Fractal dimensions of terrain profiles

Analysing terrain profiles of fields, roads, and other terrains, it was determined that terrain profiles are random and non-periodical. Mandelbrot has defined non-scaling, self-similar figures as fractals, and many investigators have tried to characterize natural forms and structures using fractal geometry. The work here investigates whether terrain profiles can be defined as fractals. Fractal dimensions of profiles were calculated. These were compared with a locus of Brownian motion further to investigate characteristics of terrain profiles. Fractals are defined to be self-similar and irregular. Measuring and analysing terrain profiles, it was established that the statistical characteristics of any part of a terrain profile are similar and that the statistical characteristics of profiles of any kind of terrain are similar irrespective of roughness. This means that terrain profiles are self-similar, and irregular. From these results, it was determined that terrain profiles are fractals. The fractal dimensions were calculated with a coarse-graining method and by Power Spectral Densities (PSD), and fractal dimensions by Scaling were between 1.1 and 1.8 and by PSD between 1.3 and 1.5. Using the locus of Brownian motion, fractal dimensions were 1.5 or slightly larger than those of the terrain profiles. Fractal dimensions for the locus of smoothed Brownian motion were nearly equal to terrain profiles. Therefore terrain profiles could be artificially generated from the locus of smoothed Brownian motion. It appears that terrain roughness is formed by random and non-periodical force.     

粗糙表面的分形特征与分形表达研究

得用触针轮廓仪和数据采集系统对磨削和车削表面的粗糙轮廓曲线进行了测量，并就粗糙表面的分形特征作了分析与讨论，同时还提出了粗糙表面的特征粗糙度概念及其定义，并将其用表面粗糙度水平的表述。     

Reconstruction of road defects and road roughness classification using Artificial Neural Networks simulation and vehicle dynamic responses: Application to experimental data

This paper reports the performance of an Artificial Neural Network based road condition monitoring methodology on measured data obtained from a Land Rover Defender 110 which was driven over discrete obstacles and Belgian paving. In a previous study it was demonstrated, using data calculated from a numerical model, that the neural network was able to reconstruct road profiles and their associated defects within good levels of fitting accuracy and correlation. A nonlinear autoregressive network with exogenous inputs was trained in a series–parallel framework. When compared to the parallel framework, the series–parallel framework offered the advantage of fast training but had a shortcoming in that it required feed-forward of true road profiles. In this study, the true profiles are not available and the test data are obtained from field measurements. Training data are numerically generated by making minor adjustments to the real measured profiles and applying them to a full vehicle model of the Land Rover. This is done to avoid using the same road profile and acceleration data for training and testing or validating the neural network. A static feed-forward neural network is trained and consequently tested on the real measured data. The results show very good correlations over both the discrete obstacles and the Belgian paving. The random nature of the Belgian paving necessitated correlations to be made using their displacement spectral densities as well as evaluations of RMS error percent values of the raw road profiles. The use of displacement spectral densities is considered to be of much more practical value than the road profiles since they can easily be interpreted into road roughness measures by plotting them over an internationally recognized standard roughness scale.     

Evaluation of fractal terrain model for vehicle dynamic simulations

Fractals are a popular method for modeling terrains that include various scales. This paper investigates the effectiveness of using fractals for generating artificial terrains which can be used for vehicle simulations. The 3-D Weierstrass–Mandelbrot function was used to generate surfaces based on experimentally measured terrains. There is an exponential relationship between the root means squared elevation of the surfaces and the fractal scaling parameter. This relationship was used to determine the required fractal parameters to generate a surface with a desired roughness. A light detection and ranging (LiDAR) sensor coupled with a global positioning system (GPS) and inertial navigation system (INS) was used to measure two off road surfaces. The experimental terrain was then compared to the simulated terrain. Based on the comparison, the fractal model can capture the general roughness of the experimentally measured terrains as determined by the dynamic response of a suspension model. However, the fractal model fails to capture some of the nuances and non-periodic events observed in experimental terrains.     

岩石节理(断裂)表面的多重分形性质

提出了新的投影覆盖法———ＰｒｏｊｅｃｔｉｖｅＣｏｖｅｒｉｎｇＭｅｔｈｏｄ（ＰＣＭ），并结合无接触激光表面测量技术直接测定出断裂表面分维值Ｄ∈［２，３）．建立了表面粗糙性投影覆盖概率函数，分析了节理（断裂）表面的多重分形性质．     

Adhesive contact in the context of multi-asperity interaction

In order to analyse the scale effect of roughness and adhesion in contact between solids, we introduce in this work a 3D model of elastic contact combined with the adhesive theory. The model of roughness with different fractal dimension is introduced to study the adhesive contact in the elastic state at different length scales.The results of this study show that the scale of roughness modifies the distribution law of the contact pressure and stiffness. The introduction of the adhesive force and fractal property of roughness clearly shows the combined influence of roughness scale and adhesive force on contact stiffness.     

Photometric measurement of terrain roughness

Information on the roughness of terrain is an important factor in designing the rolling gear of cross-country vehicles. In view of a number of advantages—especially the great information density—of the photometric measuring of terrain roughnesses, this study report discusses the applicability of this technique as a new method to complement those already in use. The report demonstrates that the resolution limit of approximately ± 5 cm in a scale 1:2000 reproduction is sufficient, for all practical purposes, because the resulting total measuring error will be distributed over the entire profile-frequency range.A comparison between an accurate ground-based technique and photometric measurement is presented to demonstrate that, in the range of the profile wavelength of 1 m, the roughness root-mean-square-values differ only by a few mm, so that these differences may be neglected.It is suggested that, for assessing terrain roughness, range-pair classification be used in addition to the usually applied power spectral density, because range-pair classification describes the terrain amplitudes occuring in terms of their absolute magnitude.The report indicates evaluation errors that may result from non-stationary roughness profiles, and it shows the possibility of conducting the photometric terrain measurement in wider and less expensive steps, and of complementing this measurement by superimposing a suitable random noise signal on the measurement curve.     

锡基巴氏合金磨损表面的分形与磨损率

利用结构函数分析法研究了滑动摩擦学系统中金属磨损表面轮廓线的分形特性。结果表明：磨损表面轮廓线在小于Ｓｍ的尺度上具有分形结构。采用结构函数法可以方便地确定粗糙表面轮廓线的分形参数，即垂直于滑动方向上磨损表面轮廓线的分维Ｄ可作为磨损表面分维的特征值，它与金属磨损率的变化有着密切的关系，最佳分维Ｄｏｐｔ值对应于材料的最低磨损率。     

岩石节理粗糙面的分形几何特性与剪切强度关系的实验研究

自然形成的岩石节理面的表面形状具有分形特性，利用分形理论来描述岩石节理的粗糙特性，并确立与岩石节理面的剪切力学特性间的关系对于指导岩体工程设计，和管理具有重要一研究利用开发设计的三维激光测距仪在精确测定岩石节理面的表面形状４ 基础上，依据变量图法建立了 岩石节理面的     

岩体结构面粗糙度系数定量表征研究进展

1978年,Barton提出的节理粗糙度系数(joint roughness coefficient,JRC)被国际岩石力学学会作为评估节理粗糙度的标准方法.然而该方法存在人为估值的主观性缺陷.就此,国内外学者围绕岩体结构面粗糙度定量化表征开展了大量的研究工作.首先,从二维节理轮廓线到三维岩体结构面,系统地阐述了其粗糙度定量化表征方法研究进展,并总结了各方法参数与JRC的关系;评价了各表征参数的本质特性及其适用性;指出了各方法参数获取过程中存在的问题,主要有:采样间隔的影响,三角形单元划分的影响,如何确定综合参数法中各参数的权重;针对这些问题,给出了笔者的一些想法、建议.与此同时,对结构面粗糙度表征的两个热点问题,即各向异性和尺寸效应的研究也进行了详细总结分析.最后,笔者认为:(1)分形维数因是描述自然界复杂几何体的一种简洁有力的工具,其仍是结构面粗糙度定量描述的有效方法;(2)3D打印技术的应用,有望在开展结构面各向异性、尺寸效应研究方面取得突破性进展.     

关联表面分形特性的润滑模型

在分配有面粗糙度与材料表面分形特性关系的基础上，将分形特性引入润滑方程，提出了关联表面分形特性的润滑模型，并分析了润滑模型中压力流量因子和剪切流量因子与表面分形维数之间关系。计算结果表明：在相同分形维数（D）下，随着微突体纵横比v的增大，压力流量因子和剪切流量因子相应增大，且其随分形维数D的变化同随油膜粗糙度比H的变化相比呈现出更强的不规则特性，出现局部最大和最小值，并且分辨率较高。     

粗糙表面法向接触刚度的分形模型

提出了以往有关粗糙表面法向接触刚度理论研究工作的缺陷与不足,并在一定的前提假设下,基于球体与平面的接触理论和粗糙表面的分形接触理论,从理论上给出了具有尺度独立性的粗糙表面法向接触刚度分形模型,并进行了数字仿真研究。     

结合面静摩擦因数分形模型的建立与仿真

提出一个确定分形维数和分形粗糙度参数(G')的分形函数,并用分形函数逼近结合面的表面粗糙度.根据分形理论和改进后的尺寸分布,推导了静摩擦因数f的解析解.数字仿真结果表明,f随量纲为一的总法向载荷(P')增加而增加.当D较小或较大时,f与(P')之间存在不同的微凸和微凹弧非线性关系,D=1.2时f与(P')的关系基本上是线性的;当D较小时,f随D增加而增加;当D较大时,f随D增加而减小;f随分形粗糙度参数(G')增加而减小,随相关系数K增加而增加,随材料特性(φ)增加而增加.     

Evaluation of fractal dimension of soft terrain surface

By using a laser profiler, the roughness of sowed and plowed surfaces was obtained. Through evaluation of the precision of fractal dimensions based on the Weierstrass–Mandelbort (W–M) function, we found the rescaled (R/S) analysis method and the structure spectral method were not suitable for the calculation of the fractal dimension on a soft terrain surface. Therefore, the fractal dimension, non-scale range and correlation coefficient for each kind of terrain were analyzed using the following fractal computational methods – (i) variate-difference method, (ii) power spectral density method, and (iii) root mean square method. The results showed that: fractal dimension of plowed terrain was large with small fluctuations, while its internal structure was complex. The power spectral density method was not robust enough to compute the fractal dimension of a soft terrain surface. The fractal dimension computed using the root mean square method was found to be more accurate for the soft terrain surface. Moreover, the correlation coefficient of linear regression when using the root mean square method was good and the range of non-scale variation was small.     

基于小波变换的撞击流压力波动信号分析

通过小波变换对信号进行多尺度细化分析,解决了傅里叶变换不能解决的许多问题．从分析噪声和信号本身奇异点的区别入手,利用小波在时频两域突出信号局部特征能力的重要性质,即噪声信号小波变换的极大值随尺度的加大而显著减少的特点,对撞击流反应器撞击区压力波动信号进行处理分析．分析结果表明:小波变换可以较好地去除信号中的噪声;通过消噪后信号的功率谱可知,撞击流压力波动信号具有自相似的分形特征,与前期的实验研究相吻合．     

粗糙表面接触力学问题的重新分析

为了克服基于统计学参数的接触模型的尺度依赖性以及现有接触分形模型推导过程中初始轮廓表征受控于接触面积或取样长度的不足,基于粗糙表面轮廓分形维数$D$、尺度系数$G$ 和最大微凸体轮廓基底尺寸$l$,建立了新的粗糙表面接触分形模型,探讨了微凸体变形机制、粗糙表面的真实接触面积和接触载荷的关系,揭示了接触界面的孔隙率和真实接触面积随端面形貌、表面接触压力等参数变化的规律,给出了不同形貌界面被压实的最大变形量. 结果表明:微凸体变形从弹性变形开始,并随着平均接触压力$p_{\rm m}$ 的增大逐步向弹塑性变形和完全塑性变形转变;接触界面的初始孔隙率$\phi_{0}$ 随$D$ 的增大而增大,压实孔隙所需要的最大变形量$\delta $ 也随之增大;接触压力$p_{\rm c}$ 增大,孔隙率$\phi$ 减小,并随着$D$ 的增大和$G$ 减小,$\phi$ 快速减小,直至填实,变为零;$D$ 较小时,$G$ 的增大对真实接触面积的增大影响较小;$D$ 较大时,$G$ 的增大对真实接触面积的增大作用明显. 研究成果为端面摩擦副的润滑与密封设计提供了理论基础.      

Techniques for averting and correcting errors in 3D terrain topology measurements

The emergence of high-fidelity vehicle and tyre models has raised the requirements for 3D terrain measurement capabilities. Inaccuracies that were once tolerable for measurement of general terrain roughness are no longer acceptable for these new applications. The data inaccuracies that arise from small inertial errors are compounded by difficulties in managing massive file sizes. These faults are most apparent when combining multiple lanes of data in the post-processing phase. This work develops two correction techniques: a general method for any terrain type and a more computationally efficient method for smooth terrain. The general terrain correction method uses overlapping points in the horizontal plane to build a discontinuity vector and applies that vector to correct adjacent lanes. While this method effectively eliminates lane-to-lane discontinuities, it is computationally inefficient for terrain that lacks localized disturbances. For simulations using smooth terrain surfaces, such as racetracks or highways, data can be down-sampled through a mean interpolation method to smooth the artefacts of inertial errors. This work provides the vehicle test engineer with guidelines to minimize inertial errors during the data collection phase as well as a post-processing tool to combine multiple lanes of high-fidelity 3D terrain surfaces.     

粗糙表面接触分形模型的提出与发展

自表面接触塑性变形模型问世以来，经过近４０年的发展，已经形成以分形几何理论为基础的Ｍ－Ｂ粗糙表面接触分形模型．Ｍ－Ｂ模型以分形参数代替统计学参数表征粗糙表面，推导出了实际接触面积与载荷的关系，以及实际弹性接触面积和实际塑性接触面积的计算公式，指出了影响接触面变形性质的因素与规律．由于分形参数的尺度独立性，可望利用Ｍ－Ｂ模型对接触面积的预测不受测量仪器分辨率和取样长度等因素的影响，故其比基于统计分析的Ｇ－Ｗ接触模型更为合理．尽管如此，Ｍ－Ｂ模型还有待完善，多方面的问题尚待进行深入研究     

Analysis of the vibration-excitation effect caused by deformable soil surfaces

Terrain roughness induces vibrations in the chassis of vehicles moving cross-country. These vibrations affect the comfort of the occupants detrimentally and, when a certain rough ride condition is exceeded, the vehicle can no longer be controlled by the driver. This study deals with the relationship between terrain roughness and the vehicle traveling over it. In the case of deforming soil, the so called original soil profile is altered significantly by the vehicle. This study determines the original as well as the modified profiles, the latter being caused by the vehicle traveling over the former. The vehicle's transfer function is used to compute the effective profile, which generates the actual vibrations of the vehicle. These investigations are presented for several soil types and for a number of vehicles whose mass and running gear design are known.