Simultaneous Resonances of Geometric Nonlinear Nonuniform Beams

Simultaneous resonances, superharmonic and subharmonic, of two-term excitation nonlinear bending vibrations in the case of moderately large curvature of nonuniform cantilever beams are reported. Cantilevers of constant width and parabolic thickness variation are considered in this research. The method of multiple scales is directly applied to the governing partial-differential equation of motion and boundary conditions. Two problems, zero- and first-order, result. Using factorization method, the linear modes of the zero-order problem are obtained in terms of hypergeometric functions. The first-order problem provides the amplitude-phase evolution relationship and consequently the simultaneous resonances response.