Geometry and differentiability requirements in multibody railroad vehicle dynamic formulations

Abstract

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.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.

Numerical analysis for dynamic contact between high‐speed wheel and elastic beam with Coulomb friction

“…Thus, various techniques using the Lagrange multiplier method or the penalty method have been developed in the multibody dynamics for the time integration of such differential algebraic equations having constraints. The literature survey shows that, even though it is difficult to find such work on the dynamic contact between a wheel and an elastic beam, the dynamic contact analysis between the rigid wheels and the beam-like structures can be found in very sophisticated works on complex railroad vehicle dynamics [1][2][3]. Although reasonable solutions may be obtained in railway vehicle dynamics by these ways with Lagrange multiplier method or penalty method for the contact constraint, the solution should require more consideration if the coupled effect between the high velocity of the contact point and the beam deformation is not small or if friction should be considered.…”

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

“…The angle ψ ti depends on the horizontal curvature of the space curve, the vertical development angle θ ti depends on the grade or vertical curvature, and the bank angle φ ti depends on the track super elevation. These three Euler angles, which can be uniquely defined in terms of arc length s i using a track preprocessor [2,8], can be used to define the following trajectory transformation matrix:…”

“…To this end, the absolute angular velocity vector of the body, the absolute angular velocity of the trajectory frame, and the angular velocity of the body with respect to the trajectory frame are written as follows [8,9]:…”

“…While the use of the trajectory coordinates leads to a more complex form of the dynamic equations of motion, such coordinates lead to simple expressions of the constraint equations in the case of specified motion trajectories. Furthermore, the use of the trajectory coordinates requires a higher order of derivatives as compared to the use of the absolute Cartesian coordinates [8]. In this study, the use of the absolute Cartesian and trajectory coordinates in the analysis of railroad vehicle systems is examined and the differentiability and smoothness requirements as the results of using these coordinates are discussed.…”

“…The differentiability requirements is one of the major disadvantages of using the trajectory coordinates, and for this reason, these requirements will be also discussed in this paper. As demonstrated by Rathod and Shabana [8], the use of the trajectory coordinates requires fourth derivates with respect to the arc length parameter regardless of whether or not the model has wheel/rail contact elements. Consequently, more information on the track geometry must be provided as an input to the dynamic simulation code in order to avoid discontinuities in the solution as demonstrated by the results presented in this study.…”