A spectral-Tchebychev technique for solving linear and nonlinear beam equations

Abstract

This paper presents a spectral-Tchebychev technique for solving linear and nonlinear beam problems. The technique uses Tchebychev polynomials as spatial basis functions, and applies Galerkin's method to obtain the spatially discretized equations of motion. Unlike alternative techniques that require different admissible functions for each different set of boundary conditions, the spectral-Tchebychev technique incorporates the boundary conditions into the derivation, and thereby enables the utilization of the solution for any linear boundary conditions without re-derivation. Furthermore, the proposed technique produces symmetric system matrices for self-adjoint problems. In this work, the spectral-Tchebychev solutions for Euler–Bernoulli and Timoshenko beams are derived. The convergence and accuracy characteristics of the spectral-Tchebychev technique is studied by solving eigenvalue problems with different boundary conditions. It is found that the convergence is exponential, and a small number of polynomials is sufficient to obtain machine-precision accuracy. The application of the technique is demonstrated by solving: (1) eigenvalue problems for tapered Timoshenko beams with different boundary conditions, taper ratios, and beam lengths; (2) an Euler–Bernoulli beam problem with spatially and temporally varying forcing, elastic boundary, and damping; (3) large-deflection (nonlinear) Euler–Bernoulli beam problems with different boundary conditions; and (4) a micro-beam problem with nonlinear electrostatic excitation. The results obtained from the spectral-Tchebychev solutions are seen to be in excellent agreement with those presented in the literature.