Variational principles for a double Rayleigh beam system undergoing vibrations and connected by a nonlinear Winkler-Pasternak elastic layer

Variational principles and variationally consistent boundary conditions are derived for a system of double Rayleigh beams undergoing vibrations and subject to axial loads. The elastic layer connecting the beams are modelled as a three-parameter nonlinear Winkler-Pasternak layer with the Winkler layer having linear and nonlinear components and Pasternak layer having only a linear component. Variational principles are derived for the forced and freely vibrating double beam system using a semi-inverse approach. Hamilton's principle for the system is given and the Rayleigh quotients are derived for the vibration frequency of the freely vibrating system and for the buckling load. Natural and geometric variationally consistent boundary conditions are derived which leads to a set of coupled boundary conditions due to the presence of Pasternak layer connecting the beams.