A new approach for steady-state dynamic response of axially functionally graded and non-uniformed beams. (15th October 2019) - Han, Hesheng; Cao, Dengqing; Liu, Lun
Abstract: The analytical solutions of steady-state dynamic responses of axially functionally graded and non-uniformed beams subjected to harmonic loadings with damping effect are presented by using the Green's function element method (GFEM). Dividing the beam into axially homogeneous and uniformed elements, and the Green's function of each element is obtained by employing the Laplace transform technology with boundary relationships, matching conditions and boundary conditions. Then, the Green's function of the beam is given in terms of the piecewise functions. Based on the superposition principle, the expression of the steady-state responses of the beam with various loadings can be directly obtained. In addition, an implicit equation is defined to determine the natural frequencies of the beam. Various beams with different boundary conditions are given as examples to illustrate the use of the approach developed. Comparisons between the results in the published literature and those obtained by GFEM show a good agreement. The influence of the elements' number on the solutions are analyzed, and the results show that GFEM is an efficient approach for studying the dynamic behavior of the beams. The damping effect and symmetric property of the solutions are discussed. Moreover, the displacements time histories of the beams are displayed.