A generalized nonlinear vibration model of continuous systems is considered. The model includes arbitrary linear and cubic differential and/or integral operators. Linear operators represent the linear parts and cubic operators represent the nonlinear parts of the model. The generalized equation of motion is analyzed by using the method of multiple scales (a perturbation method). Three-to-one internal resonances between natural frequencies are obtained. The amplitude and phase modulation equations are presented. Approximate solution is derived. Steady state solutions and their stability are discussed. Solution algorithm is applied to nonlinear vibration model of an axially moving Euler Bernoulli beam. Constant velocity case of axially moving beam is analyzed. Natural frequencies of beam are given for different velocity values. 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.
展开▼
