An analytical method for nonlinear vibration analysis of submerged tensioned anchors

Abstract

This paper systematically studies the nonlinear dynamic response and stability properties of the submerged tensioned anchor cable (STAC) considering the coupling effect of the parametric vibration and vortex-induced vibration. A refined and universal modeling approach for the STAC is suggested to describe the structural mechanical behavior more exactly. To ensure the calculation accuracy, we propose an improved Galerkin’s method to discretize the nonlinear motion equation of the STAC. With a self-consistent approach, we demonstrate that only the first three modes used to calculate the response will bring a lot of errors. Then, this paper answers the question of at least how many modal information should be used for dynamic analysis when employing the Galerkin’s discretization method. By performing a standard analytical framework for stability analysis of the STAC, the internal mechanism of the steady-state solution transition from unstable to stable is investigated. All the theoretical results have been well validated by numerical simulations. Results show that multi-valued area as well as the stable and unstable limit circles are observed under certain parameter combinations, no dangerous bifurcation phenomenon or chaos are observed.

Appendix I

1.1 The explicit expression of K

The six unique coefficients in the dynamic stiffness matrix of Eq. (28) are defined as:

$$ k_{11}^{{}} = ps\left( {p_{{}}^{2} + s_{{}}^{2} } \right){{\left[ {\left( {1 + \varepsilon_{{}}^{2} } \right)sS + \left( {1 - \varepsilon_{{}}^{2} } \right)pC} \right]} \mathord{\left/ {\vphantom {{\left[ {\left( {1 + \varepsilon_{{}}^{2} } \right)sS + \left( {1 - \varepsilon_{{}}^{2} } \right)pC} \right]} \Delta }} \right. \kern-0pt} \Delta } $$

(85)

$$ k_{12}^{{}} = ps{{\left[ { - 2\gamma_{{}}^{2} \varepsilon + 2\left( {1 - \varepsilon_{{}}^{2} } \right)psS + \gamma_{{}}^{2} \left( {1 + \varepsilon_{{}}^{2} } \right)C} \right]} \mathord{\left/ {\vphantom {{\left[ { - 2\gamma_{{}}^{2} \varepsilon + 2\left( {1 - \varepsilon_{{}}^{2} } \right)psS + \gamma_{{}}^{2} \left( {1 + \varepsilon_{{}}^{2} } \right)C} \right]} \Delta }} \right. \kern-0pt} \Delta } $$

(86)

$$ k_{13}^{{}} = ps\left( {p_{{}}^{2} + s_{{}}^{2} } \right){{\left[ { - \left( {1 - \varepsilon_{{}}^{2} } \right)p - 2s\varepsilon S} \right]} \mathord{\left/ {\vphantom {{\left[ { - \left( {1 - \varepsilon_{{}}^{2} } \right)p - 2s\varepsilon S} \right]} \Delta }} \right. \kern-0pt} \Delta } $$

(87)

$$ k_{14}^{{}} = ps\left( {p_{{}}^{2} + s_{{}}^{2} } \right){{\left[ {\left( {1 + \varepsilon_{{}}^{2} } \right) - 2\varepsilon C} \right]} \mathord{\left/ {\vphantom {{\left[ {\left( {1 + \varepsilon_{{}}^{2} } \right) - 2\varepsilon C} \right]} \Delta }} \right. \kern-0pt} \Delta } $$

(88)

$$ k_{22}^{{}} = \left( {p^{2} + s^{2} } \right){{\left[ {p\left( {1 + \varepsilon_{{}}^{2} } \right)S - s\left( {1 - \varepsilon^{2} } \right)C} \right]} \mathord{\left/ {\vphantom {{\left[ {p\left( {1 + \varepsilon_{{}}^{2} } \right)S - s\left( {1 - \varepsilon^{2} } \right)C} \right]} \Delta }} \right. \kern-0pt} \Delta } $$

(89)

$$ k_{24}^{{}} = \left( {p_{{}}^{2} + s_{{}}^{2} } \right){{\left[ {s\left( {1 - \varepsilon_{{}}^{2} } \right) - 2p\varepsilon S} \right]} \mathord{\left/ {\vphantom {{\left[ {s\left( {1 - \varepsilon_{{}}^{2} } \right) - 2p\varepsilon S} \right]} \Delta }} \right. \kern-0pt} \Delta } $$

(90)

$$ \Delta = 4ps\varepsilon - 2ps\left( {1 + \varepsilon^{2} } \right)C + \gamma^{2} \left( {1 - \varepsilon^{2} } \right)S $$

(91)

where $\varepsilon = {\text{e}}^{ - pl}$, $C = \cos sl$, $S = \sin sl$, $\gamma = \sqrt {{{Hl^{2} } \mathord{\left/ {\vphantom {{Hl^{2} } {EI}}} \right. \kern-0pt} {EI}}}$