Solution of the Boundary Layer Problems for Calculating the Natural Modes of Riser-Type Slender Structures

The present work treats the problem of the calculation of the natural frequencies and the corresponding bending vibration modes of vertical slender structures. The originality of the study lies on fact that for the derivation of natural frequencies and the corresponding mode shapes, all physical properties that influence the bending vibration of the structure were considered including the aspect of the variation of tension. The resulting mathematical formulation incorporates all principal contributions such as the bending stiffness, the weight and the tension variation. The governing equation is treated using a perturbation approach. The application of this method results to the development of two boundary layer problems at the ends of the structure. These problems are treated properly using a boundary layer problem solution methodology in order to obtain asymptotic approximations to the shape of the vibrating riser-type structure. It should be noted that in this work the term ‘boundary layer’ is not connected with fluid flows but it is used to indicate the narrow region across which the dependent variable undergoes very rapid changes. Frequently these narrow regions adjoin the boundaries of the domain of intersect, especially when the small parameter multiplies the highest derivative.