Vibration of axially moving beam with non-ideal boundary conditions

In this study, axially moving non-ideal simple and clamped support beams have been considered. Equation of motion have been obtained by usin Hamilton equations. The most general nonlinear equations of motion have been derived by considering the nonlinear effects which derived from the extension beams. Approximate solutions have been found by using time-scaled method rather than perturbation method. The first term of perturbation series are to give the linear problem. Natural frequencies have been exactly calculated by using the solution of linear problem and axially velocity values in addition to the ideal and non-ideal boundary conditions. Analytic results and perturbation solutions have been compared and shown via graphs. The nonlinear terms which occur in the second level add correction terms to the linear equation. It has been separately examined for simple and fixed support that velocity of change frequency is away to zero and also away from the twice of natural frequency, velocity of change frequency is close to zero, velocity of change frequency is close to twice of natural frequency. In the situation that velocity of change frequency is close to twice of natural frequency, basic parametric resonance situation occurs. Parametric resonance situation has analyzed and stability analysis have done for each condition.