Equations of Nonlinear Motion of Viscoelastic Beams

Abstract

A viscoelastic nonlinear beam with cubic nonlinearities is considered. In order to obtain the equations of nonlinear motion of the beam for large deformation vibrations, the Lagrangian dynamics and Hamilton principle is used. It is considered that the beam vibrates in two directions, one in longitudinal direction and the other in the transverse direction. Large amplitude vibrations cause the nonlinearities in inertia and geometry terms. Also, due to viscoelastic property of the beam, a nonlinear damping term is appeared in the equations of motion. Using the condition of inextensible beams, the equation of motion and boundary conditions of bending vibration of a Kelvin-Voigt viscoelastic beam has been obtained. Finally, if one considers the damping coefficient to be equal to zero in the obtained equation of motion of viscoelastic system then, an equation of motion for the elastic beam will be obtained.