尺度相关的分形结合面法向接触刚度模型
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摘要
基于分形理论,利用双变量Weierstrass-Mandelbrot函数模拟三维分形结合面,建立尺度相关的三维分形结合面法向接触刚度模型。推导出各等级微凸体发生弹性、弹塑性以及完全塑性变形的存在条件。确定结合面上各等级微凸体的面积分布密度函数,推导出法向接触刚度和法向接触载荷的解析表达式。计算结果表明:当结合面上的微凸体只能发生弹性变形,即自身等级小于弹性临界等级的微凸体,该部分微凸体引起的法向接触刚度和对应法向载荷关系呈非线性。当微凸体的等级大于弹性临界等级,在结合面接触过程中,微凸体弹性变形引起的法向接触刚度与对应的法向载荷关系为线性,非弹性变形引起的法向接触刚度与法向载荷关系为非线性。微凸体的等级范围对结合面的刚度影响较大,在相同的法向载荷作用下,高等级微凸体的结合面产生较高的法向接触刚度,即结合面越平整,结合面的法向刚度越高。
Based on fractal theory, a three-dimensional fractal model of normal contact stiffness of joint interfaces is developed. A modified two-variable Weierstrass-Mandelbrot function is adopted to simulate the three-dimensional joints surface. The conditions of existence of elastic deformation, elastoplastic deformation and fully plastic deformation of the single asperity are derived. The relations between size distribution function for all level asperities and size distribution function for each level asperity are given. Then the relations between the joint normal contact stiffness and normal contact load have been obtained. The results show that for the joint interface including only these asperities whose level is less than the elastic critical level, the relations between normal contact stiffness and normal contact load are nonlinear. For these asperities whose level is greater than the elastic critical level, the relations between elastic normal contact stiffness and elastic normal contact load are linear, and the relations between inelastic normal contact stiffness and inelastic normal contact load are nonlinear. For a given contact load, the normal contact stiffness of joint interface is proportional to asperity level, namely the topography of joint interface is smoother, the normal contact stiffness is higher.
Based on fractal theory, a three-dimensional fractal model of normal contact stiffness of joint interfaces is developed. A modified two-variable Weierstrass-Mandelbrot function is adopted to simulate the three-dimensional joints surface. The conditions of existence of elastic deformation, elastoplastic deformation and fully plastic deformation of the single asperity are derived. The relations between size distribution function for all level asperities and size distribution function for each level asperity are given. Then the relations between the joint normal contact stiffness and normal contact load have been obtained. The results show that for the joint interface including only these asperities whose level is less than the elastic critical level, the relations between normal contact stiffness and normal contact load are nonlinear. For these asperities whose level is greater than the elastic critical level, the relations between elastic normal contact stiffness and elastic normal contact load are linear, and the relations between inelastic normal contact stiffness and inelastic normal contact load are nonlinear. For a given contact load, the normal contact stiffness of joint interface is proportional to asperity level, namely the topography of joint interface is smoother, the normal contact stiffness is higher.
引文
[1]董冠华,殷勤,殷国富,等.机床结合部动力学建模与辨识方法的研究[J].机械工程学报,2016,52(5):162-168.DONG Guanhua,YIN Qin,YIN Guofu,et al.Research on dynamics modeling and identification of machine tool joints[J].Journal of Mechanical Engineering,2016,52(5):162-168.
[2]REN Y,BEARDS C F.Identification of effective linear joints using coupling and joint identification techniques[J].ASME Journal of Vibration and Acoustics,1998,120(2):331-338.
[3]杨红平,傅卫平,王雯,等.基于分形几何与接触力学理论的结合面法向接触刚度计算模型[J].机械工程学报,2013,49(1):102-107.YANG Hongping,FU Weiping,WANG Wen,et al.Calculation model of the normal contact stiffness of joints based on the fractal geometry and contact theory[J].Journal of Mechanical Engineering,2013,49(1):102-107.
[4]赵永武,吕彦明,蒋建忠.新的粗糙表面弹塑性接触模型[J].机械工程学报,2007,43(3):95-101.ZHAO Yongwu,LüYanming,JIANG Jianzhong.New elastic-plastic model for the contact of rough surfaces[J].Chinese Journal of Mechanical Engineering,2007,43(3):95-101.
[5]SHI J,CAO X,HU Y,et al.Statistical analysis of tangential contact stiffness of joint surfaces[J].Archive of Applied Mechanics,2015,85(12):1997-2008.
[6]SAYLES R S,THOMAS T R.Surface topography as a nonstationary random process[J].Nature,1978,271:431-434.
[7]温淑花,张学良,武美先,等.结合面法向接触刚度分形模型建立与仿真[J].农业机械学报,2009,40(11):197-202.WEN Shuhua,ZHANG Xueliang,WU Meixian,et al.Fractal model and simulation of normal contact stiffness of joint interfaces and its simulation[J].Transactions of the Chinese Society for Agricultural Machinery,2009,40(11):197-202.
[8]温淑花,张学良,文晓光,等.结合面切向接触刚度分形模型建立与仿真[J].农业机械学报,2009,40(12):223-227.WEN Shuhua,ZHANG Xueliang,WEN Xiaoguang,et al.Fractal model of tangential contact stiffness of joint interfaces and its simulation[J].Transactions of the Chinese Society for Agricultural Machinery,2009,40(12):223-227.
[9]王书亭,李杰,刘涛,等.机械固定结合面刚度特性建模[J].华中科技大学学报,2011,39(8):1-5.WANG Shuting,LI Jie,LIU Tao,et al.Interfacial stiffness characteristic modeling of mechanical fixed joints[J].Journal of Huazhong University of Science and Technology,2011,39(8):1-5.
[10]JIANG S,ZHENG Y,ZHU H.A contact stiffness model of machined plane joint based on fractal theory[J].Journal of Tribology,2010,132(1):0114011-7.
[11]PENG Y,GUO Y.An elastic-plastic adhesion model for contacting fractal rough surface and perfectly wetted plane with meniscus[J].Chinese Journal of Mechanical Engineering,2009,22(1):9-14.
[12]田红亮,钟先友,秦红玲,等.依据各向异性分形几何理论的固定结合部法向接触力学模型[J].机械工程学报,2013,49(21):108-122.TIAN Hongliang,ZHONG Xianyou,QIN Hongling,et al.Normal contact mechanics model of fixed joint interface adopting anisotropic fractal geometrical theory[J].Journal of Mechanical Engineering,2013,49(21):108-122.
[13]赵波,戴旭东,张执南,等.单峰接触研究及其在分形表面接触中的应用[J].摩擦学学报,2014,34(2):217-224.ZHAO Bo,DAI Xudong,ZHANG Zhinan,et al.Single asperity contact and its use for fractal surface contact[J].Tribology,2014,34(2):217-224.
[14]YAN W,KOMVOPOULOS K.Contact analysis of elastic-plastic fractal surfaces[J].Journal of Applied Physics,1998,84(7):3617-3624.
[15]MIAO X,HUANG X.A complete contact model of a fractal rough surface[J].Wear,2014,309(1-2):146-151.
[16]LIOU J L,CHI M T,LIN J F.A microcontact model developed for sphere-and cylinder-based fractal bodies in contact with a rigid flat surface[J].Wear,2010,268(3):431-442.
[17]JOHNSON K L,JOHNSON K L.Contact mechanics[M].Cambridge University Press,1987.
[18]KOUGHT L,ETSION I.Elastic-plastic contact analysis of a sphere and a rigid flat[J].Journal of Applied Mechanics,2002,69(5):657-662.
[19]YUAN Y,LI G,KAI L,et al.Elastoplastic contact mechanics model of rough surface based on fractal theory[J].Chinese Journal of Mechanical Engineering,2017,30(1):207-215.
[20]WANG S,KOMVOPOULOS K.A fractal theory of the interfacial temperature distribution in the slow sliding regime:Part I-elastic contact and heat transfer analysis[J].Journal of Tribology,1994,116(4):812-822.
[21]WANG S,KOMVOPOULOS K.A fractal theory of the interfacial temperature distribution in the slow sliding regime:Part II-multiple domains,elastoplastic contacts and applications[J].Journal of Tribology,1994,116(4):824-832.
[22]LIU P,ZHAO H,HUANG K,et al.Research on normal contact stiffness of rough surface considering friction based on fractal theory[J].Applied Surface Science,2015,349:43-48.
[23]蒋书文,姜斌,李燕,等.磨损表面形貌的三维分形维数计算[J].摩擦学学报,2003,23(6):533-536.JIANG S W,JIANG B,LI Y,et al.Calculation of fractal dimension of worn surface[J].Tribology,2003,23(6):533-536.
[24]索双富,葛世荣.磨削加工表面形貌的分形研究[J].中国机械工程,1996,7(1):41-42.SUO S F,GE S R.The fractal analysis of ground surface topography[J].China Mechanical Engineering,1996,7(1):41-42.
[25]陈奇.基于分形理论的汽车变速箱齿轮接触强度研究[D].合肥:合肥工业大学,2010.CHEN Qi.Research on gear contact strength analysis of automobile gearbox by fractal theory[D].Hefei:Hefei University of Technology,2010.
[2]REN Y,BEARDS C F.Identification of effective linear joints using coupling and joint identification techniques[J].ASME Journal of Vibration and Acoustics,1998,120(2):331-338.
[3]杨红平,傅卫平,王雯,等.基于分形几何与接触力学理论的结合面法向接触刚度计算模型[J].机械工程学报,2013,49(1):102-107.YANG Hongping,FU Weiping,WANG Wen,et al.Calculation model of the normal contact stiffness of joints based on the fractal geometry and contact theory[J].Journal of Mechanical Engineering,2013,49(1):102-107.
[4]赵永武,吕彦明,蒋建忠.新的粗糙表面弹塑性接触模型[J].机械工程学报,2007,43(3):95-101.ZHAO Yongwu,LüYanming,JIANG Jianzhong.New elastic-plastic model for the contact of rough surfaces[J].Chinese Journal of Mechanical Engineering,2007,43(3):95-101.
[5]SHI J,CAO X,HU Y,et al.Statistical analysis of tangential contact stiffness of joint surfaces[J].Archive of Applied Mechanics,2015,85(12):1997-2008.
[6]SAYLES R S,THOMAS T R.Surface topography as a nonstationary random process[J].Nature,1978,271:431-434.
[7]温淑花,张学良,武美先,等.结合面法向接触刚度分形模型建立与仿真[J].农业机械学报,2009,40(11):197-202.WEN Shuhua,ZHANG Xueliang,WU Meixian,et al.Fractal model and simulation of normal contact stiffness of joint interfaces and its simulation[J].Transactions of the Chinese Society for Agricultural Machinery,2009,40(11):197-202.
[8]温淑花,张学良,文晓光,等.结合面切向接触刚度分形模型建立与仿真[J].农业机械学报,2009,40(12):223-227.WEN Shuhua,ZHANG Xueliang,WEN Xiaoguang,et al.Fractal model of tangential contact stiffness of joint interfaces and its simulation[J].Transactions of the Chinese Society for Agricultural Machinery,2009,40(12):223-227.
[9]王书亭,李杰,刘涛,等.机械固定结合面刚度特性建模[J].华中科技大学学报,2011,39(8):1-5.WANG Shuting,LI Jie,LIU Tao,et al.Interfacial stiffness characteristic modeling of mechanical fixed joints[J].Journal of Huazhong University of Science and Technology,2011,39(8):1-5.
[10]JIANG S,ZHENG Y,ZHU H.A contact stiffness model of machined plane joint based on fractal theory[J].Journal of Tribology,2010,132(1):0114011-7.
[11]PENG Y,GUO Y.An elastic-plastic adhesion model for contacting fractal rough surface and perfectly wetted plane with meniscus[J].Chinese Journal of Mechanical Engineering,2009,22(1):9-14.
[12]田红亮,钟先友,秦红玲,等.依据各向异性分形几何理论的固定结合部法向接触力学模型[J].机械工程学报,2013,49(21):108-122.TIAN Hongliang,ZHONG Xianyou,QIN Hongling,et al.Normal contact mechanics model of fixed joint interface adopting anisotropic fractal geometrical theory[J].Journal of Mechanical Engineering,2013,49(21):108-122.
[13]赵波,戴旭东,张执南,等.单峰接触研究及其在分形表面接触中的应用[J].摩擦学学报,2014,34(2):217-224.ZHAO Bo,DAI Xudong,ZHANG Zhinan,et al.Single asperity contact and its use for fractal surface contact[J].Tribology,2014,34(2):217-224.
[14]YAN W,KOMVOPOULOS K.Contact analysis of elastic-plastic fractal surfaces[J].Journal of Applied Physics,1998,84(7):3617-3624.
[15]MIAO X,HUANG X.A complete contact model of a fractal rough surface[J].Wear,2014,309(1-2):146-151.
[16]LIOU J L,CHI M T,LIN J F.A microcontact model developed for sphere-and cylinder-based fractal bodies in contact with a rigid flat surface[J].Wear,2010,268(3):431-442.
[17]JOHNSON K L,JOHNSON K L.Contact mechanics[M].Cambridge University Press,1987.
[18]KOUGHT L,ETSION I.Elastic-plastic contact analysis of a sphere and a rigid flat[J].Journal of Applied Mechanics,2002,69(5):657-662.
[19]YUAN Y,LI G,KAI L,et al.Elastoplastic contact mechanics model of rough surface based on fractal theory[J].Chinese Journal of Mechanical Engineering,2017,30(1):207-215.
[20]WANG S,KOMVOPOULOS K.A fractal theory of the interfacial temperature distribution in the slow sliding regime:Part I-elastic contact and heat transfer analysis[J].Journal of Tribology,1994,116(4):812-822.
[21]WANG S,KOMVOPOULOS K.A fractal theory of the interfacial temperature distribution in the slow sliding regime:Part II-multiple domains,elastoplastic contacts and applications[J].Journal of Tribology,1994,116(4):824-832.
[22]LIU P,ZHAO H,HUANG K,et al.Research on normal contact stiffness of rough surface considering friction based on fractal theory[J].Applied Surface Science,2015,349:43-48.
[23]蒋书文,姜斌,李燕,等.磨损表面形貌的三维分形维数计算[J].摩擦学学报,2003,23(6):533-536.JIANG S W,JIANG B,LI Y,et al.Calculation of fractal dimension of worn surface[J].Tribology,2003,23(6):533-536.
[24]索双富,葛世荣.磨削加工表面形貌的分形研究[J].中国机械工程,1996,7(1):41-42.SUO S F,GE S R.The fractal analysis of ground surface topography[J].China Mechanical Engineering,1996,7(1):41-42.
[25]陈奇.基于分形理论的汽车变速箱齿轮接触强度研究[D].合肥:合肥工业大学,2010.CHEN Qi.Research on gear contact strength analysis of automobile gearbox by fractal theory[D].Hefei:Hefei University of Technology,2010.