This paper is first of the two papers dealing with the nonlinear modelling and investigation of coupled cross-flow and in-line vortex-induced vibrations (VIVs) of flexible cylindrical structures. As a continuation of the previous work (Qu and Metrikine in Ocean Eng 196:106732, 2020) where a new single wake oscillator model was proposed and studied for VIVs of rigid cylinders, the present paper focuses on applying it to flexible cylinders. In this paper, the structure is modelled as an extensible Euler–Bernoulli beam and its 3D nonlinear coupling motion is described in the absolute coordinate system. The single van der Pol wake oscillator model with nonlinear coupling to the in-line motion of the structure, in addition to the classic linear cross-flow motion coupling, is uniformly distributed along the structure to model the hydrodynamic force acting on it. The finite element method has been applied to solve the dynamics of the coupled system, and the experiments of the VIV of a top-tensioned straight riser subjected to a step flow have been taken for the validation of the model. The model has been shown to be able to capture most features of VIVs of flexible cylinders, and a good agreement between the simulation results and the experimental measurements has been observed with regard to the amplitude, frequency and excited mode of both cross-flow and in-line vibrations, as well as the mean in-line deflection due to the amplified in-line force. While it is conventionally expected that the VIV of a flexible cylinder subjected to a uniform flow is dominated by a single frequency, a multi-frequency response is observed in the simulation results over the range of flow velocities through which the transition of the dominant mode of vibration occurs.

An experimental investigation on the coupled dynamics of an offshore tunnel-rig system from the element launching to its near-seabed immersion in irregular waves is conducted, where the motions of tunnel/rigs and the tensile forces of the suspension lines are evaluated in a synchronous manner for a wide range of irregular waves. The effect of immersion depth on the coupled dynamics of the tunnel-rigs system, especially the relationship between the relative motion and suspension line tensions are experimentally examined. The system coupling mechanism is further addressed using the modal analysis and response spectra analysis. The results show that the natural frequencies of the coupled system are primarily affected by the varying immersion depth. The increasing immersion depth leads to a larger in-phase roll period of the system and a greater roll resonance amplitude. For a given immersion depth, a strong dependence of the suspension line tensions upon the relative motions is established.

The authors regret that there are a few typological errors in Eqs. (9) and (13) and (15)–(17). For Eqs. (9), (13) and (17), the signs of the first terms are wrong and they should be corrected as follows: [Formula presented] [Formula presented] [Formula presented] A symbol L is missing on the right hand sides of Eqs. (15) and (16) and they should be corrected as follows: [Formula presented] [Formula presented] The authors would like to apologise for any inconvenience caused.

To illustrate the influence of the in-line coupling on the prediction of vortex-induced vibration (VIV), the simulation results of the coupled cross-flow and in-line VIVs of flexible cylinders- obtained with three different wake oscillator models with and without the in-line coupling- are compared and studied in this paper. Both the cases of uniform and linearly sheared flow are analysed and the simulation results of the three models are compared with each other from the viewpoints of response pattern, fluid force, energy transfer and fatigue damage. The differences between the simulation results from the three models highlight the importance of the in-line coupling on the prediction of coupled cross-flow and in-line VIVs of flexible cylindrical structures.

In this paper, a new wake oscillator model with nonlinear coupling is proposed for the modelling of vortex-induced vibration. The purpose is to develop a model that is capable of reproducing both free and forced vibration experiments. To achieve this goal, an existing van der Pol wake oscillator model is first reviewed. The limitations of the model are discussed and the influence of different drag force models on the dynamic characteristics of the fluctuating lift force that matches the forced vibration experiments are studied. Based on this model, nonlinear coupling terms are introduced to improve its predictive capabilities. The tuning of this improved model to the forced vibration shows a good agreement with experiments in terms of the added damping. However, the model failed to capture the negative added mass at high reduced velocities. As a result, the new model underpredicts both the range and frequency of lock-in in free vibration tests. To eliminate this discrepancy, the model is further enhanced by introducing frequency dependent nonlinear couplings, which are achieved in the time domain by means of convolution integrals. A single set of frequency dependent, complex-valued functions – which are the Laplace transforms of corresponding convolution kernels – that reproduce the forced vibration experiments is determined over a limited range of frequencies. However, no analytical extension of these functions to the infinite frequency domain was found such that the causality principle and the energy conservation would be satisfied. The latter is a major challenge for all existing wake oscillator models that aim at reproducing the forced vibration experiments.