Nonlinear vibration of marine riser with large displacement

Abstract

This paper investigates the nonlinear free vibration of a marine riser with large displacement. The nonlinear equation of motion is derived using a variational approach based on Hamilton’s principle. The strain energy functional is composed of the strain energies due to axial deformation and bending. The strain energy associated with the rotational restraint at the bottom end of riser is also considered. The kinetic energy is derived by considering the riser and transporting fluid motions. The finite element method is applied to develop the nonlinear stiffness matrix, mass matrix, and gyroscopic matrix of the riser system. Consequently, the nonlinear natural frequencies and their corresponding mode shapes are determined by solving the eigenvalue problem incorporating the direct iteration technique. Various numerical examples are investigated to evaluate the linear and nonlinear dynamic characteristics of the riser. The parametric investigations of the horizontal top end offset, top tension, and rotational spring constant at bottom support are provided to show the modal transition phenomenon of the riser system. The nonlinear natural frequencies and their corresponding mode shapes are also presented to identify the softening and hardening behaviors of the riser.

Appendix 1

From the position vector ${\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} }}^{*}$ shown in Eq. 7, its derivative with respect to $s$ and the application of the Frenet formulas lead to

$$\frac{{{\text{d}}{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} }}^{*} }}{{{\text{d}}s}}\left( {s,t} \right) = \frac{{{\text{d}}{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} }}}}{{{\text{d}}s}}\left( s \right) + \left( {\frac{{{\text{d}}u}}{{{\text{d}}s}} + \kappa w} \right)\hat{n} + \left( {\frac{{{\text{d}}w}}{{{\text{d}}s}} - \kappa u} \right)\hat{t}.$$

(39)

According to the theory of elasticity, the dynamic axial strain is given by

$$\varepsilon_{\text{d}} = \frac{{{\text{d}}s^{ * } }}{{{\text{d}}s}} - 1,$$

(40)

where ${\text{d}}s$ and ${\text{d}}s^{ * }$ are infinitesimal arc lengths of the riser at the static state and the dynamic state, respectively. Therefore, the derivative of the parameter with respect to $s$ is related to the one with respect to $s^{ * }$ by

$$\frac{{{\text{d}}\left( {} \right)}}{{{\text{d}}s}} = \left( {1 + \varepsilon_{\text{d}} } \right)\frac{{{\text{d}}\left( {} \right)}}{{{\text{d}}s^{ * } }}.$$

(41)

Based on the differential geometry of a curve, the unit tangent vectors (see Fig. 1) at the static state $\left( {\hat{t}} \right)$ and the dynamic state $\left( {\hat{t}^{ * } } \right)$ are defined by

$$\hat{t} = \frac{{{\text{d}}{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} }}}}{{{\text{d}}s}}\left( s \right),$$

(42)

$$\hat{t}^{ * } = \frac{{{\text{d}}{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} }}^{ * } }}{{{\text{d}}s^{ * } }}\left( s \right).$$

(43)

Equations 41–43 lead to

$$\frac{{{\text{d}}{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} }}^{ * } }}{{{\text{d}}s}}\left( s \right) = \left( {1 + \varepsilon_{\text{d}} } \right)\frac{{{\text{d}}{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r} }}^{ * } }}{{{\text{d}}s^{ * } }}\left( s \right) = \left( {1 + \varepsilon_{\text{d}} } \right)\hat{t}^{ * } .$$

(44)

Using Eqs. 39 and 44 with some manipulations, the unit tangent vector at dynamic state can be expressed in terms of the displacements as:

$$\hat{t}^{ * } = \frac{1}{{\left( {1 + \varepsilon_{\text{d}} } \right)}}\left[ {\left( {\frac{{{\text{d}}u}}{{{\text{d}}s}} + \kappa w} \right)\hat{n} + \left( {1 + \frac{{{\text{d}}w}}{{{\text{d}}s}} - \kappa u} \right)\hat{t}} \right].$$

(45)

The angle $\left( {\theta_{\text{d}} } \right)$ between the unit tangent vectors $\hat{t}$ and $\hat{t}^{ * }$ can be obtained by considering direction cosine in Eq. 45 as:

$$\theta_{\text{d}} = \tan^{ - 1} \left( {\frac{{\frac{{{\text{d}}u}}{{{\text{d}}s}} + \kappa w}}{{1 + \frac{{{\text{d}}w}}{{{\text{d}}s}} - \kappa u}}} \right).$$

(46)

Utilizing Taylor series, $\theta_{\text{d}}$ can be approximated keeping the nonlinear terms of displacement up to second order as:

$$\theta_{\text{d}} \approx \frac{{{\text{d}}u}}{{{\text{d}}s}} + \kappa w + \left( {\frac{{{\text{d}}u}}{{{\text{d}}s}} + \kappa w} \right)\left( {\kappa u - \frac{{{\text{d}}w}}{{{\text{d}}s}}} \right).$$

(47)