Equivalence of Axially Moving Beam on Elastic Supports to Axially Moving Beam on Elastic Foundation with a Semi-Analytical and Semi-Numerical Method

This study investigates the dynamic equivalence between the axially moving beam on elastic supports and the axially moving beam on elastic foundation, providing the equivalence coefficient and determining the corresponding accuracy. The natural frequencies and mode shapes of the axially moving beam on the elastic foundation are obtained analytically, while those of the axially moving beam with intermediate elastic supports are derived using a semi-analytical and semi-numerical method. The equivalence relationship between the two types of beams is further established. The axially moving beam on elastic supports is analyzed segmentally, with the boundary conditions and displacement-force equilibrium at segment nodes forming a homogeneous linear equation system. The coefficient matrix of this system, termed the global matrix, yields the natural frequencies of the axially moving beam on elastic supports when its determinant equals zero. This approach further enables the derivation of modal functions for each segment and the entire beam. Following the analysis of free vibration characteristics, numerical evaluations are conducted to compare the dynamic response and stability of the two types of axially moving beams. Under moving loads, Galerkin discretization is used to obtain the first two ordinary differential motion equations for both beams, which are solved to determine their dynamic responses. The multi-scale method is applied to analyze the stability of the beams under weak disturbances in velocity, revealing instability regions when the disturbance frequency approaches specific conditions. The equivalence coefficient varied with the increasing of segment number of the axially moving beam on elastic supports. Furthermore, comparisons of the natural frequencies, dynamic responses under moving loads, and instability regions under velocity disturbances between the two types of beam allowed the determination of the equivalence coefficient and its associated precision. Considering that an accuracy of 1% is sufficient to meet most engineering requirements, the equivalence analysis presented in this study can be used to approximate complex axially moving beam on elastic supports as axially moving beam on elastic foundation, thereby facilitating simplified calculations.