An Accurate Spatial Discretization and Substructure Method With Application to Moving Elevator Cable Car Systems—Part I: Methodology

Abstract

A spatial discretization and substructure method is developed to accurately calculate dynamic responses of onedimensional structural systems, which consist of lengthvariant distributedparameter components, such as strings, rods, and beams, and lumpedparameter components, such as point masses and rigid bodies. The dependent variable of a distributedparameter component is decomposed into boundaryinduced terms and internal terms. The boundaryinduced terms are interpolated from boundary motions, and the internal terms are approximated by an expansion of trial functions that satisfy the corresponding homogeneous boundary conditions. All the matching conditions at the interfaces of the components are satisfied, and the expansions of the dependent variables of the distributedparameter components absolutely and uniformly converge if the dependent variables are smooth enough. Spatial derivatives of the dependent variables, which are related to internal forces/moments of the distributedparameter components, such as axial forces, bending moments, and shear forces, can be accurately calculated. Combining component equations that are derived from Lagrange's equations and geometric matching conditions that arise from continuity relations leads to a system of differential algebraic equations (DAEs). When the geometric matching conditions are linear, the DAEs can be transformed to a system of ordinary differential equations (ODEs), which can be solved by an ODE solver. The methodology is applied to several moving elevator cablecar systems in Part II of this work.