Linear and nonlinear vibrations of variable cross-section beams using shear deformation theory

In the present paper, the governing equations of a vibratory beam with moderately large deflection and arbitrary cross-section are derived by using the first-order shear deformation theory. The beam is homogenous, isotropic and it is subjected to the axial loads. The kinematic of the problem is according to the von-Karman strain-displacement relations and the Hooke law is used as the constitutive equations. The partial differential governing equations describing the axial and transverse vibrations of homogeneous beams contain four coupled nonlinear equations with variable coefficients which are derived employing Hamilton's principle. The Galerkin method in conjunction with the perturbation technique is applied to obtain the linear natural frequencies. A parametric study is performed and the effects of different thickness functions such as linear, polynomial and trigonometric on the results are investigated. The non-linear frequencies which contain the corrections on the linear frequencies are calculated. The corrected parts of the non-linear frequencies are functions of the axial as well as the transverse amplitudes of the vibrations. The influences of the axial load and aspect ratio on the linear and non-linear frequencies are studied too. To confirm the reliability of the vibration analysis carried out in the present paper, the analytical results are checked with the corresponding numerical results obtained from the finite element analysis. The numerical and analytical results are in a good agreement.