Abstract
This paper presents a new analytical procedure on bending vibrations of a beam subjected to an axial force. The procedure combines the procedures of modal expansion with finite element analysis. The beam is divided into finite number of segments. The mode shape function is introduced with the product of truncated power series and trigonometric function which is proposed by the senior author. The function is differentiable continuously with infinite times. The unknown coefficients of the mode shape function satisfy both the geometric and dynamical boundary conditions at the node of segment. Discretized equations of motion of the beam are derived from the Galerkin method. We call the analytical procedure as the Finite Segment Analysis (FSA). Based on the FSA, linear natural frequencies of a pre-buckled beam subjected to an axial force are calculated. Increasing the number of segments, the results by FSA are compared with the exact results and with the results of conventional Finite Element Method (FEM). Under the calculation with two segments by FSA, the natural frequencies from the lowest mode to the fourth mode coincide with five digits to the exact results.
