A Study On The Solution To Ill-conditioned Problem In Vehicle Crash Models

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(m₂/m₁≈1;m₂/m₁>>1;m₂/m₁<<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.