José L. Escalona

DEVELOPMENT OF ELASTIC FORCE MODEL FOR WHEEL/RAIL CONTACT PROBLEMS

In this investigation, a new formulation for the wheel/rail contact problem based on the elastic force approach is presented. Crucial to the success of any elastic force formulation for the wheel/rail contact problem is the accurate prediction of the location of the contact points. To this end, features of multibody formulations that allow introducing additional differential equations are exploited in this investigation in order to obtain a good estimate of the rail arc length travelled by the wheel set. In the formulation presented in this paper, four parameters are used to describe the wheel and the rail surfaces. In order to determine the location of the points of contact between the wheel and the rail, a first order differential equation for the rail arc length is introduced and is integrated simultaneously with the multibody equations of motion of the wheel/rail system. The method presented in this paper allows for multiple points of contact between the wheel and the rail by using an optimized search for all possible contact points. The normal contact forces are calculated and used with non-linear expressions for the creepages to determine the creep forces. The paper also discusses two different procedures for the analysis of the two-point contact in the wheel/rail interaction. Numerical results obtained using the elastic force model are presented and compared with the results obtained using the constraint approach.

Application of the Absolute Nodal Coordinate Formulation to Large Rotation and Large Deformation Problems

There are three basic finite element formulations which are used in multibody dynamics. These are the floating frame of reference approach, the incremental method and the large rotation vector approach. In the floating frame of reference and incremental formulations, the slopes are assumed small in order to define infinitesimal rotations that can be treated and transformed as vectors. This description, however, limits the use of some important elements such as beams and plates in a wide range of large displacement applications. As demonstrated in some recent publications, if infinitesimal rotations are used as nodal coordinates, the use of the finite element incremental formulation in the large reference displacement analysis does not lead to exact modeling of the rigid body inertia when the structures rotate as rigid bodies. In this paper, a simple non-incremental finite element procedure that employs the mathematical definition of the slope and uses it to define the element coordinates instead of the infinitesimal and finite rotations is developed for large rotation and deformation problems. By using this description and by defining the element coordinates in the global system, not only the need for performing coordinate transformation is avoided, but also a simple expression for the inertia forces is obtained. The resulting mass matrix is constant and it is the same matrix that appears in linear structural dynamics. It is demonstrated in this paper that this coordinate description leads to exact modeling of the rigid body inertia when the structures rotate as rigid bodies. Nonetheless, the stiffness matrix becomes nonlinear function even in the case of small displacements. The method presented in this paper differs from previous large rotation vector formulations in the sense that the inertia forces, the kinetic energy, and the strain energy are not expressed in terms of any orientation coordinates, and therefore, the method does not require interpolation offinite rotations. While the use of the formulation is demonstrated using a simple planar beam element, the generalization of the method to other element types and to the three dimensional case is straightforward. Using the finite element procedure presented in this paper, beams and plates can be treated as isoparametric elements.

Formulation of Three-Dimensional Joint Constraints Using the Absolute Nodal Coordinates

A wide variety of mechanical and structural multibody systems consist ofvery flexible components subject to kinematic constraints. The widelyused floating frame of reference formulation that employs linear modelsto describe the local deformation leads to a highly nonlinear expressionfor the inertia forces and can be applied to only small deformationproblems. This paper is concerned with the formulation and computerimplementation of spatial joint constraints and forces using the largedeformation absolute nodal coordinate formulation. Unlike the floatingframe of reference formulation that employs a mixed set of absolutereference and local elastic coordinates, in the absolute nodalcoordinate formulation, global displacement and slope coordinates areused. The nonlinear kinematic constraint equations and generalized forceexpressions are expressed in terms of the absolute global displacementsand slopes. In particular, a new formulation for the sliding jointbetween two very flexible bodies is developed. A surface parameter isintroduced as an additional new variable in order to facilitate theformulation of this sliding joint. The constraint and force expressionsdeveloped in this paper are also expressed in terms of generalizedCholesky coordinates that lead to an identity inertia matrix. Severalexamples are presented in order to demonstrate the use of theformulations developed in the paper.

Efficient Evaluation of the Elastic Forces and the Jacobian in the Absolute Nodal Coordinate Formulation

This paper develops a new procedure for evaluating the elastic forces, the elastic energy and the jacobian of the elastic forces in the absolute nodal coordinate formulation. For this procedure, it is fundamental to use some invariant sparse matrices that are integrated in advance and have the property of transforming the evaluation of the elastic forces in a matrix multiplication process. The use of the invariant matrices avoids the integration over the volume of the element for every evaluation of the elastic forces. Great advantages can be achieved from these invariant matrices when evaluating the elastic energy and calculating the jacobian of the elastic forces as well. The exact expression of the jacobian of the differential system of equations of motion is obtained, and some advantages of using the absolute nodal coordinate formulation are pointed out. Numerical results show that there is important time saving as a result of the use of the invariant matrices.

Modelling of structural flexiblity in multibody railroad vehicle systems

This paper presents a review of recent research investigations on the computer modelling of flexible bodies in railroad vehicle systems. The paper will also discuss the influence of the structural flexibility of various components, including the wheelset, the truck frames, tracks, pantograph/catenary systems, and car bodies, on the dynamics of railroad vehicles. While several formulations and computer techniques for modelling structural flexibility are discussed in this paper, a special attention is paid to the floating frame of reference formulation which is widely used and leads to reduced-order finite-element models for flexible bodies by employing component modes synthesis techniques. Other formulations and numerical methods such as semi-analytical approaches, absolute nodal coordinate formulation, finite-segment method, boundary elements method, and discrete elements method are also discussed. This investigation is motivated by the fact that the structural flexibility can have a significant effect on the overall dynamics of railroad vehicles, ride comfort, vibration suppression and noise level reduction, lateral stability, track response to vehicle forces, stress analysis, wheel–rail contact forces, wear and crashworthiness.

On the Use of the Restitution Condition in Flexible Body Dynamics

The difference between the classical treatment offlexible body impact and the treatment of impact in flexiblemultibody dynamics is due to several fundamental reasons. Inthe classical impact theory, simple structures such as beamsand plates are used. Infinite dimensional models can bedeveloped for these simple structural elements to study theimpact dynamics and the wave propagation problem. Flexiblemultibody impact problems, on the other hand, involve bodieswith complex geometry that cannot be modeled using infinitenumber of degrees of freedom. Furthermore, the classicalimpact theory has been mainly concerned with the impactbetween a rigid mass that moves without constraints beforeit impacts a simple flexible structure. This is not amultibody simulation scenario in which the impact occursbetween kinematically constrained bodies that are subjectedto impulsive constraint forces in addition to the impactforces. These constraint forces can influence the motion ofthe two bodies immediately after impact, and as aconsequence, the simple classical theory scenario of impactdoes not apply. It is the objective of this paper to discussthe use of the restitution condition in flexible multibodyimpact problems and demonstrate that the use of thisapproach does not exclude the classical formulation.Nonetheless, the impulse momentum balance approach can serveas an effective and efficient procedure for solving theimpact problem in finite dimensional models that do not obeythe classical wave theory. Energy results of simplestructural elements are presented in order to demonstratethe consistency of using the impulse momentum balanceapproach in solving impact problems in finite dimensionalflexible body applications.

Dynamic Analysis of a Light Structure in Outer Space: Short Electrodynamic Tether

The SET (short electrodynamic tether) is an extremely flexible deployable structure. Unlike most other tethers that orbit with their axis of smallest moment of inertia pointing towards the Earths center (natural position), the SET must orbit with its axis of smallest inertia normal to the orbit plane. The Faraday effect allows the SET to modify its orbit in this position. This is due to the interaction of the Earths magnetic field with the tether, which is an electric conductor. In order to maintain the aforementioned operating position, the SET is subjected to a spin velocity around its axis of smallest inertia. If the system were rigid, the generated gyroscopic pairs would guarantee the systems stability.The tether is not perfectly straight after deployment. This fact could make the rotation of the structure unstable. The problem is similar to the instability of unbalanced rotors. The linear study of unbalanced systems predicts the structural instability once a certain critical velocity is exceeded. Instability is due to internal damping forces. The spin velocity of the SET is greater than the critical velocity. Nevertheless, certain works that include the geometric nonlinearities show a stable behavior under such conditions. The object of this paper is to try to verify these results for the SET.The SET consists of a 100-meter tether with a concentrated mass at its end. The system has been modeled using the floating reference frame approach with natural coordinates. The substructuring technique is used to include nonlinearities in the system.

Finite-element analysis of unsupported sleepers using three-dimensional wheel–rail contact formulation

Owing to repeated traffic loads, the ballast of a railroad track can become unevenly distributed, resulting in different settlements of adjacent sleepers and possibly to a sleeper–ballast loss of contact. The sleeper–ballast loss of contact can significantly change the stiffness properties and modes of deformation of the track. This in turn can cause damage to the structure and can lead to undesirable dynamic behaviour of the vehicles that negotiate this damaged track. The focus of this investigation is on developing a more comprehensive procedure that can be used to examine the effect produced by unsupported sleepers of a flexible track on the dynamics of railroad vehicle systems. The procedure used in this study is based on a validated non-linear multi-body railroad vehicle system formulation that takes into account the coupling between the elastic deformations of the track, the sleeper movements, and the three-dimensional contact parameters. The method used in this article allows for developing a detailed finite-element track model that accounts for non-periodic and asymmetrical mechanical defects. The finite-element equations of motion of the track that includes unsupported sleepers are integrated with the non-linear constrained dynamic equations of the multi-body railroad vehicle system. Component mode synthesis methods are used to reduce the number of the governing equations of motion and eliminate high-frequency modes. The dynamic coupling between the track modal co-ordinates and the wheel–rail contact parameters is considered in this investigation by using creepage and creep force expressions that depend on the track deformations and sleeper movements. In order to demonstrate the use of the procedure described in this article, a track model which has rails and sleepers modelled as beams rigidly connected at the intersection points is used. The effect of the ballast is considered using an elastic foundation. The results obtained in this investigation are used to compare between the responses of two tracks; one with no sleeper–ballast loss of support and the other with unsupported sleepers. The results are reported for different values of the forward velocities in order to have a better understanding of the effect of the sleeper loss of support on the system dynamics. The creepages that depend on the track deformations as well as the system frequencies are analysed. The results obtained in this investigation are found to be in good agreement with the results reported in the literature on unsupported sleepers.

Modeling Two-Point Wheel/Rail Contacts Using Constraint and Elastic-Force Approaches

Two approaches are commonly used for solving the problem of wheel/rail contact in railroad dynamics. The first is the elastic approach in which the wheel is assumed to have six degrees of freedom with respect to the rail. The normal contact forces are defined using Hertz’s contact theory or in terms of assumed stiffness and damping coefficients. The second approach is the constraint approach in which nonlinear kinematic contact constraint equations are introduced, leading to a model in which the wheel has five degrees of freedom with respect to the rail. It is the objective of this investigation to present a new formulation for the wheel/rail contact problem based on the elastic force approach. Crucial to the success of any elastic force formulation for wheel/rail contact problem is the accurate prediction of the location of the contact points. To this end, features of multibody formulations that allow introducing arbitrary differential equations are exploited in this investigation in order to obtain a good estimate of the rail arc length traveled by the wheel set. In the formulation presented in this paper, four surface parameters are used to describe the wheel and the rail surfaces each with arbitrary geometry. In order to determine the location of the points of contact between the wheel and the rail, a first order differential equation for the rail arc length is introduced and is integrated simultaneously with the multibody equations of motion of the wheel/rail system. The method presented in this paper allows for multiple points of contact between the wheel and the rail by using an optimized search for all possible contact points. The normal contact forces are calculated and used with non-linear expressions for the creepages to determine the creep forces. The paper also discusses two different procedures for the analysis of the two-point contact in the wheel/rail interaction. Numerical results obtained using the elastic force model are presented and compared with the results obtained using the constraint approach.Copyright

A new numerical method for the dynamic analysis of impact loads in flexible beams

Abstract The aim of this investigation is to compare the classical St. Venant’s solution of the axial impact on flexible rods with the numerical result obtained using a finite element computational procedure involving component mode synthesis, using a new technique developed to simulate impacts. The impacts treated are those mainly governed by the propagation of elastic waves along the complete flexible body. Local effects at the vicinity of the contact surfaces are neglected. The phenomenon of succession of impacts is analytically proved finding a second period of contact in a particular case. A numerical method is developed to solve the collision based on the theoretical solution. This technique is able to identify the elastic waves which travel along the rod and a force balance is continuously made to yield the contact stress and the velocity of the surfaces in contact. Numerical solution shows an excellent agreement between the numerically obtained displacements histories with the theoretical results. The capability of the eigenvectors to describe the wave propagation is analysed as well as the effect of the elements boundaries.