CN111368470A - Modeling method for rough surface - Google Patents
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Abstract
The invention provides a modeling method of a rough surface, which comprises the following steps: establishing a rough surface infinitesimal model: determining the appearance of the surface micro-elements; determining the value of the characteristic parameter of the rough surface infinitesimal; establishing a relative surface equation of the coarse surface infinitesimal; and establishing a plurality of rough surface micro-elements distributed in an array on the surface to be modeled to obtain a surface equation of the rough surface comprising the rough surface micro-elements. The rough surface model established by the invention can embody the non-stable random change of the rough surface contour height; meanwhile, the scale of the rough surface is controllable, the multiple rough surfaces are convenient to lap, a large-range rough surface model can be formed by combination, and the actual data volume of the model is small; the technical scheme of the invention is more suitable for representing different roughness of a single surface partition.
Description
Technical Field
The invention relates to the technical field of tribology or surface engineering or the field of rough surface modeling, in particular to a modeling method of a rough surface.
Background
The influence of the surface roughness on the performances of the fatigue life, the corrosion resistance and the like of the material and the coupling synergistic effect of the surface roughness and the surface texture are always the research hotspots of academic and engineering circles, and the establishment of a rough surface model according to the surface roughness is the key of the research. Common rough surface characterization methods include a sine and cosine curve characterization method and a fractal function characterization method, wherein a rough surface model made by the sine and cosine curve characterization method is too regular and cannot reflect non-stationary random changes of the contour height of a rough surface; the rough surface model made by the fractal function characterization method also has the problems, and although the random coefficient is introduced into the variant fractal function, the data volume of the rough surface model is large, which is not beneficial to the theoretical calculation of the model.
Chinese patent discloses a rough surface parameter characterization method with a surface microstructure, which adopts fractal function W-M function to characterize the rough surface without a texture area, specifically
Wherein A isnAnd BnRelatively independent and obey [0, 2 pi ]]Uniformly distributed random numbers. Although the characterization method is a surface random design according to the surface roughness, the random mode tends to be uniform and does not conform to the non-stationary random law of the rough surface; the roughness scale is small, and the data volume of the rough surface model established by the method is large, so that the method is not suitable for representing a large-range surface, and the calculation difficulty of a theoretical model is greatly increased; meanwhile, for the design of multiple roughness of one surface sub-region, the technical scheme has high implementation difficulty.
Disclosure of Invention
Aiming at the defects in the prior art, the invention provides a rough surface modeling method, and the established rough surface model can embody the non-stable random change of the rough surface contour height; meanwhile, the scale of the rough surface is controllable, the multiple rough surfaces are convenient to lap, a large-range rough surface model can be formed by combination, and the actual data volume of the model is small; the technical scheme of the invention is more suitable for representing different roughness of a single surface partition.
The present invention achieves the above-described object by the following technical means.
A method of modeling a rough surface, comprising the steps of:
establishing a rough surface infinitesimal model: determining the appearance of the surface micro-elements; determining the value of the characteristic parameter of the rough surface infinitesimal; establishing a relative surface equation of the coarse surface infinitesimal;
and establishing a plurality of rough surface micro-elements distributed in an array on the surface to be modeled to obtain a surface equation of the rough surface comprising the rough surface micro-elements.
Furthermore, the coarse surface microelements are square surfaces with the same area range, and the square surfaces comprise microstructures and microstructure edge areas; the micro-structures are micro-bulges or micro-pits; the microstructure edge area is the area between the microstructure edge and the square surface profile.
Furthermore, the microstructure is a micro-protrusion or a micro-pit with a section curve of a parabola or a semicircle or a triangle or a flat-topped parabola or a flat-topped semicircle or a trapezoid.
Further, the characteristic parameters of the surface roughness elements comprise a square surface area range dkAnd height or depth h of the microstructurekWherein K is a sequence number of the rough surface infinitesimal, K is a set of the sequence numbers of the rough surface infinitesimal, and K ∈ K is {1, 2, 3.. Kmax}；kmaxThe maximum serial number of the surface micro element is rough.
Further, the square surface area range dkThe values are specifically as follows:
determining the range d of the square surface area according to the average width Rsm of the profile unit of the roughness on the surface to be modeledk，dkAnd d is Rsm, K ∈ K, wherein Rsm is the average width of the profile elements of the roughness.
Further, the height or depth h of the microstructurekThe determination specifically comprises the following steps: randomly determining the height or depth h of the microstructure from the arithmetic mean deviation Ra of the roughness profile and the root mean square value Rq of the deviation of the roughness relative to the mean line of the profilekThe method comprises the following steps:
the height or depth of the microstructure satisfies the expectation of mu and the variance of sigma2The normal distribution of (1) is given by μ -Ra and σ -Rq
Simultaneous equation red
Solving zeta by approximationkThe height or depth h of the microstructure can be obtainedkSet of (2)
Wherein: μ is an expectation of a normal distribution; σ is the mean square error of normal distribution; h is the set of heights or depths of the microstructures, H ═ Hk|k＝1，2，3...kmax}; f (h) is a probability function of a normal distribution of height or depth h; zetakIs hkThe value coefficient of (a); p is a radical ofkIs hkThe value symbol coefficient of (a); signkIs pkA numeric random number; poskIs a normal distribution probability; pos is the set of normally distributed probabilities,
further, establishing a relative surface equation of the coarse surface infinitesimal, specifically:
establishing a three-coordinate Cartesian coordinate system o of kth rough surface infinitesimalk-xkykzk，
The surface equation of the kth rough surface infinitesimal is:
wherein:
ok-xkykzka three-coordinate cartesian coordinate system for the kth rough surface infinitesimal;
(xk，yk) Is that any point on the k-th rough surface infinitesimal curved surface is at ok-xkykProjection coordinates of the plane;
z＝fk(xk，yk)，(xk，yk)∈Γka surface equation representing the kth rough surface infinitesimal;
z＝hk(xk，yk)，(xk，yk)∈Ωkexpressed as the surface equation of the microstructure in the kth rough surface element;
Γkthe k-th rough surface element is ok-xkykA projection area of the plane;
Ωkthe microstructure in the kth rough surface element is ok-xkykA projected area of the plane.
Further, a plurality of rough surface microelements which are distributed in a matrix are established on the surface to be modeled to obtain a surface equation of the rough surface which comprises the rough surface microelements, and the method specifically comprises the following steps:
assuming that a plurality of rough surface microelements are distributed on the surface to be modeled according to an N-row and M-column array, the total number k of the rough surface microelementsmaxN × M; wherein: n is the total row number of the coarse surface micro-element array; m is the total row number of the coarse surface micro-element array;
establishing a three-coordinate Cartesian coordinate system o-xyz of the surface to be modeled, wherein the kth rough surface element is positioned in the ith row and the jth column of the rough surface element array, and the k
The coordinates (x, y, z) of any point of the surface to be modeled on the o-xyz coordinate system are converted into o of the kth rough surface elementk-xkykzkThe coordinates of the coordinate system are (x)k，ykZ) of said xk＝x-(j-1)d，yk＝y-(i-1)d；
The surface equation for a rough surface comprising a number of rough surface microelements is:
z＝f(x，y)，(x，y)∈Γ＝f(xk+(j-1)d，yk+(i-1)d)＝fk(xk，yk)，(xk，yk)∈Γk，k＝1，2，3...kmax；
wherein:
(x, y, z) is the coordinate of any point on the surface to be modeled on an o-xyz coordinate system;
(xk，ykz) is o converted into the kth rough surface infinitesimal at any point on the surface to be modeledk-xkykzkCoordinates of a coordinate system;
and F is a projection area of the surface to be modeled on an o-xy plane.
The invention has the beneficial effects that:
1. according to the modeling method of the rough surface, the characteristic parameters of the micro elements of the rough surface designed according to the roughness obey normal distribution, and the non-stationary random variation rule of the profile height of the rough surface is met.
2. The rough surface modeling method has the advantages that the scale of the rough surface is controllable, the multiple rough surfaces are convenient to lap, a large-range rough surface model can be formed by combination, and the actual data volume of the model is small.
3. The modeling method of the rough surface is more suitable for characterization of different roughness of a single surface partition.
4. The modeling method of the rough surface can be butted with the existing finite element software and is used for analyzing the influence of the surface roughness on the performances of tribology, mechanics, thermodynamics and the like or analyzing the coupling synergistic effect of the surface roughness and the surface microtexture.
Drawings
Fig. 1 is a flow chart of a method for modeling a rough surface according to the present invention.
FIG. 2 is a schematic diagram of the rough surface micro-elements of the present invention.
Fig. 3 shows the microstructure of the present invention as a microprotrusion with a parabolic cross-sectional profile.
Fig. 4 shows the microstructure of the present invention as a dimple whose cross-sectional curve is parabolic.
FIG. 5 is a schematic diagram of an array of asperity elements on a surface to be modeled in accordance with the present invention.
Fig. 6 shows the rough surface of the microstructure of the present invention, which has a parabolic section and has a roughness Rsm of 0.3 μm, Ra of 0.2 μm, and Rq of 0.05 μm.
FIG. 7 is a cross-sectional profile of a surface to be modeled according to the present invention.
Detailed Description
The invention will be further described with reference to the following figures and specific examples, but the scope of the invention is not limited thereto.
As shown in fig. 1, the method for modeling a rough surface according to the present invention includes the following steps:
s01: establishing a rough surface infinitesimal model, which specifically comprises the following steps: determining the appearance of the surface micro-elements; determining the value of the characteristic parameter of the rough surface infinitesimal; and establishing a relative surface equation of the micro elements of the rough surface.
As shown in fig. 2, the rough surface microelements are square surfaces with the same area range, and the square surfaces include microstructures and microstructure edge areas; the micro-structures are micro-bulges or micro-pits; the microstructure edge area is the area between the microstructure edge and the square surface profile. The microstructure is a micro-bulge or a micro-pit with a section curve of a parabola or a semicircle or a triangle or a flat-topped parabola or a flat-topped semicircle or a trapezoid.
The characteristic parameters of the rough surface elements comprise a square surface area range dkAnd height or depth h of the microstructurekWherein K is a sequence number of the rough surface infinitesimal, K is a set of the sequence numbers of the rough surface infinitesimal, and K ∈ K is {1, 2, 3.. Kmax}；kmaxThe maximum serial number of the surface micro element is rough.
The square surface area range dkThe values are specifically as follows: determining the range d of the square surface area according to the average width Rsm of the profile unit of the roughness on the surface to be modeledk，dkAnd d is Rsm, K ∈ K, wherein Rsm is the average width of the profile elements of the roughness.
Height or depth h of said microstructurekThe determination specifically comprises the following steps: randomly determining the height or depth h of the microstructure from the arithmetic mean deviation Ra of the roughness profile and the root mean square value Rq of the deviation of the roughness relative to the mean line of the profilekThe method comprises the following steps:
the height or depth of the microstructure satisfies the expectation of mu and the variance of sigma2The normal distribution of (1) is given by μ -Ra and σ -Rq
Simultaneous system of equations
Solving zeta by approximationkThe height or depth h of the microstructure can be obtainedkSet of (2)
Wherein: μ is an expectation of a normal distribution; σ is the mean square error of normal distribution; a patch is a collection of heights or depths of microstructures, H ═ Hk|k＝1，2，3...kmax}; f (h) is a probability function of a normal distribution of height or depth h; zetakIs hkThe value coefficient of (a); p is a radical ofkIs hkThe value symbol coefficient of (a); signkIs pkA numeric random number; poskIs a normal distribution probability; pos is the set of normally distributed probabilities,
as shown in fig. 3 and 4, the equation of the relative surface of the rough surface infinitesimal is established, specifically:
establishing a three-coordinate Cartesian coordinate system o of kth rough surface infinitesimalk-xkykzkThe surface equation of the kth rough surface infinitesimal is as follows:
wherein:
ok-xkykzka three-coordinate cartesian coordinate system for the kth rough surface infinitesimal;
(xk，yk) Is that any point on the k-th rough surface infinitesimal curved surface is at ok-xkykProjection coordinates of the plane;
z＝fk(xk，yk)，(xk，yk)∈Γka surface equation representing the kth rough surface infinitesimal;
z＝hk(xk，yk)，(xk，yk)∈Ωkexpressed as the surface equation of the microstructure in the kth rough surface element;
Γkthe k-th rough surface element is ok-xkykA projection area of the plane; gamma-shapedk＝{(xk，yk)|xk∈[0，dk)，yk∈[0，dk)}
ΩkThe microstructure in the kth rough surface element is ok-xkykA projected area of the plane.
qkThe characteristic coefficient of the microstructure in the kth rough surface element,
S02: and establishing a plurality of rough surface micro-elements distributed in an array on the surface to be modeled to obtain a surface equation of the rough surface comprising the rough surface micro-elements.
As shown in FIG. 5, assuming that a plurality of rough surface microelements are distributed on the surface to be modeled according to an N-row and M-column array, the total number k of rough surface microelementsmaxN × M; wherein: n is the total number of rows of the coarse surface micro-element array, and N is not less than 10 and is an integer; m is the total row number of the coarse surface micro-element array, and M is not less than 10 and is an integer;
establishing a three-coordinate Cartesian coordinate system o-xyz of the surface to be modeled, wherein the kth rough surface element is positioned in the ith row and the jth column of the rough surface element array, and the k
The above-mentionedThe coordinates (x, y, z) of any point of the surface to be modeled on the o-xyz coordinate system are converted to o for the kth rough surface elementk-xkykzkThe coordinates of the coordinate system are (x)k，ykZ) of said xk＝x-(j-1)d，yk＝y-(i-1)d；
The surface equation for a rough surface comprising a number of rough surface microelements is:
z＝f(x，y)，(x，y)∈Γ＝f(xk+(j-1)d，yk+(i-1)d)＝fk(xk，yk)，(xk，yk)∈Γk，k＝1，2，3...kmax；
wherein:
(x, y, z) is the coordinate of any point on the surface to be modeled on an o-xyz coordinate system;
(xk，ykz) is o converted into the kth rough surface infinitesimal at any point on the surface to be modeledk-xkykzkCoordinates of a coordinate system;
Γ is a projection area of the surface to be modeled on an o-xy plane, and { (x, y) | x ∈ [0, M × d), y ∈ [0, N × d }.
As shown in fig. 6, the microstructure in the rough surface microelements is a section parabola-shaped micro-protrusion or micro-pit, the roughness value is Rsm is 0.3 μm, Ra is 0.2 μm, and Rq is 0.05 μm, a plurality of rough surface microelements are distributed on the surface to be molded according to an array of 10 rows and 10 columns, and the model process is calculated by MATLAB programming to obtain the rough surface. As shown in fig. 7, the cross-sectional profile of the rough surface unit.
The present invention is not limited to the above-described embodiments, and any obvious improvements, substitutions or modifications can be made by those skilled in the art without departing from the spirit of the present invention.
Claims (8)
1. A method of modeling a rough surface, comprising the steps of:
establishing a rough surface infinitesimal model: determining the appearance of the surface micro-elements; determining the value of the characteristic parameter of the rough surface infinitesimal; establishing a relative surface equation of the coarse surface infinitesimal;
and establishing a plurality of rough surface micro-elements distributed in an array on the surface to be modeled to obtain a surface equation of the rough surface comprising the rough surface micro-elements.
2. The modeling method of the rough surface according to claim 1, wherein the rough surface microelements are square surfaces with the same area range, and the square surfaces comprise microstructures and microstructure edge areas; the micro-structures are micro-bulges or micro-pits; the microstructure edge area is the area between the microstructure edge and the square surface profile.
3. The method for modeling a rough surface according to claim 2, wherein the microstructure topography is a micro-protrusion or a micro-indentation having a cross-sectional curve that is a parabola or a semicircle or a triangle or a plateau parabola or a plateau semicircle or a trapezoid.
4. Method for modelling a rough surface according to claim 2, characterized in that said characteristic parameters of the surface microelements comprise the square surface area range dkAnd height or depth h of the microstructurekWherein K is a sequence number of the rough surface infinitesimal, K is a set of the sequence numbers of the rough surface infinitesimal, and K ∈ K is {1, 2, 3.. Kmax}；kmaxThe maximum serial number of the surface micro element is rough.
5. Method for modelling a rough surface according to claim 4, characterized in that said square surface area dkThe values are specifically as follows:
determining the range d of the square surface area according to the average width Rsm of the profile unit of the roughness on the surface to be modeledk，dkAnd d is Rsm, K ∈ K, wherein Rsm is the average width of the profile elements of the roughness.
6. Method for modelling a rough surface according to claim 4, characterized in that the height or depth h of the microstructure is such that it is equal tokThe determination specifically comprises the following steps: randomly determining the height or depth h of the microstructure from the arithmetic mean deviation Ra of the roughness profile and the root mean square value Rq of the deviation of the roughness relative to the mean line of the profilekThe method comprises the following steps:
the height or depth of the microstructure satisfies the expectation of mu and the variance of sigma2The normal distribution of (1) is given by μ -Ra and σ -Rq
Simultaneous system of equations
k∈K，posk∈ Pos, solving for ζ by approximationkThe height or depth h of the microstructure can be obtainedkSet of (2)
Wherein: μ is an expectation of a normal distribution; σ is the mean square error of normal distribution; h is the set of heights or depths of the microstructures, H ═ Hk|k＝1，2，3...kmax}; f (h) is a probability function of a normal distribution of height or depth h; zetakIs hkThe value coefficient of (a); p is a radical ofkIs hkThe value symbol coefficient of (a); signkIs pkA numeric random number; poskIs a normal distribution probability; pos is the set of normally distributed probabilities,
7. the method for modeling a rough surface according to claim 2, wherein the equation for the relative surface of the rough surface is established as follows:
establish the kth coarseThree-coordinate cartesian coordinate system o of coarse surface infinitesimal elementsk-xkykzk，
The surface equation of the kth rough surface infinitesimal is:
wherein:
ok-xkykzka three-coordinate cartesian coordinate system for the kth rough surface infinitesimal;
(xk，yk) Is that any point on the k-th rough surface infinitesimal curved surface is at ok-xkykProjection coordinates of the plane;
z＝fk(xk，yk)，(xk，yk)∈Γka surface equation representing the kth rough surface infinitesimal;
z＝hk(xk，yk)，(xk，yk)∈Ωkexpressed as the surface equation of the microstructure in the kth rough surface element;
Γkthe k-th rough surface element is ok-xkykA projection area of the plane;
Ωkthe microstructure in the kth rough surface element is ok-xkykA projected area of the plane.
8. The modeling method of a rough surface according to claim 2, characterized in that a plurality of rough surface microelements of matrix distribution are established on the surface to be modeled to obtain a surface equation of the rough surface including the plurality of rough surface microelements, specifically:
assuming that a plurality of rough surface microelements are distributed on the surface to be modeled according to an N-row and M-column array, the total number k of the rough surface microelementsmaxN × M; wherein: n is the total row number of the coarse surface micro-element array; m is the total row number of the coarse surface micro-element array;
establishing a three-coordinate Cartesian coordinate system o-xyz of the surface to be modeled, wherein the kth rough surface element is positioned in the ith row and the jth column of the rough surface element array, and the k
The coordinates (x, y, z) of any point of the surface to be modeled on the o-xyz coordinate system are converted into o of the kth rough surface elementk-xkykzkThe coordinates of the coordinate system are (x)k，ykZ) of said xk＝x-(j-1)d，yk＝y-(i-1)d；
The surface equation for a rough surface comprising a number of rough surface microelements is:
z＝f(x，y)，(x，y)∈Γ＝f(xk+(j-1)d，yk+(i-1)d)＝fk(xk，yk)，(xk，yk)∈Γk，k＝1，2，3...kmax；
wherein:
(x, y, z) is the coordinate of any point on the surface to be modeled on an o-xyz coordinate system;
(xk，ykz) is o converted into the kth rough surface infinitesimal at any point on the surface to be modeledk-xkykzkCoordinates of a coordinate system;
and F is a projection area of the surface to be modeled on an o-xy plane.
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