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Research Papers

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

[+] Author and Article Information
Ioannis K. Chatjigeorgiou

School of Naval Architecture and Marine Engineering, National Technical University of Athens, 9 Heroon Polytechniou Avenue, Athens GR157-73, Greece

J. Offshore Mech. Arct. Eng 130(1), 011003 (Dec 04, 2007) (7 pages) doi:10.1115/1.2786476 History: Received December 16, 2006; Revised May 25, 2007; Published December 04, 2007

## Abstract

The present work treats the bending vibration problem for vertical slender structures assuming clamped connections at the ends. The final goal is the solution of the associated eigenvalue problem for calculating the natural frequencies and the corresponding mode shapes. The mathematical formulation accounts for all physical properties that influence the bending vibration of the structure including the variation in tension. The resulting model incorporates all principal characteristics, such as the bending stiffness, the submerged weight, and the function of the static tension. The governing equation is treated using a perturbation approach. The application of this method results in 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. Here, the term “boundary layer” is used to indicate the narrow region across which the dependent variable undergoes very rapid changes. The boundary layers adjacent to the clamped ends are associated with the fact that the stiffness (which is small) is the constant factor multiplied by the highest derivative in the governing differential equation.

###### FIGURES IN THIS ARTICLE
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Copyright © 2008 by American Society of Mechanical Engineers
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## Figures

Figure 1

The first four natural frequencies for a clamped-clamped slender structure as a function of the nondimensional bending stiffness K for a tension ratio T=0.01

Figure 2

The first four mode shapes for a clamped-clamped slender structure for K=2.6236×10−6 and tension ratio T=0.001

Figure 3

The first three mode shapes for a clamped-clamped slender structure for K=2.6236×10−6 and tension ratio T=0.01

Figure 4

The first three mode shapes for a clamped-clamped slender structure for K=2.6236×10−9 and tension ratio T=0.0001

Figure 5

The first three mode shapes for a clamped-clamped slender structure for K=2.6236×10−9 and tension ratio T=0.001

Figure 6

Effect of end supports on the first mode shape; K=2.6236×10−9 and tension ratio T=0.0001

Figure 7

Effect of end supports on the first mode shape; K=0.0026 and tension ratio T=0.115

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