Nonlinear vibrations of a slightly curved beams carriying a concentrated mass

In this study, nonlineer vibrations of slightly curved beams carryingconcentrated mass and resting on an elastic foundation are investigated. The curvatureis taken an arbitrary function. The ratio of the maximum value of the curvature functionto the projection lenght of the beam assumed to be less than 1/10. The ends of the beamare on immovable simple supports. These supports result in the extension of the beamduring vibration. The extensions are taken into account, hence the equation of themotion includes further non-linear terms. Forcing and damping terms are added to theequations.The integro-differential equations of motion are solved by using the method ofmultiple scales, a perturbation method. The amplitude and phase modulation equationsare derived. From these equations, first free vibrations are examined. Nonlinearfrequency-amplitude graphics under different curvature functions, mass ratios, andelastic foundation coefficients are obtained. Then by adding the terms of damping andforcing, external excitation frequency-amplitude graphics are obtained.Finally, effect of magnitude and position of the concentrated mass as well as theeffect of elastic foundation on vibrations of the curved beam are examined. Also theeffect of curvature on vibrations are compared between sinus and paraphol functions.