Free vibration analysis of axially functionally graded tapered Bernoulli–Euler microbeams based on the modified couple stress theory

A new approach for free vibration of axially functionally graded beams with non-uniform cross-section

Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams are studied through a finite element approach. The exact shape functions for uniform homogeneous Timoshenko beam elements are used to formulate the proposed element. The accuracy of the present element is considerably improved by considering the exact variations of cross-sectional profile and mechanical properties in the evaluation of the structural matrices. Carrying out several numerical examples, the convergence of the method is verified and the effects of taper ratio, elastic constraint, attached mass and the material non-homogeneity on the natural frequencies and critical buckling load are investigated.

Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions

Free vibration and stability of tapered Euler–Bernoulli beams made of axially functionally graded materials

This paper presents a new approach for investigating the vibration behaviors of axially functionally graded Timoshenko beams with non-uniform cross-section. By introducing an auxiliary function, we can change the coupled governing equations with variable coefficients for the deflection and rotation to a single governing equation. Moreover, all physical quantities can be expressed in terms of the solution of the resulting equation. Making use of power series for unknown function, we can transform the single equation to a system of linear algebraic equations and will get a characteristic equation in natural frequencies for different boundary conditions. An advantage of the suggested approach is that the derived characteristic equation is a polynomial equation, where the lower and higher-order natural frequencies can be determined simultaneously from the multi-roots. Several examples of estimating natural frequencies for axially grade beams and non-uniform beams are presented, which show that our method has fast convergence and obtained numerical results have high accuracy.

S. Candan

Papers

Apparently first closed-form solution for vibrating: inhomogeneous beams