Free Vibration Frequencies of a Variable Cross-Section Timoshenko-Ehrenfest Beam using Fourier-p Element

An efficient method has been developed to analyze the natural frequencies of a Timoshenko-Ehrenfest beam with variable cross-sections undergoing free flexural vibration. The method uses a single element to discretize the beam and combines polynomial and trigonometric shape functions to represent the displacement fields, with the trigonometric shape functions representing the internal degrees of freedom (DOF) of the beam. By incorporating trigonometric shape functions as enriching functions alongside polynomials, the potential issues related to ill-conditioning associated with higher-order polynomials are avoided. The differential equations of motion for the free vibration are derived using Lagrange's equation. The generalized eigenvalue problem is then formulated. Non-dimensional natural frequencies are computed considering different parameter variations such as taper ratios, shear deformation, rotary inertia parameters, as well as end conditions. Incorporating rotary inertia and shear deformation in the analysis of Timoshenko-Ehrenfest beam lowers its natural frequencies compared to simpler beam theories like Euler-Bernoulli. It is observed that as taper ratios increase, the beam's mass decreases toward the tip, causing higher frequency parameter values for the fundamental mode. Increasing rotary inertia parameter leads to a decreasing trend in frequency parameters especially in higher modes. The combined effect of rotary inertia and tapering significantly reduces frequency parameter values for all modes, with the rotary inertia counteracting the tapering influence. The proposed Fourier-p element model accurately predicts Timoshenko-Ehrenfest beam frequencies with high precision compared to other existing models.