A general approach to vibrations of a cubic-nonlinear continuous system with external and parametric excitation

A general model of continuous system with an arbitrary cubic nonlinearity covering many practical vibration problems has been proposed. Linear parts of the equation are also represented by arbitrary operators. Analytical solution is considered by using the method of multiple scales (a perturbation method).Firstly, the amplitude and phase modulation equations are found for the case of primary resonances of the external excitation of the model. The coefficients of the amplitude and phase modulation equations are calculated in their most general form and an approximate analytical solution is considered. Steady state solutions and their stability are discussed in the general sense. Finally the algorithm of the general model is applied to two special problems. These problems are chosen from axially moving continua and they are examples of gyroscopic systems. First model is axially moving Euler Bernoulli beam problem and the second model is axially moving viscoelastic beam problem. Natural frequencies of the systems are found for different beam parameters and different velocity values. Frequency-amplitude graphs are drawn for different beam and excitation parameters. Stable and unstable regions are shown.Secondly, three-to-one internal resonances between natural frequencies are obtained. The amplitude and phase modulation equations are presented. Approximate analytical general solution is derived by using the coefficients of amplitude and phase modulation equations. Steady state solutions and their stability are discussed. Solution algorithm is applied to nonlinear vibration model of an axially moving Euler Bernoulli beam and axially moving viscoelastic beam. Constant axial velocity case of beams is analyzed. Steady state solutions and their stability are determined numerically. Frequency response relations are obtained. Energy transfer of one mode to another via a three-to-one internal resonance is observed. Jump phenomena of the system are shown graphically by choosing different vibration and beam parameter values.Finally principal parametric resonances are obtained with/without internal resonances. For the case of internal resonances, three to one internal resonances are obtained. The amplitude and phase modulation equations are presented. Approximate analytical solution is derived. Steady state solutions and their stability are discussed. Solution algorithm is applied to nonlinear vibration model of an elastic and viscoelastic pipes conveying fluid .Constant velocity case of fluid is analyzed. Natural frequencies of the pipe conveying fluid are obtained. The graphs of natural frequencies versus fluid velocity are drawn for different pipe parameters. Frequency-response relations are also obtained for different pipe and excitation parameters. Stable and unstable regions are also plotted.