A Frozen-Time Method for Discontinuous Internal Forces of a Multi-Span Beam Carrying Moving Subsystems

Abstract

Abstract A flexible structure that carries moving subsystems is subject to dynamical interaction forces with time-varying locations due to the elastic coupling between the subsystem and the structure. Those interaction forces render the internal shear force and moment distribution in the structure piecewisely continuous with jump discontinuities. Conventional modal expansion method that utilizes continuously differentiable eigenfunctions encounters very slow convergence and inevitable overshoot and undershoot at the jumps. To overcome this issue, we propose a novel method called modified series solution method (MSSM) to precisely determine the jump discontinuities in the internal shear force and bending moment of the host beam structure. The new solution contains three terms, including a conventional series dynamical solution and two static solutions evaluated at the current “frozen” time. The proposed augmented solution of the beam can not only precisely predict the time-varying jump discontinuities in the internal shear force and bending moment distribution without introducing any overshoots or undershoots, but also has a fast convergence rate the same as the displacement and the slope. It provides a new framework to analyze the discontinuous shear force and moment of a host structure induced by viscoelastically coupled subsystems.