On geometric nonlinear vibrations of nonuniform beams

Nonlinear bending vibrations in the case of moderately large curvature are reported for a nonuniform cantilever beam of rectangular cross section and a sharp end. This is a beam of constant width and parabolic thickness variation. The method of multiple scales is directly applied to the governing partial-differential equation of motion and boundary conditions. The linear modes are obtained in terms of hypergeometric functions by using the factorization method. In the absence of internal resonance (weakly nonlinear systems) the nonlinear modes are taken to be perturbed versions of the linear modes. The nonlinear mode shapes and frequencies of the beam are reported.