Nonlinear Vibrations of Deepwater Catenary Riser Subjected to Wave Excitation

Abstract

The purpose of this research is to present the mathematical model for the nonlinear vibration analysis of a deepwater catenary riser subjected to a wave excitation. The mathematical model of the catenary riser is derived from the work-energy principle, including the geometrical nonlinearity and nonlinear loading from a square drag term of hydrodynamic force. The finite element method is used to form the equation of motion for solving the numerical solution. The nonlinear equation of motion can be written in the appropriate form, which is convenient to perform the numerical integration. The Newmark time integration incorporated direct iteration is used to solve the nonlinear vibration response of the catenary riser. The nonlinear forced vibration analysis is carried out for both small and large vibration amplitudes. From numerical investigation, it was found that the vibration response of the riser due to wave force is a harmonic motion and the vibration amplitude increases as the increasing in wave amplitude. In addition, the effect of nonlinear geometry has a significant influence on the vibration amplitude of the riser.

Appendix A

According to Eq. (16), the linear element stiffness matrix $\left[{{\varvec{k}}}_{L}\right]$ can be calculated by

$$ \left[ {{\varvec{k}}_{L} } \right] = \left[ {{}_{b}{\varvec{k}}_{L} } \right] + \left[ {{}_{a}{\varvec{k}}_{L} } \right] $$

(A.1)

where the linear element bending stiffness matrix $\left[ {{}_{b}^{{}} {\varvec{k}}_{L} } \right]$ and the linear element axial stiffness matrix $\left[ {{}_{a}^{{}} {\varvec{k}}_{L} } \right]$ are

$$ \begin{aligned} \left[ {{}_{b}^{{}} {\varvec{k}}_{L} } \right] & = EI_{P} \mathop \smallint \limits_{0}^{l} \left\{ {\left[ {{\varvec{N}}_{d} } \right]^{T} \left[ {\begin{array}{*{20}c} {\kappa_{s}^{4} } & {\kappa_{s}^{2} \kappa_{s}^{^{\prime}} } \\ {\kappa_{s}^{2} \kappa_{s}^{^{\prime}} } & {\kappa_{s}^{^{\prime}2} } \\ \end{array} } \right]\left[ {{\varvec{N}}_{d} } \right] + \left[ {{\varvec{N}}_{d} } \right]^{T} \left[ {\begin{array}{*{20}c} {\kappa_{s}^{2} } & 0 \\ {\kappa_{s}^{^{\prime}} } & 0 \\ \end{array} } \right]\left[ {{\varvec{N}}_{d}^{^{\prime\prime}} } \right]} \right\}{\text{d}}s_{s} \\ & \quad + EI_{P} \mathop \smallint \limits_{0}^{l} \left\{ {\left[ {{\varvec{N}}_{d}^{^{\prime\prime}} } \right]^{T} \left[ {\begin{array}{*{20}c} {\kappa_{s}^{2} } & {\kappa_{s}^{^{\prime}} } \\ 0 & 0 \\ \end{array} } \right]\left[ {{\varvec{N}}_{d} } \right] + \left[ {{\varvec{N}}_{d}^{^{\prime\prime}} } \right]^{T} \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & 0 \\ \end{array} } \right]\left[ {{\varvec{N}}_{d}^{^{\prime\prime}} } \right]} \right\}{\text{d}}s_{s} \\ \end{aligned} $$

(A.2)

$$ \begin{aligned} \left[ {{}_{a}^{{}} {\varvec{k}}_{L} } \right] & = \mathop \smallint \limits_{0}^{l} \left\{ {\left[ {{\varvec{N}}_{d} } \right]^{T} \left\{ {N_{as} \left[ {\begin{array}{*{20}c} {\kappa_{s}^{2} } & 0 \\ 0 & {\kappa_{s}^{2} } \\ \end{array} } \right] + EA_{P} \left[ {\begin{array}{*{20}c} {\kappa_{s}^{2} } & 0 \\ 0 & 0 \\ \end{array} } \right]} \right\}\left[ {{\varvec{N}}_{d} } \right]} \right\}{\text{d}}s_{s} \\ & \quad + \mathop \smallint \limits_{0}^{l} \left\{ {\left[ {{\varvec{N}}_{d} } \right]^{T} \left\{ {N_{as} \left[ {\begin{array}{*{20}c} 0 & { - \kappa_{s} } \\ {\kappa_{s} } & 0 \\ \end{array} } \right] + EA_{P} \left[ {\begin{array}{*{20}c} 0 & { - \kappa_{s} } \\ 0 & 0 \\ \end{array} } \right]} \right\}\left[ {{\varvec{N}}_{d}^{^{\prime}} } \right]} \right\}{\text{d}}s_{s} \\ & \quad + \mathop \smallint \limits_{0}^{l} \left\{ {\left[ {{\varvec{N}}_{d}^{^{\prime}} } \right]^{T} \left\{ {N_{as} \left[ {\begin{array}{*{20}c} 0 & {\kappa_{s} } \\ { - \kappa_{s} } & 0 \\ \end{array} } \right] + EA_{P} \left[ {\begin{array}{*{20}c} 0 & 0 \\ { - \kappa_{s} } & 0 \\ \end{array} } \right]} \right\}\left[ {{\varvec{N}}_{d} } \right]} \right\}{\text{d}}s_{s} \\ & \quad + \mathop \smallint \limits_{0}^{l} \left\{ {\left[ {{\varvec{N}}_{d}^{^{\prime}} } \right]^{T} \left\{ {N_{as} \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ \end{array} } \right] + EA_{P} \left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & 1 \\ \end{array} } \right]} \right\}\left[ {{\varvec{N}}_{d}^{^{\prime}} } \right]} \right\}{\text{d}}s_{s} \\ \end{aligned} $$

(A.3)

The first-order nonlinear element stiffness matrix $\left[ {{\varvec{k}}_{N1} } \right]$ resulting from the nonlinear geometric is

$$ \begin{aligned} \left[ {{\varvec{k}}_{N1} } \right] & = \frac{3}{2}EA_{P} \mathop \smallint \limits_{0}^{l} \left\{ {\left[ {{\varvec{N}}_{d} } \right]^{T} \left[ {\begin{array}{*{20}c} {2\kappa_{s}^{2} \alpha } & { - \kappa_{s}^{2} \beta } \\ { - \kappa_{s}^{2} \beta } & 0 \\ \end{array} } \right]\left[ {{\varvec{N}}_{d} } \right] + \left[ {{\varvec{N}}_{d} } \right]^{T} \left[ {\begin{array}{*{20}c} { - \kappa_{s} \beta } & { - 2\kappa_{s} \alpha } \\ 0 & {\kappa_{s} \beta } \\ \end{array} } \right]\left[ {{\varvec{N}}_{d}^{^{\prime}} } \right]} \right\}{\text{d}}s_{s} \\ & \quad + \frac{3}{2}EA_{P} \mathop \smallint \limits_{0}^{l} \left\{ {\left[ {{\varvec{N}}_{d}^{^{\prime}} } \right]^{T} \left[ {\begin{array}{*{20}c} { - \kappa_{s} \beta } & 0 \\ { - 2\kappa_{s} \alpha } & {\kappa_{s} \beta } \\ \end{array} } \right]\left[ {{\varvec{N}}_{d} } \right] + \left[ {{\varvec{N}}_{d}^{^{\prime}} } \right]^{T} \left[ {\begin{array}{*{20}c} 0 & \beta \\ \beta & {2\alpha } \\ \end{array} } \right]\left[ {{\varvec{N}}_{d}^{^{\prime}} } \right]} \right\}{\text{d}}s_{s} \\ \end{aligned} $$

(A.4)

and the second-order nonlinear element stiffness matrix $\left[ {{\varvec{k}}_{N2} } \right]$ is

$$ \begin{aligned} \left[ {{\varvec{k}}_{N2} } \right] & = \frac{3}{2}EA_{P} \mathop \smallint \limits_{0}^{l} \left\{ {\left[ {{\varvec{N}}_{d} } \right]^{T} \left[ {\begin{array}{*{20}c} {\kappa_{s}^{2} \alpha^{2} } & { - \kappa_{s}^{2} \alpha \beta } \\ { - \kappa_{s}^{2} \alpha \beta } & {\kappa_{s}^{2} \beta^{2} } \\ \end{array} } \right]\left[ {{\varvec{N}}_{d} } \right] + \left[ {{\varvec{N}}_{d} } \right]^{T} \left[ {\begin{array}{*{20}c} { - \kappa_{s} \alpha \beta } & { - \kappa_{s} \alpha^{2} } \\ {\kappa_{s} \beta^{2} } & {\kappa_{s} \alpha \beta } \\ \end{array} } \right]\left[ {{\varvec{N}}_{d}^{^{\prime}} } \right]} \right\}{\text{d}}s_{s} \\ & \quad + \frac{3}{2}EA_{P} \mathop \smallint \limits_{0}^{l} \left\{ {\left[ {{\varvec{N}}_{d}^{^{\prime}} } \right]^{T} \left[ {\begin{array}{*{20}c} { - \kappa_{s} \alpha \beta } & {\kappa_{s} \beta^{2} } \\ { - \kappa_{s} \alpha^{2} } & {\kappa_{s} \alpha \beta } \\ \end{array} } \right]\left[ {{\varvec{N}}_{d} } \right] + \left[ {{\varvec{N}}_{d}^{^{\prime}} } \right]^{T} \left[ {\begin{array}{*{20}c} {\beta^{2} } & {\alpha \beta } \\ {\alpha \beta } & {\alpha^{2} } \\ \end{array} } \right]\left[ {{\varvec{N}}_{d}^{^{\prime}} } \right]} \right\}{\text{d}}s_{s} \\ \end{aligned} $$

(A.5)

The parameters $\alpha$ and $\beta$ in Eqs. (A.4) and (A.5) are defined as follows.

$$ \alpha = u_{td}^{^{\prime}} - \kappa_{s} v_{nd} $$

(A.6)

$$ \beta = v_{nd}^{^{\prime}} + \kappa_{s} u_{td} $$

(A.7)

The matrices $\left[ {\varvec{m}} \right]$, $\left[ {\varvec{g}} \right]$, and $\left[ {{\varvec{k}}_{i} } \right]$ in Eq. (17) are the element mass matrix, element gyroscopic matrix, and element centrifugal force matrix, respectively. These matrices can be expressed as follows.

$$ \left[ {\varvec{m}} \right] = \mathop \smallint \limits_{0}^{l} \left[ {{\varvec{N}}_{d} } \right]^{T} \left( {m_{P} + m_{i} } \right)\left[ {{\varvec{N}}_{d} } \right]{\text{d}}s_{s} $$

(A8)

$$ \left[ {\varvec{g}} \right] = \mathop \smallint \limits_{0}^{l} \left[ {{\varvec{N}}_{d}^{^{\prime}} } \right]^{T} 2m_{i} V_{is} \left[ {{\varvec{N}}_{d} } \right]{\text{d}}s_{s} $$

(A.9)

$$ \left[ {{\varvec{k}}_{i} } \right] = \mathop \smallint \limits_{0}^{l} \left[ {{\varvec{N}}_{d}^{^{\prime}} } \right]^{T} m_{i} V_{is}^{2} \left[ {{\varvec{N}}_{d}^{^{\prime}} } \right]{\text{d}}s_{s} $$

(A.10)

The matrices $\left[ {{\varvec{m}}_{a} } \right]$, $\left[ {\varvec{c}} \right]$, and $\left\{ {\varvec{f}} \right\}$ in Eq. (18) are the element added mass matrix, the element hydrodynamic dam** matrix, and the element hydrodynamic excitation vector, respectively. These matrices can be expressed as follows.

$$ \left[ {{\varvec{m}}_{a} } \right] = \mathop \smallint \limits_{0}^{l} \left[ {{\varvec{N}}_{d} } \right]^{T} C_{a}^{*} \left[ {{\varvec{N}}_{d} } \right]{\text{d}}s_{s} $$

(A.11)

$$ \left[ {\varvec{c}} \right] = \mathop \smallint \limits_{0}^{l} \left[ {{\varvec{N}}_{d} } \right]^{T} \left[ {\begin{array}{*{20}c} {C_{eqn}^{*} } & 0 \\ 0 & {C_{eqt}^{*} } \\ \end{array} } \right]\left[ {{\varvec{N}}_{d} } \right]{\text{d}}s_{s} $$

(A.12)

$$ \left\{ {\varvec{f}} \right\} = \mathop \smallint \limits_{0}^{l} \left[ {{\varvec{N}}_{d} } \right]^{T} \left\{ {\begin{array}{*{20}c} {C_{Dn}^{*} V_{Hn}^{2} + C_{M}^{*} \dot{V}_{Hn} } \\ {C_{Dt}^{*} V_{Ht}^{2} + C_{M}^{*} \dot{V}_{Ht} } \\ \end{array} } \right\}{\text{d}}s_{s} $$

(A.13)