An improved analog equation method for non-linear dynamic analysis of time-fractional beams with discontinuities

In this paper, an improved version of the analog equation method (AEM) is proposed, ideally suitable for non-linear dynamic analysis of time-fractional beams with discontinuities. Various sources of non-linearity will be considered as well as different discontinuities, such as those associated with external point supports (shear force discontinuity) and local flexural flexibility (rotation discontinuity). The main idea of the proposed approach is to reformulate the classical AEM by considering an analog equation with an unknown space-time dependent fictitious load and a spatial generalised operator, which includes the classical 4th-order differential operator and generalised functions modelling the discontinuities along the beam. Consistently, the solution to the analog equation is represented by an integral form involving appropriate Green’s functions of the generalised operator. As in the classical AEM formulation, the integral representation of the solution is then approximated dividing the beam in finite elements and considering a constant value of the fictitious load within every beam element. Substituting the so-built approximate solution in the original equation of motion yields a system of fractional differential equations governing the discrete values of the fictitious load, solved by employing a Newmark integration scheme in conjunction with a G-1 algorithm to account for the fractional-derivative memory effects. Computational efficiency and accuracy will be demonstrated against the classical AEM formulation.