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Wrinkles in a longitudinally stretched thin film


Aalim Motasim Aalim Mustafa, Vibration of an axially moving curved web, Master’s thesis, Dept. of Mechanical Engineering, King Fahd University of Petrolium & Minerals, Dhahran, Saudi Arabia, May 2015

ABSTRACT: This thesis presents a study on vibration of an axially moving web following a curved path. The web is considered as a simply supported beam travelling axially on a curved guide that consists of a combination of linear and nonlinear elastic supports. The main objective of this work is to investigate the effect of the path curvature on the moving beam vibration and investigate the effect of different parameters on the system’s dynamic response. These parameters include axial speed, applied tension, degree of curvature of the path and stiffness of the path supports. The Galerkin decomposition with a first mode-shape of a straight a pinned-pinned basis function is utilized to realize a mathematical model that describes the static and dynamic behaviors of the axially moving curved beam. Numerical solutions of the developed model are obtained using a fourth-order Runge-Kutta algorithm under MATLAB environment. Fundamental frequencies are calculated results for axially moving curved beams and compared with those for axially moving straight beam. Amplitude-frequency curves are developed to study forced vibration of the axially moving curved beam under an external force excitation. Poincaré sections and bifurcation diagrams are obtained for three cases: primary, sub-harmonic, and super-harmonic resonance excitations. It is found that the natural frequency of an axially moving beam travelling on a curved elastic support is higher than that of its axially moving straight beam for all considered cases of different path curvatures and different degrees of support stiffness. Forced vibrations of an axially moving beam on a curved elastic support are considered under harmonic excitation. Using the excitation amplitude as a controlling parameter over a wide range of variation, while keeping the excitation frequency fixed, it is found that the system exhibits many types of bifurcations, including period doubling bifurcation, period four bifurcation and many jumps. Compared to an axially moving beam resting on a straight elastic support, the axially moving curved beam showed earlier bifurcation and more swarming bifurcation diagram.

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