An analytical-numerical approach to vibration analysis of periodic Timoshenko beams

Abstract The subject of this article is analysis of transverse vibrations of beams which geometric and material properties vary periodically along the longitudinal axis. The aim is to present averaged models that take into account the shear deformation and geometric non-linearity, and to analyse transverse vibrations of such beams in moderately large deflection range. As the theoretical foundations, we use Timoshenko beam theory with von Karman-type non-linearity. This results in obtaining new differential equations with constant coefficients, some of which explicitly depend on the beam inhomogeneity period size. Then, a reasonably simplified model is proposed to describe the vibrations of the considered beams in the low frequency range. The differential equations are transformed into a system of algebraic equations according to the Galerkin method. The response of the beam to transverse harmonic load is investigated by means of a pseudo arc-length continuation scheme. Non-linear coupling between vibration modes and the possibility of superharmonic resonance occurrence are taken into account. As an example of application, few special cases of beam geometry and boundary conditions are examined and compared. The results have the potential application to structural vibration control.