NONLINEAR VIBRATIONS OF A SLIGHTLY CURVED BEAM RESTING ON A NONLINEARELASTIC-FOUNDATION

In this study, non-linear vibrations of slightly curved beams are inve stigated. The curvature is taken as an arbitrary function of the spati al variable. The initial displacement is not due to buckling of the be am, but is due to the geometry of the beam itself. The ends of the cur ved beam are on immovable simple supports and the beam is resting on a non-linear elastic foundation. The immovable end supports result in t he extension of the beam during the vibration and hence introduces fur ther non-linear terms to the equations of motion. The integro-differen tial equations of motion are solved analytically by means of direct ap plication of the method of multiple scales (a perturbation method). Th e amplitude and phase modulation equations are derived for the case of primary resonances. Both free and forced vibrations with damping are investigated. Effect of non-linear elastic foundation as well as the e ffect of curvature on the vibrations of the beam are examined. It is f ound that the effect of curvature is of softening type. For sufficient ly high values of the coefficients, the elastic foundation may suppres s the softening behaviour resulting in a hardening behaviour of the no n-linearity. (C) 1998 Academic Press Limited.