Vibration of an axially moving beam supported by a slightly curved elastic foundation

The vibration of an axially moving beam following a slightly curved path was studied. The simply supported beam was travelling axially on a curved frictionless foundation with nonlinear elastic characteristics. The main objective of this work was to investigate the effect of the moving beam path curvature on its vibration, and the effect of different parameters on the system's dynamic response. These parameters include axial speed, applied tension, and stiffness of the supporting foundation. A Galerkin decomposition approach with four-term truncation accuracy was used to realize a mathematical model that describes the dynamic behavior of the axially moving beam on a slightly curved foundation. Numerical solutions showed that the natural frequency of the axially moving beam travelling on the curved elastic support was higher than that of an axially moving straight beam for all cases considered of different path curvatures and different degrees of support stiffness. Forced vibrations of an axially moving beam on a curved elastic support were also considered under harmonic excitation. Bifurcation diagrams were obtained for the primary resonance excitation using the excitation amplitude as a controlling parameter, while keeping the excitation frequency fixed. It was found that the amplitude-frequency diagram for the axially moving beam on the curved path exhibited many types of bifurcations, including period doubling bifurcation, period four bifurcation and many jumps, compared to that of an axially moving beam resting on a straight elastic support.