A velocity transformation method for the nonlinear dynamic simulation of railroad vehicle systems

The solution of the constrained multibody system equations of motion using the generalized coordinate partitioning method requires the identification of the dependent and independent coordinates. Using this approach, only the independent accelerations are integrated forward in time in order to determine the independent coordinates and velocities. Dependent coordinates are determined by solving the nonlinear constraint equations at the position level. If the constraint equations are highly nonlinear, numerical difficulties can be encountered or more Newton–Raphson iterations may be required in order to achieve convergence for the dependent variables. In this paper, a velocity transformation method is proposed for railroad vehicle systems in order to deal with the nonlinearity of the constraint equations when the vehicles negotiate curved tracks. In this formulation, two different sets of coordinates are simultaneously used. The first set is the absolute Cartesian coordinates which are widely used in general multibody system computer formulations. These coordinates lead to a simple form of the equations of motion which has a sparse matrix structure. The second set is the trajectory coordinates which are widely used in specialized railroad vehicle system formulations. The trajectory coordinates can be used to obtain simple formulations of the specified motion trajectory constraint equations in the case of railroad vehicle systems. While the equations of motion are formulated in terms of the absolute Cartesian coordinates, the trajectory accelerations are the ones which are integrated forward in time. The problems associated with the higher degree of differentiability required when the trajectory coordinates are used are discussed. Numerical examples are presented in order to examine the performance of the hybrid coordinate formulation proposed in this paper in the analysis of multibody railroad vehicle systems.     

A comparative study of joint formulations: application to multibody system tracked vehicles

This paper is focused on the dynamic formulation of mechanical joints using different approaches that lead to different models with different numbers of degrees of freedom. Some of these formulations allow for capturing the joint deformations using a discrete elastic model while the others are continuum-based and capture joint deformation modes that cannot be captured using the discrete elastic joint models. Specifically, three types of joint formulations are considered in this investigation; the ideal, compliant discrete element, and compliant continuum-based joint models. The ideal joint formulation, which does not allow for deformation degrees of freedom in the case of rigid body or small deformation analysis, requires introducing a set of algebraic constraint equations that can be handled in computational multibody system (MBS) algorithms using two fundamentally different approaches: constrained dynamics approach and penalty method. When the constrained dynamics approach is used, the constraint equations must be satisfied at the position, velocity, and acceleration levels. The penalty method, on the other hand, ensures that the algebraic equations are satisfied at the position level only. In the compliant discrete element joint formulation, no constraint conditions are used; instead the connectivity conditions between bodies are enforced using forces that can be defined in their most general form in MBS algorithms using bushing elements that allow for the definition of general nonlinear forces and moments. The new compliant continuum-based joint formulation, which is based on the finite element (FE) absolute nodal coordinate formulation (ANCF), has several advantages: (1) It captures modes of joint deformations that cannot be captured using the compliant discrete joint models; (2) It leads to linear connectivity conditions, thereby allowing for the elimination of the dependent variables at a preprocessing stage; (3) It leads to a constant inertia matrix in the case of chain like structure; and (4) It automatically captures the deformation of the bodies using distributed inertia and elasticity. The formulations of these three different joint models are compared in order to shed light on the fundamental differences between them. Numerical results of a detailed tracked vehicle model are presented in order to demonstrate the implementation of some of the formulations discussed in this investigation.     

On the Computer Formulations of the Wheel/Rail Contact Problem

In this investigation, four nonlinear dynamic formulations that can be used in the analysis of the wheel/rail contact are presented, compared and their performance is evaluated. Two of these formulations employ nonlinear algebraic kinematic constraint equations to describe the contact between the wheel and the rail (constraint approach), while in the other two formulations the contact force is modeled using a compliant force element (elastic approach). The goal of the four formulations is to provide accurate nonlinear modeling of the contact between the wheel and the rail, which is crucial to the success of any computational algorithm used in the dynamic analysis of railroad vehicle systems. In the formulations based on the elastic approach, the wheel has six degrees of freedom with respect to the rail, and the normal contact forces are defined as function of the penetration using Hertzs contact theory or using assumed stiffness and damping coefficients. The first elastic method is based on a search for the contact locations using discrete nodal points. As previously presented in the literature, this method can lead to impulsive forces due to the abrupt change in the location of the contact point from one time step to the next. This difficulty is avoided in the second elastic approach in which the contact points are determined by solving a set of algebraic equations. In the formulations based on the constraint approach, on the other hand, the case of a non-conformal contact is assumed, and nonlinear kinematic contact constraint equations are used to impose the contact conditions at the position, velocity and acceleration levels. This approach leads to a model, in which the wheel has five degrees of freedom with respect to the rail. In the constraint approach, the wheel penetration and lift are not permitted, and the normal contact forces are calculated using the technique of Lagrange multipliers and the augmented form of the system dynamic equations. Two equivalent constraint formulations that employ two different solution procedures are discussed in this investigation. The first method leads to a larger system of equations by augmenting all the contact constraint equations to the dynamic equations of motion, while in the second method an embedding procedure is used to obtain a reduced system of equations from which the surface parameter accelerations are systematically eliminated. Numerical results are presented in order to examine the performance of various methods discussed in this study.     

A thermodynamic approach to non-linear viscoelasticity

We show that Extended Irreversible Thermodynamics (EIT) may be regarded as a unifying thermodynamic scheme which generates a variety of non-linear viscoelastic equations of state. In particular, we show how the so-called eight-constants Oldroyd model, with all its invariant properties, may be derived from the postulates of EIT. It is also shown that the structure of the thermodynamic based rheological equations of state is in complete agreement with that of the representation theorems formulation of constitutive relationships originally proposed by Rivlin and Ericksen. We discuss how more complex rheological models can be derived from higher order approximations of the theory. Finally, the connections between our approach and other related works are briefly discussed.     

A matrix formulation for the dynamic analysis of planar mechanisms using point coordinates and velocity transformation

This paper presents a matrix formulation for the dynamic analysis of planar mechanisms consisting of interconnected rigid bodies. The formulation initially uses the rectangular Cartesian coordinates of an equivalent constrained system of particles to define the configuration of the mechanical system. This results in a simple and straightforward procedure for generating the equations of motion. The equations of motion are then derived in terms of relative joint coordinates through the use of a velocity transformation matrix. The velocity transformation matrix relates the relative joint velocities to the Cartesian velocities. For the open-loop case, this process automatically eliminates all of the non-working constraint forces and leads to an efficient integration of the equations of motion. For the closed-loop case, suitable joints should be cut and few cut-joints constraint equations should be included for each closed loop. Two examples are used to demonstrate the generality and efficiency of the proposed method.     

A divide and conquer algorithm for constrained multibody system dynamics based on augmented Lagrangian method with projections-based error correction

This paper presents a?new parallel algorithm for dynamics simulation of general multibody systems. The developed formulations are iterative and possess divide and conquer structure. The constraints equations are imposed at the acceleration level. Augmented Lagrangian methods with mass-orthogonal projections are used to prevent from constraint violation errors. The proposed approaches treat tree topology mechanisms or multibody systems which contain kinematic closed loops in a?uniform manner and can handle problems with rank deficient Jacobian matrices. Test case results indicate good accuracy performance dependent on the expense put in the iterative correction of constraint equations. Good numerical properties and robustness of the algorithms are observed when handling systems with single and coupled kinematic loops, redundant constraints, which may repeatedly enter singular configurations.     

Higher order model for soft and hard elastic interfaces

The present paper deals with the derivation of a higher order theory of interface models. In particular, it is studied the problem of two bodies joined by an adhesive interphase for which “soft” and “hard” linear elastic constitutive laws are considered. For the adhesive, interface models are determined by using two different methods. The first method is based on the matched asymptotic expansion technique, which adopts the strong formulation of classical continuum mechanics equations (compatibility, constitutive and equilibrium equations). The second method adopts a suitable variational (weak) formulation, based on the minimization of the potential energy. First and higher order interface models are derived for soft and hard adhesives. In particular, it is shown that the two approaches, strong and weak formulations, lead to the same asymptotic equations governing the limit behavior of the adhesive as its thickness vanishes. The governing equations derived at zero order are then put in comparison with the ones accounting for the first order of the asymptotic expansion, thus remarking the influence of the higher order terms and of the higher order derivatives on the interface response. Moreover, it is shown how the elastic properties of the adhesive enter the higher order terms. The effects taken into account by the latter ones could play an important role in the nonlinear response of the interface, herein not investigated. Finally, two simple applications are developed in order to illustrate the differences among the interface theories at the different orders.     

Geometry and differentiability requirements in multibody railroad vehicle dynamic formulations

The dynamic equations of multibody railroad vehicle systems can be formulated using different sets of generalized coordinates; examples of these sets of coordinates are the absolute Cartesian and trajectory coordinates. The absolute coordinate based formulations do not require introducing an intermediate track coordinate system since all the absolute coordinates are defined in the global system. On the other hand, when the trajectory coordinates are used, a track coordinate system that follows the motion of a body in the railroad vehicle system is introduced. This track coordinate system is defined by the track geometry and the distance traveled by the body along the track centerline. The configuration of the body with respect to the track coordinate system is defined using five coordinates; two translations and three Euler angles. In this paper, the formulations based on the absolute and trajectory coordinates are compared. It is shown that these two sets of coordinates require different degrees of differentiability and smoothness. When an elastic contact formulation is used to study the wheel/rail dynamic interaction, there are significant differences in the order of the derivatives required in both formulations. In fact, as demonstrated in this study, in the absence of a contact constraint formulation, higher order derivatives with respect to geometric parameters are still required when the equations are formulated using the trajectory coordinates. The formulation of the constraints used in the analysis of the wheel/rail contact is discussed and it is shown that when the absolute coordinates are used, only third order derivatives need to be evaluated. The relationship between the track frame used in railroad vehicle dynamics and the Frenet frame used in the theory of curves to describe the curve geometry is also discussed in this paper. Based on the analysis presented in this paper, the advantages and drawbacks of a hybrid method which employs both the absolute and trajectory coordinates and planar contact conditions in order to reduce the number of contact constraints and relax the differentiability requirements are discussed. In this method, the absolute coordinates are used to formulate the equations of motion of the railroad vehicle system. The absolute coordinate solution can be used to determine the trajectory coordinates and their time derivatives. Using the trajectory coordinates, the motion of the body in the vehicle with respect to the track coordinate system can be predicted and used in the formulation of the planar contact model.     

A transient solution method for the finite element incompressible Navier-Stokes equations

A numerical procedure for solving the time-dependent, incompressible Navier-Stokes equations is presented. The present method is based on a set of finite element equations of the primitive variable formulation, and a direct time integration method which has unique features in its formulation as well as in its evaluation of the contribution of external functions. Particular processes regarding the continuity conditions and the boundary conditions lead to a set of non-linear recurrence equations which represent evolution of the velocities and the pressures under the incompressibility constraint. An iteration process as to the non-linear convective terms is performed until the convergence is achieved in every integration step. Excessively artificial techniques are not introduced into the present solution procedure. Numerical examples with vortex shedding behind a rectangular cylinder are presented to illustrate the features of the proposed method. The calculated results are compared with experimental data and visualized flow fields in literature.     

A geometric solution to the general single contact frictionless problem by combining concepts of analytical dynamics and b-calculus

This study presents a systematic approach, leading to a new set of equations of motion for a class of mechanical systems subject to a single frictionless contact constraint. To achieve this goal, some fundamental concepts of b-geometry are utilized and adapted to the general framework of Analytical Dynamics. These concepts refer to the theory of manifolds with boundary and provide a suitable and strong theoretical foundation. First, the boundary is defined within the original configuration manifold of the system by the equality in the unilateral constraint. Then, an appropriate vector bundle is considered, involving only smooth vector fields, even at the boundary. After determining the essential geometric properties (i.e., the metric and the connection) near the boundary, Newton’s law of motion is applied. In this way, the equations of motion during the contact phase are derived as a system of ordinary differential equations. These equations possess a special form inside a thin boundary layer. In particular, the essential dynamics of the systems examined is found to be governed by a single second order ordinary differential equation, which is investigated fully. Moreover, a critical comparison of the present formulation with the classical formulations applied to systems with a frictionless contact is performed. Finally, the effect of the dominant parameters on the dynamics during the contact phase and the steps for the application process to mechanical systems are illustrated by two selected examples, referring to contact of a particle and a rigid body with a plane.     

Local boundary integral equations for orthotropic shallow shells

A meshless local Petrov–Galerkin (MLPG) formulation is presented for bending problems of shear deformable shallow shells with orthotropic material properties. Shear deformation of shells described by the Reissner theory is considered. Analyses of shells under static and dynamic loads are given here. For transient elastodynamic case the Laplace-transform is used to eliminate the time dependence of the field variables. A weak formulation with a unit test function transforms the set of governing equations into local integral equations on local subdomains in the plane domain of the shell. Nodal points are randomly spread in that domain and each node is surrounded by a circular subdomain to which local integral equations are applied. The meshless approximation based on the moving least-squares (MLS) method is employed for the implementation. Unknown Laplace-transformed quantities are computed from the local boundary integral equations. The time-dependent values are obtained by the Stehfest’s inversion technique.     

A formulation of anisotropic continuum elastoplasticity at finite strains. Part I: Modelling

A constitutive model for anisotropic elastoplasticity at finite strains is developed together with its numerical implementation. An anisotropic elastic constitutive law is described in an invariant setting by use of structural tensors and the elastic strain measure C_e. The elastic strain tensor as well as the structural tensors are assumed to be invariant in relation to superimposed rigid body rotations. An anisotropic Hill-type yield criterion, described by a non-symmetric Eshelby-like stress tensor and further structural tensors, is developed, where use is made of representation theorems for functions with non-symmetric arguments. The model also considers non-linear isotropic hardening. Explicit results for the specific case of orthotropic anisotropy are given. The associative flow rule is employed and the features of the inelastic flow rule are discussed in full. It is shown that the classical definition of the plastic material spin is meaningless in conjunction with the present formulation. Instead, the study motivates an alternative definition, which is based on the demand that such a quantity must be dissipation-free, as the plastic material spin is in the case of isotropy. Equivalent spatial formulations are presented too. The full numerical treatment is considered in Part II.     

The Elastic-Plastic Torsion Problem: A Direct Boundary Approach∗

A new mechanics theorem pertaining to the elastic-plastic torsion problem is shown; i.e., the complementary energy evaluated on some admissible set of functions is minimized with respect to the transient (free) boundary between the elastic and plastic zones. The elastic-plastic torsion problem is then formulated as an unconstrained shape optimal design problem which treats the free boundary directly as a unknown. Unlike other formulations there is no inequality constraint involved in this formulation. The derivation of shape design sensitivity and numerical results for circular cross section and Sokolovsky's oval are presented.     

A Flow-on-Manifold Formulation of Differential-Algebraic Equations

We derive a flow formulation of differential-algebraic equations (DAEs), implicit differential equations whose dynamics are restricted by algebraic constraints. Using the framework of derivatives arrays and the strangeness-index, we identify the systems that are uniquely solvable on a particular set of initial values and thus possess a flow, the mapping that uniquely relates a given initial value with the solution through this point. The flow allows to study system properties like invariant sets, stability, monotonicity or positivity. For DAEs, the flow further provides insights into the manifold onto which the system is bound to and into the dynamics on this manifold. Using a projection approach to decouple the differential and algebraic components, we give an explicit representation of the flow that is stated in the original coordinate space. This concept allows to study DAEs whose dynamics are restricted to special subsets in the variable space, like a cone or the nonnegative orthant.     

On Isotropic Invariants of the Elasticity Tensor

Isotropic invariants of the elasticity tensor always yield the same values no matter what coordinate system is concerned and therefore they characterize the linear elasticity of a solid material intrinsically. There exists a finite set of invariants of the elasticity tensor such that each invariant of the elasticity tensor can be expressed as a single-valued function of this set. Such a set, called a basis of invariants of the elasticity tensor, can be used to realize a parametrization of the manifold of orbits of elastic moduli, i.e. to distinguish different kinds of linear elastic materials. Seeking such a basis is an old problem in theory of invariants and seems to have been unsuccessful until now. In this paper, by means of the unique spectral decomposition of the elasticity tensor every invariant of the elasticity tensor is shown to be a joint invariant of the eigenprojections of the elasticity tensor, and then by utilizing some properties of the eigenprojections a basis for each case concerning the multiplicity of the eigenvalues of the elasticity tensor is presented in terms of joint invariants of the eigenprojections. In addition to the foregoing properties, the presented invariants may also be used to form invariant criteria for identification of elastic symmetry axes.     

Fractional-order relaxation laws in non-linear viscoelasticity

Viscoelastic constitutive equations are constructed by assuming that the stress is a nonlinear function of the current strain and of a set of internal variables satisfying relaxation equations of fractional order. The dependence of the relaxation equations on the strain can also be nonlinear. The resulting constitutive equations are examined as mapping between appropriate Sobolev spaces. The proposed formulation is easier to implement numerically than history-based formulations.      

CONTROLLABILITY OF A CLASS OF HYBRID DYNAMIC SYSTEMS (Ⅰ)—BASIC PROPERTIES AND PRELIMINARY RESULTS

IntroductionHybriddynamicsystemscombinebothcontinuoustimedynamicsanddiscreteeventdynamics.Thetwodynamicsinteractoneachothersuchthatthesystemsperformmorecomplexdynamicbehaviors .Switchedlinearsystemsareanimportantclassofhybriddynamicsystemswhichconsistofafamilyoflineartime_invariantsystemsandaswitchinglawspecifyingtheswitchingbetweenthem .Inrecentyears ,therehasbeenincreasinginterestinthecontrolproblemsofswitchedsystemsduetotheirsignificancebothintheoryandapplications[1～ 2 1] .Thusfar,therese…     

A note on formulations of static shakedown analysis with bounded kinematic hardening

Two different formulations for the two-surface model of bounded kinematic hardening can be found in literature on shakedown analysis with the von Mises yield criterion. This short paper explains that these two formulations are not equivalent, although there exists literature asserting that they are equivalent. More specifically, the formulation using the constraint on the stress is not a sufficient condition for the two-surface model. Consequently, the static shakedown analysis using this formulation over-evaluates the shakedown factor in general.