An algorithm for dynamic vehicle-track-structure interaction analysis for high-speed trains

TL;DR

The paper tackles dynamic vehicle-track-structure interaction (VTSI) for high-speed trains crossing bridges by modeling the train and bridge as independent systems and coupling them through time-varying kinematic constraints, with wheel-rail contact forces represented as $oldsymbol \lambda$ (Lagrange multipliers) in a differential-algebraic setting. It investigates two remedies to spurious oscillations caused by constraint derivatives: cubic B-spline interpolation of the constraint to achieve $C^{2}$ continuity and the Bathe two-step time integration method, both complemented by linear complementarity problem (LCP) formulations to handle possible wheel contact separation. The approach is highly modular, designed to integrate into existing bridge analysis software, and is validated against a generic DAE solver (Sundials) as well as preliminary experimental data; resonance and track irregularities are explicitly examined. The results show that the Bathe method robustly eliminates oscillations in most cases, while B-spline interpolation reduces oscillations with some convergence and boundary-condition limitations, and contact separation is effectively modeled via LCP, enabling detachment events. The work provides a practical, adaptable framework for safety and comfort assessment in high-speed rail VTSI and lays groundwork for broader validation with measured data and more complex track models.

Abstract

The objective of the present work is to develop a robust, yet simple-to-implement algorithm for dynamic vehicle-track-structure-interaction (VTSI) analysis, applicable to trains passing over bridges. The algorithm can be readily implemented in existing bridge analysis software with minimal code modifications. It is based on modeling the bridge and train separately, and coupling them together by means of kinematic constraints. The contact forces between the wheels and the track become Lagrange multipliers in this approach. A direct implementation of such an approach results in spurious oscillations in the contact forces. Two approaches are presented to mitigate these spurious oscillations - (a) a cubic B-spline interpolation of the kinematic constraints in time, and (b) an adaptation of an alternate time-integration scheme originally developed by Bathe. Solutions obtained using this algorithm are verified using a generic differential algebraic equation (DAE) solver. Due to high train speeds and possible track irregularities, wheels can momentarily lose contact with the track. This contact separation is formulated as a Linear Complementarity Problem (LCP). With this formulation, including contact separation in the analysis amounts to replacing a call to a linear equation solver by a call to an LCP solver, a modification of only two steps of the procedure. The focus of this paper is on the computational procedure of VTSI analysis. The main contribution of this paper is recognizing computational issues associated with time-varying kinematic constraints, clearly identifying their cause and developing remedies.

Figures (28)

• Figure 1: Conceptual model of a train passing a bridge
• Figure 2: One of the train wheels is on joint 2, hence it will be arbitrarily assigned to one of the two bridge elements connected to that joint, or
• Figure 3: Simplified car model used for illustration
• Figure 4: Beam element with equivalent nodal forces
• Figure 5: Modeling the kinematic constraint