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This paper presents the transverse vibration of Bernoulli-Euler homogeneous isotropic damped beams with general boundary conditions. The beams are assumed to be subjected to a load moving at a uniform velocity. The damping characteristics of the beams are described in terms of fractional derivatives of arbitrary orders. In the analysis where initial conditions are assumed to be homogeneous, the Laplace transform cooperates with the decomposition method to obtain the analytical solution of the investigated problems. Subsequently, curves are plotted to show the dynamic response of different beams under different sets of parameters including different orders of fractional derivatives. The curves reveal that the dynamic response increases as the order of fractional derivative increases. Furthermore, as the order of the fractional derivative increases the peak of the dynamic deflection shifts to the right, this yields that the smaller the order of the fractional derivative, the more oscillations the beam suffers. The results obtained in this paper closely match the results of papers in the literature review.

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The large deflection of a prismatic Euler-Bernoulli cantilever beam under a combination of end-concentrated coplanar inclined force and tip-concentrated moment is investigated. The angle of inclination of the applied force with respect to the horizontal axis remains unchanged during deformation. The cantilever beam is assumed to be naturally straight, slender, inextensible and elastic. The large deflection of the cantilever beam induces geometrical nonlinearity; hence, the nonlinear theory of bending and the exact expression of curvature are used. Based on an elliptic integral formulation, an accurate numerical solution is obtained in terms of an integration constant that should satisfy the boundary conditions associated with the cantilever beam. For some special cases this integration constant is exactly found, which leads to closed form solution. The numerical solution obtained is quite simple, accurate and involves less computational time compared with other techniques available in literature. The details of elastica and its corresponding orientation curves are presented and analyzed for extremely large load combinations. A comparative study with pre-obtained results has been made to verify the accuracy of the presented solution; an excellent agreement has been obtained.