# Thermal contact conduction modeling method for contact interface of channel and tapered roller

## A technology of tapered rollers and contact interface, applied in the field of heat conduction, can solve the problems such as the influence of contact parameters

Pending Publication Date: 2022-04-12

CHONGQING UNIV

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## AI-Extracted Technical Summary

### Problems solved by technology

So far, fractal contact mechanics models have not been implemented directly for tapered roller/ditch interfaces because surface geometry and contact form have an influence on contact parameters

In addition, there is no relevant...

### Method used

[0286] Surface randomness has a strong influence on the value of TCC, as shown in FIG. 14 . In order to simulate the real topography of the rough surface, the rough surface is generated using fractal theory and Monte Carlo method. The position of each microbump is truly random by generating truly random An, Bn, Cn, r1 and r2 values using a Monte Carlo method. Furthermore, the height of each micro-protrusion is truly random bec...

## Abstract

The invention discloses a thermal contact conduction modeling method for a contact interface of a channel and a tapered roller, which comprises the following steps of: representing surface topography: generating a rough surface by using a fractal theory and a Monte Carlo method to obtain fractal dimensions; constructing a contact model: introducing a coincidence factor aiming at the geometrical shape and the contact form of the tapered roller/channel interface, and correcting the contact area and the contact load by utilizing the coincidence factor; and constructing a thermal contact conduction model: constructing the thermal contact conduction model based on the surface topography characterization and the contact model by combining matrix thermal contact conduction and shrinkage heat basic conduction. According to the thermal contact conduction modeling method for the contact interface of the channel and the tapered roller, the analysis requirement of thermal contact conduction of the tapered roller/channel interface can be met.

Application Domain

Geometric CADDesign optimisation/simulation +3

Technology Topic

EngineeringRough surface +6

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## Examples

- Experimental program(1)

### Example Embodiment

[0104] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments to better understand the invention and can be implemented, but the exemplary embodiments are not limited to the present invention.

[0105] The channel and cone roller contact interface thermal contact conduction mode method, including the following:

[0106] 1, characterize surface morphology

[0107] The rough surface is generated using fractal theory and Monte Carlo method to obtain fractal dimension.

[0108] Further, the WeiersTrass-Mandelbrot fractal surface with fractal dimension D is expressed as:

[0109]

[0110] Where C n Is independent and obedient distribution n (μ = 0, σ 2 Real random number of = 1); a n And B n It is independent and uniform distribution of [0, 2π]; 2 <3; λ is a constant greater than 1; n is a natural sequence number;

[0111] X (n) = l / 2 + l / 2 · cos (r 1 · 2π)

[0112] Y (n) = l / 2 + l / 2 · cos (r 2 · 2π)

[0113] Among them, R 1 R 2 Indicates two real random numbers; l represents the length of the nominal;

[0114] For each microphone, the following relationship is met:

[0115] 0≤x (n) + b (n) / 2 ≤ L

[0116] 0≤y (n) + b (n) / 2 ≤ L

[0117] Among them, B (n) represents the diameter of the microjection.

[0118] Specifically, the identification method of fractal dimension D is:

[0119] (1) Randomly select a base point X in the surface area 0 = (X 0 Y 0 ), Determine Z 0 = Φ (x 0 );

[0120] (2) The remaining data points are in x 0 = (X 0 Y 0 ) The center of the polar grid is sampled. which is:

[0121] X 0 + h = x 0 + R (cos θ, sinθ)

[0122] Wherein, h represents the height; R represents the radius; θ represents an angle;

[0123] (3) z = φ (x 0 + R j (cos θ j SINθ j )) Is determined by 0 ≤ i ≤ m-1 and 0 ≤ j ≤ N, where i represents the i-th microjection; m represents the number of maximum microjjjjjjjjjjj

[0124] (4) Assumption Δ ij = | Z ij -z 0 , The average value is calculated:

[0125]

[0126] (5) Draw the data point on the logarithmic coordinate, select the positive integer as N* And 5≤n * ≤n, so that the data is concentrated in the first N * A strong linear trend is displayed in a data point to get a {(logR j LOGδ j : 0≤j ≤n * )} H The slope line, the identified franked number is:

[0127]

[0128] Among them, L H Indicates the slope.

[0129] The measured cyclic groove is like figure 1 As shown, the number of microices has many times and the dimensions are different. Then calculate the power spectrum (PS) and power spectral density (PSD), such as figure 2 (a) and 2 (b) shown. It can be seen that PS and PSD are axis symmetrical. Therefore, the bearing sleeve groove is the same-nature, not an anisotropy. figure 2 (c) The surface topography of the identified fractal dimension specification is shown in accordance with equation (1).

[0130] 2, build contact model

[0131] Introducing the compounding factor for the geometry and contact form of the tapered roller / channel interface, and the contact area and contact load are corrected using the overflow factor.

[0132] Deformation of each contact point is a continuous process. If the deformed is not large, some contact status of the contact point is elastic or elastic. If the deformation is large enough, the contact state of its contact point is plastic.

[0133] 2.1, coincidence factor

[0134] The coincidence factor χ is designed for tapered roller / channel interface. The introduction of the compounding factor is:

[0135]

[0136] Where S h Represents nominal area; σs represents the total surface area, X h Represents a comprehensive curvature coefficient;

[0137] ΣS = S 1 ± S 2

[0138] Where, '+' indicates an external contact, '-' indicates inner contact; 1 S 2 The surface area of the object 1 and the object 2 is respectively, respectively; and the surface area of the object 1 and the object 2 is shown in:

[0139] S 1 = Π (r 1 + R 2 )

[0140] S 2 = ΠR 3 B 1

[0141] Among them, R 1 R 2 The top and bottom radius of the tapered roller are respectively, respectively; L represent bus length; R 3 A radius of the bearing sleeve; B 1 Represents the width of the bearing ring;

[0142] The total surface area of the two surface σs is obtained:

[0143] ΣS = S 1 ± S 2 = Π (r 1 + R 2 ) L ± πR 3 B 1

[0144] Comprehensive curvature coefficient x h Related to the radius of curvature, expressed as:

[0145]

[0146] The theoretical contact area is obtained when contact occurrence:

[0147]

[0148] The resulting compound factor χ is expressed as:

[0149]

[0150] Where f represents the contact load; E represents an equivalent elastic modulus.

[0151] Contact load f = 1000n; tapered roller top radius r 1 50mm; thickness B 1 90mm; bus length L is 75mm; inner cylindrical radius r 3 = 80mm; integrated elastic modulus is E = 155 GPa, calculate the factor χ, such as image 3 Indicated. Be 1 R 3 When fixed, the compound factor χ with R 2 Increased increase, but the coincidence factor χ is always less than 1. For internal contact, coincidence factor χ → 1; r 2 The increase can be expanded to match the factor χ, so the design of the factor χ is trusted.

[0152] 2.2, contact area and contact load

[0153] 2.2.1, critical contact area

[0154] Assume that each microphone is a half-ball. The contact of the hemisphere of the rigid plane and the radius R Figure 4 Indicated. Crown plastic deformation δ c1 for:

[0155]

[0156] Among them, H represents a microscopic hardness; K and E represent an equivalent thermal conductivity and an equivalent elastic modulus.

[0157] Crown plastic contact area is:

[0158]

[0159] According to the Monte Carlo method, the diameter of each microphone is obtained:

[0160]

[0161] Among them, B min And B max Indicates minimum and maximum diameter of each microphone; i Indicates the true random number.

[0162] Critical elastic deformation δ c2 for:

[0163]

[0164] Critical elastic contact area is:

[0165]

[0166] 2.2.2, single-to-microphone contact parameters

[0167] (1) Elastic deformation

[0168] When Δ c1 When it is calculated according to Hz Theory, a single microphone contact parameter is calculated:

[0169] A = (r δ) 1/2

[0170] A e = Πa 2

[0171]

[0172] Among them, a represents the diameter of the microjection; δ represents the amount of micro intensive body; e Expressive contact area when deformation; f e Indicates the contact load at the time of elastic deformation; E represents an equivalent elastic modulus; R represents the diameter of the top end of the microjjjjj

[0173] (2) Complete plastic deformation

[0174] Δ> δ c2 When calculating a single microphone contact parameter:

[0175] A = (2R δ) 1/2

[0176] A p = Πa 2

[0177] Fly p = ΠHA 2

[0178] Among them, A p Contact area during complete plastic deformation; f p Contact loads indicating complete plastic deformation.

[0179] (3) Elastic contact

[0180] It is necessary to establish an elastoplastic deformation model. Assume that the deformation is small (Δ / r << 1) and the deformation is mainly concentrated in the contact area. The contact area does not deform at a certain height h, such as Figure 4 Indicated. Therefore, a control volume is formed, and the upper surface diameter is D, the lower surface diameter is b, and the height is h.

[0181] The height of the control volume in the critical state is h i ,Expressed as:

[0182] hide i = Kδ

[0183] Where k is constant; δ represents the amount of deformation of the micro intensity.

[0184] The upper surface diameter is:

[0185] di i = 2 (Rδ p ) 1/2

[0186] The lower surface diameter is:

[0187] B = 2 [2R (H i + δ p )] 1/2

[0188] The volume of the control volume is:

[0189]

[0190] Δ> δ p At the time, the additional plastic relative normal approximation is expressed as δ-δ p The height of the control volume is represented as:

[0191] h = h i - (δ-δ p )

[0192] Thus the volume of the control volume is:

[0193]

[0194] Assume that the volume of the control volume constant, that is, V = V i It can solve the upper surface diameter:

[0195] di 2 = 4RδC

[0196] in:

[0197]

[0198] Among them, δ p The amount of plastic deformation of the micro-integrated body is shown.

[0199] When the value of K is large enough, select C is:

[0200]

[0201] Then the contact area is expressed as:

[0202]

[0203] in:

[0204]

[0205] Therefore, the elastic contact area A ep Isometric elastic contact area a e. If the nominal is relatively close, it is equal to the critical nominal relative (δ = δ p ), Then a ep = ΠRδ = a e If the nominal is relatively near, it is relatively close to the name of the critical (δ >> Δ) p ), Then a ep = 2πRδ = a p. In order to get contact pressure, it is necessary to define a rubber hardness H ep :

[0206]

[0207] Therefore, that is, if the nominal is close to δ = δ p Elastoplastic hardness h ep Equal to F / A e , And f / a e = 0.4H, h ep = 0.4h; if the nominal relative approximation is much larger than the critical nominal relative approximation (δ >> δ p ), Rubber hardness h ep Is equal to material hardness, hu ep = H. So defining the elastoplastic hardness H ep It is reasonable.

[0208]

[0209] 2.2.3, total contact parameters

[0210] Further, the distribution of microevers is:

[0211]

[0212] Among them, a L Indicates the maximum contact area; a represents the contact area of each microphone; φ represents the contact integral function;

[0213] Correct N (a) by coincidence factor χ, obtained:

[0214] n * (a) = χn (a)

[0215] The total contact load is obtained by considering the overbred factor:

[0216]

[0217] which is:

[0218] As a L c1 Time:

[0219]

[0220] As a c1 ≤a L c2 Time:

[0221]

[0222] As a L A c2 Time:

[0223]

[0224] Among them, a c1 Representation of critical plastic contact area; A c2 Critical elastic contact area; F e , F p And f ep The contact load when the single pair of micro-converged elastic deformation contact, complete plastic deformation contact, and elastic deformation contact;

[0225] Total contact area is obtained by considering the compounding factor χ:

[0226]

[0227] which is:

[0228] As a L c1 Time:

[0229]

[0230] As a c1 ≤a L c2 Time:

[0231]

[0232] As a L A c2 Time:

[0233]

[0234] Among them, A e , A p And a ep The contact area when a single pair of elastic deformation contact, complete plastic deformation contact, and elastoplastic deformation contact, respectively, respectively.

[0235] 3. Construct a thermal contact conduction model

[0236] Combined with matrix thermal contact conduction and shrinkage heat infrastructure, a thermal contact conduction model is constructed based on surface morphology characterization and contact model.

[0237] Figure 5 (a) The temperature at which the contact interface is significantly decreased, resulting in limiting the resistance of the heat flow. according to Figure 5 (b), the total TCC H of the tapered roller / channel interface is:

[0238] h = h OR + h OI

[0239] Where H OR Hidden OI The TCC of the tapered roller / outer ring channel and the tapered roller / inner ring channel interface, respectively.

[0240] 3.1, matrix TCC

[0241] The heat flow is limited by the block TCC, and the block TCC is caused by heat conduction of contact with micropions;

[0242] The height of the micropions that define the size B is:

[0243] z i = G (D-1) B i (2-D)

[0244] Where g represents the scale constant; B i Indicates the diameter of the micro-integrated body;

[0245] Then a single microphone TCR is:

[0246]

[0247] Where k represents an equivalent thermal conductivity; c represents a constant;

[0248] Then the overall TCR of the microphone is:

[0249]

[0250] Among them, B m Indicates the diameter of the mth microjection; M 1 Indicates an integer, and di i Based on the size of B i The micro-convergence of the corresponding contact point size.

[0251] 3.2, shrink TCC

[0252] The 3.1 section discusses the microphone Tcc. According to the classic theory of contact heat transfer, there is a contraction Tcc. That is, a shrinking TCR is also present at the tapered roller / channel contact interface. The complete TCC network includes two parts: shrinking TCR and substrate TCR. To this end, the shrinking TCR of a single contact point is calculated, and then the equivalent shrinkage Tcc of the contact point is derived by the distribution law of the fractal surface contact point. Considering the actual contact of the two objects, the contact interface is analyzed by the thermal runway model. The contact between a pair of contact points is like Image 6 As shown, the shrinking Tcc of each contact point is:

[0253]

[0254] Among them, q indicates the heat flow through the channel; Δt c Indicates temperature drop; B i Represents the radius of microphone; c i Indicates contact radius; k represents an equivalent thermal conductivity; ψ (c / b) represents a shrink factor, in this embodiment:

[0255] (C) i / B i ) = 1-1.40918c i / B i +0.33801 (C i / B i ) 3 +0.06792 (C i / B i ) 5 + ...

[0256] 3.3, total TCC

[0257] Base TCR R bi Contract TCR R ci ,Such as Figure 7 Indicated. Total TCRR in contact with micro ti for:

[0258] rim ti = R bi + R ci

[0259] All I contact points are parallel, their equivalent shrinkage TCR R ti Defined as a series TCRR ti Dip connected in parallel N t The ratio:

[0260]

[0261] Where N t Indicates the number of parallel connections;

[0262] Get tcc h from tapered roller / outer ring channel interface OR for:

[0263]

[0264] Get tcc h from tapered roller / inner ring channel interface OI for:

[0265]

[0266] Among them, R cIndicates contact thermal resistance;

[0267] Thus, the total TCC H of the tapered roller / channel interface is:

[0268] h = h OR + h OI

[0269] Where H OR Hidden OI The TCC of the tapered roller / outer ring channel and the tapered roller / inner ring channel interface, respectively.

[0270] 4, model verification

[0271] 4.1, measurement method and equipment

[0272] Figure 8 The design setting of measuring tapered roller / channel interface Tcc is shown, which consists of an outer ring, an inner ring, a tapered roller, an electric heater, a heat sink, a thermocouple, a bolt, and a cooling system. The surface of the sample was taken to obtain an asstructive experimental surface having different rough morphology. Then, a 24 Ni-Cr-Ni-Si thermocouple, the thermocouple has a measurement range of [200 ° C, 400 ° C], and the thermocouple is calibrated prior to measurement, ensuring that the measurement error is ± 1.2 ° C. 4 bolts are used to adjust contact loads. The contact pressure range is LMPA to 7MPa. The power of the electric heater is from 1 kW to 5 kW. Electric heater enhances the temperature of the outer ring. In addition, the compression pump is used to force the cooling gas to flow in the cooling tube. The cooling tank is filled with cooling water, and then the heat of the cooling gas in the cooling tube is taken away. Measure the TCC of the tapered roller / channel interface. The measurement principle is:

[0273]

[0274] Among them, ΔT is a temperature difference of the contact interface, and its value is obtained by the external push method; Q indicates the average thermal flux flowing through the contact interface, equal to the average thermal flux flowing through the outer ring, the cone roller, and the inner ring, Q = ( Qi ir + Q tr + Q or ) / 3. The heat flux Q of the outer ring, tapered roller and inner ring ir Q tr Sum or Substant and Temperature gradient and thermal conductivity K ir K tr And K or Product calculation, namely:

[0275]

[0276]

[0277]

[0278] 4.2, comparison

[0279] In order to test the predictive ability of the proposed TCC model, the sample was used to use AISI E52100 and ANSL304 TCC experiments. The model will be proposed to the Majumdar-Tien model, the CMY model, the Mikic elastic model, the Mikic elasticity model, the ZHAO model, and the experimental data, such as Figure 9 Indicated. Majumdar-Tien Model], the CMY plastic model and the Mikic plastic model were overestimate TCC value. The MIKIC elastic model underestimates TCC results. The predicted TCC obtained by Zhao Mode is lower than the experimental data. The maximum deviation between the proposed model, the prediction of the sample # 1 and # 2, and the maximum deviation between the measured TCC is 5.65% and 8.70%, respectively. The larger the actual contact area, the more microphes that are in contact, and then the heat flow channel is significantly increased, and the final TCC increases. When reaching a certain pressure, the rise in the TCC slowed down. In addition, the nonlinear relationship obtained by this model is different from the linear relationship obtained by the TCC model of Majumdar-Tien. This embodiment takes into account the coincidence factor, in a conventional study, there is no consideration of the above nonlinear relationship. Finally, linear relationships are obtained by Majumdar-Tien's TCC model. Therefore, the overflow factor design of the cone roller / channel interface is reasonable.

[0280] 5, results and discussion

[0281] 5.1, the impact of the coincidence factor

[0282] Figure 10 The impact of the compounding factor on TCC is shown. The difference between TCC having a compound factor and the TCC without coincidence increases as the contact pressure increases. Figure 11 The contact performance of the rough surface of the conventional radius of curvature is described is different from the contact performance of the surface of the curvature radius. The radius of the curvature of the two curved surfaces affects the number of contacts and the total contact area. For internal contact, the larger the radius of curvature, the smaller the number and the total contact area of the contact. For external contact, the larger the radius of curvature, the larger the number of contacts and the total contact area.

[0283] 5.2, influence of fractal dimension

[0284] Figure 12 The TCC showing the predicted and experiments is non-linear variation with D. At different fractal dimensions, TCC gradually increases because D will affect the height of each protrusion. In fact, G and D represent the rough surface structure. The height of each microphone is in the form of D's influence. Further, the height of the micropions is reduced with D. The larger the fractal dimension D, the more flat and slightly. Accordingly, the number of micro-projections increases. The above two reasons have led to an increase in TCC. D on the surface morphology Figure 13 Indicated. In fact, D has an impact on low frequency and high frequency components. Big D means that the high-frequency component of the surface profile dominates, which corresponds to a small height. In addition, when D becomes large, more micro-convex is in contact. That is, the contact area formed by these microjes is large, resulting in a smaller TCR formed in these microphes.

[0285] 5.3, the effect of surface randomness

[0286] Surface randomness affects the value of TCC, such as Figure 14 Indicated. In order to simulate the true terrain of the rough surface, a rough surface is generated using fractal theory and Monte Carlo method. Generate real random A by using Monte Carlo Method n , B n , C n R 1 R 2 Value, each microphone position is truly random. In addition, the height of each microphone is truly random because the height of each microphone is generated with Monte Carlo method. By generating true random R i To ensure high randomness. In fact, Monte Carlo method is a method of simulating a random process for obtaining diameter and height of micropions to simulate changes in microphone. Therefore, the impact of surface randomness should be considered when developing TCC models.

[0287] 5.4, influence of speed and dynamic viscosity

[0288] For tapered roller bearings, the rotational rotation of the inner ring relative to the outer ring is inevitable. A lubricating oil film is generated at a tapered roller / inner ring channel and a tapered roller / outer ring channel interface. The TCC of the oil film is defined as:

[0289]

[0290] Among them, K 1 Indicates thermal conductivity; h 0 It is the thickness of the oil film.

[0291] When the speed is increased, the oil film thickness h 0 Increase. The TCC then decreases as the speed increases, such as Figure 15 Indicated. Due to the presence of centrifugation, the lubricant is transferred from the inner ring to the outer ring, and then the average thickness of the outer ring H. 0 Increase. Figure 16 It indicated that when the rotational speed was 7000 r / min, the radial load was 100 N, the TCC of the oil film varied as the contact pressure. Different dynamic viscosity of the oil film TCCS, such as Figure 17 Indicated. The oil film Tccs at the conical roller / inner coil interface and the tapered roller / outer ring channel interface is significantly reduced as the dynamic viscosity lubricant increases. It is important to choose a suitable, viscosity lubricant according to the actual conditions.

[0292] 6 Conclusion

[0293] This embodiment establishes a fractal geometry - Monte Carlo model of fractal rough surface. The most significant difference between the model is to emphasize the random, disorder, multi-scale structure characteristics of the rough surface. It uses fractal parameters independent of the scale to ensure randomness of each microphone. The uniqueness of this method is to use random number to determine the height and position of microphone on the rough surface. The model produces a non-Gaussian surface with any direction distribution, which is more in line with the actual situation than many other models.

[0294] The compounding factor χ can reflect the effects of geometric shapes and contact forms, considering E, R 1 R 2 R 3 , F, B 1 , B 2 And l on the impact of the coincidence factor. The problem of solving the traditional method cannot analyze the performance of surface contact performance, the proposed fractal contact mechanical model has also laid a certain theoretical basis for TCC modeling.

[0295] For the inner contact of the tapered roller / inner ring channel, the coincidence factor is:

[0296]

[0297] For the outer contact of the tapered roller / outer ring channel, the coincidence factor is:

[0298]

[0299] This embodiment constructs a fractal network TCC model of a tapered roller / channel interface based on fractal theory and Monte Carlo method, and incorporates the body TCC and the contraction TCC into the TCC model. Constructs the impact of the compounding factor to reflect the geometric shape and contact form, and then in contact modeling. In addition, the effects of coincidence factors, fractal dimension, surface randomness, speed, and dynamic viscosity on TCC were also studied. conclusion as below:

[0300] (1) The proposed fractal network TCC model is effective for the TCC modeling of the tapered roller / channel interface, and is more accurate than the traditional model. Volume TCC and contraction TCC are considered. The maximum TCC deviation of the sample # 1 and # 2 is 5.65% and 8.70%, respectively. The effects of coincidence factors, fractal dimensions, surface randomness, speed, and dynamic viscosity on TCC were studied. TCC has a growth trend with the increase in fractal dimension and contact pressure. The TCC decreases with the speed and dynamic viscosity. When considering the surface randomness, TCC is greater than the TCC that does not consider the surface randomness.

[0301] (2) Established fractal geometry - Monte Carlo model of fractal rough surface. The most prominent difference is that it emphasizes the random, disorder and multi-scale structural characteristics of the rough surface. Surface randomness is achieved by revising traditional fractal functions, and each microphone is determined by the true random number. The uniqueness of this method is that the Monte Carlo model uses a true random number to determine the size and position of unevenness on the rough surface. The model can generate non-Gaussian surfaces distributed in any direction, which is more in line with the actual situation than many other models.

[0302] (3) The overflow factor χ of the tapered roller / channel interface is constructed, which is effective to characterize the contact performance of the conical roller / channel interface. The coincidence factor χ can describe the influence of geometry and contact form on TCC. In addition, the coincidence factor of the cone roller / channel interface is less than 1, and then the contact modeling is made, solving the problem of the conventional method cannot analyze the performance of the surface contact performance.

[0303] The above embodiments are merely illustrative of the preferred embodiments of the present invention, and the scope of the invention is not limited thereto.Those skilled in the art will be within the scope of the invention on the scope of the invention.The scope of protection of the present invention is subject to the claims.

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