Introduction

During deployment/retrieval, vessel motion, or harsh environmental conditions, a drilling riser needs to be suspended from a floating unit (Williams and Kenny 2010). Compared with the riser fixed on a subsea wellhead, the hang-off drilling riser encounters more significant challenges and undergoes more severe dangers, for instance, more drastic dynamic responses, more massive rotation and deflection, a higher probability of striking, and severer fatigue damage or buckling. Especially when the frequency of vessel motion, wave loads, or vortex-induced vibration is close to the hang-off drilling riser's natural frequency, riser resonance is likely to be excited (Li et al. 2005; Zhang and Gao 2010; Wang et al. 2022), and disastrous consequences, for instance, environmental pollution and economic loss (Fan et al. 2017; Chang et al. 2018) are likely to be caused. To extract sufficient mechanical characteristics and fully analyze the dynamic behaviors for prevention of the riser's potential threats, modal state vectors such as natural frequency, modal displacement, slope, bending moment, and shear of the riser are thus urgently needed to be studied in detail.

The dynamic behaviors and characteristics, just including natural frequency and modal displacement or even slope of a drilling riser, have been widely studied in the literature. Dareing and Huang (1976) firstly studied the eigenvalues and eigenvectors of marine drilling risers and discussed the resonance of the risers under wave excitation. Patel and Jesudasen (1987) studied the dynamic displacements and stresses of free-hanging risers excited by direct loading and vortex shedding. Kuiper and Metrikine (2005) studied the dynamic stability of a submerged, free-hanging riser conveying fluid via the natural frequencies and displacements. Yang and Li (2009) researched the parametric resonance of free-hanging marine risers through eigenvalues and amplitudes. Dai et al. (2009) analyzed a riser's dynamic responses via the displacements, bending moments, and shears in current. Wang et al. (2012) and Xu and Wang (2012) studied the time-domain displacements of free-hanging risers subjected to waves, currents, and wind forces. The typical dynamic responses and characteristics such as frequencies, displacements, bending moments, and stress of hang-off risers are also investigated in the studies (Wu et al.2014; Liu et al. 2018; Wang et al. 2015a, b; Li et al. 2016; Pestana et al. 2016; Song et al. 2018; Meng et al. 2018) with consideration of marine environment loads. Whereas the dynamic responses and characteristics of risers have been studied extensively, few studies were devoted to the intrinsic modalities of displacements, cross section slopes, bending moments, and shears of free-hanging risers.

Chen et al. (2009) investigated the natural frequencies and modal shapes, i.e., modal displacements, of free-hanging risers with different boundary conditions. Meng and Chen (2012) studied the eigenvalues of free vibration of fluid-conveying steel catenary riser and analyzed its natural frequencies and modal displacements. Lei et al. (2015) studied the variations of natural frequencies and modal displacements of a free-hanging riser under two kinds of conditions, i.e., the hanging total mass and no hanging mass at all at the bottom end. Zhou et al. (2017; 2018) extracted the natural frequencies, modal displacements, and displacement-based characteristics of drilling risers underworking and free-hanging conditions. Zhang et al. (2022) analyzed the lock-in frequency, single-frequency mode, and multi-frequency modes of a long-hanged and weighted riser in internal fluid flow and external currents. Jiang et al. (2019) studied the coupled frequencies for cross-flow and in-line motions of fluid-conveying risers in uniform cross-flow. Lu et al. (2018) predicted a riser's modal shapes and exciting and damping regions of excited modes by a proposed modal-space-based direct method for the prediction of vortex-induced vibration. Zhou et al. (2020) studied the variation of natural frequencies and modal displacements versus cracks in hang-off risers. Yu et al. (2018) analyzed the intrinsic relationships between natural frequency and oscillating frequency to explain the occurrence of the dominant mode. Wang et al. (2016) studied the response frequencies and cross section trajectories of displacements of a free-hanging water intake riser.

In brief, most of the studies are about time-domain analysis, frequency-domain analysis, and free modal analysis. The first two kinds of studies mainly study the dynamic behaviors including vibration frequencies, displacements, and bending moments of marine risers under normal connected conditions and hang-off (disconnected) conditions. The dynamic parameters extracted under marine loads reflect the complex motions of the risers directly rather than the inherent structural state characteristics. A modal decomposition method or finite difference method (FDM) may be needed to get the inherent state vectors. The further calculation and additional processing likely lead to more complex problems, time-consuming, or even errors especially when the boundaries at both ends are solved using FDM. For the third category of the studies, just modal frequencies and displacements are directly and commonly studied using the numerical methods, e.g., finite element method (FEM), lumped mass method (LMM), differential transformation method (DTM), etc. To study the modal slopes, bending moments, and shears, a state-vector approach is developed based on the precise integration method (PIM). The modal state vectors, such as natural frequencies, modal displacements, slopes, bending moments, and shears, are studied in detail under four classic cases according to the riser's hard and soft hang-offs and the deployment and retrieval processes.

Methodology

Physical and mechanical models

A marine drilling riser system consists of many different components and subsystems, e.g., the top tensioning system, diverter, flex and ball joints, telescopic joint, riser joints, spider and gimbal, lower marine riser package (LMRP), and blowout preventer (BOP) stack (ABS 2017). During deployment or retrieval, the top of the riser is directly suspended on the spider and gimbal from the drill floor and is tensioned by the LMRP and/or BOP near the bottom wellhead (Mao et al. 2019). As sketched in Fig. 1, the hang-off riser is recommended to be modeled from the drill floor to the suspended LMRP and/or BOP.

Fig. 1
figure 1

Physical model of a typical hang-off marine riser

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The following assumptions and classic units are utilized to describe the mechanical behaviors of the specialized hang-off drilling riser's transverse vibration (Fan et al. 2017; Wang et al. 2015a, b; Pestana et al. 2016).

  • The riser material is deemed homogeneous, isotropic, and linearly elastic.

  • The riser's main pipe is considered to be flooded by zero-velocity seawater during deployment and/or retrieval, and the riser body cannot be extended along the axial direction.

  • The vertical component of the current is neglected, and the effects of upper and lower flex/ball joints and auxiliary lines are also neglected.

  • All riser joints are assumed to be bare, i.e., not covered by buoyancy modules. The axial geometry and mass of the riser are assumed to be uniform and unchanged.

  • The riser with variable tension and high slenderness has large deflection under small strain and can be modeled by the tensioned Euler–Bernoulli beam.

Figure 2a is a diagram of the hang-off riser's simplified mechanical model based on the above assumptions. Taking the suspension point location, the vertically downward direction, and the horizontal vibration deformation as the origin o, \(\hat{x}\) positive, and \(\hat{x}o\hat{y}\) working plane, respectively, the Cartesian coordinate system is defined and sketched in Fig. 2a. Figure 2b is the element diagram of the riser that can be discretized by a series of tensioned Euler–Bernoulli beam segments according to the last one of the above assumptions, where 0, 1, 2, \(\cdots\), k, k + 1, \(\cdots\), n are the number of the nodes. The force equilibrium diagram of each micro-segment of the riser's free vibration is shown in Fig. 2c, where the axial tension, bending moment and shear force are denoted as \(\hat{T}\), \(\hat{M}\) and \(\hat{S}\), respectively, the length of each micro-segment is written as \(\hat{s}\), \(\hat{\theta }\) is the cross section slope, \(\delta \hat{T}\), \(\delta \hat{M}\), \(\hat{S}\) and \(\delta \hat{\theta }\) are the increments of axial tension, bending moment, shear and slope, respectively, and \(m_{{\text{e}}} \left( {\hat{x}} \right)\) and \(\hat{w}\) are the effective mass and net weight per unit length of the riser.

Fig. 2
figure 2

a Simplified physical model of the riser; b the discretized riser body; c mechanical model of each segment

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The vertical force equilibrium equation of the segment is,

$$\left( {\hat{T} + \delta \hat{T}} \right)\sin \left( {\hat{\theta } + \delta \hat{\theta }} \right) - \hat{T}\sin \hat{\theta } - \left( {\hat{S} + \delta \hat{S}} \right)\cos \left( {\hat{\theta } + \delta \hat{\theta }} \right) + \hat{S}\cos \hat{\theta } - m_{{\text{e}}} \delta \hat{s}\frac{{\partial^{2} \hat{y}}}{{\partial t^{2} }} = 0$$
(1)

Referring to the small strain assumption, there are \(\sin \theta \approx \theta\), \(\sin \delta \theta \approx \delta \theta\) and \(\cos \theta \approx \cos \left( {\theta { + }\delta \theta } \right) \approx 1\). Because of the inextensibility assumption of the riser, \(\hat{s}\) is deemed as \(\hat{x}\). Neglecting the infinitesimal \(\delta T\delta \theta\), Eq. (1) is simplified as,

$$\frac{{\partial \hat{S}}}{{\partial \hat{x}}} - \frac{{\partial \left( {\hat{T}\hat{\theta }} \right)}}{{\partial \hat{x}}}{ + }m_{{\text{e}}} \frac{{\partial^{2} \hat{y}}}{{\partial t^{2} }} = 0$$
(2)

Based on the Euler–Bernoulli beam assumption, the relationships between displacement \(\hat{y}\), bending moment \(\hat{M}\), shear force \(\hat{S}\), and slope \(\hat{\theta }\) are,

$$\frac{{\partial \hat{y}\left( {\hat{x},t} \right)}}{{\partial \hat{x}}} = \hat{\theta }\left( {\hat{x},t} \right),_{{}}^{{}} \frac{{\partial \hat{\theta }\left( {\hat{x},t} \right)}}{{\partial \hat{x}}} = \frac{{\hat{M}\left( {\hat{x},t} \right)}}{{EI\left( {\hat{x}} \right)}},_{{}}^{{}} \frac{{\partial \hat{M}\left( {\hat{x},t} \right)}}{{\partial \hat{x}}} = \hat{S}\left( {\hat{x},t} \right)$$
(3)

where \(EI\left( {\hat{x}} \right)\) is the bending stiffness of the riser.

The effective axial tension \(\hat{T}\) is (Wang et al. 2015a, b),

$$\hat{T}\left( {\hat{x}} \right) = W_{{\text{LMRP/BOP}}} + \int_{{\hat{x}}}^{L} {\hat{w}\left( {\hat{x}} \right)d\hat{x}}$$
(4)

where \(W_{{\text{LMRP/BOP}}}\) is the wet weight of LMRP and/or BOP, and L is the total length of the riser.

State-vector approach based on PIM

The lateral deflection of the free vibration riser may be expressed as the product of spatial function \(\hat{Y}\left( {\hat{x}} \right)\) and a time function \(\hat{\phi }\left( t \right) = \sin \left( {\omega t + \varphi } \right)\), where the free vibration of the riser is a simple harmonic vibration. That is,

$$\hat{y}\left( {\hat{x},t} \right) = \hat{Y}\left( {\hat{x}} \right)\hat{\phi }\left( t \right) = \hat{Y}\left( {\hat{x}} \right)\sin \left( {\omega t + \varphi } \right)$$
(5)

where φ is the initial phase, and ω is the natural frequency.

Then, Eqs. (2)–(4) are derived as,

$$\left\{ \begin{gathered} \frac{{d\hat{Y}\left( {\hat{x}} \right)}}{{d\hat{x}}} = \hat{\theta }\left( {\hat{x}} \right) \hfill \\ \frac{{d\hat{\theta }\left( {\hat{x}} \right)}}{{d\hat{x}}} = \frac{{\hat{M}\left( {\hat{x}} \right)}}{{EI\left( {\hat{x}} \right)}} \hfill \\ \frac{{d\hat{M}\left( {\hat{x}} \right)}}{{d\hat{x}}} = \hat{S}\left( {\hat{x}} \right) \hfill \\ \frac{{d\hat{S}\left( {\hat{x}} \right)}}{{d\hat{x}}}{ = }m_{{\text{e}}} \omega^{2} \hat{Y}\left( {\hat{x}} \right) - \hat{w}\left( {\hat{x}} \right)\hat{\theta }\left( {\hat{x}} \right) + \hat{T}\left( {\hat{x}} \right)\hat{M}\left( {\hat{x}} \right) \hfill \\ \end{gathered} \right.$$
(6)

Using the nondimensionalized technique,

$$\begin{gathered} x = \frac{{\hat{x}}}{L},_{{}}^{{}} Y\left( x \right) = \frac{{\hat{Y}\left( {\hat{x}} \right)}}{L},_{{}}^{{}} \theta \left( x \right) = \hat{\theta }\left( {\hat{x}} \right),_{{}}^{{}} e\left( x \right) = \frac{{EI\left( {\hat{x}} \right)}}{EI\left( 0 \right)},_{{}}^{{}} M\left( x \right) = \frac{{\hat{M}\left( {\hat{x}} \right)L}}{EI\left( 0 \right)} \hfill \\ S\left( x \right) = \frac{{\hat{S}\left( {\hat{x}} \right)L^{2} }}{EI\left( 0 \right)},_{{}}^{{}} a\left( x \right) = \frac{{\hat{T}\left( {\hat{x}} \right)L^{2} }}{EI\left( 0 \right)},_{{}}^{{}} w\left( x \right) = \frac{{\hat{w}\left( {\hat{x}} \right)L^{3} }}{EI\left( 0 \right)},_{{}}^{{}} \lambda \left( x \right) = \frac{{m_{{\text{e}}} \left( {\hat{x}} \right)\omega^{2} L^{4} }}{EI\left( 0 \right)} \hfill \\ \end{gathered}$$
(7)

and denoting the state vector Z as Z = [Y;θ;M;S], the state-vector governing equation of the riser is expressed as (Pestana et al. 2016),

$${\mathbf{Z^{\prime}}} = {\mathbf{HZ}}$$
(8)

where

$${\mathbf{H}} = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 \\ 0 & 0 & {\frac{1}{e\left( x \right)}} & 0 \\ 0 & 0 & 0 & 1 \\ {\lambda \left( x \right)} & { - w\left( x \right)} & {\frac{a\left( x \right)}{{e\left( x \right)}}} & 0 \\ \end{array} } \right]$$

For the kth segment, the solution of Eq. (8) is,

$${\mathbf{Z}}_{k} = \exp \left( {{\mathbf{H}}_{k}^{{\text{c}}} l_{k} } \right){\mathbf{Z}}_{k - 1}$$
(9)

where lk is the dimensionless length of the kth segment. \({\mathbf{Z}}_{k - 1}\) and \({\mathbf{Z}}_{k}\) are the state vectors at the (k-1)th and kth nodes, respectively. \({\mathbf{H}}_{k}^{{\text{c}}}\) is expressed as,

$${\mathbf{H}}_{k}^{{\text{c}}} { = }\left[ {\begin{array}{*{20}c} 0 & 1 & 0 & 0 \\ 0 & 0 & {\frac{1}{{e_{k} }}} & 0 \\ 0 & 0 & {\frac{{w_{k} l_{k}^{2} }}{{12e_{k} }}} & 1 \\ {\lambda_{k} } & { - w_{k} } & {\frac{{a_{k - 1} }}{{e_{k} }} - \frac{{w_{k} l_{k} }}{{2e_{k} }}} & { - \frac{{w_{k} l_{k}^{2} }}{{12e_{k} }}} \\ \end{array} } \right]$$
(10)

where

\(e_{k} = \frac{{EI\left( {\hat{x}_{k} } \right)}}{EI\left( 0 \right)}\),\(w_{k} = \frac{{\hat{w}\left( {\hat{x}_{k} } \right)L^{3} }}{EI\left( 0 \right)}\), \(\lambda_{k} = \frac{{m_{{\text{e}}} \left( {\hat{x}_{k} } \right)\omega^{2} L^{4} }}{EI\left( 0 \right)}\) and \(a_{k - 1} = \frac{{\hat{T}\left( {\hat{x}_{k - 1} } \right)L^{2} }}{EI\left( 0 \right)}\).

Denoting \(\xi_{k}\) as the secondary calculated length,

$$\xi_{k} = {\raise0.7ex\hbox{${l_{k} }$} \!\mathord{\left/ {\vphantom {{l_{k} } m}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$m$}}$$
(11)

where

\(m = 2^{N} ,\;N = 1,2,3, \cdots\).

According to the theorem of Padé approximation (Zhang et al. 2022), there is,

$$\exp \left( {{\mathbf{H}}_{k}^{{\text{c}}} \xi_{k} } \right) = {\mathbf{I}} + {\mathbf{R}}_{a}$$
(12)

where

$${\mathbf{R}}_{a} = \left( {{\mathbf{I}} + {\mathbf{D}}_{a} } \right)^{ - 1} \left( {{\mathbf{N}}_{a} - {\mathbf{D}}_{a} } \right)$$

where

$$\begin{gathered} {\mathbf{D}}_{a} = \sum\limits_{r = 1}^{N} {\frac{{\left( {2N - r} \right){!}r{!}}}{{\left( {2N} \right){!}r{!}\left( {N - r} \right){!}}}\left( { - {\mathbf{H}}_{k}^{{\text{c}}} \xi } \right)^{r} } \hfill \\ {\mathbf{N}}_{a} = \sum\limits_{r = 1}^{N} {\frac{{\left( {2N - r} \right){!}r{!}}}{{\left( {2N} \right){!}r{!}\left( {N - r} \right){!}}}\left( {{\mathbf{H}}_{k}^{{\text{c}}} \xi } \right)^{r} } \hfill \\ \end{gathered}$$

Note

$$\begin{array}{*{20}c} {{\mathbf{T}}_{k} = \exp \left( {{\mathbf{H}}_{k}^{{\text{c}}} l_{k} } \right) = \left[ {\exp \left( {{\mathbf{H}}_{k}^{{\text{c}}} \xi_{k} } \right)} \right]^{m} = \left( {{\mathbf{I}} + {\mathbf{R}}_{a} } \right)^{m} } \\ { = \left( {{\mathbf{I}} + 2{\mathbf{R}}_{a} + {\mathbf{R}}_{a} {\mathbf{R}}_{a} } \right)^{{2^{N - 1} }} } \\ \end{array}$$
(13)

Using the data storing skill of the PIM, the following computer language is cycled N times,

$${\text{for}}\left( {iter = 0;\,\,iter < N;\,\,iter + + } \right)\;\;\;{\mathbf{R}}_{a} { = }2{\mathbf{R}}_{a} { + }{\mathbf{R}}_{a} {\mathbf{R}}_{a}$$
(14)

The kth transfer matrix Tk is,

$${\mathbf{T}}_{k} = {\mathbf{I}} + {\mathbf{R}}_{a}$$
(15)

The total transfer matrix T between the state vectors Z0 and Zn at the top and bottom ends is,

$${\mathbf{Z}}_{n} = {\mathbf{TZ}}_{0}$$
(16)

where

\({\mathbf{T}} = {\mathbf{T}}_{n} \cdots {\mathbf{T}}_{2} {\mathbf{T}}_{1}\).

Boundary conditions and different cases

The hang-off configuration can support just the LMRP, after an emergency disconnect or a planned disconnect, or it can support the BOP during its deployment. During installation and/or retrieval, the riser string is suspended on the spider that equipped with dogs, which is known as the hard hang-off condition. In severe weather, the riser string is hung off on the gimbal, allowing the top of the riser to rotate freely to avoid excessive bending moment, which is known as the soft hang-off condition. Therefore, there are two kinds of corresponding boundary considerations at the top end of the riser.

  1. (1)

    Under the hard hang-off condition, the displacement Y0 and slope θ0 are zeros, i.e., the state vector of the riser at the top boundary is,

    $${\mathbf{Z}}_{0} = \left[ {{0;}0{;}M_{0} {;}S_{0} } \right]$$
    (17)
  2. (2)

    Under the soft hang-off condition, the displacement Y0 and bending moment M0 are zeros, i.e., the state vector of the riser at the top boundary is,

    $${\mathbf{Z}}_{0} = \left[ {{0;}\theta_{0} {;}0{;}S_{0} } \right]$$
    (18)

Regardless of the top constraints, the bottom end of the riser is usually seen as a free end, where the rotational stiffness of the lower flex/ball joint is neglected and the LMRP and/or BOP is simplified as a lumped mass (Liu et al. 2018; Zhou et al. 2020). Therefore, the bottom boundary condition expressed by state vectors is,

$$\left\{ {\begin{array}{*{20}l} {M_{n} = 0} \hfill \\ {S_{n} - a_{{\text{b}}} \theta_{n} + \varepsilon_{{\text{M}}} \lambda_{n} Y_{n} = 0} \hfill \\ \end{array} } \right.$$
(19)

where

\(a_{{\text{b}}} = \frac{{W_{{\text{LMRP/BOP}}} L^{2} }}{EI\left( 0 \right)},\;\varepsilon_{{\text{M}}} = \frac{{M_{{\text{LMRP/BOP}}} }}{{m_{{\text{e}}} L}}\), and \(M_{{\text{LMRP/BOP}}}\) is the effective mass of LMRP and/or BOP.

Substituting Eqs. (17)-(19) into Eq. (16), the determinant of total transfer matrix T can be expressed as a function of only one unknown parameter ω. The equation is called the characteristic equation. The natural frequency and modal shape state vectors are the eigenvalues and feature vectors of the characteristic equation, in fact, and can be obtained by solving the eigenvalue problem.

Under the hard and soft hang-offs, four classic cases are studied for a clear description of the processes of deployment and retrieval, as sketched in Fig. 3. Case I and Case II are the hard hang-off riser during deployment and retrieval, respectively. Case III and Case IV are the soft hang-off riser during deployment and retrieval, respectively. It should be noted that the bottom suspended mass during the retrieval process is the mass of the LMRP after an emergency disconnection or a planned disconnection from the BOP.

Fig. 3
figure 3

Four cases of the riser under hard and soft hang-offs during deployment and retrieval

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Verification of the state-vector approach

To extract the natural frequencies and modal state vectors, a calculation program based on the above methodology is written and implemented in MATLAB. To verify the calculations, another program using DTM is also developed in MATLAB, and a FEM program is coded by Ansys parametric design language (APDL) in the Ansys software.

A typical deep-water drilling riser system is taken as an example. The key properties of the riser system are given in Table 1.

Table 1 Properties of the hang-off drilling riser system
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A total of 15 riser joints are considered to be lowered during the deployment. The longest suspension length of the riser is thus 342.9 m. The riser body is discretized uniformly by a series of segments. When the riser hang-off length is maximum, the natural frequencies of the riser under the four classic cases, as shown in Fig. 3, are extracted and listed in Table 2.

Table 2 Natural frequencies calculated by different methods
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Table 2 shows that the natural frequencies calculated by the method in this study, the DTM, and the FEM are consistent. The calculated results are proven to be correct and feasible in the investigation of the modal characteristics of the four-case drilling risers.

Taking the riser under the hard hang-off conditions, i.e., Case II and Case III, as an example, the normalized modal displacements are extracted by the state-vector approach and the traditional FEM and are plotted in Fig. 4.

Fig. 4
figure 4

Comparison of modal displacements under hard hang-off conditions using different methods where the first letter of subscript ‘d’ or ‘r’ means the deployment or retrieval, and the second letter of subscript ‘h’ means the hard hang-off condition.

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Figure 4 illustrates that the modal characteristics extracted by the state-vector approach in this study are consistent with those obtained by the FEM. The developed numerical models are thus verified again to be valid for the investigation of the detailed response characteristics of the riser's free vibration in the different classic cases.

Results and analysis

Because only modal displacement can be directly obtained by the traditional methods, i.e., DTM and FEM in this study, the developed state-vector approach is utilized to obtain not only the modal displacement but also the modal slope, bending moment, and shear of the drilling riser.

State vectors of the riser in the final/initial stage of deployment/retrieval

In the final/initial stage of deployment/retrieval, the state vectors such as modal displacement, slope, bending moment, and shear of the riser under the four classic cases are shown in Figs. 5–8, respectively.

Fig. 5
figure 5

The normalized modal displacements of the riser in the final/initial stage of deployment/retrieval where the second letter of subscript ‘s’ means the soft hang-off condition.

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Figure 5 shows the modal displacements of the deployed and retrieval risers near the top end are closer to each other, and the modal displacements of the hard hang-off and soft hang-off risers near the bottom end are closer to each other.

Figure 6 shows that the slopes under the deployment are relatively larger than those under the retrieval when the data are below the top end, that the slopes under the deployment are relatively smaller than those under the retrieval when the data are above the bottom end, and that the slopes near the bottom (top) end are almost unaffected (greatly affected) by the hang-off modes.

Fig. 6
figure 6

The normalized modal slopes of the riser in the final/initial stage of deployment/retrieval

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Figure 7 shows that the amplitudes of the first-order bending moments of the soft hang-off risers are larger than those of the hard hang-off risers near the bottom end. The higher-order bending moments for the four cases near the bottom end are almost the same, equal to zeros. The amplitude of bending moment of the riser in Case I is larger than that of the riser in Case III at the top end.

Fig. 7
figure 7

The normalized modal bending moment of the riser in the final/initial stage of deployment/retrieval

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Figure 8 shows that the amplitudes of shear in the hard hang-off risers are larger than those of shear in the soft hang-off risers near the top end. The first-order shears away from both ends are almost equal to zeros. The amplitudes of shear in the soft hang-off modes are larger than those of shear in the hard hang-off modes near the bottom end.

Fig. 8
figure 8

The normalized modal shear of the riser in the final/initial stage of deployment/retrieval

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State vectors of the riser in the different stages of deployment/retrieval

During deployment and/or retrieval, the vibration characteristics of the riser vary with the suspension length. To study the detailed relationships of the dynamic characteristics versus the riser suspension length, five deployment or retrieval stages: 20%, 40%, 60%, 80%, and 100% of the total riser length are considered. The first four-order normalized modal displacement, slope, bending moment, and shear of the hard hang-off risers are studied, and the results are shown in Figs. 9–12, respectively.

Fig. 9
figure 9

The first four-order normalized modal displacements in the hard hang-off condition

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As shown in Fig. 9, the maximum amplitude of the modal displacements during the deployment is relatively larger than that of the modal displacements during the retrieval. The maximum difference of modal displacements between the deployment and the retrieval almost occurs at the bottom end, and the difference increases with the increase of the suspension length.

As shown in Fig. 10, the amplitude of first-order slopes during deployment is relatively first bigger, then smaller, and then bigger than that of first-order slopes during retrieval. The difference of the first-order slopes between the deployment and retrieval at the bottom end decreases with the riser suspension length increasing. However, the amplitudes of high-order slopes during the deployment are obviously smaller than those of slopes during the retrieval near the bottom end.

Fig. 10
figure 10

The first four-order normalized modal slopes in the hard hang-off conditions

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As shown in Fig. 11, the first-order bending moments of the riser below about 50 m depth are almost equal to 0. Below the top end, the amplitude of first-order (higher-order) bending moments during the deployment is relatively bigger (smaller) than that of the first-order (higher-order) bending moments during the retrieval.

Fig. 11
figure 11

The first four-order normalized modal bending moments in the hard hang-off conditions

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As shown in Fig. 12, the first-order shear forces of the riser below about 50 m depth are almost equal to 0. Below the top end, the amplitude of first-order (higher-order) shear during the deployment is relatively bigger (smaller) than that of the first-order (higher-order) shear during the retrieval.

Fig. 12
figure 12

The first four-order normalized modal shears in the hard hang-off conditions

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Natural frequencies of the riser with different hang-off lengths

As shown in Fig. 13, the first four natural frequencies in the four classic cases in Fig. 3 decrease rapidly and then slowly with the increase of the riser's hang-off length.

Fig. 13
figure 13

The first four-order natural frequencies of the risers versus different hang-off lengths

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By comparing Case I and Case III or Case II and Case IV, the natural frequencies in the deployment are relatively smaller first and then larger than those in the retrieval, which means that the effect of BOP mass is dominant when the hang-off length is short and that the effect of riser length is dominant when the hang-off length is longer.

By comparing Case I and Case II or Case III and Case IV, the differences of natural frequencies between the hard and soft hang-offs become smaller with the increase of the riser hang-off length, which means that the effect of constraints at the top end should be considered when the hang-off length is short.

Resonance analysis during deployment and retrieval

As is well known, all-natural modes can participate in forced vibrations of marine risers, and if one of the wave frequencies is tuned to the natural frequency the riser will respond to that frequency and one natural mode will be amplified, which is called riser resonance (Dareing and Huang 1976; Samuel et al. 2006; Persent et al. 2009). During the deployment and retrieval, the riser structure is subjected to sizeable periodic wave loads, and resonance excited by waves may likely occur if the hang-off riser's natural frequencies are close to the frequencies of the random waves.

Under the resonance, a natural mode will dominate the overall dynamic behavior of the riser and two or more modes may be excited simultaneously. Referring to the literature by Dareing and Huang (1976), the first two modes are the most vulnerable if the marine drilling riser is in the North Sea. The first two-order critical resonant suspension lengths of the riser under irregular waves are thus investigated and discussed herein.

According to the theory of double-peaked sea irregular wave spectra (Zhou et al. 2020), the sea spectrum Sω can be represented by the function of the significant wave height Hs, which is,

$$S_{\omega } = \frac{{173H_{s}^{2} }}{{\Gamma^{4} \omega^{5} }}\exp \left( { - \frac{691}{{\Gamma^{4} \omega^{4} }}} \right)$$
(20)

where

\(\Gamma = \frac{5.127}{{B^{0.25} }}\) and \(B = \frac{3.12}{{H_{s}^{2} \omega^{4} }}\).

Figure 14a is the curves of the double-peaked wave spectra Sω varying with different significant wave heights Hs. By defining that 95% of the total energy is concentrated in the red region, as shown in Fig. 14b, the lower and upper critical frequencies ωL and ωU can be obtained for each kind of environmental conditions with different significant wave heights, respectively.

Fig. 14
figure 14

a different double-peaked sea wave spectra with different significant wave heights; b a sketch of resonant and non-resonant regions with the lower and upper resonance frequencies ωL and ωU

Full size image

Reconsidering the natural frequencies of the hang-off risers in the four classic cases, the critical resonant suspension lengths Lc of the risers corresponding to the ωL and ωU of the different significant wave heights Hs are then calculated and listed in Table 3.

Table 3 The critical lengths of the hang-off risers versus different waves
Full size table

By comparing Cases I and II or Cases III and IV, the L 1c of the hard hang-off riser is longer than that of the soft hang-off riser. However, the L 2c of the hard hang-off riser is longer first and then shorter than that of the soft hang-off riser with the increase in significant wave heights Hs. By comparing Cases I and III, i.e., in the hard hang-off conditions, the L 1c (L 2c ) of the riser during the deployment is shorter (longer) than that of the riser during the retrieval. However, by comparing Cases II and IV, i.e., in the soft hang-off conditions, the L 1c of the riser during the deployment is slightly greater than that of the riser during the retrieval, and the L 2c of the riser during the deployment is significantly longer than that of the riser during the retrieval.

Because all L 1c is less than about 3Ls = 3 × 22.86 = 68.58 m, the deploying of the first three riser joints should be paid more attention to and reconsidered when they cross the splash zone. Because L 2c varies from 91.62 to 287.72 m, the detailed analysis and control of the gimbal and the mass of LMRP/BOP corresponding to the four classic cases are crucial to the prevention of the riser's second-order resonance. Besides, because of the difference of Lc between ωL and ωU varying with the four cases, the deploying or retrieving speed of the riser joints is another key factor to prevent resonance.

Conclusions

Not only the natural frequencies but also the modal displacements, cross section slopes, bending moments, and shears of a deepwater drilling riser, affected by the hang-off modes and the mass of LMRP/BOP during the deployment/retrieval, are studied in detail by a proposed state-vector approach in this study. State-vector analysis leads to non-trivial findings, of general relevance for practice.

  1. 1)

    The modal shapes including displacements and slopes are significantly affected by the hang-off modes and the suspension mass of LMRP/BOP. The state vectors of modal forces such as bending moments and shears are dispersed along the riser span in the soft hang-off mode, and their maximum amplitudes occur at some certain distances above the LMRP/BOP.

  2. 2)

    The difference between the deployment and the retrieval increases with the increase of the riser suspension length. The first-order bending moments and shears of the riser are close to 0 below about 50 m depth. The first-order (higher-order) modal force amplitude during the deployment is relatively bigger (smaller) than that during the retrieval.

  3. 3)

    The suspension of LMRP/BOP plays a leading role in mass first and then in tension-enhanced dynamic stiffness with the increase of the riser suspension length. The top constraints alter the natural frequencies significantly when the riser suspension length is short.

  4. 4)

    Compared with the hard hang-off mode, the soft hang-off mode makes the riser's critical resonant suspension length Lc shorter. The BOP mass makes the first-order Lc shorter in the hard hang-off mode but makes the second-order Lc longer in the hard hang-off mode and the first two-order Lc longer in the soft hang-off mode. Moreover, the first-order resonance is likely to be excited when the first two riser joints are suspended.

In brief, the study on the state-vector analysis and the resonance analysis of the free-hanging riser under the random waves is of great significance to the riser's deployment and retrieval operations in ocean engineering. As the gimbal constraint, the mass of LMRP/BOP, and the deploying or retrieving speed of riser joints are the key factors to prevent the hang-off riser's resonance, a more detailed and reasonable plan including how to design and control those key factors and cross the splash zone is much needed in future work.