A state space approach for the eigenvalue problem of marine risers

Abstract

A numerical state-space approach is proposed to examine the natural frequencies and critical buckling limits of marine risers. A large axial tension in the riser model causes numerical limitations. These limitations are overcome by using the modified Gram–Schmidt orthonormalization process as an intermediate step during the numerical integration process with the fourth-order Runge–Kutta scheme. The obtained results are validated against those obtained with other numerical methods, such as the finite-element, Galerkin, and power-series methods, and are found to be in good agreement. The state-space approach is shown to be computationally more efficient than the other methods. Also, we investigate the effect of a high applied tension, a high apparent weight, and higher-order modes on the accuracy of the numerical scheme. We demonstrate that, by applying the orthonormalization process, the stability and convergence of the approach are significantly improved.

Appendix

1.1 Convegence of the methods used to compute the riser eigenvalues

We present convergence of the methods used to compute the first natural frequency presented in Table 2. For higher order modes and higher values of tension, more steps are needed in each method to achieve convergence.

1.1.1 Galerkin method

In the Galerkin method, we use the mode shapes of a straight beam given by the boundary-value problem in [38]. Then, substituting the mode shapes and applying the orthogonality condition on Eq. (7) reduces it to a set of n algebraic equations that need to be solved. The determinant of the matrix containing the coefficient of the algebraic equations gives the characteristic equation of the frequencies. Convergence of the frequency is shown in Fig. 7.

Fig. 7

Convergence of the first natural frequency of the riser versus the number of mode shapes: (‘’) relative error and (‘’) convergence of $\omega_{1}$

1.1.2 Chebyshev tau method

In this method, the governing Eq. (7) is solved using a spectral decomposition in shifted Chebyshev polynomials [32]. The converged results are shown in Fig. 8.

Fig. 8

Convergence of the first natural frequency of the riser equation versus the number of Chebyshev polynomials: (‘’) relative error and (‘’) convergence of $\omega_{1}$

It was shown in [14] that 320 Chebyshev polynomials are required to achieve convergence for the first natural frequency for tension dominated structures.

1.1.3 Finite element

We use lagrange interpolation with quadratic shape functions in the FE procedure. The mesh is divided into three domains where an adaptive mesh is applied near each end. The solver is a built-in parallel computing algorithm given by the PARDISO solver [39]. A relative tolerance of 10⁻¹⁰ is used to compute the amplitude of the mode shape before the next mesh refining step. Convergence of the eigenvalue is shown in Fig. 9 (see Figs. 7, 8, 9).

Fig. 9

Convergence of the first natural frequency of the riser equation from finite element solution versus number of elements used in the analysis: (‘’) relative error and (‘’) convergence of $\omega_{1}$