Principal parametric resonances of a general continuous system with cubic nonlinearities

A generalized vibrational model of cubic nonlinear continuous system with arbitrary parametric excitation is considered. The method of multiple scales (a perturbation method) is used to find an approximate analytical solution. The primary parametric resonance of the parametric excitation is considered. The amplitude and phase modulation equations are derived. Steady state solutions and their stability are discussed. The solution algorithm is applied to the problem of nonlinear vibrations of viscoelastic pipes conveying fluids. Natural frequencies of viscoelastic pipes are found. Frequency response curves and bifurcation points are drawn. Stable and unstable regions of trivial and nontrivial solutions are obtained.