Vibration Analysis of Axially Functionally Graded Non-Prismatic Timoshenko Beams Using the Finite Difference Method

This paper presents an approach to the vibration analysis of axially functionally graded non-prismatic Timoshenko beams (AFGNPTB) using the finite difference method (FDM). The characteristics (cross-sectional area, moment of inertia, elastic moduli, shear moduli, and mass density) of axially functionally graded beams vary along the longitudinal axis. The Timoshenko beam theory covers cases associated with small deflections based on shear deformation and rotary inertia considerations. The FDM is an approximate method for solving problems described with differential equations. It does not involve solving differential equations; equations are formulated with values at selected points of the structure. In addition, the boundary conditions and not the governing equations are applied at the beam’s ends. In this paper, differential equations were formulated with finite differences, and additional points were introduced at the beam’s ends and at positions of discontinuity (supports, hinges, springs, concentrated mass, spring-mass system, etc.). The introduction of additional points allowed us to apply the governing equations at the beam’s ends and to satisfy the boundary and continuity conditions. Moreover, grid points with variable spacing were also considered, the grid being uniform within beam segments. Vibration analysis of AFGNPTB was conducted with this model, and natural frequencies were determined. Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of AFGNPTB, considering the damping. The results obtained in this study showed good agreement with those of other studies, and the accuracy was always increased through a grid refinement.

Timoshenko beam and plate non-stationary vibrations

The problems of Timoshenko beams and plates lateral vibrations under the influence of unsteady loads are considered. Both beam and plate is supposed to be unlimited. In case of the plate the problem has been simply studied. The approach to the solution was based on dominant function method and principle of superposition. Integral models of solutions with cores as dominant functions were built which could be analytically found with the help of the Fourier and Laplace integral transforms. Two original analytical methods for Fourier and Laplace transforms were offered and realized. The examples of calculations were given.