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In this paper, the effect of an open edge crack on the instability of rotating non-uniform beams subjected to uniform distributed tangential compressive load is studied. The local stiffness due to the presence of crack is considered in the global stiffness matrix of the structure using the finite element method. The cracked beam element is modeled as two equal sub-beam elements connected by a massless rotational spring. Based on the fracture mechanics, the strain energy release rate and the stress intensity factors are employed to investigate the stiffness of the rotational spring. Then, the modified shape functions are developed to reflect the crack stiffness in the finite element analysis. To validate the accuracy of the finite element model and results obtained, comparisons have been made between the results obtained and those available in the literature. The effects of several parameters, including the linear and nonlinear thickness variations, angular velocity, crack location and size, on the instability of cracked rotating non-uniform cantilevers are also examined. The results show that the location of crack significantly influences the critical magnitude of the follower force that destabilizes the cantilevers. In addition, geometric non-uniformity reduces the stability of the cracked cantilevers. For the same amount of cantilever mass, different patterns of mass distribution result in different stability diagrams.

Since the bridge is often treated as the uniform beam for simplicity in most numerical studies of vehicle-bridge interaction, this study proposes a non-uniform vehicle-bridge interaction system by incorporating a three-mass vehicle model in a non-uniform bridge for wider applications, in which non-uniform beam elements of constant width and varying depth are considered. For clarity, the inclined ratios of the entire bridge and one beam element are separately defined in order to describe the non-conformity in computation while both mass and stiffness matrices are re-formulated to comply with the finite element sign convention. As the natural frequencies of a non-uniform bridge cannot be accessed directly, the vehicle scanning method is first adopted to obtain the bridge frequencies. Then, the parametric study is conducted by considering vehicle damping, bridge damping, and pavement irregularity. In addition to the vehicle frequency, the numerical results show that the proposed vehicle-bridge interaction system is able to scan the first four bridge frequencies with desired accuracy subject to pavement irregularity. Concerning the pitching effect of the vehicle, it is shown that the locations for installing sensors are actually affected by both the geometry and the cross-sectional geometry of the bridge in the concern of achieving high resolution of frequency identification.

Non-uniform beam structures are widely applied in various engineering fields, and the accurate estimation of modal parameters is significant to the design and optimization of non-uniform beams. This work aims to investigate the transverse free vibration of non-uniform beams with exponentially varying rectangular and circular cross-sections. When solving the governing equations for non-uniform beams in transverse free vibration, utilizing the Adomian Decomposition Method (ADM) can achieve semi-analytical expressions for natural frequencies and modal shape functions. To verify the efficiency of ADM, comparisons are made among the results obtained by the present proposed method, previous studies, the Finite-Element-Method (FEM) model and the experimental modal testing. Variations of natural frequencies and modal shapes under representative constraint boundary conditions are illustrated considering the effects of exponential factors. The results indicate that the modal parameters of non-uniform beams depend on the exponential factors significantly. Especially, for non-uniform beams with varying rectangular cross-sections, the exponential factor of the thickness direction has a greater impact on the modal parameters compared to the width direction. In addition, the positions of modal nodes and antinodes in non-uniform beams will shift compared to uniform beams due to the exponential non-uniformity.