The Method of Matrix Differential Equation to Calculate the Natural Frequencies of Variable Cross-section Beams on the Non-uniform Elastic Foundations

Abstract

The tradition methods to calculate the natural frequencies of beam on the elastic foundation were based on high order differential equation by defection. This paper derived the general equation based on the one order matrix differential equation represented by the mixed variable (a vector formed by deflection, angle of rotation, bending moment and shearing force) to calculate the natural frequencies of the variable cross-section beams on the non-uniform elastic foundations. To calculate the natural frequencies of this kind of beams, we divide the beams into several sections, calculate the transfer matrices of each section, multiply the matrices in proper order and obtain the overall transfer matrix of the mixed variable from one end to the other end of the variable cross-section beam on non-uniform elastic foundations. According to the conditions of boundary at both ends of the beam, we can calculate out the natural frequencies in MATLAB easily.