Research On Three-Dimensional High-order Continuous Aligment Design Of High Speed Railway Based On Space Curve Natural Equations

Speed is the lifeline and core competitiveness of high-speed railways.With the substantial increase of speed,there are higher requirements for the stability and safety of train operation.The railway lines work to guide trains forward,so under the condition of high speed,trains rely more and more on the geometric alignment of the lines to provide stable,safe and comfortable operation conditions.Restricted by the development of computer technology and the complexity of expressing three-dimensional lines,the lines subject to the traditional design method are divided into planes,vertical sections and cross sections,and are designed on different planes respectively,then to form a three-dimensional curve through the combination of horizontal and vertical lines.This method considers the impacts of the horizontal and vertical lines on their sections respectively,which has not considered the coupling effect in the design process,separated the internal relationship of the linear space,with no strict analytical model.With the increase of train speed,the two-dimensional design is liable to cause design errors and omissions.In order to clarify the shortcomings of the two-dimensional design method and improve the quality of alignment design for adapting to the development of high-speed railways,this paper starts with the fact that a high-speed railway is a three-dimensional curve in the space,combines differential geometry,takes the natural equations curvature,torsion of the three-dimensional curve as the main parameters to carry out three-dimensional descriptions of the line spatial alignment,complete three-dimensional expression of the train motion model,establish an index for evaluating three-dimensional alignment control,realize the three-dimensional design of the spatial alignment and the three-dimensional display of the design results model,and complete in-depth research on the key problems therein,with the main research contents and innovative results as follows:(1)Established a three-dimensional linear expression equation and completely expressed the spatial curve by the curvature and torsion as well as the spatial attitude parameters of the Frenet frame at the origin of the curve.Established a bi-axle car body motion coordinate system based on the Frenet frame of the particle kinematics,then established a train motion model in a three-dimensional space accordingly and then researched the motion law of the train in the space with curvature,torsion and Frenet frame as the main parameters based on such a model.As is shown in the research,the curvature and torsion determine the shape of the curve and the attitude of the Frenet frame at the origin of the curve ultimately determines the direction of the curve;the curvature and the rolling of the Frenet frame are the key factors to calculate the lateral and vertical acceleration of the car body.The curvature change rate and torsion are the main factors to cause jerkiness.Under the condition of high speed,a discontinuous curvature will lead to the sharp increase of the jerkiness.The three-dimensional train running model shows that the three-dimensional design method can more accurately calculate the train motion state in space,and is more conducive to design lines higher in speed.According to the motion characteristics of high-speed trains,the spatial alignment is divided into high-quality and low-quality curves under the condition of whether the osculating plane changes or not,which has provided a theoretical basis for selecting and optimizing the spatial alignment parameters.(2)Referring to the design criteria and experience of two-dimensional alignment and taking passenger comfort as the evaluation criterion,summarized the limit values of lateral and vertical acceleration and jerkiness of the car body in the two-dimensional design.With such limit values as limit conditions,finally determined the constraint indexes of the three-dimensional alignment design with the speed ranging from 300km/h to 1000km/h based on calculation.Considering the continuity of the line alignment,completed the overall evaluation of the two-dimensional design alignment from the perspective of the three-dimensional design by using such constraint indexes.As can be found from the research,the three-dimensional alignment parameters can more accurately control the minimum curve radius(maximum curvature),which is conducive to flexible arrangement of lines and control of construction investment;the alignment of high-speed railway lines should satisfy at least G2 geometric continuity,the two-dimensional design will cause the geometric continuity to attenuate at the change of curve elements,from G2 to G1 or even lower,and it is difficult to achieve high-order continuity by modifying two-dimensional parameters.The analysis from the perspective of the three-dimensional design reveals the drawbacks of horizontally and vertically separated design ftindamentally:In the two-dimensional alignment design,the parameters are complex and not mutually independent in the influence on spatial alignment,which thus makes it difficult for the two-dimensional design to accurately control the curvature,torsion,curve continuity and other key parameters of alignment,so it is easy to cause design errors and omissions.(3)Established the framework of the three-dimensional alignment design,compared the advantages and disadvantages of the common methods for three-dimensional curve design and found that the common three-dimensional curves have deficiencies in the design of high-speed railway alignment in such aspects as curvature and torsion control,generation of straight line segments,generation of quadratic curves and optimization of alignment,so according to high-speed railway requirements for the geometric characteristics of alignment and the train operation characteristics,a three-dimensional alignment design method has been proposed with non-geometric factors taken into consideration for designers to solve the space curve based on the control points laid in the three-dimensional space.Firstly,put forward the concept of generalized osculating plane,established the expression method of three-dimensional curve space characteristics suitable for high-speed railway alignment design,then established the penalty function of normal fuzzy distribution,and carried out grouped and fitted calculation of the control points according to the spatial characteristics based on the dynamic programming algorithm;then,checked and optimized the compliance and high-order continuity of the alignment through first-order logical reasoning and finally obtained a high-speed railway central line composed of as many high-quality linear units as possible.The research shows that the alignment design of high-speed railways should not only consider the geometric factors of the alignment,but focus on the constraints of the alignment design and the movement characteristics of the train running and select reasonable parameters for the line curve,including curvature,torsion,alignment continuity,curve combinations;if only the geometric properties of the curve are considered,it is not enough to complete high-quality alignment design.(4)Studied the method for rapid three-dimensional modeling of CPU-GPU collaborative urban rail transit lines.Proposed an algorithm for drawing the line model of "CPU discrete-GPU modeling":CPU carries out the discrete operation of the center line,calculates the boundary conditions of the discrete center line,the spatial position and attitude of the components,and relevant other data,and transmits the data package of very simple discrete attributes directly to GPU.CPU and memory only need to manage and save a very small number of line attributes.The results show that the time-consumed by this method is only 0.55%-1.30%of the time using traditional CPU-based method,and the performance indicators such as occupancy of memory and CPU are greatly improved,which help to lessen the pressure of design platform and improve the designing efficiency.The research results in this Paper provide a new perspective for improving the design quality of the traditional curves subject to horizontal and vertical separation during design,provide a new theoretical support for the three-dimensional alignment design of high-speed and higher-speed railways,and provide a practical and theoretical basis for the follow-up research and development of the platform system for three-dimensional alignment design of railways.98 figures,32 tables,150 references.