Method for establishing thermal contact conduction model of tapered roller / groove interface

Abstract

The invention discloses a method for creating a thermal contact conduction model of a tapered roller / groove interface, which applies the Cantor set theory to characterize the morphology of a rough surface, and adopts the structure function method to identify fractal parameters; a fractal contact mechanics model is established by constructing a contact coefficient, and the influence of geometric shape and contact form on the contact area and contact load is considered; a fractal network TCC model is proposed based on the surface morphology characterization and the fractal contact mechanics model, and a matrix TCC and a shrinkage TCC are incorporated into the TCC model; experiments show that the maximum deviations between the predicted and measured TCCs are 5.65% and 8.70%, respectively, which proves that the designed tapered roller / groove interface contact coefficient can effectively reflect the contact state of the tapered roller and the groove, and the contact fractal mechanics model can effectively calculate the contact parameters of the tapered roller / groove interface.

Description

Technical Field

[0001] The invention belongs to the field of heat conduction technology, and specifically relates to a method for creating a thermal contact conduction model of a tapered roller / groove interface. Background Art

[0002] Modeling and experimental studies of thermal contact conductivity (TCC) have emerged in an endless stream, and TCC has attracted much attention because it is indispensable in engineering thermal analysis. For example, the introduction of TCC in thermal analysis of machine tools shows that TCC has a direct impact on the temperature field distribution of the entire machine tool. In fact, TCC is related to surface morphology, contact load, material properties, actual contact area, etc. Moreover, the nonlinear mapping between TCC and the above parameters is difficult to establish because TCC modeling is an interdisciplinary problem involving geometry, including contact mechanics and heat transfer. Surface morphology and contact load determine the contact area, which affects the thermal resistance flowing through the contact interface. Bearings are important functional components in precision machine tools and are obvious heat-generating elements. Frictional heat is generated at the contact interface between the rolling element and the bearing ring groove. TCC affects the normal heat conduction, resulting in heat accumulation at the contact interface. Summary of the invention

[0003] In view of this, the purpose of the present invention is to provide a method for creating a thermal contact conduction model of a tapered roller / groove interface, and the constructed thermal contact conduction model can meet the analysis requirements of thermal contact conduction of the tapered roller / groove interface.

[0004] In order to achieve the above object, the present invention provides the following technical solutions:

[0005] A method for creating a thermal contact conduction model for a tapered roller / groove interface to characterize the surface morphology: using the Cantor set theory to characterize the rough surface morphology and using the structure function method to identify the fractal dimension;

[0006] Construct a fractal contact mechanics model: by constructing the contact coefficient and combining the geometric shape and contact form to obtain the contact area and contact load;

[0007] Construct a thermal contact conduction model: Combining substrate thermal contact conduction and shrinkage thermal contact conduction, a thermal contact conduction model is constructed based on surface morphology characterization and fractal contact mechanics model.

[0008] Furthermore, the method of Cantor set theory to characterize the rough surface morphology is:

[0009] The single microprotrusion in the initial 0th generation is divided into 2s-1 segments along the horizontal direction, and then the middle segment is removed to obtain the microprotrusions of the first generation; the total length of the remaining microprotrusions in the first generation microprotrusions is 1 / f of the total length of the initial 0th generation microprotrusions r , the total depth of the first generation of concave parts is 1 / f of the initial 0th generation micro-protrusion depth z ;

[0010] The single microprotrusion in the first generation is divided into (2s-1) segments along the horizontal direction, and then the middle segment is removed to obtain the microprotrusion of the second generation; the total length of the remaining microprotrusions in the second generation microprotrusions is 1 / f of the total length of the microprotrusions of the first generation r The total depth of the second generation concave part is 1 / f of the depth of the first generation micro-protrusion z ;

[0011] In this way, micro-protrusions of different sizes are generated i times, and the total horizontal length and depression depth of the i-th generation micro-protrusions are obtained as follows:

[0012] l i =(1 / f r ) i-1 =(1 / f r ) i 10

[0013] h i =(1 / f z )h i-1 =(1 / f z ) i h0

[0014] Among them, l i represents the total horizontal length of the i-th generation micro-protrusions; h i represents the depression depth of the i-th generation micro-protrusion; l i-1 represents the total horizontal length of the (i-1)th generation microprotrusions; h i-1 represents the depression depth of the i-1th generation micro-protrusion; l0 represents the total horizontal length of the 0th generation micro-protrusion; h0 represents the depression depth of the 0th generation micro-protrusion; f r The ratio coefficient of the total length of the next generation of micro-protrusions to the previous generation of micro-protrusions; f z The ratio coefficient of the depth of the next generation of micro-protrusions to the previous generation of micro-protrusions; s Indicates the parameters that need to be identified;

[0015] The total number of generations i is N = s i micro-protrusions, and the length of a single micro-protrusion of the i-th generation is expressed as:

[0016] x i =(1 / sf r ) i 10

[0017] The gap width between two adjacent micro-protrusions on the i-th generation is:

[0018]

[0019] The height of the i-th generation micro-protrusion is expressed as:

[0020] z i =(1 / f z ) i (f z -1)h0

[0021] By changing the parameters s and f r and f z The proportional coefficient of can be used to obtain Cantor set fractal surfaces with different structures. The fractal dimension of the Cantor set surface is related to s 、f r and f z The relationship is expressed as:

[0022]

[0023] Among them, D s represents the fractal dimension of the two-dimensional section; D p represents the fractal dimension of the 3D rough surface, and D p =D s +1;

[0024] parameter s It is expressed as:

[0025]

[0026] Further, the method for identifying the fractal dimension is:

[0027] Fractal dimension D of rough surface profile p It can be obtained directly through its structural function:

[0028]

[0029] Where z(x) represents the horizontal length x The height of the rough profile at ; τ represents the increment in the x direction; G represents the characteristic parameter related to the surface morphology; γ represents the parameter that determines the scale ratio of the spectral density; <> represents the statistical mean value of z(x); Γ represents the first kind of elliptic integral;

[0030] The physical meaning of the structure function is arbitrary x The mean square of the surface roughness height difference in the direction with τ as the increment, that is, the structure function is calculated as:

[0031]

[0032] Wherein, N represents the measurement point data; n represents the number of intervals; Δt represents the length interval in the x direction; z k+n represents the roughness height at position k+n; z k represents the roughness height at position k;

[0033] The relationship between the structure function S(τ) and the parameter τ on the double logarithmic coordinates is a straight line. The slope of the straight line can be used to obtain the fractal dimension D. p ,get:

[0034] D p =(4-k s ) / 2

[0035] Among them, k s represents the slope of the structure function S(τ);

[0036] The characteristic parameter G can be obtained by the intercept of the straight line with the z-axis; the actual contact between two rough surfaces is equivalent to the contact between a flat rigid surface and an equivalent rough surface; if the roughness heights of the two contact surfaces are z1(x) and z2(x), respectively, the equivalent surface roughness height function is:

[0037] z(x)=z1(x)-z2(x)

[0038] Thus we get:

[0039] S(τ)=<[z1(x+τ)-z1(x)]> 2 -2<[z1(x+τ)-z1(x)]><[z2(x+τ)-z2(x)]>+<[z2(x+τ)-z2(x)]> 2

[0040] Since the rough surfaces in contact with each other are statistically unrelated, the product term in the formula is zero, and the structure function of the equivalent rough surface is:

[0041] S(τ)=S1(τ)+S2(τ)

[0042] Among them, S1(τ) and S2(τ) represent the structure functions of the two contact surfaces respectively, that is, the structure function of the equivalent rough surface is the sum of the structure functions of the two contact surfaces.

[0043] Furthermore, the contact coefficient is constructed as follows:

[0044] The contact between two Cantor set fractal surfaces is equivalent to the contact between an imaginary entity and an ideal smooth rigid entity. When an external load is applied, the imaginary entity undergoes elastic-plastic deformation. According to the theory of plastic deformation, the critical load is F c =HA c , corresponds to the transition from elastic deformation to plastic deformation, where H and A c represent hardness and critical contact area respectively;

[0045] The ideal smooth rigid plane will first contact the last generation of micro-protrusions; if the normal load is greater than the critical load of the i-th generation, the deformation of the i-th generation of micro-protrusions will change from elastic deformation to plastic flow; then the ideal smooth rigid plane will contact the previous generation of micro-protrusions, and this process will continue until the n-th generation. c -1 generation ends, the critical load of this generation is greater than the applied load, and the nth generation c The micro-protrusions of the -1 generation will be in elastic contact with the ideal smooth rigid surface, while the nth generation c The micro-protrusions of the first generation are the oldest generation that have undergone plastic deformation; therefore, for a given normal load, the intersection of each generation of micro-protrusions on the entire rough surface can be divided into two categories: [n c ,∞] generation range of micro-protrusions undergo plastic deformation, [0,n c -1] generation range of micro-protrusions undergo elastic deformation; the nth c -1 generation range of micro-protrusions actually in contact with the ideal plane, n c The value of can be expressed as:

[0046]

[0047] Where, trun(x) means that the parameter x is truncated to an integer, ξ represents the stage frequency, and F represents the normal contact load.

[0048] For the contact interface of tapered roller / groove, the contact coefficient should be considered. The contact coefficient and the comprehensive curvature coefficient X of the two contact bodies h , equivalent elastic modulus E, contact load F and other factors, and the contact coefficient is used to correct the critical value. c ':

[0049] n c '=n c · χ

[0050] Among them, n c ' represents the modified critical generation; χ represents the contact coefficient, and:

[0051]

[0052] Among them, S h represents the theoretical contact area; ∑S represents the sum of the contact areas of the two objects, that is:

[0053] ∑S=S1±S2

[0054] Where S1 and S2 represent the contact areas of object 1 and object 2, respectively;

[0055] The contact interface of the tapered roller / groove belongs to the contact of different curvatures, that is, the contact area of ​​object 1 and object 2 can be expressed as:

[0056] S1=2π(R1+R2)l

[0057] S2=πR3B1

[0058] Among them, R1 and R2 represent the top and bottom radii of the tapered roller respectively; l represents the thickness and length of the generatrix; R3 represents the radius of the bearing ring groove; B1 represents the width of the bearing ring; thus:

[0059] ΣS=S1±S2=2π(R1+R2)l±πR3B1

[0060] Comprehensive curvature coefficient X h It is expressed as:

[0061]

[0062] Among them, '+' indicates external contact, and '-' indicates internal contact;

[0063] The theoretical contact area when tapered roller / bearing ring contact occurs is:

[0064]

[0065] The contact coefficient is thus obtained:

[0066]

[0067] Furthermore, the solution for the contact area and contact load is:

[0068] The contact area of ​​a single micro-bump on the i-th generation is:

[0069]

[0070] There are s on the fractal interface of the i-th generation Cantor set 2i micro-protrusions, the total contact area of ​​the i-th generation micro-protrusions is expressed as:

[0071]

[0072] The total gap area between individual micro-protrusions on the i-th generation generated in each i-1 generation is obtained as:

[0073]

[0074] Among them, g i represents the gap width between two adjacent micro-protrusions on the i-th generation;

[0075] There are s in the i-1th generation of the Cantor set fractal profile. 2(i-1) The total contact area of ​​the i-1th generation micro-protrusion is expressed as:

[0076]

[0077] The total contact area of ​​the contact interface after correction by the contact coefficient χ is:

[0078]

[0079] When the maximum Hertz contact pressure reaches 0.6H, plastic deformation occurs, and the critical plastic deformation u c1 It is expressed as:

[0080]

[0081] Among them, H represents the microhardness of the softer material, k represents the equivalent thermal conductivity, and k = 1 / (1 / k1+1 / k2); E represents the equivalent elastic modulus, that is:

[0082]

[0083] E1 and E2 represent the Young's elastic modulus of the two contact bodies; v1 and v2 represent the Poisson's ratio of the two contact bodies; k1 and k2 represent the thermal conductivity of the two contact bodies;

[0084] When the minimum contact load applied to the micro-protrusion reaches H, complete plastic flow occurs on the contact interface; the critical plastic deformation u c1 and critical elastic deformation u c2 The relationship is expressed as:

[0085] u c2 =110u c1

[0086] When u i >u c2 When the contact load is

[0087] When u i c1 When the contact load is

[0088] The total contact load corrected by the contact coefficient χ is:

[0089]

[0090] Among them, u i Represents the contact deformation of the asperity.

[0091] Furthermore, the calculation method of substrate thermal contact conduction is:

[0092] The thermal contact resistance (TCR) of a pair of micro-bumps is expressed as:

[0093] ​

[0094] The sum of the substrate TCR of the micro-bump contact deformation is:

[0095]

[0096] Where m1 represents an integer, and λ i The bottom dimension is x i The contact point size corresponding to the micro-bump.

[0097] Furthermore, the calculation method of shrinkage heat contact conduction is as follows: According to the classical theory of contact heat conduction, when the heat flow passes through the contact interface, the heat flow shrinks in the contact area. Since the actual contact area is smaller than the nominal contact area, there is a shrinkage TCC. Considering the actual contact situation of the two objects, the contact interface is analyzed using the hot runner model, and the contact model of the shrinkage TCC of each contact point is obtained as follows:

[0098]

[0099] Where Q is the heat flow through the channel; ΔT c represents the temperature difference between the two interfaces; b represents the radius of the contact point; c is the hot runner radius; ψ(c / b) represents the contraction factor; k represents the equivalent thermal conductivity;

[0100] For the distribution of the contact points at the center of the micro-bump, Ψ(c / b) is approximately:

[0101]

[0102] Get the contraction TCR:

[0103]

[0104] Among them, λ i The bottom dimension is x i The contact point size corresponding to the micro-protrusion; d i =x i +x ci , x ci represents the gap size, and:

[0105]

[0106] in,

[0107] Furthermore, the bottom dimension is x i The volume of the micro-protrusion is:

[0108]

[0109] Where S represents the bottom area of ​​the micro-convex body; h represents the height of the micro-convex body;

[0110] The volume of the truncated microprotrusion is expressed as:

[0111]

[0112] Among them, λ i represents the diameter of the contact point of the deformed micro-protrusion at a given contact height d';

[0113] The deformed micro-protrusions satisfy the volume conservation law, that is:

[0114]

[0115] The size λ of the contact point corresponding to the micro-protrusion can be obtained i for:

[0116]

[0117] Right now:

[0118]

[0119] Where d'=d+h m , d represents the distance between the contact rigid plane and the average plane, h m represents the average height of the microprotrusions. Furthermore, the total TCR of a single pair of contacting microprotrusions is ti for:

[0120] r ti =r bi +r ci

[0121] Among them, r bi represents matrix TCR; r ci indicates contraction of TCR;

[0122] The contraction TCCs of all i contact points are connected in parallel, and their equivalent contraction TCRRs are ti Defined as r ti Number of parallel connections n t Ratio of:

[0123]

[0124] Among them, n t Represents the number of parallel connections, defined as:

[0125]

[0126] Among them, x max Indicates the maximum lateral dimension of the microconvex body; xi represents the lateral size of the i-th generation micro-protrusion; x i+1 represents the lateral size of the i+1th generation micro-asperities;

[0127] Get the TCC h of the outer ring / tapered roller interface OR and TCC h of the inner ring / tapered roller interface OI for:

[0128]

[0129] The TCC of the tapered roller / groove interface is expressed as:

[0130]

[0131] Among them, A a is 0 th represents the total contact area of ​​the contact interface, and

[0132] The beneficial effects of the present invention are:

[0133] The method for creating a thermal contact conduction model of a tapered roller / groove interface of the present invention applies the Cantor set theory to characterize the morphology of a rough surface and adopts the structure function method to identify fractal parameters; a fractal contact mechanics model is established by constructing a contact coefficient, and the influence of geometric shape and contact form on the contact area and contact load is considered; a fractal network TCC model is proposed based on the surface morphology characterization and the fractal contact mechanics model, and the matrix TCC and the shrinkage TCC are incorporated into the TCC model; experiments have shown that the maximum deviations between the predicted and measured TCCs are 5.65% and 8.70%, respectively, which proves that the designed tapered roller / groove interface contact coefficient can effectively reflect the contact state of the tapered roller and the groove, and the contact fractal mechanics model can effectively calculate the contact parameters of the tapered roller / groove interface. BRIEF DESCRIPTION OF THE DRAWINGS

[0134] 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:

[0135] Figure 1 This is a schematic diagram of the 3D Cantor set fractal surface;

[0136] Figure 2 Schematic diagram of the Cantor set fractal outline (s=3);

[0137] Figure 3 Schematic diagram of the 3D rough topography of the bearing ring groove; (a) topographic measurement equipment; (b) measured topography;

[0138] Figure 4 Surface morphology behavior; (a) power spectrum; (b) power spectrum density;

[0139] Figure 5 It is a schematic diagram of the elastic-plastic deformation model;

[0140] Figure 6 is the curve diagram of contact coefficient χ;

[0141] Figure 7 Schematic diagram of the mechanism of TCC;

[0142] Figure 8 Schematic diagram of the uneven geometric model; (a) stacking model; (b) diagram for determining m1; (c) deformation roughness;

[0143] Fig. 9 is a schematic diagram of discrete contact points;

[0144] Fig.10 is a schematic diagram of the hot runner;

[0145] Fig.11 Schematic diagram of two TCR networks contacting rough surfaces;

[0146] Fig.12 It is the structural diagram of the experimental device;

[0147] Fig.13 Model comparison diagram; (a) AISI E52100; (b) AISI304;

[0148] Fig.14 The influence curve of contact coefficient on TCC; (a) AISI E52100; (b) SS304;

[0149] Fig.15 Schematic diagram of plane / plane and curved surface / curved surface contact model; (a) traditional contact model (b) curved surface contact model;

[0150] Fig.16 is the influence curve diagram of fractal dimension;

[0151] Fig.17 It is the curve diagram of the effect of parameter s on TCC;

[0152] Fig.18 is the proportionality factor f r For the critical algebra n c Impact curve diagram;

[0153] Fig.19 is the proportionality factor f r Impact graph on TCC;

[0154] FIG20 is a graph showing the effect of rotation speed on TCC;

[0155] FIG21 is a graph showing the TCC of the oil film at the contact interface;

[0156] FIG. 22 is a graph showing the effect of dynamic viscosity on TCC. DETAILED DESCRIPTION

[0157] 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.

[0158] The method for creating a thermal contact conduction model of a tapered roller / groove interface in this embodiment includes the following contents.

[0159] 1. Characterization of surface morphology

[0160] The Cantor set theory is used to characterize the rough surface morphology, and the structure function method is used to identify the fractal dimension.

[0161] 1.1. Cantor Set Theory

[0162] The morphology of rough surfaces exhibits fractal, self-similarity and multi-scale behavior. The morphology of rough surfaces can be generated using Cantor set theory, such as Figure 1 The process of Cantor's collection theory of generating surface fractal morphology is as follows: a single micro-protrusion in the 0th generation is divided into 2s-1 segments along the horizontal direction, and then the middle segment is removed to obtain the micro-protrusions of the 1st generation; the total length of the remaining micro-protrusions in the 1st generation of micro-protrusions is 1 / f of the total length of the initial 0th generation of micro-protrusions r , the total depth of the first generation of concave parts is 1 / f of the initial 0th generation micro-protrusion depth z ; Divide the single microprotrusion in the first generation into 2s-1 segments along the horizontal direction, and then remove the middle segment to obtain the second generation of microprotrusions; The total length of the remaining microprotrusions in the second generation of microprotrusions is 1 / f of the total length of the first generation of microprotrusions r The total depth of the second generation concave part is 1 / f of the depth of the first generation micro-protrusion z ; In this way, micro-protrusions of different sizes are generated i times, and the total horizontal length and depression depth of the i-th generation micro-protrusions are obtained as follows:

[0163] l i =(1 / f r ) i-1 =(1 / f r ) i 10

[0164] h i =(1 / f z )h i-1 =(1 / f z ) i h0

[0165] Among them, l i represents the total horizontal length of the i-th generation micro-protrusions; h i represents the depression depth of the i-th generation micro-protrusion; l i-1 represents the total horizontal length of the i-1th generation microprotrusions; h i-1 represents the depression depth of the i-1th generation micro-protrusion; l0 represents the total horizontal length of the 0th generation micro-protrusion; h0 represents the depression depth of the 0th generation micro-protrusion; f r The ratio coefficient of the total length of the next generation of micro-protrusions to the previous generation of micro-protrusions; f z represents the ratio coefficient of the depth of the next generation of micro-protrusions to the previous generation of micro-protrusions; s represents the parameter to be identified;

[0166] The total number of generations i is N = s i A slight protrusion, such as Figure 2 As shown. The length of a single protrusion of the i-th generation is expressed as:

[0167] x i =(1 / sf r ) i 10

[0168] The gap width between two adjacent micro-protrusions on the i-th generation is:

[0169]

[0170] The height of the i-th generation micro-protrusion is expressed as:

[0171] z i =(1 / f z ) i (f z -1)h0

[0172] By changing the parameters s and f r and f z The proportional coefficient of can be used to obtain Cantor set fractal surfaces with different structures; the fractal dimension of the Cantor set surface is related to s and f r and f z The relationship is expressed as:

[0173]

[0174] Among them, D s represents the fractal dimension of the two-dimensional section; D p represents the fractal dimension of the 3D rough surface, and D p =D s +1. Fractal Dimension D p , proportionality factor f r and f z And the initial scale parameters l0 and h0 are obtained by measuring the morphology of the rough surface.

[0175] The parameter s is expressed as:

[0176]

[0177] The 3D roughness morphology of the bearing ring groove was measured by Keyence optical microscope, e.g. Figure 3 (a) shows the surface morphology obtained. Figure 3 (b) As shown, the curvature radius of the bearing ring groove is obvious, and there are a large number of micro-protrusions with different sizes.

[0178] Calculate the power spectrum (PS) and power spectral density (PSD), such as Figure 4 As shown. It can be seen that PS and PSD are axisymmetric, so the bearing ring groove is an isotropic surface, not an anisotropic surface. It is reasonable to use a 2D profile to approximate the 3D topography. In addition, the contour data in the vertical direction of machining should be used as 2D contour data and then used to identify the fractal dimension.

[0179] 1.2. Identifying fractal dimension

[0180] s, f r and f z A specific set of parameters is used to simulate a specific rough surface. The basic parameters used in the Cantor set fractal model can be obtained through surface measurement. The fractal dimension D of the rough surface profile p (1<D p <2) can be directly obtained through its structure function:

[0181]

[0182] Wherein, z(x) represents the height of the roughness profile at the horizontal length x, which is obtained by measuring the surface morphology; τ represents the increment in the x direction; G represents the characteristic parameter related to the surface morphology; γ represents the parameter that determines the spectral density scale ratio, and in this embodiment, γ=1.5; <> represents the statistical average value of z(x); Γ represents the first kind of elliptic integral; therefore, the physical meaning of the structure function is the mean square of the surface roughness height difference in any x direction with τ as the increment, that is, the structure function is calculated as:

[0183]

[0184] Where N represents the measurement point data; n represents the number of intervals; Δt represents the length interval in the x direction; z k+n represents the roughness height at position k+n; z k represents the roughness height at position k.

[0185] The relationship between the structure function S(τ) and the parameter τ on the double logarithmic coordinates is a straight line. The slope of the straight line can be used to obtain the fractal dimension D. p,get:

[0186] D p =(4-k s ) / 2

[0187] Among them, k s Represents the slope of the structure function S(τ).

[0188] The characteristic parameter G can be obtained by the intercept of the straight line with the z-axis; the actual contact between two rough surfaces is equivalent to the contact between a flat rigid surface and an equivalent rough surface; if the roughness heights of the two contact surfaces are z1(x) and z2(x), respectively, the equivalent surface roughness height function is:

[0189] z(x)=z1(x)-z2(x)

[0190] Thus we get:

[0191] S(τ)=<[z1(x+τ)-z1(x)]> 2 -2<[z1(x+τ)-z1(x)]><[z2(x+τ)-z2(x)]>+<[z2(x+τ)-z2(x)]> 2

[0192] Since the rough surfaces in contact with each other are statistically unrelated, the product term in the formula is zero, and the structure function of the equivalent rough surface is:

[0193] S(τ)=S1(τ)+S2(τ)

[0194] Among them, S1(τ) and S2(τ) represent the structure functions of the two contact surfaces respectively, that is, the structure function of the equivalent rough surface is the sum of the structure functions of the two contact surfaces. Therefore, the fractal dimension D of the equivalent rough surface profile is calculated. p , it is necessary to measure the morphology of the rough surface, and then obtain the structure functions of S1(τ) and S2(τ), calculate the structure function of the equivalent rough surface, the structure function is a straight line in the double logarithmic coordinate system, find the slope of the straight line, and calculate the fractal dimension D of the equivalent surface profile s , and find the intercept of the straight line and the z-axis to obtain the characteristic parameter G. For other special cases, the fractal dimension D s = 1 when the structure function S(τ)→+∞. Fractal dimension D s = 1.5 when the structure function S(τ) = πGτ / (2lnβ). Fractal dimension D s When =2, the structure function S(τ)→+∞.

[0195] 2. Constructing fractal contact mechanics model

[0196] The contact area and contact load are obtained by constructing the contact coefficient and combining the geometric shape and contact form.

[0197] In order to determine the contact parameters, the contact state of the contact point must be studied, that is, the three modes of elastic, elastoplastic and plastic deformation should be considered. In previous studies, the complete deformation of the contact point was ignored. That is, the deformation of the contact point was considered to be elastic, elastoplastic or plastic. In fact, the deformation of the contact point is a continuous process. If the deformation is not large, some of the contact states of the contact point are elastic. If the deformation is not large, some of the contact states of the contact point are elastic or elastoplastic. If the deformation is large enough, the contact state of other contact points is plastic.

[0198] 2.1 Contact coefficient

[0199] The contact between two Cantor set fractal surfaces is equivalent to the contact between an imaginary solid and an ideal smooth rigid solid, such as Figure 5 As shown in Figure 1. When an external load is applied, the imaginary entity undergoes elastic-plastic deformation. The Cantor set fractal surface is generated by stacking smaller micro-protrusions on slightly larger micro-protrusions in a self-similar manner, and then each protrusion bears the normal contact load. For a single micro-protrusion of any generation, it undergoes elastic deformation, and then as the external load increases, the deformation changes from elastic deformation to plastic deformation. According to the theory of plastic deformation, the critical load is F c =HA c , corresponds to the transition from elastic deformation to plastic deformation, where H and A c represent hardness and critical contact area respectively.

[0200] An ideal smooth rigid plane will first contact the last (or youngest) micro-protrusion, and the generation of the youngest micro-protrusion tends to infinity. th The critical load of the generation (F>F cii ), the deformation of the i-th micro-protrusion changes from elastic deformation to plastic flow. Considering the volume conservation of the plastically deformed micro-protrusion, it is assumed that the i-th micro-protrusion with plastic deformation th The micro-protrusions of each generation will flow into the gaps between the micro-protrusions and then contact the ideal smooth rigid surface of the previous generation (the i-th generation). This process continues until the n-th generation. c -1 generation ends, the critical load of this generation is greater than the applied load. c The micro-protrusions of the -1 generation will be in elastic contact with the ideal smooth rigid surface, while the nth generation c The microprotrusions of the first generation are the oldest generation that have undergone plastic deformation.

[0201] Therefore, for a given normal load, the intersection of each generation of micro-protrusions on the entire rough surface can be divided into two categories. c,∞] generation range of micro-protrusions undergo plastic deformation, [0,n c -1] generation range of micro-protrusions undergo elastic deformation. c -1 generation range of micro-protrusions actually in contact with the ideal plane, n c The value of can be expressed as:

[0202]

[0203] Where trun(x) means that the parameter x is truncated to an integer, ξ represents the stage frequency, and F represents the normal contact load.

[0204] Due to the influence of the truncation function trun(x), n c is a positive integer. When the normal contact load F increases, N≤lnξ / ln(1 / f r 2 )≤N+1, n c The critical algebra remains unchanged. When the contact load F increases to a certain value, lnξ / ln(1 / f r 2 )=N+1, n c The critical number of generations increases to n c +1. However, the above formula is only applicable to solving the critical algebra n of the contact between two rough surfaces. c The tapered roller / groove contact interface is the contact between the tapered roller surface and the cylindrical surface. Changes in surface geometry and contact form result in changes in the number of contact micro-protrusions. The contact parameters, including the actual contact load and contact area on the tapered roller / groove contact interface, will change. That is, the critical generation c It is related to the surface geometry and contact form. For the tapered roller / groove contact interface, the contact coefficient should be considered. The contact coefficient is related to the comprehensive curvature coefficient X of the two contact bodies. h , equivalent elastic modulus E, contact load F and other factors. Therefore, by considering the shape and material of the two contact bodies and the contact force, a modified critical factor is introduced. c ', using the contact coefficient to correct the critical generation n c ':

[0205] n c '=n c · χ

[0206] Among them, n c ' represents the modified critical generation; χ represents the contact coefficient, and:

[0207]

[0208] Among them, S hrepresents the theoretical contact area; ∑S represents the sum of the contact areas of the two objects, that is:

[0209] ∑S=S1±S2

[0210] Where S1 and S2 represent the contact areas of object 1 and object 2, respectively;

[0211] The contact interface of the tapered roller / groove belongs to the contact of different curvatures, that is, the contact area of ​​object 1 and object 2 can be expressed as:

[0212] S1=2π(R1+R2)l

[0213] S2=πR3B1

[0214] Among them, R1 and R2 represent the top and bottom radii of the tapered roller respectively; l represents the thickness and length of the generatrix; R3 represents the radius of the bearing ring groove; B1 represents the width of the bearing ring; thus:

[0215] ∑S=S1±S2=2π(R1+R2)l±πR3B1

[0216] s Comprehensive curvature coefficient X h Related to the radius of curvature, including ('+' is external contact, '-' is internal contact), K 12 =0, and K 22 = 0. Then the comprehensive curvature coefficient X h It is expressed as:

[0217]

[0218] Among them, '+' indicates external contact, and '-' indicates internal contact;

[0219] The contact interface of the tapered roller / bearing ring deforms under the external load, and the contact area is rectangular. The theoretical contact area when contact occurs is

[0220]

[0221] The contact coefficient is thus obtained:

[0222]

[0223] The contact load F = 1000N, the top radius R1 and bottom radius R2 of the tapered roller are 50mm and 65mm respectively, the thickness B1 is 50mm, the generatrix length l is 75mm, the cylinder radius R3 = 80, and the comprehensive elastic modulus is E = 155GPa. Calculate the contact coefficient χ, such as Figure 6As shown in Figure 2, the contact coefficient χ is always less than 1 in the entire range of R2. That is, the modified critical number n of the tapered roller / groove contact interface is c ' is less than the critical number n of contact between two rough surfaces without curvature radius c '. In addition, when R1 and R3 are fixed, the contact coefficient χ increases with R2. In particular, for external contact, the contact coefficient χ increases with R2, but the contact coefficient χ is still less than 1. For internal contact, the contact coefficient χ→1. The increase of R2 can increase the contact coefficient χ by improving the contact conditions. When R1=R2, the contact coefficient χ of the internal contact is smaller than the contact coefficient of the external contact. The number of contact points of the internal contact will be greater than the number of contact points of the external contact, then the contact load of the internal contact will be smaller than the contact load of the external contact. The above conclusions are consistent with the conclusions obtained from Hertz's elastic contact theory. The contact coefficient χ can reflect the reasonable influence of surface geometry and contact form on contact performance. Therefore, the design of the contact coefficient χ of this embodiment is credible.

[0224] 2.2 Contact area and contact load

[0225] The contact area of ​​a single micro-bump on the i-th generation is:

[0226]

[0227] There are s on the fractal interface of the i-th generation Cantor set 2i micro-protrusions, the total contact area of ​​the i-th generation micro-protrusions is expressed as:

[0228]

[0229] The total gap area between individual micro-protrusions on the i-th generation generated in each i-1th generation is obtained as:

[0230]

[0231] Among them, g i represents the gap width between two adjacent micro-protrusions on the i-th generation;

[0232] The (i-1) fractal profile in the Cantor set th There are s 2(i-1) The total contact area of ​​the i-1th generation micro-protrusion is expressed as:

[0233]

[0234] The total contact area of ​​the contact interface after correction by the contact coefficient χ is:

[0235]

[0236] When the maximum Hertz contact pressure reaches 0.6H, plastic deformation occurs, and the critical plastic deformation u c1 It is expressed as:

[0237]

[0238] Among them, H represents the microhardness of the softer material, k represents the equivalent thermal conductivity, and k = 1 / (1 / k1+1 / k2); E represents the equivalent elastic modulus, that is:

[0239]

[0240] E1 and E2 represent the Young's elastic modulus of the two contact bodies; v1 and v2 represent the Poisson's ratio of the two contact bodies; k1 and k2 represent the thermal conductivity of the two contact bodies;

[0241] When the minimum contact load applied to the micro-protrusion reaches H, complete plastic flow occurs on the contact interface; the critical plastic deformation u c1 and critical elastic deformation u c2 The relationship is expressed as:

[0242] u c2 =110u c1

[0243] When u i >u c2 When the contact load is

[0244] When u i c1 When the contact load is

[0245] The total contact load corrected by the contact coefficient χ is:

[0246]

[0247] Among them, u i Represents the contact deformation of the asperity.

[0248] 3. Construct thermal contact conduction model

[0249] Combining substrate thermal contact conduction and shrinkage thermal contact conduction, a thermal contact conduction model is constructed based on surface morphology characterization and fractal contact mechanics model.

[0250] according to Figure 7 , the total TCC h of the tapered roller / groove interface is:

[0251] h=h OR +h OI

[0252] In the formula, h​OR Indicates the TCC of the tapered roller / outer ring groove interface; h OI Represents the TCC at the tapered roller / inner ring groove interface. Heat flow is limited by the bulk TCC, which is caused by heat conduction from the contact micro-bumps. In addition, the shrinkage TCC also has an effect on heat flow, which is caused by the shrinkage of heat flow when flowing through the contact interface.

[0253] 3.1. Matrix TCC

[0254] The calculation method of substrate thermal contact conduction is:

[0255] The thermal contact resistance (TCR) of a pair of micro-bumps is expressed as:

[0256]

[0257] according to Figure 8 (a), the sum of the substrate TCR of the micro-bump contact deformation is:

[0258]

[0259] Where m1 represents an integer, and λ i The bottom dimension is x i The contact point size corresponding to the micro-bump.

[0260] The bottom dimension is x i The volume of the micro-protrusion is:

[0261]

[0262] Where, S represents the bottom area of ​​the micro-convex body; h represents the height of the micro-convex body;

[0263] The deformed micro-bumps are approximate regardless of their exact shapes, assuming that their deformation mainly occurs near the contact point, e.g. Figure 8 (c) is shown. Then the volume of the truncated micro-protrusion is expressed as

[0264]

[0265] Among them, λ i It represents the diameter of the contact point of the micro-protrusion after deformation at a given contact height d', such as Figure 8 (b) as shown.

[0266] The deformed micro-protrusions satisfy the volume conservation law, that is:

[0267]

[0268] The size λ of the contact point corresponding to the micro-protrusion can be obtained i for:

[0269]

[0270] Right now:

[0271]

[0272] Where d'=d+h m , d represents the distance between the contact rigid plane and the average plane, h m Represents the average height of micro-asperities.

[0273] 3.2. Shrinking TCC

[0274] This embodiment discusses the overall TCC of the micro-protrusions. According to the classical theory of contact heat conduction, when heat flows through the contact interface, the heat flow shrinks in the contact area. Since the actual contact area is smaller than the nominal contact area, a shrinkage TCC exists. That is, a shrinkage TCR also exists at the tapered roller / groove contact interface. The complete TCC network consists of two parts: the shrinkage TCR and the bulk TCR. To this end, the shrinkage TCR of a single contact point is calculated, and then the distribution law of the contact points on the fractal surface is used to derive the equivalent shrinkage TCC of all contact points.

[0275] Considering the actual contact situation between two objects, the contact interface is analyzed using the hot runner model. Fig.10 As shown. The contact model of shrinkage TCC at each contact point is obtained as:

[0276]

[0277] Where Q is the heat flow through the channel; ΔT c represents the temperature difference between the two interfaces; b represents the radius of the contact point; c represents the radius of the hot runner; Ψ(c / b) represents the contraction factor; k represents the equivalent thermal conductivity;

[0278] For the distribution of the contact points at the center of the micro-bump, Ψ(c / b) is approximately:

[0279]

[0280] Once the size λ of each contact point is determined i , and get the contraction TCR:

[0281]

[0282] Among them, λ i The bottom dimension is x i The contact point size corresponding to the micro-protrusion; d i =x i +x ci , x cirepresents the gap size, and:

[0283]

[0284] in, A a is the total contact area of ​​the 0th generation contact interface, and

[0285] 3.3 Contact interface TCC

[0286] As heat flows through the contact interface, the heat flow lines contract at discrete contact points. bi Produced by the microprotrusions themselves, contracting the TCR r ci There are also Fig.11 Therefore, the total TCR of a single pair of contacting microprotrusions is ti for:

[0287] r ti =r bi +r ci

[0288] Among them, r bi represents matrix TCR; r ci indicates contraction of TCR;

[0289] The contraction TCCs of all i contact points are connected in parallel, and their equivalent contraction TCRRs are ti Defined as r ti Number of parallel connections n t Ratio of:

[0290]

[0291] Among them, n t Represents the number of parallel connections, defined as:

[0292]

[0293] Among them, x max Indicates the maximum lateral dimension of the microconvex body; x i represents the lateral size of the i-th generation micro-protrusion; x i+1 Represents the lateral size of the i+1th generation micro-protrusion.

[0294] Get the TCC h of the outer ring / tapered roller interface OR and TCC h of the inner ring / tapered roller interface OI for:

[0295]

[0296] The TCC of the tapered roller / groove interface is expressed as:

[0297]

[0298] Among them, A a is the total contact area of ​​the 0th generation contact interface, and

[0299] 4. Experimental verification

[0300] 4.1 Experimental setup

[0301] The experimental setup is designed to measure the TCC of the tapered roller / groove interface, such as Fig.12 As shown. The experimental device consists of an outer ring, an inner ring, a tapered roller, an electric heater, a radiator, a thermocouple, a bolt, a compression pump, a throttle valve, a cooling pipe, and a cooling water tank. The tapered roller / outer ring groove interface and the tapered roller / inner ring groove interface are involved, and TCC is generated at the outer ring / tapered roller interface and the inner ring / tapered roller interface. The sample surface is shot peened to obtain an isotropic experimental surface with different rough morphologies. 24 Ni-Cr-Ni-Si thermocouples measure the contact interface temperature. The thermocouple measurement range is [200℃, 400℃]. The thermocouple is calibrated before measurement to ensure a measurement error of ±1.2℃. Each hole with a diameter of 1mm and a depth of 5mm is drilled on the side of the specimen with a spacing of 10mm to install the thermocouple. 4 bolts are used to adjust the contact load at the inner ring / tapered roller interface and the outer ring / tapered roller interface. The contact pressure range of the inner ring / tapered roller interface and the outer ring / tapered roller interface is lMPa-7MPa. The power of the electric heater is from 1kW to 5kW. The electric heater increases the temperature of the outer ring, and the heat sink ensures that the temperature of the inner ring is constant and lower than the temperature of the outer ring. Moreover, the compression pump is used to force the cooling gas to flow in the cooling pipe, and the throttle valve is used to adjust the flow rate of the cooling gas. The cooling water tank is filled with cooling water, which then takes away the heat from the cooling gap in the cooling pipe. Therefore, the heat flows from the outer ring to the inner ring, and the TCC of the tapered roller / groove interface is measured. The measurement principle is:

[0302]

[0303] Where ΔT is the temperature difference of the contact interface, and its value is obtained by extrapolation; Q represents the average heat flux flowing through the contact interface, which is equal to the average heat flux flowing through the outer ring, tapered roller and inner ring, Q = (Q ir +Q tr +Q or ) / 3. Q ir ,Q tr and Q or The outer ring, tapered roller and inner ring are obtained by least squares fitting. and Temperature gradient and thermal conductivity k ir ,ktr and k or The product calculation is:

[0304]

[0305]

[0306]

[0307] In order to test the predictive ability of the proposed TCC model based on Cantor set theory, TCC experiments were carried out on AISI E52100 and ANSl304. The uncertainty of heat flux is determined by the uncertainty of thermal conductivity, temperature measurement, length measurement and heat loss. The uncertainty of thermal conductivity used for calculation is 2%, the accuracy of data acquisition system is ±0.2%, the measurement range is 200℃~400℃, the measurement error is ±1.2℃, and the temperature measurement uncertainty is 0.6%. The uncertainty of length measurement is 0.4%. Heat loss is the key factor affecting TCC measurement. In the experiment, when the gas pressure is less than 0.02Pa, the residual gas heat transfer in the vacuum chamber is 0.001W, and the thermal radiation of the sample is 0.011W. The thermal conductivity of constantan thermocouple wire is 0.0023W, so the total heat loss is 0.0143W, and the minimum heat flux through the sample is 0.65W. The uncertainty caused by heat loss is 2.2%, the uncertainty of heat flux measurement is 5.2%, and the maximum uncertainty of temperature measurement in the experiment is less than 9%. According to the uncertainty transfer rule, the uncertainty of TCC measurement is (0.052 2 +0.09 2 ) 0.5 =10.4%.

[0308] 4.2 Comparison of results

[0309] The proposed model is compared with the Majumdar-Tien model, CMY model, Mikic elastic model, Mikic elastic model, Zhao model and experimental data, e.g. Fig.13As shown in the figure. Majumdar-Tien model, CMY plastic model and Mikic plastic model overestimate TCC values. Mikic elastic model underestimates TCC results. The predicted TCC obtained by Zhao model is lower than the experimental data. For the proposed model, the maximum deviation between the predicted and measured TCC for samples #1 and #2 is 5.65% and 8.70%, respectively. The results show that the predicted TCC is in good agreement with the experimental results, and the prediction accuracy of the model proposed in this embodiment is significantly higher than that of the classical model. TCC increases with the increase of contact pressure because the actual contact area increases with the increase of contact pressure. The larger the actual contact area, the more micro-protrusions are in contact, and then with the increase of contact pressure, the more elastic and plastic deformation of micro-protrusions occurs, and then the heat flow channel increases significantly, and finally TCC increases. When a certain pressure is reached, the rising trend of TCC slows down. In addition, the nonlinear relationship between TCC and contact pressure is obtained by this model, which is different from the linear relationship obtained by Majumdar-Tien's TCC model. The nonlinear relationship between contact coefficient χ and contact load is considered in this study. In previous studies, the above nonlinear relationship was not considered, which led to the contact coefficient χ being linearly related to the contact load. Finally, a linear relationship was obtained through the TCC model of Majumdar-Tien. Therefore, the design of the contact coefficient χ of the tapered roller / groove interface is reasonable. In addition, with the introduction of the contact coefficient χ, the fractal contact mechanics model of the tapered roller / groove interface can effectively calculate the contact parameters.

[0310] 4.3 Results and Discussion

[0311] 4.3.1. Influence of contact coefficient

[0312] This example studies the effect of the contact coefficient on TCC. Fig.14 As shown in Figure 2. When the contact coefficient χ is considered, the TCC is smaller than the case without the contact coefficient χ because the contact coefficient χ is less than 1. In addition, the difference between the TCC with the contact coefficient χ and the TCC without the contact coefficient χ increases with the increase of the contact pressure, which proves the necessity of considering the contact coefficient χ. There are differences between the traditional rough surface contact model without curvature radius and the curved surface contact model with curvature radius, such as Fig.15 The contact between two rough surfaces without curvature radius is equivalent to the contact between rigid planes. The equivalent rough plane is as follows: Fig.15 (a). The contact between two surfaces with a radius of curvature is shown in Fig.15As shown in (b), the curvature radius of the two surfaces affects the number of contact micro-protrusions and the contact area. The contact area is mainly determined by the comprehensive curvature coefficient, the equivalent elastic modulus and the load between the two contacting bodies. The geometry of the contact surface, the material of the two contacting bodies, and the contact force all affect the contact area. It is expected that the number of contact micro-protrusions of the external contact between the two surfaces is smaller than the number of contact micro-protrusions of the contact between the two surfaces. Moreover, the number of contact micro-protrusions of the internal contact between the two surfaces is greater than the number of contact micro-protrusions of the external contact between the two surfaces.

[0313] 4.3.2 Influence of fractal dimension

[0314] The predicted and experimental TCC vary nonlinearly with the fractal dimension, e.g. Fig.16 As shown. Under different contact pressures, TCC gradually increases with fractal dimension. The reason is that fractal dimension has an effect on the height of micro-protrusions. With the increase of fractal dimension, the micro-protrusions on the rough surface become flatter. That is, with the increase of fractal dimension, the number of micro-protrusions increases while their height decreases. The above two reasons lead to the increase of TCC. In addition, Fig.16 The relationship between TCC and contact pressure is shown in Fig.14 Much the same as shown.

[0315] 4.3.3 Impact of parameter s

[0316] The effect of parameter s on TCC is as follows: Fig.17 As shown in the figure, the larger the parameter s is, the larger the TCC value is. In addition, the rate of change of TCC gradually decreases, and the orientation is finally stable. The parameter s corresponds to the number of micro-protrusions in each generation. According to the Cantor set rough surface model, the increase of parameter s will have a series of effects on the surface morphology. The increase of parameter s leads to i th Microprotrusion length x i The decrease and i th Number of microprotrusions 2i A sharp increase. th The total length of the micro-protrusion does not change with the parameter s. Moreover, the parameter s has no significant effect on the height z of the micro-protrusion. i No impact, th Microprotrusion length x i The decrease of parameter s will make the shape of the microprotrusion become slender and steep. The increase of parameter s leads to the increase of body TCRr bi The decrease of , and then the total TCC increases with the parameter s. In addition, TCC increases with the increase of contact pressure, which is consistent with the conclusion shown in Section 5.3.2.

[0317] 4.3.4. Proportional coefficient f r Impact

[0318] Proportional coefficient f r For the critical algebra nc The impact of Fig.18 As shown. Proportional coefficient f r Impact on TCC, such as Fig.19 As shown. The i-th microprotrusion of each generation generated from the 0th generation microprotrusion has a proportional coefficient f r The larger the value, the smaller the length of the i-th generation microprotrusion. r The increase in the height z i and the number of the i-th generation micro-protrusions. For the same sampling length l0, the number and height of the i-th generation micro-protrusions generated by the 0th generation micro-protrusions will not change, but their plane size will change due to the proportionality coefficient f r decreases with the increase of. Therefore, the proportionality factor f r The increase of leads to an increase in the TCC of a pair of micro-protrusions. r The total contact area of ​​any i-th generation micro-bump is changed, and the critical generation number n of the micro-bump elastic-plastic transition is changed. c Under the same load, the proportionality factor f r The increase in the critical generation n c Transfer from the next generation to the previous generation, such as Fig.18 As shown, the number of contact micro-bumps is reduced. In addition, the critical generation n c The reduction of will reduce the number of micro-protrusions involved in heat transfer. Then the total TCC shows a downward trend. r When the critical generation n increases from 1.25 to 2, c The change is slow, the TCC effect of a single pair of bumps gradually increases, the effect of the reduction of the number of bumps weakens, and TCC shows a slight upward trend. r When increasing from 1.25 to 2, TCC increases from about 220W / m2K to about 520W / m2K.

[0319] 4.3.5 Influence of rotation speed

[0320] For tapered roller bearings, the rotation of the inner ring relative to the outer ring is inevitable. A lubricating oil film is generated at the tapered roller / inner ring groove and tapered roller / outer ring groove interfaces. The TCC of the lubricating oil film should be considered. The rotation speed has an effect on the TCC of the lubricating oil film. The thickness of the lubricating oil film at the contact area corresponding to the oil inlet is h min , the lubricating oil film thickness in other parts of the contact area is approximately the average center oil film thickness h0. The thickness of the center oil film is used to calculate the TCC of the oil film. For tapered roller bearings, the oil film is relatively thin and the contact load is relatively large. The TCC model of the contact point oil film is equivalent to a plane heat transfer model, and the TCC of each contact area is:

[0321]

[0322] Where k and A represent thermal conductivity and elliptical contact area, respectively.

[0323] The average central oil film thickness h0 is expressed as:

[0324] h0=2.62U 0.7125 G 0.53 W -0.16 (1-0.61e -0.73k )

[0325] Among them, U and W represent two introduced intermediate variables, which are related to the rotational speed and the contact load; G = λE, where λ represents the viscosity coefficient and E represents the equivalent elastic modulus; k represents the ratio of the contact length to the contact width.

[0326] According to Fourier's law:

[0327]

[0328] Among them, T1 and T2 represent the temperatures of two objects.

[0329] The heat flux is expressed as:

[0330]

[0331] The TCC of an oil film is defined as:

[0332]

[0333] As the speed increases, the TCC of the oil film decreases as the oil film thickness increases, e.g. Fig. 20 As shown. The lubricating oil flow between the rolling element and the inner and outer rings increases with the increase of the rotational speed. When the rotational speed is not particularly high, the TCC of the oil film at the inner ring / tapered roller interface is slightly larger than the TCC of the oil film at the outer ring / tapered roller interface, because the maximum local elastic extrusion between the tapered roller and the outer ring is greater than the maximum local elastic extrusion between the tapered roller and the inner ring. The oil film thickness at the outer ring / tapered roller interface is greater than the oil film thickness at the inner ring / tapered roller interface. As the rotational speed continues to increase, the TCC of the oil film at the outer ring / tapered roller interface is smaller than the TCC of the oil film at the inner ring / tapered roller interface, because the centrifugal force at the outer ring / tapered roller interface is greater than the centrifugal force at the outer ring / tapered roller interface. Due to the existence of centrifugal force, the contact area of ​​the outer ring / tapered roller interface increases. When the rotational speed is 7000r / min and the radial load is 100N, the TCC of the oil film changes with the change of load, as shown in Fig.21As shown in the figure. The TCC of the oil film at the outer ring / tapered roller interface and the inner ring / tapered roller interface increases. At the same speed, the TCC of the oil film at the inner ring / tapered roller interface is greater than the TCC of the oil film at the outer ring / tapered roller interface. Moreover, when the axial load is small, the contact pressure has a greater effect on the TCC of the oil film, and when the contact pressure is large, the contact pressure has a smaller effect on the TCC of the oil film.

[0334] 4.3.6 Effect of dynamic viscosity

[0335] The TCC of oil films at different dynamic viscosities was obtained, such as Fig. 22 As shown. The TCC of the oil film at the inner ring / tapered roller interface and the outer ring / tapered roller interface decreases significantly with the increase of dynamic viscosity lubricant. At the same speed, the TCC of the oil film at the inner ring / tapered roller interface is slightly greater than the TCC of the oil film at the outer ring / tapered roller interface. Since the dynamic viscosity has a great influence on the TCC of the oil film, it generally does not change in actual work. Therefore, it is important to select a suitable lubricant with reasonable viscosity according to the actual working conditions.

[0336] 5. Conclusion

[0337] This embodiment proposes a TCC modeling method for the tapered roller / groove interface, using the Cantor set theory to characterize the surface morphology, then using the structure function method to identify fractal parameters, and by constructing the contact coefficient to consider the influence of the contact body's geometry and contact form on TCC, a contact mechanics model is proposed. Finally, by considering volume TCC and shrinkage TCC, a TCC model is proposed. This embodiment also discusses the fractal dimension, parameter s, and scale factor f r , speed and dynamic viscosity on TCC. The main conclusions are summarized as follows:

[0338] (1) The fractal network TCC model proposed in this example can effectively predict the TCC of the tapered roller / groove interface, and the proposed model is much more accurate than the traditional model. For the proposed model, the maximum deviations between the predicted and measured TCCs of samples #1 and #2 are 5.65% and 8.70%, respectively, which proves that the fractal rough surface morphology characterization model based on the Cantor set theory can describe the random, disordered, self-affine and multi-scale characteristics of the rough surface.

[0339] (2) This embodiment constructs the contact coefficient χ of the tapered roller / groove interface and introduces it into the fractal contact mechanics model. The designed contact coefficient χ of the tapered roller / groove interface effectively reflects the contact state between the tapered roller and the groove. The contact coefficient χ can describe the influence of geometry and contact form on TCC. In addition, the contact coefficient χ of the tapered roller / groove interface is less than 1. When the contact coefficient χ is considered, since the contact coefficient χ is less than 1, the TCC is less than the TCC without the contact coefficient χ.

[0340] (3) This embodiment introduces the contact coefficient χ and establishes a fractal contact mechanics model of the tapered roller / groove interface. The elastic deformation, elastoplastic deformation and plastic deformation of the contact micro-protrusions are taken into account. The fractal contact model is suitable for the calculation of the contact parameters of the tapered roller / groove interface. The contact parameters such as the actual area and contact load of the tapered roller / groove interface are determined by the fractal contact model with the contact coefficient. This solves the problem that the traditional fractal contact mechanics model cannot calculate the contact parameters of the tapered roller / groove interface.

[0341] (4) This embodiment studies the contact coefficient, fractal dimension, parameter s, and scale factor f r , speed and dynamic viscosity on TCC. TCC increases with fractal dimension, parameter s and contact pressure. In addition, TCC decreases with increasing speed, and TCC decreases with the dynamic viscosity of the lubricant. In addition, TCC decreases with the proportional coefficient f r decreases with the increase of the proportional coefficient f r increases with the increase of .

[0342] 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 creating a thermal contact conduction model of a tapered roller / groove interface, characterized in that: Characterize surface morphology: Use Cantor set theory to characterize rough surface morphology, and use structure function method to identify fractal dimension; Construct a fractal contact mechanics model: by constructing the contact coefficient and combining the geometric shape and contact form to obtain the contact area and contact load; Constructing a thermal contact conduction model: Combining substrate thermal contact conduction and shrinkage thermal contact conduction, a thermal contact conduction model is constructed based on surface morphology characterization and fractal contact mechanics model; The method of Cantor set theory to characterize the rough surface morphology is: The single microprotrusion in the initial 0th generation is divided into 2s-1 segments along the horizontal direction, and then the middle segment is removed to obtain the microprotrusions of the first generation; the total length of the remaining microprotrusions in the first generation microprotrusions is 1 / f of the total length of the initial 0th generation microprotrusions r The total depth of the first generation of depressions is 1 / f of the initial 0 generation micro-protrusion depth. z ; The single microprotrusion in the first generation is divided into 2s-1 segments along the horizontal direction, and then the middle segment is removed to obtain the second generation of microprotrusions; the total length of the remaining microprotrusions in the second generation of microprotrusions is 1 / f of the total length of the first generation of microprotrusions r The total depth of the second generation concave part is 1 / f of the depth of the first generation micro-protrusion z ; In this way, micro-protrusions of different sizes are generated i times, and the total horizontal length and depression depth of the i-th generation micro-protrusions are obtained as follows: the i =(1 / f r )l i-1 =(1 / f r ) i l0 h i =(1 / f z )h i-1 =(1 / f z ) i h0 Among them, l i represents the total horizontal length of the i-th generation micro-protrusions; h i represents the depression depth of the i-th generation micro-protrusion; l i-1 represents the total horizontal length of the i-1th generation microprotrusions; h i-1 represents the depression depth of the i-1th generation micro-protrusion; l0 represents the total horizontal length of the 0th generation micro-protrusion; h0 represents the depression depth of the 0th generation micro-protrusion; f r The ratio coefficient of the total length of the next generation of micro-protrusions to the previous generation of micro-protrusions; f z The ratio coefficient of the depth of the next generation of micro-protrusions to the previous generation of micro-protrusions; s Indicates the parameters that need to be identified; The total number of generations i is N = s i micro-protrusions, and the length of a single micro-protrusion of the i-th generation is expressed as: x i =(1 / sf r ) i l0 The gap width between two adjacent micro-protrusions on the i-th generation is: The height of the i-th generation micro-protrusion is expressed as: <h2 style=";text-align:left;direction:ltr">z<h2 style=";text-align:left;direction:ltr"> i <h2 style=";text-align:left;direction:ltr"> (1 / f)<h2 style=";text-align:left;direction:ltr"> z <h2 style=";text-align:left;direction:ltr"> )<h2 style=";text-align:left;direction:ltr"> i <h2 style=";text-align:left;direction:ltr"> (f<h2 style=";text-align:left;direction:ltr"> z <h2 style=";text-align:left;direction:ltr"> -1)h0 By changing the parameters s and f r and f z The proportional coefficient of can be used to obtain Cantor set fractal surfaces with different structures. The fractal dimension of the Cantor set surface is related to s 、f r and f z The relationship is expressed as: Among them, D s Represents the fractal dimension of the two-dimensional section; D p represents the fractal dimension of the 3D rough surface, and Dp = D s +1; The parameter s is expressed as: The method to identify the fractal dimension is: The fractal dimension Dp of the rough surface profile can be directly obtained through its structure function: Where z(x) represents the height of the roughness profile at the horizontal length x; τ represents the increment in the x direction; G represents the characteristic parameter related to the surface morphology; γ represents the parameter that determines the spectral density scale ratio; < > represents the statistical mean of z(x); Γ represents the first kind of elliptic integral; The physical meaning of the structure function is the mean square of the surface roughness height difference with τ as an increment in any x direction, that is, the structure function is calculated as: Where N represents the measurement point data; n represents the number of intervals; Δt represents the length interval in the x direction; z k+n represents the roughness height at position k+n; z k represents the roughness height at position k; The relationship between the structure function S(τ) and the parameter τ on the double logarithmic coordinates is a straight line. The slope of the straight line can be used to obtain the fractal dimension D. p ,get: D p =(4-k s ) / 2 Among them, k s represents the slope of the structure function S(τ); The characteristic parameter G can be obtained by the intercept of the straight line with the z-axis; the actual contact between two rough surfaces is equivalent to the contact between a flat rigid surface and an equivalent rough surface; if the roughness heights of the two contact surfaces are z1(x) and z2(x), respectively, the equivalent surface roughness height function is: z(x)=z1(x)-z2(x) Thus we get: S(τ)=([z1(x+τ)-z1(1]]) 2 -2([z1(x+τ))-z1(x])><[[z2(x+τ)-z2(x)]>+<[z2(x+τ)-z2(x)]> 2 Since the rough surfaces in contact with each other are statistically unrelated, the product term in the formula is zero, and the structure function of the equivalent rough surface is: S(τ)=S1(τ)+S2(τ) Among them, S1(τ) and S2(τ) represent the structure functions of the two contact surfaces, that is, the structure function of the equivalent rough surface is the sum of the structure functions of the two contact surfaces; The contact coefficient is constructed as follows: The contact between two Cantor set fractal surfaces is equivalent to the contact between an imaginary entity and an ideal smooth rigid entity. When an external load is applied, the imaginary entity undergoes elastic-plastic deformation. According to the theory of plastic deformation, the critical load is F c =HA c , corresponds to the transition from elastic deformation to plastic deformation, where H and A c represent hardness and critical contact area respectively; The ideal smooth rigid plane will first contact the last generation of micro-protrusions; if the normal load is greater than the critical load of the i-th generation, the deformation of the i-th generation of micro-protrusions will change from elastic deformation to plastic flow; then the ideal smooth rigid plane will contact the previous generation of micro-protrusions, and this process will continue until the n-th generation. c -1 generation ends, the critical load of this generation is greater than the applied load, and the nth generation c The micro-protrusions of the -1 generation will be in elastic contact with the ideal smooth rigid surface, while the nth generation c The micro-protrusions of the first generation are the oldest generation that have undergone plastic deformation; therefore, for a given normal load, the intersection of each generation of micro-protrusions on the entire rough surface can be divided into two categories: [n c ,∞] generation range of micro-protrusions undergo plastic deformation, [0,n c -1] micro-protrusions within the range of n undergo elastic deformation; c -1 generation range of micro-protrusions actually in contact with the ideal plane, n c The value of can be expressed as: Where, trun(x) means that the parameter x is truncated to an integer, ξ represents the stage frequency, and F represents the normal contact load. For the contact interface of tapered roller / groove, the contact coefficient should be considered. The contact coefficient and the comprehensive curvature coefficient X of the two contact bodies h , equivalent elastic modulus E, contact load F and other factors, and the contact coefficient is used to correct the critical algebra n c ': n c ′n c ·h Among them, n c ' represents the modified critical number; χ represents the contact coefficient, and: Among them, S h represents the theoretical contact area; ∑S represents the sum of the contact areas of the two objects, that is: ∑S=S1±S2 Where S1 and S2 represent the contact areas of object 1 and object 2, respectively; The contact interface of the tapered roller / groove belongs to the contact of different curvatures, that is, the contact area of ​​object 1 and object 2 can be expressed as: S1=2π(R1+R2)l S2=πR3B1 Among them, R1 and R2 represent the top and bottom radii of the tapered roller respectively; l represents the thickness and length of the generatrix; R3 represents the radius of the bearing ring groove; B1 represents the width of the bearing ring; thus: ∑S=S1±S2=2π(R1+R2)l±πR3B1 Comprehensive curvature coefficient X h It is expressed as: Among them, '+' indicates external contact, and '-' indicates internal contact; The theoretical contact area when tapered roller / bearing ring contact occurs is: The contact coefficient is thus obtained: The solution for the contact area and contact load is: The contact area of ​​a single micro-bump on the i-th generation is: There are s on the fractal interface of the i-th generation Cantor set 2i micro-protrusions, the total contact area of ​​the i-th generation micro-protrusions is expressed as: The total gap area between individual micro-protrusions on the i-th generation generated in each i-1th generation is obtained as: Among them, g i represents the gap width between two adjacent micro-protrusions on the i-th generation; There are s in the i-1th generation of the Cantor set fractal profile. 2(i-1) The total contact area of ​​the i-1th generation micro-protrusion is expressed as: The total contact area of ​​the contact interface after correction by the contact coefficient χ is: When the maximum Hertz contact pressure reaches 0.6H, plastic deformation occurs, and the critical plastic deformation u c1 It is expressed as: Among them, H represents the microhardness of the soft material, k represents the equivalent thermal conductivity, and k = 1 / (1 / k1+1 / k2); E represents the equivalent elastic modulus, that is: E1 and E2 represent the Young's elastic modulus of the two contact bodies; v1 and v2 represent the Poisson's ratio of the two contact bodies; k1 and k2 represent the thermal conductivity of the two contact bodies; When the minimum contact load applied to the micro-protrusion reaches H, complete plastic flow occurs on the contact interface; the critical plastic deformation u c1 and critical elastic deformation u c2 The relationship is expressed as: in c2 =110u c1 When u i >u c2 When the contact load is When u i <u c1 When the contact load is The total contact load corrected by the contact coefficient χ is: Among them, u i represents the contact deformation of the asperity; The calculation method of substrate thermal contact conduction is: The contact thermal resistance TCR of a pair of micro-bumps is expressed as: The sum of the substrate TCR of the micro-bump contact deformation is: Where m1 represents an integer, and λ i The bottom dimension is x i The contact point size corresponding to the micro-protrusion; z i represents the height of the i-th generation micro-protrusion; The calculation method of shrinkage heat contact conduction is as follows: According to the classical theory of contact heat conduction, when the heat flow passes through the contact interface, the heat flow shrinks in the contact area. Since the actual contact area is smaller than the nominal contact area, there is a shrinkage TCC. Considering the actual contact situation of the two objects, the contact interface is analyzed using the hot runner model, and the contact model of the shrinkage TCC of each contact point is obtained as follows: Where Q is the heat flow through the channel; ΔT c represents the temperature difference between the two interfaces; b represents the radius of the contact point; c is the hot runner radius; Ψ(c / b) represents the contraction factor; k represents the equivalent thermal conductivity; For the distribution of the contact points at the center of the micro-bump, Ψ(c / b) is approximately: Get the contraction TCR: Among them, λ i The bottom dimension is x i The contact point size corresponding to the micro-protrusion; d i =x i +x ci , x ci represents the gap size, and: in, A a Represents the total contact area of ​​the 0th generation contact interface; A g represents the total contact area of ​​the contact interface after correction by the contact coefficient χ; φ represents A g With A a The ratio between .

2. The method for creating a thermal contact conduction model of a tapered roller / groove interface according to claim 1, characterized in that: The bottom dimension is x i The volume of the micro-protrusion is: Where, S represents the bottom area of ​​the micro-convex body; h represents the height of the micro-convex body; The volume of the truncated microprotrusion is expressed as: Among them, λ i represents the diameter of the contact point of the deformed micro-protrusion at a given contact height d'; The deformed micro-protrusions satisfy the volume conservation law, that is: The size λ of the contact point corresponding to the micro-protrusion can be obtained i for: Right now: Where d'=d+h m , d represents the distance between the contact rigid plane and the average plane, h m represents the average height of the micro-convex body; z i represents the height of the i-th generation micro-protrusion.

3. The method for creating a thermal contact conduction model of a tapered roller / groove interface according to claim 1, characterized in that: Total TCR of a single pair of contact micro-bumps r ti for: r ti ＝r bi +r ci Among them, r bi represents basal TCR; r ci indicates contraction of TCR; The contraction TCCs of all i contact points are connected in parallel, and their equivalent contraction TCRRs are ti Defined as r ti Number of parallel connections n t Ratio of: Among them, n t Represents the number of parallel connections, defined as: Among them, x max Indicates the maximum lateral dimension of the microconvex body; x i represents the lateral size of the i-th generation micro-protrusion; x i+1 represents the lateral size of the i+1th generation micro-asperities; Get the TCC h of the outer ring / tapered roller interface OR and TCC h of the inner ring / tapered roller interface OI for: The TCC of the tapered roller / groove interface is expressed as: Among them, A a is the total contact area of ​​the 0th generation contact interface, and

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