CN104077440A - Junction surface contact area and rigidity confirming method based on surface fitting - Google Patents

Abstract

Disclosed is a junction surface contact area and rigidity confirming method based on surface fitting. The junction surface contact area and rigidity confirming method based on the surface fitting includes: firstly, using confocal microscopy and three coordinate measuring to measure micro bulge morphology, external waviness and shape error of a mechanical junction surface, performing fitting on obtained point cloud data by using a binary high order function, and accordingly obtaining a final analysis formula of a contact surface; secondly, judging positions and contact directions of micro bulge contact points, then performing Hertz contact calculation on each single contact point, and calculating contact deformation and contact area under functions of force; finally, calculating overall contact area and contact rigidity of the contact surface so as to obtain total contact area and contact rigidity in each direction. Compared with a traditional analysis method, the junction surface contact area and rigidity confirming method based on the surface fitting has the advantage of approaching real morphology. Compared with a finite element method, the junction surface contact area and rigidity confirming method based on the surface fitting can enlarge the contact area which can be calculated.

Description

A kind of definite faying face contact area based on surface fitting and the method for rigidity

Technical field

The invention belongs to mechanical engagement face mechanics field, be specifically related to a kind of definite faying face contact area based on surface fitting and the method for rigidity.

Background technology

In the complex mechanical systems such as lathe, gas turbine, motor car engine, have a large amount of faying faces, they have destroyed the continuity of mechanical system structure, have affected to a great extent the overall performance of mechanical system.The connection performance of faying face presents certain non-linear, can effectively improve the overall performance predictive ability of Design Stage in connection with face characterisitic parameter model drawing-in system complete machine modeling process.Therefore, accurately building Contact characteristics parameter model will provide fundamental basis for complicated machinery parts faying face optimal design.

The real contact area at faying face place and stress distribution are the basic reasons that determines the characteristics such as mechanical bond surface resistance, thermal resistance, contact stiffness.The order of accuarcy calculating will directly affect the foundation of faying face parameter characteristic model.Two surface in contacts of faying face are hackly under micro-scale, be commonly referred to be by the surface of macroshape error, percent ripple and three kinds of yardsticks of roughness and be formed by stacking, therefore in the time that two machining surfaces are in contact with one another, the touching act at faying face place occurs over just on some discrete micro-bulges.And the contact condition of these micro-bulges has determined the characteristics such as the resistance thermal resistance at faying face place.Therefore, be necessary to propose a kind of mathematical method and come rough surface modeling, with accurately calculations incorporated face contact stiffness and contact area, realize the Accurate Prediction of interface properties.

Due to rough surface morphology complexity, micro-bulge out-of-shape, the method that Chinese scholars adopts in the time of the touching act of research faying face microcosmic is at present: by the regular solid simulation such as ball or rotary paraboloid rough peak, actual micro-bulge touching act is reduced to the touching act of these regular geometric bodies.This research to microcosmic touching act is to be based upon in the situation of many assumed conditions to carry out, although model developing, perfect always, but still all there is following problem in existing contact model: 1, micro-bulge shape is assumed to spherical or other simple shapes, loses contact with reality substantially; 2, in the time of the touching act on the whole surface of research, suppose that surface profile height distributes to obey certain function and distributes as normal distribution, employing surfaceness statistical parameter as mean radius of curvature etc., does not have from real surface; 3, be confined to the contact stiffness at coarse scale layer viewpoint faying face place, do not consider that geometric configuration is percent ripple, the affect rule of macroshape error on joint surface contact stiffness.Existing Finite Element Method can adopt real surface measurement data, carries out the research of contact problems, but because FEM (finite element) calculation need to be divided superfine grid, has seriously limited size and the counting yield of reference area.

Summary of the invention

In order to overcome the shortcoming of above-mentioned prior art, the object of the present invention is to provide a kind of definite faying face contact area based on surface fitting and the method for rigidity, can greatly expand reference area, and improve counting yield.

In order to achieve the above object, the technical scheme that the present invention takes is:

Definite faying face contact area based on surface fitting and a method for rigidity, comprise the following steps:

1) measurement of real topography and function representation

1.1) faying face pattern level divide: by Machine Joint Surfaces by ascending roughness, percent ripple, three levels of shape error divided into of wavelength, roughness is regarded in the fluctuating that peak and peak-to-peak spacing is less than to 1mm as, corresponding single projection is regarded micro-bulge as, and supposes that micro-bulge shape is identical; Regard the distance between crest and crest as percent ripple at 1mm to the surface undulation within the scope of 10mm; Surface shape error processing is used as in the surface undulation that distance between crest and crest is exceeded to 10mm scope;

1.2) micro-bulge Function Fitting: adopt Laser Scanning Confocal Microscope observation micro-bulge pattern, the cloud data obtaining is used for matching micro-bulge, carry out matching with binary higher order functionality, press tool sharpening trajectory direction, by machining locus direction be decided to be u to, perpendicular to the direction of machining locus be decided to be v to, each micro-bulge fits to:

$f_{\mathrm{ai}}=f_{\mathrm{ai}}(u,v)=a_i+\underset{t=0}{\overset{1}{\mathrm{\Σ}}}{b}_i{u}^t{v}^{1-t}+\underset{t=0}{\overset{2}{\mathrm{\Σ}}}{c}_i{u}^t{v}^{2-t}+......,((u,v)\∈{\mathrm{\Ω}}_i,i=\mathrm{1,2,3}......)$

Wherein: i represents i micro-bulge on rough surface, Ω _irepresent i the surf zone that micro-bulge is shared;

1.3) percent ripple and shape error matching: adopt three-coordinates measuring machine to measure percent ripple and the shape error of surface of contact, the cloud data of acquisition carries out matching with binary higher order functionality, and surface shape error and percent ripple fit to:

$f_e=f_e(u,v)=a_e+\underset{i=0}{\overset{1}{\mathrm{\Σ}}}{b}_e{u}^i{v}^{1-i}+\underset{i=0}{\overset{2}{\mathrm{\Σ}}}{c}_e{u}^i{v}^{2-i}+......$

1.4) function representation of final curved surface: micro-bulge function is distributed on surface shape error function and is added, obtain the function expression of final real surface pattern:

$f=f(u,v)=\left\{\begin{array}{cc}{f}_{a1}(u,v)+f_e(u,v)& ((u,v)\∈{\mathrm{\Ω}}_1)\\ f_{a2}(u,v)+f_e(u,v)& ((u,v)\∈{\mathrm{\Ω}}_2)\\ f_{a3}(u,v)+f_e(u,v)& ((u,v)\∈{\mathrm{\Ω}}_3)\\ ........& \end{array}\right.$

2) two surface in contact contacting points position judgements and single pre-service to micro-bulge contact

2.1) spacing function expression between two surfaces: two surface in contact functions are subtracted each other, obtain contactinterval formula:

δ＝f ₁(u，v)-f ₂(u，v)

Wherein, f ₁(u, v) represents upper surface fitting function, f ₂(u, v) represents lower surface fitting function,

2.2) contacting points position obtains: contact point (u _c, v _c) satisfied condition is δ (u _c, v _c) be function minimum point, and δ (u _c, v _c), ask method can try to achieve all possible contact point according to minimal value, then according to δ (u _c, v _c) < 0 judges, can draw have point of contact;

2.3) contact point direction solves and contact point coordinate transform: the volume coordinate of establishing contact point is (x (u _c, v _c), y (u _c, v _c), z (u _c, v _c)), contact direction solves by differential geometric method:

$\stackrel{\&RightArrow;}{r_u^{\&prime;}}=\left(\begin{array}{ccc}{x}_{u_c}^{\&prime;}& y_{u_c}^{\&prime;}& z_{u_c}^{\&prime;}\end{array}\right),\stackrel{\&RightArrow;}{r_v^{\&prime;}}=\left(\begin{array}{ccc}{x}_{v_c}^{\&prime;}& y_{v_c}^{\&prime;}& z_{v_c}^{\&prime;}\end{array}\right),\stackrel{\&RightArrow;}{n}(u_c,v_c)=\stackrel{\&RightArrow;}{r_u^{\&prime;}}\&times;\stackrel{\&RightArrow;}{r_v^{\&prime;}}=\stackrel{\&RightArrow;}{n}{(x_n,y_n,z_n)}^T$

Wherein divide two tangent vectors of contact point, for the normal vector at contact point place, afterwards near curved surface contact point place is carried out to coordinate transform, making section direction is the xoy plane of new coordinate axis, contact point is new origin, by the two surface Taylor series expansions at contact point place, ignore high-order infinitesimal, retain the quadratic component of function, thereby obtain the approximate expression of two surfaces under new coordinate system:

$\left\{\begin{array}{c}{z}_1=A_1{\stackrel{\&OverBar;}{x}}^2+B_1{\stackrel{\&OverBar;}{y}}^2+C_1\stackrel{\&OverBar;}{\mathrm{xy}}\\ z_2=A_2{\stackrel{\&OverBar;}{x}}^2+B_2{\stackrel{\&OverBar;}{y}}^2+C_2\stackrel{\&OverBar;}{\mathrm{xy}}\end{array}\right.$

Two formulas are subtracted each other, and convert by coordinate system rotation, eliminate xy cross term, and the expression formula that obtains contact interval is:

$z_1-z_2=A{\stackrel{\&OverBar;}{x}}^2+B{\stackrel{\&OverBar;}{y}}^2$

Wherein, A, B is constant, just can make single requirement that the contact analytic formula of micro-bulge is met to Hertz contact calculating through such conversion;

3) the Hertz contact mechanics at single butt contact place calculates

3.1) contactinterval coefficient solves: according to Hertz arbitrary shape curved surface contact theory, for oval arbitrfary point, A, B can be tried to achieve by principal curvatures and the principal direction of curvature at two micro-bulge contact point places, with R, R ' represents respectively two principal curvaturess of one of them micro-bulge at contact point place, w represent two and micro-bulge contact point between principal direction of curvature angle, coefficient A, B meets the represented equation of following formula, oneself knows the angle between each micro-bulge principal curvatures size and principal direction of curvature thereof, can try to achieve A, the concrete numerical value of B:

$\left\{\begin{array}{c}A+B=\frac{1}{2}(\frac{1}{R_1}+\frac{1}{R_1^{\&prime;}}+\frac{1}{R_2}+\frac{1}{R_2^{\&prime;}})\\ B-A=\frac{1}{2}{\left(\begin{array}{c}{(\frac{1}{R_1}-\frac{1}{R_1^{\&prime;}})}^2+{(\frac{1}{R_2}-\frac{1}{R_2^{\&prime;}})}^2+\\ 2(\frac{1}{R_1}-\frac{1}{R_1^{\&prime;}})(\frac{1}{R_2}-\frac{1}{R_2^{\&prime;}})\cos 2w\end{array}\right)}^{\frac{1}{2}}\end{array}\right.$

3.2) contact area length semiaxis solves: the contact area of complex-curved some contact is approximately oval, and length semiaxis is respectively:

$\left\{\begin{array}{c}a=m\sqrt[3]{\frac{3\mathrm{\&pi;}}{4}\frac{P(k_1+k_2)}{(A+B)}}\\ b=n\sqrt[3]{\frac{3\mathrm{\&pi;}}{4}\frac{P(k_1+k_2)}{(A+B)}}\end{array}\right.$

Wherein m, n relevant to ratio (B-A)/(A+B) is, numerical value adopts the document Jamari J of contact theory, Schipper D J.An elastic – plastic contact model of ellipsoid bodies[J] .Tribology letters, 2006,21 (3): 262-271, a, b is respectively the length semiaxis of contact area, k ₁+ k ₂for the synthetical elastic modulus at contact point place, k is represented by following formula:

wherein, E, v represents respectively elastic modulus and the Poisson ratio of material,

3.3) displacement in contact point place contact method vector direction solves: the average that is decided to be both direction contact displacement perpendicular to the displacement of contact direction:

3.4) Maximum Contact stress and contact area solve: Maximum Contact stress and contact area can be drawn by following equation:

$\left\{\begin{array}{c}{q}_0=\frac{3}{2}\frac{P}{\mathrm{\&pi;ab}}\\ s=\mathrm{\&pi;ab}\end{array}\right.$

Wherein P represents the power that single butt contact is suffered, q ₀for the Maximum Contact stress at contact point place, s is the contact area at single butt contact place,

4) the tangential contact stiffness of the total real contact area of faying face and normal direction calculates

4.1) total contact area is calculated: total contact area equals the summation of each contact point contact area,

4.2) all directions contact stiffness calculates: can obtain making a concerted effort finally by cumulative the suffered power at each point place and decompose by each change in coordinate axis direction, obtain the normal component of force P perpendicular to whole surface of contact _sumzand be parallel to two component P of surface of contact _sumxand P _sumy, the displacement at each point place is by resolution of vectors and average and can obtain the average displacement in all directions rigidity numerical value in certain direction, can obtain as follows:

The expression formula of normal contact stiffness is: rigidity in two tangential direction is: $k_z=\frac{\&PartialD;P_{\mathrm{sumx}}}{\&PartialD;{\stackrel{\&OverBar;}{d}}_x},k_z=\frac{\&PartialD;P_{\mathrm{sumy}}}{\&PartialD;{\stackrel{\&OverBar;}{d}}_y}.$

The present invention has following beneficial effect: taken into full account three-dimensional appearance and the impact of micro-bulge true three-dimension shape on contact area and contact stiffness of the shape error of faying face, start with from actual uneven surface pattern, break away from the shortcoming of original recipe based on a large amount of hypothesis, compared with finite element method, can greatly expand reference area, and improve counting yield.

Brief description of the drawings

Fig. 1 is the classification schematic diagram of Machine Joint Surfaces rough surface.

Fig. 2 is single to micro-bulge contact and coordinate transform schematic diagram thereof.

Fig. 3 is Hertz curved surface contact contact schematic diagram.

Embodiment

Describe the present invention below in conjunction with accompanying drawing:

Definite faying face contact area based on surface fitting and a method for rigidity, comprise the following steps:

1) measurement of real topography and function representation

1.1) faying face pattern level is divided: as shown in Figure 1, by Machine Joint Surfaces by ascending roughness, percent ripple, three levels of shape error divided into of wavelength, roughness is regarded in the fluctuating that peak and peak-to-peak spacing is less than to 1mm as, corresponding single projection is regarded micro-bulge as, and supposes that micro-bulge shape is identical; Regard the distance between crest and crest as percent ripple at 1mm to the surface undulation within the scope of 10mm; Surface shape error processing is used as in the surface undulation that distance between crest and crest is exceeded to 10mm scope;

1.2) micro-bulge Function Fitting: adopt Laser Scanning Confocal Microscope observation micro-bulge pattern, the cloud data obtaining is used for matching micro-bulge, carry out matching with binary higher order functionality, press tool sharpening trajectory direction, by machining locus direction be decided to be u to, perpendicular to the direction of machining locus be decided to be v to, each micro-bulge fits to:

$f_{\mathrm{ai}}=f_{\mathrm{ai}}(u,v)=a_i+\underset{t=0}{\overset{1}{\mathrm{\&Sigma;}}}{b}_i{u}^t{v}^{1-t}+\underset{t=0}{\overset{2}{\mathrm{\&Sigma;}}}{c}_i{u}^t{v}^{2-t}+......,((u,v)\&Element;{\mathrm{\&Omega;}}_i,i=\mathrm{1,2,3}......)$

Wherein i represents i micro-bulge on rough surface, Ω _irepresent i the surf zone that micro-bulge is shared;

1.3) percent ripple and shape error matching: adopt three-coordinates measuring machine to measure percent ripple and the shape error of surface of contact, the cloud data of acquisition carries out matching with binary higher order functionality, and surface shape error and percent ripple fit to:

$f_e=f_e(u,v)=a_e+\underset{i=0}{\overset{1}{\mathrm{\&Sigma;}}}{b}_e{u}^i{v}^{1-i}+\underset{i=0}{\overset{2}{\mathrm{\&Sigma;}}}{c}_e{u}^i{v}^{2-i}+......$

1.4) function representation of final curved surface: micro-bulge function is distributed on surface shape error function and is added, obtain the function expression of final real surface pattern,

$f=f(u,v)=\left\{\begin{array}{cc}{f}_{a1}(u,v)+f_e(u,v)& ((u,v)\&Element;{\mathrm{\&Omega;}}_1)\\ f_{a2}(u,v)+f_e(u,v)& ((u,v)\&Element;{\mathrm{\&Omega;}}_2)\\ f_{a3}(u,v)+f_e(u,v)& ((u,v)\&Element;{\mathrm{\&Omega;}}_3)\\ ........& \end{array}\right.$

2) two surface in contact contact point mechanics judgements and single pre-service to micro-bulge contact

2.1) spacing function expression between two surfaces: two surface in contact functions are subtracted each other, obtain contactinterval formula:

δ＝f ₁(u，v)-f ₂(u，v)

Wherein, f ₁(u, v) represents upper surface fitting function, f ₂(u, v) represents lower surface fitting function,

2.2) contacting points position obtains: contact point (u _c, v _c) satisfied condition is δ (u _c, v _c) be function minimum point, and δ (u _c, v _c) < 0, ask method can try to achieve all possible contact point according to minimal value, then according to δ (u _c, v _c) < 0 judges, can draw have point of contact;

2.3) contact point direction solves and contact point coordinate transform: the volume coordinate of establishing contact point is (x (u _c, v _c), y (u _c, v _c), z (u _c, v _c)), contact direction solves by differential geometric method:

$\stackrel{\&RightArrow;}{r_u^{\&prime;}}=\left(\begin{array}{ccc}{x}_{u_c}^{\&prime;}& y_{u_c}^{\&prime;}& z_{u_c}^{\&prime;}\end{array}\right),\stackrel{\&RightArrow;}{r_v^{\&prime;}}=\left(\begin{array}{ccc}{x}_{v_c}^{\&prime;}& y_{v_c}^{\&prime;}& z_{v_c}^{\&prime;}\end{array}\right),\stackrel{\&RightArrow;}{n}(u_c,v_c)=\stackrel{\&RightArrow;}{r_u^{\&prime;}}\&times;\stackrel{\&RightArrow;}{r_v^{\&prime;}}=\stackrel{\&RightArrow;}{n}{(x_n,y_n,z_n)}^T$

Wherein divide two tangent vectors of contact point, for the normal vector at contact point place, afterwards near curved surface contact point place is carried out to coordinate transform, as shown in Figure 2, making section direction is the xoy plane of new coordinate axis, contact point is new origin, by the two surface Taylor series expansions at contact point place, ignores high-order infinitesimal, retain the quadratic component of function, thereby obtain the approximate expression of two surfaces under new coordinate system:

$\left\{\begin{array}{c}{z}_1=A_1{\stackrel{\&OverBar;}{x}}^2+B_1{\stackrel{\&OverBar;}{y}}^2+C_1\stackrel{\&OverBar;}{\mathrm{xy}}\\ z_2=A_2{\stackrel{\&OverBar;}{x}}^2+B_2{\stackrel{\&OverBar;}{y}}^2+C_2\stackrel{\&OverBar;}{\mathrm{xy}}\end{array}\right.$

Two formulas are subtracted each other, and convert by coordinate system rotation, eliminate xy cross term, and the expression formula that obtains contact interval is:

$z_1-z_2=A{\stackrel{\&OverBar;}{x}}^2+B{\stackrel{\&OverBar;}{y}}^2$

Wherein, A, B is constant, just can make single requirement that the contact analytic formula of micro-bulge is met to Hertz contact calculating through such conversion;

3) the Hertz contact mechanics at single butt contact place calculates

3.1) contactinterval coefficient solves: according to Hertz arbitrary shape curved surface contact theory, for oval arbitrfary point, A, B can be tried to achieve by principal curvatures and the principal direction of curvature at two micro-bulge contact point places, as shown in Figure 3, with R, R ' represents respectively two principal curvaturess of one of them micro-bulge at contact point place, w represent two and micro-bulge contact point between principal direction of curvature angle, coefficient A, B meets the represented equation of following formula, and oneself knows the angle between each micro-bulge principal curvatures size and principal direction of curvature thereof, can try to achieve A, the concrete numerical value of B:

$\left\{\begin{array}{c}A+B=\frac{1}{2}(\frac{1}{R_1}+\frac{1}{R_1^{\&prime;}}+\frac{1}{R_2}+\frac{1}{R_2^{\&prime;}})\\ B-A=\frac{1}{2}{\left(\begin{array}{c}{(\frac{1}{R_1}-\frac{1}{R_1^{\&prime;}})}^2+{(\frac{1}{R_2}-\frac{1}{R_2^{\&prime;}})}^2+\\ 2(\frac{1}{R_1}-\frac{1}{R_1^{\&prime;}})(\frac{1}{R_2}-\frac{1}{R_2^{\&prime;}})\cos 2w\end{array}\right)}^{\frac{1}{2}}\end{array}\right.$

3.2) contact area length semiaxis solves: the contact area of complex-curved some contact is approximately oval, and length semiaxis is respectively:

$\left\{\begin{array}{c}a=m\sqrt[3]{\frac{3\mathrm{\&pi;}}{4}\frac{P(k_1+k_2)}{(A+B)}}\\ b=n\sqrt[3]{\frac{3\mathrm{\&pi;}}{4}\frac{P(k_1+k_2)}{(A+B)}}\end{array}\right.$

Wherein m, n relevant to ratio (B-A)/(A+B) is, numerical value adopts the document Jamari J of contact theory, Schipper D J.An elastic – plastic contact model of ellipsoidbodies[J] .Tribology letters, 2006,21 (3): 262-271, a, b is respectively the length semiaxis of contact area.K ₁+ k ₂for the synthetical elastic modulus at contact point place, k is represented by following formula:

wherein, E, v represents respectively elastic modulus and the Poisson ratio of material,

3.3) displacement in contact point place contact method vector direction solves: the average that is decided to be both direction contact displacement perpendicular to the displacement of contact direction:

3.4) Maximum Contact stress and contact area solve: Maximum Contact stress and contact area can be drawn by following equation:

$\left\{\begin{array}{c}{q}_0=\frac{3}{2}\frac{P}{\mathrm{\&pi;ab}}\\ s=\mathrm{\&pi;ab}\end{array}\right.$

Wherein P represents the power that single butt contact is suffered, q ₀for the Maximum Contact stress at contact point place, s is the contact area at single butt contact place,

4) the tangential contact stiffness of the total real contact area of faying face and normal direction calculates

4.1) total contact area is calculated: total contact area equals the summation of each contact point contact area,

4.2) all directions contact stiffness calculates: can obtain making a concerted effort finally by cumulative the suffered power at each point place and decompose by each change in coordinate axis direction, obtain the normal component of force P perpendicular to whole surface of contact _sumzand be parallel to two component P of surface of contact _sumxand P _sumy, the displacement at each point place is by resolution of vectors and average and can obtain the average displacement in all directions rigidity numerical value in certain direction, can obtain as follows:

The expression formula of normal contact stiffness is: rigidity in two tangential direction is: $k_z=\frac{\&PartialD;P_{\mathrm{sumx}}}{\&PartialD;{\stackrel{\&OverBar;}{d}}_x},k_z=\frac{\&PartialD;P_{\mathrm{sumy}}}{\&PartialD;{\stackrel{\&OverBar;}{d}}_y}.$

Claims (1)

1. the definite faying face contact area based on surface fitting and a method for rigidity, is characterized in that, comprises the following steps:

1) measurement of real topography and function representation

1.1) faying face pattern level divide: by Machine Joint Surfaces by ascending roughness, percent ripple, three levels of shape error divided into of wavelength, roughness is regarded in the fluctuating that peak and peak-to-peak spacing is less than to 1mm as, corresponding single projection is regarded micro-bulge as, and supposes that micro-bulge shape is identical; Regard the distance between crest and crest as percent ripple at 1mm to the surface undulation within the scope of 10mm; Surface shape error processing is used as in the surface undulation that distance between crest and crest is exceeded to 10mm scope;

1.2) micro-bulge Function Fitting: adopt Laser Scanning Confocal Microscope observation micro-bulge pattern, the cloud data obtaining is used for matching micro-bulge, carry out matching with binary higher order functionality, press tool sharpening trajectory direction, by machining locus direction be decided to be u to, perpendicular to the direction of machining locus be decided to be v to, each micro-bulge fits to:

$f_{\mathrm{ai}}=f_{\mathrm{ai}}(u,v)=a_i+\underset{t=0}{\overset{1}{\mathrm{\&Sigma;}}}{b}_i{u}^t{v}^{1-t}+\underset{t=0}{\overset{2}{\mathrm{\&Sigma;}}}{c}_i{u}^t{v}^{2-t}+......,((u,v)\&Element;{\mathrm{\&Omega;}}_i,i=\mathrm{1,2,3}......)$

Wherein i represents i micro-bulge on rough surface, Ω _irepresent the shared surf zone of the micro-bulge;

1.3) percent ripple and shape error matching: adopt three-coordinates measuring machine to measure percent ripple and the shape error of surface of contact, the cloud data of acquisition carries out matching with binary higher order functionality, and surface shape error and percent ripple fit to:

$f_e=f_e(u,v)=a_e+\underset{i=0}{\overset{1}{\mathrm{\&Sigma;}}}{b}_e{u}^i{v}^{1-i}+\underset{i=0}{\overset{2}{\mathrm{\&Sigma;}}}{c}_e{u}^i{v}^{2-i}+......$

1.4) function representation of final curved surface: micro-bulge function is distributed on surface shape error function and is added, obtain the function expression of final real surface pattern,

$f=f(u,v)=\left\{\begin{array}{cc}{f}_{a1}(u,v)+f_e(u,v)& ((u,v)\&Element;{\mathrm{\&Omega;}}_1)\\ f_{a2}(u,v)+f_e(u,v)& ((u,v)\&Element;{\mathrm{\&Omega;}}_2)\\ f_{a3}(u,v)+f_e(u,v)& ((u,v)\&Element;{\mathrm{\&Omega;}}_3)\\ ........& \end{array}\right.$

2) two surface in contact contacting points position judgements and single pre-service to micro-bulge contact

2.1) spacing function expression between two surfaces: two surface in contact functions are subtracted each other, obtain contactinterval formula:

δ＝f ₁(u，v)-f ₂(u，v)

Wherein, f ₁(u, v) represents upper surface fitting function, f ₂(u, v) represents lower surface fitting function,

2.2) contacting points position obtains: contact point (u _c, v _c) satisfied condition is δ (u _c, v _c) be function minimum point, and δ (u _c, v _c) < 0, ask method can try to achieve all possible contact point according to minimal value, then according to δ (u _c, v _c) < 0 judges, can draw have point of contact;

2.3) contact point direction solves and contact point coordinate transform: the volume coordinate of establishing contact point is (x (u _c, v _c), y (u _c, v _c), z (u _c, v _c)), contact direction solves by differential geometric method:

$\stackrel{\&RightArrow;}{r_u^{\&prime;}}=\left(\begin{array}{ccc}{x}_{u_c}^{\&prime;}& y_{u_c}^{\&prime;}& z_{u_c}^{\&prime;}\end{array}\right),\stackrel{\&RightArrow;}{r_v^{\&prime;}}=\left(\begin{array}{ccc}{x}_{v_c}^{\&prime;}& y_{v_c}^{\&prime;}& z_{v_c}^{\&prime;}\end{array}\right),\stackrel{\&RightArrow;}{n}(u_c,v_c)=\stackrel{\&RightArrow;}{r_u^{\&prime;}}\&times;\stackrel{\&RightArrow;}{r_v^{\&prime;}}=\stackrel{\&RightArrow;}{n}{(x_n,y_n,z_n)}^T$

Wherein divide two tangent vectors of contact point, for the normal vector at contact point place, afterwards near curved surface contact point place is carried out to coordinate transform, making section direction is the xoy plane of new coordinate axis, contact point is new origin, by the two surface Taylor series expansions at contact point place, ignore high-order infinitesimal, retain the quadratic component of function, thereby obtain the approximate expression of two surfaces under new coordinate system:

$\left\{\begin{array}{c}{z}_1=A_1{\stackrel{\&OverBar;}{x}}^2+B_1{\stackrel{\&OverBar;}{y}}^2+C_1\stackrel{\&OverBar;}{\mathrm{xy}}\\ z_2=A_2{\stackrel{\&OverBar;}{x}}^2+B_2{\stackrel{\&OverBar;}{y}}^2+C_2\stackrel{\&OverBar;}{\mathrm{xy}}\end{array}\right.$

Two formulas are subtracted each other, and convert by coordinate system rotation, eliminate cross term, the expression formula that obtains contact interval is:

$z_1-z_2=A{\stackrel{\&OverBar;}{x}}^2+B{\stackrel{\&OverBar;}{y}}^2$

Wherein, A, B is constant, just can make single requirement that the contact analytic formula of micro-bulge is met to Hertz contact calculating through such conversion;

3) the Hertz contact mechanics at single butt contact place calculates

3.1) contactinterval coefficient solves: according to Hertz arbitrary shape curved surface contact theory, for oval arbitrfary point, A, B can be tried to achieve by principal curvatures and the principal direction of curvature at two micro-bulge contact point places, with R, R ' represents respectively two principal curvaturess of one of them micro-bulge at contact point place, w represent two and micro-bulge contact point between principal direction of curvature angle, coefficient A, B meets the represented equation of following formula, oneself knows the angle between each micro-bulge principal curvatures size and principal direction of curvature thereof, can try to achieve A, the concrete numerical value of B:

$\left\{\begin{array}{c}A+B=\frac{1}{2}(\frac{1}{R_1}+\frac{1}{R_1^{\&prime;}}+\frac{1}{R_2}+\frac{1}{R_2^{\&prime;}})\\ B-A=\frac{1}{2}{\left(\begin{array}{c}{(\frac{1}{R_1}-\frac{1}{R_1^{\&prime;}})}^2+{(\frac{1}{R_2}-\frac{1}{R_2^{\&prime;}})}^2+\\ 2(\frac{1}{R_1}-\frac{1}{R_1^{\&prime;}})(\frac{1}{R_2}-\frac{1}{R_2^{\&prime;}})\cos 2w\end{array}\right)}^{\frac{1}{2}}\end{array}\right.$

3.2) contact area length semiaxis solves: the contact area of complex-curved some contact is approximately oval, and length semiaxis is respectively:

$\left\{\begin{array}{c}a=m\sqrt[3]{\frac{3\mathrm{\&pi;}}{4}\frac{P(k_1+k_2)}{(A+B)}}\\ b=n\sqrt[3]{\frac{3\mathrm{\&pi;}}{4}\frac{P(k_1+k_2)}{(A+B)}}\end{array}\right.$

Wherein m, n relevant to ratio (B-A)/(A+B) is, numerical value adopts the document Jamari J of contact theory, Schipper D J.An elastic – plastic contact model of ellipsoid bodies[J] .Tribology letters, 2006,21 (3): 262-271, a, b is respectively the length semiaxis of contact area, k ₁+ k ₂for the synthetical elastic modulus at contact point place, k is represented by following formula:

wherein, E, v represents respectively elastic modulus and the Poisson ratio of material,

3.3) displacement in contact point place contact method vector direction solves: the average that is decided to be both direction contact displacement perpendicular to the displacement of contact direction:

3.4) Maximum Contact stress and contact area solve: Maximum Contact stress and contact area can be drawn by following equation:

$\left\{\begin{array}{c}{q}_0=\frac{3}{2}\frac{P}{\mathrm{\&pi;ab}}\\ s=\mathrm{\&pi;ab}\end{array}\right.$

Wherein P represents the power that single butt contact is suffered, q ₀for the Maximum Contact stress at contact point place, s is the contact area at single butt contact place,

4) the tangential contact stiffness of the total real contact area of faying face and normal direction calculates

4.1) total contact area is calculated: total contact area equals the summation of each contact point contact area,

4.2) all directions contact stiffness calculates: can obtain making a concerted effort finally by cumulative the suffered power at each point place and decompose by each change in coordinate axis direction, obtain the normal component of force P perpendicular to whole surface of contact _sumzand be parallel to two component P of surface of contact _sumxand P _sumy, the displacement at each point place is by resolution of vectors and average and can obtain the average displacement in all directions rigidity numerical value in certain direction, can obtain as follows:

The expression formula of normal contact stiffness is: rigidity in two tangential direction is: $k_z=\frac{\&PartialD;P_{\mathrm{sumx}}}{\&PartialD;{\stackrel{\&OverBar;}{d}}_x},k_z=\frac{\&PartialD;P_{\mathrm{sumy}}}{\&PartialD;{\stackrel{\&OverBar;}{d}}_y}.$