Finite element free and forced vibration analysis of gradient elastic beam structures

Abstract

The dynamic stiffness matrix of a gradient elastic flexural Bernoulli–Euler beam finite element is analytically constructed with the aid of the basic and governing equations of motion in the frequency domain. The simple gradient theory of elasticity is used with just one material constant (internal length) in addition to the classical moduli. The flexural element has one node at every end with three degrees of freedom per node, i.e., the displacement, the slope, and the curvature. Use of this dynamic stiffness matrix for a plane system of beams enables one by a finite element analysis to determine its dynamic response harmonically varying with time external load or the natural frequencies and modal shapes of that system. The response to transient loading is obtained with the aid of Laplace transform with respect to time. A stiffness matrix is constructed in the transformed domain, the problem is formulated and solved by the finite element method, and the time domain response is finally obtained by a time domain inversion of the transform solution. Because the exact solution of the governing equation of motion in the frequency domain is used as the displacement function, the resulting dynamic stiffness matrices and the obtained structural response or natural frequencies and modal shapes are also exact. Examples are presented to illustrate the method and demonstrate its advantages. The effects of the microstructure on the dynamic behavior of beam structures are also determined.