Application of the difference equation method to the vibrations analysis of infinite Rayleigh beams by the isogeometric approach

An efficient method of vibration investigations of infinite Rayleigh beams applied to the isogeometric analysis called NURBS (non-uniform rational B-splines) is proposed. The research objective is to examine the influence of rotational inertia effects on the dynamic behaviour of the discrete systems approximated by NURBS and compare obtained results with the finite element method (FEM) and exact ones. In NURBS methodology transverse displacements are approximated by quadratic, cubic and quartic B-splines basis functions. For all types of approximations stiffness and consistent mass matrices with rotational inertia effects are found. The equilibrium conditions for an arbitrary interior element are expressed in the form of one difference equation equivalent to the infinite set of equations derived by numerical NURBS formulation for this dynamic problem. The convergence of these equations to the exact differential equations of motion is presented. Assuming the wavy nature of vibration propagation phenomenon the analytical dispersive equations are obtained for various orders of B-spline basis functions. The parametrical analysis of rotational inertia effects on the wave propagation is carried out. The influence of the adopted discretization and the mass distribution is taken into account, as well. The analytical NURBS results are compared with the FEM and exact ones.