Element-lumped-based beam vibration modelling with multi-flexible constraints and implementation into nonlinear train-track-bridge interactions

Abstract Beam-like structure with multi-flexible constraints (MFCB) has been applied in various engineering structures fields. However, conventional solutions often generate complex mode functions or large eigenmatrices when facing arbitrary numbers and positions of MFCs or cracks, resulting in significant deviations or even failures. To this end, two novel analytical solutions for free and forced Euler–Bernoulli and Timoshenko MFCB vibration are newly presented, namely the element-lumped method (ELM), and are implemented in nonlinear train-track-bridge interactions (TTBI). Initially, the MFCB model is divided into finite, intact segments at constraint locations. Each segment is represented by an independent matrix form of the vibration equations, which are discretized and decoupled using the Galerkin method and MSM. The overall key matrix of the MFCB model is then constructed by assembling the element submatrices, considering the continuity, boundary conditions, or motion equations. Subsequently, the MFCB eigenproblem is solved by homogeneous linear equations, and the forced vibration equation is derived based on the principle of modal orthogonality. The comparison of the FEM results and the TMM and GFM solutions from previous literature confirms the accuracy of ELM, while the ELM effectively overcomes the theoretical derivation complexity of TMM and GFM, as well as the time-domain analysis limitations of the DSM. The classical constrained beams are proven to be degenerate solutions of MFCB, and the limitations inherent in flexible constraint approximation are elucidated. Moreover, the dynamic characteristics differences of incorporating periodic cubic nonlinear support into traditional TTBI of rail transit are revealed, the anti-torsional effect of fasteners increases wheel-rail wear and derailment risk is highlighted. The proposed methods are simple, stylized, robust, and general for complex hybrid systems, and could provide theoretical guidelines for the dynamics modelling of MFCB-like structures.