A general solution procedure for the forced vibrations of a continuous system with cubic nonlinearities: Primary resonance case

Nonlinear vibrations of a general model of continuous system is considered. The model consists of arbitrary linear and cubic operators. The equation of motion is solved by the method of multiple scales (a perturbation method). The primary resonances of external excitation is analysed. The amplitude and phase modulation equations are presented. Approximate analytical solution is derived. Steady-state solutions and their stability are discussed. Finally, the solution algorithm is applied to two different engineering problems. One of the application is the transverse vibration of an axially moving Euler–Bernoulli beam and the other is a viscoelastic beam.