Logarithmic Wavelet Spectrum Method for Computing Fractal Dimension of Machined Surface Profiles

Abstract

Abstract: A multi-scale wavelet decomposition for surface profiles was achieved herein by the multi-scale analysis capability of wavelet; and a logarithmic wavelet spectrum method for calculating the fractal dimensions of machined surface profiles and the effective decomposition scale concept were proposed. Fractal profiles with different fractal dimensions and different sampling intervals were generated by M-B function. The fractal dimensions of simulated profiles whose dimensions were known were calculated by logarithmic wavelet spectrum method, which was compared with the calculation results of other five kinds of methods such as the power spectral density (PSD) method. The results show that logarithmic wavelet spectrum method may process multi-scale fractal features nicely, and it has the highest accuracy with errors less than 0.15% when sym4 wavelet is adopted. Finally, logarithmic wavelet spectrum method was applied to calculate the fractal dimensions of actual machined surface profiles and the results illustrate its practicability.

Key words: machined surface profile, fractal dimension, wavelet decomposition, effective decomposition level, logarithmic wavelet spectrum

摘要： 为了提高接触表面的建模精度，利用小波的多尺度分析能力，对表面轮廓进行多尺度小波分解，提出了计算机械加工表面轮廓分形维数的对数小波谱法以及有效分解尺度概念，并认为轮廓只在有效分解尺度上具有分形特征；通过M-B函数模拟生成不同分形维数、不同采样区间的分形轮廓；应用对数小波谱法计算了模拟轮廓的分形维数，进而与功率谱密度法（PSD法）等5种方法的计算结果进行了分析比较，结果表明：对数小波谱法能很好地处理分形的多尺度特征，并且选用sym4小波时计算精度最高，误差在0.15%以内；最后应用对数小波谱法对一实际机械加工表面轮廓分形维数进行了计算,说明了其实用性。

关键词: 机械加工表面轮廓, 分形维数, 小波分解, 有效分解尺度, 对数小波谱

CLC Number:

TH161.14

Zhang Xueliang, Wang Yusong, Wen Shuhua, Fan Shirong, Chen Yonghui, Lan Guosheng. Logarithmic Wavelet Spectrum Method for Computing Fractal Dimension of Machined Surface Profiles[J]. China Mechanical Engineering.

张学良, 王余松, 温淑花, 范世荣, 陈永会, 兰国生. 机械加工表面轮廓分形维数对数小波谱计算方法[J]. 中国机械工程.

References

[1]杨红平，傅卫平，王雯，等.小波系数表征机械加工表面分形特征的计算方法[J].仪器仪表学报, 2010, 31(7):1454-1459.

Yang Hongping，Fu Weiping，Wang Wen，et al. Calculation Method for Fractal Characteristics of Machining Topography Surface Based on Wavelet Coefficients[J]. Chinese Journal of Scientific Instrum-ent，2010，31(7)：1454-1459.

[2]张学良. 机械结合面动态特性及应用[M]. 北京：中国科学技术出版社, 2002:20-24.

[3]温淑花，张宗阳，张学良，等. 固定结合面刚度分形模型[J]. 农业机械学报，2013,44(2):255-260.

Wen Shuhua, Zhang Zongyang, Zhang Xueliang, et al. Stiffness Fractal Model for Fixed Joint Interfaces[J]. Transactions of the Chinese Society for Agricultural Machinery, 2013,44(2):255-260.

[4]Majumdar A, Bhushan B. Fractal Model of Elastic-plastic Contact between Rough Surfaces[J]. ASME Journal of Tribology,1991,113(1):1-11.

[5]张学良，王南山，温淑花，等.机械结合面切向接触阻尼能量耗散弹塑性分形模型[J].机械工程学报，2013,49 (12):43-49.

Zhang Xueliang,Wang Nanshan,Wen Shuhua, et al. Elastoplastic Fractal Model for Tangential Contact Damping Energy Dissipation of Machine Joint Interfaces [J]. Journal of Mechanical Engineering, 2013,49(12):43-49.

[6]田红亮，郑金华，赵春华，等. 界面损耗因子与法向阻尼的计算方法[J].上海交通大学学报，2015,49(5):687-694.

Tian Hongliang, Zheng Jinhua, Zhao Chunhua, et al. Calculating Method of Surface Dissipation Factor and Normal Damping[J]. Journal of Shanghai Jiao Tong University, 2015,49(5):687-694.

[7]张学良，陈永会，温淑花，等.考虑弹塑性变形机制的结合面法向接触刚度建模[J].振动工程学报，2015,28 (1):91-99.

Zhang Xueliang, Chen Yonghui, Wen Shuhua, et al. The Model of Normal Contact Stiffness of Joint Interfaces Incorporating Elastoplastic Deformation Mechanism[J]. Journal of Vibration Engineering, 2015,28 (1):91-99.

[8]李成贵,张国雄,袁长良.分形维数与表面粗糙度参数的关系[J].工具技术, 1997, 31(12):36-38.

Li Chenggui，Zhang Guoxiong，Yuan Changliang.The Relationship between the Fractal Dimension and Surface Roughness Parameters[J].Tool Engineering，1997，31(12)：36-38.

[9]李成贵,刘杰,郎青山.机加工表面轮廓分形维数的两种计算方法[J].航空精密制造技术,1997, 33(4):25-27.

Li Chenggui，Liu Jie，Lang Qingshan.Two Methods for the Calculation of the Fractal Dimension of a Machined Surface Profile[J].Aviation Precision Manufacturing Technology，1997，33(4)：25-27.

[10]王建军,魏宗信.粗糙表面轮廓分形维数的计算方法[J].工具技术, 2006, 40(8):73-75.

Wang Jianjun，Wei Zongxin.Computation Method for Fractal Dimension of Rough Surface Profile[J].Tool Engineering，2006，40(8)：73-75.

[11]葛世荣,索双富.表面轮廓分形维数计算方法的研究[J].摩擦学学报, 1997, 17(4):354-362.

Ge Shirong，Suo Shuangfu.The Computation Methods for the Fractal Dimension of Surface Profiles[J].Tribology，1997，17(4)：354-362.

[12]葛世荣, Tonder K. 粗糙表面的分形特征与分形表达研究[J].摩擦学学报, 1997, 17(1):73-80.

Ge Shirong，Tonder K.The Fractal Behavior and Fractal Characterization of Rough Surfaces[J]. Tribology，1997，17(1)：73-80.

[13]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 Hua，Ge Shirong.Comparison of Fractal Characterization Effects of Structure Function and Mean Square Root[J].Journal of China University of Mining & Technology，2004，33(4)：396-399.

[14]王安良,杨春信. 机械加工表面形貌分形特征的计算方法[J].中国机械工程, 2002, 13(8):714-718.

Wang Anliang，Yang Chunxin.The Calculate Method for the Fractal Characterization of Profiles[J].China Mechanical Engineering，2002, 13(8):714-718.

[15]王安良,杨春信.小波变换方法评价曲线的分形特征[J].机械工程学报, 2002, 38(5):80-85.

Wang Anliang，Yang Chunxin.Wavelet Transform Method Evaluate the Fractal Characterization of Profiles[J]. Chinese Journal of Mechanical Engineering，2002，38(5)：80-85.

[16]Flandrin P. On the Spectrum of Fractional Brown-ian motions [J]. IEEE Transactions on Informa-tion Theory,1989,35(1):197-199. [17]杨福生.小波变换的工程分析与应用[M].北京:科学出版社, 1999.

[18]Wornell W. Wavelet-Based Representations for the 1/f Family of Fractal Processes [J]. Proceedings of the IEEE,1993,81(10):1428-1450.

[19]Jackson R L, Streator J L. A Multi-scale Model for Contact between Rough Surfaces[J]. Wear, 2006, 261:1337-1347.

[20]樊计昌,刘明军,王夫运,等.浅析小波最大分解层[J].科技导报(北京), 2008, 26(10):40-42.

Fan Jichang，Liu Mingjun，Wang Fuyun，et al. Analysis on Maximum Wavelet Decomposition Level from Theory and Application[J].Science & Technology Review(Beijing)，2008，26(10)：40-42.

[21]Majumdar A, Tien C L. Fractal Characterization and Simulation of Rough Surfaces[J]. Wear, 1990, 136 (2):313-327.