Martin B. Hamper

Coupled finite element and multibody system dynamics modeling of a three-dimensional railroad system

During the last two centuries, railroad vehicles have been an important means of transportation of both people and cargo, due to their economic and comfort advantages. Railroad vehicles are a highly economical means of transporting large quantities of cargo over long distances, and also provide a safe and comfortable means of passenger transport. Over the last 30 years or so, the finite element method (FEM) has become more widely used to model railroad systems including the rails, sleepers and substructure. Multibody system dynamics (MBS) software programs are used to model the contact between the wheels and the rails in an effort to study the contact forces and the general dynamics of railroad vehicles. Coupling both the FEM and MBS is a very useful technique to build a reliable model that includes the advantages of both methods. In this work, a full three-dimensional finite element model is created that uses beam, solid and spring elements to model the rails, fasteners, sleepers and substructure. The model treats the rails and the substructure as deformable bodies. Mode shapes of the finite element model are extracted for use in a MBS code to analyze the deformation of the track and substructure under dynamic loading conditions. The results of this new model agree well with results published in the literature.

Use of Finite Element and Finite Segment Methods in Modeling Rail Flexibility: A Comparative Study

Safety requirements and optimal performance of railroad vehicle systems require the use of multibody system (MBS) dynamics formulations that allow for modeling flexible bodies. This investigation will present three methods suited for the study of flexible track models while conclusions about their implementations and features are made. The first method is based on the floating frame of reference (FFR) formulation which allows for the use of a detailed finite element mesh with the component mode synthesis technique in order to obtain a reduced order model. In the second method, the flexible body is modeled as a finite number of rigid elements that are connected by springs and dampers. This method, called finite segment method (FSM) or rigid finite element method, requires the use of rigid MBS formulations only. In the third method, the FFR formulation is used to obtain a model that is equivalent to the FSM model by assuming that the rail segments are very stiff, thereby allowing the exclusion of the high frequency modes associated with the rail deformations. This FFR/FS model demonstrates that some rail movement scenarios such as gauge widening can be captured using the finite element FFR formulation. The three procedures FFR, FSM, and FFR/FS will be compared in order to establish differences among them and analyze the specific application of the FSM to modeling track flexibility. Convergence of the methods is analyzed. The three methods proposed in this investigation for modeling the movement of three-dimensional tracks are used with a three-dimensional elastic wheel/rail contact formulation that predicts contact points online and allows for updating the creepages to account for the rail deformations. Several conclusions will be drawn in view of the results obtained in this investigation.

Use of ANCF Surface Geometry in the Rigid Body Contact Problems: Application to Railroad Vehicle Dynamics

In the analysis of multibody system (MBS) dynamics, contact between two arbitrary rigid bodies is a fundamental feature in a variety of models. Many procedures have been proposed to solve the rigid body contact problem, most of which belong to one of two categories: off-line and on-line contact search methods. This investigation will focus on the development of a contact surface model for the rigid body contact problem in the case where an on-line three-dimensional non-conformal contact evaluation procedure, such as the elastic contact formulation - algebraic equations (ECF-A), is employed. It is shown that the contact surface must have continuity in the second order spatial derivatives when used in conjunction with ECF-A. Many of the existing surface models rely on direct linear interpolation of profile curves which leads to first order spatial derivative discontinuities. This, in turn, leads to erroneous spikes in the prediction of contact forces. To this end, an absolute nodal coordinate formulation (ANCF) thin plate surface model is developed in order to ensure second order spatial derivative continuity to satisfy the requirements of the contact formulation employed. A simple example of a railroad vehicle negotiating a turnout, which includes a variable cross-section rail, is tested for the cases of the new ANCF thin plate element surface, an existing ANCF thin plate element surface with first order spatial derivative continuity, and the direct linear profile interpolation method. A comparison of the numerical results reveals the benefits of using the new ANCF surface geometry developed in this investigation.Copyright

Development of the Finite Segment Method for Modeling Railroad Track Structures

In the finite segment method, the dynamics of a deformable body is described using a set of rigid bodies that are connected by elastic force elements. This approach can be used, as demonstrated in this investigation, in the simulation of some rail movement scenarios. The purpose of this investigation is to develop a new track model that combines the absolute nodal coordinate formulation (ANCF) geometry and the finite segment method. The ANCF finite elements define the track geometry in the reference configuration as well as the change in the geometry due to the movement of the finite segments of the track. Using ANCF geometry and the finite segment kinematics, the location of the wheel/rail contact point is predicted online and used to update the creepage expressions due to the finite segment displacements and rotations. The location of the wheel/rail contact point and the updated creepage expressions are used to evaluate the creep forces. A three-dimensional elastic contact formulation (ECF-A), that allows for wheel/rail separation, is used in this investigation. The rail displacement due to the applied loads is modeled by a set of rigid finite segments that are connected by set of spring-damper elements. Each rail finite segment is assumed to have six rigid body degrees of freedom. The equations of motion of the finite segments are integrated with the railroad vehicle system equations of motion in a sparse matrix formulation. The resulting dynamic equations are solved using a predictor-corrector numerical integration method that has a variable order and variable step size. As shown in this paper, the finite segments may be used to model specific phenomena that occur in railroad vehicle applications, including rail rotations and gage widening. The procedure used in this investigation to implement the finite segment method in a general purpose multibody system (MBS) computer program is described. Four simple models are presented in order to demonstrate the implementation of the finite segment method in MBS algorithms. The limitations of using the finite segments approach for modeling the track structure and rail flexibility are also discussed.Copyright

Use of ANCF Surface Geometry in the Rigid Body Contact Problem: Application to Railroad Vehicle Dynamics

In the analysis of multibody system (MBS) dynamics, contact between two arbitrary rigid bodies is a fundamental feature in a variety of models. Many procedures have been proposed to solve the rigid body contact problem, most of which belong to one of two categories: off-line and on-line contact search methods. This investigation will focus on the development of a contact surface model for the rigid body contact problem in the case where an on-line three-dimensional non-conformal contact evaluation procedure, such as the elastic contact formulation - algebraic equations (ECF-A), is employed. It is shown that the contact surface must have continuity in the second order spatial derivatives when used in conjunction with ECF-A. Many of the existing surface models rely on direct linear interpolation of profile curves which leads to first order spatial derivative discontinuities. This, in turn, leads to erroneous spikes in the prediction of contact forces. To this end, an absolute nodal coordinate formulation (ANCF) thin plate surface model is developed in order to ensure second order spatial derivative continuity to satisfy the requirements of the contact formulation employed. A simple example of a railroad vehicle negotiating a turnout, which includes a variable cross-section rail, is tested for the cases of the new ANCF thin plate element surface, an existing ANCF thin plate element surface with first order spatial derivative continuity, and the direct linear profile interpolation method. A comparison of the numerical results reveals the benefits of using the new ANCF surface geometry developed in this investigation.Copyright

A Spatial Geometry Approach to Variable Cross-Section Rail Modeling: Application to Switches and Turnouts

Contact between the wheel and rail can have a significant effect on the dynamics of vehicle/track interaction models. Many existing rail surface models rely on curve based geometry which may lead to some geometric inaccuracy in the case of variable cross-section rails. This investigation will focus on the development of a new spatial geometry based rail surface description which reduces this geometric inaccuracy. It has been shown in literature that certain CAD geometry types, such as B-Spline curves and surfaces, may be converted to equivalent absolute nodal coordinate formulation (ANCF) finite elements without a loss of geometric accuracy. To this end, a new ANCF surface description of variable cross-section rails is developed. This investigation also demonstrates the feasibility of using, in the future, 3D surface scanning techniques as well as profile curve measurements to develop a rail surface geometry model using the new ANCF surface which can be systematically integrated with complex multibody system (MBS) models. A realistic railroad vehicle example of a turnout, which includes variable cross-section rails, is tested for the case of the new ANCF surface. A study of the numerical results reveals the benefits of using the ANCF surface geometry developed in this investigation.Copyright

Rolling Condition and Gyroscopic Moments in Curve Negotiations

In vehicle system dynamics, the effect of the gyroscopic moments can be significant during curve negotiations. The absolute angular velocity of the body can be expressed as the sum of two vectors; one vector is due to the curvature of the curve, while the second vector is due to the rate of changes of the angles that define the orientation of the body with respect to a coordinate system that follows the body motion. In this paper, the configuration of the body in the global coordinate system is defined using the trajectory coordinates in order to examine the effect of the gyroscopic moments in the case of curve negotiations. These coordinates consist of arc length, two relative translations and three relative angles. The relative translations and relative angles are defined with respect to a trajectory coordinate system that follows the motion of the body on the curve. It is shown that when the yaw and roll angles relative to the trajectory coordinate system are constrained and the motion is predominantly rolling, the effect of the gyroscopic moment on the motion becomes negligible, and in the case of pure rolling and zero yaw and roll angles, the generalized gyroscopic moment associated with the system degrees of freedom becomes identically zero. The analysis presented in this investigation sheds light on the danger of using derailment criteria that are not obtained using laws of motion, and therefore, such criteria should not be used in judging the stability of railroad vehicle systems. Furthermore, The analysis presented in this paper shows that the roll moment which can have a significant effect on the wheel/rail contact forces depends on the forward velocity in the case of curve negotiations. For this reason, roller rigs that do not allow for the wheelset forward velocity cannot capture these moment components, and therefore, cannot be used in the analysis of curve negotiations. A model of a suspended railroad wheelset is used in this investigation to study the gyroscopic effect during curve negotiations.Copyright

Rolling condition and gyroscopic moments in curve negotiations

In vehicle system dynamics, the effect of the gyroscopic moments can be significant during curve negotiations. The absolute angular velocity of the body can be expressed as the sum of two vectors; one vector is due to the curvature of the curve, while the second vector is due to the rate of changes of the angles that define the orientation of the body with respect to a coordinate system that follows the body motion. In this paper, the configuration of the body in the global coordinate system is defined using the trajectory coordinates in order to examine the effect of the gyroscopic moments in the case of curve negotiations. These coordinates consist of arc length, two relative translations and three relative angles. The relative translations and relative angles are defined with respect to a trajectory coordinate system that follows the motion of the body on the curve. It is shown that when the yaw and roll angles relative to the trajectory coordinate system are constrained and the motion is predominantly rolling, the effect of the gyroscopic moment on the motion becomes negligible, and in the case of pure rolling and zero yaw and roll angles, the generalized gyroscopic moment associated with the system degrees of freedom becomes identically zero. The analysis presented in this investigation sheds light on the danger of using derailment criteria that are not obtained using laws of motion, and therefore, such criteria should not be used in judging the stability of railroad vehicle systems. Furthermore, The analysis presented in this paper shows that the roll moment which can have a significant effect on the wheel/rail contact forces depends on the forward velocity in the case of curve negotiations. For this reason, roller rigs that do not allow for the wheelset forward velocity cannot capture these moment components, and therefore, cannot be used in the analysis of curve negotiations. A model of a suspended railroad wheelset is used in this investigation to study the gyroscopic effect during curve negotiations.Copyright

Modeling Rail Flexibility Using Finite Element and Finite Segment Methods

Safety requirements and optimal performance of railroad systems require the utilization of multibody System (MBS) formulations that allow for modeling flexible bodies. This investigation will present three methods suited for the study of flexible track models while conclusions about their implementations and features are made. A validated method combining Floating Frame of Reference (FFR) and Finite Element (FE) to model flexible rails is utilized for comparison. In this procedure, component mode synthesis is used to extract a number of low-frequency modes of vibration which describe the deformation of the rail. Likewise, a method that discretizes the flexible body as a finite number of rigid elements that are linked by springs and dampers is applied for railroad simulations. This method, called Finite Segment or Rigid Finite Element (FS), can in turn be combined with FFR through the extraction of mode shapes of the FS model. Convergence of the methods is analyzed. A comparison will be made between these three procedures establishing differences among them and analyzing the specific application of FS to modeling track flexibility. The three aforementioned procedures may be applied to three-dimensional track models and will be used together with three-dimensional wheel/rail contact formulation that predicts contact points online and allows for updating the creepages to account for the rail movements and deformations. Several comparisons and conclusions will be drawn in view of the results obtained in this investigation.Copyright