基于分形理论的路面不平度分级与模拟研究
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摘要
路面不平度涉及人、车、路三方面的因素。在车辆工程研究领域,路面不平度是车辆运行环境中的主要因素,是车辆行驶的主要激励和振动源,路面不平使车辆在行驶中产生行驶阻力和振动,行驶阻力消耗车辆的功率并且影响车辆动力系统和传动系统的寿命,而在路面冲击下产生的振动,则直接影响了车辆平顺性、乘坐舒适性、操纵稳定性、车辆行驶速度以及承载系的可靠性和寿命。在道路工程领域,路面不平度也是国内外路面工程质量的评定指标之一,不平度检测贯穿于路面施工质量检测、评定、验收及运营期路面质量检测维护等环节。研究分析路面不平度、对其进行合理的评价与分级、建立路面输入模型对车辆工程和道路工程领域均具有重要意义。
本文针对当前路面不平度分析、模拟和分级研究的现状,结合正在各领域广泛应用的分析理论、计算方法、数字模型等理论和技术,围绕路面不平度分形特征的确定、路面不平度分形分级方法、路面不平度分形模型等关键技术内容进行了深入和系统的研究。具体工作如下：
首先设计了基于虚拟仪器技术的路面不平度数据采集系统,改善了接触式路面不平度检测仪的功能,并结合非接触式激光不平度检测车采集了大量路面不平度数据。然后对实测路面不平度数据进行了传统特性参数分析,结果发现：要完整描述路面的不平度特性需要较多的参数；而且这些参数随着测量尺度的变化而表现出不稳定性,参数值相同的表面其特性却可以相差很大,即存在尺度相关性和非唯一性。也就是说,传统参数无法唯一的表征路面不平度特性,因此,有必要寻找新的描述表面特性的参数。
分形理论是自1975年以来迅速发展起来的一个新的数学分支理论。它在信号描述、信号处理及其它众多领域中具有广泛的应用。它与表面不平度相结合的研究已成为一个重要的交叉科学研究分支。众多研究表明分形维数能够描述复杂现象的本质特性,本文利用分形理论对实测路面不平度的特性进行了系统研究。为确定路面不平度的分形特性,首先以W-M函数生成的理想分形曲线来对比分析不同的分形维数测定方法,得到各分形维数测定方法在分析分形曲线时的适用范围及优缺点。进而应用不同的测定方法分析实测路面不平度的标度律直线关系、无标度区间和分形维数值,通过计算可知,变差法、结构函数法和均方根法在双对数坐标上的标度律直线关系均较好,但变差法和结构函数法的无标度区间存在不稳定性,均方根法下的无标度区间较稳定；又由于均方根法的物理意义明确,对表面轮廓曲线有很好的表征作用,因此,可以确定均方根法为计算路面不平度分形维数的有效方法。采用确定的均方根法计算所选路面样本的分形维数,根据计算结果来表达路面样本的不平整程度发现,仅用分形维数这一参数不能唯一确定某一路面不平度状况,即分形维数在表征实测路面不平度中存在相对性。
为进一步确定能够唯一表征路面不平度的参数,从现有路面不平度分级的国际标准出发,根据各等级路面的功率谱数据,通过傅立叶逆变换法获得满足各个等级的路面高程序列,然后分析路面不平度的常用表征参数与分形参数之间的关系,结果表明,傅立叶逆变换方法模拟的不同等级路面表现出显著的分形特征,其分形维数差别很小,没有明显分界；但等级越差的路面在均方根法所得路面的测度与尺度的对数分布图纵轴上的截距越大,进一步通过对傅立叶逆变换所得路面数据的表征参数的计算,发现这一截距值与轮廓均方根偏差呈较好的线性关系,鉴于路面的特征分形参数值较小且其变化不利于路面分级使用,提出将分形维数与轮廓均方根偏差结合起来成为路面不平度表征的综合参数,即提出以路面不平度指数作为路面的表征参数,并计算各等级路面的路面不平度指数。
从国际标准对路面不平度的分级原理分析出发,通过实测路面不平度数据分析国际标准对路面不平度分级的局限性；计算实测路面的表征参数,统计分析轮廓均方根偏差、分形维数、路面不平度指数与按国际标准划分的路面等级的关系,建立了以路面不平度指数为表征参数的路面不平度分级方法,该方法对于路面不平度分级方法和标准的改进具有一定参考价值。
在路面不平度实测离散数据的基础上进行分形插值模拟,并通过时频域、分形特性参数检验分形插值的模拟效果,对影响分形插值精度的因素进行分析。研究结果表明基于迭代函数系统的分形插值函数对路面不平度进行模拟是可行的,其对于路面不平度的客观表征、数据的压缩和路面不平度测量仪器的制造等具有重要的参考价值。路面不平度分形插值模型可以作为车辆振动响应仿真分析的路面输入激励,该输入模型更加接近实际路面。
本文最后通过仿真和试验的方式验证分形路面模型的适用性。评价本文建立的路面不平度分形模型与实际随机输入的路面的近似程度。以ADAMS软件为平台,建立了多路面输入下的整车模型,路面输入分别选择实测随机路面数据、傅立叶逆变换数据、分形插值模型数据和软件中内置的路面模型,整车模型的建立与实测车辆一致,然后进行整车动态响应仿真试验,通过比较四种路面下的车辆振动响应参数可知,分形模型能较好的逼近实测路面,其振动响应参数最接近实测路面。进一步通过实车实验采集车辆振动响应参数,对仿真结果进行了检验。
The roughness of road surface involves the three factors of people, cars, and roads. In the field of vehicle engineering road roughness is the main factor in the environment of vehicle operating and the main sources of incentive and vibration. Road roughness generates resistance and vibration while vehicles are running. The resistance depletes vehicles' power and influences the life-span of vehicle dynamical system and transmission system. And the vibration impacts vehicles'smooth-going and taking comfortableness, handling stability, traveling speed, bearing system's reliability and life-span. In the field of road engineering, road roughness is one of the evaluation indexes of the road project quality at both domestic and international countries, and roughness-test runs though the various links such as quality testing, assessment, confirmation in the period of road's construction and quality inspection, maintenance in the period of road's operation. It is very significant for the fields of vehicle engineering and road engineering to analyze on the road surface roughness, reasonably carry on appraising and grading the roughness, and establish the input model of road surface.
In this paper, the key technological contents, such as characteristic determination of road surface roughness, its fractal grade-way roughness and fractal model, were carried on deep and systematic research, based on the current situation of analysis, simulation and grading on the road roughness, combined with various of theories and technologies which have been widely used in various fields, such as analytic theory, computing technology, mathematical model.The concrete work is as follows:
Firstly road roughness data collecting system was designed based on virtual instrument technology, to improve the function of the contact-type detectors of the roughness, and a great deal of road roughness data was collected combined with non-contact type of laser roughness car. Then we carried on the traditional analysis of characteristic parameter on the measured data of road roughness. Result showed that more parameters were required when the whole characteristics of the road were described, and these parameters had instability with the changes in metrical scale, even with the same parameters surface characteristics are different greatly, so metrical scale-related and non-uniqueness existed. That is say, the traditional parameters are unable to represent characteristic of the road roughness uniquely, so it is essential to find a new parameters.
Fractal theory has rapidly developed into a new branch of mathematics theory since the year of 1975.It has extensive application in describing the signal, the signal dealing and other fields. The research, which confined with surface roughness and fractal theory, has already become an important research branch in crossing science. Numerous researches indicate that fractal dimension can describe essential characteristic of the complicated phenomenon, so in this paper, fractal theory was used to systematically research into characteristics of measured road roughness. First of all, the ideal fractal curves was generated though Weierstrass-Mandelbrot function to analyze comparatively on different methods by which fractal dimensions was confirmed, then the different methods' applicable range, advantages and disadvantages were confirmed to analyze fractal curves. Through different computational methods to confirm fractal dimension, we analyzed the scale law linear relation of longitudinal sections of measured road roughness, which is regarded as two-dimensional curve. Then we found there is relevant scale law linear relation in log-log coordinate of the three methods, besides root-mean-square method, remainder-variation method and structure function method. After calculating and analyzing on both non-scale sector and fractal dimension of the three methods, we found that root-mean-square method is quite stable in the non-scale sector instead of remainder-variation method and structure function method. On the other hand, root-mean-square method has both clear physical significance and signal function to the surface profile curve, so root-mean-square method is definitely a efficacious method to calculate the fractal dimension of road roughness. After the fractal dimension of selected road roughness was calculated by the method of root-mean-square, we expressed the degree of the road roughness based on the results, then we found that only the parameters of fractal dimension can not determine the condition of a road surface roughness, in other words, the fractal dimension has relativity when it is used to signify measured road roughness.
In order to further confirm the only parameter which signify the road roughness uniquely, started off with existing international grade standard of road roughness, based on the power spectrum-datum of road surface in every grades, high-procedures tabulates for road surface in each grades were obtained through Fourier inverse transformation law. Then we analyzed the relations between the daily signal parameters of road roughness and its fractal parameters. The result showed that there was very small difference and unclear divide between fractal dimension calculated through root-mean-square method and the prominent fractal characteristics of road surface in different grade simulated by Fourier inverse transformation, but the vertical axis'intercept was greater in Logarithmic distribution of measure and scale, which were from worse grade of road and calculated through root-mean-square method. After we calculated the signal parameters, we found that this intercept and root-mean-square deviation of the outline showed a good linear relationship. Because the fractal parameters of road characteristics were very small and their changes were adverse to use of road grade, we proposed to regard the combination between fractal dimension and root-mean-square deviation of profile as a integrated parameter to signify the road roughness, that is, road roughness index was regarded as a signal parameter of road roughness, and the indexes of every grade road were calculated.
Based on the analysis of international standards on classification principles of road surface roughness, its limitations were analyzed by actually road surface roughness. The characteristic parameters of measured road were calculated. Root-mean-square roughness, fractal dimension and the relationship between the roughness index and the road grade based on international standards were analyzed statistically. A road surface roughness classification was established, that regards roughness index as the characteristic parameter.
Fractal road model was finished by fractal interpolation function based on the measured discrete data of road surface roughness. Then the results of fractal model through time-frequency domain and fractal parameters were tested, and the factors which affect the accuracy of fractal interpolation were analyzed. The results indicate that it is feasible to use fractal interpolation function which is based on iterated function system to simulate road surface roughness. It has an important reference on objective characterization of road surface roughness, data compression and the manufacture of road surface roughness measuring instruments. The fractal interpolation model of road surface roughness can be used as road input in the simulation of automobile vibration response and this model is closer to the actual road.
Finally, the applicability of the fractal road mode was verified through simulation and experiment and the approximation degree, between the fractal model of road surface roughness and the actual random input road, was evaluated. On the platform of ADAMS,the vehicle model of multiple input roads has been established and it is consistent with the actual vehicle. The selected input data are actual measured random data, inverse Fourier transform data, data of fractal interpolation model and the built-in road surface model in the software. Then vehicle dynamic response simulation test has been done. After comparing the vehicle dynamic response parameters under the four above input roads, the result indicates that fractal model can better approach the actual measured road. Its dynamic response parameter is closest to measured road. Further collecting the vehicle vibration response parameters through real vehicle experiment and comparing with the simulation results, we found that the simulation result is in good agreement with the real vehicle test result.
本文针对当前路面不平度分析、模拟和分级研究的现状,结合正在各领域广泛应用的分析理论、计算方法、数字模型等理论和技术,围绕路面不平度分形特征的确定、路面不平度分形分级方法、路面不平度分形模型等关键技术内容进行了深入和系统的研究。具体工作如下：
首先设计了基于虚拟仪器技术的路面不平度数据采集系统,改善了接触式路面不平度检测仪的功能,并结合非接触式激光不平度检测车采集了大量路面不平度数据。然后对实测路面不平度数据进行了传统特性参数分析,结果发现：要完整描述路面的不平度特性需要较多的参数；而且这些参数随着测量尺度的变化而表现出不稳定性,参数值相同的表面其特性却可以相差很大,即存在尺度相关性和非唯一性。也就是说,传统参数无法唯一的表征路面不平度特性,因此,有必要寻找新的描述表面特性的参数。
分形理论是自1975年以来迅速发展起来的一个新的数学分支理论。它在信号描述、信号处理及其它众多领域中具有广泛的应用。它与表面不平度相结合的研究已成为一个重要的交叉科学研究分支。众多研究表明分形维数能够描述复杂现象的本质特性,本文利用分形理论对实测路面不平度的特性进行了系统研究。为确定路面不平度的分形特性,首先以W-M函数生成的理想分形曲线来对比分析不同的分形维数测定方法,得到各分形维数测定方法在分析分形曲线时的适用范围及优缺点。进而应用不同的测定方法分析实测路面不平度的标度律直线关系、无标度区间和分形维数值,通过计算可知,变差法、结构函数法和均方根法在双对数坐标上的标度律直线关系均较好,但变差法和结构函数法的无标度区间存在不稳定性,均方根法下的无标度区间较稳定；又由于均方根法的物理意义明确,对表面轮廓曲线有很好的表征作用,因此,可以确定均方根法为计算路面不平度分形维数的有效方法。采用确定的均方根法计算所选路面样本的分形维数,根据计算结果来表达路面样本的不平整程度发现,仅用分形维数这一参数不能唯一确定某一路面不平度状况,即分形维数在表征实测路面不平度中存在相对性。
为进一步确定能够唯一表征路面不平度的参数,从现有路面不平度分级的国际标准出发,根据各等级路面的功率谱数据,通过傅立叶逆变换法获得满足各个等级的路面高程序列,然后分析路面不平度的常用表征参数与分形参数之间的关系,结果表明,傅立叶逆变换方法模拟的不同等级路面表现出显著的分形特征,其分形维数差别很小,没有明显分界；但等级越差的路面在均方根法所得路面的测度与尺度的对数分布图纵轴上的截距越大,进一步通过对傅立叶逆变换所得路面数据的表征参数的计算,发现这一截距值与轮廓均方根偏差呈较好的线性关系,鉴于路面的特征分形参数值较小且其变化不利于路面分级使用,提出将分形维数与轮廓均方根偏差结合起来成为路面不平度表征的综合参数,即提出以路面不平度指数作为路面的表征参数,并计算各等级路面的路面不平度指数。
从国际标准对路面不平度的分级原理分析出发,通过实测路面不平度数据分析国际标准对路面不平度分级的局限性；计算实测路面的表征参数,统计分析轮廓均方根偏差、分形维数、路面不平度指数与按国际标准划分的路面等级的关系,建立了以路面不平度指数为表征参数的路面不平度分级方法,该方法对于路面不平度分级方法和标准的改进具有一定参考价值。
在路面不平度实测离散数据的基础上进行分形插值模拟,并通过时频域、分形特性参数检验分形插值的模拟效果,对影响分形插值精度的因素进行分析。研究结果表明基于迭代函数系统的分形插值函数对路面不平度进行模拟是可行的,其对于路面不平度的客观表征、数据的压缩和路面不平度测量仪器的制造等具有重要的参考价值。路面不平度分形插值模型可以作为车辆振动响应仿真分析的路面输入激励,该输入模型更加接近实际路面。
本文最后通过仿真和试验的方式验证分形路面模型的适用性。评价本文建立的路面不平度分形模型与实际随机输入的路面的近似程度。以ADAMS软件为平台,建立了多路面输入下的整车模型,路面输入分别选择实测随机路面数据、傅立叶逆变换数据、分形插值模型数据和软件中内置的路面模型,整车模型的建立与实测车辆一致,然后进行整车动态响应仿真试验,通过比较四种路面下的车辆振动响应参数可知,分形模型能较好的逼近实测路面,其振动响应参数最接近实测路面。进一步通过实车实验采集车辆振动响应参数,对仿真结果进行了检验。
The roughness of road surface involves the three factors of people, cars, and roads. In the field of vehicle engineering road roughness is the main factor in the environment of vehicle operating and the main sources of incentive and vibration. Road roughness generates resistance and vibration while vehicles are running. The resistance depletes vehicles' power and influences the life-span of vehicle dynamical system and transmission system. And the vibration impacts vehicles'smooth-going and taking comfortableness, handling stability, traveling speed, bearing system's reliability and life-span. In the field of road engineering, road roughness is one of the evaluation indexes of the road project quality at both domestic and international countries, and roughness-test runs though the various links such as quality testing, assessment, confirmation in the period of road's construction and quality inspection, maintenance in the period of road's operation. It is very significant for the fields of vehicle engineering and road engineering to analyze on the road surface roughness, reasonably carry on appraising and grading the roughness, and establish the input model of road surface.
In this paper, the key technological contents, such as characteristic determination of road surface roughness, its fractal grade-way roughness and fractal model, were carried on deep and systematic research, based on the current situation of analysis, simulation and grading on the road roughness, combined with various of theories and technologies which have been widely used in various fields, such as analytic theory, computing technology, mathematical model.The concrete work is as follows:
Firstly road roughness data collecting system was designed based on virtual instrument technology, to improve the function of the contact-type detectors of the roughness, and a great deal of road roughness data was collected combined with non-contact type of laser roughness car. Then we carried on the traditional analysis of characteristic parameter on the measured data of road roughness. Result showed that more parameters were required when the whole characteristics of the road were described, and these parameters had instability with the changes in metrical scale, even with the same parameters surface characteristics are different greatly, so metrical scale-related and non-uniqueness existed. That is say, the traditional parameters are unable to represent characteristic of the road roughness uniquely, so it is essential to find a new parameters.
Fractal theory has rapidly developed into a new branch of mathematics theory since the year of 1975.It has extensive application in describing the signal, the signal dealing and other fields. The research, which confined with surface roughness and fractal theory, has already become an important research branch in crossing science. Numerous researches indicate that fractal dimension can describe essential characteristic of the complicated phenomenon, so in this paper, fractal theory was used to systematically research into characteristics of measured road roughness. First of all, the ideal fractal curves was generated though Weierstrass-Mandelbrot function to analyze comparatively on different methods by which fractal dimensions was confirmed, then the different methods' applicable range, advantages and disadvantages were confirmed to analyze fractal curves. Through different computational methods to confirm fractal dimension, we analyzed the scale law linear relation of longitudinal sections of measured road roughness, which is regarded as two-dimensional curve. Then we found there is relevant scale law linear relation in log-log coordinate of the three methods, besides root-mean-square method, remainder-variation method and structure function method. After calculating and analyzing on both non-scale sector and fractal dimension of the three methods, we found that root-mean-square method is quite stable in the non-scale sector instead of remainder-variation method and structure function method. On the other hand, root-mean-square method has both clear physical significance and signal function to the surface profile curve, so root-mean-square method is definitely a efficacious method to calculate the fractal dimension of road roughness. After the fractal dimension of selected road roughness was calculated by the method of root-mean-square, we expressed the degree of the road roughness based on the results, then we found that only the parameters of fractal dimension can not determine the condition of a road surface roughness, in other words, the fractal dimension has relativity when it is used to signify measured road roughness.
In order to further confirm the only parameter which signify the road roughness uniquely, started off with existing international grade standard of road roughness, based on the power spectrum-datum of road surface in every grades, high-procedures tabulates for road surface in each grades were obtained through Fourier inverse transformation law. Then we analyzed the relations between the daily signal parameters of road roughness and its fractal parameters. The result showed that there was very small difference and unclear divide between fractal dimension calculated through root-mean-square method and the prominent fractal characteristics of road surface in different grade simulated by Fourier inverse transformation, but the vertical axis'intercept was greater in Logarithmic distribution of measure and scale, which were from worse grade of road and calculated through root-mean-square method. After we calculated the signal parameters, we found that this intercept and root-mean-square deviation of the outline showed a good linear relationship. Because the fractal parameters of road characteristics were very small and their changes were adverse to use of road grade, we proposed to regard the combination between fractal dimension and root-mean-square deviation of profile as a integrated parameter to signify the road roughness, that is, road roughness index was regarded as a signal parameter of road roughness, and the indexes of every grade road were calculated.
Based on the analysis of international standards on classification principles of road surface roughness, its limitations were analyzed by actually road surface roughness. The characteristic parameters of measured road were calculated. Root-mean-square roughness, fractal dimension and the relationship between the roughness index and the road grade based on international standards were analyzed statistically. A road surface roughness classification was established, that regards roughness index as the characteristic parameter.
Fractal road model was finished by fractal interpolation function based on the measured discrete data of road surface roughness. Then the results of fractal model through time-frequency domain and fractal parameters were tested, and the factors which affect the accuracy of fractal interpolation were analyzed. The results indicate that it is feasible to use fractal interpolation function which is based on iterated function system to simulate road surface roughness. It has an important reference on objective characterization of road surface roughness, data compression and the manufacture of road surface roughness measuring instruments. The fractal interpolation model of road surface roughness can be used as road input in the simulation of automobile vibration response and this model is closer to the actual road.
Finally, the applicability of the fractal road mode was verified through simulation and experiment and the approximation degree, between the fractal model of road surface roughness and the actual random input road, was evaluated. On the platform of ADAMS,the vehicle model of multiple input roads has been established and it is consistent with the actual vehicle. The selected input data are actual measured random data, inverse Fourier transform data, data of fractal interpolation model and the built-in road surface model in the software. Then vehicle dynamic response simulation test has been done. After comparing the vehicle dynamic response parameters under the four above input roads, the result indicates that fractal model can better approach the actual measured road. Its dynamic response parameter is closest to measured road. Further collecting the vehicle vibration response parameters through real vehicle experiment and comparing with the simulation results, we found that the simulation result is in good agreement with the real vehicle test result.
引文
[1]赵济海,王哲人.路面不平度的测量分析与应用[M].北京：北京理工大学出版社,2000：11-16
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