NONLINEAR VIBRATIONS OF NON-UNIFORM BEAMS

Abstract eng:
The determination of natural frequencies of non-uniform beams attracted the attention of various researchers in the past, especially during the seventies [1] and the eighties [2]. Recently, the investigation of the natural frequencies of non-uniform beams is undergoing a sort of revival, also in connection with the functionally graded material/beam [3]. This work represents a sequel of [4], where the simple frequency formulas for the determination of natural frequencies in non-uniform bars are proposed. The present paper deepens the discussion on both free and forced nonlinear vibrations of the same type of structural elements. It is worth emphasizing some of the new additions to [4]. The differential equation of motion for the nonuniform beam – which takes into account the variation of the flexural stiffness and/or the geometric stiffness (due to normal force variation) and/or the unit mass along the beam length –, differ only by the addition of an ad-hoc viscous damping, a longitudinal transversal load, and of a geometrically nonlinear term. Further, instead of the Lindstedt-Poincaré method, the Multiple Time Scales (MTS) method is used to obtain both free and forced nonlinear dynamic responses of the beam, which permits having an analytical (and simple, indeed) expression for the (approximate) solution. The frequency response curves are obtained, and the frequency-dependent amplitude of the nonlinear oscillation is determined. The results of the present work show that the non-uniformity of the beam largely influences the linear natural frequencies of the beam, while it is less important for the nonlinear behaviour. The general theory has been illustrated by means of two alternative examples. In the first one, the beam has constant cross-section but varying axial load. In the second one, the tapered beam, the axial force vanishes, but the cross-section varies according to a power law.

• Export as: BibTeX | MARC | MARCXML | DC | EndNote | NLM | RefWorks
• Add to your basket