Nonlinear response and resonance analysis of beam with non-uniform cross-section under harmonic and impulse excitations: an analytical approach

An analytical solution for the nonlinear transverse and axial responses of an elastic beam with a non-uniform cross-section subjected to harmonic and impulse excitations is presented by employing the multiple-scale method. Due to the moderately large deformation assumption, the geometrical nonlinearity effect is considered although the constitutive equations are linear. The equations of motion which are a system of coupled nonlinear partial differential equations with variable coefficients are obtained using the Hamilton principle and by considering the first-order shear deformation theory and the nonlinear von Karman relations. Both the in-plane and transverse accelerations are considered in the formulation and the relations between nonlinear frequencies and phases with the amplitudes of the axial and transverse displacements are determined. A parametric study is conducted to study the effect of geometrical parameters on the beam's nonlinear response and resonance behavior. To investigate the accuracy and precision of the obtained results, a comparison with the finite element results is performed.