Corotational modeling and NURBS-based kinematic constraint implementation in three-dimensional vehicle-track-structure interaction analysis

TL;DR

The paper tackles the challenge of three-dimensional dynamic vehicle-track-structure interaction (VTSI), where a multibody vehicle interacts with a deformable bridge and track through time-varying kinematic constraints. It introduces a corotational formulation that attaches reference Frenet frames to the bridge path, enabling large rigid-body motion while treating in-frame deformations as small and linear, and adopts NURBS-based discretization of the track to suppress spurious oscillations in contact forces and accelerations. Two complementary non-dissipative strategies are developed to solve the resulting index-3 differential-algebraic equations: enforcing constraints at the acceleration level to reduce the system to index-1, and a constraint-projection method that corrects velocities and accelerations at each time step. The approach is demonstrated on simplified and realistic models, showing improved stability and modularity that facilitates integration with existing finite element software, and is applicable to other multibody systems coupled to deformable structures by time-varying constraints. Overall, the work advances practical 3D VTSI analysis by combining corotational kinematics, isogeometric track modeling, and robust DAE solution techniques with clear pathways for software integration and extension to broader applications.

Abstract

An algorithm for three-dimensional dynamic vehicle-track-structure interaction (VTSI) analysis is described in this paper. The algorithm is described in terms of bridges and high-speed trains, but more generally applies to multibody systems coupled to deformable structures by time-varying kinematic constraints. Coupling is accomplished by a kinematic constraint/Lagrange multiplier approach, resulting in a system of index-3 Differential Algebraic Equations (DAE). Three main new concepts are developed. (i) A corotational approach is used to represent the vehicle (train) dynamics. Reference coordinate frames are fitted to the undeformed geometry of the bridge. While the displacements of the train can be large, deformations are taken to be small within these frames, resulting in linear (time-varying) rather than nonlinear dynamics. (ii) If conventional finite elements are used to discretize the track, the curvature is discontinuous across elements (and possibly rotation, too, for curved tracks). This results in spurious numerical oscillations in computed contact forces and accelerations, quantities of key interest in VTSI. A NURBS-based discretization is employed for the track to mitigate such oscillations. (iii) The higher order continuity due to using NURBS allows for alternative techniques for solving the VTSI system. First, enforcing constraints at the acceleration level reduces an index-3 DAE to an index-1 system that can be solved without numerical dissipation. Second, a constraint projection method is proposed to solve an index-3 DAE system without numerical dissipation by correcting wheel velocities and accelerations. Moreover, the modularity of the presented algorithm, resulting from a kinematic constraint/Lagrange multiplier formulation, enables ready integration of this VTSI approach in existing structural analysis and finite element software.

Figures (22)

• Figure 1: The corotational approach in VTSI: motion of a body is described with respect to a moving Frenet frame. Displacements and rotations of the Frenet frame can be large, but displacements and rotations relative to the Frenet frame are small. Here, $XYZ$ is a Global frame, $tnb$ is a Frenet frame, and $xyz$ is a rigid body frame. The apostrophe indicates the frames moved to a new, corotated position.
• Figure 2: Bridge curved in plan and a simplified vehicle model (generalized coordinates of the vehicle model with respect to the Frenet frame are shown in blue; position vectors of the vehicle parts are shown in red)
• Figure 3: Paths in simplified and realistic bridge models used in numerical examples
• Figure 4: An example of a realistic train model (one car is shown)
• Figure 5: Schematic of a realistic bridge and track model