Free vibrations of non-uniform beams on a non-uniform Winkler foundation using the Laguerre collocation method

Abstract

Natural frequencies and free vibration are important characteristics of beams with non-uniform cross section. Hence, the solution for free vibrations of non-uniform beams is presented using a Laguerre collocation method. The elastically restrained beam model is based on the Euler-Bernoulli theory. Also, the non-uniform beam is rested on a non-uniform foundation (Winkler type). The Laguerre collocation method is introduced for solving the differential equation. This approach reduces the governing differential equation to a system of algebraic equations, and finally, the problem is greatly simplified. Properties of Laguerre polynomials and the operational matrix of derivation are first presented. Eventually, the proposed method is applied for solving the governing differential equation subject to initial conditions, and the results are compared with other results from the literature.

The exact characteristic equation of frequency and mode shape for transverse vibrations of non-uniform and non-homogeneous Euler Bernoulli beam with general non-classical boundary conditions at both ends

“…Celik [10] used Chebyshev Wavelet Collocation Method to consider vibration of non-uniform beams with various boundary conditions and with flexible end conditions. Ghannadiasl et al [11] studied the elastically restrained non-uniform beam modeled based on the Euler-Bernoulli theory using a Laguerre collocation method. The third group: Huang and Li [12] presented a novel approach based on transforming the differential equation of motion for transverse vibration of non-uniform Euler-Bernoulli beams into a Freed Holm integral equation.…”

A Unique Approach to Investigate the Free Vibrations of Non-uniform and Functionally Graded Euler-Bernoulli Beams

“…The vibration of non-uniform beams with different boundary conditions and flexible ends has studied by Selik [6] using the Chebyshev wavelet collocation method. The free vibration of beams with a nonuniform cross-section and placed on a fixed and linear Winkler bed based on the Euler-Bernoulli beam theory and using the Lager collocation method has been studied by Qanadiasl et al [7]. 3 rd groups: A new approach based on the transformation of the differential equation of transverse vibration of Euler-Bernoulli beams with non-uniform cross-section to Fredholm integral equations has introduced by Huang and Lee [8] Liu et al [9] have created new simple models for studying the free vibrations of Euler-Bernoulli FG conical beams using the spline finite point method.…”