Nonlineer vibrations of axially moving multi-supported string

In this study, axially moving string system supported from both ends and in middle part simple was discussed. Supports at the end and middle of the beam were simple supports. Supports at the end and middle of the string were simple supports. It is suggested that string velocity is harmonically changed around the average speed. Nonlinear equations of motion and boundary conditions are derived using Hamilton?s Principle. Equations of motion and boundary conditions are converted to nondimensional form. Thus equations become independent from geometry and material properties. The equations of motion were obtained using Hamilton?s Principle. General Nonlinear equation of motion was obtained by considering nonlinear effect caused by beam stretching for multi-supported beam. Approximate solutions were obtained using the Method of Multiple Scales, a perturbation method. First term in perturbation series compose linear problem. Natural frequencies were calculated by solving the linear problem for different displacement of support in the middle. Correction terms are needed because of the nonlinear terms appears in the second order. The effect of nonlinear terms on natural frequency was calculated for different parameters. Base parametric resonance occurred where the velocity change frequency was equal to approximately two fold of natural frequency. Basic parametric resonance condition was examined in detail. By performing stability analysis of solutions, stable and unstable regions were identified. The effects of axial velocity on the bifurcation points of transverse flexibility coefficient, support number and position have been investigated. Phase modulation equations were obtained. The bifurcation (pitchfork) points where amplitude starts were determined.