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风电法兰摩擦因数预测与表面工艺优化∗

  • 施敏捷

    机 构：

    太原科技大学重型机械教育部工程研究中心 太原 030024

    ×
  • ， 王建梅

    机 构：

    太原科技大学重型机械教育部工程研究中心 太原 030024

    ×
  • ， 侯定邦

    机 构：

    太原科技大学重型机械教育部工程研究中心 太原 030024

    ×
  • ， 宁可

    机 构：

    太原科技大学重型机械教育部工程研究中心 太原 030024

    ×

太原科技大学重型机械教育部工程研究中心 太原 030024；

Friction Factor Prediction and Surface Technology Optimization of Wind Power Flange

  • SHI Minjie

    Affiliation：

    Engineering Research Center Heavy Machinery Ministry of Education, Taiyuan University of Science and Technology, Taiyuan 030024 , China

    ×
  • ， WANG Jianmei

    Affiliation：

    Engineering Research Center Heavy Machinery Ministry of Education, Taiyuan University of Science and Technology, Taiyuan 030024 , China

    ×
  • ， HOU Dingbang

    Affiliation：

    Engineering Research Center Heavy Machinery Ministry of Education, Taiyuan University of Science and Technology, Taiyuan 030024 , China

    ×
  • ， NING Ke

    Affiliation：

    Engineering Research Center Heavy Machinery Ministry of Education, Taiyuan University of Science and Technology, Taiyuan 030024 , China

    ×

Engineering Research Center Heavy Machinery Ministry of Education, Taiyuan University of Science and Technology, Taiyuan 030024 , China；

作者简介:

施敏捷,男,1995年出生,硕士研究生。主要研究方向为摩擦学与界面科学。E-mail:smj20190289@163.com

王建梅(通信作者),女,1972年出生,博士,教授,博士研究生导师。主要研究方向为摩擦学与界面科学、重大装备关键基础件研发。E-mail:wjmhdb@163.com

中图分类号:TH117

文献标识码:A

DOI:10.11933/j.issn.1007-9289.20210722001

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目录contents

    摘要

    风力发电机法兰承受较大切向载荷,法兰结合面的静摩擦因数直接影响法兰连接性能,目前法兰静摩擦因数预测理论和表面工艺优化的研究较少。 为建立风电法兰结合面静摩擦因数预测模型,探究表面工艺参数对法兰静摩擦因数的影响。 基于分形理论构建法兰结合面静摩擦因数的分形预测模型,并采用摩擦试验对预测模型的准确性进行验证。 设计正交试验来探究表面粗糙度、表面处理工艺、表面涂层多种表面工艺参数对静摩擦因数的影响,建立表面工艺参数与静摩擦因数映射数学模型,基于该数学模型得到产生最大静摩擦因数的工艺组合方案。 研究结果表明:静摩擦预测模型具有较高的准确性, 为法兰精确化设计提供了理论基础。 对正交试验结果进行极差和方差分析得出:表面粗糙度对静摩擦因数影响最小,表面处理工艺次之,表面涂层的影响最大,产生最大静摩擦因数的表面工艺为:表面粗糙度为 Ra 6. 3、表面喷丸处理、涂覆 Paint B 油漆,建立的表面工艺参数与静摩擦因数映射数学模型能够准确获得最优表面工艺参数,缩短了法兰表面工艺设计周期。

    Abstract

    The flange of wind turbine bears large tangential load, and the static friction factor of flange joint surface directly affects the flange connection performance. At present, there are few studies on the prediction theory of flange static friction factor and surface process optimization. In order to establish the prediction model of static friction factor of wind power flange joint surface, the influence of surface process parameters on flange static friction coefficient is explored. The fractal prediction model of static friction factor of flange joint surface is constructed based on fractal theory, and the accuracy of the prediction model is verified by friction experiment. Orthogonal experiments are designed to explore the effects of surface roughness, surface treatment process and surface coating on the static friction factor. The mapping mathematical model between surface process parameters and static friction factor is established. Based on the mathematical model, the process combination scheme with the maximum static friction factor is obtained. The results show that the static friction prediction model has high accuracy, which provides a theoretical basis for the accurate design of flange. The range and variance analysis of the orthogonal test results show that the surface roughness has the least influence on the static friction factor, the surface treatment process takes the second place, and the surface coating has the greatest influence. The surface processes that produce the maximum static friction factor are roughness Ra 6. 3, surface shot peening, coating Paint B. The mapping mathematical model between surface process parameters and static friction factor can accurately obtain the optimal surface process parameters and shorten the flange surface process design cycle.

  • 0 前言

  • 海上风电运行环境十分复杂,高温、高湿、高盐雾和长日照等外部环境非常苛刻,启停频繁、超重载、交变载荷等内部工作环境恶劣,这对风力发电机上进行传动和紧固的摩擦副提出了更高要求[1-4]。 Q345E、球墨铸铁是风力发电机法兰常用的摩擦副材料,在实际工程中,法兰承受切向载荷,切向载荷由法兰结合面的摩擦力进行平衡,结合面的静摩擦因数直接影响摩擦力的大小,从而影响摩擦副承受切向载荷的能力[5]。

  • GREENWOOD和WILLIAMSON [6] 于1966年首次提出微观-宏观接触模型,并利用表面形貌特征 (粗糙高度分布和表面分离)建立了数学-统计弹性模型,称为GW模型,该模型在大载荷下接触载荷接触面积曲线是不连续的。针对这一现象,2000年ZHAO等[7-8]建立连续的弹塑性接触模型,该模型解决了接触的不连续问题。 WEBSTER等[9] 提出粗糙表面数值模型,该模型利用表面形貌数据进行接触建模,分析接触区域的压力分布和变形。分形的概念由MANDELBROT等[10] 提出, 经过GANTI和BHUSHAN [11]的发展,认为工程表面具有分形特征。基于此,MAJUMDAR和BHUSHAN [12]总结并提出粗糙表面接触的二维分形模型,即M-B模型。该模型首次将粗糙表面轮廓参数与接触载荷和接触面积相关联,以获得与比例无关的接触模型。 PAN等[13-19] 通过研究静摩擦因数随塑性指数、法向载荷的变化规律得出,静摩擦因素随着塑性指数、法向载荷的增大呈现下降趋势。以上国内外学者建立的摩擦因数预测模型主要通过仿真数据定性地分析摩擦副参数对静摩擦因数的影响,缺乏理论结合实际加工工艺的摩擦因数研究。

  • 海上风力发电机的防腐以涂层防腐和金属热喷涂为主,表面涂层或金属热喷涂后必然会影响摩擦副表面的静摩擦因数,从而影响到其承受切向载荷的能力[20]。 ElAGINA等[21]等对表面涂覆氮化钛涂层的摩擦副进行了研究,研究表明氮化钛涂层能显著提高接触面摩擦因数。 WANG等[22] 对喷丸后不同涂层材料对摩擦副抗滑移系数的影响进行了研究,研究结果表明,由于喷丸处理后摩擦副之间的摩擦力由涂层间的范德华力提供,涂层之后的法兰面拥有更高的摩擦因数,不同涂层材料的静摩擦因数有差异性。 SHAN等[23] 对含涂层材料进行了摩擦特性对比试验,结果表明含涂层材料比普通材料的摩擦因数40%。郝建民[24] 发现喷丸之后的试件相比未喷丸试件有着更高的摩擦因数和抗滑移性能; 摩擦副表面的静摩擦因数受到多工艺参数的影响。国内外学者对多工艺参数耦合的法兰静摩擦因数研究较少,为此,本文通过研究不同表面工艺参数对静摩擦因数的影响,建立了工艺参数与静摩擦因数的映射数学模型,基于该模型得到产生最大静摩擦因数的工艺参数,为实际工程提供理论指导。

  • 1 静摩擦因数预测

  • 1.1 分形理论

  • 在M-B接触模型中(见图1),假设两个粗糙面之间的接触为一个理想的刚性光滑表面和一个粗糙表面的接触,推导静摩擦因数与载荷的关系。真实接触面积 A ∗ r 与接触载荷 P ∗ 满足如下关系[12]。

  • 图1 M-B接触模型

  • Fig.1 M-B contact model

  • P*=4πG*(D-1)3g1(D)Ar*D2×(D-2)Ar*D(3-2D)-αc*(3-2D)/2+KΦg2(D)Ar*D/2αc*(2-D)/2D≠1.5
    (1)
  • P*=πG*1/2Ar*33/4ln⁡Ar*3αc*+3KΦ4Ar*33/4αc*D=1.5
    (2)
  • P*=PAaE;G*=GAa;Ar*=ArAaαc*=αcAa=G*2(KΦ/2)2/(D-1)g1(D)=D3-2D2-DDD/2g2(D)=2-DD(2-D)/2E=1-ν12E1+1-ν22E2-1
    (3)

  • 式中, A a 为名义接触面积;E 为综合弹性模量,E 1、 E 2 为配对材料的弹性模量,v 1、v 2 为配对材料的泊松比;K 为与硬度 H 及屈服强度的相关因数;Φ 是屈服强度R P 0.2 与弹性模量的相关因数,Φ=R P 0.2 /E;A r 是实际接触总面积。

  • 根据黏着摩擦理论,总摩擦力可表示为:

  • F=Fa+Fe=Arτb+Fe
    (4)

  • 式中, F 为总摩擦力,F a 为黏着力,F e 为犁沟力,τb 为较软金属的剪切强度。一般情况下犁沟力较小, 可以忽略不计[25]。摩擦因数与载荷的关系可以简化为

  • μ=FP=ArτbP
    (5)

  • 选取不同表面粗糙度试件( Ra 1.6、Ra 3.2、 Ra 6.3)各一块,使用三丰SJ-420表面粗糙度仪测绘不同粗糙度表面轮廓,该粗糙度仪轮廓绘制精度可达0.001 μm,能准确绘制试件表面的表面轮廓,绘制的轮廓图见图2,每组图由试件表面的微观照片与轮廓数据图组成,每组图中的第一幅图是试件的微观照片,其他3幅是具体的轮廓数据图。

  • 根据轮廓数据计算出分形维数 D 和尺度系数 G, 运用结构函数法对轮廓数据进行处理[26],结构函数满足如下关系式

  • σ(τ)=[z(x+τ)-z(x)]2=Gτ4-2D
    (6)

  • 式中, τ 为坐标位移增量;z(x) 为W-M函数。对 (6)式两边取对数,在双对数坐标中对lgσ(τ)-lgτ 进行直线拟合,直线拟合图如图3所示。

  • 根据拟合直线的斜率 k 和截距,计算得到分形维数 D 和尺度系数 G,见表1。表2为试件参数。将表1、表2数据,以及三个不同法向载荷(9kPa、 15kPa、25kPa)代入式(1)、(3)~(5),得到试件随表面粗糙度和法向载荷变化的静摩擦因数预测值, 如表3和图4所示。

  • 由图4可知:静摩擦因数随着载荷增加呈现下降趋势。不同表面粗糙度静摩擦因数大小排序为: Ra 6.3>Ra 3.2>Ra 1.6。

  • 1.2 摩擦试验验证

  • 采用平板摩擦试验对静摩擦因数预测值的准确性进行验证,为保证试验数据的准确性,每组试验按照图5试验方向进行3次试验,最终静摩擦因数结果取3次试验的平均值。

  • 平板摩擦试验采用Labthink生产的兰光MXD02摩擦因数测定仪对静摩擦因数进行测定,该摩擦因数测定仪的精度为0.5级,可以实现试件静、动摩擦因数的准确测量。试验按照标准GB10006进行, 试验时间42s,速度100mm/min。

  • 图2 表面轮廓图

  • Fig.2 Surface profile

  • 表1 分形系数

  • Table1 Fractal coefficient

  • 图3 lgσ(τ)-lgτ 直线拟合图

  • Fig.3 lgσ(τ)-lgτ line fitting diagram

  • 表2 试件参数

  • Table2 Specimen parameters

  • 表3 静摩擦因数预测值

  • Table3 Static friction coefficient predictive value

  • 图4 静摩擦因数预测值

  • Fig.4 Static friction coefficient predictive value

  • 图5 试验方向

  • Fig.5 Experimental direction

  • 试验原理如图6所示,试验结果如图7所示。试验初始时,试件未出现相对滑动,摩擦因数随着时间不断上升,到达摩擦因数最高点时,试件出现相对滑动,摩擦因数开始略有下降,由于试件表面微凸体分布不均,剪切力不连续,摩擦因数曲线图呈现上下波动的现象,确定静摩擦因数为试件即将出现相对滑动时的摩擦因数值。

  • 图6 试验原理图

  • Fig.6 Experimental schematic

  • 对比静摩擦因数预测值和试验值可知:静摩擦因数预测值接近试验值,相对误差值在10%之内。对于分形预测模型来说,该静摩擦因数预测值具有较高的准确性。摩擦副静摩擦因数随载荷变化如图8所示,预测值与实测值随载荷变化趋势大致相同: 随着法向载荷增加,静摩擦因数呈现下降趋势,这是由于表面不平整度与微凸体高度差相关,随着法向载荷增加,微凸体被压缩,表面不平整度降低,摩擦切向阻力变小,从而导致摩擦因数下降。

  • 图7 摩擦因数试验曲线图

  • Fig.7 Friction coefficient experimental graph

  • 表4 理论值与实测值对比

  • Table4 Comparison of theoretical value and measured value

  • 图8 试验值对比预测值

  • Fig.8 Experimental value comparison predictive value

  • 2 表面工艺正交试验

  • 摩擦副表面静摩擦因数受表面粗糙度、表面处理工艺、表面涂层这三种表面工艺参数影响较大,为研究不同参数对静摩擦因数的影响程度,建立表面工艺参数与静摩擦因数映射数学模型,寻找产生最大静摩擦因数的工艺组合方案。本文对不同表面粗糙度(Ra 1.6、Ra 3.2、Ra 6.4)、表面处理方式(表面喷丸、表面喷锌、无表面处理)、表面涂层种类(PaintA、Paint B、无涂层)处理后的试件进行摩擦因数试验,为了提高试验效率和更好量化影响因素对试验结果的影响程度,本次试验采用正交试验法[27] L 9(3 4) 进行试验规划,正交试验表如表5所示。

  • 表5 正交试验表

  • Table5 Orthogonal test table

  • 按照表5加工试件,其中上试件尺寸为20mm× 40mm×3mm,下试件尺寸为240mm×130mm×3mm, 加工好的试样要求平整、无皱纹和可能改变摩擦性质的伤痕,试样试验表面要求无灰尘、指纹和任何可能改变表面性质的外来物质,试件如图9所示。试验之前,试件在标准环境23℃、50%RH下放置16h以上。按照正交试验表顺序进行试验,每组试验进行3次,结果取其平均值,正交试验结果如表6所示,并对正交试验结果进行极差和方差分析,结果见表7与表8。

  • 图9 已加工试件

  • Fig.9 Machined specimen

  • 表6 正交试验结果

  • Table6 Table of orthogonal test results

  • 表7 极差分析表

  • Table7 Table of orthogonal test results

  • 表8 方差分析

  • Table8 Analysis of variance

  • 对正交试验结果进行极差分析,极差越大表明该因素对结果的影响越大。从表7和图10中可以得出:极差大小排序为C>B>A,即表面涂层对正交试验结果影响最大,表面处理工艺次之,表面粗糙度影响最小,设表面粗糙度对静摩擦因数的影响系数为1,表面处理工艺和表面涂层对结果的影响系数分别为1.97、3.13。产生最大静摩擦的工艺参数为:表面粗糙度Ra 6.3(A1)、表面喷丸处理(B1)、涂覆Pain B油漆(C2)。

  • 为了进一步验证正交试验分析的准确性,对正交试验结果进行了方差显著性检验。根据F分布分位数表查询得到: F 0.05(2,2)=19,F 0.1(2,2)=9, 方差分析结果见表8。可以得出,表面涂层对静摩擦因数影响最大,表面处理工艺次之,表面粗糙度对静摩擦因数影响最小,这与极差分析的结果一致,证明了极差分析的准确性。表面涂层 F> F 0.05(2,2)=19,证明表面涂层这个因素对静摩擦因数结果有着显著影响,通过观察加工好的试件,表面喷丸涂覆Paint B油漆有着更为粗糙的表面形貌,在摩擦过程中剪切力较大,从而有着更大的静摩擦因数。

  • 图10 极差分析图

  • Fig.10 Range analysis chart

  • 3 结论

  • (1) 基于分形理论建立了法兰结合面在不同粗糙度和法向载荷时的摩擦模型,通过摩擦因数试验对摩擦因数预测值的准确性进行了验证。验证结果表明:摩擦因数预测值与试验值相接近,最大相对误差小于10%,该预测模型具有较高的准确性,为法兰精确化设计提供了理论基础。

  • (2) 对不同表面粗糙度、表面处理工艺、表面涂层多种表面工艺参数的试件进行了正交试验,对试验结果进行极差和方差分析,根据极差分析建立了多个表面工艺参数耦合下的法兰静摩擦因数映射数学模型,该数学模型能够准确获得最优表面工艺参数,缩短了法兰设计制造周期。

  • (3) 通过构建的数学模型,得到最佳工艺组合, 即当表面粗糙度为Ra 6.3、表面进行喷丸处理、涂覆Paint B油漆,法兰结合面的静摩擦因数最大,该工艺组合相较传统表面工艺,法兰承载能力得到了较大的提升,为法兰的尺寸优化和轻量化设计留出了空间。

  • 参考文献

    • [1] 梁义,周云龙,盛忠起,等.机械能助渗锌铝渗层的防腐耐磨性能分析[J].中国表面工程,2020,33(2):65-74.LIANG Y,ZHOU Y L,SHENG Z Q,et al.Anticorrosion and wear properties of mechanical energy assisted Zn-Al coating[J].China Surface Engineering,2020,33(2):65-74.(in Chinese)

    • [2] DÍAZ H,SOARES C G.Review of the current status technology and future trends of offshore wind farms[J].Ocean Engineering,2020,209:107381.

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    • [4] EOM S H,KIM S S,LEE J B.Assessment of anti-corrosion performances of coating systems for corrosion prevention of offshore wind power steel structures [J].Coatings,2020,10(10):970.

    • [5] DENG H,CHAO L,SONG X,et al.Tensile resistance and design model of an external double-layered flange connection[J].Journal of Constructional Steel Research,2019,161:309-327.

    • [6] GREENWOOD J A,WILLIAMSON J B P.Contact of nominally flat surfaces[J].Proceedings of the Royal Society of London.Series A.Mathematical and Physical Sciences,1966,295(1442):300-319.

    • [7] ZHAO Y,MAIETTA D M,CHANG L.An asperity microcontact model incorporating the transition from elastic deformation to fully plastic flow[J].Journal of Tribology,2000,122(1):86-93.

    • [8] ZHAO Y,CHANG L.A model of asperity interactions in elasticplastic contact of rough surfaces[J].Journal of Tribology,2001,123(4):857.

    • [9] WEBSTER M N,SAYLES R S.A numerical model for the elastic frictionless contact of real rough surfaces [J].ASME J Tribology,1986,108(3):314-320.

    • [10] MANDELBROT,BENOIT B.The fractal geometry of nature[J].American Journal of Physics,1998,51(3):468.

    • [11] GANTI Suryaprakash,BHUSHAN Bharat.Generalized fractal analysis and its applications to engineering surfaces[J].Wear,1995,180(1-2):17-34.

    • [12] MAJUMDAR A,BHUSHAN B.Role of fractal geometry in roughness characterization and contact mechanics of surfaces[J].Trans ASME J Tribology,1990,112(2):205-216.

    • [13] PAN W,LI X,WANG L,et al.Influence of surface topography on three-dimensional fractal model of sliding friction.[J].AIP Advances,2017,7(9):095321-1-095321-12.

    • [14] GARCÍA J M,MARTINI A.Measured and predicted static friction for real rough surfaces in point contact [J].Journal of Tribology,2012,134(3):031501.

    • [15] CHANG W R,ETSION I,BOGY D B.Static friction coefficient model for metallic rough surfaces [J].Journal of Tribology,1988,110(1):57-63.

    • [16] 王建梅,陶德峰,黄庆学,等.多层圆筒过盈配合的接触压力与过盈量算法研究[J].工程力学,2013(9):278-283.WANG J M,TAO D F,HUANG Q X,et al.Algorithm research on contact pressure and magnitude of interference for multi-layer cylinder’s interference fit [J].Engineering Mechanics,2013,30(9):270-275.(in Chinese)

    • [17] WU J,WANG Z X,WANG G Q.The key technologies and development of offshore wind farm in China[J].Renewable and Sustainable Energy Reviews,2014,34:453-462.

    • [18] WANG J,NING K,XU J,et al.Reliability-based robust design of wind turbine’s shrink disk[J].ARCHIVE Proceedings of the Institution of Mechanical Engineers Part C:Journal of Mechanical Engineering Science,2018,232(15):2685-2696.

    • [19] 盛选禹,雒建斌,温诗铸.基于分形接触的静摩擦因数预测 [J].中国机械工程,1998,9(7):16-18.SHENG X Y,LUO J B,WENG S Z.Static friction coefficient model based on fractal contact [J].China Mechanical Engineering,1998,9(7):16-18.(in Chinese)

    • [20] SETH P,RITA F.Corrosion protection systems and fatigue corrosion in offshore wind structures:current status and future perspectives[J].Coatings,2017,7(2):25.

    • [21] ELAGINA O Y,KOMADYNKO A C,POLESHCHUK E D,et al.Prospects for using Titanium Nitride coatings for the contact surfaces of friction clutches[J].Journal of Friction and Wear,2020,41(1):25-30.

    • [22] WANG Y B,WANG Y,CHEN K,et al.Slip factor of high strength steel with inorganic zinc-rich coating[J].Thin-Walled Structures,2020,148:106595.

    • [23] SHAN L,WANG Y,LI J,et al.Tribological behaviours of PVD TiN and TiCN coatings in artificial seawater [J].Surface and Coatings Technology,2013,226:40-50.

    • [24] 郝建民,耿刚强,张智龙.电弧喷涂铝对高强度螺栓摩擦型连接面抗滑移系数的影响[J].长安大学学报(自然科学版),2003(4):85-87.HAO J M,GENG G Q,ZHANG Z L.Effect of arc spraying aluminium on anti-glide coefficient of friction type high-strength bolt adjacent plane[J].Journal of Xi’ an Highway University,2003.(in Chinese)

    • [25] CHEN W W,WANG Q J.A numerical static friction model for spherical contacts of rough surfaces,influence of load,material,and roughness [J].Journal of Tribology,2009:131(2):021402.

    • [26] 朱华,葛世荣.结构函数与均方根分形表征效果的比较[J].中国矿业大学学报,2004,33(4):396-399.ZHU H,GE S R.Comparison of fractal characterization effects of structure function and mean square root [J].Journal of China University of Mining and Technology,2004,33(4):396-399.(in Chinese)

    • [27] WANG Z F,WANG H.Inflatable wing design parameter optimization using orthogonal testing and support vector machines [J].Chinese Journal of Aeronautics,2012,25(6):887-895.

  • 基本信息

    中图分类号: TH117
    文献标识码: A
    DOI: 10.11933/j.issn.1007-9289.20210722001

    基金信息

    国家自然科学基金(52035006, 51875382)；
    第六批山西新兴产业领军人才(202006)；
    山西省回国留学人员科研 (2020—125) 资助项目；
    Supported by National Natural Science Foundation of China (52035006, 51875382)；
    the Sixth Batch of Funding Projects for Leading Talents of Emerging Industries in Shanxi (202006)；
    Research Project Supported by Shanxi Scholarship Council of China (2020—125).；

    引用信息

    稿件历史

    收稿日期: 2021-07-22

  • 参考文献

    • [1] 梁义,周云龙,盛忠起,等.机械能助渗锌铝渗层的防腐耐磨性能分析[J].中国表面工程,2020,33(2):65-74.LIANG Y,ZHOU Y L,SHENG Z Q,et al.Anticorrosion and wear properties of mechanical energy assisted Zn-Al coating[J].China Surface Engineering,2020,33(2):65-74.(in Chinese)

    • [2] DÍAZ H,SOARES C G.Review of the current status technology and future trends of offshore wind farms[J].Ocean Engineering,2020,209:107381.

    • [3] VICTOR I,ALI M,ATHANASIOS K.Materials selection for XL wind turbine support structures:A corrosion-fatigue perspective [J].Marine Structures,2018,61:381-397.

    • [4] EOM S H,KIM S S,LEE J B.Assessment of anti-corrosion performances of coating systems for corrosion prevention of offshore wind power steel structures [J].Coatings,2020,10(10):970.

    • [5] DENG H,CHAO L,SONG X,et al.Tensile resistance and design model of an external double-layered flange connection[J].Journal of Constructional Steel Research,2019,161:309-327.

    • [6] GREENWOOD J A,WILLIAMSON J B P.Contact of nominally flat surfaces[J].Proceedings of the Royal Society of London.Series A.Mathematical and Physical Sciences,1966,295(1442):300-319.

    • [7] ZHAO Y,MAIETTA D M,CHANG L.An asperity microcontact model incorporating the transition from elastic deformation to fully plastic flow[J].Journal of Tribology,2000,122(1):86-93.

    • [8] ZHAO Y,CHANG L.A model of asperity interactions in elasticplastic contact of rough surfaces[J].Journal of Tribology,2001,123(4):857.

    • [9] WEBSTER M N,SAYLES R S.A numerical model for the elastic frictionless contact of real rough surfaces [J].ASME J Tribology,1986,108(3):314-320.

    • [10] MANDELBROT,BENOIT B.The fractal geometry of nature[J].American Journal of Physics,1998,51(3):468.

    • [11] GANTI Suryaprakash,BHUSHAN Bharat.Generalized fractal analysis and its applications to engineering surfaces[J].Wear,1995,180(1-2):17-34.

    • [12] MAJUMDAR A,BHUSHAN B.Role of fractal geometry in roughness characterization and contact mechanics of surfaces[J].Trans ASME J Tribology,1990,112(2):205-216.

    • [13] PAN W,LI X,WANG L,et al.Influence of surface topography on three-dimensional fractal model of sliding friction.[J].AIP Advances,2017,7(9):095321-1-095321-12.

    • [14] GARCÍA J M,MARTINI A.Measured and predicted static friction for real rough surfaces in point contact [J].Journal of Tribology,2012,134(3):031501.

    • [15] CHANG W R,ETSION I,BOGY D B.Static friction coefficient model for metallic rough surfaces [J].Journal of Tribology,1988,110(1):57-63.

    • [16] 王建梅,陶德峰,黄庆学,等.多层圆筒过盈配合的接触压力与过盈量算法研究[J].工程力学,2013(9):278-283.WANG J M,TAO D F,HUANG Q X,et al.Algorithm research on contact pressure and magnitude of interference for multi-layer cylinder’s interference fit [J].Engineering Mechanics,2013,30(9):270-275.(in Chinese)

    • [17] WU J,WANG Z X,WANG G Q.The key technologies and development of offshore wind farm in China[J].Renewable and Sustainable Energy Reviews,2014,34:453-462.

    • [18] WANG J,NING K,XU J,et al.Reliability-based robust design of wind turbine’s shrink disk[J].ARCHIVE Proceedings of the Institution of Mechanical Engineers Part C:Journal of Mechanical Engineering Science,2018,232(15):2685-2696.

    • [19] 盛选禹,雒建斌,温诗铸.基于分形接触的静摩擦因数预测 [J].中国机械工程,1998,9(7):16-18.SHENG X Y,LUO J B,WENG S Z.Static friction coefficient model based on fractal contact [J].China Mechanical Engineering,1998,9(7):16-18.(in Chinese)

    • [20] SETH P,RITA F.Corrosion protection systems and fatigue corrosion in offshore wind structures:current status and future perspectives[J].Coatings,2017,7(2):25.

    • [21] ELAGINA O Y,KOMADYNKO A C,POLESHCHUK E D,et al.Prospects for using Titanium Nitride coatings for the contact surfaces of friction clutches[J].Journal of Friction and Wear,2020,41(1):25-30.

    • [22] WANG Y B,WANG Y,CHEN K,et al.Slip factor of high strength steel with inorganic zinc-rich coating[J].Thin-Walled Structures,2020,148:106595.

    • [23] SHAN L,WANG Y,LI J,et al.Tribological behaviours of PVD TiN and TiCN coatings in artificial seawater [J].Surface and Coatings Technology,2013,226:40-50.

    • [24] 郝建民,耿刚强,张智龙.电弧喷涂铝对高强度螺栓摩擦型连接面抗滑移系数的影响[J].长安大学学报(自然科学版),2003(4):85-87.HAO J M,GENG G Q,ZHANG Z L.Effect of arc spraying aluminium on anti-glide coefficient of friction type high-strength bolt adjacent plane[J].Journal of Xi’ an Highway University,2003.(in Chinese)

    • [25] CHEN W W,WANG Q J.A numerical static friction model for spherical contacts of rough surfaces,influence of load,material,and roughness [J].Journal of Tribology,2009:131(2):021402.

    • [26] 朱华,葛世荣.结构函数与均方根分形表征效果的比较[J].中国矿业大学学报,2004,33(4):396-399.ZHU H,GE S R.Comparison of fractal characterization effects of structure function and mean square root [J].Journal of China University of Mining and Technology,2004,33(4):396-399.(in Chinese)

    • [27] WANG Z F,WANG H.Inflatable wing design parameter optimization using orthogonal testing and support vector machines [J].Chinese Journal of Aeronautics,2012,25(6):887-895.

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