汽车碰撞事故模型病态问题处理方法的研究
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摘要
本论文以国家自然科学基金重点资助项目《城市路网动态交通管理与控制关键理论及其模拟技术研究》(50338030)的子项目“道路交通事故快速处理技术研究”为依托,针对相互碰撞的两车质量比处于特定范围时基于动量定理的汽车碰撞事故模型所出现的严重病态问题作者进行了三年深入细致的理论研究。为了准确地推算汽车碰撞前的行驶速度,利用矩阵摄动理论对模型病态问题的实质进行了分析,结果表明:病态问题的根本原因在于模型中某些方程间存在严重的线性相关现象。通过数学变换消除了线性相关现象,用数学变换后的新方程替代原始模型中的某些方程建立了重组模型,原始模型与重组模型共同组成了新汽车碰撞事故计算方法。对典型汽车碰撞案例的分析结果表明,当两车质量比处在特定范围时,重组模型的计算精度远远高于原始模型,从而验证了所建立的模型病态问题处理方法的有效性。与目前已有的处理方法相比,本论文所建立的模型病态问题处理方法的创新之处在于只须对模型中的某些方程进行简单的数学变换,即可消除导致模型病态问题的线性相关现象,通过重组模型并应用传统的数学方法—行(列)均衡法彻底解决了模型病态问题,而仅应用行(列)均衡法并不能有效地解决这一问题。本文的研究成果为解决基于动量定理建立的汽车碰撞事故模型病态问题提供了可以借鉴的经验。
Establishing model and applying it is a critical step in vehicle collision accident reappearance.For the Conservation of Momentum to be the most telling theory for analyzing vehicle collision accident,models in internal and external accident reconstruction softwares and methods widely used at present were established on the grounds of the law.Vehicle collision accident reappearance model in reference [1] was set up on the grounds of the the Conservation of Momentum and vehicle collision mechanics character.The model is a linear equation group in the form of Ax=b with the feature of simplicity and practicality,and vehicle collision accident can be analyzed quickly and accurately by means of its sofeware.But,In application of the model,a serious ill-condition problem was discovered when the mass ratio(m2 /m1) of two vehicles in a collision accident fell in some specific range. More significant errors in a calculation are resulted from small errors in certain parameters in the model,and the ill-condition problem can’t be solved completely only by the row equilibrium.In order to calculate the vehicle collision impact in an accident accurately,matrix disturbance theory was applied in analyzing the ill-condition problem nature in the model. The analysis showed that the ill-condition problem was caused by the serious linear interrelation between some equations in the model.Some equations with the serious linear interrelation were found by the Linear Zone and the Interior Product Zone theory.According to the geometry explanation of the ill-conditioned problem in a linear equation group,the linear interrelation problem was solved by the equality shift to some equations with the serious linear interrelation.Some new models were established by substitution of new shifted equations for those in the original model,and these new models could provide more accurate calculation than the original model when the mass ratio is within the specific range so that the ill-conditioned problem was solved completely.Combination of the new model and the original model constituted the vehicle collision impact calculation in the paper,which made up for the original model weakness with a bigger error than the new models when the mass ratio of two vehicles in a collision accident fell in some specific range. The analyses results of a vehicle collision impact case show that the method for solving the ill-conditioned problem and the vehicle collision impact calculation based on the method are effective and practical.
This dissertation combines the national science fund item that is the key theory and simulation technique of city road traffic management and control,and the author has researched into the model ill-conditioned problem discovered in meticulous theory study and typical case analysis for 3.The basic contents as follows:
(1)The author established a systematic and complete theory of solving the ill-conditioned problem in vehicle collsion accident models.The basic contents as follows:The norm of a vector and a matrix.The analysis of ill-conditioned problem in a linear equation group. the geometry explanation of the ill-conditioned problem in a linear equation group.The judgement of the ill-conditioned problem in a linear equation group.The solution to the ill-conditioned problem in a linear equation group. The analysis of the linear equation group capability against perturbation.
(2)Taking condition number as the index for judging the ill-conditioned problem and choosing 4 types of buses as typicle vehicles that fell in the mass ratio(m_2/m_1≈1;m_2/m_1>>1;m_2/m_1<<1) of two vehicles in a collision accident,the author reaserched into the condition number range of the model by simulation to vehicle collision,which examined the effectiveness of solution to the ill-conditioned problem.
(3)Through analysing the ill-condied problem by the geometry explanation of the ill-conditioned problem in a linear equation group and the Linear Zone and the Interior Product Zone theory,the author found the ill-conditioned problem cause that resulted from the serious linear interrelation between some equations in the model.
(4)The angle between two equations in the original model was calculated by the Linear Zone and the Interior Product Zone theory. From the calculation,the author found some equations with the serious linear interrelation so that the aim of solving the ill-conditioned problem was determined.
(5)Through a mathematical transform to some equations with the serious linear interrelation,The angle(θ_(ij))between any two equations was increased in order to eliminate the serious linear interrelation. The linear interrelation problem was solved by the equality shift. A reconstruction model was subsequently established by substitution of new shifted equations for those in the original model. The condition number examination to reconstruction models shows that the condition number of reconstruction models is much smaller than one of the original model,and is smaller than 100,which the solution to the ill-condioned problem meets the necessary requirement of the solution to the ill-condioned problem .
(6)The capability against perturbation in reconstruction models was analyzed by the perturbation theory so as to examine the effect of solution to the ill-conditioned problem.The basic contents as follows:According to the perturbation theory,the author took the relative error of a model result as the index for Analysis of capability against perturbation in models and established the analysis method.Choosing the coefficient—α_(i0) andφ_i that bring forth error easily in a practical application as the perturbation coefficient,the author established aδA_(0i) foemula of A_(0i) and aδb_i foemula of bi in models when the coefficient—α_(i0) andφ_i possess error—Δα_(i0) andΔφ_i.On the base of these foemulas,the author set up relative error demarcation foemulas of models that were balanced by the row equilibrium ,which laid the theoretical foundations for examining the solution effect of the ill-condioned problem.The calculated result of a real collision case shows that the method for solving a model ill-conditioned problem in this dissertation is very effective on increasing the capability against perturbation in models.
The method makes up for the original model weakness that a calculatical accuracy is very low when the mass ratio of two vehicles in a collision accident falls in some specific range.The method for solving a model ill-conditioned problem in the pactical engineering field dependeds on the physical character of a model.In comparison with references,this dissertation creation follows:
The author analysed the ill-conditioned problem nature and found some equations with the serious linear interrelation by the Linear Zone and the Interior Product Zone theory. the linear interrelation problem was solved by the simple mathematical transform to these equations with the serious linear interrelation.The ill-conditioned problem of the original model was solved by reconstructing models.The vehicle collision reappearance model ill-conditioned problem in reference [1] can be solved completely by applying the method for solving a model ill-conditioned problem in this dissertation and the row equilibrium,which supplys a successful experience for solving the ill-conditioned problem in vehicle collision reappearance models based on the Conservation of Momentum.Although buses were taken as typical vehicles in the dissertation,new vehicle collision calculation can be applied to any other vehicles.
Establishing model and applying it is a critical step in vehicle collision accident reappearance.For the Conservation of Momentum to be the most telling theory for analyzing vehicle collision accident,models in internal and external accident reconstruction softwares and methods widely used at present were established on the grounds of the law.Vehicle collision accident reappearance model in reference [1] was set up on the grounds of the the Conservation of Momentum and vehicle collision mechanics character.The model is a linear equation group in the form of Ax=b with the feature of simplicity and practicality,and vehicle collision accident can be analyzed quickly and accurately by means of its sofeware.But,In application of the model,a serious ill-condition problem was discovered when the mass ratio(m2 /m1) of two vehicles in a collision accident fell in some specific range. More significant errors in a calculation are resulted from small errors in certain parameters in the model,and the ill-condition problem can’t be solved completely only by the row equilibrium.In order to calculate the vehicle collision impact in an accident accurately,matrix disturbance theory was applied in analyzing the ill-condition problem nature in the model. The analysis showed that the ill-condition problem was caused by the serious linear interrelation between some equations in the model.Some equations with the serious linear interrelation were found by the Linear Zone and the Interior Product Zone theory.According to the geometry explanation of the ill-conditioned problem in a linear equation group,the linear interrelation problem was solved by the equality shift to some equations with the serious linear interrelation.Some new models were established by substitution of new shifted equations for those in the original model,and these new models could provide more accurate calculation than the original model when the mass ratio is within the specific range so that the ill-conditioned problem was solved completely.Combination of the new model and the original model constituted the vehicle collision impact calculation in the paper,which made up for the original model weakness with a bigger error than the new models when the mass ratio of two vehicles in a collision accident fell in some specific range. The analyses results of a vehicle collision impact case show that the method for solving the ill-conditioned problem and the vehicle collision impact calculation based on the method are effective and practical.
This dissertation combines the national science fund item that is the key theory and simulation technique of city road traffic management and control,and the author has researched into the model ill-conditioned problem discovered in meticulous theory study and typical case analysis for 3.The basic contents as follows:
(1)The author established a systematic and complete theory of solving the ill-conditioned problem in vehicle collsion accident models.The basic contents as follows:The norm of a vector and a matrix.The analysis of ill-conditioned problem in a linear equation group. the geometry explanation of the ill-conditioned problem in a linear equation group.The judgement of the ill-conditioned problem in a linear equation group.The solution to the ill-conditioned problem in a linear equation group. The analysis of the linear equation group capability against perturbation.
(2)Taking condition number as the index for judging the ill-conditioned problem and choosing 4 types of buses as typicle vehicles that fell in the mass ratio(m_2/m_1≈1;m_2/m_1>>1;m_2/m_1<<1) of two vehicles in a collision accident,the author reaserched into the condition number range of the model by simulation to vehicle collision,which examined the effectiveness of solution to the ill-conditioned problem.
(3)Through analysing the ill-condied problem by the geometry explanation of the ill-conditioned problem in a linear equation group and the Linear Zone and the Interior Product Zone theory,the author found the ill-conditioned problem cause that resulted from the serious linear interrelation between some equations in the model.
(4)The angle between two equations in the original model was calculated by the Linear Zone and the Interior Product Zone theory. From the calculation,the author found some equations with the serious linear interrelation so that the aim of solving the ill-conditioned problem was determined.
(5)Through a mathematical transform to some equations with the serious linear interrelation,The angle(θ_(ij))between any two equations was increased in order to eliminate the serious linear interrelation. The linear interrelation problem was solved by the equality shift. A reconstruction model was subsequently established by substitution of new shifted equations for those in the original model. The condition number examination to reconstruction models shows that the condition number of reconstruction models is much smaller than one of the original model,and is smaller than 100,which the solution to the ill-condioned problem meets the necessary requirement of the solution to the ill-condioned problem .
(6)The capability against perturbation in reconstruction models was analyzed by the perturbation theory so as to examine the effect of solution to the ill-conditioned problem.The basic contents as follows:According to the perturbation theory,the author took the relative error of a model result as the index for Analysis of capability against perturbation in models and established the analysis method.Choosing the coefficient—α_(i0) andφ_i that bring forth error easily in a practical application as the perturbation coefficient,the author established aδA_(0i) foemula of A_(0i) and aδb_i foemula of bi in models when the coefficient—α_(i0) andφ_i possess error—Δα_(i0) andΔφ_i.On the base of these foemulas,the author set up relative error demarcation foemulas of models that were balanced by the row equilibrium ,which laid the theoretical foundations for examining the solution effect of the ill-condioned problem.The calculated result of a real collision case shows that the method for solving a model ill-conditioned problem in this dissertation is very effective on increasing the capability against perturbation in models.
The method makes up for the original model weakness that a calculatical accuracy is very low when the mass ratio of two vehicles in a collision accident falls in some specific range.The method for solving a model ill-conditioned problem in the pactical engineering field dependeds on the physical character of a model.In comparison with references,this dissertation creation follows:
The author analysed the ill-conditioned problem nature and found some equations with the serious linear interrelation by the Linear Zone and the Interior Product Zone theory. the linear interrelation problem was solved by the simple mathematical transform to these equations with the serious linear interrelation.The ill-conditioned problem of the original model was solved by reconstructing models.The vehicle collision reappearance model ill-conditioned problem in reference [1] can be solved completely by applying the method for solving a model ill-conditioned problem in this dissertation and the row equilibrium,which supplys a successful experience for solving the ill-conditioned problem in vehicle collision reappearance models based on the Conservation of Momentum.Although buses were taken as typical vehicles in the dissertation,new vehicle collision calculation can be applied to any other vehicles.
引文
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