Chaotic, Hyperchaotic Vibrations and Stability of Porous Euler–Bernoulli Beams Considering Physical and Geometrical Nonlinearities

A mathematical model of flexible (according to the theories of T. von Karman and Green–Lagrange) physically nonlinear porous size-dependent Euler-Bernoulli beams subjected to transversal alternating loading is developed. The required differential equations are derived from the Hamilton–Ostrogradsky principle. Iterative algorithm is developed to compute chaotic and hyperchaotic vibrations in a mechanical system with “almost” infinite degrees of freedom. These algorithm include the Finite Difference Method (FDM) combined with Birger's Method of Variable Elasticity Parameters (MVEP), which takes into account physical nonlinearity. Chaos is analysed according to Gulick's definition. The instability of beam structures, including both metallic continuous and porous functionally graded Euler–Bernoulli beams, is studied within the framework of the Lavrentiev–Ishlinsky and Rayleigh–Taylor concepts.