Numerical free vibration analysis of higher-order shear deformable beams resting on two-parameter elastic foundation

Free vibration analysis of higher-order shear deformation beam resting on one- and two-parameter elastic foundation is studied using differential transform method (DTM) as a part of a calculation procedure. First, the governing differential equations of beam are derived in a general form considering the shear-free boundary conditions (zero shear stress conditions at the top and bottom of a beam). Using DTM the derived equations governing beams, followed by higher-order shear deformation model, Timoshenko model and Bernoulli-Euler model are transformed to algebraic forms and a set of recurrence formulae is then derived. Upon imposing the boundary conditions of the beam at hand, a set of algebraic equations are obtained and expressed in matrix form. Finally, the transverse natural frequencies of the beam are calculated through an iterative procedure. Several numerical examples have been carried out to demonstrate the competency of the present method and the results obtained by DTM are in good agreement with those in the literature. Afterward, the free vibration of beams followed up by different models (i.e. Bernoulli-Euler, Timoshenko and different higher-order models) are taken into consideration.