wheel grinding surface into a two-dimensional annular plane along the axial side view, convert the three-dimensional expression of the grinding wheel grinding surface profile into a two-dimensional expression of the grinding wheel grinding surface side view, and express any abrasive grain position on the grinding wheel grinding surface in two dimensions; wherein, the difference between the radius of any point A on the two-dimensional annular plane and the inner circle radius of the two-dimensional annular plane is equal to the length of the curve segment between point A and the inner end point of the curve intercepted by the plane passing through the grinding wheel grinding surface point A′ and the grinding wheel axis on the grinding wheel grinding surface, and any point A on the two-dimensional annular plane is obtained by expanding the point A′ on the grinding wheel grinding surface along the axial side view; 23) Expand the two-dimensional annular plane into an isosceles trapezoidal plane along the circumferential direction and perform numerical discretization, and convert the two-dimensional side view expression of the grinding wheel grinding surface into a grinding wheel matrix expression; Step 3: Construct the micro-geometric parameter model of the grinding wheel abrasive Elements are filled in the grinding wheel matrix according to the random distribution law of abrasive particles. The actual spatial position of the abrasive particles is expressed by the number of rows and columns of the elements in the grinding wheel matrix. The abrasive particle information is expressed by the size of the elements. The abrasive particle information includes the protruding height of the abrasive particles, the shape of the abrasive particles and the direction of the abrasive particles.

2. The method for building a grinding wheel multi-information fusion model considering the random characteristics of abrasive particles according to claim 1 is characterized in that: The workpiece is an involute spur gear, and the grinding surface of the involute spur gear is the tooth surface. The grinding surface profile equation of the grinding wheel obtained by the tooth surface is: in: is the involute angle parameter, and are the upper and lower limits of the parameters respectively; r b is the base circle radius of the gear: <h2 style=";text-align:left;direction:ltr">r<h2 style=";text-align:left;direction:ltr"> b <h2 style=";text-align:left;direction:ltr"> (mzcosα) Where: m is the gear module, z is the number of gear teeth, and α is the pressure angle.

3. The method for building a grinding wheel multi-information fusion model considering the random characteristics of abrasive particles according to claim 2 is characterized in that: In step 21), the three-dimensional expression W1 of any abrasive grain position on the grinding surface of the grinding wheel is: in: is the gear involute angle parameter corresponding to the abrasive position; θ is the rotation angle parameter of the abrasive position on the circumference of the grinding wheel.

4. The method for building a grinding wheel multi-information fusion model considering the random characteristics of abrasive particles according to claim 3 is characterized in that: In step 22), the two-dimensional expression W2 of the position of any abrasive grain on the grinding surface of the grinding wheel is: W2=f(R,θ) Where R is the radius of the abrasive particle in the two-dimensional annular plane; and: Where: Y D Y is the distance between the grinding wheel center and the origin of the gear coordinate system in the Y direction; end is the Y coordinate value of the tooth top; is the involute angle parameter corresponding to the starting point of the involute; and: in: is the involute angle parameter of the tooth addendum.

5. The method for building a grinding wheel multi-information fusion model considering the random characteristics of abrasive particles according to claim 4 is characterized in that: In the step 23), the conversion relationship between the side view two-dimensional expression W2 of the grinding wheel grinding surface and the grinding wheel matrix expression W3 is: Where: i w and j w Respectively represent the number of rows and columns of the grinding wheel matrix; Δi w and Δj w are the row spacing and column spacing between adjacent abrasive particles, respectively, which are determined by the number of abrasive particles per unit area; δ(i w ) is the distance between the grinding wheel matrix element boundary and the maximum width position of the matrix; and: The maximum number of rows in the grinding wheel matrix is: i wmax =1+r b (((z+2) / zcosα) 2 ―1) / 2Δi w For the solved i wmax Perform rounding; The maximum number of columns in the grinding wheel matrix is: For the solved j wmax Round up.

6. The method for building a grinding wheel multi-information fusion model considering the random characteristics of abrasive particles according to claim 5 is characterized in that: In the step 3, when filling elements in the grinding wheel matrix, from i w =1,j w =1 starts filling the matrix elements; If satisfied: d(i w ) / Δj w ≤j w ≤j wmax ―2δ(i w ) / Δj w Then there are abrasive particles at the position corresponding to the corresponding matrix element, otherwise there are no actual abrasive particles at the position corresponding to the corresponding matrix element.

7. The method for building a grinding wheel multi-information fusion model considering the random characteristics of abrasive particles according to claim 6 is characterized in that: The protruding height of the abrasive particles follows a Gaussian distribution: Where: h i is the protrusion height of any abrasive grain; a, μ and σ are the peak value of the Gaussian distribution curve, the protrusion height value corresponding to the peak value and the standard deviation, respectively, which are obtained by measuring the microscopic morphology of the working surface of the grinding wheel.

8. The method for building a grinding wheel multi-information fusion model considering the random characteristics of abrasive particles according to claim 6, characterized in that: Abrasive grain orientation angle The scope meets: in: and The abrasive particle orientation angles are minimum and maximum values ​​of ; and: Where: x g ,y g is the coordinate point of the abrasive particle space position; and y oc are the coordinates of the two intersection points of a circle with the spatial position of the abrasive grain as the center and the protruding height of the abrasive grain as the radius and the tooth profile involute, which are obtained by combining the grinding wheel grinding surface profile equation and the parametric equation of the abrasive grain radius circle, and the parametric equation of the abrasive grain radius circle is: Where: x oc ,y oc is the coordinate point on the abrasive particle radius circle; is the parameter variable of the equation of the wear particle radius circle, and and They are The minimum and maximum values ​​of .

9. The method for building a grinding wheel multi-information fusion model considering the random characteristics of abrasive particles according to claim 7, characterized in that: The shapes of abrasive particles are conical and quadrangular pyramidal. The cross section of abrasive particles is triangular, and the vertex of the triangle is located on the radius circle of the abrasive particles. Its position is determined by the orientation of the abrasive particles: x tv =x g +h i cosθ i and tv =and g +h i sinθ i Among them, x tv and tv The vertex coordinates of the abrasive cross-section triangle; θ i is the orientation angle of any abrasive particle; h i is the protrusion height of any abrasive grain; The upper boundary line of the triangular cross-section abrasive is represented by the abrasive vertex angle and the abrasive orientation direction: Where: x tt and tt are the coordinate values ​​of the boundary lines on the triangular cross section; t tt is the parameter variable of the upper boundary line equation, t tt_max t tt The maximum value of is the direction vector of the upper boundary line; and The lower boundary line of the triangular cross-section abrasive is represented by the abrasive vertex angle and the abrasive orientation direction: Where: x tb and tb are the coordinates of the lower boundary line of the triangular cross section; t tb is the parameter variable of the lower boundary line equation, t tb_max t tb The maximum value of is the direction vector of the lower boundary line, and: Where: θ v is the vertex angle of the triangular cross section.

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