Vibrations of axially moving multiple support beam system

In this study, axially moving beam system supported from both ends and in middle part simple was discussed. Supports at the end and middle of the beam were simple supports. It is suggested that beam velocity is harmonically changed around the average speed. 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 twofold 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. Unstable regions for total difference type resonances were investigated.